TEACHING MATHEMATICS: RETROSPECTIVE AND PERSPECTIVES

Transcription

1 8. starptautiskā konference MATEMĀTIKAS MĀCĪŠANA: VĒSTURE UN PERSPEKTĪVAS RAKSTU KRĀJUMS 2007.gada maijs, Rīga VIII International conference TEACHING MATHEMATICS: RETROSPECTIVE AND PERSPECTIVES PROCEEDINGS May 10-11, 2007, Riga Latvijas Universitāte Rīga, 2007

2 Teaching mathematics: retrospective and perspectives. Proceedings 8 th international conference / Editors Agnis Andžāns, Dace Bonka, Gunta Lāce. Rīga, University of Latvia / Mācību grāmata, pp. International Programme Committee Prof. Jānis Mencis, University of Latvia, Latvia (chair) Prof. Agnis Andžāns, University of Latvia, Latvia (deputy chair) Prof. Mihails Belovs, University of Latvia, Latvia (deputy chair) Ms. Dace Bonka, University of Latvia, Latvia (scientific secretary) Prof. Mati Abel, University of Tartu, Estonia Prof. Benedikt Johannesson, TALNAKÖNNUN, Iceland Prof. Ričardas Kudžma, Vilnius University, Lithuania Prof. Matti Lehtinen, Helsinki Defence College, Finland Prof. Vidmantas Pekarskas, Kaunas Technological University, Lithuania Prof. Līga Ramāna, University of Latvia, Latvia Organizing Committee Jānis Mencis (chair), Agnis Andžāns, Mihails Belovs, Dace Bonka, Jānis Buls, Jānis Cepītis, Ojārs Judrups The logo was designed by Lāsma KalniĦa and Diāna Mežecka. ISBN left to the authors Reă. apl. No Iespiests SIA Mācību grāmata, RaiĦa bulv. 19, Rīgā, LV-1586, tel./fax

3 CONTENT Jüri Afanasjev. PUPILS PERFORMANCE IN MATHEMATICS DURING THE FIRST THREE SCHOOL YEARS... 6 Agnis Andžāns, Līga Ramāna. TENURE TRACK OF MODERN ELEMENTARY MATHEMATICS Eve Aruvee, Olga Panova. TEACHING CALCULUS WITH MATHCAD GOOD AND WEAK SIDES Svetlana Asmuss, Aleksandrs Šostaks, Ingrīda UĜjane. PROGRAMME OF PROFESSIONAL STUDIES IN MATHEMATICAL STATISTICS: MODERNIZATION AND PERSPECTIVES OF DEVELOPMENT Algirdas Ažubalis. LITHUANIAN MATHEMATICS TEACHERS ABOUT THEIR PROFESSIONAL EXPERIENCE ( ) Tatjana Bakanovien, Arkadijus Kiseliovas. RESEARCH OF VIEWPOINT AND NEEDS OF TEACHERS WORKING WITH CHILDREN GIFTED IN MATHEMATICS Mihails Belovs. PROGRAMME OF ALGEBRA FOR BACHELORS Dace Bonka. THE FIRST 15 YEARS OF YOUNG MATHEMATICIANS CONTEST Inese Bula, Halina LapiĦa. SLIDE SHOW IN LEARNING PROCESS Jānis Buls. A FUNCTION IN CALCULUS Andrejs Cibulis. A REVIEW ON RESEARCH WORK OF PUPILS OF LATVIA ( ) Sarmīte ČerĦajeva, Ilze Jēgere. OBSERVATION OF AGE FACTOR TEACHING MATHEMATICS TO PART TIME STUDENTS Laura Freija, Sandra Zabarovska. WHAT IS MORE IMPORTANT THAN SCALARS? Rasma Garleja, Ilmārs Kangro. COMPETENCY OF MATHEMATICAL THINKING AND ITS VERSION OF APPLICATIONS Barbro Grevholm. WHY IS THE EDUCATION OF TEACHERS OF MATHEMATICS PROBLEMATIC? Edvins Ăingulis. INNER MATHEMATICAL MODELLING IN TEXTBOOKS FOR 5TH 6TH GRADES Maksim Ivanov, Elts Abel. THE ANALYSIS OF THE RESULTS OF TEAMS FROM BALTIC STATES AND FINLAND AT INTERNATIONAL COMPETITIONS IN MATHEMATICS

4 Hannes Jukk, Lea Lepmann, Tiit Lepmann. TEACHERS BELIEFS ABOUT THE COGNITIVE AND APPLICATION-ORIENTED COMPETENCIES IN SCHOOL MATHEMATICS Zane Kaibe, Laila Rācene. THE PORTRAIT OF CORRESPONDENCE MATHEMATICS SCHOOL AS IT IS SEEN BY MATHEMATICS FANS Romualdas Kašuba. ON GOOD BUT ACCESSIBLE PROBLEMS Kirsti Kislenko. A COMPARATIVE PERSPECTIVE ON THE ESTONIAN AND NORWEGIAN STUDENTS BELIEFS IN MATHEMATICS Elfrīda KrastiĦa, Iveta Nikolajeva. THE DEVELOPMENT OF STUDENTS RESEARCH SKILLS IN MATHEMATICS Dzidra Krūče. ELEMENTS OF TRIGONOMETRY IN ANIMATED PICTURES Aija Kukuka. A MATHEMATICS CONTEST AN OPPORTUNITY FOR EVERYONE Aira Kumerdanka. SOME PROBLEMS OF THE TEACHER IN-SERVICE TRAINING Gunta Lāce. ACHIEVEMENTS OF 5TH 8TH GRADERS AT THE SECOND ROUND OF THE 57TH LATVIAN MATHEMATICAL OLYMPIAD. FACTS AND LESSONS Joana Lipeikiene. GRAPHIC PECULIARITIES OF OPEN SOURCE COMPUTER ALGEBRA SYSTEMS Juozas Juvencijus Mačys. ON A SIMPLE PROOF OF SOME RESULTS IN NUMBER THEORY Jānis Mencis, Visvaldis Neimanis. DEVELOPMENT OF SECONDARY SCHOOL TEACHER OF MATHEMATICS STUDY PROGRAMME Diāna Mežecka. DEAR, WHAT DO YOU MEAN? Leonas Narkevičius. STUDENTS MATHEMATICS COMPETITIONS IN LITHUANIA Bohumil Novák. MATHEMATICS AS AN ENVIRONMENT FOR DEVELOPING STUDENTS PERSONALITY Michal Novak. DERIVATIVES USING PHP A SUITABLE TEACHING MATERIAL NOT ONLY FOR COMBINED OR DISTANCE STUDENTS Raitis Ozols. PARADOXICAL MATHEMATICAL PROBLEMS

5 Anu Palu, Eve Kikas. MATHEMATICAL TASKS CAUSING DIFFICULTY FOR PRIMARY SCHOOL STUDENTS Vidmantas Pekarskas. ON THE HISTORY OF LITHUANIAN MATHEMATICAL TERMS Kaarin Riives-Kaagjärv. ON UNDERSTANDING PRINTED TEXTS IN THE STUDY OF MATHEMATICS Tatyana Shamshina, Ilona Zasimchuk. STUDY METHODICAL COMPLEX FOR MATHEMATICAL EQUALIZATION COURSE FOR THE FIRST YEARS STUDENTS Jaak Sikk. MATHEMATICS TEACHING INNOVATION IN TECHNOLOGICAL EDUCATIONS Jaak Sikk, Eve Aruvee. NEW ENTRANCE MATHEMATICS TESTS AT ELS Regina Dalia Sileikiene, Vilija Dabrisiene. METHODOLOGY FOR TEACHING POLYNOMIAL APPROXIMATION OF FUNCTIONS IN CAS ENVIROMENT Eugenijus Stankus. ON PROOFS IN TEACHING MATHEMATICS Zane Škuškovnika, Agnis Andžāns. ON THE EVOLUTION OF MATHEMATICAL CONTESTS AT JUNIOR LEVEL Ingrida Veilande. THE MEAN VALUE METHOD IN MATHEMATICAL GAMES Anna Vintere, Aivars ĀboltiĦš. MODERNIZATION OF MATHEMATICS STUDY PROCESS AT LUA LIST OF PARTICIPANTS

6 PUPILS PERFORMANCE IN MATHEMATICS DURING THE FIRST THREE SCHOOL YEARS Jüri Afanasjev, University of Tartu, Abstract. In the frames of the international IPMA project (International Project on Mathematical Attainment) the same 269 Estonian primary school pupils on their first, second and third school years were tested. The pupils movement between the achievement level groups (four status groups, determined as test results quartile groups) established at the beginning of the first and the ends of these three school years was studied. The stability of a pupil s progress measured as his position on this ladder of success increases with years. The chance to change one s position on the ladder of success in mathematics diminishes with every school year. The pupils becoming fixed in the extremities of success (weak and strong groups) is considerably bigger than their becoming fixed in the groups in between (fairly weak, fairly strong). The girls positions on the status ladder are a little bit more stable than the boys positions. Keywords: achievement stability, gender difference, longitudinal study, performance in mathematics, primary school, status ladder. In this paper we continue the analysis previously presented in the articles (Afanasjev J. 2005; Afanasjev J., Palu A., 2006,) on first school stage pupils progress in mathematics. The source data were collected in the frames of the international project IPMA (International Project on Mathematical Attainment) during which pupils were tested in 22 schools over Estonia from 2002 to The pupils involved in the project took 4 different tests: Test 0 (T0) in the beginning of the first school year; T1 in the end of the first school year; T2 and T3 accordingly in the end of second and third school years. The items of tests came from IPMA coordination body. Texts of the tests are available on The items in the tests were cumulative, all the items of T0 were included in T1, all the items of T1 were included in T2 etc. To analyze pupils longitudinal performance, only the results of the pupils having taken all the tests can be used. So all the tests analyzed in this paper (T0, T1, T2 and T3) were taken by 269 pupils from 20 schools; 44,2% of the tested pupils were boys. The average age of pupils can be seen in the table 1. 6

7 Table 1. Age of tested pupils Test n Minimum Maximum Mean Std. Deviation T ,3 9,4 7,4 0,3 T ,3 10,4 8,4 0,3 T ,3 11,4 9,4 0,3 T ,3 12,4 10,4 0,3 The evaluation of pupils progress is based on their position in the socalled status groups of the test results. They were found on the scale of quartiles of distribution rows of the summary results of a given test: weak pupil s results were less than 25%, "fairly weak" results from 25% to 50%, "fairly strong" results between 50% and 75% and "strong" up from 75%. The general results of the tests are given in table 2 and the quartiles of each test in table 3. Test Table 2. Tests` summary results Possible maximum Min Max Mean Std. Deviation Percent of mean from possible maximum T ,1 2,3 61,4 T ,0 3,1 84,9 T ,1 4,2 85,3 T ,9 6,0 79,8 Percentiles T0 statusgroup Table 3. Tests` results quartiles Test T0 T1 T2 T3 25% 5,0 15,0 32,0 44,5 50% 7,0 18,0 35,0 49,0 75% 8,0 19,5 37,0 52,0 Table 4. Pupils' movement in status groups between the tests T0 and T3 T3 status group weak fairly fairly strong Total weak strong weak Number % within T0 status group % within T3 status group 46,0 19,0 20,6 14,3 100,0 43,3 19,7 22,4 10,8 23,4 7

8 fairly weak fairly strong strong Total % of Total 10,8 4,5 4,8 3,3 23,4 Numbert % within T0 status group % within T3 status group 25,7 32,9 25,7 15,7 100,0 26,9 37,7 31,0 13,3 26,0 % of Total 6,7 8,6 6,7 4,1 26,0 Number % within T0 24,1 18,5 18,5 38,9 100,0 status group % within T3 status group 19,4 16,4 17,2 25,3 20,1 % of Total 4,8 3,7 3,7 7,8 20,1 Number % within T0 status group % within T3 status group 8,5 19,5 20,7 51,2 100,0 10,4 26,2 29,3 50,6 30,5 % of Total 2,6 5,9 6,3 15,6 30,5 Number % within T0 status group 24,9 22,7 21,6 30,9 100,0 T2 statusgroup Table 5. Pupils' movement in status groups in tests T2 and T3 T3 status group weak weak fairly weak fairly strong strong Total Number % within T2 status group % within T3 status group 71,0 22,6 6,5 0,0 100,0 65,7 23,0 6,9 0,0 23,0 % of Total 16,4 5,2 1,5 0,0 23,0 8

9 fairly weak fairly strong Number % within T2 status group % within T3 status group 25,5 31,4 29,4 13,7 100,0 19,4 26,2 25,9 8,4 19,0 % of Total 4,8 5,9 5,6 2,6 19,0 Number % within T2 status group % within T3 status group 9,0 20,9 32,8 37,3 100,0 9,0 23,0 37,9 30,1 24,9 % of Total 2,2 5,2 8,2 9,3 24,9 Number % within T2 4,5 19,1 19,1 57,3 100,0 status group strong % within T3 6,0 27,9 29,3 61,4 33,1 status group Total % of Total 1,5 6,3 6,3 19,0 33,1 Number % within T2 status group 24,9 22,7 21,6 30,9 100,0 From table 4 we see that there is a general significant relationship (χ 2 test, p<0,05, Spearman ρ=0,38) between the pupils positions in status groups. In both the first and the last tests (T0, T3) 39% of all pupils stayed in the same status group. In the weak group there were 11%, in the fairly weak 9%, in the fairly strong 4% and in the strong group 16% of them. It is interesting to note that stable extremities (weak and strong together) in the test are considerably larger than the groups in between. At the same time, the strong status group seems to be more stable than the weak status group (in the first group more than half, 51% of the pupils, kept their positions, in the second group less than half, 46% of the pupils). At the same time some noticeable changes take place between the status groups. For example, the entire 14% of pupils from the weak group of the first test T0 moved to the strong group, and from the strong group 9% moved to the weak group. A similar situation revealed itself when the 9

10 following pairs of other tests (T0-T1; T0-T2, T1-T2, T1-T3, except the pair T2-T3) were closely observed. This allows us to create a hypothesis that there is a bigger chance for the pupils who seem weak in the beginning of the first school year to improve their status position during the first school years than for the pupils of the strong group to fall into a weaker one. The pupils positions in the status groups stabilize gradually. For example, in the last two test pairs T2-T3 (the final tests of second and third grade, table 5) no pupils of the weak group have moved into the strong group. Let us consider the pupils position stability in all tests. Here (table 6) we discern four so-called entirely stable groups of pupils: stably weak (tests in the weak group in all four) and, following the same criteria, stably fairly weak, stably fairly strong and stably strong. We find that pupils keeping a fully stable position form only 13,4% of all pupils. It is interesting to note that such pupils are much less numerous among boys than among girls (10,9 and 15,3 per cent accordingly). It is important to remark that fully stable pupils belong mostly to both extremities. Such stably strong and weak groups of pupils form 6,3 and 5,9 per cent of all pupils accordingly. Such pupils can be regarded as having needs for a special educational programme. Table 6. Fully stable pupils grouping Percent from fully stable Status group in all tests Frequency Percent pupils fully stable weak 16 5,9 44,4 fully stable fairly weak 0 0 0,0 fully stable fairly strong 3 1,1 8,3 fully stable strong 17 6,3 47,2 Total fully stable 36 13,4 100,0 Not fully stable ,6 Total ,0 In conclusion of what we have described here and in (Afanasjev, Palu 2006) we would like to emphasize the following: Although the pupils progress in mathematics during the first three school years is generally quite stable, we can point to noticeable mobility in pupils status scale of mathematical progress. The shifts move in both ways, upwards and downwards. The shifts are bigger during the first school year when almost two thirds of pupils change status group. Almost one fifth of 10

11 pupils who were in the weakest group in the beginning of the school year moved into the strongest group by the end of the same year. This makes questionable the possibility of predicting the pupils future success in mathematics before school. The stability of a pupil s success grows in the course of the years; the chance to change one s position on the scale of success in mathematics diminishes with every school year. At the same time the pupils becoming fixed in the extremities (weak group and strong group) is much stronger than their becoming fixed in the groups in between (fairly strong and fairly weak). Such pupils, belonging stably to the extremities, are to be considered as having needs for a special educational programme. References 1. Afanasjev J. (2005). On First-Year Pupils Initial Knowledge of Mathematics. In Teaching Mathematics: Retrospective and Perspectives; 5th International Conference, Liepaja Gingulis E. et al. (editors). Liepāja: Liepājas pedagoăijas akadēmija, 2005, pp Afanasjev J.; Palu A. (2006). Esimese ja teise klassi õpilaste edenemine matemaatikas. /The first and second forms pupils' performance in mathematics/ Koolimatemaatika XXXIII. Abel E., Lepmann L. (Toim.) Tartu: Eesti Matemaatika Selts, 2006, lk Afanasjev J., Palu A. (2005). Esimese klassi õpilaste teadmised ja edenemine matemaatikas. /The first form pupils' knowledge and performance in mathematics/ Koolimatemaatika XXXII, TÜ kirjastus, Tartu, 2005, lk Anon1. IPMA Coordinators Manual. URL= Anon2. The 3 rd coordinators report for the IPMA project. URL=

12 TENURE TRACK OF MODERN ELEMENTARY MATHEMATICS Agnis Andžāns, Līga Ramāna, University of Latvia, Abstract. The concept of elementary mathematics has changed during the times. The main features of it are considered. Keywords: elementary mathematics, mathematical competitions. The content of elementary mathematics is changing constantly; in general it becomes broader all time. For example, nobody will deny that the algorithm for dividing natural numbers is a part of elementary mathematics; nevertheless, in Middle Ages people who were able to use it were awarded scientific degrees. The problems investigated by world's most famous mathematicians only 250 years ago are now a part of classical elementary geometry (Eiler's circle, Gaus s line, etc.). It is a tradition that the words "elementary mathematics" are connected with school only. It's not correct. Of course, the greatest part of the mathematics taught at schools can be considered as elementary (there are exceptions; in some French high schools, for example, the basic concepts of Hilbert space are studied). Nevertheless, the part of elementary mathematics beyond the school programs is much broader. Let's make the concept of elementary mathematics more precise. Of course, no definition in the mathematical sense is possible. Trying to list the parts of elementary mathematics we include Euclidean planimetry and stereometry, linear operations with plane and space vectors, scalar, pseydoscalar and vectorial products, the greatest part of combinatorial geometry, elementary number theory, equations and systems solvable in radicals, algebraic inequalities, elementary functions and their properties, the simplest properties of sequences and the combinatorics of finite sets. There are many mathematicians however who include also elements of graph theory, simplest combinatorial algorithms, simplest functional equations in integers, etc. There are parts of mathematics which definitely should not be included: we can mention the methods which are effectively used only by a small amount of mathematicians as well as methods which, although used widely, demand a specific and advanced mathematical formalism. We can give the following approximate description of elementary mathematics. 12

13 Elementary mathematics consists of: 1) the methods of reasoning recognized by a broad mathematical community as natural, not depending on any specific branch of mathematics and widely used in different parts of it, 2) the problems that can be solved by means of such methods. Evidently, such a conception of elementary mathematics is historically conditioned. In the greatest part of scientific investigations elementary methods alone are not enough; they are used together with the specific methods of corresponding branch. Elementary methods are often used in obtaining the estimation, in proving lemma, in the analysis of a singular case, etc. Nevertheless, there is a number of cases when the whole basic idea of a solution lies in the unexpected use of an elementary method. We can note that sometimes only after some period of time the solution is recognized as the application of an elementary method. Along with classical period of the development of elementary mathematics the 20 th century has been the golden age for it. At first, some significant single mathematical discoveries were made using more or less elementary methods. We mention here some examples only: 1) Combinatorial Nullstellensatz, proved by N.Alon in 1999, 2) elementary proof of prime distribution theorem by P.Erdös and A.Selberg in 1940-ies, 3) RSA-cryptosystem developed in 1987 by Rivest, Shamir and Adleman, 4) effective and fruitful advances in inequalities (Jensen, Karamata etc.), 5) basics of Ramsey theory. This list can easily be prolonged very far; see, e.g., [1]. At second, many new areas of mathematics, especially in the discrete and algorithmnic parts of it, are still today exploring elementary methods as the main tool. Obviously it can be explained at least partially with the fact that the natural questions there have not yet been exausted, and natural approaches are therefore effective. At third, the movement of mathematical contests, especially of mathematical olympiads, has made an important service to elementary mathematics. Becoming a mass activity, the system of math competitions created a large and constant demand for original problems on various levels of difficulty. Clearly school curricula couldn t settle the situation, and the organizers of the competitions turned to their own research fields where they found rich and still unexhausted possibilities. Clearly only problems accessible by relatively simple methods were suitable, and so a search for such methods and examples expanded rapidly. As a feedback, simple and 13

14 clear solutions of serious scientific questions were sometimes achieved generalizing a simple olympiad problem (see, e.g., [2]). One of important results that originated from the olympiad mathematics was the identification of the so called general combinatorial methods (mean value method, invariant method, extremal element method, interpretation method); see, e.g., [3]. We must mention that, formally speaking, olympiad mathematics doesn t fit completely into elementary mathematics; e.g., in Romania elements of abstract algebra, calculus, function theory are included into olympiad curricula. See [4] for examples. Elementary mathematics was first officially recognized as an independent branch of mathematics in 1995 when the Latvia s Council of Science established the formal structure of science in Latvia, Modern elementary mathematics and didactics of mathematics being one of 12 parts of mathematics. Since then, master and doctoral degrees are awarded in this area. References 1. K.Kokhas e.a. Problems of Sankt-Petersburg Mathematical Olympiads (in Russian). SPb, A.KaĦepājs, R.Kreicbergs. On a sorting problem. Acta Universitatis Latviensis. 2005, Vol. 688, pp A.Andžāns. General Combinatorial Ideas in Contest Problems. In: Proc. NORMA/98 Conference, Kristiansand, pp R.Gologan e.a. Romanian Mathematical Competitions Bucuresti,

15 TEACHING CALCULUS WITH MATHCAD GOOD AND WEAK SIDES Eve Aruvee, Olga Panova, Estonian University of Life Sciences, Abstract. Humans have different styles of thinking. A calculus course at the university must develop students analytical and visual thinking. For better visualization of mathematics we have various computer programs (Matlab, Mathematica, Mathcad, etc). This paper reviews the experiences of using the computer package Mathcad. Keywords: Mathcad, teaching methods, visualization Introduction The higher mathematics course in our university for engineering specialities (11 points) is divided into three parts: linear algebra and analytical geometry, mathematical analysis I (differential and integral calculus of functions of one variable, differential equations), and mathematical analysis II (differential and integral calculus of functions of several variables, number and function series). The last part of the course consists of 24 hours of lectures, 40 hours of practical work and 16 hours in computer class with Mathcad. We have been using Mathcad program packages for four years. Mathcad is being used by today's top engineers to perform, document and share calculations and design work. The unique Mathcad visual format and scratchpad interface integrate standard mathematical notation, text and graphs in a single worksheet making it ideal for gain of knowledge, calculation, and collaboration in engineering. It has been used by engineers for more than 20 years already. Benefits of using the Mathcad package Teaching calculus with Mathcad has a number of benefits. 1. It is easy to learn and use no special programming skills are required. Its primary worth is the intuitive intelligibility the workscreen looks like work done with paper and pencil, and we can enter equations, graphs and texts at any place, using very clear and easy tools. The text appears as in a book. Here is an example of entered text: 15

16 x dx 2. It increases productivity, saving students time and reducing error. Students get quick answers to calculations. In our practice periods we do not require technical work and can focus on theoretically essential questions. We have now enough time to give additional explanations and solve more complicated tasks. The following is one example where we get a quick answer: x + 2 ( ) 2 x 2 x dx 2 x + ln( x) 1 ( ) + 2 ln x2 1 3 atan( x) 1 4 ( 4 x 2) ( ) x With Mathcad we can solve complicated equations, for example 5 4 x + x 8 = 0 : i i i i x 5 + x 4 8 solve, x i = i i i It improves verification and validation of critical calculations. If we have some notion about what a result might be and we want to verify our expectation, then visualization could be a very good helping tool. With Mathcad it is easy to produce various graphical representations. For example, visualization of a matrix If A : = is a matrix then its graph is: 16

17 Suppose we want to visualize the Fourier transform of the function, where k =0,1,...,n and x = -π, -π + π/100,...,π: π + x if π x 0 f(x) : = 2 π x if 0 < x < π 2 π Let n:=50 k:=0.. n and x : = π, π +..π 100 π 1 a k := f ( x) cos( k x) dx π π π 1 b k := f ( x) sin( k x) dx π π n a 0 S( x, n) := + a 2 ( k cos( k x) + b k sin( k x) ) k = 1 We can express this situation in different graphs and can see what happens when we change the parameter n n=1 S( x, 1) f( x) x

18 n=3 S( x, 3) f( x) x S( x, 10) f( x) n=10 Bad sides of Mathcad x The need of acquiring a licence of use. 2. As we know, the indefinite integral involves a constant of integration, which Mathcad does not give. Students must always remember and add this constant; otherwise it is very easy for a mistake to occur. 3. One must know how to go about solving different problems. For example, suppose it is required to find the coordinates of the centre of gravity of a homogeneous material lamina which is 2 2 bounded by the lines x + y 6x 16 = 0 and 1 2 y = x 2x To calculate double integral, the students must first solve a system of equations, express y from equation of circle. We get double integral 18

19 6 0 x 2 + 6x x2 2x+ 4 z dy dx Then we calculate double integral boundary points from inverse function. Finally we can solve the double integral y z dx dy y 3 4 and we get the result x=3 and y=3, y y 2 Classical teaching or teaching using computer? z dx dy As our technical support is good, we could increase the number of teaching hours in computer classes. But the question is: What is the right balance between classical paperwork and work with computer in a university calculus course? We certainly think that the students should know the relevant definitions and how to solve problems by the traditional method while they should also be able to solve more complicated tasks with the use of Mathcad. We think that Mathcad could occupy up to 30 40% of practical work. Conclusion Teaching calculus with Mathcad can be, to students, a useful and attractive alternative. It helps to understand various functions and operations through visualization. It makes it possible to solve complicated tasks wherein there is much technical work References 1. P. J. Pritchard. Mathcad: a tool for engineering problem solving. Boston, A. И. Плис, Н. A. Cливина. Mathcad 2000: мaтeмaтический практикум для экономистов и инженеров. Москва,

20 PROGRAMME OF PROFESSIONAL STUDIES IN MATHEMATICAL STATISTICS: MODERNIZATION AND PERSPECTIVES OF DEVELOPMENT Svetlana Asmuss, Aleksandrs Šostaks, Ingrīda UĜjane, University of Latvia, Abstract. The programme of professional studies in Mathematical Statistics remained the most popular among all mathematical programmes at the University of Latvia. Nevertheless we felt the necessity to make an essential improvement of the programme. Recently we applied for the ESF financed projects and have won three of them: for modernization of the programme, for better organization of the probation work, and for professional orientation of students. In this paper we discuss what has been done by now in the process of realization of the projects. Key words: mathematical statistics, ESF financed projects, modernization, probation work, professional orientation. The programme of professional studies in Mathematical Statistics runs at the Department of Mathematics, University of Latvia since 1997/1998 academic year. In 2001 it was accredited for the period till December During this period some changes were done in the programme. Although not very essential, these changes were important in order to keep the educational process up to date. During the whole period the programme remained the most popular among all mathematical programmes at the University of Latvia. Now, anticipating the new, regular accreditation of the programme, we see the necessity to rework it more substantially. In the sequel we discuss the main reasons and problems for this rework and as we see the perspectives of our programme. First, the general tendency in the EU is to shorten the time of educational process. On the other hand, the recently accepted rules of the Minister Cabinet of the Republic of Latvia establish Probation work in the volume of 26 national credit points 1. These two requests became contradictory in case of our programme. The previously accredited programme was 5 years (or 10 semesters) long and foresaw 2 month long time of probation work (4 credit points). In order to satisfy these contradictory tendencies and at the same time not to decrease the quality of 1 2 national credit points are equivalent to 3 EU credit points. 20

21 our graduates we, the stuff realizing the programme, had many serious and stormy discussions about the future of the programme. The students were also involved into discussions, in particular, by questionnaires. Our final decision was to accredit the programme as 4,5 years long. The problem to rework in such a drastic way was far to be easy. However, in addition to these, more or less, technical problems, when reworking the programme, we had to take into account that since 2001 (the time of previous accreditation) the situation has changed and it became necessary to introduce new theoretical courses as well as to apply new software. Besides, the reworked programme had to have at least 70 credit points common with Bachelor Programme in Mathematics: this was necessary in order that our graduates could continue Master Studies in Mathematics. To solve this difficult and important problem of modernization of the programme we got a support of the European Social Fund, which in November 2006 confirmed our project Matemātiėa - statistiėa studiju programmas modernizēšana Latvijas Universitātē (Modernization of the Mathematical statistics study programme at the University of Latvia). The Project foresees to make a certain revision of the study programme, to improve and update the courses of the programme and to work out lecture notes and tests for these courses. As the result, there will be an essentially reworked and improved programme. In the meantime much work of analysing the previous programme and developing a modified programme was done. A four and a half years long 180 national credit points valued programme of Mathematical statistics is worked out. This programme foresees the following modules of the courses and requested activities of our students: a module of Pure Mathematics; a module of Mathematical Statistics and Probability Theory; a module of Analysis of Social and Economic Processes; a module of Software Provision for Statistical and Numerical Problems; a module of General Subjects; a probation work and, finally, a Diploma Thesis. The module of Pure Mathematics aims at two mutually connected goals and tasks. First, to guarantee the students, after successfully mastering the module, to have knowledge and professional skills in different areas of Pure Mathematics (Mathematical Analysis, Algebra, Geometry, Mathematical Logic, etc,) which is necessary in order to acquire speciality courses in probability, statistics, econometrics, decision making and others. Second, the courses included in this module should give a student enough knowledge in order that he/she could be able to continue studies at the second, master, level in one of the four offered directions in Mathematics at our university (Probability Theory and Mathematical Statistics; Mathematical Simulation; Mathematical Didactics and Modern Elementary 21

22 Mathematics; Pure Mathematics, oriented on Algebra, Topology and Analysis). The aim of the module of Probability Theory and Mathematical Statistics is to educate specialists of high level of proficiency in Mathematical Statistics, whose knowledge and professional skills would be sufficient in order to make statistical analysis in all areas of modern economic, social and scientific activities. After successfully passing the courses of this module a student will get enough knowledge and skills in Probability Theory and Mathematical Statistics in order to apply them successfully in his/her professional work. We emphasize that all the courses are given at a serious mathematical level. This guarantees that the students who passed these courses will be able not only to apply ideas, methods and constructions considered in these courses in specific situations of various nature with which students will encounter in their professional activity, but develop new, original methods and constructions, which will be more adequate for a given situation than the traditional ones. The courses constituting the module of Mathematical Simulation and Analysis of Processes in Economics provide students with principles and methods in simulation and decision making. Basic models used in mathematical economics are discussed and corresponding mathematical tools will be studied. And again, as in the case of courses in the previous module, a special attention is played on the scientific correctness of the applied mathematical methods, which, in our opinion is crucial for the successful professional work of a specialist in mathematical statistics and guarantees that our graduates will manage not only standard problems that they will encounter in professional activity, but also dealing in peculiar situations. One cannot imagine a high level professional without having good skills in the work with computers. Of course, a high demand for having such skills concerns also specialists in Mathematical Statistics. Therefore an essential part of our programme occupies the fourth module: the module of the courses about the software provision for statistical and numerical problems. After mastering these courses, a student will acquire programming language PASCAL and will able to operate briefly with such software packages as Mathematica", "Maple V", MATLAB, SPSS, MINITAB, as well as with some other packages which are popular in mathematics and its applications. The module of courses of general subjects is relatively small and foresees a course in professional English and a course in Natural Sciences. A Diploma work (Diploma thesis) constitutes an essential part of the programme. By defending the Diploma a student demonstrates that his/her 22

23 theoretical knowledge and developed skills are sufficient in order to start professional activities. The subjects of Diploma works for graduates of our programme usually are related to Probability Theory, Mathematical Statistics, Applied Statistics and Actuary Sciences as well as to Mathematical Simulation of Social and Financial Process. It is common (however not obligatory) that the subject of the Diploma is closely related to the work which a student has fulfilled in the result of the probation work. However, in some cases students develop thesis in subjects in Probability Theory and Mathematical Statistics unrelated to the topic of the probation work and also in the areas of Pure Mathematics such as Algebra, Topology, Analysis, Geometry and Differential Equations. One of the principal criteria to accept the work for defence is its mathematical correctness. As it was mentioned above, the curriculum of Mathematical statistics programme foresees an essential increase of the role of probation work. The probation period should increase form 4 months in the previous programme to 26 weeks in the modernized programme. The problem to organize fruitfully the probation work for students was quite difficult even in the previous programme requesting 2 months long probation and became much more difficult now. Here we mention only two of the numerous problems connected with organization of probation. First, we send our students to enterprises of a very different profile (insurance companies, banks, statistical bureaus, etc.) and therefore the tasks which the students have to fulfil are essentially different. Hence the problem to supervise student s work for our staff is individual and quite difficult. Second, in many cases the probation work is connected with statistical data processing. Often these data are restricted and there arises a problem to let the students receive and use these data. Our second project Otrā līmeħa augstākās profesionālās programmas matemātiėis - statistiėis studentu prakse (Probation for the student of the second level higher professional programme Mathematical statistics ) confirmed by the European Social Fund will help to organize a half year long probation work for our students. The main goal of the project is to improve essentially the organization of the probation work and its quality. In order to achieve this goal we foresee the following activities: to work out a unique standard exposing the tasks which should be achieved in the result of probation; to develop collaboration with enterprises where we could successfully organize probation for our students, both to increase the number of such enterprises and by informing potential supervisors from the enterprises, how to arrange student s work. Besides, we feel that it is necessary to improve the professional training of our students before probation work. 23

24 In order to develop our contacts with enterprises we have worked out three questionnaires: one for the administration of the enterprises, one for supervisors of our students from the enterprises, and the last one, for the students which they are expected to fill out after the defence of the probation work. Besides we have worked out two booklets about our programme containing information concerning the probation work: one for the students and one for their potential employers. The aim of these questionnaires and booklets is to investigate the strong and the weak points of the professional education of our students, and to propagate the scientific level and professional skills of the graduates of the programme. In January 2007 also our third project supported by the EU, LU augstākā profesionālā Matemātiėa - statistiėa programma: vidusskolēnu un studentu profesionālā orientācija (Mathematical statistics higher professional programme at the University of Latvia: professional propagation for the secondary school pupils and for the University students) was confirmed. The objectives of this project are agitation of secondary school graduates to enrol our programme as well as the professional work with our students in order to diminish the number of students who discontinue studies. Every year up to 35 students enrol our programme. Taking into account that unavoidably some students drop out, especially after the first and the second terms, yearly 20 to 25 students graduate our programme. However this is not sufficient enough in order to satisfy the needs for specialists of this profession in our state. Therefore we feel the necessity of better propagation of our programme among secondary school pupils and first and second year students, in particular, to inform them more about the role of mathematical statistics in modern economy and the career perspectives for our graduates. The abovementioned project is aimed to realize these ideas. In order to do this we have planned three directions of activities. First, to work out, to publish and to spread informative materials (booklets, posters, et. al.) about our programme. The first version of such materials was already spread out in Spring 2007 anticipating the enrolment of students at the University of Latvia in June Now, after approval of the programme by the Senate of the University of Latvia at the end of May 2007, a slightly reworked version of these materials is being published. Second, to work out a web-page with information about our programme and the perspectives of its graduates (such web-page will appear in the Internet in October 2007.) Third, to organize seminars for secondary school pupils and discussion clubs for the students of our programme, in order to give more information about the advantages of the programme and the perspectives of career for our graduates thus giving more motivation to master the programme. First 24

25 such seminars and discussions were successfully organized in Spring 2007, next are planned for November To conclude: During the end of the 20th century and especially, in the 21st century the life is changing very fast. In particular it concerns fast changes in education and in science. Mathematical Statistics is to be mentioned among those sciences (or, if one prefers, among those education programmes), which have changed especially strong during last decades. The demands for precise, mathematically correctly interpreted, dates and for simulating on the basis of these data the possible development of the society (economy, finance, nature, pollution, medicine, etc,) have essentially increased during the last years. And to satisfy these demands is the task of statistics and mathematics in particular of mathematical statistics. In addition, we should remind that in Latvia, being a part of the Soviet Union, statistics was developed in a very deformed form, and the necessity to make a strong rework in this field became very important after Latvia won independence in 1991 and especially after Latvia jointed the EU in Concerning our programme, launched in 1997, we felt the necessity to make serious, and not only cosmetic changes, during several years, especially after And we are very thankful for the ESF projects of the European Union for supporting us and confirming our three above mentioned projects. We hope (actually we promise!) that in the result of successful fulfilling of these projects we shall essentially improve the structure of our programme and the scientific and professional level of our graduates. 25

26 26 LITHUANIAN MATHEMATICS TEACHERS ABOUT THEIR PROFESSIONAL EXPERIENCE ( ) Algirdas Ažubalis, The General Jonas Žemaitis Military Academy of Lithuania, Abstract. The paper describes how mathematics teachers shared their experience with their non-experienced colleagues in the period between the years 1945 and 1990 the hard times for Lithuania. 11 teachers that began their pedagogical activities before the World War II and 19 teachers that came to school during the War or after it provided most publications to pedagogical periodicals. Some of them published in didactics of mathematics, although it was not an easy task because of the difficulties typical for the Soviet period. Some of them took part in pedagogical readings and published articles in collections of such readings. The losses suffered by Lithuanian didactics of mathematics caused by the Soviet occupation are mentioned as well: some active mathematics teachers were deported to Siberia, some of them were forced to emigrate to the West. Keywords: mathematics teachers, pedagogical periodicals, pedagogical readings, publications in didactics of mathematics. The period under examination included two occupations and the burden of the World War II, so it was complicated for Lithuania. So, it was hard for Lithuanian school as well. Deportations to Siberia in 1941, the holocaust, massive leaving of intelligent people for the West in 1944, postwar repressions, and invitation of the top-qualification teachers for employment at higher schools reduced the number of high-qualification teachers at general education school considerably. So, experienced teachers were encouraged to share their professional experience. It was arranged in three ways: a) by publishing articles in pedagogical periodicals; b) by pedagogical readings; c) by publishing various individual publications [1, 2]. Hereinafter we ll describe the contribution of the teachers who shared their professional experience in didactics of mathematics most actively within the period under discussion. We ll divide the teachers into two groups: a) those who started their activities in prewar independent Lithuania; and b) those who started their activities during or after the war. In the first group, the first in the alphabetic order is Romualdas Balaišis ( ). He published 14 articles in pedagogical periodicals. He published also a book on investigation of equations [3]. Juozas Baltūsis ( ) was engaged as a teacher in the years Before the war, he published two arithmetic coursebooks and a mathematical book for

27 young businessmen. After the war he published 2 articles in pedagogical periodicals. In one article he analyzed equation solving, in the other the links of mathematics with life. He took part in pedagogical readings and published an article on the links of mathematics with manufacture in the collection of reports of the readings. Juozas Gailevičius ( ) was engaged as a teacher. Before the war, he published 15 articles as well as a collection of mathematical problems for real gymnasia. In his articles, he wrote about urgent problems of teaching mathematics in that period, such as integration of teaching mathematics and physics, tasks for final examinations in mathematics, out-of-class teaching of mathematics, enunciations of some parts of geometry and trigonometry. In the postwar period he published one article on the correct use of mathematical terms in the teaching process. Jonas Janulionis ( ) was engaged as a teacher in the years After the war, he published 2 articles on working with pupils gifted for mathematics in pedagogical periodicals. Juozas Janulionis ( ) was employed as a teacher in the years After the war he published 3 articles in pedagogical periodicals. In them he shared his experience in making and using visual aids, developing arithmetical problems of true-life contents. Antanas Lakiūnas ( ) worked as a teacher in the years After the war he published 9 articles in pedagogical periodicals. In them he discussed upon working with pupils that are hanging back and with the gifted pupils, shared his experience in out-of-class work, and analyzed the methodics of repeating the course in mathematics. Vincas Plaušinaitis ( ) was engaged as a teacher in the years Before the war he published 2 articles in pedagogical periodicals. In them he wrote about teaching to solve problems involving common fractions and textual problems. After the war he published 6 articles. In them he analyzed the issues related to encouraging an activity of pupils and using the true-life materials at lessons. He wrote on the experience in teaching some parts of algebra and geometry as well. Zigmas Rupeika ( ) was engaged as a teacher in the years He translated 4 coursebooks from Russian, published his own coursebook in trigonometry [16] the only original Lithuanian mathematical coursebook for higher grades in the postwar period. He graduated from Vilnius Pedagogical Institute after equivalency examinations. Jankelis Terespolskis ( ) was engaged as a teacher. In the postwar period he published (together with A.V.Klebanskis) a book on arranging a mathematical room. Elžbieta Zavišait ( ) was engaged as a teacher in the years She published an article on teaching geometry, namely, teaching stereometry for higher grades, in the collection of reports of pedagogical readings (in 1961). Jonas Kostas 27

28 Žemaitis ( ) was engaged as a teacher. Before the year 1944, his surname was Žemaitaitis, then he changed it in order to withhold the fact that his wife and daughters were deported to Altai. He published 4 articles in pedagogical periodicals prior to the war and the same number of articles in the postwar period. It should be mentioned that a considerable number of pedagogical journals were published before the year 1940: usually 3 of them were published and in some years 4 or even 5. The authors were paid considerable fees. Since the year 1945, journal Tarybin mokykla ( The Soviet School ) was published and in the year 1955, the journal Tarybinis mokytojas ( The Soviet Teacher ) appeared. They often paid fair fees for articles. However, the volumes of those periodicals were not large, so few authors could publish their materials in them. Coursebooks were prepared and published in Moscow; then, they were translated from Russian into languages of the Soviet Republics. So, it was almost impossible to publish an original mathematical coursebook for the secondary school for Lithuanian authors. Only the activities of translators were left to them. Since the year 1960, the situation related to publishing methodic books became somewhat easier; however, it was still hard enough. The strict Soviet censorship regulated these activities, there was a lack of paper, the facilities of the printing-houses were poor, and thus the process of book publication was very long. The Soviet censorship checked the contents of books and periodicals to be published: they should include no positive opinions on prewar independent Lithuania, and only negative opinions on this matter were desirable; positive opinions on Western states were not desirable as well. Simultaneously, the censorship took an interest in biographies of the authors as well. If a teacher was a deportee in the Stalin period or a state prisoner, he (she) usually couldn t publish. Such person hardly found a job at a school according to the acquired qualification and was afraid of losing it, if his (her) sins come out. So, many prewar authors ceased to write articles in didactics of mathematics in the postwar period. A part of them were deported to Siberia, some of them perished there. A larger part of authors avoided deportation before the war; however, they left for West after the war. Hereinafter, we ll describe the most active teachers that started their activities in the postwar period and provided materials in didactics of mathematics (in alphabetical order). Bronislovas Aleksandravičius (born in 1916) was engaged as a teacher in the years He published articles on integration of teaching mathematics with industrial training. Vytautas Baipšys (born in 1924) was engaged as a teacher in the years He published a number of articles on teaching arithmetic in 28

29 pedagogical and other periodicals. Ren Barščiauskas (born in 1925) was engaged as a teacher in the years R.Barščiauskas took part in pedagogical readings; his reports on the links of teaching mathematics with the life were published in collections of reports of the readings. Stasys Bulota (born in 1928) was engaged as a teacher in the years He made a number of reports at pedagogical readings, published a number of articles on improving the quality of knowledge of pupils in pedagogical periodicals. Petras Čeliauskas (born in 1929) was engaged as a teacher in the years He developed two aids for programmed teaching [4, 5], published a number of articles on programmed teaching of mathematics in pedagogical periodicals. Aleksandras Dokšus (born in 1927) was engaged as a teacher in the years He published 5 articles on teaching arithmetic and programmed teaching using elementary technical means in pedagogical periodicals. Motiejus Gudynas (born in 1923) was engaged as a teacher in the years He published a number of articles on arrangement of home tasks in mathematics as well as the links between the theory of mathematics and solution of mathematical problems in pedagogical periodicals. Academician Prof. Dr. Hab. Bronius Grigelionis was his pupil. Irena Jaškien (born in 1931) was engaged as a teacher in the years She published a number of articles on teaching mathematics at a secondary school for adults in pedagogical periodicals. Sigitas Mačionis (born in 1935) is engaged as a teacher since the year For many years, he was a supervisor of pedagogical practice for students of Vilnius Pedagogical Institute. He took part in pedagogical readings. The functiograph made by him was exposed at the USSR Exhibition of the Achievements of National Economy in Moscow. He published a number of articles on application of a functiograph and a development of self-sufficiency of pupils in teaching geometry in pedagogical periodicals. Genovait Malanskait -Tamulevičien (born in 1933) was engaged as a teacher in the years and She published a number of articles on using visual aids and a development of self-sufficiency of pupils in the process of teaching mathematics in pedagogical periodicals. Petras Martusevičius (born in 1921) was engaged as a teacher in the years He developed three aids for programmed teaching [7-9]. Algimantas Neniškis (born in 1925) was engaged as a teacher in the years He published a number of articles in pedagogical periodicals and theses of reports at various scientific conferences. Eugenija Nevronien ( ) was engaged as a teacher in the years She developed two aids for programmed teaching [10, 11]. Romualdas Paliokas ( ) was engaged as a teacher in the years He took part in pedagogical readings held in Lithuania 29

30 and the USSR, published his reports in collections of the readings; he also delivered reports at conferences of psychologists of Lithuania and the Baltic States. Theses of the reports were published as well. He published several articles on developing mathematical skills in pedagogical periodicals. Juozas P stininkas (born in 1932) was engaged as a teacher in the years His pupils often were prize-winners at contests of young mathematicians. He took part in pedagogical readings held in Lithuania and the USSR, published his reports in collections of the readings. He published a number of articles in pedagogical periodicals and one article in the journal «Математика в школе» ( Mathematics at School ). He wrote about developing self-sufficiency of pupils in the process of teaching mathematics. In 1993, he was awarded a title of a mathematics teacherexpert. Stasys Pikelis ( ) was engaged as a teacher in the years He published a collection of complicated tasks in mathematical rudiments with solutions [12] for self-education in mathematics and preparation for entering a higher school, published a number of articles in pedagogical periodicals. He wrote about using a slide-rule for calculations, teaching trigonometric functions, preparation of pupils for contests of young mathematicians. Prof. Dr. Hab. Vaclovas Bliznikas ( ) was a pupil of S.Pikelis. Ričardas Razmas ( ) was engaged as a teacher in the years He published three collections of distributable didactic materials for teaching mathematics [13-15]. Aloyzas Šilauskas (born in 1924) was a mathematics teacher in the years He published a number of articles in pedagogical periodicals where shared his experience in developing the concept of a number in teaching mathematics. Vladas Vitkus (born in 1941) is engaged as a teacher since the year He issued books for pupils of lower forms that take an interest in mathematics [17-19]. THE CONCLUSIONS 1. The pedagogical periodicals provided the principal opportunity to share the professional experience. Almost all above-mentioned teachers wrote for them. A considerable part of them took part in pedagogical readings as well. 2. In the period under discussion, teachers who started their activities in the prewar period took part in issuing books in didactics of mathematics to much less extent because of the elder age and the difficulties of that period, such as the censorship, lack of paper, poor capacities of out-of-date printing houses. Generally, it was very hard to publish an original book in didactics of mathematics in that period: first of all, translations of books published in 30

31 Moscow were issued. So, a considerable part of the best professionals in didactics of mathematics were forced to be translators only. REFERENCES 1. Ažubalis A. Matematika lietuviškoje mokykloje (XIX a. pr m.). Vilnius, Ažubalis A. Matematikos didaktika Lietuvos pedagogin je periodikoje ( m.). Vilnius, Balaišis R. Lygčių tyrimas. Kaunas, Čeliauskas P. Medžiaga savarankiškiems darbams su perforuotomis plokštel mis. Algebra VI klasei. Vilnius, Čeliauskas P. Metodin s pastabos darbui su perforacin mis plokštel mis, d stant IX klas s kurso temą Tiesin s lygtys ir nelygyb s. Vilnius, Klebanskis V., Terespolskis J. Vidurin s mokyklos matematikos kabinetas. Kaunas, Martusevičius P. VIII klas s geometrijos kurso pratybos. Vilnius, Martusevičius P. Pratybos VII klas s geometrijos kursui. Vilnius, Martusevičius P. Sutrumpintos daugybos formul s. Vilnius, Nevronien E. Aritmetikos savarankiški darbai. Kaunas, Nevronien E. Logaritmin funkcija ir logaritmai. Vilnius, Pikelis S. Elementarin s matematikos uždavinynas. Kaunas, Razmas R. Algebros ir elementarinių funkcijų didaktin medžiaga X klasei. Kaunas, Razmas R. Algebros ir elementarinių funkcijų didaktin medžiaga IX klasei. Vilnius, Razmas R. Algebros ir elementarinių funkcijų didaktin medžiaga XI klasei. Kaunas, Rupeika Z. Plokštumos trigonometrija. Kaunas, Vitkus V. Jaunajam matematikui. Kaunas, Vitkus V. Jaunajam V klas s matematikui. Vilnius, Vitkus V. Jaunajam VI klas s matematikui. Vilnius,

32 RESEARCH OF VIEWPOINT AND NEEDS OF TEACHERS WORKING WITH CHILDREN GIFTED IN MATHEMATICS 32 Tatjana Bakanovien, Arkadijus Kiseliovas, Siauliai university, Abstract. Globalisation, development of information, changes in society as well as changes of stereotypes and quality of life all these are the challenges which also should be met by the educational system. In the course of the process, and while scientific theories are undergoing changes, requirements for the teacher s profession are changing, too. Requirements posed by society for a teacher in fact are very high; however, it is seldom attended to their needs, opinions which might especially be valuable while improving quality of education. A lot of attention is paid to exceptional children; however, the problem of education of gifted children is investigated insufficiently. However, our teachers meet such exceptional children every day, and, to their mind, the time is now for finding out about education and needs of gifted children. The aim of the paper is to investigate attitude of teachers working with pupils who are gifted in mathematics towards their education as well as needs in work with such children. While analysing the results, we have noticed that the teachers feel lack of information, methodical material and methodical seminars concerning gifted children, also, they were short of time and financial reward. They link the education of pupils who are good in mathematics with increase of amount of tasks and their complexity as well as with participation at various mathematicsrelated contests and Olympiads. The respondents pay insufficient attention towards development of special curricula dedicated to children who are good at mathematics. Keywords: gifted education, identification, mathematically gifted children. Throughout the 20th century Lithuanian school was guided by a classical paradigm which approached training as the conveyance of the society s generalized experience (scientific knowledge, values and skills of mental and practical activities) to the trainees. Currently, a paradigm of free training that fosters the child s natural strengths is being entrenched (Dautaras J., et. al., 2006). Scientific literature indicates that an educator s profession requires pedagogical, psychological and professional knowledge, competencies and skills as well as the individual s special qualities allowing for an efficient and creative pedagogical work (Tamošiūnas T., 2002). In the given context, the teacher must continually enhance his/her qualification, and the need for the life-lasting learning is emerging. However, it would be

33 illogical to put requirements to the working teacher alone. In parallel with same, investigations into their demands and opinions, that would allow improvement in the quality of education, should be carried out. This is useful for both, the learners and the whole system of education. A special attention is given to the groups of manifold disjuncture. However, the reference here should be made not only to the children with special needs; the gifted students with certain peculiarities and positive deviances should also be taken into consideration. Any child with certain divergences has special needs. Most of the scientists (Newland, 1976; Leites, 1988; Feldhusen, 1989; Gross, 1995; Rost, 2000 et al.) engaged in the research of gifted children have pointed out special teaching/learning needs of the latter. Special teaching/learning needs appear due to the capability to abstract, to concentrate, to observe and to establish connections, to master the subject fast, etc. These are exactly the qualities causing the rise of special teaching/learning needs (Narkevičien B., et al., 2002). The aim of the article is to reveal the needs of teachers of primary classes and mathematics working with gifted children and their attitude towards the education of such children. The subject of the study is the needs of teachers of primary classes and mathematics working with mathematically gifted children and their attitude towards the education of such children. The results of the study and discussion Questionnaires were applied for carrying out the survey. A questionnaire was compiled with reference to The Analysis of Especially Gifted Students in Lithuania accomplished by KTU in 2002 as per order of the Ministry of Education and Science. The study sample consisted of teachers of primary classes and mathematics working with mathematically gifted children. The selection of the study participants proved to be complicated enough due to a vague concept of a gifted student. Therefore, the sample involved the teachers selected at random. The questionnaire involved the following question: Have you ever worked with gifted children? which ascertained the teachers involvement with such a group of students. The items presented in the questionnaire were aimed at finding out the teachers organizational peculiarities evidenced in educating mathematically gifted students. The point was to ascertain the way they identified such students and organized their training, and the demands the teachers had in the given field. Over 83,2 percent of the respondents assumed they have had mathematically gifted students in their classes. More than 67,8 percent 33

34 pointed out the irrelevance of any additional testing in order to identify a mathematically gifted student. They were unaware of any necessity to do such testing. Almost 52 percent of the teachers ascertained they were able to identify a mathematically gifted student in the run of a few lessons. However, it should be questioned here if skills were not confounded with academic progress. According to Freeman, teachers fail to recognize as many as half of the children s skills established by study tests. Only those skills that manifest themselves at first sight are easily identified, whereas other abilities unfold only under appropriate conditions. The analysis of the study findings in terms of the teachers work experience showed different assessments of the identification of gifted students by the respondents. Educators who had 10 to 20 years of work experience at school agreed with the statement that an additional testing was necessary more willingly, whereas a rarer agreement with the given statement was expressed by educators with a less work experience (up to 10 years) and those having work experience of over 20 years (p < 0,05, p = 0,07). The following tendency was observed: the teachers with a greater work experience were more apt to second statement that gifted children could be identified in the run of a few lessons. Another block of questions was intended for the analysis of the reality of training (the methods employed by the respondents for implementing the children s education) mathematically gifted children. Some scientists suggest that gifted students or those with particular abilities should not be taught in the same manner as their peers (Rost D. H. 1999). The respondents replies are represented in the Table 1: Table 1. The methods of education of mathematically gifted learners Yes I don t No know % % % The gifted should be taught in separate groups 7,9 10,1 82,0 The gifted should be given additional home tasks 77,4 4,2 18,4 More tasks should be given in class 93,7 6,3 The gifted should be delivered more complicated 73,7 10,5 15,8 tasks They should work conjointly with others 51,1 7,9 40,5 performing the same tasks They should be allowed to skip a lesson provided 11,6 5,3 83,2 the subject is clear to them They should assist others in acquiring the subject 65,8 15,8 17,9 Special curricula should be compiled 9,5 8,4 82,1 They should get extra teaching after classes 27,9 11,1 61,1 Teachers should prepare special assignments 88,4 2,1 9,5 34

35 Parents should take care of gifted children s 35,3 24,7 40,0 training Extra classes should be organized 54,2 8,4 37,4 The gifted should be prepared for Olympiads and competitions in mathematics 87,9 4,2 7,9 The study findings lead to the conclusion that teachers are aware of the necessity of gifted children s training but the techniques employed by them bring forward the assumption that educators lack sufficient competencies or motivation for such performance. Most teachers stimulate independent work of gifted students (additional home assignments, a bigger number of tasks in classes, individual tasks, preparation for Olympiads and competitions). The comparative analysis of the results provided a statistically significant difference with respect to the type of school (p < 0,05). All the respondents working in primary schools expressed the opinion that individual curricula for the students good at mathematics are not obligatory. The study showed that, in most cases, educators work with gifted students does not differ from that with the whole class, with the only exception that the former are delivered more complicated tasks or the number of assignments is increased. Teachers from gymnasiums expressed even stronger rejection against the statement that the students good at mathematics should be treated as a usual class (p < 0,05). Teachers from secondary schools are inclined to surrender the right of training the students good at mathematics to their parents (p < 0,05). The needs of all the participants of educational processes are very impotant. But the main focus should be given to the students and the teachers opinions. Our survey presented teachers the questions aimed at establishing what was the work with the students good at mathematics lacking. The findings are represented in the Table 2 below: Table 2. Teachers needs working with mathematically gifted learners (1 totally disagree; 2 disagree; 3 I don t know; 4 agree; 5 - totally agree) Statements Information about gifted children 8,9 18,4 34,7 35,3 2,6 Methodological material 7,9 17,4 26,3 42,6 5,8 Improved preparation at a higher 19,0 21,7 27,0 23,3 9,0 educational institution Methodological seminars 6,8 19,5 20,0 44,2 9,5 Time for the work with the gifted 2,6 12,6 10,5 50,0 24,1 Remuneration for the work 1,1 1,6 15,3 51,6 30,5 Measures for skill identification and 2,1 9,5 22,6 44,2 21,6 evaluation Parents concernment - 17,1 24,6 43,9 14,4 Nothing 31,6 37,5 19,1 10,5 1,3 35

36 Dispersion of the remaining demands is varied: some of the respondents focused on financial matters, while others stressed methodological issues. Some respondents noted the lack of time for the work with the students good at mathematics. Some teachers indicated they were missing good preparation at a higher educational institution. Meanwhile, in Lithuania the education of would-be teachers on the issues of gifted children s training is insufficient, although experience of other countries have proved that such training is very important and a proper additional preparation is necessary (Narkevičien B. 1999). Teachers competencies are influenced by their experiences. The comparison of the respondents demands in all the groups in terms of their work experience revealed their similarity, although several differences were observed. Educators, whose work experience at school did not exceed 10 years, considered parents concernment to be one of the most marked demerits, whereas other respondents were inclined to overestimate the given factor. The respondents with work experience at school exceeding 30 years were more missing methodological seminars in comparison with other educators. The statement that nothing was lacking for the work with the students good in mathematics found the least favour with the educators having little work experience (up to 10 years) and the greatest work experience (over 30 years). Conclusions: 1. In the teachers view, no special testing is necessary for identifying mathematically gifted students. They are confident in their experience and in that few lessons are enough to find out a mathematically gifted student. 2. Education of mathematically gifted students is restricted to their independent work (a bigger number of home assignments, more complicated tasks, etc.) and their preparation for participation in Olympiads and competitions in mathematics. A little agreement with the statement that special curricula should be compiled for such students was evidenced. 3. The demands of the teachers working with mathematically gifted students are quite similar: they feel the lack of reward for their job and time most of all. Majority of the teachers agreed on the lack of parents interest in the training of their children. 4. The analysis of the results leads to the conclusion that the teachers working with mathematically gifted students must be provided with an adequate preparation at a higher educational institution. 36

37 References 1. Dautaras J., Rukštelien N. Mokymosi visą gyvenimą motyvacija: pedagogų požiūris. Pedagogika 83, Tamošiūnas T. Pedagogo profesiniai geb jimai: ekspertų nuostatų analiz, p. 194.// Pedagogika 61, Grakauskait Karkockien D. (2003). Kūrybos psichologija. Vilnius. 4. Kardelis K. (2002). Mokslinių tyrimų metodologija ir metodai. Kaunas. 5. Narkevičien B. ir kt., (2002). Itin gabių vaikų ugdymo situacijos Lietuvoje analiz. Vilnius Kaunas. [ ] 6. Narkevičien B. Gabiųjų mokymas svetur//mokykla Nr. 4 5, Rost H. D. Kaip nustatyti ypatingus gabumus// Mokykla Nr. 4 5,

38 PROGRAMME OF ALGEBRA FOR BACHELORS Mihails Belovs, University of Latvia, Abstract. The necessity of changing the algebra programme is considered in the paper. Solving practical tasks is the main part of the new programme. Mostly in the first term linear algebra is taught. Other elements of abstract algebra are left for the next terms. Description and analysis of the programme is given. Keywords: conceptual thinking, correlation, linear algebra, practical problems, programme of algebra. Introduction The necessity of changing the algebra programme for students of the Department of Physics and Mathematics in University of Latvia is motivated with a number of reasons. One may mention: The reduction of the number of credit points assigned for the course from 12 to 8. It is motivated with a new European standard. The background knowledge of a first year students. It is a consequence of the new school programme in mathematics. Quite another social status of mathematical education. The main task is not to fit the programme in the new credit point limits, but to synchronise new programme of algebra with an average first year student. This is not the question of knowledge only. Historical background Algebra course has been taught in University of Latvia for more than 50 years. Earlier algebra was the most abstract study course of the first term. There has been a lot of abstract (for the first term students) conceptions and technically complicated (not at all obvious) proofs in the course. Classical books [1,2,3,4,5] and collections of activities and methodical literature [6,7,8,9,10,11] have been used widely while studying. The course was based on the mathematical background of pupils and the high social status of physical and mathematical education. Problem Now, classical abstract algebra contradicts the level of first term students. The mathematical education at school can be described with the following: 38

39 No more conceptual thinking. Pupils get to use the intuition. No more ability of mathematical proving. No idea why such is requested at all. Algorithmic approach in school mathematics. This all goes contra the classical course of abstract algebra. Trying to keep this course we will just harm students. Programme must be correlated with the current level of a first term student. We must go back and teach the conceptual thinking to the student; learn why it is important to explore basic notions of the course; face the student with the fact he cannot answer a question only because he is not familiar with the corresponding notion. First term students are not sillier they were before. They are different. We have to help them. The contact of a professor and a student is of high importance in this task. First term students are not ready to study scientific literature on their own. They must be taught to learn independently. They must be taught the critical thinking. (Critical thinking is not just a negation of wide beliefs and a stating of your own. It is, at first, an analysis, seeking an answer to the question why, not how.) The social status of physical and mathematical education now is lower than it used to be. As a result a number of average students is reduced. A part of first term students is not sure in their choice. This causes additional difficulties for the first year teaching. Solution of the problem The contradiction between the current programme of algebra and the level of an average first term student can be resolved by changing the course. Paradigms for this change are as follows: The practical aspect of algebra must be emphasised. Mathematics is not an abstract glasperlenspiel, but an instrument constructed for solving concrete problems. Main part of the course is formed by linear algebra. Formal algebra formulae are interpreted with geometrical language. With a help of geometrical interpretation many formal algebraic constructions become easier and even obvious (for example, the theory of linear equation systems). Practical problems are closer to a modern student 39

40 of a first term. They attract his attention, and afterwards help him to tackle abstract algebra problems. Programme gives just an overview of basic abstract algebra notions. Elements of abstract algebra are left for higher terms. Teaching and understanding of elements of abstract algebra always has been concerned with pedagogical problems. The reason is in the high level of formal abstraction (axiomatic approach, low intuitive recognition). But the main problem is a lack of practical examples and problems (at lest, I am not able to find such). At present it is extremely hard to motivate first term students in the necessity of studying abstract algebra. This is a consequence of a lower prestige of physical and mathematical education. The postponing of abstract algebra towards high terms is a result of correlation of the programme with the current state of the school education. Main results of new algebra course will be: The knowledge of matrix algebra. The ability to use standard methods of linear algebra (linear equation system solving and analysis, mapping of linear space, solving of geometrical problems) Basic knowledge of general algebra. In advance it is proposed to Improve conceptual thinking. Improve proving abilities. The proposed programme is similar to the approach given in [12,13,14]. Additional literature is [15,16,17,18,19,20]. Structure of the programme 40 The study process could follow the plan: 1. Matrix algebra. 2. Systems of linear equations. 3. Determinants. 4. Inverse matrix. 5. Linear space R n. 6. Subspaces of a linear space. 7. Dimension and base of a linear space. 8. Abstract linear space. 9. Algebra of complex numbers. 10. Algebra of operators. 11. Inverse operator. Change of a base. 12. Eigenvalues and eigenvectors of a linear operator. 13. Transforming a matrix into a diagonal form. 14. Euclidean space.

41 15. Orthogonal projection and the method of least squares. 16. Unitary space, self-adjoint operator. 17. Quadratic forms. 18. Elements of general algebra. 19. Ring of polynomials. Let us note the following: Systems of linear algebraic equations and the matrix form of Gauss method illustrate the practical aspect of matrix algebra. The theory of determinants is given in a brief form. Cramer s rule shows the practical aspect of inverse matrixes. In main, concrete space R n is considered. Geometrical theory of systems of linear equations and matrix mappings is central. The role and necessity of axiomatic theory is of high importance when studying abstract linear spaces. (Student, at first, must realise the difference between the axiomatic theory and its constructive realisation). Transformation of a matrix into a diagonal form is a main task of the theory of linear operators. Orthogonal projections onto the subspaces and the least square method are central in the theory of Euclidean space. The realisation of the programme is based on the methodical tools and queries designed through the project [20]. Methodical tools and queries are created for independent work of the students during the second half of the day. Another important task is a design of a new collection of activities. Summary Earlier algebra was the most abstract course in the first semester. At present such an abstract form has come into conflict with the students preparation, because their conceptual thinking skills are not developed. Our main objective now is to facilitate first year students conceptual thinking competence. In order to achieve this objective the student should be in the centre and the focus should shift from pure theory to solving practical tasks. References: 1. Курош А.Г. Курс высшей алгебры. Наука, 1968 etc. 2. Ван дер Варден Б.Л. Алгебра. Наука, Кострикин А.И. Введение в алгебру. Наука, Скорняков Л.А. Элементы алгебры. Наука,

42 5. StrazdiĦš I. Diskrētās matemātikas pamati. Rīga, LU, EĦăele N. Ievads algebrā. Rīga, LU, EĦăele N. Ievads matemātisko struktūru teorijā. Rīga, LU, EĦăele N. Lineāras algebras papildu nodaĝas. Rīga, LU, Окунев Л.Я. Сборник задач по высшей алгебре. Наука, Проскуряков И.В. Сборник задач по линейной алгебре. Наука, 1969 etc. 11. Фадеев Д.К., Соминский И.С. Сборник задач по высшей алгебре. Наука, 1968 etc. 12. Strang G. Linear algebra and its applications. Academic Press, Канатников А.Н., Крищенко А.П. Линейная алгебра. Москва, Шевцов Г.С. Линейная алгебра. Теория и прикладные аспекты. Москва, Кострикин А.И. Введение в алгебру I, II. Москва, Воеводин В.В., Воеводин Вл. В. Энциклопедия линейной алгебры. Электронная система Линеал. Санкт-Петербург, Lorenz F. Linear algebra I. Berlin, Shores T.S. Applied linear algebra and matrix analysis. Springer, Кострикин А.И. Сборник задач по алгебре. Москва, Projekts Nr. 2005/0116/VDP1/ESF/PIAA/04/APK/ /0019/0063 (LU reăistrācijas Nr. ESS2005/15) Matemātikas studiju satura strukturēšanas un to atrukturēšanas un to akadēmiskās vides pilnveide Latvijas Universitātē. 42

43 THE FIRST 15 YEARS OF YOUNG MATHEMATICIANS CONTEST Dace Bonka, University of Latvia, Abstract. Young mathematicians contest (YMC) is a correspondence math contest for junior students. From one district in Latgale YMC expanded and became popular overall Latvia. Thematics of YMC problems and principles of problem set s composition along with contest s history and management are considered. Keywords: Young mathematicians contest, math contests for junior students, general mathematical methods. History of Young mathematicians contest This contest is a brain-child of PreiĜi 1st secondary school teacher of mathematics Mārīte Seile. That time contest Professor Littledigit s Club (PLC) already was popular in Latvia. However there were some psychological and economical reasons, why PLC was not wide accessible to pupils in rural areas: problems of PLC are too difficult for not-advanced student; problems of PLC were published in a newspaper which was not so popular and accessible in the back-country because of economical crisis which heavily affected region of Latgale in those times; another psychological aspect which detered students from solving PLC problems was that PLC comes from the big Riga. It aroused inferiority complex to students how can I a rural student compete with students from RIGA? So Ms. M.Seile established a new similar, but easier contest in Latgale Young mathematicians contest (YMC). The central aim of this contest is a development of junior students self-assertion along with improving their mathematical culture. So there is at least one very easy problem which is solvable for every 4 th 7 th Grade student. Another aspect publishing the names of winners in a regional newspaper also raises self-assurance of students and pride of their parents. The first problems of YMC were published in January 1993 in the PreiĜi region newspaper Novadnieks. Until May rounds of this contest took place. Soon after that it became quite popular in and beside this district. In school year 1994/95 YMC comprised already all Latgale region. Since 1996, the management of YMC is taken up by Dace Bonka, the educational assistant of A.Liepas Correspondence Mathematics School 43

44 (CMS). So, in fact problems are now composed in Riga, though they are published in the same newspapers of Latgale and return address is Rudzāti secondary school in PreiĜi region. Math teacher of this school V.SpriĦăe is a local organizer of YMC. By progress of ICT a web-site of CMS was established in 1999 and problems of correspondence contests are published also on the Internet (see [1]). So YMC contest became popular overall Latvia. In the course of time young generation become more certain and convinced, hence abovementioned inferiority complex is not so pronounced as some years ago. Organization of Young mathematicians contest YMC is a mathematical contest for junior students up to 7 th Grade (14 years). The contest is organized by correspondence problems are published in several regional newspapers in Latgale and on the Internet. Students can solve them during about 4 weeks and then send solutions by mail or to contest s organizers who check students papers. During each school year there are 5 rounds; a set of 5 problems is proposed in each round. Each round problems extended solutions are published in the newspapers and on the Internet after allowed time for solution has expired. It is a possibility to acquire new knowledge outside school. There are approx. 200 contestants each year. Individual as well as collective papers have been sent in. Names of the winners of each round are published in the newspapers and on the Internet. The final winners of all 5 rounds are awarded at the end of the school year. Awards are provided with support of Latvian National Foundation in Sweden. Principles of composition YMC problem set Math contests should cover broad spectrum of mathematics, as more as better. It is particularly important also because olympiad and contest problems from previous years are broadly used afterwards in everyday teaching practice. Now we describe main criteria accordingly to which the set should be well balanced: it should cover main areas of school mathematics: algebra, geometry, number theory and combinatorics. Combinatorics is understood in a broad sense including not only counting but also existence and nonexistence of combinatorial objects. Particularly, the general combinatorial methods (mathematical induction, invariants, mean value, extremal element, interpretation) must be reflected; 44

45 it should contain both problems of deductive nature and problems of algorithmical nature; there should be problems of prove it! type along with problems in which the answer must be found by the solver; discrete mathematics and continuous mathematics both are to be represented. See [2] for more about the principles how problem sets for Latvian contests are developed. Some specific aspects have to been taken into account in process of composition of the problem set for junior students: there are 1 2 quite easy problems and some quite complicated problems; texts of problems must be interesting and exciting; mathematical concepts should not exceed school curricula. Main topics represented in problems of YMC There are two main approaches for classification: accordingly to the content of problem and accordingly to the method of solution. Nevertheless it is often quite difficult for contests problem to decide to which topic does it possess because mainly these problems are rather complicated and correspond to more than one topic. As YMC is a contest for students up to grade 7 and junior students don t have yet deep knowledge in mathematics, contest s problems also should not be based on serious mathematical facts and theorems. So the greatest part (~35%) among all YMC s problems are algebraic and arithmetical problems. Operations with integers and various text problems (e.g. about uniform motion, about fractions and percents, etc.) are the most popular among them. Geometry as a separate subject starts at school only in the grade 7. So YMC problems may consider only basic geometrical concepts. Appr. 24% of all YMC problems are geometrical, and problems on figure cutting and plane tessellations make ~42% of them. Elements of number theory are represented appr. in 15% of all YMC s problems. Very important are combinatorial and algorithmical problems, which make appr. 25% of all problems. The main topics represented in YMC s problems are listed further. Algebra and Arithmetics Operations with integers Numerical rebuses Properties of addition and multiplication Equations, systems, inequalities Percents 45

46 Numerical sequences Number theory Divisibility of natural numbers Prime numbers Geometry Problems on cutting figures and plane tessellations Basic properties of simplest geometrical objects (square, rectangle, circle, regular triangle) Combinatorial geometry (constructing/ analysis of systems of geometrical objects) Combinatorics Counting problems Graphs Combinatorial systems (tournaments etc.) Mathematical games Searching and sorting problems (weightings etc.) Another algorithmical problems (analysis of algorithm, discovering of algorithm) Logical problems General mathematical methods Mean value method (especially Dirichlet principle) Method of invariants Method of extremal element Method of interpretation The last topic General mathematical methods is considered separate though in fact it is integrated in all previous ones: general mathematical methods and approaches are useful not only by solving combinatorial problems, but also by solving problems of all other topics. A special attention is paid to those problems, which are solvable applying the abovementioned general methods (they make appr. 12% of all contest s problems). They were identified as thinking tools reflecting the mankind s general experience in everyday problem solving rather than mathematical methods. In an informal way they can be characterized correspondingly as build a staircase!, there should be made significant investments to reach significant results in at least one area, look for extremes!, look for eternal values! and if you can t cross the obstacle, go around it!. For more about these methods see [3]. Acknowledgment. The publication was prepared with the support of ESF. 46

47 References A.Andžāns, L.Ramāna. What Problem Set Should be Called Good for a Mathematical Olympiad? In: Matematika ir matematikos destymas, Technologija, Kaunas, 2002, D.Bonka, A.Andžāns. General Methods in Junior Contests: Successes and Challenges. In: Proceedings of TSG4 of the 10 th Congress on Mathematical Education, University of Latvia, Rīga, 2004, Andžāns, I. BērziĦa, D. Bonka, B. Johannessons. Matemātikas sacensības klasēm. LU, Rīga, Andžāns, D. Bonka, Z. Kaibe, L. Rācene, B. Johannessons. Matemātikas sacensības klasēm. Rīga: Mācību grāmata,

48 48 SLIDE SHOW IN LEARNING PROCESS Inese Bula, Halina LapiĦa, University of Latvia, Abstract. The main idea of the article is to identify the reasons of the necessity for using slide shows in the learning process of Mathematics. The positive and negative aspects of using slides have been examined. Keywords: slide, slide show, teaching of mathematics with slide show. 1. Introduction From time to time the changes in the era and the development of the technology make the learning process change and develop as well. The classical approach of Mathematics, however, remains immutable. Nevertheless, if looking at the length of the epoch, we can observe that hand in hand with the development of orthography and the book production and printing, the writing down and design of mathematical symbols have changed as well. Why is there a necessity for new teaching books in classical Mathematics Algebra, Geometry, and Mathematical Analysis? The possible answer can be found in the need of today s people to acquire more and more information. As a result, the new learning material is even more concentrated if compared with the previous publications and more new symbols are used. More and more knowledge is required from students of both universities and secondary schools. And, if the student comes from a Faculty of Economics or Information Technologies, the time devoted for Mathematics studies reduces substantially, since his or her own studies seem to be of a greater interest. And in the field of Economics or IT, a student also has to learn more and more to become a good specialist. As a result, something has to be sacrificed. Hence, in those study programs, where the knowledge of Mathematics is important but not of the primary importance, the study hours are reduced. Especially, in the current situation when the study programs last for 3 rather than 4 years What should a lecturer do, whose course size is reduced by one third or a half, while the amount of knowledge is supposed not to diminish? How to intensify the contents of the lectures? One of the possibilities is to use slide shows during the lectures. In this way, lecturers do not have to spend time on writing the material on the blackboard. Still, the slides should be prepared before the lecture, which unfortunately is a time-consuming process. The lecturer should also evaluate which information and how much

49 of it should be inserted. The practical side of the lectures should also be not forgotten. A student cannot learn how to calculate limits, derivatives, and integrals only by looking at what the lecturer is doing! 2. What is it? With the term slide show in learning process we understand the lecture material presented with the help of: Microsoft PowerPoint presentations or Microsoft Word presentations, or another Microsoft presentations (for example, Excel), or PDF or PS file presentations, or another presentations with computer, or transparencies, or movie. During one lecture several variants can be used. Slides can be used only partly or all the material may be presented on the slides. 3. Questionnaire about the usage of slides In spring, 2007, a questionnaire about the usage of slides was conducted in University of Latvia, the Faculty of Physics and Mathematics. The respondents of the survey were lecturers from the Departments of Computer Science and Mathematics. Department of Mathematics 16 respondents (lecturers) Department of Computer Science 14 respondents (lecturers) Do you use slides during the lectures? Yes 5 No 11 Yes 11 No 3 Is the usage of slides helpful presenting the lecture? Yes 16 Yes 14 Is the usage of slides helpful for students to understand the lecture? Yes 16 Yes 12 No 2 The positive aspects of slide usage It is possible to expand the material taught to students at the lecture It is possible to explain the material better (by using visualizations, diagrams, graphs, chart, block schemes) Structural peculiarities The printout is available The negative aspects of slide usage It is a cinema! Students can not manage to write down the information Students prefer studying at home to lectures 49

50 Prof. J.Borzovs: for people the movements of hands are closely linked to thinking. Necessary Home works and exercises In spring, 2007, a questionnaire about the lectures in Mathematical Analysis I (the lecturer I. Bula) was conducted in University of Latvia, the Faculty of Physics and Mathematics. The respondents of the survey were first course students from the Department of Computer Science. Questionnaire about the lectures of Mathematical analysis I 85 respondents Which lectures do you like best? with slides without slides In which case do you understand the new topic of the lecture better? with slides without slides answers: 50%:50% Which is the more efficient way to use slides? a) slides should be handed in before the lecture; 54 b) slides should be handed in after the lecture; 6 c) slides should be available in general; 30 d) slides should not be available for students. 0 Is there a need for exercises done by hand during the lectures with slides exercises are necessary exercises are not necessary 84 1 Should the whole lecture material be presented with slides? a) yes; 24 b) no, only definitions and theorems should be on slides; 30 c) no, only illustrative material should be on slides; 23 50

51 Students have commented on the usage of slide shows; however, their opinions are quite contradictious: The whole lecture material should be on WebCT More exercises are needed Without slides it is much easier to follow lectures on the blackboard More practical works are needed Mathematical analysis is not a subject where you can learn from slides Better design is needed for slides Without slides I do not have a feeling of safety and self-confidence 4. Conclusions Our experience and the questionnaire conducted confirmed that slides can be used in teaching Mathematics for those students whose main subject of interest is in another field. However, the material presented on the slides should be well thought-out. It is also important what comments you are giving to slides. Too shortened material would make it difficult for a student to remember the layout of the lecture. On the contrary, slides with too much information posted on them would complicate the perception of the most constitutive information, which is presented on the slide. The design of slides should be simple and modest, so that it would not sidetrack the attention of students from the written text. If the slides are available for students and they can use them during the lecture, then students should be able to manage to supplement the slide print-outs with the remarks of the lecturer, solutions of the exercises, as well as the key to the used symbols. References 1. I. Bula. Slide show in learning process, VIII International conference Teaching Mathematics: Retrospective and Perspectives, May 10-11, 2007, Riga, Abstracts, P

52 A FUNCTION IN CALCULUS Jānis Buls, University of Latvia, Abstract. We recall there are situations when explanation is given as the definition. This leads to perplexity. We demonstrate there is no need to restrict ourselves with explanation of a function even in calculus. Keywords: definition, explanation, function, ordered pair. 1. Introduction Sets are unordered; this is stipulated by the axiom of extensionality. But mathematics needs ordering, and ordered pair is a tool of introducing it into set-based mathematics. A little ingenuity is used in defining an ordered pair as a set: specifically, if A and B are sets, then the applications of the Pairing Principle establish that {{A},{A, B}} is also a set, and it is this set which is called the ordered pair (A, B). Sets of ordered pairs are called relations and relations with a certain uniqueness property are called functions (see, e.g.,[4]). We define the Cartesian product A B to be {( x, y) x A and y B}. Let A, B and C be sets. Then we define the ordered triple (A, B, C) to be ((A, B), C). Definition. An ordered triple f ( X, Y, F ) called a function if for each ( x y ) ( x z) F =, where F X Y, is,,,, there is y = z. This definition is quite clear, nevertheless there is a praxis to restrict ourselves with explanation in calculus (see, e.g.,[5]). Explanation. A function f on the real-number line is a rule which associates to each real number x a uniquely specified real number written f(x) and pronounced f of x. Unfortunately, there are situations when this explanation is given as the definition. This leads to perplexity. The term rule is not a basic notion in mathematics. 52

53 2. Rules One never uses the notation (x,y) F in respectable society when f is a function. We often use the notation f : x a y. For instance, f : function for which f ( x) = x. We write x a x is another way of saying that f is the f : X o Y or X o Y to mean that f is a function out of X into Y. Here Dom(f)={x y Y ( f : x a y) } is called the domain of f, and Ran(f)={y x X ( f : x a y) } is called the range of f, or the image of f. We note that the domain of f is to be distinguished from the start space X and the range of f is to be distinguished from the target space Y. The words map and mapping are synonyms for functions. A function f : X o Y is called a total function, or a function from X to Y, if Dom(f)=X. Then the notation f : X Y or X Y is used. The definition of a function identifies a function with the specific ordered triple, and so does the program, popular among mathematicians in the last century, postulating that all mathematical objects should be sets. In practice, of course, almost everybody thinks of a function out of X into Y as a rule f which assigns an element f(x) of Y to some x X. What about the term rule? There are several meanings: 1) law or custom which guides or controls behavior or action; 2) decision made by an organization; 3) something that is the usual practice; habit; 4) government; authority; 5) a strip of wood, metal, etc, used to measure. Obviously if we like to clarify the term rule we ought to consider the term law too. There are some meanings which correspond to the term rule : 1) the laws as a system of science; 2) rule of action or procedure, especially in the arts or a game; 3) factual statement of what always happens in certain circumstances. f f 53

54 There is some uncertainty in usage of the term rule. This term is too general. There is no strict limits for usage of this term. This term is not the term of the formal mathematics. Therefore, mathematicians avoid use of this term in definitions. Nevertheless the term rule is used in explanations. The problems concerned to an inverse function are discussed in [2,3]. 3. Basic notions If we restrict ourselves with naïve set theory we can increase the number of basic notions. We recommend the ordered pair as a basic notion. Then we can add n-tuple as a basic notion too (see, e.g., [6]). These notions are intuitive clear (see, e.g., [1]). Now we can define the Cartesian product and a function. Subsequently, we demonstrate several examples such as y=x, x,if x < 0, y = x = x,if x 0; x sin x, if x 0, y = x, if x 2; y = = max{ t t Z and t x}. References 1. V. Detlovs. Matemātiskā loăika. Zvaigzne, Rīga, R. Kudžma. Inverse function. Which one? Proceedings of the 4th International Conference Teaching Mathematics: Retrospective and Perspective, Tallin, 2003, R. Kudžma. Inverse function and semiotics. Proceedings of the 5th International Conference Teaching Mathematics: Retrospective and Perspective, Liepāja, 2005, M. Ó Searcóid. Elements of Abstract Analysis. Springer-Verlag, J. Marsden, A. Weinstein. Calculus I. Springer-Verlag, Л. А. Скорняков. Элементы алгебры. Москва, «Наука»,

55 A REVIEW ON RESEARCH WORK OF PUPILS OF LATVIA ( ) Andrejs Cibulis, University of Latvia, Abstract: Pupils contest (research) papers are considered. The attention is focused on pupils mathematical achievements. Keywords: convex polygons, disentanglement puzzles, pentomino twins, polyominoes, pupils research. The National Conference for Young Scientists is organized annually in Latvia. Twelve contest papers in mathematics were presented in 2006 and only seven in The winner is always asked to defend his/her research work in elimination contest, and if the person successfully meets the requirements of the jury then he/she is nominated to participate in the European Union Contest for Young Scientists. To gain some idea on the content of the pupils research let us mention some of those presented at 2006 Conference: Calculation of Centroid of Zemgale; Compatibility Problem for Polyhexes; History of Mathematics in Textbooks; Invariants and Quasi-invariants of Function; Methods of Solving Problems of Mathematical Olympiads of Latvia ( ); Partitioning of the Rectangle 6 10 into Pentomino Twins; Polyominoes. The topics of 2007 are as follows: Analysis of Curves; Analysis of Disentanglement Puzzles; Application of Graph Theory to the Tram Network Optimization in Riga; Construction Tasks in Geometry; Golden Section as an Intermediate Link of School Subjects in the Learning Process; Graphical Method of Finding Extrema; Research of Mathematical Model of Fractals. The four best submissions of the pupils of Latvia have been briefly considered in this review. A. Vihrovs (Riga, Form 11) studied a difficult problem in plane geometry: Can a convex plane figure F be embedded in a right triangle T whose area A(T) does not exceed the doubled area of the figure F? The problem was positively solved in his paper for triangles, rhombi, parallelograms, trapezoids as well as for circles and regular polygons. For the circle and some regular polygons the minimum of the ratio A(T) : A(F) has been determined. The proof for any trapezium has been obtained independently and by techniques other than those of Prof. M. Werner [1]. The solution of this problem for arbitrary quadrilaterals is recommended as a continuation of this work. The Power-point presentation of his research and that of some other pupils is available in [2]. 55

56 In two research papers (2006) the attention was paid to pentomino twins (p-twins) problem. A pentomino is a geometric shape formed by joining five unit squares edge to edge. There are twelve different pentominoes, see Figure 4. Pentomino twins are two equal shapes (polyominoes) that can be assembled of pentominoes. Pentominoes are known to have been first featured in H. E. Dudeney's The Canterbury Puzzles, Dover, New York, 1958, pp The problem of pentomino twins for the rectangle 6 by 10 has been solved by Olga Mihailova (Krāslava, Form 12) in her research paper (2006). The problem can be reformulated as follows: divide the rectangle into equal parts each coverable by pentominoes so that all 12 of them are used. Twins problem for the rectangles 4 15 and 5 12 has been solved in [3]. The idea of how to find p-twins for the rectangle 6 10 is very simple. If we have the eligible printout of all the solutions of the rectangle 6 10 then it suffices only to examine which ones give p-twins. The realization of this idea is not such a simple task. The printout of all the solutions, namely 2339, was obtained by computer programme elaborated by Atis Blumbergs (Nereta). The analysis of the solutions was made (as scrupulous handwork) to find the ones that can be divided into equal parts, i. e. giving p-twins. Some statistical data from Olga s work are as follow: 6 The length of the shortest common border of p-twins (It is a straight line); 24 The length of the longest common border (See bold line in Figure 1); There are 527 solutions (out of 2339) giving twins; There are 28 solutions giving double twins (An example of such twins is shown in Figure 2); There are 64 different separating lines or common borders of p-twins. This is also a number of all p-twins for the rectangle Figure 1 Figure 2 Figure 3 56

57 Challenge task. Find a pentomino twins solution for the rectangle 6 10 having a straight separating line. Do it when the separating line is as shown in Figure 3. Let us emphasize that the straight separating line occurs 16 times in 2339 solutions for the rectangle 6 10, but the line in Figure 3 only once. Inga Saknīte and Madars Virza (Valmiera, Form 11) investigated substantially more difficult Pentomino twins problem, namely: find all pentomino twins fitted on a chessboard 8 8. The four free squares are the ones creating most of difficulties now. Two examples of pentomino twins having the straight line as a common border are shown in Figures 4 5. Further we are going to deal with pentomino twins covering 60 squares of a chess-board 8 8 and use the notion p-twins as above. Figure 4 Figure 5 Figure 6 I. Saknīte started investigating the Pentomino twins problem in 2005 and worked alone for some months. Realizing that the problem has too many solutions and it would be a wild goose chase without a computer she addressed M. Virza having good computer skills. Elaborating a computer programme for this problem is, in no case, an easy task. However, M. Virza was able to create computer programmes by means of which one can find all p-twins. The first algorithm found all the ways how to divide the chessboard into two equal parts. The number of such partitions (92 263) was known earlier. The number of partitions of 2n 2n chessboard into two congruent edgewise-connected sets, taking partitions equal under rotation or reflection only once, one can find in the famous book of sequences, see [4]: 1, 6, 255, 92263, Double twins also exist, see Figure 6. After all the separating lines (or joint borders for 32-mino twins) were obtained, the second algorithm examined both the parts trying to cover 57

58 them with 12 different pentominoes. Notice that two squares of these 32- mino twins remain uncovered. Hence, it is obvious that one and the same separating line may generate many p-twins (see Figures 4 5). In its turn, it seems to be not so evident that two different separating lines may generate the same p-twins. The example illustrating such a situation is given in Figure 3 where the second separating line is a straight line. The third algorithm stated that only 440 borders out of correspond to p-twins. Three of such borders are shown in Figures 7 9. For the above-mentioned reasons the correspondence between these 440 borders and p-twins is not one-to-one. Let us observe that the borders in Figures 7 8 consist of the segments of equal length. Exercise. Prove or disprove that there are no other borders having this property. Additional efforts were made to select twins that would be good for creating mathematical toys. Pentomino twins having a unique solution are of special interest. To find such twins one has to examine the solutions of two types: 1) with a fixed boundary; 2) with fixed free squares; 3) with a fixed distribution of pentominoes. It is not difficult to prove the uniqueness of the solution of p-twins with boundary shown in Figure 9. It is not so easy to find p-twins for the chessboard with fixed four squares c3, c6, f3 and f6. But it is extremely difficult to find p-twins consisting of pentominoes F, N, T, U, X, Z and I, L, P, V, W, Y respectively. In the last case the solver knows neither a border nor positions of free squares. For producing a qualitative mathematical toy I would like to recommend colouring these two groups of pentominoes in two colours. Figure 7 Figure 8 Figure 9 Challenge task. Find p-twins on the chess-board with a common border of the minimum length and the maximum length. The length of the common border shown in Figure 6 is equal to 6. However, it is not a minimum length. 58

59 The research work Analysis of Disentanglement Puzzles by Anna Jansone (Form 11, Riga State Gymnasium No. 1) was the best one in These puzzles can be broadly described as rope and string puzzles. The book [5] is highly recommended as an introduction to disentanglement puzzles, see also [6]. The lion s share of A. Jansone s work is the description and investigation of the disentanglement puzzles from the collection of Ernests Fogels (Figure 10). She solved also some more difficult puzzles as those from Fogel s collection, for example, the exchange puzzle Narrow Escape of International Puzzle Party 25 (Helsinki, 2005) of Dirk Weber designed by Markus Götz. Remark. The purpose of the International Puzzle Party is to provide an annual forum for serious puzzle collectors for the exchange and sale of puzzles, books and related items, as well as for fun and fellowship. In short, it is a puzzling fun for puzzling enthusiasts. International Puzzle Parties have been held every year since Some conclusions made by A. Jansone in her research paper (see also the presentation available in [2]): The Fogels collection of disentanglement puzzles is arranged, i. e. these puzzles have been solved and supplied with instructions in the form of photos. Several other disentanglement puzzles not found in the Fogel s collection have been investigated. During the research 14 books (in German, Russian, English, and Latvian) have been studied. A great experience is achieved in solving disentanglement puzzles, thus few new puzzles have been created. A joy one expears during a research is invaluable, especially when one succeeds in solving puzzles that seemed having no solution at a first glance. Remark. E. Fogels ( , a Latvian scientist), devoted the last years of his life to the Riemann hypothesis. For more information see [7], [8]. However, there is no information about his hobby Disentangling Topological Puzzles in the known four biographic sources for E. Fogels cited in [9]. 59

60 Figure 10. Puzzles from the collection of E. Fogels. References [1] Mögling Werner, Über Trapezen umbeschriebene rechtwinklige Dreiecke, Wiss. Beitr. M.-Luther-Univ., Halle-Wittenberg, 1989, Nr. 56, [2] [3] Cibulis A., Pentamino. I daĝa, Rīga, Latvijas Universitāte, 2001, 95 lpp. [4] [5] Zhang W., Exploring Math Through Puzzles, Key Curriculum Press, 1996, 120 lpp. [6] Horak M., Disentangling Topological Puzzles by Using Knot Theory, Mathematics Magazine, 2006, vol. 79, No. 5, pp [7] [8] [9] 60

61 OBSERVATION OF AGE FACTOR TEACHING MATHEMATICS TO PART TIME STUDENTS Sarmīte ČerĦajeva, Ilze Jēgere, Latvian University of Agriculture, Abstract. The study process is being improved continuously but the main aim of studies remains permanent students must be given an opportunity to acquire knowledge of good quality and skills to provide their further professional career. By successful co-operation of two independent parts tutors and students the aim can be reached. Learning process consists of interaction between teachers and students sharing aims, interests and activities. The teacher provides a qualitative study process with his/her life experience, proficiency and complex of approaches. The student is a person eager to gain the teacher s experience and knowledge in order to develop his/her own knowledge, creativity and ability to take independent decisions and actions. Significant attention should be paid to the quality of math studies as the role of mathematics increases not only in scientific research but also in study process using software programs. The teacher has to evaluate level of knowledge of a particular student and a group, has to evaluate students abilities to work independently and in cooperation with group mates, has to assess the gaps in students background. Thereafter, the lecturer can develop appropriate syllabus and teaching methodology. This is the main problem of pedagogical practice. Most university applicants have problems with the bulky revision material. Therefore it is important to organize the use of visual aids and create positive atmosphere in the way of communication. Thus students are willing to achieve good results. In order to organize study process the tutor should take into consideration particular psychological individualities of applicants and their behavior during classes. Keywords: mathematics, the study of mathematics in higher educational institutions, the style of learning, adult education, educational discourse. There is a significant inadequacy between the growing number of demands for qualified employees and potentialities of education, further education included, to provide sufficient number of specialists in the most common professions on demand. Labor market circumstances claim for ability to upgrade one s qualifications or even re-qualify in a short period of time. The economics now is being redirected from producing to development of service industry, because of the following reasons: 61

62 according to researches in Europe, every year 10% of institutions and enterprises discontinue their activity; 10% of traditional jobs disappear and they are replaced with new technologies; labor force is ageing, whereas technologies are in continuous progress Lack of qualified labor force is the most current problem of the world. Implementation of new study curricula with broader potentials for specialization and additional branches of the basic profession is the only possibility to provide welfare and competitiveness in the labor market. Nowadays, for that reason, we should focus on implementation of a united curriculum for further education. The complicated situation of the Latvian economy, rapid changes in politics and education, unemployment, disability of Soviet-born personalities to adjust the new lifestyle will hopefully motivate the non- Soviet young generation to create a democratic and human community. When creating an open-minded and democratic community, every single individual must get the learning experience that enables adequate and free activities. Within the framework of the UNESCO World Education Forum in Dakar, 2000, the program Education for Everyone was ratified. The program makes provision for continuous development and changes in education, giving special emphasis to the necessity of life-long education. In the modern world the term educational institution is replaced with the term the learning society. Thus the basic pedagogical terms like education, learning, knowledge and skills turn into new ones. Nowadays learning comes under acquisition of new knowledge and skills to keep up with the demands and needs of the changing era. Hence, the approach to basic skills becomes more extensive - traditional reading, writing, speaking and doing sums skills are supplemented with a wide range of extra skills like ICT, foreign languages, entrepreneurship and social skills (for example, self-learning skills). These changes also state the continuity of life-long learning. Learning processes are not only related to schools, but also to: workplaces and professional environment where the newly acquired knowledge and skills are applied; home and social life conditions by extending experience of social skills and comprehension of current events in the society By this way, learning tends to carry out roles and functions of social nature outside schools and universities. The modern individual realizes that learning gives opportunities to improve the quality of life and his/her job. Mr. P. Jarvis points out, that reflexive learning characterizes the modern life. This condition makes teachers/tutors understand the importance on 62

63 self-expression and interaction skills instead of acquisition of knowledge in various subjects. First of all, the curriculum should focus on learning foreign languages, development of self-assessment skills as well as ability to interact successfully with others and seek for original solutions. The chairman of UNESCO Mr. Kochimiro Macura already expressed the issues during his visit in Latvia at the end of 2001: 1) promotion of life-long learning; 2) creation of new knowledge; 3) research initiatives. More expanded version of these ideas can also be found in Life-Long Learning Memorandum edited by the European Commission, which was broadly discussed in the Latvian society in According to this issue, educational institutions are evaluated by their input to improve the quality of life. Isolation of each education level and definition of aims within separate levels are the most important obstacles that prevent the reformation of education system. According to the statistical data, 0, 6% of residents do not continue their education after the age of 15 and 0, 2% is analphabets. Taking into account these numbers, it is important to create a flexible lifelong education system in order to increase the role of schools in educational environment. The necessity for life-long learning also changes the learning discourse in schools the emphasis is being directed towards independent learning. All school curricula should be supplemented with a course that enables the development of independent learning skills such as ability to maintain/control self-learning, ability to solve problems and approach issues analytically and ability to assess the result. One more important issue of life-long education is qualification upgrades of academic staff in terms of pedagogical knowledge and skills. In this relation it is important to point out the role of the international project Don t be behind your knowledge, keep pace with your potentialities, started in Enthusiasts and teaching staff of adult education unions from the Baltic and Nordic countries took part in the project. Since W. Humboldt, university education, in its traditional meaning, has been related to research-based scientific activities, excluding acquisition of students professional skills. Nowadays the term research has been extended in order to include practical experience and effect of the knowledge acquired. In 1999 during the UNESCO Congress of the World Science it was even suggested to base the fundamental scientific researches on the priorities that enable the national welfare and provide continuous growth of national economy. Education needs are not determined by the labor market only, but also by the measures taken against negative stimulants (drug addiction and alcoholism). R. J. Hagigherst s research on 63

64 relations between various stages of age and needs for education displays the following data: 1) year olds relate their education potential to career and employment prospects; 2) year olds consider their education as the way of selfexpression; 3) year olds find their education to be the way of expressing beliefs and convictions; 4) people above their 50ies prefer ways of education that stimulate thoughts, analysis and evaluation. At later stages of life people limit their activities and tend to focus on education of interests only. In order to stimulate adults to learn, several principles of adult education should be taken into account: 1) during the learning discourse the learners are provided with conditions to express themselves, gain experience, think and read independently; 2) the focus is laid on critical reflection, that is based on the processes such as altered assumptions, newly found alternatives to the ideas, activities and opinions assumed previously, and insight to inconformity of ideas. Development of critical reflection helps individuals to expand awareness, and decline one-sided opinions to become more difficult to be manipulated with: 1) improvement of personal experience, stimulus-based motivation for further education by initiating the inner conflict between the present experience and new needs; 2) provision of possibility of learning to learn; competence and independent cognition potentialities are widened Learning is becoming a way of life step by step. Analysis of the statistical data in connection with the education level in the European countries proves the fact of conscious and fruitful life-long learning resulting in productive professional and social activities of every individual. There are full and part time courses of mathematics available in the Latvian University of Agriculture. The most important difference between part and full time learning courses is the variety in age, learning breaks and learning regimes of the part time students. Each of the groups will be described below. A postgraduate who takes up a part time course is able to attend all classes, is eager to keep his/her usual learning regime which includes teacher's explanations, continuous learning process with his/her fellow students; when classroom activities are focused on doing sums on the 64

65 blackboard, the teacher sets up home task, the acquired material is marked gradually by sections and the marks affect the total result of the term. Through this particular learning style the new material is based on elements of knowledge and skills that have been acquired before. Usually the introduction to a new topic takes approximately one part of the lesson. A student who has had a learning break is eager to spend a longer period of time on revision, to focus on the discourse in general as well as each new section in particular. A part time student usually does not associate a new topic/section with the knowledge and skills that he/she has acquired before because the learning break has affected not only the real knowledge but also his/her views about the knowledge. As this type of student has forgotten the most part of learning strategies, much time is spent on persuading the student that any work can be done with proper input. The student who works and learns at the same time and is able to attend only part of the classes offered, desires to get short and detailed information. He/she frequently ignores homework, but he/she wants to receive concise, comprehensible material with samples suitable to individual work. The part time student finds an advisor in the teacher who will satisfy the student s needs. Testing should not be frequently planned and it should definitely be at the time suitable to the student. A student above twenty years is quite experienced; attitude towards learning is more serious, motivated and focused on acquiring knowledge. Learning goals may be related to profession, earning good money, selfexpression, widening view points about the world and keeping up with friends and peers. Main issue of the modern pedagogy is interaction between the student and the teacher; it includes psychological and practical preparation, realization and evaluation of the activity. Proficiency of the teacher consists of his/her ability to approach the students so that they feel aroused, interested and the atmosphere gets positive in the class. During the process of changes in education, the approaches and methods chosen by the teacher are much more important than the program and textbooks because a new program, textbook or a software program will not improve the quality of education if the teacher does not appear to be professional. Methods like cooperative method, individual work and self-evaluation are often applied in the learning discourse. As the students are interested in their own education, they are active and do not hesitate to ask questions during the classes. The new environment and the aim to learn to use math in real life help to stimulate motivation, independence and ability to focus on doing mathematics. 65

66 Four approaches are applied to teach mathematics at university: 1) group work to find, comprehend, analyze and exchange information in the way of projects; 2) frontal work the teacher explains information in classroom; 3) cooperative or pair/group work; 4) individual work practice of different skills, assessment and self-evaluation. Specific character of part time learning discourse and use of ICT require improvement of the existing learning program. Departments of Mathematics and Physics have started the project Modernization of mathematics and physics discourse in the LUA supported by ESF. It is planned to increase the quality of the subjects study programs, methodological materials through activating ICT and approaches based on necessary competence for the labor market as well as creating challenging atmosphere for learning. The best result anticipated practical IT classes that will give to students the opportunities to get acquainted with the available software programs to do their tasks in mathematics. In addition, it will improve the skills to use ICT. In order to activate practical use of knowledge in the specialty needed and competence in mathematics considering the aspect of professional use, in the framework of the project it has been planned to improve several study courses of advanced level, as Mathematical modeling of systems. The courses mentioned above are mainly focused on students of engineering programs. Next courses, such as Use of mathematical approaches in economics and business mathematics, will be offered to the students of economics and social sciences. 66

67 WHAT IS MORE IMPORTANT THAN SCALARS? Laura Freija, Sandra Zabarovska, University of Latvia, Abstract. This report considers the Latvian education system in context of teaching vectors. We also describe the reasons why vectors are great in mathematics and consider some examples of problems. We suggest solutions for issues that we have discovered when interacting with students. Keywords: cognition, geometry, vectors. Introduction The title of the report is too pretentious in order to be provocative. The answer is supposed to be vectors. In reality it is not possible to decide which is more important vectors or scalars. Here we list several reasons, why vectors are indeed very important for education of gifted students. Let us begin with a more general description of education system. General framework Mankind has so far discovered four kinds of cognition: emotional, empiric, deductive and in the 20 th century modeling. The duty of a school in forming personalities is to introduce all four kinds of cognition to the students. Emotional cognition is developed mainly through literature and art subjects, the empiric cognition is introduced via natural sciences and algebra, modeling in programming and the deductive cognition almost only in geometry. It is rather difficult to teach and comprehend geometry, therefore the grades are lower. In order to give the impression of rising level of education both algebra and geometry were merged into one subject mathematics. Because geometry is so difficult, teachers prefer to concentrate more on algebra so geometry is fading from school curricula. The actual level of education is therefore dramatically falling, because one kind of cognition is left out. To prevent this, geometry must be introduced to students in every possible way. The topic of vectors is one of the possibilities because vectors form the link between different parts of mathematics, especially algebra and geometry, therefore allowing geometry exercises in algebra. Reasons and examples There are other great reasons to study vectors. One of them is that there are many problems, that can be solved using various methods, and using 67

68 vectors may be pleasant. There are even problems that can be solved using vectors, but not otherwise (at least on school level). Another good reason to use vectors in studies is to cause positive emotions. They are an important element in learning. Vectors can be a cause for positive emotions in students, because there are problems that can be explained shortly using vectors, but take a great effort otherwise. Consider example Nr.1. Example Nr.1. Prove that at least one of numbers ac+bd, ae+bf, ag+bh, ce+df, cg+dh, eg+fh is larger or equal to zero if a, b, c, d, e, f, g, h are nonzero numbers. If the numbers are considered as coordinates of four vectors (a,b), (c,d), (e,f), (g,h), the given expressions can be considered as inner products of the four vectors taken pairwise. Then if the initial points of the vectors are placed together, the sum of successive angles between these vectors is 360 therefore one of the angles is smaller or equal to 90 degrees. If it is φ then cosφ is larger or equal to 0. The inner product of vectors that create angle φ is (x,y)(w,z)= (x,y) (w,z) cosφ obviously larger or equal to 0. Solving this problem using algebraic methods requires two pages (loose handwriting) of very careful writing. Examples Nr.2 and Nr.3 illustrate one more reason to use vectors. Problems in geometry may require a high level of abstraction and understanding. Sometimes that kind of problems can be solved using vectors. Students who know vectors well can solve problems that they couldn t easily solve by synthetic geometric reasoning. Example Nr.2. Three parallelograms are constructed using the sides of a given triangle. Prove that we can build up a triangle with sides equal and parallel to the line segments a, b and c. (See Fig.1.) a a s p c c 68 b b Fig.1. r Fig.2. The line segments can be turned into vectors (see Fig.2.). Circuit is confined so the sum of these vectors is 0 r. Sum of vectors p, r and s is 0 r. So the sum of vectors a, b and c is 0 r.

69 Solving it geometrically requires imagining how to place segments a, b and c together to form a triangle. It is much harder to imagine correct transformations and shifts than to imagine the solution with vectors. Example Nr.3. M, N, K, L are midpoints of pentagon s ABCDE sides AB, BC, CD, DE correspondingly. Quadrangle MNKS is a parallelogram. Prove that SL AE. (See Fig.3.) In order to prove that the lines AE and SL are parallel, we can prove that vectors AE and SL are collinear, expressing them as linear combinations of other vectors. B N C A M S K D L Fig.3. E It appears that the expressions are proportional, what is sufficient. The details are left to the reader. A synthetic solution requires a lot of additional lines to be drawn and is definitely much longer and more complicated. Curriculum the history All the reasons listed above make us feel uneasy about the changes in mathematics curricula in Latvia during the last 30 years. In 1976 teachers had a unified obligatory curriculum for all the students. In this program vectors were considered in much detail and had references to foundations laid in students in middle school. Program itself was precise and there was an associated problem book with problems corresponding to each item. In curriculum of 1986 demands for students were still sufficient but the program was not as elaborate as the previous one. The main topics on vectors were presented in the last grade of middle school and they appeared again in the last year of high school. The curricula of 1995 are still used today. There is no one obligatory curriculum but there are requirements for graduates, and teachers use them as a guide when choosing one of several recommended curricula. On one hand it sounds positive - teachers have more freedom. But it also requires a lot more work and preparation on part of the teacher, because the 69

70 formulations are very general. With the workload our teachers have it's very likely that preparations will not be done properly. There are two types of programs - for basic course and for extended course. The basic course curricula contain very little on vectors. The extended course curricula consider vectors on a satisfactory level. That means that students who do not take the extended course, will receive minimal knowledge about vectors, at best. At the worst, they might not learn about them at all (for example, if the student changes schools). Students who take the extended course, however, are rarely capable of solving vector problems of olympiad-level difficulty. A few months ago the government has confirmed new requirements for high school but unfortunately it does not improve the vector situation. There is a new curriculum under construction. Hopefully it will have more on the topic of vectors. Issues from experience Just as for the extended course, one of the tasks of Latvian University s A.Liepas CMS (Correspondence Mathematics School) is to prepare students for solving problems on a higher level of difficulty, often found in mathematics olympiads. CMS has developed a teaching aid on vectors; the first part of it is published. On the basis of this book four booklets are formed for high school students which are sent to them via internet/ordinary mail during the school year. They contain elements of theory and problems for independent solution. The students have 1 month to cope with each of these booklets. It should be so that students who take the extended course can solve CMS s problems properly and students of the basic course can do it after they study the theory booklets. Unfortunately the assessment of students' solutions show, that even those taking the extended course are not capable of solving these problems. While in the first set, where the problems were easier, most students got about half of the maximum possible score, the results fell rapidly with every next set. It was the same with the number of participants. 100 students subscribed to participate in the event. For actual number of participants see table Table 1. 1st set 2nd set 3rd set 4th set Students Total maximum Average results 43, , , ,1666

71 Average results excluding most excellent 42, , , ,2 Studying the results we discovered the following issues: 1. Many students don t use the theory booklets, but struggle using only what they know in advance. 2. Students feel uncertain if the way of solution isn t a standard one. 3. Some classes of problems are especially hard: a) Those dealing with n vectors, n points etc., instead of a fixed number of them; b) Multilevel problems; c) Problems, in which the proof is required. Suggested solutions Issue Nr.1 is an attitude issue, so its solution might be in CMS s cooperation with participants teachers. One way to solve issue Nr.2 is to make the problems more familiar. Right now most of them are very non-standard problems. Problem Nr.3a might have been caused by failures in the theory booklets all the examples in them dealt with fixed number of elements. Issues Nr.2, Nr.3b, Nr.3c could be solved if students would see more different versions of problem solutions in school. Mostly students see just algebraic way of coping with vector problems which uses operations with vector coordinates. The issue with problems requiring proof is that students aren t familiar enough with the deductive thinking. The proofs are under the threat to be omitted from the official curricula at all. This issue needs to be reckoned with on the government level and solved by returning geometry back to schools. In our opinion, along with the operations with linear expressions of vectors the following topics are highly welcome for students interested in mathematics: Scalar/ inner product of the vector; Pseudoscalar/ outer product of the vectors; Mixed product of the vectors; Rotation; Physical interpretations of the vectors and their applications to mathematical problem solving. 71

72 Conclusions The issue that students do not perform great in solving vector related problems is not caused by vectors being a difficult concept to grasp. It is rather caused by the education system in general particularly the careless attitude towards mathematics and geometry. To improve the situation it is not enough to have rare attempts at changing the system. A great amount of work on the highest levels is necessary to return proper teaching of mathematics to schools. The ultimate goal is for students to understand mathematics. Vectors are powerful means to achieving that goal. References 1. A.Andžāns, L.Ramāna, B.Johannessons. Vektori. 1.daĜa. Rīga. LU Akadēmiskais apgāds, V.Klopskis, Z.Skopecs, M.Jagodovskis. Ăeometrija klasē. 1.daĜa. Palīglīdzeklis skolotājiem. Rīga: Zvaigzne, V.Klopskis, Z.Skopecs, M.Jagodovskis. Ăeometrija klasē. 2.daĜa. Palīglīdzeklis skolotājiem. Rīga: Zvaigzne, Latvijas PSR Izglītības ministrija. Matemātikas programma klasei. Rīga. Zvaigzne, Izglītības satura un eksaminācijas centrs. Matemātika. Ieteicamās mācību programmas klasei, Rīga. Rīgas 15.arodvidusskola,

73 COMPETENCY OF MATHEMATICAL THINKING AND ITS VERSION OF APPLICATIONS Rasma Garleja 1, Ilmārs Kangro 2 1 University of Latvia, evf@lanet.lv, 2 Rēzekne Higher Education Institution, kangro@ru.lv Abstract. The article analyses the role of learnability and utility of study process elements (computer mathematical systems (CMS) and teaching/learning aids for CMS) in developing mathematical thinking competency (MTC) (namely, variety and validity of cognitive methodology as a part of MTC). The main techniques of MTC include a) possibilities to practically analyse the relevant problem and to evaluate suitability of the accomplished action; b) to employ the full extent of one's intellectual capabilities in the course of these actions. Keywords: Computer mathematical systems, learnability, learning style, mathematical thinking competency, teaching/learning aids, utility. Introduction Mathematical thinking competency (MTC) is a component of human individual mental experience, which depends on one s knowledge, experience, understanding of a certain issue and skills to apply this knowledge and experience in a particular situation [3]. From such a perspective MTC is crucial in studying various objects or phenomena because it involves both theoretical and practical approach to acquiring new, specialised knowledge [21]. Methodology In the light of theoretical and practical knowledge applications we analysed development of MTC components [3] variety and validity of cognitive methodology, the principal implementation techniques of which are [3]: a) possibilities to practically analyse the relevant problem and to evaluate suitability of the accomplished action; b) to employ the full extent of one's intellectual capabilities in the course of these actions. By associating acquisition of theoretical and practical knowledge with the new information technologies (e.g., computer mathematical systems (CMS) and CMS-based teaching/learning aids [6]), three sign problems (student s language mathematics language training (including teacher s) language) [17], the modern solutions require investigation and application of the common and different characteristics among [2, 4]: 73

74 1) CMS (Maple, MatLab, etc.), teaching/learning aids and mathematics; 2) CMS, utility and learnability of teaching/learning aids; 3) CMS, teaching/learning aids language and mathematical language. The usefulness of teaching/learning aids (instrument) can be determined for a wide variety of users (subject), which enables the best possible evaluation of the theoretical and practical implementation aspects of the relevant problem [18], i.e., taking into account the users intellectual capabilities learning styles [7], but in the process of evaluating the usefulness of teaching/learning aids (especially those for application of CMS in the teaching/learning process) application and training possibilities thereof [14]. The accomplished study on developing theoretical and practical application aspects of MTC (period: academic years 2003/04 to 2005/06, research basis: Rezekne Higher Education Institution, Faculty of Engineering (1st and 2nd year students) and Economy (1 st year students), study fields: higher professional education study programme (Engineer- Programmer, Economist), higher professional education Bachelor study programme (Environmental Engineer), study courses: Higher Mathematics in Engineering Science, Mathematics for Economists, Mathematical methodology in Environmental Science and Computers, Theory of probability and mathematical statistics, number of respondents 592) shows a proportional parity of practical approach learning style (Table 2: 1) active, 4) pragmatic, 5) active pragmatic) and theoretical approach learning style (Table 2: 2) reflexive, 3) theoretical, 6) reflexive theoretical) 38% and 36% respectively. A striking finding is the considerable proportion of respondents with an equally pronounced theoretical and practical approach (Table 2: 7) active reflexive). The explanation (understanding, interpretation) of the concepts of CMS and application possibilities, training possibilities of teaching/learning aids is based on: 1) investigation and application of the common and different features of CMS (Maple, MatLab, etc.), teaching/learning aids language and mathematical language [13]; 2) Focal points of teaching mathematics with regard to engineering and technical problem-solving [13, 14]; 3) Possibilities and role of mathematical software (CMS) in acquisition of engineering mathematics [13]; 4) The role of modelling in combining two major cognitive stages of the scientific research empiric and theoretical (usage of conceptual models) [21]; 5) Practical applications of structural elements of mathematical models and programming language [4]. The student classification in terms of the chosen application possibilities and training possibilities was based on the study process analysis [6] depending on the stages used in the solution process of the task (problem) (Table 1) (1. Problem statement; 2. Solution process planning 74

75 (development); 3. Final outcome), where selection and implementation of all three stages corresponds to the following classification: use of training possibilities or learnability (LAB-Y), but selection and implementation of two or just one stage use of application possibilities or utility (UT-Y). The establishment and acquisition of the study course was based on the invariants of the respective scientific discipline [12]: 1) Study subject related (special); 2) Logical; 3) Psychological. It must be added that motivation to act plays a significant role in evaluation of efficiency and competence of an individual s performance [1, 15]. The strengths and weaknesses of the respective (each) learning style [8], their possibilities and significance from the theoretical and practical aspect of the relevant task (problem) were investigated and complied with, namely, problem (task) statement, development of solution plan and its implementation (tasks, exercises [9, 10], application of CMS and teaching/learning aids, implementation of Intellectual actions gradual development theories (IAGDT) stages [19]), where the following means were used [6]: 1. Types of information coding (a) verbally symbolic; b) visual; c) objectively practical, d) sensitively sensory); 2. Cognitive patterns; 3. Semantic structures. Results In the context of theoretical or practical (approach to) acquisition of new knowledge CMS were used [4, 6]. The symbol-based data tracing form of CMS (Maple, MatLab, Derive, etc.) preserves the conventional mathematical data tracing form and enables a clear visualisation of the relevant algorithm when operating with numerical and symbolic values. The representatives with theoretical and reflexive learning style are particularly interested in the relationship between mathematical and CMS software base knowledge (interdependence) and their compliance with the relevant problem (task), their use in solving algorithms and result explanation, etc. Whereas the representatives with active and pragmatic learning style are highly interested in usefulness of the problem (task): practical solving stages and practical implantation of solving algorithm components by using CMS (creating charts, figures, calculations, etc.). Student classification in terms of selection of training possibilities learnability (LAB-Y) and usage possibilities utility (UT-Y) at the beginning of the semester (approximately until the mid-semester ( beginning )) and at the end of the semester ( end ) respectively was carried out. A statistically significant percentage difference between the groups (LAB-Y) and (UT-Y) during the periods beginning and end was established using Fischer s multifunctional ϕ criterion [6, 20] 75

76 (applying ϕ criterion to each learning style individually). Table 2 shows that the empirical value ϕ for all learning styles exceeds the critical 76 emp value ϕ = 1, 64 (significance level α = 0, 05 ). By applying techniques of cr developing mathematical thinking competency a) practical implementation of the task and evaluation of suitability of the accomplished action; b) use of one s own intellectual capabilities in this process, it was established: 1. The research findings about the necessity of a gradual transition stage in the acquisition of mathematical concepts, namely, sign, concept object, meaning concept image, were confirmed (e.g., IAGDT stages [19]) [11]; 2. The necessity of CMS applications in task (problem) solving (algorithm construction, solving process, verification of the obtained result, numerical implementation of analysis and explicit visualisation); 3. Usefulness of a combination of both processes (CMS applications in the acquisition of mathematical concepts and implementation of IAGDT stages). 4. CMS and CMS-based teaching/learning aids enable full implementation of all IAGDT stages, granting the required attention to preintellectual action stages as well (motivation, establishment of action basis, the materialised outcomes, external manifestation of action, and internal manifestation of action). Furthermore, in each IAGDT stage it is possible to provide practical or theoretical description of and solution to the problem (task) required for the respective learning style. 5. The enumerated techniques of developing MTC a) practical implementation of the problem and evaluation of suitability of the accomplished action and b) use of one s intellectual capabilities complement each other and simplify their mutual implementation because investigation and use of the dominating (theoretical or practical) cognitive type (learning style), in its turn, enables successful problem-solving and evaluating suitability of the accomplished action. References 1. Bandura, A. (1997) Self efficacy. The exercise of control. New York: W. H. Freeman. 2. Bespalko, (2002) Education and learning with computers participation (The Science of the Third Millenium). Moscow Voronesh: «МОДЭК». (in Russian). 3. Garleja, R. & Kangro, I (2005-1) The formation of competency of mathematical thinking during studying mathematics In.: Proc. of the Int. Conference: Teaching Mathematics: Retrospective and Perspectives, 6 th international conference, Vilnius, May 2005, pp , Vilniaus universitetas, 2005, (in Russian).

77 4. Garleja, R. & Kangro, I (2005-2) Complex management of mathematical knowledge and skills. In.: Teaching Mathematics: Retrospective and Perspectives. 5 th international conference (Ed. E. Ăingulis) Liepāja: LPA, Pp (in Latvian). 5. Garleja, R. & Kangro, I (2006-1) Development of the basic components of competence of mathematical thinking during studying mathematics. In.: Proc. of the Int. Conference: Teaching Mathematics: Retrospective and Perspectives, 7 th international conference, Tartu, May 2006, pp , University of Tartu and Estonian Mathematical Society, 2006, (in Russian). 6. Garleja, R. & Kangro, I (2006-2) The importance of computer mathematical systems during formation of the competence of mathematical thinking in Rezekne Higher Educational Institution. In: ATEE SPRING UNIVERSITY TEACHER OF THE 21st CENTURY: Quality Education for Quality Teaching. (Ed. Erik de Vreede) Rīga: Izglītības soĝi, Pp , (in Latvian). 7. Garleja, R. & Kangro, I. (2004) Determining an individual cognitive style in the study process, in V. Ivbulis (ed.) Humanities and Social Sciences. Latvia, Education Management in Latvia, University of Latvia, 2(42), pp Honey, P. & Mumford, A. (1995) Using your learning styles. Maidenhead, Berkshire: Honey. 9. Kalis H., Kangro I. (2004) Mathematical methods in the engineering sciences. The textbook, Rēzekne: Rēzeknes Augstskolas izdevniecība, 2004, (in Latvian). 10. Kangro I. (2005) Tests in mathematics. Rēzekne: RA izdevniecība, (in Latvian). 11. Kangro, I. (2006-1) Theoretical and practical aspects of research into student s mathematical thinking. In: Pedagogy and Teachers Education. (Ed. I. Žogla) Rīga: Latvijas Universitāte, Pp (Scientific Papers University of Latvia, vol. 700), (in Latvian). 12. Kangro, I. (2006-2) Problems of organization of mathematic studies. In: Education Management. (Ed. A. Kangro) Rīga: Latvijas Universitāte, Pp (Scientific Papers University of Latvia, vol. 697), (in Latvian). 13. Kent, P. & Noss, R. (2000) The visibility models: using technology as a bridge between mathematics and engineering, Internationa Journal of Mathematical Education in Science & Technology, Jan/Feb2000, Vol. 31, issue Noss, R. (1999). Learning by design: Undergradyate scientsts learning mathematics, International Journal of Mathematical Education in Science & Technology, May/Jun99,Vol. 30, issue 3, Sternberg, J. & Kolligan, J. (1990) Competence considered. New York: Vail- Ballou Press, Binghamton. 16. Whitton, d. (2000) Revisions to Bloom s Taxonomy, Primary Educator, 2000, Vol. 6 Issue 1, p16, 6p. 17. Кудрявцев Л. Д. (1980) Modern mathematics and its teaching М.: Главная редакция физико-математической литературы., (in Russian). 18. Рабардель, П. (1999) Люди и технологии (когнитивный подход к анализу современных инструментов). М.: Институт психологии РАН. 19. Решетова, З. (2002) Formation of system thinking in training, М: ЮНИТИ- ДАНА., (in Russian). 77

78 20. Сидоренко, Е. В. (2002) Methods of mathematical processing in psychology, СПб.: ООО «Речь». (in Russian). 21. Щедровицкий, Г. (1995) Selected Works, Москва:.Культ.Полит. (in Russian). Appendix Table 1. Task (problem) solving stages 1. Problem (task) statement 1.1. Basic knowledge [4, 16] according to the relevant problem (task) Essence, cause and symptoms of the problem (task) Transition from a practical problem (reality) to a mathematical problem using assumptions, generalisations and formalisation [4] Mathematisation of the problem, task the relationship with mathematical concepts and selection of an adequate mathematical apparatus. [4] 2. Construction of problem (task) solving algorithm (development of mathematical algorithm) [5] 2.1. Division of the problem (task) into individual components Highlighting individual components, temporary excluding the other ones Construction of an auxiliary task (with a simpler structure) from the highlighted individual components General formulation of the solving algorithm Formulation of the most difficult algorithm construction stages and the inherent difficulties. 3. Final outcome: finding a solution to the problem (task) (implementation of solving algorithm) and evaluation of results 3.1. Practical implementation of the algorithm (using CMS) 3.2. Solution analysis: Compliance of the solution with the set objective: Significance of the solution: a) theoretical aspect (illustration and strengthening of the existing theoretical knowledge, obtaining new theoretical knowledge); b) practical aspect (checking the theoretical knowledge in practice); Application limits of the solution: a) in relation to possibilities of finding a theoretical solution (e.g., correctness of the differential equation solution, parametric stability, etc.); b) in relation to possibilities of finding a practical solution (finding an analytical and numerical solution, role and possibilities of CMS); 3.3. Significance and emotional intensity of the accomplished action and its results. Table2. Empirical value Learning style ϕ emp ϕemp of learning styles 1) 2) 3) 4) 5) 6) 7) 1,83 3,17 3,92 1,81 1,69 3,32 2,28 78

79 WHY IS THE EDUCATION OF TEACHERS OF MATHEMATICS PROBLEMATIC? Barbro Grevholm, University of Agder, Abstract. In many countries mathematical teacher education has been a subject of societal debate and criticism has been expressed. In this paper I present the problematique of teacher education in mathematics and discuss issues and concerns related to it, trying to understand why there is a problem. A longitudinal study of teacher education in mathematics in Sweden is used as an illustrating example and recent evaluations of teacher education in Denmark, Norway and Sweden serve as sources for finding criticism and problems. A model for teacher education seen as the development of a professional identity is presented and discussed. This leads to the suggestion of solving the problems with teacher education by experiencing it as a life long learning and the development of a professional identity. Keywords: Educational changes, mathematics teacher education, educational changes, evaluation of teacher education, research on teacher education. Background and problematique Mathematics is a subject that arouses strong feelings in almost everybody. It can create joy to learn or it can create anxiety and unpleasant feelings (Skolverket, 2003). The teacher has a crucial importance for the motivation and progress of pupils learning. Thus the education of good teachers of mathematics is a matter of concern for both the pupils who succeed and those who reject mathematics. In the efforts of society to educate teachers of mathematics we can discern problems with several aspects. The access to well educated mathematics teachers is rarely enough, the recruitment to the education is problematic and the effect of the education is often questioned. The way society has reacted to this situation has been to implement repeatedly new reforms in teacher education. Why do we have these problems and what are the characteristics of them? A longitudinal study of a Swedish teacher s group education in mathematics In a longitudinal study of student teachers I followed a group of prospective mathematics teachers for eight years (Grevholm, 2000, 2005). This group consisted of 48 students when they were taken up in the programme and after 4.5 years when the education was finished 25 of them 79

80 took their degree as teachers. Several crucial aspects of the teacher education were studied. The pre-knowledge in mathematics of the student teachers influences their learning outcomes in many ways. The motivation to study and engagement in the activities are conclusive. The recruitment of student teachers is difficult as there are often not enough applicants with the pre-requisites that are demanded and needed in order to be successful in the studies. As it can be seen from the numbers mentioned above the proportion of students who are able to fulfil the education in the expected time is low (50 %). When the student teachers take up their first jobs in school they are struck by what one could call a reality chock (Grevholm, 2003a). Videorecordings from the classrooms of these teachers and interviews with them give evidence of several kinds of problems. Mentors they have been promised as beginning teachers disappear too early, the lack of experienced colleagues leave them with little support in finding relevant teaching material and tasks. The preparations of lessons are time-consuming for newcomers in the teaching profession and they are left with limited opportunities to find time to have a private life. This makes many of them leave the profession as teachers rather soon in their career. Many student teachers have problems with the mathematics courses and fail several times in the examinations. Although they claim that the study programme is demanding and they work as hard as they can, it is not enough to create success for them. Mathematics and mathematics education takes up only around 20 % of the study time in the programme and still it seems as if the learning outcome in mathematics creates much problem (Grevholm, 2003b). Thus mathematics comes to play the role of sorting instrument of the students. The student teachers, who feel less successful in mathematics, carry this experience with them out in the classrooms and it might influence their own pupils in less favourable ways (Pehkonen, 2001). Once the student teachers graduate and start working in a classroom they find that their professional language is too limited. When they try to assist pupils during lessons they have problems to express themselves in such ways that pupils understand. The teachers also find it difficult to vary their explanations if the pupil does not follow the first explanation (Grevholm, 2003a). As a result of the longitudinal study, changes took place in the teacher education programme, in order to meet better the student teachers demands to develop a professional language. Natural study groups were introduced where student teachers were given appropriate tasks to discuss in group without a teacher present. This way they could exercise using a professional language and developing it (Grevholm, 2004a). Classroom management is crucial for beginning teachers. Videorecordings from the study referred to here, give evidence of different kinds 80

81 of problems. To be able to help all pupils that need help at the same time is a hard task when you are not skilled in using many different ways of working. To find appropriate tasks on different levels of difficulty adequate for each pupil is demanding and calls for access to and knowledge about different materials and methods. Pupils who get bored because of inadequate tasks can create disciplinary problems and newcomers in the teaching profession have difficulties to handle such situations. The competence to judge and diagnose pupils learning is crucial in order to be able to plan lessons that are appropriate for a specific group of pupils. Beginning teachers naturally are not as skilled in this respect as a more experienced teacher can be. Examples of beginning teachers offering too hard tasks to pupils illustrate how this can lead to a confrontation and feeling of failure for both pupil and teacher (Grevholm, 2003a). Other examples show how chaos can be created in the classroom if work-forms and material are not well chosen for the class. National evaluations of teacher education Recently the teacher educations in Sweden, Denmark and Norway have been evaluated and findings from the reports indicate the same kind of crucial issues (Högskoleverket, 2005, Danmarks evalueringsinstitut, 2003, NOKUT, 2006). The evaluation of the Swedish teacher education was done in 2004 after a new teacher education was implemented in The reform has contributed to the creation of educations with varied construction and varied content. The evaluators discuss quality and conditions for quality in the education. They notice that most of the teachers who teach, supervise and examine in teacher education do not have a research education. The teachers who have the most qualified scientific education, for example the professors, are rarely working in teacher education. The new teacher education offers the student teachers great opportunities to choose and form their own education and take responsibility for it, which places great demands on them. Based on this evaluation, Högskoleverket has decided to make a new evaluation within two years. If still serious problems exist in some of the institutions, Högskoleverket will question the right of these universities to examine student teachers. The evaluators recommend that the plans for the education are made more explicit and clear, the practice periods must be guaranteed in such amounts as is decided, to intensify the work with recruiting teacher educators who have a research education, to scrutinise and strengthen the levels of demands in the education, to assure quality of the examination theses and to invest in developing strategies for 81

82 work with quality assurance and evaluations inside teacher education (Högskoleverket, 2005). The Danish report (Danmarks evalueringsinstitut, 2003) is evaluating the outcome of a new regulation for teacher education, which was implemented by the seminars (where teacher education takes place in Denmark) from Much work in the teacher education seminars has gone into this implementation and mainly the evaluators conclude that the education works well in principle and in most areas the content is relevant for the work as teacher in compulsory school. But in Denmark, as well as in the other countries, the education has double aims, to educate teachers for compulsory school and to prepare the student teachers for further studies on bachelor or higher levels. Here the evaluators claim that the seminars have not come far enough in giving the students a well defined basis for further studies. The evaluators recommend raising the level of prerequisites to enter the education, and making changes so that student teachers see themselves as students not as pupils. The demands for students to participate in the education are not adequate and the evaluation reveals that many students participate rarely. Four main school subjects will still be included and the pedagogical subjects are recommended to be given in the two first years. They also recommend strengthening the subject didactics in the studies. From 2001 there are demands to include relevant research in the education. In Norway the evaluation contains a self evaluation part and external evaluation (NOKUT, 2006). Teacher education is regulated by a Rammeplan from 2003 (guiding framework) and thus the evaluation takes place in the middle of a process of change. The main impression expressed is that the education has varied quality, but the conditions to offer teacher education are also varied. A number of influencing factors vary between institutions, such as students participation in the studies, teacher educators engagement in subject didactic teaching and teacher educators participation in the choice of research and development works. It is a great challenge for the teacher education to integrate praxis, subject studies, and subject didactical and pedagogical theories. Theory and praxis seems to take place in different cycles and there is a lack of connections between different parts of the education. The evaluators suggest improved communication between different actors in the education and discussions that can contribute to common beliefs about what professionalism in teacher education and in the teacher profession is. Data from mathematics in the teacher education is sometimes taken as example in the evaluations. Thus, it seems, the same kinds of problems sustain and again the suggested solution often seems to be to reform the teacher education. But is it possible to solve problems related to 82

83 recruitment, pre-knowledge in mathematics, motivation, activity during the education and success in learning outcomes through reforms of the education? Or are we actually in a situation where the demands on teacher education are too complex and too many to be handled in a reasonable way during the basic education of teachers? I have claimed elsewhere (Grevholm, 2006) that there is a need to see mathematics teacher education as life long learning, and as the development of a professional identity. Some crucial aspects in mathematics teacher education In all three examples above, Denmark, Norway and Sweden, the plans for teacher education stress the connection of the studies to research and fostering of a competence for change and development. The recommendation is to strengthen the research basis for teacher education. This can be done in many different ways and I have discussed such opportunities in other connections (Grevholm, 2004b, 2004c). The development in order to link teacher education closer to research is influenced by the historical fact that teacher education has rather recently become part of the academic structure (and still is not in Denmark). But there is a need to make changes more efficient in this area. Relevant research, for example in didactics of mathematics, must be part of the foundation for mathematics teacher education. Teacher education is a professional education but it is also expected to prepare for further studies and possibly lead to doctoral studies. These double aims are, as can be seen from above, problematic. What scientific field could be relevant for a teacher, who chooses research studies? I have argued elsewhere for didactics of mathematics as the natural scientific area for a mathematics teacher (Grevholm, 2006). For this to be achievable, the teacher education must lay a foundation in didactics of mathematics solid enough for further research studies. The evaluations referred to above indicate that the subject didactical studies still are weak parts of teacher education. Another crucial issue that can be traced in all three countries is the need for better connection between the different parts of the education. The conception of a didactic divide between disciplinary and pedagogical knowledge has been used as an analytic tool to describe the rationale behind designs of reforms in teacher education (Bergsten & Grevholm, 2004). Even if the backgrounds to renewal of teacher education relate strongly to societal changes, including changes in the school system and general views on teaching and learning, it is obvious, that when it comes to the content and organisation of teacher education programmes, the didactic divide between disciplinary and pedagogical knowledge has played a major role. The idea 83

84 of integration as a solution to problems of connecting different parts of the teacher education is tempting to follow but hard to implement, as can be seen from the evaluations reviewed above. After many attempts in several countries and over a long period of time it might be time to question if this idea can provide a solution to the problems after all. Another solution is to accept that teacher education is so complex and the object for so many different demands that it is not possible in a basic academic education to meet all these demands. A changed perspective is needed. The teacher education must be seen as life long learning and the development over time of a professional identity. A model for teacher education in mathematics Based on the findings in the longitudinal study, a model was created in the form of a concept map, showing how teacher education can be perceived as a development of a professional identity (see Figure 1). This identity development is complementing the private identity of the teacher and it is governed by social demands, culture and the national identity (Grevholm, Even, Szendrei, & Carillo, 2004). In the case of the teacher profession the outside demands on the education are especially visible in the societal debate. The five main elements in the model constitute core parts of the professional identity that is developed in teacher education. They are knowledge in mathematics related to teaching, competence to judge and diagnose pupils learning in mathematics, knowledge about classroom management, methods and material, a personal view on and beliefs about knowledge and learning in mathematics, and a professional language for a mathematics teacher. All these five main elements are interrelated and closely linked to each other. The model also indicates the basis for the five main areas and the sources for the knowledge and competencies, and how they are interrelated in a complex system. Student teachers experiences, earlier knowledge, observations, reflections, practice, research and theoretical studies during the education contribute to the development of the five aspects of the teacher identity. One answer to why mathematics teacher education is problematic is the amount of different demands from both outside and inside the education that can be noticed. I have argued here and earlier for the view that because of all these incompatible demands, it is only possible to create a basis for the professional work in the pre-service education of teachers (Grevholm, 2006). After the pre-service education, a lifelong learning for the teacher must follow. The development of the professional identity will go on throughout the whole professional life of the teacher. Thus in-service 84

85 teacher education and different kinds of competence development for teachers are crucial and must be provided generously by society. Teacher education in mathematics develops governs Societal demands, culture and national identity a private identity complements a professional identity expressed with contains means means means terminology a professional language contains ability to communicate consists of uses research results in mathematics education a personal view on and beliefs about knowledge and learning stored in concept structures founded in influence theories about learning way of working theories about knowledge knowledge about class room management, methods and material builds on work forms tools, aids and artefacts competence to judge and diagnose pupils' learning in mathematics builds on pupils' development confirmed in orally pupils' results expressed written is part of knowledge in mathematics related to teaching builds on builds on uses knowledge about learning in mathematics the structure of mathematics founded in own experience and observations Figure 1. A concept map showing how mathematics teacher education can be seen as the development of a professional identity, with five main elements and their sources (Grevholm et al, 2004). What are the consequences of problems in teacher education? The Swedish Government published in 2004 a report aiming at changing attitudes to and raising the interest for mathematics and developing mathematics education (SOU 2004:97). One of four main suggestions in this report is dealing with mathematics teacher education. Four suggestions are given: To improve recruitment to the mathematics teacher education, to develop the basic mathematics teacher education at all levels, to support qualifying competence development and further education, and to increase resources for research on teacher education and competence development (ibid, p 113). So far not much has been done in the direction indicated in SOU 2004:98, but recently the government presented a programme for competence development, Lärarlyftet, which aims at raising the status and 85

86 competence of teachers (Skolverket, 2007). A substantial amount of money will be invested in this programme until Unqualified teachers can get complementing education for qualification and active qualified teachers can get further education with 80 % of their salary during the studies. A number of teachers will be able to go into research education under these conditions. It must be expected that many of these teachers will be mathematics teachers. A recent investigation (Statskontoret, 2007) reports that only 35 % of the teachers in Swedish upper secondary schools have a teacher education degree for that level and education for the subject (mathematics) they are teaching (ibid, p 33). Thus the need for complementing mathematics teacher education is huge. It remains to be seen if the mathematics teachers will take this chance for in-service education. Why the interest for teacher educations? In Norway preparations are made in order to offer research resources for a comparative study of teacher education in the Nordic countries. In this connection, Peder Haug, professor in pedagogy, presented some important problems in teacher education from a Nordic perspective (2007). He claims that there is a huge interest for teacher education in the Nordic countries. In society an intense debate is going on and the same questions are central in all places. First, all teacher educations in the Nordic countries are under reorganisation to come into alignment with the Bologna-declaration. The second group of themes, he claims, comes from research and evaluation of the internal activity in teacher education, where a number of questionable areas have been identified. This concerns the relation theory and praxis, lacking professional direction, weak subject didactical orientation, tension between subject and pedagogy, missing research connection and low study activity from students. The third area debated is related to the conditions in the school we are educating teachers for and this is maybe the most important driving factor for quality in teacher education. International surveys, like TIMSS and PISA, show that the performance of pupils in the Nordic countries is maybe not what we are hoping for. The political view is that the Nordic countries depend on a high level of knowledge and competence in the global market in order to keep the welfare situation. Human capital is the dominating perspective. Again, in this description by Peder Haug (2007), we recognise the demands put on teacher education from outside and inside, not only for teacher education to work well but also for the school to do so. He expresses that if the wish is to better understand teacher education and professional qualification, it is relevant to carry out comparative studies. 86

87 The main reason is that in the Nordic countries teacher education offers manifold and good conditions to answer many and wide questions raised about it. References 1. Bergsten, C. & Grevholm, B. (2004). The didactic divide and eduation of teachers of mathematics in Sweden. Nordic Studies in Mathematics Education, vol. 9, No 2, Danmarks evaueringsinstitut (2003). Læreruddannelsen. Køpenhamn: EVA. 3. Grevholm, B. (2000). Teacher education in transition. The case of Sweden. Paper presented at WGA7, International Conference of mathematics Education 9, Makuhari, Tokyo, Japan. 4. Grevholm, B. (2003a). Teachers' work in the mathematics classroom and their education. What is the connection? In C. Bergsten & B. Grevholm (red), Proceedings from Madif3, 2002(pp ). Linköping: SMDFs skriftserie. 5. Grevholm, B. (2003b). Student teachers' conceptions of equations and inequalities. Paper presented at the Pre ICME10 conference in Växjö. Retrieved February 2006 from 6. Grevholm, B., Even, R., Szendrei, J. & Carillo, J. (2004). From a study of teaching practices to issues in teacher education. Thematic Working Group 12, CERME3. In M. A. Mariotti et al (eds.), Proceedings of CERME3. Electronic publication. Pisa: University of Pisa. 7. Grevholm, B. (2004a). Mathematics worth knowing for a prospective teacher. In B. Clarke, D. Clarke, G. Emanuelsson. B. Johansson, F. K. Lester, D.V. Lambdin, A. Wallby, & K. Wallby (Eds.). International Perspectives on Learning and Teaching Mathematics (pp ). Göteborg: National Center for Mathematics Education. 8. Grevholm, B. (2004b). A research based vision of mathematics teacher education. In M. Lepik et al (Eds.), Teaching mathematics: Retrospective and perspectives. Proceedings from Tallin conference, May 24-25, 2003, (pp ). Tallin University. 9. Grevholm, B. (2004c). What does it mean for mathematics teacher education to be research based? In R. Stræsser, G. Brandell, B. Grevholm, O. Helenius (Eds.), Educating for the Future. Proceedings of an International Symposium on Mathematics Teacher Education (pp ). Stockholm: The Royal Swedish Academy of Sciences. 10. Grevholm, B. (2005). Research on student teachers learning in mathematics and mathematics education. In H. Fujita, Y. Hashimoto, B. R. Hodgson, P. Y. Lee, S. Lerman & T. Sawada (Eds). Proceedings of the Ninth International Congress on Mathematics Education (pp & CD). Dordrecht: Kluwer Academic Publishers. 11. Grevholm, B. (2006). Matematikdidaktikens möjligheter i en forskningsbaserad lärarutbildning. In S. Ongstad (Ed.), Fag og didaktikk i lærerutdanning. Kunnskap i grenseland (pp ). Oslo: Universitetsforlaget. 87

88 12. Haug, P (2007). Vesentlige problemstillingar i lærerutdanninga-eit nordisk perspektiv. Oslo: NOKUT. 13. Högskoleverket (2005). Utvärdering av den nya lärarutbildningen vid svenska universitet och högskolor. Högskoleverkets rapportserie 2005:17 R. 14. NOKUT (2006). Evaluering av allmennlærerutdanningen i Norge Hovedrapport. Oslo: Nasjonalt organ for kvaviltet i utdanningen. 15. Pehkonen, E. (2001). Lärares och elevers uppfattningar som en dold faktor i matematikundervisningen. In B. Grevholm (Ed.), Matematikdidaktik ett nordiskt perspektiv (pp ). Lund: Studentlitteratur. 16. Skolverket (2003). Lusten att lära med fokus på matematik: nationella kvalitetsgranskningar (Skolverkets rapport nr 222). Stockholm: Skolverket. 17. Skolverket (2007). Lärarfortbildning BCDDB SOU 2004:97 (2004). Att lyfta matematiken intresse, lärande, kompetens. Statens Offentliga utredningar. Stockholm. 19. Statskontoret (2007). Lärares utbildning och undervisning i skolan. Stockholm: Statskontoret, Rapport 2007:8. 88

89 INNER MATHEMATICAL MODELLING IN TEXTBOOKS FOR 5 TH 6 TH GRADES Edvins Ăingulis, Liepāja Academy of Pedagogy, edvins.gingulis@inbox.lv Abstract. The paper considers 9 most significant types of inner mathematical modelling we come across in the school mathematics course. After that analysis is made of application of inner modelling used in the textbooks currently used in grades 5-6 in Latvia. A short insight is also given into two textbooks currently used in the Federative Republic of Germany. It was ascertained that the most frequently used models in grades 5-6 are those linking arithmetics with geometry. Keywords: arithmetical models in geometry, geometrical models in arithmetics, inner modelling, mathematical modelling, types of models. In order to train students for application of mathematics in other branches of science and in practice, it is important to develop mathematical modelling skills. One of the fields, which presents opportunities for exercising in mathematical modelling, is mathematics itself. In this case we speak about inner modelling. Inner mathematical modelling is significant also, because it promotes the students awareness that mathematics is uniform. Within the content of the school mathematics course we consider arithmetic (Ar), algebra (Al) and geometry (G); therefore there are 9 cases of inner mathematical modelling (see Fig. 1). They are: Ar Ar, Ar Al, Ar G, Al Ar, Al Al, Al G, G Ar, G Al, G G. Аl Аr G Why? Fig. 1. Cases of inner mathematical modelling Problem 1. (Ar Ar). Which of the fractions or is larger?

90 90 Solution. Express the fractions in the form of difference: 2 1, then compare and and. As = >, the second fraction is larger. Problem 2. (Al Al). Solve the system of equations ( x y+ z)( x+ y z) ( y z+ x)( y+ z x) ( z x+ y)( z+ x y) { Solution. Express as: x y+ z= t, x+ y z= u, y+ z x= v. We get: ut = 9 uv. vt= = 1 4 If we multiply the left and the right sides of the given equations, we get: uvt= ± 6. We divide both sides of the last equation sequentially 2 3 with both sides of the given equations and we get: u= ± 6, v=±,t= ±. So the answer is: ( 3 ; 3 ; 1 ), ( 3 ; 3 ; 1 ) Problem 3. (G G). Each point in the space has been painted white, blue or red. Prove that it is possible to find two points, which are at a distance of 1cm one from another and have the same colour. Solution. Instead of space we deal with its model a regular tetrahedron, the length of its edge is 1cm. At least two of the vertexes of the tetrahedron are painted in the same colour; these vertexes can be chosen as the points to be found. The author has made a study of how often and in what way inner mathematical modelling is represented in eight mathematics textbooks for grades 5 6: among them six ([1] [4], [7] and [8]) are used in Latvia, but two ([5] and [6]) are used in Bundesland Mecklenburg Vorpommern in Germany. It is interesting to note that usage of colours and figures/drawings differs significantly in books by different authors and publishing houses. So, for example, [3] and [4] are the only black-and-white textbooks, but [7] and [8], according to the author, are too colourful: excessive and not sufficiently considered colour use prevents recognition of the essence and creates difficulties in using the book. Colourful illustrations, which in their nature do not correlate with the text of the textbook and presentation of the system, are used in [1], [2], [7] and [8]. The textbooks by German authors for grades = 9 = 4 = 1.

91 5-6 contain no special guides for pupils to get deeper into mathematics, but in the textbook [1] the students are guided by a spotty cat, in [2] by a baby fox, in [3] and [4] there is a bunny rabbit and a young squirrel studying together with the students, in [7] it is done by a fairy, in [8] each page is ornamented by two cubes. From 8 textbooks only four ([3], [4], [5] and [6]) have revised editions, and this manifests itself both in a better general quality of these books and in a more reflected usage of inner mathematical modelling. Assessment was made of the frequency of different types of inner modelling in the above-mentioned textbooks and a conclusion was made that evidently due to the specific features of the content of the 5-6 grade mathematics course types of inner modelling Ar Ar, Ar Al, Al Al and Al G are not represented at all, but the models included in the textbooks most often are Ar G and G Ar. The textbooks from Germany [5] and [6] present a larger variety of G G models, possessing a higher difficulty level. The reason for this might be rooted in the more lasting and serious traditions of technical education, resulting in inclusion of richer material from geometry and depiction of geometric shapes also in the content of mathematical education in grades 5-6. In all textbooks modelling is used both in presentation of the new material and in problem solving. It is impossible to draw a border between both situations, as presentation of the new material usually starts with solution of some introductory problem. Record-keepers in using inner mathematical modelling of all types are textbooks by a team of authors supervised by Jānis Mencis (sen.) [3] and [4]. With the help of drawings or models Ar G carrying out arithmetic operations with natural and rational numbers are explained. Usually scales of temperature are used for this purpose, but not necessarily, especially in the cases when operations with expressions consisting of letters, fractions and mixed numbers are illustrated and explained. Problem 4. Deal with a difference between the types of transforming expressions! (Captions in Fig. 2 mean 13 fish, 6 fish, 7 cats, 15 eggs in a basket.) 91

92 Fig. 2 [1, 85] Problem 5. By examining Fig. 3, explain how multiplication and division of the fractions are carried out! Fig. 3 [4, 56] Problem 6. Examine Fig. 4 and explain the two ways of calculating the 1 product 2 3 : 5 1) by changing the mixed fraction into an improper fraction; 2) by multiplying the integer part and the fractional part separately! Fig. 4 [3, 252] 92

93 Expressive drawings/figures and diagrams help in memorising new terms. Problem 7. (After these drawings the rules follow how to calculate the unknown addend, the unknown subtrahend and the unknown minuend). Remember! Addend + addend = sum Minuend subtrahend = difference Fig. 5 [1, 91]. Problem 8. Relationships between measuring units can also be depicted visually (see Fig. 6). Fig. 6 [6, 117]. The textbooks quite often use wording of the problem in a ready-made figure, which in its nature is a usage of G Ar model. Problem 9. Examine Fig. 7 and calculate the mass of the pumpkin, on the condition that both pumpkins are equal! Fig. 7 [3, 197] 93

94 Problem 10. How large is the unshaded portion in Fig. 8? What percentage does it constitute? Fig. 8 [3, 227] Problem 11. Examining the drawing, compile a problem. Solve it by writing an equation! Fig. 9 [4, 276]. The textbooks by Jānis Mencis (sen.) and the team are the only ones with problems, which include a geometrical model diagram or schematic depiction - of some phenomenon, but the learners are requested to examine the given drawing/figure carefully and after that to compile and solve a problem corresponding to it. References 1. Lude I. Matemātika 5.klasei. Rīga: Pētergailis, Lude I. Matemātika 5.klasei. Rīga: Pētergailis, Matemātika 5.klasei. J.Menča redakcijā. Rīga: Zvaigzne ABC, Matemātika 6.klasei. J.Menča redakcijā. Rīga: Zvaigzne ABC, Mathematik. Lehrbuch für die Klasse 5 Mecklenburg-Vorpommern. - Berlin: PAETEC Verlag für Bildungsmedien, Mathematik. Lehrbuch für die Klasse 6 Mecklenburg-Vorpommern. - Berlin: PAETEC Verlag für Bildungsmedien, Mencis J. (jun.) Matemātika 5.klasei. Rīga: Zvaigzne ABC, Mencis J. (jun.) Matemātika 6.klasei. Rīga: Zvaigzne ABC,

95 THE ANALYSIS OF THE RESULTS OF TEAMS FROM BALTIC STATES AND FINLAND AT INTERNATIONAL COMPETITIONS IN MATHEMATICS Maksim Ivanov, Elts Abel, University of Tartu, Abstract. The International Mathematical Olympiad and the team competition Baltic Way are most important international contests in mathematics for the states around the Baltic Sea giving an opportunity to compare the level of the knowledge and preparation of the teams in modern elementary mathematics. The purpose of this article is to compare the results of Estonian, Latvian, Lithuanian and Finnish teams at the International Mathematical Olympiads ([2]) and at the international team competitions Baltic Way ([3]) in the last twelve years. It gives us a possibility to compare the results of these countries with other participating countries, to compare the results of these countries with each other and also to find out the trends of changes in math achievements. Keywords: IMO, Baltic Way, modern elementary mathematics. In first part of this article the results of the Baltic States and Finland on the International Mathematical Olympiad (so-called IMO) are considered in more details. Although the IMO is officially an individual competition, every year the so-called unofficial country ranking or unofficial table of national performance is made, with the ranking in terms of the sum of the scores achieved by each country s students. Information of rankings is received from the official web-pages ([2]) of this competition. One of the basic elements of the comparison is the mean score in percentage (named by MS(%)), which is found as follows: the sum of team scores for the considered period of time of all the problems or the problems of the certain mathematical field is divided by the greatest possible score under the same conditions. This definition can be simply explained by the next formula MS (%) = TEAM_SCORE MAXIMUM_SCORE 100(%). As it has already been told, the statistics about the performance of teams at the IMO is presented for the last twelve years (it means since 1995 to 2006). It is possible to consider this interval of time optimal for the analysis as the Baltic States for the first time have taken part in the Olympiad in 95

96 1993. A year before they have taken part as observers. Finland participates there since In the following Table 1 the best Estonian, Latvian, Lithuanian and Finnish team s achievements are shown. The best place for each country in the unofficial table of national performance and the best mean score in percentage are presented. The best result in the official ranking and the best MS(%) during this period was achieved by the Latvian team in Table 1. The best places and mean scores of the teams in the unofficial country rankings (IMO) since 1995 the best place the best MS(%) EST 37 (2001) 37.3 % (2004) LAT 22 (1997, 1999) 49.2 % (1997) LIT 32 (1996) 37.3 % (2006) FIN 34 (1997) 40.1 % (1995) For viewing the tendencies of performance of these countries at the IMO it is convenient to present their achievements graphically and in comparison with each other both for one year and for all the period. As the number of participating teams at the IMO every year was not the same, the so-called ratio of places in percentage (named by RP(%)) has been found as follows NUMBER_OF_TEAMS - PLACE_OF_TEAM + 1 RP (%) = 100(%). NUMBER_OF_TEAMS Such representation of the ratio of places enables us more clearly to see the dynamics of the received places and to notice some tendencies. Table 2. The dynamics of the received places in the unofficial country rankings at IMO 100,0 90,0 80,0 70,0 60,0 50,0 40,0 30,0 20,0 10,0 0, EST LAT LIT FIN 96

97 From these tables it follows that the teams of Latvia and Finland in the first half of the considered period of time have solved problems of the IMO much better than other two countries. During the six first years the Latvian and Finnish teams have won 5 times among these 4 countries. In the second half of this period the teams of Estonia and Lithuania began to act more successfully, both of them have achieved several victories. In conclusion, it is impossible to specify any concrete leader at the IMO among the Baltic States and Finland, and every year it is possible to expect more successful performance of a team from any country. The next direction is the analysis of the results through mathematical fields by comparing the team scores of the Baltic States and Finland in four traditional mathematical fields (Algebra, Combinatorics, Geometry and Number theory) ([1]). It is possible to notice that problems in geometry and algebra are used at the IMO more often. Table 3. Number of problems of different mathematical fields for the period Algebra A 18 Combinatorics C 13 Geometry G 25 Number theory N 16 In the Table 4 the mean scores in percentage of all the teams (ALL), having participated in the IMO in the last twelve years ( ), the top ten teams (TOP-10) and the four teams of the respective countries have been shown. Using this table some interesting remarks can be made. Table 4. The mean scores in percentage based on the different mathematical fields MS (%) ALL TOP-10 EST LAT LIT FIN Algebra Combinatorics Geometry Number theory Firstly, from the Table 4 one can see that on average all the participants of the IMO receive most of all points for solving geometrical problems (the mean score in percentage for problems of geometry is equal to 40.1). The same law holds for the top teams and one can say that these results define the level of difficulty of given problems. But there is not any Baltic country, for which the mean score for geometrical problems is the highest. For example, for Finnish and Lithuanian teams it is one of the lowest. 97

98 Secondly, combinatorial problems are the most difficult for top teams, the mean score for this field is only 66.6, but the Baltic States and Finland get the best points in this field of mathematics. Thirdly, the least mean scores in percentage of all the teams (27.6) is received at solving algebraic problems. The same situation with solving algebraic problems takes place for Estonian, Latvian and Lithuanian teams. The previous table data can be presented in the form of a difference between the mean scores of all the teams and the teams of considered countries. This difference gives a possibility to estimate the backlog or advantage of one of the countries from an average level at problem solving in certain field of mathematics. In other words, it gives a possibility to estimate how many percentage points the mean score of one country s team differs from the MS(%) of all the teams. Table 5. The differences between the mean scores of all the teams and the teams of considered countries MS (%) ALL EST ALL LAT ALL LIT ALL FIN Algebra Combinatorics Geometry Number theory There are only two negative numbers in this table, which means that only the mean scores in combinatorics of Finnish and Latvian teams exceed the mean scores of all the competitors at the IMO. At the same time Table 5 shows that the biggest differences from the mean score of all the participating teams in the IMO are in geometry (planimetry) for the Estonian, Lithuanian and Finnish teams, but for the Latvian team this difference is somewhat bigger in algebra than in geometry. Therefore it is possible to say that the most serious problems for students of the Baltic States and Finland are first of all problems in geometry and also in algebra. Using the obtained data, the results of the previous table can be presented so that the levels of difficulty of mathematical fields decrease for each country. Hence, in the first column for each country it is shown that the field of mathematics where the greatest backlog from the mean score is visible. 98

99 Table 6. Levels of difficulty (LD) of mathematical fields for each country LD 1 LD 2 LD 3 LD 4 EST G A N C LAT A G N C LIT G A N C FIN G N A C In conclusion of this part of the article the table which reflects the general tendencies of the Baltic States and Finland teams in the solving IMO problems is given. In Table 7 one can see in which fields of mathematics our students lose more percentage points in comparison with the average level. Table 7. The general tendencies of LD for the Baltic States and Finland LD 1 LD 2 LD 3 LD 4 EST & LAT & LIT & FIN G A N C If we look at the levels of difficulty of the IMO problems, it is possible to find out some more reasons of the results for each country. As it has already been told, the mean score in percentage of top teams defines a level of the difficulty of problems. Let us say that the problem is simple if the mean score in percentage of the top teams is not less than 90, the problem is medium if the mean score of the top teams is less than 90 and not less than 60 and difficult if the mean score is less than 60 percents. In Table 8 all the problems of the Olympiad since 1995 are distributed by the levels of difficulty. Table 8. Distribution of the IMO problems by the levels of difficulty MS (TOP-10) 90% MS (TOP-10) 60% MS (TOP-10) < 60% simple medium difficult A C G N The greatest number of the so-called simple problems is in geometry (10 problems). Medium and difficult problems are distributed between all the fields of mathematics uniformly. If to analyze the levels of difficulty of the IMO problems, the Estonian, Lithuanian and Finnish teams lose most of the points in solving simpler geometrical problems in comparison with the mean score (namely, the difference is more than 20 percentage points). 99

100 Table 9. The mean scores in percentage based on the levels of difficulty of geometrical problems Geometry ALL EST LAT LIT FIN simple medium difficult Table 10. The differences between the mean scores in solving geometrical problems Geometry ALL EST ALL LAT ALL LIT ALL FIN simple 21,9 4,1 24,6 24,6 medium 12,6 10,7 18,8 21,6 difficult 6,7 0,6 5,8 6,3 In the case of the problems of the medium level of difficulty the losses in comparison with the mean score of all teams are also very significant. Table 11. The topics of the simpler geometrical problems at the IMO Problems Angles and metric relations in the circle 46,3 % 2 - Metric relations in triangles 17,2 % 3 - Geometrical transformations 12,4 % 4 - Other topics (metric relations in polygons ect.) 12,1 % 5 - Knowledge of some specific theorems 5,0 % 6 - Similarity and calculations based on it 4,9 % 7 - Geometrical inequalities 2,1 % Analyzing the set of the simpler geometrical problems, presented at the IMO, one can find more information about the important basic concepts and methods which cause most of the difficulties for our students. In this article in Table 11 we offer only a percentage ratio of the geometrical topics used at the Olympiad and the list of problems with marks of using any topic. From Table 11 one can see that since 1995 every simple geometrical problem has contained elements of the first topic Angles and metric relations in the circle

101 In the next part of the article the similar information about the competition Baltic Way is presented. Different from the IMO, the Baltic Way is the team mathematical competition. The majority of the participating countries in this competition do not enter into the number of the IMO leaders. Further some statistical data of the results of the competition Baltic Way are analyzed and compared with the results of the IMO. In Table 12 the mean scores in percentage of all the teams, having participated in the Baltic Way in last twelve years, the top three teams (TOP-3) and the four teams of the respective countries are shown. In Table 13 similarly the differences between the mean scores of all the teams and the teams of Estonia, Latvia, Lithuania and Finland are presented. Table 12. The mean scores in percentage based on the different mathematical fields MS (%) ALL TOP-3 EST LAT LIT FIN Algebra 63,8 80,4 61,0 68,7 67,7 68,0 Combinatorics 59,6 74,8 59,7 62,3 52,0 60,7 Geometry 55,9 79,5 67,7 57,7 53,0 50,3 Number theory 64,3 83,2 63,7 73,3 54,3 67,0 Table 13. The differences between the mean scores of all the teams and the teams of considered countries MS (%) ALL EST ALL LAT ALL LIT ALL FIN Algebra 2,8 4,9 3,9 4,2 Combinatorics 0,1 2,8 7,6 1,1 Geometry 11,8 1,8 2,9 5,5 Number theory 0,6 9,0 10,0 2,7 It is not possible to observe the tendencies similar to the results of the IMO. More likely in many cases they are opposite. So, for example, the geometry which was the most serious problem for the Estonian and Lithuanian students at the IMO, does not represent any special difficulty at the Baltic Way. At the same time the geometrical problems seem to be more difficult for the Latvian team at the Baltic Way in comparison with other fields of mathematics. One of the principal causes of such differences can consider various numbers of problems, a principle of their selection, essential differences in a level of difficulty of problems at these competitions. As the second reason, it may be different strategy for selecting the members of teams. If every participant of the IMO should be able to solve equally well problems from all the given fields of mathematics, then the participants of the competition 101

102 Baltic Way may be selected so that only one student solves all the problems from one or more fields. In conclusion the authors of this article hope that this small research will become the precondition for analyzing the preparation courses for the IMO and the Baltic Way and will help to raise efficiency of them. References 1. Djukic, D., Jankovic, V., Matic, I., Petrovic, N. The IMO Compedium A Collection of Problems Suggested for the International Mathematical Olympiads: Springer Science + Business Media, Anon1. International Mathematical Olympiad Competition results and exam problems. URL= 3. Anon2. Rahvusvahelised matemaatika olümpiaadid. URL= 102

103 TEACHERS BELIEFS ABOUT THE COGNITIVE AND APPLICATION-ORIENTED COMPETENCIES IN SCHOOL MATHEMATICS Hannes Jukk, Lea Lepmann, Tiit Lepmann, University of Tartu, Abstract. This research tries to develop understanding of Estonian mathematics teachers beliefs about the importance of the tasks which improve different cognitive and application-oriented competencies. With repertory grid technique it has become evident that the teachers are using only purely mathematical features characterizing the tasks and they mention very few cognitive characteristics. In total, 170 teachers are separated into three groups on the basis of their beliefs about competencies. Keywords: beliefs of teachers, cognitive competencies. In many countries the school mathematics curricula presents the list of cognitive skills along with purely subject-oriented skills (competencies). Usually they are called process competencies and have been outlined as a separate part in the curriculum. For example, in NCTM 2000 [7] five general process strands are given: problem solving, reasoning and proof, communication, connections and representation. The Canadian curriculum [6] presents 7 components of mathematical processes: communication, connections, estimation and mental mathematics, problem solving, reasoning, technology and visualization. In German Bildungsstandards [1] we can find 6 common competencies: argumentation, modelling, representation, problem solving, communication and the use of formal language. Until recently in the Estonian curriculum the main stress has been laid on the subject competencies. Cognitive competencies are mentioned only in a couple of phrases in the general part of the curriculum: the student s creativity, clear and exact ability to express oneself based on logical thinking and intuition, should be developed. As we see, in describing cognitive competencies and measuring the level of mathematical education, somewhat different standards and terms are used in different curricula. Cognitive standards externalize what we appreciate most in the mathematics learning process. The development of these competencies is to a more extensive degree connected with the teaching process than the development of subject competencies. It seems that a relatively general description of cognitive competencies in the curriculum is 103

104 not a sufficiently good target for the teacher in planning and carrying out his or her work, and the developing of these competencies takes place in a relatively spontaneous manner. The major part of the competencies obtained thanks to school mathematics is developed through tasks. There is no task that accesses a single cognitive competence. There is a considerable overlap among them, and when using mathematics, it is usually necessary to develop simultaneously many of the competencies. The selection of the tasks for the use at the lesson reflects the teacher s knowledge about the fact which competencies s/he considers as being important for the development and which opportunities in the tasks for this purpose s/he sees. Several research papers reveal that in the case of mathematics teachers we see four different views on mathematics and its structure: schematic oriented ( toolbox ), process oriented, system oriented and application oriented [4]. Different teachers give importance to different cognitive competencies. For example, sometimes the teacher thinks that the students should be guided most directly to the right solution. In this case the cognitive opportunities remain unused [8]. This way the students develop result-oriented beliefs that the teacher explains all the rules for the solution of the task and it takes only a couple of minutes to solve the task [5]. Proceeding from what is said above, we studied the Estonian mathematics teachers beliefs about cognitive and practical competencies in teaching mathematics. It was done in the framework of the Estonian Science Foundation grant No Our aim was to reveal the level of importance teachers give to the tasks which develop different cognitive and practicalimplementational competencies. At the same time we tried to study how do Estonian teachers describe the competencies which one or another task allows to develop. As a result, we can give an approximate assessment to the teachers professionalism in the selection of the tasks and in setting the goals for teaching mathematics. Research design. In the first stage of our research (30 teachers) we tried to reveal which features are used by teachers when describing the tasks. We used the repertory grid technique [2]. With this purpose we chose two opposing cognitive tasks from one topic (the linear equation). For each task the teachers had to write a maximum number of such features which one task had, but the other did not have. The selected tasks were the following: 1. Solve the linear equations: 3-4x = x - 12 and 10(x - 5) 10(2x + 5) = 2-8(4x - 1) 2. Give an expression containing a variable, the value of which you can calculate. Compile a text task suitable for this expression. Give also the 104

105 question of this task in words. Demonstrate how to find an answer to the question you asked. It became evident that the teachers use, as a rule, only purely mathematical features in characterizing the tasks. Very few cognitive characteristics were mentioned and the teachers were not able to use suitable vocabulary. This is why we gave up this methodology in our further research. After that we compiled a possible list of competencies which can be developed by mathematical tasks, and we studied the teachers opinions how the tasks carrying these competencies should be represented in the system of the tasks in the textbook. In our questionnaire we presented 56 statements about the possible roles of the tasks and asked the teachers how big the number of tasks, developing a respective competency, could be in their opinion when dealing with some more extensive integral theme in the 3 rd stage of the school (7 th 9 th Forms). Altogether 170 teachers of mathematics presented their opinions on a five-point Likert scale: 5 the majority of the tasks of the theme 1 single tasks of the theme. Grouping of tasks according to the competencies to be developed. Each task supports the development of several competencies. In our research we distinguish the following seven groups of competencies: the task offers opportunities for communication, the task supports developing of creative thinking, the task offers opportunities for developing logical thinking, the task is predominantly oriented only to obtaining the knowledge of the subject, solving the problem-task presupposes only routine activities, the task offers possibilities for creating connections, the task allows to find implementations in everyday life and arouses interest. As we see afterwards, the teachers consider, as expected, the routine tasks to be more important in teaching the subject. The corresponding competencies gained 3.9 points on average, which in our scale corresponds to the statement more than a half. On the contrary, the teachers expect the smallest number (less than a half) of non-routine tasks in the textbook as supporting creative thinking. Also, the support to the tasks developing communication and logical thinking and the tasks arousing interest remains weak. Considering the peculiarities of the age of the students in the age group under discussion and the real situation in schools (big classes, overburdened curricula, etc.), the obtained research result may seem quite natural, but it is clear that the main goal of our teaching at school cannot be only the teaching of factual knowledge. We have repeatedly declared that 105

106 the future society needs innovative, creatively and logically thinking citizens. The activities to achieve this goal should start as early as possible. Grouping of teachers. For grouping of teachers on the basis of their beliefs of competencies we used cluster analysis. It became evident that relatively saliently three groups of teachers were formed: the subject teacher (78 teachers, 47% of respondents), the developer (55, 33%) and the provoker of interest (34, 20%). The common important feature of these groups is that the teachers of all the groups assess very highly the tasks aimed at learning the material (the task demands: the knowledge of the learnt facts, notions and connections; affirms the learnt material; demands the skills of calculation and conversion; applies the learnt notions, procedures, the connections and formulae inside the subject; has exact data and a single definite answer). But, however, the subject teacher limits himself to the high evaluation of only such competencies. The task is highly evaluated if: the task has one definite way of solution, it demands the recognition of the learnt material, it demands the application of the learnt material in a typical situation, applies the learnt models, etc. While the subject teacher considers the tasks as tools for teaching mathematical knowledge, the developer sees in them, in addition to the above said, also the values developing the student s logical and creative thinking. The teachers of this type give a relatively high estimation, in addition to the above said, to the following features: the task demands independent thinking, independent conclusions, synthesizing the existing material, giving reasons, making generalizations, categorizing objects and notions, differentiating significant information, selecting the learnt method, interpreting the content of the model, connecting the new information with the existing material; leads to the discussion, develops creativity, demonstrates connections inside the subject. As we see, the teachers who highly evaluate logical and also creative thinking are gathered in one group. The provoker of interest is naturally characterized by the fact that s/he places the tasks, rising the students interest in the subject, to the highest level. The teacher finds such material primarily in the tasks with a vital content. It is interesting to add that the teachers of this type give lower estimation than others to the tasks, which are directly aimed at teaching the subject and also developing the student. Now we shall characterize the background of the given groups of teachers. Significant connections were revealed between the type of the teacher and the following background features: the teacher s age; the stage of the school, where the teacher teaches (I teach only the classes of the basic school, the classes of the basic and secondary school, only the classes 106

107 of the secondary school); the knowledge level of the classes taught (I teach mainly in the classes formed on the basis of the results of a contest, mainly in ordinary classes, in both); dealing with the preparation for mathematics contests (in general I am not pressed for time, I do not deal especially with this kind of work) and the attitude to the examinations (absolutely necessary, may or may be not, not necessary). The teachers beliefs are essentially influenced by the environment where they work [3]. It became clear that the higher the level of the class taught the more probable that its teacher belongs to the type of the developers. On the contrary, in the classes with a lower level of knowledge, the interest provoker and the subject teacher groups are dominating. It could also be expected that the developers are more strongly represented among the teachers who either constantly or temporarily prepare students for olympiads. At the same time, the teachers, who are not involved in this work at all, are mainly the subject teachers. Research shows that the strongest support is given to the exterior assessment of the school in the form of the national school leaving examinations with similar tasks in the basic school by the subject teachers. The least importance is given to these examinations by the provokers of interest. We see that the subject teachers are most strongly represented among the teachers who teach only in secondary school (67% of the respondents in this group). At the same time, the developers constitute only 17% in this group. On the contrary, the developers are most strongly represented among the teachers who teach either in the classes of the basic school or in the basic and secondary school. The situation, where only the teachers in the secondary school have devoted more of their attention than other teachers to teaching the subject, may be caused by the fact that at the Estonian secondary school, the school leaving national mathematics examination plays an extraordinarily important role. This examination is at the same time valid for entering higher educational establishments in many specialities and it has become a decisive orientation in studying mathematics for both teachers and students. The preparation for this examination is usually to a great extent just learning the subject. Research shows that among the young teachers almost half of them expect that tasks, in addition to teaching the subject, should develop the students thinking. With years such optimism decreases. The factor of developing thinking gains importance in the case of teachers with a longer experience of work. In the retirement age teaching the subject becomes most important. 107

108 Conclusion. Already in the first stage of the present research the hypothesis was confirmed that a very big number of Estonian mathematics teachers is not able to use the selection of tasks to set the goals for teaching mathematics and to plan to develop thinking proceeding from the cognitive component. In mathematics tasks the teachers see primarily an instrument for teaching the subject. At the same time they are not able to see the opportunities for developing the students more general cognitive abilities. Also, Estonian teachers lack the terminology for describing such more general competencies. The result of the research allows us to conclude that there is an evident need that our curricula express more profoundly than today the general cognitive goals of teaching mathematics. Our research also showed that among teachers we can distinguish three different types. A significant common feature for all the groups was the fact that the teachers in their expectations to the tasks place particular competencies connected with teaching the subject to a high position. Studying the background of these three groups, it was revealed that the fact to which group one or another teacher belongs is not generally determined by the so-called teacher s inner factors. A great role in the teacher s attitude to mathematics tasks is played by the level of the students taught, in which stage of the school the teacher works, etc. It is also possible that there may be significantly more general factors connected with the organisation of education, for example, the national exterior control system. 108 References 1. Bildungsstandards der KMK für den mittleren Schulabschluss Bruder R., Lenink K., Prediger S. (2003) Ein Instrumentarium zur Erfassung subjektiver Theorien über Mathematikaufgaben. Preprint Nr Darmstadt. 3. Furinghetti F. (1996) A theoretical framework for teachers conceptions. Current State of Research on Mathematical Beliefs III. Proceedings of the MAVI-3 Workshop. Helsinki, Grigutsch S., Raatz U., Törner G. (1998) Einstellungen gegenüber Mathematik bei Mathematiklehrern. Journal für Mathematikdidaktik, 19(1), Maass K. (2006) Mathematik zwischen Schülervorstellungen und Bildungszielen erste Ergebnisse einer empirischen Studie. Beiträge zum Mathematikunterricht Vorträge auf der 40 Tagung für Didaktik der Mathematik in Osnabrück. Franzbecker, Mathematics Programs of Studies. Grades Principles & Standards for School Mathematics Ticha M. (2006) Sind die Reflexionen des Unterrichts zutäglich? Beiträge zum Mathematikunterricht Vorträge auf der 40 Tagung für Didaktik der Mathematik in Osnabrück. Franzbecker,

109 THE PORTRAIT OF CORRESPONDENCE MATHEMATICS SCHOOL AS IT IS SEEN BY MATHEMATICS FANS Zane Kaibe, Laila Rācene, The University of Latvia, Abstract. Our report contains an analysis and general conclusions from the questionnaire developed by The University of Latvia A.Liepa Correspondence Mathematics School in the beginning of the year Keywords: Correspondence Mathematics School, motivation, questionnaire Introduction The University of Latvia A.Liepa Correspondence Mathematics School (CMS) is an exclusive organization in Latvia by its form and scale. It provides students and teachers with more profound teaching and point of view in mathematics. Its course differs substantially from traditional school mathematics, and it stimulates students and teachers to think out of the box. According to its name, the great part of activities of Correspondence Mathematics School are correspondence courses, which take place by the instrumentality of internet and traditional mail. It includes also the preparation to present-way olympiads. So, to carry out its activities, to fulfill its audience s desires and to involve new active participants, it is essential for Correspondence Mathematics School to know, what are the interests and needs of its audience and what kind of activities they are attracted to. Abovementioned considerations were the main reasons, why an inquiry form that contained 15 questions about evaluation of the activities and home page of Correspondence Mathematics School was developed in the beginning of January, Some questions of this inquiry form were composed as tests, while some were set up in a free form. The inquiry form was sent by to those who had subscribed to it. The questionnaire was available also at the home page of Correspondence Mathematics School [1]. 109

110 Analysis of the questionnaire Respondents By the end of the term in the middle of the February 128 correctly filledin forms were received. Among all respondents 48% were teachers, as many students and 4% were students' relatives (see Fig. 1). This data is gratifying, and we are glad about respondents involvement. Besides we did not have this kind of information before. Respondents: 4% 48% 48% students teachers students relatives 110 Fig. 1. Respondents Activities As mentioned before, most of CMS activities are held by correspondence, also the preparation for the Olympiads. We might say that each year the most important task of the CMS is to organize all three rounds of State Olympiad and Open Mathematical Olympiad. Although in the course of time we have developed a good collaboration with teachers and students, most of the time we could only guess what is their attitude to these activities. In collaboration with Ministry of Education and Science and other organizations, every year three rounds of State Olympiad are held. In the first two rounds of State Olympiad (Preparatory Olympiad and Regional Olympiad) each pupil can attend. Best participants of Regional Olympiad (about 300 participants in the country) are invited to the third (final) round of the State Olympiad. According to the data, 80% of respondents have involved themselves or others in Preparatory Olympiad. In Regional Olympiad this number is only a bit larger 87%. These results are impressive, but they are not a big

111 surprise Preparatory and Regional Olympiads are local and easily available for all students and teachers also at distant regions. 41% of respondents have involved themselves or others in the State Olympiad, and it lets us think that most of the respondents are interested in mathematics and have proven themselves as gifted in their studies and organisational activities. According to data, most of the respondents think that problems in Preparatory and Regional Olympiads are medium hard, but State Olympiad s problems are hard. Still the respondents say that they like all three stages of the Olympiad. CMS is organizing also Open Mathematical Olympiad. Unlike in State Olympiad, everyone can participate here. That is why it is surprising, that only 59% of respondents have involved themselves or others in Open Mathematical Olympiad. This indicator could be higher, because Open Olympiad gathers not only those with skills and knowledge, but also those with upcoming growth and enthusiasm. 50% of respondents think that the problems in Open Mathematical Olympiad are middle hard, but 48% of respondents think that the problems are hard. Even in this case, most of respondents (62%) like Open Mathematical Olympiad. Activity in the Open Mathematical Olympiad and the stages of the State Olympiad: 100% 50% 80% 87% 41% 59% 0% Preparatory Olympiad State Olympiad Regional Olympiad Open Mathematical Olympiad Fig. 2. Activity in the Open Mathematical Olympiad and the stages of the State Olympiad Other contests organized by CMS Young Mathematicians contest, Professor Littledigit s Club, Correspondence activities for high-schoolers (9-12 graders) have involved accordingly 39%, 43% and 26% of respondents. These results are acceptable for now. Considering that the 111

112 number of active students in these contests is growing each year, they take more and more important place in the education sistem (see Fig. 3). Activity in other competitions of CMS 50% 39% 43% 40% 26% 30% 20% 10% 0% Young Mathematicians contest Professor Littledigit s Club Correspondence activities for high-schoolers Fig. 3. Activity in others competitions of CMS Respondents suggestions for CMS As mentioned before, the were some questions in the inquiry set up in a free form, where respondents could pass opinions and suggestions for CMS to improve our organizational performance. Most of these suggestions are analyzed in the further text. Mathematical activities for graduates not to allow them to forget about maths. Most active students, especially those who study at University of Latvia, are actively helping to check the pupils solutions and are on duty in mathematical Olympiads. In addition, previous laureates and contestants who work in schools as math teachers prepare their students very well and attend courses for teachers. But obviously they miss their own problems to solve. Russian speaking persons on duty in the olympiads. In all rooms, where there are any Russian speaking pupils, we try to have at least one Russian speaking person on duty. These persons are volunteers, so we can not require the knowledge of Russian language from them. 112

113 More advertising of CMS in the mass media. Idea is very good and approvable. We think, that it would be worth doing in the beginning of the school year, together with first round of all mathematical contests. There are schools in Latvia with very good students but not so good coordinators. Symbolic participation fee in the Open Mathematical Olympiad. We are looking negatively to the suggestion to introduce small participation fee in the Open Mathematical Olympiad, with the justification that 1) students must learn that nothing is for free, 2) there are students who apply to Olympiad and do not participate. As long as we have resources to organize Olympiads, it would not be advisable to introduce participation fee; this kind of activity should not be commercialised. More detailed solutions of problems for the beginners. Respondents reveal that more detailed solutions should be available for those who are only getting to know non-standard math problems. It is true that solutions given at CMS homepage are short and summarized. However, CMS publishes also books with detailed solutions of Olympiad and contest problems. Besides, these books are free of charge and available in the office or CMS. While writing these books we are thinking that everyone should be able to understand them not only teachers, but also students of the appropriate age. More effortless problems and/or differentiation of problems in the stages of the State Olympiad. Many respondents mention in their suggestions the need for easier problems and, probably, even problem level differentiation. As the greatest reason not to participate in Olympiads it was indicated that in earlier years a student had had bad results. Nevertheless, for many years in each Olympiad problem set at least one problem has been multistage. In addition it has become a tradition to have one repeated problem from the last year. We see that the students who regularly attend Olympiads and contests and put a lot of effort in their development gain good results. There was a suggestion to set up an interactive question-answer service in the homepage. As we receive very many questions from activists to whom we reply individually, we like this idea. We think that until the beginning of 2008 it will be implemented. 113

114 Conclusions Through the questionnaire we realised that respondents are active, knowing and appreciate our job. Our attention was drawn to things that need to be improved. We were given valuable advices for qualitative further work. Broad response and positive comments from respondents gave us good stimulus and ideas for further development References 114

115 ON GOOD BUT ACCESSIBLE PROBLEMS Romualdas Kašuba, Vilnius University, Abstract. We know that when Ptolemy I asked Euclid to show him the easiest way of learning geometry the answer was, O king, there is no king s way in geometry! These words about the non-existence of royal ways in geometry may mean also mathematics or other remarkable area of human deeds. In every area all this is exactly and perfectly true. But it is also true that all of us are resolutely very much interested to look for such semi royal ways, look for such shortcuts, which quickly lead our students and us to considerable success or remarkable progress. We are very fond to investigate such problems, which are of highest possible level and where it is possible to come from the street and to achieve something of importance. This is somehow paradox and in the same time quite human situation, when we driving perhaps the second day would like to understand the feelings of participants of Formula I. In mathematics or strictly speaking in problem solving such a thing of wonderful nature is possible in an astonishing higher degree. This is very interesting from the point of popularising and attracting the students showing how starting from zero you may get such results as if you were a skilled specialist or experienced problem solver. The situation is very much like with reclaim where we all understand that there are no such miracles as announced upon the sun but there are many amazing, outstanding and interesting circumstances which are rather similar to them. Real things have great attracting force and that s why they will always awake great interest. Still we ought to remain honest and not to create the impression that it happens every day. We ought also to add that dealing in such outstanding situations we must always remember that all small matters otherwise of no importance suddenly may become very useful and play a remarkable role. In the organizing of non-standard events there are usually no small details everything is important. It is just as it is in all-important arts or in miracles where there are not unimportant things. This offers additional possibilities for creative teaching and human pedagogic. Attempts to demonstrate some aspects of this or at least to tell some concrete details of such problems or say anything what could be used trying to achieve some progress dealing with otherwise serious things are undertaken. Keywords: attractiveness, divisibility, non-standard approach, proof and teaching, psychology. You are old, Father William, the young man said, And your hair has become very white; And you incessantly stand on you head Do you think at your age it is right? 115

116 This cite of the famous verse due to Lewis Carroll speaks not only about the family problems. From the first sight we do not see any indication that it might be something more than the usual conversation between father and son. In fact these lines are reinterpretation of another verse, which were the more usual conversation between father and son. It speaks also not so much or not at all about the eternal generation conflict. From the mathematical point of view it might mean only that sometimes or rather often two different elements (or human persons) would eagerly like to be regarded as belonging to the different subsets (or different generations). But it is known that each good formulated or fine sounding question might be regarded being a question which deals with matters taking place in many areas or practically in each important area. You could state and believe that it is a question, which deals with the teaching of mathematics. You may understand that the son is the student, which asks his father or the teacher why he still tries to teach us mathematics or problem solving and why father still believes that he able to do it, why he regards that it is so important, why he incessantly tries to motivate the student and even whether the teacher himself believes that all this adventure is important worth doing, worth undertaking. This is also the question why do you teach us in the same way as it was being done in Euclid s days and not in somehow more modern way? These are good questions. They are as good as all fundamental questions are. All really good questions are of eternal nature; these entire questions will be of great importance at least for the next few thousand years. From the essential nature almost every normal person or keen observer permanently or without pauses deals with the matters of the sort What should I do, Why should I do or What s this good for? Again trying to apply poetical terms all these questions may he called the question of type To follow you I were very much content - how should I know which way you went? So it is also about how difficult is to do anything of importance everywhere especially in the teaching where you seldom if ever are able to see that results in explicit form. But frankly speaking every experienced teacher sometimes has the feeling that his efforts are going in the good direction. And still one of the most fundamental things in teaching, which is also seldom formulated in words, is a question What do you have to show us? Demonstrate something especially for us. These are questions tending to the direction demonstrate us the mirror, please and could serve as the explanation, why the most motivated part of 116

117 the mankind children are so fond of attending to circus. The circus is the place where they can see something really interesting taking place and not only the comments that it would be so nice to see something special. We are seldom able to generate miracles ourselves. But we are essentially (much) more able to tell about it. These are not empty words. We indeed are able. These are not only modern sounding democratical phrases. We are always able at least to try to do it. It always pays. One from the most attractive things is let me feel that I can come from the street as a freshman and nevertheless achieve considerable progress in short time. It is really striking. Speaking figural starting on Monday to learn violin I can almost give a performance on the next Sunday. Amicus Plato sed magis amica veritas (est). Plato is my friend, but the truth is my greater friend. Many of us have heard these words many times. But it would be curious and a bit amazing to add something about the context so Plato was speaking about the existence of Atlantis and many people till this day believe that Plato was who invented it in his mind. In mathematical competitions some problems are of course formulated in such away, that you couldn t expect that they will be one day discussed in daily newspapers or be as famous as the last phrase. For example, I can t ever imagine that the following problem real problem 17 from the mathematical Baltic Way team competition in Turku 2007 which we could formulate using only one sentence would ever cause the public discussion although it is short enough to be cited everywhere. Its formulation is indeed not much longer as 2 words it is really short: Determine all positive integers n such that 3ⁿ+1 is divisible by n². One would quickly ask why we must solve these problems and why must we do that? What will I gain finding several if any of such numbers? What will it give me as the citizen of the modern society? These questions are not those the narrow specialists like very much. In order to make them attractive I am expected to explain in such a convincible way that you start to believe that without understanding how to solve it I will loose something of importance. And not only I will loose. You also will. Many of us would. Each of us will. If we compare it with another real and more difficult problem 20 this is the last problem and last problems in any competition are usually not the easiest ones, we will state that that problem is much more attractive from the standard point of view. And this is a nice circumstance. It means that reading the condition you are not so much afraid that it will cause you additional difficulties in understanding what s going on. Let us present you that formulation and you will see that on the stage - from the first glimpse - the usual when not banal performance is announced. At least nothing very 117

118 special some question with numbers - in addition even with not very large numbers. A 12-digit positive integer consisting only of digits 1, 5 and 9 is divisible by 37. Prove that the sum of its digits is not equal to 76. This is not identical with advertising industry with common proposal to take the Chevrolet for 1 Euro do you still remember the phrase of Hugo Steinhaus Chevrolet non olet assisted with some foot step notice about existence of some additional conditions. As it is usual in the life an attracting part first she walks (enters) in beauty like the night - and the assisting reality appears a bit later. We must agree that the formulations of the type you can t do that are very attractive. They are top challenging and the listener, having heard it is at once being brought to the natural state of admiration what s that?- such a simple thing and unrealizable. And he wishes to review the situation and to clear everything and better at once. That is the part of secret of an excellent formulation. Here is exactly the situation of that kind. We do not see any reason why the divisibility by 37 is so bad for the 12-digital numbers and, in addition, so bad, that it doesn t allow the sum of its digits to be 76. The first impulse of the young man such as from conversation with father William is not that it could be so and even more it ought to be possible. Just look for some multiplies of 37. The least multiply of 37 or 37 is not suitable it has nothing to do with the integers 1, 5 and 9 just as the second multiple of 37 being 74 also doesn t. But the third multiple of 37 is 111 and that multiple is especially good for constructing the numbers we are looking for. Taking some small multiplies of 111 or numbers, which are written using 3 identical digits, and afterwards putting it as fragments together we may in few seconds land at the number This number is almost an answer because it is indeed divisible by 37 as a number which is constructed from multiplies of 111 and its sum of digits is = 72 which is rather close to the expected 76. So it s not so easy to believe that this is impossible. After that experiment with the certain numbers although we didn t find the solution and it is not a wonder because we were told that the solution doesn t exist - we have learned a lot. Now we start with the usual considerations using a lot of sentences with preposition if. So imagine there is such a number. Then we subtract from that number the clear multiply of 37 that is the number

119 Then we ll get a new number which is written only with 0, 4, 8 still is divisible by 37 with the sum of digits = 64, which is also impossible to have as well as the initial one. But this impossible number is clearly divisible by 4 and that division wouldn t disturb the divisibility by 37. So it is impossible also to get the number the decimal expression of which consists exceptionally from 0 s, 1 s and 2 s with the sum of digits 64 : 4 = 16 and which would be divisible by 37. The first act of solution is over, psychologically we are involved to the process of solving and what could follow? One act for Problem 20 the last problem couldn t be enough. What could be done afterwards? After that the modest reflection we will involve such simple things like the criteria of divisibility by 9. Frankly speaking we will involve not the criteria of the divisibility by 9 itself but some criteria of really very same kind. And the criteria of the divisibility by 9 can be formulated in the very same manner as criteria of divisibility by 99, 999, 9999 etc. You will understand the general nature of it at once or immediately after we will present you the formulation for three nines or for 999. So any positive integer N has the same remainder when divided by 999 as the sum of all its 3-digits parts starting from the right to the left has. For example the number has the same rest when divided by 99 as the number, which we get adding its 3-digital fragments starting from right to the left or taking the sum = 200 and this rest is of course 2. And at that place instead of 999 we can take any divisor of 999 and 37 is clearly one of them! The criteria remains true, we notice only that our 12- digital number consists exactly from 4 such 3-digital fragments giving us as its sum also the 4-digital number with the maximal digit 8 and with the sum of digits being again 16. This is so because there were no carries by adding these fragments! We remind that the greatest digit now of the 12- number we got is 2, Applying of that criteria gives us the clear and direct possibility to end the solution as if we were in Grade 6 stating that there is no 3-digital number with the greatest digit 8 which would be multiply of 37 simply by presenting the list of all 3-digital multiplies of 37 there are only 25 of them and we can t resist the temptation simply to list them all showing how prosaic the finish from methodical point of view sometimes might be. 119

120 These 3-digital multipliers are: 111, 148, 185, 222, 259, 296, 333, 370, 407, 444, 481, 518, 555, 592, 629, 666, 703, 740, 777, 814, 851, 888, 925, 962, 999. Exactly 3 of them have the sum of digits 16. These are 259, 592, 925. They all have the digit 9, which is in our case impossible and forbidden so such an initial number doesn t exist. It should be added that with the computer it would last some seconds but as you see it could also be done in two steps also without the computer as if we were living in the desert island somewhere in the middle of billowy ocean. Still there are extreme situation of the life in every area and from that philosophical point of view these are most interesting because they set boundaries in the usual sense so that knowing them you know that all other things are running between or in the middle of them. The same but as always in much stronger degree take place in the world of good but simply looking problems there are elephants in that world too. As perhaps most striking example we will firstly present the formulation and only afterwards add several prosaic details. In mathematical competition some competitors are friends. Friendship is always mutual. Call a group of competitors a clique if each two of them are friends. (In particular, any group of fewer than two competitors is a clique.) The number of members of a clique is called its size. Given that, in this competition, the largest clique is even, prove that the competitors can be arranged in two rooms such that the largest size of a clique contained in one room is same as the largest size of a clique contained in the other room. How do you find that problem? It is not difficult to understand that in each finite gathering not only in competitions everywhere and from any gathering you are able to find the largest group of people knowing each other. And if this largest group contains an even number of persons the only thing, which is of importance there that you can every gathering to split in two with the size of clique being the same. So from the philosophical point of view you state only that the object, which is even in some sense, may be divided into two equal parts (in the same sense). So how did you find it? Methodically speaking, we would like to ask your opinion about the following: the students of which grade are able to understand what this problem is about? Or starting from what form is it possible to understand? Understanding that everything is relative it s true in the same degree as each aphorism is we would like to show another problem which was once being given in the Sankt-Petersburg City Olympiad and to ask which one of them do you think is more difficult? 120

121 How many 10-digit numbers divisible by are there whose decimal representation contains only the digits 3, 4, 5, and 6? Now about the problem of dividing the cliques into equal parts, which seems theoretically understandable it ought to be said that this modestly looking problem is Problem 3 from International Mathematical Olympiad in Vietnam International Mathematical Olympiad sounds very convincing still we think that the name World Students Cup in Mathematics would be not worse. In that case it could be added that in that Cup 520 students took place. We could ask many things about all these events, e.g. how many students according to your opinion were able to divide the competitors in that democratic way? We do not like to keep you in the state of uncertainty and kindly inform you, that 2 (in words: two) participants solved it one of them as was clearly indicated from Serbia and another one from the country, whose name seems not to be indicated as yet but it would be also interesting to know. There are enough problems, which are as natural or otherwise speaking democratic or powerful in formulation and not so difficult in solution. As the example we could mention the problem 7 from the Baltic Way The photographer took some pictures at a party with 10 people. Each of the 45 possible pairs of persons appears together on exactly one photo, and each photo depicts 2 or 3 persons. What is the smallest number of photos taken? The accessibility of problems of such kind is that if you want or are motivated to learn to solve them you might try to reduce the numbers in the formulation and start solving. For example, above you oughtn t to be afraid to reduce the party twice and regard the same problem in the party with 5 persons and then only with 10 possible pairs of them and remaining the rest to be just as it was. If it is too much let us simply take 4 persons with the 6 possible pairs. The problem is already defined for 3 persons. After reducing everything seems and usually is much easier. And finally we would like to try to say some words about the problems whose formulation itself is already as nice as it is possible to imagine. Let us refer to these as to problems, which are perfect in formulation. The problems, which are perfect in formulation, are already unforgettable just as they are. Finally let us take a rather idealistic note and sing some lines from the song all are equal, all are different and all are unique. This saying is of course, correct, while at least while all real twin brothers have nothing against it. But the one thing is to know some things theoretically, e.g., my wife is beautiful or my mother is excellent - which is already not bad. 121

122 Other thing is that in the main cases these theoretical knowledge has a real content you have in your mind the concrete picture of your indeed remarkable wife and also the same concrete picture of your 25 hours per day busy mother. Let us say the same with some number examples. We all have heard something about the outstanding Indian mathematician Ramanujan and his towards infinity oriented formulas. It was told and repeated that every natural number was his friend. It was being reported that once he was asked what is so particular about the number 1729? in the sense that there is nothing special about that number. And the answer of Ramanujan stated that exactly this number is the first natural number, which can be represented as a sum of two natural cubes in a two different ways that is 1729 = 10³ + 9³ and in the same time of course 1729 = 12³ + 1³ In the same time it is difficult to believe that before Ramanujan nobody noticed it, for example, Leonhard Euler. I would like to precede that line citing some lines from the small article Ditigal powers from the January issue of otherwise also outstanding magazine Mathematical digest, No. 142, 2006, p.8. It will also about the uniqueness of some other numbers. Again the similar question what is so important about the number ? Different from the number 1729, which our conscience regards as rather small number and which we psychologically are less afraid of, that given number is large enough not to be regarded being too small. What s so important on that number? What is so really outstanding? Standard observation could indicate, e.g., that this is an even number and has a rest 1 by division 3, so that this is a number 6n + 4. You will be very soon indeed convinced that is something rather special about that number because = How do you think one can find such a number? Is it possible without applying the computer? The only question which ought to be discussed is of course 0 to power 0 and it is being told that we have to admit that is a trifle 0 0 dodgy, since 0 (like 0/0) is undefined. But if we naively assume that 0 is zero (ex nihilo nihil nothing comes from nothing - with perhaps the only exception 0! = 1), then the above equation is true as may be easily checked. Is this the first such number? It is claimed that there is only one other number with the property. It has four digits, none of them equal to zero so for purists it would be probably the only absolutely correct number with this 122

123 property. And of course it is interesting to find it without applying any calculator. This is not difficult. All this is some continuation of an article [1] and is based on some ideas about the possibly simplest presentation of some well-known problems. Some of such problems are discussed and the explanations about their solutions are published in the English translation of author s book. [2]. This book appeared in the LAIMA series, which is common Icelandic-Latvian project. This book contains the first half of the Lithuanian version. The translation of the second half is also announced to appear in the current year. The attempts to solve the problems in the simplest possible way are being made since the ancient times. This was important during all periods of human history. But in our days when patience is expected to be shown by everybody to explain all matters every time when asked this seems to be even more important. References 1. Romualdas Kašuba. About so-called democratic problems proposed at International Mathematical Olympiads (IMO). VII International Conference, May, 2006 Teaching Mathematics:Retrospective and perspectives, Proceedings of the Conference, Tartu, 2006, pp R.Kašuba, What to do when you don t know what to do?; Rīga: Mācību grāmata, 2006, 129 p., ISBN

124 A COMPARATIVE PERSPECTIVE ON THE ESTONIAN AND NORWEGIAN STUDENTS BELIEFS IN MATHEMATICS Kirsti Kislenko, Agder University and Tallinn University, Abstract. The focus of this paper is to compare the Norwegian and Estonian students beliefs about mathematics. The study was carried out in the urban areas in Norway and Estonia, and it used a web-based questionnaires. Except the result that the Estonians hold a slightly more formalistic view of mathematics, there did not appear significant differences in-between the nations. Keywords: affective domain, comparative study, cross-cultural study, mathematics education, students beliefs. Introduction Mathematics has been a highly respected discipline in school for centuries and its high status makes students success in mathematics very important. This has implied the situation where affect has become an interest of researchers in mathematics education as it has been pointed out that students beliefs about mathematics are strongly related to their learning outcomes in mathematics. Some studies about students and teachers attitudes have been carried out in Estonia during the last 10 years (e. g. Lepmann, 2000; Lepmann & Afanasjev, 2005), and many of them aimed the perspective of an international comparison (e.g. Pehkonen, 1994; Pehkonen, 1996; Pehkonen & Lepmann, 1994). In the light of these studies researchers from Norway and Estonia carried out a study to investigate Norwegian and Estonian students views towards mathematics. In the Norwegian study, which was a part of the LCM-project within the Norwegian Research Council s KUL 2 program, the data collection was finished in spring 2005, and the data collection in Estonia took place in spring One of the aims of the study was to investigate students beliefs about mathematics in one urban area in Norway and Estonia. The focus of this paper is to enlighten the differences and similarities in the Norwegian and Estonian students views towards mathematics. 2 KUL - Kunnskap, utdanning og læring, translated as Knowledge, education and learning. 124

125 Research participants The study included a total of 859 students; 266 students from grade 7, 317 students from grade 9, and 279 students from grade 11 all of whom responded to a questionnaire about their beliefs and attitudes towards mathematics. Six schools, from the same urban area in Norway, that were partners in the KUL-LCM project took part in this study. The study consisted of 276 students; 32 students from grade 7, 85 students from grade 9, and 159 students from grade 11. All the schools were located in the same local area; there were one primary school (up to 7 th grade), one primary plus lower secondary school (up to 10 th (included) grade), two lower secondary schools (from 8 th to 10 th (included) grade), and two upper secondary schools (upper secondary starts with grade 11 (students are years of age), and continues for three years). Seven schools from one urban area in Estonia participated in the study. The number of the students who participated was 580 whereas there were 232 seventh grade students, 232 students from the ninth grade and 116 students from grade 11. Seven schools in the study divided as, first, one primary plus lower secondary school (from 1 st to 9 th (included) grade), six primary plus lower secondary plus upper secondary schools (from 1 st to 12 th (included) grade); secondly, 6 state schools, including 2 elite schools (look the description below), and one private school. In Norway students in primary and lower secondary level either go to the so called local school (urban area is distributed to small regions in which every one has its local school) or the school closest to their home (for example, students, who live at the border of one region, and to whom the neighbourhood s local school is closer than their own local school have the possibility to choose in-between these two schools). In Estonia there are no restrictions about a school choice one can choose to attend any desired school. Children can either go to the local school (there is always a place guaranteed for the child), or to the so called elite schools (state schools which have a high status in the community; they have compulsory entrance exams), or any other school in the area if there are vacant places. Data sources and data collection The instrument of the study was a web-based Likert-scale questionnaire, which was made available for the students on an Internet webpage. The language used in the questionnaire was respectively either Norwegian or Estonian (both studies used the same questionnaire that was translated in the respective language). Each student was given a unique code that he/she used to log into the questionnaire page. In Norway the students answered the 125

126 questionnaire mainly during their spare. In Estonia majority of the students completed the questionnaire in their mathematics or computers lessons and they had the possibility to get explanations of the completing procedure from their teacher or from the researcher. The questionnaire contained 98 statements divided into 11 groups. The first four groups included my thoughts about : mathematics (A, 16 statements); learning mathematics (B, 13 statements); my own mathematical abilities (C, 10 statements); my own experiences (D, 3 statements). Groups E ( activities in mathematics lesson, 15 statements), F ( learning a new topic in mathematics, 7 statements), and H ( teaching tools in mathematics, 6 statements) included the statements in relation to the activities and the structure of the ordinary mathematics lesson. Nine statements in relation to the environment in class i.e. classmates attitudes towards mathematics constituted group G. Statements about what are the important things to learn in mathematics (I, 7 statements), teachers expectations i.e. what the teacher finds important when assessing ones work in mathematics (J, 10 statements), and two statements about the relationship in-between the future profession and mathematics (K, 2 statements) constituted last three groups. Some open-ended questions were also included in the questionnaire, mostly in the form: something else you want to mention or why/why not. Data analysis The following data analysis procedure was applied to both databases the Norwegian one, and the Estonian one separately. For carrying out a deeper analysis, I chose the statements from the first four groups (A-D, 42 statements) as these represented thoughts about mathematics, learning mathematics and ones own abilities in mathematics. The statements in groups A, B, C used a 5-point Likert-scale type responses from totally agree to totally disagree (for clarification, the Norwegian questionnaire used in groups B and C a 4-point Likert-scale type response as the choice undecided was excluded). Group D statements used the choices from never to very often i.e. 4-point Likert-scale type responses. First, exploratory factor analysis was employed for determining the preliminary structure of factors. Secondly, based on exploratory analysis, a confirmatory factor analysis was carried out. Finally, after several considerations based on the fit indices, theoretical underpinnings, personal judgement, and the idea that the best is to leave in as many statements as possible, 6 factors were extracted which all reflected a reasonably good fit. All the names of the factors referred to the literature; I mainly followed the work of Streitlien 126

127 et al. (2001), and Brew et al. (2006). All the items loading to these six factors are presented below. Results Here I present the results of the studies from the perspective of the comparison. As the data collected are ordinal that puts some limits to the analysis then the percentages of the answers of every item are taken as a main basis for the analysis. Factor 1: Interest The percentages of the answer of every item are presented in table. The columns refer respectively: totally agree (Ta), partially agree (Pa), undecided (U), partially disagree (Pd), totally disagree (Td). Item Country Ta Pa U Pd Td A2 Mathematics is exciting and NOR interesting EST A3 Mathematics is one of the NOR subjects I like the least EST A8 Mathematics is one of the NOR subjects I like the best EST A9 I never get tired of doing NOR mathematics EST A10 I like to do and think about NOR mathematics also out of school EST A13 Mathematics is boring NOR EST C3 Mathematics does not suit me NOR EST In all the answers the Norwegian students tended to be more radical i.e. they marked more totally agree/disagree than the Estonians, and the uncertainty percentage was lower (except item A10) as well. What comes to agreeing and disagreeing then marginal differences could not be detected, except item A13. Approximately 50 % of the Norwegian students claimed that mathematics is boring against 37 % of the Estonians. Moreover, the majority of the Norwegians agreed with A13 but the majority of the Estonians disagreed with the statement. Looking closer, more than fifth of the Estonians could not make up their mind about this item. The biggest difference in the total disagreement appeared in the item C3 where more than 30 % of the Norwegian students strongly disagreed that mathematics doesn t suit them against 12 % of the Estonians (when calculating the means of every item for seeing the tendencies this result seemed to be confirmed). But as there was a high uncertainty amongst the Estonian students (23 %), and the Norwegians did not have the possibility 127

128 of uncertainty at all then this result can easily be misleading and is not taken as a valid conclusion. Factor 2: Hard-working Item Country Ta Pa U Pd Td B5 I have to solve many tasks to NOR become good in mathematics EST B7 I have to work hard in mathematics NOR even if I do not want it EST B8 To become good in mathematics is NOR dependent on hard work EST B9 I have to solve many tasks to NOR remember the method EST C4 If I want to be able in mathematics I have to spend NOR plenty of time solving tasks EST C6 I could become clever in NOR mathematics if I would learn all rules EST It seemed that the slight differences in the answers were due to the uncertainty option. Otherwise, both students acknowledge that hard work is needed in learning mathematics. The majority of the students (84 % in Norway, and 73 % in Estonia) claimed that to become good in mathematics is dependent on hard work, and agreed (79 %, and 72 %, respectively) that they have to work hard even if they do not want it. Being diligent i.e. solving tasks appeared to be equally important for both of the groups (items B5, B9, and C4). Factor 3: Self-confidence Item Country Ta Pa U Pd Td A4 Mathematics is difficult NOR EST A14 Mathematics is easy NOR EST C1 I am able in mathematics NOR EST C2 I can solve most of the NOR mathematical tasks if I concentrate EST C8 Mathematics is easy for me NOR EST C10 It is only bad luck if I do not do NOR well on a mathematics task EST

129 The only notable difference in items A4 and A14 seemed to be a switch in radicalism. When in Interest factor the Norwegians tended to be more radical then here 21 % of the Norwegians against 33 % of the Estonians totally agreed with the item mathematics is difficult, and 14 % against 21 %, respectively, strongly disagreed that mathematics is easy. In general, both groups of students claimed mathematics to be difficult (agreement with item A4 was 65 %, and 71 %, respectively), and not easy (agreement with item A14 was 34 %, and 28 %, respectively). Even though around half of the respondents (56 % in Norway, and 45 % in Estonia) disagreed that mathematics is easy for them, both groups agreed with high certainty (77%, and 72 %, respectively) that they can solve most of the mathematical tasks if they concentrate. Factor 4: Usefulness Item Country Ta Pa U Pd Td A1 Mathematics is important NOR EST A5 Mathematics is useful for NOR me in my life EST A6 It is important to be good in NOR mathematics in school EST A7 I need mathematics in order to study what I would like after NOR finishing school EST A11 Mathematics helps me to NOR understand life around me better EST A16 Good mathematical knowledge NOR makes it easier to learn other subjects EST The only notable difference considered item A11, where the Norwegian students agreed with 38 % of certainty, against 53 % of the Estonians, that mathematics helps them to understand life around them better. Moreover, when approximately 16 % of the Norwegians strongly disagreed with the same item, the total disagreement amongst the Estonians was only around 4 %. Otherwise, the tendencies in all other answers were practically the same in both groups. Generally, both nations highly acknowledged the importance (agreement with A1 was 91 %, and 92 %, respectively) and usefulness (agreement with A5 was 81 %, and 88 %, respectively) of mathematics. Students did not only recognize the importance of mathematics but the importance of being good in it as well (agreement with A6 was 82 %, and 73 %, respectively). More than 60 % (62 %, and 67 %, respectively) noted that good mathematical knowledge makes learning other subjects easier (A16). There was a slight difference (around 12 %) in the 129

130 certainty of the item A7, where the Norwegians tended to be more uncertain about the necessity of mathematics in relation to their future study plans. Factor 5: MAD - Mathematics as an Absolute Discipline Item Country Ta Pa U Pd Td B1 The most important in mathematics NOR is to know many rules EST B2 It is important to be fast finding a NOR right answer in mathematics EST B3 It is just one right answer in NOR mathematical tasks EST B4 When I make mistakes in mathematics it shows that I do not have NOR enough knowledge in mathematics EST B6 Right answer is more important NOR than the procedure I have used EST C7 It is innate to be good in NOR mathematics EST It was obvious that the Norwegian students more radically disagreed: 25 % of the Norwegians against 6 % of the Estonians totally disagreed that one has to be fast when finding an answer (B2), 29 % against 11 %, respectively, strongly disagreed that there is only one right answer (B3), and 28 % against 9 %, respectively, strongly disagreed that mathematical abilities are innate (C7). Despite the fact that the Norwegian students missed the choice undecided, it is remarkable that the difference in the answers appeared in one end of the scale i.e. total disagreement, and the partial agreement and disagreement were similar to the Estonians. Nevertheless, taking the agreement as a basis of the comparison no notable differences could be determined. In both groups majority of students valued knowing rules in mathematics (agreement with the item B1 was 71 %, and 53 %, respectively). Even though almost 30 % of the Norwegians totally disagreed with the idea of innate abilities in mathematics, more than 46 % of students in both countries as well agreed that it is innate to be good in mathematics (C7). Factor 6: Anxiety The columns refer respectively: never (N), seldom (S), often (O), very often (Vo). Item Country N S O Vo D1 I am afraid of making mistakes NOR when I do mathematics EST D2 I become nervous when we have NOR tests in mathematics EST D3 I am afraid to show my teacher that I do NOR not understand mathematical problems EST

131 In general, students from the both countries felt rather secure in mathematics. Students on average were seldom afraid of making mistakes in mathematics, and more than three-quarter of respondents from the both nations (80 %, and 75 %, respectively) were seldom or never afraid to show their teacher that they do not know how to solve a mathematical problem. Generally, there was not a remarkable difference between the Norwegian and the Estonian students answers. The only notable difference concerned tests in mathematics (D2) where the Norwegians claimed to become nervous very often with 20 % certainty against the Estonian 10 %. Synthesis When interpreting these results one has to stay critical as, first, the sample used in both countries was a convenient one, and the number of the sample size was different. Secondly, the questionnaire used in Norway missed in several groups the choice undecided that, strictly talking, made the comparison in between the countries not really appropriate (as data collection tools the questionnaires - were actually different). Nevertheless, keeping in mind the aforementioned problems one can still make some general conclusions based on the presented frequency tables. What comes to the interest of mathematics the Norwegian students were more radical in their answers. They disagreed more strongly that mathematics is the subject they like the best/the least; that they like to do mathematics out of school; and that mathematics does not suit them. Even though the TIMSS2003 study detected the differences in the enjoyment of mathematics in between Estonia and Norway, where Norwegian agreed considerably more with the statement I enjoy learning mathematics (Mullis et al., 2004), my study did not confirm this phenomenon. Even quite an opposite, the Norwegians agreed significantly more that mathematics is boring. It has been pointed out in the literature that one of the reasons for the students finding mathematics boring might be an approach that emphasises the abstract nature and heavy symbolic representation of mathematics (e.g. Tickly & Wolf, 2000). Erkki Pehkonen and his colleagues found out in their study that Estonian students favour a strict discipline and rigour in relation to mathematics teaching, and hold more formalistic view about doing mathematics than the students from Sweden, Finland, and the United States (Pehkonen, 1994). Therefore, one can speculate that as the Estonian students are somehow used to, and even value the formalistic view of mathematics (Pehkonen, 1994), the Estonian students in my study maybe did not find an abstract nature and symbolic representation in mathematics boring but considered it as a part of mathematics itself. This is coherent with the results in factor Mathematics as an Absolute Discipline 131

132 where the Estonian students significantly less totally disagreed most of the items that actually reflect the formalistic view of mathematics. Nevertheless, almost half of the students in both nations agreed to the idea that mathematical abilities are innate. It has been pointed out in the literature that importance and effort in mathematics are highly related (Kloosterman & Coughan, 1994; Ma, 1997). So it seems in my study as well as students from both countries found mathematics highly important and useful, and agreed that to become good in mathematics demands hard work. There were not notable differences in the Norwegian and Estonian students answers about the usefulness and importance of mathematics, and being diligent in mathematics, and this result is coherent with the conclusion of the TIMSS2003 study where they concluded that Estonian and Norwegian students value mathematics similarly (Mullis et al., 2004). In 2003 in the TIMSS study Norwegian students exposed significantly higher self-confidence than students from Estonia (Mullis et al., 2004). My results did not support it as students from the both countries in my study hold a rather similar view of their self-confidence. Even though approximately half of the students from Norway and Estonia disagreed that mathematics is easy for them they agreed that they are able in mathematics and can solve most of the mathematical tasks if they concentrate. Most of the students claimed that they feel secure in mathematics lessons and I could not detect any significant differences in-between the Norwegians and Estonians. One might see a slight difference in their answers regarding tests in mathematics the Norwegians become more often nervous when they have tests than the Estonians. Generally, based on my study one could conclude that the Norwegian and Estonian students views of mathematics are similar. Only remarkable outcome to mention was the fact that the Estonians tended to hold more formalistic view about mathematics than the Norwegians. As studies have shown that Estonian teachers agree more than the Finns with the formalist aspects of mathematics teaching (Pehkonen & Lepmann, 1994), and it has been pointed out that students beliefs reflect the mathematics teaching they have experienced (Pehkonen, 2003), it seems that the outcome of the Estonians favouring formalistic view of mathematics is not surprising. References 1. Brew, C., Riley, P., Walta, C. (2006). Fear Factor: What does maths anxiety measure in pre-service primary teachers? Paper presented at the Annual Conference of the Australian Association for Research in Education, Adelaide, Nov 27-30,

133 2. Kloosterman, P., & Cougan, M. C. (1994). Students beliefs about learning school mathematics. The Elementary School Journal, 94 (4), Lepmann L. (2000). Eesti ja vene õpilaste arusaamad matemaatikaõpetusest. Koolimatemaatika XXVI (pp ). Tartu: Tartu Ülikooli Kirjastus. 4. Lepmann, L., & Afanasjev, J. (2005). Conceptions of mathematics in different ability and achievement groups among 7th grade students. In C. Bergsten and B. Grevholm (Eds.), Proceedings of NORMA 01, the 3rd Nordic Conference on Mathematics Education (pp ). Linköping: Linköping University. 5. Ma, X. (1997). Reciprocal relationships between attitude toward and achievement in mathematics. Journal of Educational Research, 90, Mullis, I. V. S., Martin, M. O., Gonzalez, E. J., & Chrostowski, S. J. (2004). TIMSS 2003 international mathematics report. Findings from IEA s trends in international mathematics and science study at the fourth and eighth grades. Chestnut Hill, MA: TIMSS & PIRLS International Study Center, Boston College. 7. Pehkonen, E. (1994). On differences in pupil s conceptions about mathematics teaching. The Mathematics Educator, 5(1), Pehkonen, E. (1996). Some findings in the international comparison of pupils mathematical views. In E. Pehkonen (Ed.), Current state of research on mathematical beliefs III. Proceedings of the MAVI-3 Workshop. Helsinki, August , Pehkonen, E. (2003). Læreres og elevers oppfatninger som en skjult faktor i matematikk -undervisningen. In B. Grevholm (Ed.), Matematikk for skolen (pp ). Bergen: Fagbokforlaget. 10. Pehkonen, E., & Lepmann, L. (1994). Teachers conceptions about mathematics teaching in comparison (Estonia - Finland). In M. Ahtee and E. Pehkonen (Eds.), Constructivist viewpoints for school teaching and learning in mathematics and science (pp ). Helsinki: University of Helsinki, Department of Teacher Education. Research Report Streitlien, Å., Wiik, L., & Brekke, G. (2001). Tankar om matematikkfaget hos elever og lærarar. Oslo: Læringssenteret. 12. Tikly, C., & Wolf, A. (2000). The maths we need now: demands, deficits and remedies. Bedford Way Paper 12. London: Institute of Education. 133

134 THE DEVELOPMENT OF STUDENTS RESEARCH SKILLS IN MATHEMATICS Elfrīda KrastiĦa, Iveta Nikolajeva, Daugavpils University, Abstract. Today the accent in education is put on students research skills. To develop them, students are offered activities in different projects both during project weeks and in other periods, they have possibilities to write scientific research papers etc. How the developed students research skills are? Keywords: problem solving, research activities, research skills, skills beyond content level. Introduction Modern young generation which is living in the epoch of continuous changes especially needs the ability to solve global, local and everyday s problems. Problem solving as a form of practical thinking is closely related to research activity of thinking. Problem solving develops research skills and promotes students creative and critical thinking (Robert Fisher, 1990). Therefore modern learning process is oriented to the development of students research skills not only on a secondary level, when students do their research works, but these skills are to be developed in basic school already. Problem-based, research learning has been analyzed in papers of many authors (Gage N.L. & Berliner D.C., 1999; Fisher R., 1990; Fisher R., 2005; Žogla I., 2001; Maslo E., 2003; Piažē Ž., 2000). Now it is actual to evaluate previous experience in order to find possible ways how to develop students research skills, taking into account not only content but also beyond-content level. Why it is necessary for a student to develop research skills? Research skills are the skills which are connected with finding out facts, analysis, drawing conclusions and acquiring systematic knowledge (The Dictionary of pedagogical terms, 2000). Students research activity, in its turn, is connected with the solution of creative, research work. Research tasks, apart from traditional standard tasks, cannot be solved following a previously acquired algorithm. Their solution requires developed logical thinking and creative approach. Students research brings discoveries to students themselves, and they acquire initial skills of research. These skills 134

135 are useful not only for studies in higher school, but also in practical life as well. While acquiring mathematics, students develop their mathematic skills, acquire mathematical models and come to understanding about their role in describing and analysis of nature and society. The methods of mathematical reasoning, which are based on strict logical motivations and precise calculations, enable us to obtain different descriptions of nature and technical processes and evaluate their usefulness. The mathematical way of thinking the production and analysis of a model, inner necessity to argument, creating a strict deductive view of the world is an essential part of general education. The aim of learning is to awake in students a creative impulse, to encourage their research activity. Organization of students scientific research activity in Latvia Students scientific research activity is one of the fields of their out of class education, the traditions of which have been developing for several decades. The Ministry of Science and Education has adopted Regulations of scientific research of students in secondary schools in In several schools students are taught optionally a special course Basics of scientific research etc. In some schools research centers are established. Secondary school students write scientific research papers, defend them at scientific research work (SRW) conferences at schools. The best ones are promoted to be defended on regional and state level conferences. The winners of scientific research work contests have a possibility to take part in summer camp Alfa. Research skills necessary for research activity To write a SRW, a student should acquire certain skills, which are used on different stages of research process. The problem, aim and objectives of the research Observations and collection of information Analyses of information and processing of acquired data Formation of research description Conclusions and presentation Fig. 1. Stages of a research process (Hakele R., 2002) To write a research paper one needs to have skills beyond the content level. Here a student needs: to see the problem under research, to put forward the idea, to formulate the topic of the research; 135

136 to put forward the aim and objectives, to formulate the question or hypothesis of the research; to plan the process of the research, its structure, to choose methods, tools, to set a schedule; to choose and work with sources of information, to select necessary information, to compare views of different authors on the problem of research, to formulate own opinion; to collect, systematize, interpret data of research, to process it with IT, to verify credibility of the result; to formulate conclusions appropriate to the aim and the tasks of the research, to substantiate own opinion, to put a proposal; to format the scientific research work according to the requirements; to prepare deliberative, logical presentation of the research and defend it; to evaluate the results of own or other students research works. The acquirement of research skills is realized in optional classes. As practice proves the skills of problem solving and research methods do not appear out of blue sky they have to be acquired (France I.). This demands time; that is why it is important to ensure a systematic students preparation to research work in learning process. The introduction of a new education standard, starting with years 2005/2006 in the area Basics of technology and science, foresee to acquire basics of practical research, which involves obtaining information, planning research, solving problems, carrying out experiments, processing and evaluation of results as well as ability to present the results obtained (State standards of basic education, 2005). The results of acquiring research skills To illustrate the situation in acquiring research skills by students we will analyze the results of students from a certain grammar school. As the analysis of the results of international research OECD SSNP 2003 (Geske A., Grinfelds A., Kangro A., Kiselova R., 2004) shows, the average level of the mathematical competence almost completely coincides with problem solving competence, so we chose to analyze students learning achievements in basic and secondary school. Students learning achievements in mathematics in basic school in grades 7 9 are the following. 136

137 42% 33% 36% 23% Annual average mark 2003./2004. Annual average mark 2004./2005. Annual average mark 2005./2006. Exam in Fig. 2. Number of students in per cent who have marks 8-10 in exam or as an annual mark. The analysis of the results in basic school (See fig. 2.) shows, that if the difficulty level grows, the results worsen due to different objective and subjective reasons. However, the results of exams demonstrate that 36% of students have acquired research skills well. Students learning achievements in mathematics in secondary school in grades are the following. 34% 12% 14% 14% 17% Annual average mark 2003./2004. Annual average mark 2004./2005. Annual average mark 2005./2006. National test National exam Fig. 3. The number of students in per cent, having annual mark or national test or exam mark Secondary school students achievements in mathematics (See Fig. 3.), on the contrary, have the tendency to improve, and the annual results shown in this diagram are stable. National test proves that 17% of students have acquired research skills on high level. National exam in mathematics is not 137

138 compulsory and therefore the results are higher. So results let us foresee how well prepared the future students of higher schools are. Evaluating students SRW in which they prove their research skills, we see that only 35% of them have got a higher mark. In comparison with the results in mathematics, these are higher, as numbers grow with the help of humanitarian research papers. As it turned out, 17% of secondary school students wrote SRW in exact science (mathematics, physics, chemistry, computer studies) in 2006/2007. The choice of topics for students SRW shows how popular the exact sciences at school are. They need good mathematical preparation. All in all the results coincide with the results of national test in mathematics. If a student cannot prove oneself in mathematics studies, he/she will not acquire a stable interest to do research in exact sciences. Conclusions The development of students research skills is one of the tasks in the area Basics of technology and science in basic education standard: to encourage the acquirement of basics of research skills. As the analyses of school practice shows, many school have no relevant material bases in exact sciences to fulfill this task. In general during the transition period methodical preparation of teachers has not been carried out. Preparation of teachers for guiding students research work is in the focus of in-service training courses. Taking into account that teachers visit courses once in three years, we cannot predict that methodical competence of teachers in the sphere of research works will rise this year. To help the students to acquire research skills, we can try to do it in the content of concrete subjects: acquiring skills to solve practical and research tasks; putting in order, analyzing data and predicting the result to be obtained; giving mathematically grounded conclusions; acquiring skills to study the reasons of phenomena and processes and nature laws in nature studies, biology, geography, physics and chemistry; as well as acquiring individual and cooperation experience in research and practical activities, using information technologies (State standards of basic education, 2005). These general guidelines should be specified in academic programs. The acquirement of research skills beyond content level foresees to develop students learning experience, in which students acquire experience of intellectual work, learning to think logically, critically, independently, productively; choose the relevant technology in searching information, its analysis, organization and transformation; use mathematical skills and research work experience in acquiring other knowledge and practical life (State standards of basic education, 2005). 138

139 Purposeful development of students research skills development beyond content level needs a special program starting from class 1. The implementation of it should be carried out in various subject lessons at once. To promote strengthening of students research skills in self regulated activities, a special guidebook for secondary school students is necessary with detailed information and relevant tasks, which can be used in optional courses and in various subjects as well. References 1. BeĜickis I., Blūma D., Koėe T. (2000) Pedagoăisko terminu skaidrojošā vārdnīca (Explanatory dictionary of pedagogical terms). Rīga, Zvaigzne ABC. 2. Bybee R. W. et. al. (2005) Doing Science: The process of scientific inquiry. BSCS. 3. Fisher R. (1990) Mācīsim bērniem domāt (Teaching children to think). Rīga, RaKa. 4. Fisher R. (2005) Mācīsim bērniem mācīties (Teaching children to learn). Rīga, RaKa. 5. Gage N.L. & Berliner D.C. (1999) Pedagoăiskā psiholoăija (Pedagogical Psychology). Rīga, Zvaigzne ABC. 6. Geske A., Grinfelds A., Kangro A., Kiselova R. (2004) Mācīšanās nākotnei (Learning for future). Rīga, LU Institute of education research. 7. Hakele R. (2005) Skolēna zinātniski pētnieciskā darbība (Scientific research activity of student). Rīga, RaKa. 8. Maslo E. (2003) Mācīšanās spēju pilveide (Development of learning skills). Rīga, RaKa. 9. Piažē Ž. (2000) Bērna intelektuālā attīstība (Intellectual development of child). Rīga, Pētergailis 10. Valsts Pamatizglītības standarts (2005) (State standards of basic education). Rīga, ISEC. 11. Žogla I. (2001) Didaktikas teorētiskie pamati (Theoretical basics of didactics). Rīga, RaKa. 139

140 ELEMENTS OF TRIGONOMETRY IN ANIMATED PICTURES Dzidra Krūče, Liepāja Akademy of Pedagogy, Abstract. A software for acquiring the basic elements of trigonometry is described. Keywords: animation, graph of a function, trigonometry. There was time when the school course treated trigonometry separately in algebra and geometry. For example, the geometry course defined trigonometric functions only for acute angles as qoutients of the sides of a right triangle, but when dealing with bigger angles trigonometry was included in the topic about functions in the algebra course. However, a clearly expressed feature of a trigonometry separating it from other topics is the issue about the motion of figures, contrary to the treatment of such figures being static as in geometry. In geometry the angle is defined as a part of the plane, bordered by two rays extending from one point. For trigonometry this definition is not sufficient. Trigonometry needs a direction of the angle, we should deal with angles bigger than 360 o, but they can not be statically displayed; motion is required. It is difficult to give the definitions of trigonometric functions and to explain the properties of them statically. The need for motion can be partially tackled with the help of animation pictures. It is difficult to present animation in a course-book; therefore other teaching aids are used, such as modern technologies, which are available in every school now. Changes of sine and cosine function values depending on the size of the angle (arc) can be clearly seen in the further four figures. The programme has been designed using Macromedia Flash Player 6. The running of the programme was interrupted four times, thus obtaining four figures. First of all, the user enters the value of the angle (positive or negative), this time it is 360 o, then presses the button Zīmēt (Draw), the moving radius starts turning in the positive direction and its end makes an arc. Simultaneously the sine and cosine lines are changing, at the bottom of the figure we also see how the sinα and cosα values of the angle change gradually. To interrupt the process press another button Pārtraukt (Stop). At first the process was interrupted in the first quadrant (Fig. 1). When examining the animation, it can be seen that the cosine function in the quadrant 1 decreases, starting with the value 1, and the sine function 140

141 increases starting with the value 0. We stopped the process at α = 57 o, sin α = 0.84 and cos α = Fig. 1 After that the process was interrupted in quadrant 2 (Fig. 2). We see that in this quadrant the sine function decreases, but remains positive, the cosine function continues decreasing, but is already negative. When the process was stopped we had α = 129 o, sin α = 0.78 and cos α = Fig. 2 Fig. 3 shows that the process had been interrupted in quadrant 3. Here we see that the cosine function keeps growing starting with -1, but it remains negative, the sine function continues decreasing, but is already negative. When the process was stopped we had α = 221 o, sin α = 0.66 and cos α =

142 Fig. 3 Fig. 4 shows that the process had been stopped in quadrant 4. We see that the cosine function continues growing, but it is already positive, the sine function increases starting with -1, but remains negative. When the process was stopped we had α = 318 o, sinα = 0.67 and cosα = Fig. 4 If we enter the angle α > 360 o we will be able to observe periodical repetition of the values of sine and cosine functions. By pressing button 3 Dzēst (Erase) everything is erased and we can repeat the process again. The teacher can make use of such animation model for demonstrations as well as the student can do this for acquisition of the material or for fast revision. Properties of tangent and cotangent functions can be displayed in a similar way. In a quick animation form we can also draw the graphs of inverse trigonometric functions. Figure 5 shows how the graph of arctg x is designed. 142

143 First of all we draw graph of y = tg x (button 1), then a bisector y = x (button 2) (Fig. 5). After that in an animated way symmetrically to the graph of y = tg x with respect to the bisector we draw the graph of y = arctg x (Fig. 5). Fig. 5 Even and odd trigonometric functions, reduction formulas, solution of trigonometric inequalities, relation of the sine curves and tangent curves with the sine and tangent lines etc. can be visually shown with the help of animation. The Elements of Trigonometry in Animated Pictures can be used by students before the examination for repeating the course in trigonometry. 143

144 A MATHEMATICS CONTEST AN OPPORTUNITY FOR EVERYONE Aija Kukuka, Liepājas Pedagoăijas akadēmija, Abstract. A mathematical contest organized at Liepāja Pedagogical Academy is described. Keywords: mathematics olympiads, mathematics olympiad participants, mathematical potential of pupils Introduction As Hegel once exclaimed: What at one time was thought of as recondite knowledge, understood only by wise old men, today is obvious and understood by every school boy. This statement is an expression of the cultural differences and requirements of each generation and culture as well as a reflection of individual self-development. Personality is an intricate structure which needs to be studied in a systematic way. Talent is an important attribute of personality. In mathematics talent is recognized by three main characteristics: speed and facility, retention and profundity. A systematic method for nurturing mathematical talent is still in its infancy; yet there are aspects of this process that are understood. Many leading scientists and practitioners claim that mathematical ability is closely linked to problem solving [1]. The major goal of a mathematical education is to nurture creativity. [1, 77]. Traditionally one of the most common ways of involving young people in mathematics has been by the International Mathematics Olympiads. These contests have many goals but one of them is to inspire young people to become interested in mathematics [2]. In Latvia a tradition of School Mathematics Olympiads is a well established one; it has existed for over fifty years. The participants are trained on a high professional level and their achievements are comparable with international standards. Yet without the amateur leagues which are open to all pupils, whose participants are either unable or not interested in undergoing the professional training required for professional level participation, an important source for finding professional level talent would be lost. These otherwise quite ordinary pupils enjoy solving mathematical problems but are not interested in going beyond the curricular requirements. These pupils are often interested in problems of a concrete nature and especially enjoy solving puzzles. In order to be able to identify the abilities of these students, it is necessary to apply various approaches: introduce 144

145 contests into class activities and develop math clubs. The contest idea could be extended to include municipal, regional and even international contests. The aims The Department of Mathematics and Information Science at Liepāja Pedagogical Academy (LPA) has organized mathematics Olympiads for pupils for many years. For the last two years LPA has adapted contest rules that were first devised in Estonia. On the basis of this model, LPA has developed the following goals for its Olympiads. 1. To encourage interest in mathematics among the youth. 2. To discover and identify mathematically gifted students in Liepāja and Liepāja region. 3. To encourage the youngsters to become interested in mathematics and to reinforce learning acquired in school by providing creative opportunities for its application. 4. To encourage network development among young people with similar interests. 5. To introduce pupils to the training programs at Liepāja Pedagogical Academy. Schedule of Olympiad The venue of the Olympiad is at Liepāja Pedagogical Academy, Lielā Street 14 in Liepāja. It is a regular event and takes place in the autumn and spring, usually in November and April. Participants The contestants are from Liepāja and from the Liepāja district schools. Each team has four participants. It is constructed according to the following rules: 1) An 8 th and 9 th grade team in which 2 pupils are selected from each of the grades. 2) A 10 th to 12 th grade team where one pupil is selected from each of the participating grades and the fourth pupil is the free-choice of the school. If substitutes are required, it is possible to replace the pupils from higher grades with the pupils from the lower grades. The rules also allow that a school can send more than one team per category. The limit depends upon the number of parallel classes. For example, if a school has two parallel classes for the 8 th and 9 th grades, then it can send two teams from this category; if it has three classes, it can send three teams. The same procedure applies to the teams of the higher grades. 145

146 The Agenda of the Olympiad At the Olympiad each team receives a set of two examination questions whose difficulty is appropriate to the teams grade level. The first part of the examination consists of an individual competition. There are 16 questions in this part which are attached a variety of potential answers. The participants have 30 minute time-limit to complete this part of the Olympiad. Each participant marks a chart that lists all the possible answers from which he must identify the correct ones. The second part of the Olympiad is a team contest. In this part of the examination there are 20 problems. These are short-answer ones which must be answered within a 60 minute time-frame. Each team submits two pieces of paper: one consisting of the short answers, the other mapping out the strategy of problem-solving. The examination is based upon preparation, creativity, analytical thinking and elementary knowledge of mathematical theory that in generally available in textbooks and most library reference collections. Assessment criteria The Olympiad examinations are graded by LPA faculty and students. Supervising teachers who accompanied the teams and who have mathematical training may also volunteer as graders. The grading takes place immediately after the competition and must be completed within two to two and a half hours, so that the winners may be awarded the same day. Grading Criteria 1) For the first part of the Olympiad, each participant receives two points for each correct answer, loses one point for each incorrect answer, and gets a score of zero for each unanswered question. The overall score for the team is arrived at by multiplying the total number of points that the team participants have earned by ) In the second part of the Olympiad, the answers are graded by assigning a value of 0 to 3 possible points for each answer. The ranking of the teams depends upon the total number of points earned on both parts of the examination. Distributing the Awards The awards are distributed on the same day as the competition holds. The previous winners award six winning teams. Six outstanding participants are selected from each grade category and are awarded by their counterparts previous winners. Additionally, two travelling awards are distributed; one for the 8 th to 9 th grade group; and the other for the 10 th to 12 th grade group. 146

147 For this award, the school whose teams provided the most correct answers will be regarded as the most deserving. Each school s score will be arrived at by adding the number of correct answers of all its teams and averaging the sum by the number of participants from the school. The team with the highest average score will receive the award. In case of a tie, the school which has sent more participants will receive the award. The Olympiad has been organized already for two years in accordance with this scenario Autumn of the school year 2005/2006 Spring of the school year 2005/2006 Autumn of the school year 2006/2007 Spring of the school year 2006/2007 Fig. 1. Total number of participants for two years Pupils are interested in participating in the Olympiad because they understand that it gives them an opportunity to apply what they have learned at school in a new and challenging way. They also seem to enjoy the format of an individual competition balanced by team competition. Of the 221 students interviewed during the autumn Olympiad in the academic year 2006 /2007, 119 expressed that they had participated for the first time, 41 for the second time, but 58 for the third time. Yet the most active Olympiad participants are pupils from the elementary grades. The statistics is given in Fig. 2. The knowledge and skills the pupils have acquired at school is most visibly reflected for this category; for the higher grades formal education is seen not to be as vital. 147

148 Autumn of the school year 2005/ Spring of the school year 2005/2006 Autumn of the school year 2006/2007 Spring of the school year 2006/ Class number 8 Class number 9 Class number 10 Class number 11 Class number 12 Fig. 2. Olympiad participants according to grade levels Figure 3 indicates that the participation of girls in the Olympiads are about the same for the middle grades as for the boys; in the lower grades the participation of girls exceeds that of boys by 30 to 40 competitors Autumn of the school year 2005/2006 Spring of the school year 2005/2006 Autumn of the school year 2006/2007 Spring of the school year 2006/ T he Boy s of 8 T h e G ir ls o f 8 T he Boy s of 9 T h e G ir ls o f 9 T he Boy s of 10 T h e G irls o f 1 0 T he Boy s of 11 T h e G irls o f 1 1 T he Boy s of 12 T h e G irls o f 1 2 Fig. 3. Categories according to gender (girls and boys) In 2006 /2007 some new activities were iniated in order to make the event more meaningful for the pupils and teachers: 148

149 1) Special seminars were organized for the teachers concerning dynamic approaches to mathematics teaching. 2) Pupils were given the opportunity to participate in creativity workshops: Plumbing the Process ; The Young Scientist ; The Young Programmer at School ; Mathematics for Everyday Life ; Determining Mathematical Ability ; Creating Animations. 3) The students from Saldus district were also invited. 4) The team competition included mental arithmetic and the game Sudoku. After the 2006 / 2007 academic year autumn Olympiad 151 out of 221 interviewed pupils expressed themselves positively about the mathematics creativity workshops, while 37 had a negative attitude. In the 2005/2006 academic year, the travelling award for the elementary grades went to Juris Māters Kazdanga elementary school, and the middle school award went to the Liepāja Municipal middle school No. 12. In the 2006/2007 academic year, the travelling award for the elementary grades went to Kalvene elementary school, and the one for middle schools went to VaiĦode middle school. References 1. Гингулис Э.Ж.. Развитие математических способностей учащихся. Чебоксары: Чувашский государственный педагогический университет им. И.Я.Яковлева, Фарков А.В. Математические олимпиады в школе: 5 11 классы. Издание четвертое. М.: Айрис-пресс,

150 SOME PROBLEMS OF THE TEACHER IN-SERVICE TRAINING Aira Kumerdanka, Centre for Curriculum Development and Examinations, Abstract. The quality of teaching and learning in science, mathematics and IT in secondary education in Latvia improves prepare knowledgeable young people having a competitive advantage in today s world but the teacher is not always ready for changes. It is one of the fundamental problems in the introduction of contemporary education at school. Learning to learn competence is the ability to pursue and persist in learning, ability to organize own learning but the teachers are afraid to dare experiment, apply various teaching methods, practical and research works. Many new and creative ideas remain unused, because the teacher is not prepared to work with teacher s support materials, therefore one of the project s major activities is organisation of in-service training for teachers in mathematics, chemistry, physics, biology and natural science. Teachers wait changes in natural sciences and mathematics teaching process at the secondary school, they have ideas that it should be done differently, but it is only general agreement, discussion on the best approach to the teaching process that may result in the maximum effect of this work. Keywords: Standards, curriculum, technologies and classroom equipment, changes in education, various teaching methods, practical and researh works, teacher support materials, in-service training of teachers. In the Republic of Latvia topical issue is secondary school mathematics guideline (standards, curriculum) because we have joined the educational system of the European Union. As a transition to completely new approach in teaching physics, mathematics and natural sciences is planned in 2008, it is essential to prepare teachers for that transition. The framework of The European Union structural funds Latvian national program Development and Improvement of Subject Curricula in Natural Science, Technology and Mathematics in Secondary Education Project Curriculum Development and In-service Training of teachers in Science, Mathematics and Technology project in mathematics envisages: modernizing content of studies; working out the material for teachers` support; ensuring mathematics teachers` possibility to develop professional mastery; 150

151 classrooms in mathematics will be modernized according to European standards in 50 pilot schools of the Republic of Latvia [1]. To improve the quality of teaching and learning in science, mathematics and IT in secondary education in Latvia, in order to: prepare knowledgeable young people having a competitive advantage in today s world; create preconditions for the country s development in science-based and technology-based sections of economy; promote the development in the European Union in accordance with the goals set in the Lisbon Strategy (2000). Yet, an educated and competent young person can develop him-herself if there is interaction between the school s training environment, the curriculum and the teacher. If we are able to change the school s environment and the curriculum by changing school subject standards, curricula, acquiring new technologies and classroom equipment, unfortunately, the teacher is not always ready for changes [2]. It is also one of the fundamental problems in the introduction of contemporary education at school. Taking into consideration the high quality of the specialists in Latvia we are sure to manage the goals. The problems can only arise in teachers` further education. It is so because a large number of teachers have to be involved in a short period of time and teachers have to be persuaded that in mathematics, it is not only collections of facts and formulas that are important, but also mathematical understanding. According to Recommendation of the European Parliament mathematical competence is the ability to use addition, subtraction, multiplication, division and ratios in mental and written computation to solve a range of problems in everyday situations. Scientific competence is the ability and willingness to use the body of knowledge and methodology to explain the natural world. Digital competence is understanding and knowledge of the nature, role and opportunities of IST in everyday contexts. Learning to learn competence is the ability to pursue and persist in learning, ability to organize own learning, including effective management of time both individually and in groups. Yet, teachers are afraid to dare experiment, apply various teaching methods [3], practical and research works, as the limited duration of classes is not always sufficient to master the basic requirements of the standard, moreover, new university students are required even more knowledge and skills than the school standard provides for. In order to solve the problem, the implementation of the project involves university teachers participating in designing a new curriculum and experting, mathematics, science, engineering sciences curricula advertising; municipalities supporting the equipping of pilot schools classrooms in 151

152 accordance with the up-to-date requirements as well as manufacturers, with whom educational institutions cooperate for the building of motivation in young people as to the importance of mathematics and natural sciences in various sectors of economy. In order to carry out the new standard, it is not sufficient that they are approved as the regulations of the Cabinet of Ministers, other materials, too, are required, to enable to implement this document in the training process. Therefore the Project does not only offer an example of the subject curriculum, but also other Teacher support materials including: multiple and active teaching and learning methods; problem solving; research works; assessment; ICT. The materials are organized in accordance with topics for Grades 10, 11 and 12. Each of the topics offers an example of class plans, ideas of tasks for the development of various cognition levels in students, research tasks, regular and final assessment works as well as CDs with visual materials and an interactive CD for student self-training. It could be a valuable support for teachers while designing their training materials as well as saving their spare time. The Project stresses that the main problem is not the content of studies, but the form of studies, how to achieve cooperation between teacher and students. Many new and creative ideas remain unused, if the teacher is not prepared to work with such materials, therefore one of the project s major activities is organisation of further education for teachers in mathematics, chemistry, physics, biology and natural science. In spite of the fact that in mathematics everything has to be proven strictly we stress that it is very important in forming mathematical understanding to accentuate three from eight competences.[ 4 ] I would like to mention that in teachers` further education the following competences are essential: mathematical and basic competences in science and technology, digital competence and learning to learn. The programme of the professional development courses for general secondary education teachers is designed co-ordinately, interconsistently and successively with natural science curriculum professional development course programme, complying with the basic approaches of contemporary didactics of natural science, mathematics and technology, based on the state-of-the-art findings with regard to the development of teacher further education syllabi. The programme has been developed in accordance with 152

153 the modular principle developed within the Project (refer to Fig. 1), in order to contribute to the introduction of the new curricula. General guidelines on contemporary Science at school Research works Multiple methods of teaching and learning ICT usage Fig. 1. Modules of further education curricula. Each of the modules provides for educational curricula in accordance with contents and the goal set in the secondary education subject standard in Mathematics and Subject curriculum in mathematics in grades 10, 11 and 12 to develop the skills of applying mathematical methods in the cognition of the world and multiform work, while expanding the understanding about the role of mathematical models in the description of natural and social processes and developing mathematical judgment skills. The content included in the programme is divided into topics in accordance with modules specifying the assignments set for the mastering of each topic as well as the expected result. The Teacher s Professional Development Programme will be module-based, each one including the following: Module I General Guidelines on Contemporary Science at School offers teachers topics related to the development of scientific 153

154 comprehension, explanation of approach of contemporary curricula and studies, planning and assessment of curricula using the results set in the standard and in the programme about the relation and interconformity of contemporary curricula of mathematics and natural science. Module II Multiple Methods of Teaching and Learning offers teachers to improve knowledge about the organisation of multiform learning process, taking into consideration contemporary education theories about teaching and learning with the application of various methods and forms. Teachers will familiarise with innovations in subject curricula and the fact how to activate student cognition in order to master the curriculum in accordance with the goals set by the standard. Module III Scientific Inquiry in the Laboratory offers teachers to familiarise with students research organisation for the development of scientific cognition skills, in order to successfully fulfil the goal of the subject taught to develop scientific thinking, to solve problem situations, to formulate judgements and to offer substantiations. Module IV ICT Usage is to help teachers introduce contemporary technologies in mathematics learning process, make teachers acquainted with various ways of applying information technologies (IT) in the mathematics learning process, offers the possibility to master and use IT advantages in mathematics learning process, in order to successfully implement the subject s task: to improve mathematical language and information application skills. For the implementation of the project, further educators materials are developed for classes as well as hand-outs for participants. Materials are designed based on the mathematics subject standard and programme project, using Teacher support materials, which will provide an overall support to teachers in the implementation of contemporary curriculum, fulfilment of the standard and the programme. The programme s total volume is 72 hours, including 64 contact hours (classes and practical works), 8 hours for independent work. For the teachers convenience and fuller awareness, the project has designed a website with the lists of mathematics teachers in the regions of Latvia and the dates of the planned seminars. The information in the website is being updated on a weekly basis. At school, much has been talked about the change of goals of training, and I have known before the new guidelines in education, however, those were the examples dealing with mathematics material planning that helped my understanding most. I have acquired a huge load of energy for the implementation of new ideas. It was interesting to myself so it will be 154

155 interesting to children, too. It was useful for us to act as students with those various methods. I realised that research is possible in mathematics, too, and math problems should not be very complex. Until now, I have been apprehensive about doing research during a class; I doubted whether it was needed. I now have a different attitude to that. These and also other citations are the source of energy for the working group and it creates a belief in further education training instructors that teachers have been waiting for changes in natural sciences and mathematics teaching process at the secondary school, they have had some ideas that it should be done differently, but it is only general agreement, discussion on the best approach to the teaching process that may result in the maximum effect of this work. References 1. L.Burton. Learning as research../international Perspectives on Learning and Teaching Mathematics, Goteborg University, 2004, p.. 2. J.Mencis. How deep is mathematics teachers perception about. methodology of teaching mathematics?/ Abstracts of VIII International conference Teaching mathematics: retrospective and perspectives,riga,2007,32 p. 3. Recommendation of the European Parliament and of the Council (2005:3) Cf

156 ACHIEVEMENTS OF 5 TH 8 TH GRADERS AT THE SECOND ROUND OF THE 57 TH LATVIAN MATHEMATICAL OLYMPIAD. FACTS AND LESSONS Gunta Lāce, University of Latvia, guntinja@mikronet.lv Abstract. Results of regional mathematical olympiad 2006/2007 in 22 districts are analyzed. The amount of data is sufficient to make conclusions about this year and to build up hypothesis. To have more general results the analysis should be carried out also in the following years. Keyword: olympiad. The same as every year regional mathematical olympiad in Latvia has gathered a large number of problem solving enthusiasts. This proves that students are interested in mathematics and want to participate at mathematical competitions. What were the results of the pupils this year? Every student had to solve 5 exercises, with every exercise worth 10 points. The maximal obtainable result was 50 points. Average 35% 30% 25% 20% 15% 10% 5% 0% 5.grade 6.grade 7.grade 8.grade The average (in the whole country) result in the fifth grade was 30%, i.e. 15 points. If a pupil gets 30% of the maximal possible result in the exam his evaluation is weak. There are differences between districts, because in some districts only the best pupils participate in the olympiad, consequently 156

157 the average result is higher compared to the results in districts where qualification criterions for the participants are lower and therefore more pupils take part in the event. This topic should be discussed by the pedagogues and other involved organizers. The exercises are chosen from various themes. The best results were achieved in combinatorial geometry. For example, the seventh graders were solving the following exercise: The square consists of 6x6 smaller squares. Is it possible to color some smaller squares so that: *) every row and every column contains exactly four colored squares *) it is possible to reach every colored square from every other by walking along a path, which, in every turn, trespasses a common side of some two colored squares. In this exercise the pupils averaged 49% of possible results; 25% of participants got at least 90% of the possible amount of points. Pupils are keen experimentators and get solution. The only problem is the understanding of the term every. Sometimes the better result is not achieved because of laziness. Pupils attempt few times to get the demanded. If they don t succeed, they use some false invariant and prove the opposite. Relatively good results are achieved in exercises based on logics, which can be solved simply checking all the possible cases (with the case selection). Exceptionally good, if the question is connected with common daily problems. For example, the eighth graders had the best results in the following exercise: Fifty first-graders wrote exam in Latvian. Some of them know all letters except "l" (which they omit when writing), others know all letters except "d" (which they omit when writing). The teacher asked 10 students to write "gads", 18 others - to write "gals", and all remaining students - to write "galds". Each of words "gads" and "gals" were written in 15 papers. How many students fulfilled the task correctly? In this exercise the pupils averaged 37% of possible results; 25% of participants got at least 70% of the possible amount of points. The problems in solving this type of exercises arise when pupils do not make the mathematical model for the given situation. They judge emotionally. For example, the sixth graders had to solve the exercise about liars. Each of three dwarves Alfa, Beta and Gamma are either always lying or always telling the truth. Once upon a time professor Littledigit heard one of them announcing - "Alfa and Beta are both liars", but other - "Beta and Gamma are both liars" 157

158 (professor didn't hear which dwarf pronounced either statement). How many of the dwarfs are liars? Most of the pupils solve this exercise making the following conclusions: Beta definitely is a liar, because two dwarfs told that she is and both should not be telling lies or The first statement was made by Gamma because nobody would tell about himself to be a liar, or similar. The worst results this year were in algebra and number theory exercises. Let us start with algebra. This subject is taught at school starting from 7th grade. No additional knowledge which is not in school programme was necessary. Some technical skills should be used which cause problems, for example tricky splitting into multipliers. The pupils in non-standard situation will use only the skills common to them, the skills which are trained within typical exercises. This is the main problem. For example, the 7th graders had very bad results solving the following exercise: Which positive integers n can be written in form n=x/y, where x=a 5, y=b 5 (with both a and b being positive integers). In this exercise the pupils averaged 10.5 % of possible results. It can be easily solved if one understands the properties of exponentiation and knows how to use them. In February when the olympiad is organized the 7th graders only recently have learned about the properties of exponentiation. They have used them in standard situations very little, therefore they don t see the possibility to use them in a non-typical situation. The exception, of course, are the pupils with a special training. There are problems not only with algebra, but also with the exercises where the solver needs an algebraic way of thinking: Thirteen not necessary distinct integers are written in a row, with their sum being positive. Every sum of three consecutively written numbers is negative. * show at least one example how it can be done; * prove: at least five of these numbers are positive. In this exercise the pupils averaged 24 % of possible results. Students are able to find an example if they have correctly understood the phrase "every sum of three consecutively written numbers". However proving the other part is more difficult, because it makes the students to use element's position in the sequence, which requires rather developed abstract thinking. The results in the exercises of number theory are also weak. Only about fifteen lessons from elementary school curriculum are devoted to number theory, most of them are in fifth or sixth grade. In those lessons students are 158

159 introduced to divisibility tests and prime decomposition of integers. They also learn how to find greatest common divisior and least common multiple of two (or more) numbers. In my opinion, the problem is that this knowledge is very underused in further lessons. For example, when we are finding the common denumerator of two fractions, the teacher does not tell, that really we are finding the least common multiple of denumerators and the students are only using a technical algorithm. Likewise students are often cancelling fractions without thinking of prime decomposition of numerator and denumerator, etc. Also when performing operations with numbers, students don't have to think about "contents" of those numbers - they are just using algorithms. Results in the exercises of number theory 30% 25% 20% 15% 10% 5% 0% 5.grade 6.grade 7.grade 8.grade Such results are not pleasing. Why is it so? What should we do to get better results? We should prepare, practice for the olympiad! Nobody awaits that someone without practicing can outrun professional athlete in marathon. In an olympiad in mathematics the same principle stands. The best results obtain the ones who solve exercises regularly. The preparation for the olympiad is not sufficient. One can observe a negative trend - more than 80% are not preparing for the olympiad at all. It can be concluded from the fact that they are getting less than a half of maximum possible points for solving a problem that has been presented at olympiad a year or two ago. The teachers are informed that problem set includes a "repeated" problem in order to motivate students to examine the problems and solutions of past years' olympiads, which are published on the internet. 159

160 Results in the "repeated" problem 45% 40% 35% 30% 25% 20% 15% 10% 5% 0% 5.grade 6.grade 7.grade 8.grade The work with talented mathematicians should be first of all carried out in the classes of mathematics. I will compare it once again with marathon. The preparation for the competition would have no result if it is a walk in the park with friends who are not interested in sports. Then there is no point in complaining that there was not enough time for serious practicing. Unfortunately in mathematics usually it is exactly this way. And there are objective reasons for that. The work with talented pupils is very well organized in Riga state 1st gymnasium. The pupils from this school show outstanding results in regional olympiad, moreover, better in 8th grade than in 7th. However their experience can not be used in standard class because this gymnasium gathers the best mathematicians from all city. Therefore the higher level skills can be trained during classes, for example, various ways of splitting polynomial into multipliers. It can not be done or is very difficult to do in standard school, where in a typical class there are only one to three talented pupils. Sometimes there is none. In the considered age the pupils are not ready to learn from the book, therefore the teacher should teach him or her. Unfortunately meanwhile no attention is paid to the rest of the class. Of course they could solve easier exercises which accord to their level, but this is rarely possible. What should we do? Regular additional classes for the work with the most talented pupils should be organized. Such classes demand serious work from the teacher, 160

161 therefore this process should be additionally financed. Schools very seldom finance the classes where only few pupils participate. One of the possibilities is to work with the best pupils from the whole region. We have such experience in Latvia. The regional school of mathematics is organized in Valmiera since The best pupils had also very good results in olympiad. However it should be stressed that the benefit from the additional training appears only if the additional knowledge and skills are also strengthened during regular classes. It is not enough to train only once a month in Saturday. The question about the motivation of teachers should be discussed and solved on governmental level. The lack of motivation can be seen in the results for the repeated exercise. The teachers also admit it. The work with the talented pupils is very serious and hard and it is not paid at all or the salary for it is very small. This question should be solved by government or local authority. The teachers don t know what to do in these special classes. There are lot of materials in latvian which can be used in preparation for the olympiad, however the special programme is necessary where it is determined what should pupils from grade 5., 6., 7 etc. learn. Also the question about the elitarism is open. At the moment the exercises in the regional olympiad are elitist, because 75% of participants do not get more than 15 points. For these 75% there is a weak motivation to continue work and study, because the success is the best motivator. Perhaps only the remaining 25% should participate in olympiads. However the mentioned 75% usually are in a way better in mathematics than their classmates. Perhaps additional regional competitions should be organized with a little bit easier exercises. We have such experience in Latvia. Anually an olympiad is organized in Zemgale area. Enthusiasts and finances are necessary, or better both of them. At the moment we have more questions than certain answers. I hope that continuation of the work will let me find the way to make the olympiads more available also for the rest of the best, meanwhile holding the requirements generally the same. Acknowledgement. The publication was prepared with the support of ESF References 161

162 162 GRAPHIC PECULIARITIES OF OPEN SOURCE COMPUTER ALGEBRA SYSTEMS Joana Lipeikiene, Vilnius pedagogical university Institute of Mathematics and Informatics, Abstract. Open Source (OS) software can be downloaded, localized, modified and redistributed for free. Educators also can use a variety of OS information technology tools. Legality, localization and adoption problems can be solved. But the quality of OS mathematical software remains the problem, although some features of OS computer algebra systems (CAS) approach the level of commercial CAS. Earlier we investigated the general features of OS CAS. The present paper deals with graphic facilities of general purpose OS CAS and possibilities to use them for visualization in teaching mathematics. Six freely distributed Windows versions of OS CAS Eigenmath, Descartes, Deadline, Maxima, Octave and Scilab were analyzed and used in future teachers training at the Vilnius Pedagogical University. The visualization features of the systems were investigated comparing them with graphic facilities of commercial CAS such as Derive, Maple or Matlab. The paper also refers to students opinion and observations of the students work, discusses encountered difficulties. Keywords: Computer algebra systems (CAS), graphic facilities of CAS, Open Source (OS) software, user interface. 1. Introduction Commercial computer algebra systems, such as Derive, Maple, Mathcad, Mathematica, Matlab [1], are globally used at the universities for teaching mathematics. Commercial computer algebra systems have a great variety of graphic facilities, but the main of them are possibilities to get quickly and easily one or more graphs in 2D and 3D, implicit plots, parametric and contour plots, graphs in polar coordinates, rotation of graphs, animation and much more. These systems usually have special graphic functions for visualization of integrals, Taylor approximation, function investigation, and other visualization tools useful for teaching mathematics. One of the great changes in education was the Open Source Software idea. Open Source (OS) software shortly can be characterized as computer software whose source code is available under a copyright license that permits users to study, change, and improve the software, and to redistribute it in modified or unmodified form [2]. Because of possibilities to use legal software, localize and adopt the programs without negotiations with developers, OS software is of increasing interest in education. OS

163 mathematical software also could be useful for higher mathematical education. But the quality of OS CAS isn t equal to that of commercial CAS yet [3-6]. The lack of some necessary features and drawbacks make difficulties to use OS mathematical systems in teaching mathematics. There are many special purpose mathematical programs. The present paper considers only general purpose CAS (Table 1). In [5,6] such features of OS CAS as mathematical capabilities to solve at least Calculus and Linear algebra problems, convenience of user interface, access to Help, stability of a system and programming facilities, were investigated during practical use in the classroom. The exploration showed that some features of OS CAS are approaching the level of commercial CAS. The main goal of the paper is the exploration of graphic facilities of OS CAS. Six OS mathematical systems were chosen Eigenmath, Descartes, Deadline, Maxima, Octave and Scilab. All these systems were analyzed and used in teacher training at the Vilnius Pedagogical University, and the paper refers to students opinion and observation of students work. 2. Open Source Computer algebra systems We investigated six OS CA systems. Latest stable versions (for Windows) of the systems were downloaded from the Internet: Descartes 0.7 (descartes.sourceforge.net), Deadline 2.36 ( Eigenmath 132 (eigenmath. sourceforge.net), Maxima 5.13 (maxima.sourceforge.net), Octave ( Scilab 3.4 ( The first system Descartes is not a computer algebra system, more likely a simple plotting tool, but it was used together with computer algebra systems for comparison. It is a small image, data and function plotter, easyto-learn interactive 2d plot program. DeadLine is a program for equations solving, graph plotting and obtaining an in-depth analysis of a function. Most equations are supported, including algebraic equations, trigonometric equations, exponential equations, parametric equations. DeadLine solves equations graphically and numerically. It displays the graph of the function and a list of the real roots of the equation. One can evaluate the function and the first two derivatives, find extreme of the function and integrate numerically. Eigenmath is a small open mathematical system, the interface of which reminds the interface of one of commercial systems Derive. As in Derive, expressions in Eigenmath are typed in a special line, and the correction of expressions is convenient after you return the expression to the line. Though the system is quite small, it has main operations of symbolic calculator, elementary functions, basic concepts of algebra, possibility to solve 163

164 problems with matrixes and vectors. It has also some 2D-graphic means, but they are limited in comparison with Derive. Maxima interface is similar to the commercial CAS, especially to Maple. It can be used as numerical and symbolic calculator, has many facilities in all fields of mathematics. Correction of expressions is similar to Maple, has good programming means, solve various systems of equations and inequalities, some differential equations. Tutorial of Maxima writes that one can get help with command? function_name, but this does not work properly. The menu has the item Help. There arwe good visualization facilities 2D and 3D graphics. In Maxima two open graphic programs Gnuplot and Openmath are used, and a user has some possibilities to choose the type of a graph. Charts, created with Maxima, are similar to those created with Maple. 3D graphics is restricted, for example, there is no possibility to draw more than one surface in one graphic window. Scilab distinguishes from Open Source mathematical systems as Matlab differs from other commercial CAS: Scilab and Matlab are not symbolic calculators but numeric systems devoted to data processing, mathematical modeling and automation of calculations in various fields of research and applications. Friendly user interface of Scilab reminds Matlab. A convenient help shows that Scilab is a real Scientific Laboratory. Octave is very similar to Matlab and Scilab, but Scilab has more modern user interface. 3. Investigation of OS CAS with students All these systems were used during two semesters for teaching the subject Computer Mathematics Systems. During one semester students tried to use the systems and to evaluate their possibilities to solve various problems of calculus and linear algebra: they calculated limits, derivatives, integrals, investigated functions, applied integrals to calculate areas, volumes, length of curves, areas of surfaces, used matrixes, solved equations and systems of equations. The second semester was devoted more to explore graphic possibilities of OS systems. We have searched for the graphic features that commercial CAS have. For using graphic features of CA systems in teaching it is very important to have stability of a system (mistakes should not break a system); possibilities to get a smart plot (without choosing of ranges); easy editing of graphs (change of size, color and other features); special graphic features and possibilities. Observation of the laboratory works with students and students opinion led to evaluation of graphic features of CAS. After experiments students 164

165 filled in a questionnaire about the systems features and graded the systems according to the level of systems graphic features. 4. Features of Open Source Computer algebra systems The Table 1 presents main results of investigation of the graphics. If the system has the feature enumerated in the table, it is marked with +. In opposite case it is marked with -. Experiments showed that using several OS systems one can get graphs as with commercial CAS, because these systems have many good graphic features. Fig. 1 presents graphs made with various OS CAS. Table 1. Main features OS CAS With OS CAS one can draw 2D and 3D graphs of one or several functions, parametric and contour plots, divide graphic window for illustrating differences, rotate graphs in 3D, represent matrixes by surfaces, and use many other visualization tools. But as one can see in Table 1, these systems lack some important features. For example, no one of the systems have special functions for animation. Descartes and Deadline have only 2D graphics. As regards Eigenmath graphics, it is too restricted to use at the universities. With Eigenmath one can draw in 2D a graph of one function (it can be parametric), but graphs are fixed in an interval [-10, 10], and a user can not edit a view. Maxima draws easy one or several graphs in 2D and 165

166 3D, has editing and rotating (in 3D) facilities. Students did not like often breaks of the system after they made some mistakes in typing commands. The main drawback of Maxima s graphics is the possibility to keep open only one graphic window. Scilab is the best for data processing and data visualization. Octave is very similar to Scilab, so students chose more Scilab for friendlier interface. Results of students evaluations are presented in Fig. 2. The students have evaluated that, on the average, Maxima has 42 % of Maple graphic facilities, while Eigenmath 3 %, Descartes 5 %, and Deadline 8 %. Comparing the graphic facilities of Scilab with those of Matlab, they regarded that, on average, Scilab had 85 % of Matlab facilities. Fig.1. Various graphs made with OS CAS Fig. 2. Average students evaluation of OS CAS graphic facilities 166

167 5. Conclusions Open Source computer algebra systems (OS CAS) can be useful in teaching mathematics for visualization. With OS CAS one can draw 2D and 3D graphs of one or several functions, parametric and contour plots, divide graphic window for illustrating differences, rotate graphs in 3D, represent matrixes by surfaces, and use many other visualization tools. But graphic features of different OS CA systems differ from each other. Investigation of graphical features of the system showed that no one of investigated OS CAS equals commercial CAS, when comparing their graphic features; OS CA systems have some nice graphic facilities, approaching the level of commercial CAS; Descartes, Deadline and Eigenmath have only 2D graphics and can be used only for visualization of the functions of one variable; Maxima is not only the best OS symbolic calculator, but also has the best graphic facilities among all OS mathematical systems: 2D and 3D graphics, editing possibilities, rotation in 3D; main drawback of graphics possibility to have only one open graphic window; Scilab is the best for applications, data processing and visualization as it has diverse graphic facilities useful for applications. References 1. J. Lipeikiene. Mathematics with computer (in Lithuanian). Vilnius, 2002, p Open Source Software Amundson J. Random thoughts on Open CAS Fearnley C. J. Open Source Software for Higher Mathematics J. Lipeikiene, A. Lipeika. Open Source Computer Algebra Systems in Teaching Mathematics. Proceedings of the 7 th international conference Teaching mathematics: retrospective and perspectives, ISBN-13: , Tartu, 2006, J. Lipeikiene. Open Source CAS in Mathematical Education of Teachers. Proceedings of the 8 th International Conference on Technology in Mathematics Teaching, ISBN , Hradec Kralove, 2007, in CD. 167

168 ON A SIMPLE PROOF OF SOME RESULTS IN NUMBER THEORY Juozas Juvencijus Mačys, Institute of Mathematics and Informatics, Vilnius, jmacys@ktl.mii.lt Abstract. An elementary proof of necessary and sufficient conditions to represent a positive integer as a sum of two squares (Girard s theorem) is presented. New proofs of Fermat s and Wilson s theorems are given. Keywords: elementary proof, Girard s theorem, Fermat s theorem, Wilson s theorem. Propositions of number theory are frequently formulated very simply while their proof is sometimes very complicated or nor yet found so far. Namely due to simplicity of formulations we seek elementary proofs. For instance, in [1], the author presented an elementary proof that the equality x + y = z is impossible if x, y and z are positive integers. In this paper, we will illustrate an elementary way of proving a wellknown proposition on the representability of an integer as a sum of two squares (so called Girard s theorem see [2] and [3], p ). Theorem. A positive integer n can be represented as a sum of squares of two integers if and only if the factorization of the number n into primes contains no prime of the form 4k 1 that has an odd exponent. Note that in the expression of n by a sum of two squares one may take a 2 summand 0. Conditions that would allow expressing the number n by a sum of squares of two positive integers are much more complex. The proof of the theorem is based on the propositions presented below. Due to the lack of place the proof of some propositions is just hinted at. Proposition 1. The product of two representable numbers is representable. Proof. The proposition follows immediately from the formula ( a + b )( c + d ) = ( ac bd) + ( ad + bc), (1) that can be verified by simple algebra. By the way, we can also rely on the formula ( a + b )( c + d ) = ( ac + bd) + ( ad bc), (2) obtained by substituting b in equality (1) for b. Formulas (1) and (2) show that composite representable numbers can have several expressions. 168

169 Proposition 2. If a representable number is divisible by a representable prime, then the quotient is also representable. If a representable number is divisible by a number all the prime divisors of which are representable, then the quotient is representable as well. Note that, if a divisor has nonrepresentable prime divisors, then the quotient can be nonrepresentable either. E.g., the number 3 = 0 + (3 ) is representable. However, nonrepresentable. Proof. Assume that a 6 3 is divisible by , while the quotient + b is divisible by the prime 3 3 = 27 is 2 2 P = p + q. Then pb aq pb + aq = p b a q = p b + p a p a a q = p a + b ( )( ) ( ) a ( p + q ) is also divisible by P. Since P is a prime, pb aq or pb + aq is divisible by it. Say that at first pb + aq is divisible by P. Then the identity ( a + b )( p + q ) = ( ap bq) + ( aq + bp) (3) implies that ap bq is divisible by sides of the identity can be divided by 2 2 P = p + q either. Consequently, both P = ( p + q ), and the quotient ( a + b ) /( p + q ) will be written as a sum of two squares. Another case can be considered analogously it suffices to replace a by a in identity (3). Let us prove the second part of the proposition. According to Proposition 1, a number whose all the prime divisors are representable is 2 2 representable itself. Suppose that a + b is divisible by n = p... 1p2 pk where p i are representable and 2 2 a + b = nq. We have to prove that the 2 2 quotient q is representable. Let us divide the equality a + b = p... 1 p2 pkq by representable p 1. We have already proved that we obtain the equality 2 2 c + b = p... 2 pkq. Thus, consecutively dividing by p,..., 2 p k, we obtain the 2 2 equality e + f = q, which means that q is representable. Proposition 3. If a and b are relatively prime, then each divisor of the 2 2 number a + b is representable. Proof (compare [3], p. 65). Assume on the contrary that we have found a 2 2 nonrepresentable divisor of the number a + b. Let us choose the smallest divisor out of nonrepresentable divisors of all the sums of squares with relative prime components and denote it by x. Then take any sum of such 2 2 squares that has the divisor x. We have A + B = yx, where A and B are 169

170 relatively prime. By division A = mx ± c, B = nx ± d, where c and d are not larger than x / Then A + B = m x ± 2mxc +c + n x ± 2nxd +d = Nx + ( c + d ) is divisible by x, therefore c + d is also divisible by x, i.e., c + d = zx. If c and d have a common divisor, then x cannot be divisible by it, otherwise A and B would also have such a divisor. Having divided the last equality by 2 2 the square of the greatest common divisor, we get e + f = wx. Here e and f are relatively prime, and w x / 2, because wx = e + f 2 c 2 + d ( x / 2) + ( x / 2) = x / 2. Since x is the smallest nonrepresentable divisor, w is representable. All the prime divisors of it pi w < x are representable as well. Therefore, on the basis of Proposition 2, we can divide the equality 2 2 e + f = wx by w, and the quotient x will be expressed by a sum of squares. It is a contradiction. Proposition 4. If p is a prime and does not divide a, then p divides p 1 a 1(Fermat s theorem). If p is a prime, then p divides ( p 1)! + 1 (Simpson s theorem). Proof. For proving these well-known theorems we propose a new method based on simple facts of symmetrical polynomials theory. It is of interest that both theorems are proved simultaneously. For the odd prime p, consider a polynomial p 1 p 2 p 3 ( x + 1)( x + 2)...( x + p 1) = x + σ x + σ x σ x + σ. 1 2 p 2 p 1 Let us prove that the coefficients on the right-hand side of the polynomial (symmetric polynomials of the variables 1, 2, 3,..., p 1 ) σ 1 = ( p 2) + ( p 1), σ 2 = ( p 2)( p 1), σ 3 = ( p 3)( p 2)( p 1), σ 4 = ( p 4)( p 3)( p 2)( p 1),... σ p 2 = ( p 3)( p 2) ( p 3)( p 1) ( p 2)( p 1) are divisible by p. Then in the initial identity, x = 0 gives σ 1 = ( p 1)!. Now x = 1, x = 2,..., x = ( p 1) imply that 1 + ( p 1)! is divisible by p 1 p, 2 p 1 + ( p 1)! is divisible by p,..., ( p 1) + ( p 1)! is divisible by p. The first statement means Simpson s theorem, the rest ones Fermat s theorem (for odd p; for p = 2 the theorems are obvious). p

171 k k k So denote, in the standard way, S = ( p 1). Then S1 = σ1 = ( p 1) = p( p 1) / 2 is divisible by p. We prove by induction that p divides S k, k = 1, 2,..., p 2. Assume that p divides S1, S2,..., Sk 1. Write k + 1 k + 1 k + 1 k 1 k 1 k + 1 k + 1 k + 1 k + 1 p 1 = p ( p 1) + ( p 1) ( p 2) = so p 1 p 1 k + 1 k k 2 k 1 k k + 1 k + 1 k + 1 [( m + 1) m ] = ( C m + C m C m + 1) = m= 1 m= 1 = C S + C S C S + p 1, 1 2 k k + 1 k k + 1 k 1 k p p = C S + C S C S. We see that p divides k k k + 1 k k + 1 k k ( k + 1) S k. Since k + 1 < p, k + 1 is relatively prime with p. So p divides Sk too. Let us write well-known (and easily provable) formulas that relate and σ k ( k = 1, 2,..., p 2) : S1 = 1 σ1, S2 = σ1 S1 2σ 2, S3 = σ1s2 σ 2S1 + 3σ 3, S4 = σ1 S3 σ 2S σ 3S1 4σ 4,..., Sk = σ1 Sk 1 σ 2 Sk 2 + σ 3S k 3... ( 1) k + kσ k. Obviously, if p divides S 1, S 2,..., S k, then p divides σ, σ,..., 1 2 σ k too, i.e., p divides σ1, σ 2,..., σ p 2. The proof of Fermat s theorem and Simpson s theorem is completed. Proposition 5. Each prime number of the form 4n + 1 is representable as a sum of two squares. Proof. Based on Proposition 4, p = 4n + 1 divides the numbers 4 2 n 1, n 4n,... (4 n) 1. Let us decompose them: (2 n 1)(2 n + 1), (3 n 1)(3 n + 1),... ((4 n) n 1)((4 n) n + 1). We prove that p does not divide at least one of the first factors. Assume 2 on the contrary that p divides all the numbers 2 n 2 1, 3 n 1,..., 2n (4 n) 1. Then p divides their differences, consequently p divides the 2nd differences (differences of differences),..., and p divides the (2n)th differences. It is well known that this (2n)th difference is equal to (2n)! (a simple proof of it takes a pair of words, see [3], pp. 69, 449.) But each factor of the factorial is smaller than p, so p does not divide (2 n )!. That is a contradiction. 2n Since p does not divide at least one of the factors k 1, we have some 2n k + 1 divisible by p. However, this expression is a sum of squares, so on k S k 171

172 the ground of Proposition 3 all its factors including p are representable as a sum of squares. The proposition is proved. Other proofs of Proposition 5 see in [4]. Proof of the theorem. Necessity. We prove that, if n can be represented as a sum of two squares, then it does not contain prime factors of the form 4k + 3 (i.e., of the form 4k 1 ) that have an odd exponent. In other words, 2 2 given that n = a + b, we prove that even if n contains prime divisors of the form 4k + 3, each of them has an even exponent. If a and b have common divisors, then, dividing both sides of the equality by the square of 2 2 the greatest common divisor, we derive n1 = a1 + b1, where a 1 and b 1 are relatively prime. Let us decompose n 1 into prime factors. The prime factor of the form 4k + 3 (as well as any other number) is nonrepresentable as a sum of two squares: an odd square divided by 4 yields the remainder 1 while an even one yields the remainder 0, thus, the remainder of division of a sum of squares by 4 may be 0, 1 and 2, but not 3. Proposition 3 implies that n 1 contains no prime factors of the form 4k + 3. Consequently, 2 n = d n 1 can contain a prime factor of the form 4k + 3 only with an even exponent. The necessity is proved. 2 2 Sufficiency. Assume the number n = a + b not containing primes 2 2 4k 1 with an odd exponent. If n = 1, we achieve n = If n is larger than 1, decompose it into prime factors. Denote by m the largest number 2 whose square divides n. Then n = m q, where q is either 1 or a product of 2 2 odd primes that has no primes of the form 4k 1. However, 2 = and the primes of the form 4k 3 by Proposition 4 are also representable as a sum of two squares. Proposition 1 is true not only for two, but also for any number of factors, so we can express q by the sum of two squares q = c + d. Therefore n = m q = ( mc) + ( md). The sufficiency is proved and the proof of the theorem thereto is completed. References 1. Мачис Ю. Ю. Новое доказательсвто леммы Эйлера. Proceedings of 7th International Conference Teaching Mathematics: Retrospective and Perspectives, Tartu, 2006, pp Dickson L. E. History of the theory of numbers. Vol. 2. New York: Chelsea, 1966, p Эдвардс Г. Последняя теорема Ферма. Москва: Наука, Тихомиров В. Теорема Ферма Эйлера о двух квадратах. Квант, 10 (1991),

173 DEVELOPMENT OF SECONDARY SCHOOL TEACHER OF MATHEMATICS STUDY PROGRAMME Jānis Mencis, Visvaldis Neimanis, University of Latvia, Abstract. Already for more than 50 years the Faculty of Physics and Mathematics of the University of Latvia has been preparing secondary school teachers of mathematics. The Chair of General Mathematics is responsible for working out, organizing and coordinating the study programmes. In process of preparing teachers of mathematics, certain traditions and experience have been formed up, which is used for regular improvement and complement of study programmes according to innovations in science of mathematics and pedagogy, as well as taking into account modern didactics technologies. There is a necessity to evaluate previous achievements and carefully consider possible changes in the process of education due to the fact that Latvia has joined the European Union. The essence of the question is simple: how long should the educational programme be four, five or three years long? Keywords: Secondary school mathematics teachers; bachelor of mathematics; programme of Professional Studies for Secondary School Mathematics Teachers. Introduction University of Latvia Faculty of Physics and Mathematics educates Secondary school teachers of mathematics for more than 50 years. To improve the quality of mathematics pedagogy education in 1972 the speciality No 2013 standard curricula was modified and created specialization mathematics didactics. Every year in this specialization student group was enrolled who wanted to become teachers of mathematics. In 1993 based on the existing curricula bachelor of mathematics and teacher of mathematics professional study programmes were created. Both programmes with total scope 200 credit points were taught in 4 years acquiring bachelor of mathematics diploma and higher professional education certificate respectively. With transition to 5 years professional studies period in University of Latvia in 1998 Secondary school teacher of mathematics professional study programme was developed where both previous programmes were merged with several changes (increased scope of pedagogy and psychology study 173

174 courses, different didactics special courses planned during the whole study period) (1). Current situation Curricula of the previous programme of Professional Studies for Secondary School Mathematics Teachers: Courses s/u exa m Calculus A 1,2,3, Differential equations A 4 4 Methods of mathematical physics B 5 4 Complex function theory Elements of functional analysis Methods of optimization Algebra A 1,2 4 8 Mathematical logic A 1 2 Discrete I semester II III IV V VI VI VI II X I Credits A 6 4 A 8 2 A 7 4 mathematics B 6 2 Number theory B 3 3 Elements of combinatorics B 8 2 General methods of elementary mathematics B 9 2 Theory of A 4 4 probability Mathematical A 5 4 statistics Analytic geometry A 1 4 Differential B 6 4 geometry Fundamental of B 7 4 geometry Programming and A 1,2, computers Numerical methods A 3, Physics B 5, Foreign language B X 174

175 Humanities C 6,7,10, Psychology A 3 2 Development psychology A 4 2 Pedagogy A 5 3 Methods of teaching A 6, mathematics Methods of teaching informatics B 9 2 1,2,3,3, Special courses B 4,5,7,8, 9.9,10, Pedagogical praxis 12 weeks A 8,9 6 6 Course thesis A 4 Diploma thesis A 10 Recent changes in legislation foresee that a person can acquire pedagogical education and professional further education necessary for teacher profession in a second level professional higher education study programme that contains bachelor programme compulsory study courses and after acquisition of which bachelor degree in pedagogy and teacher qualification or bachelor degree in the respective science field and teacher qualification are awarded (length of studies four to five years) (2). We also have to take into account that: a) the amount of credit points for pedagogical practice has increased from 12 CP to 26 CP; b) in the compulsory part of the programme more general educational courses have to be included; c) the amount of credit points in one semester cannot be more than 20 CP, therefore, we should focus on students` independent work during the second half of the day. 175

176 Problem solution The project for the new programme of Professional Studies for Secondary School Mathematics Teachers: Calculus Analytic geometry 3 3 Algebra Differential equations 4 4 Mathematical statistics and Theory of probability Mathematical logic 2 2 Elements of combinatorics 2 2 Methods of mathematical physics 3 3 Physics Number theory 3 3 General methods of elementary mathematics 2 2 Programming and computers Practicum of Elementary Mathematics 2 2 Fundamental of geometry 2 2 Numerical methods Oral and Written Communication (English language) Management theory 2 2 Economics 4 4 General Psychology 2 2 Science of Law 2 2 Communication Psychology 2 2 Intercultural Communication and 2 2 Intercultural Learning Rhetoric 2 2 To achieve amount of 160 CP, it is necessary to add the pedagogical practice, term papers, the bachelor paper, study courses in pedagogy and methodology of teaching mathematics. Efficiency of such plan can be motivated from different aspects but we will stress the full conformity with requirements for the teacher profession, summarized in the table below. 176

177 Skills: Setting targets and planning attainment Selecting learning and education instruments (contents, methods, form) Selecting and creating teaching methodological materials Using different learning and education instruments, including IT Identifying and solving problem situation Motivating and leading student work and teaching to learn Developing creative projects, make research, lead student research work Evaluating student learning attainments Fulfilling own pedagogical skills Observing student personality development Critical thinking in study process Education theory Mathematics teaching methodology Mathematics teaching methodology Analysis didactics of elementary mathematics Mathematics teaching methodology Special methods of elementary mathematics Computer in study process Critical thinking in study process Pedagogical praxis Thesis Term paper Mathematics teaching methodology Pedagogical praxis Pedagogical praxis Education theory Knowledge: Learning and education theory and methodology Class management Study organization (teaching methods, classes, study programme) Attainment evaluation and self evaluation Organization of learning environment Methods of pedagogical research Mathematics teaching methodology Analysis didactics of elementary mathematics Education theory Mathematics teaching methodology Critical thinking in study process Pedagogical praxis General psychology 177

178 Psychology (general, development, personality, social) Physiology of age Study course methodology and didactics Information technologies State language and foreign language Conclusion General psychology Development psychology Development psychology Mathematics teaching methodology Analysis didactics of elementary mathematics Special methods of elementary mathematics Methods of geometrical illustration Didactic basics of instrumental mathematics Practicum in elementary mathematics Programming and computers Informatics teaching methodology Computer in study process Foreign language Term paper Thesis Pedagogical praxis Let us note that changes in study programmes require accurate work, and University of Latvia give it all the necessary support work group is established to improve teacher education in UL. Parallel to the formal courses the programme includes also respective competence creating courses (3). The programme is suitable also for part time students and also the short version (2 years) is offered for those that already hold higher education but are lacking pedagogical qualification. References 1. Jānis Mencis, Visvaldis Neimanis. Genesis of Professional study Programme for Secondary School Mathematics Teachers. Theses in 6 th international scientific conference Teaching Mathematics: Retrospective and Perspectives, Vilnius, 2005, (May ), pp Cabinet of Ministers regulations No Recommendation of the European Parliament and of the Council (2005:3) Cf

179 DEAR, WHAT DO YOU MEAN? Diāna Mežecka, University of Latvia, Abstract. The teaching aid, containing problems and solutions of the math competition between students of Riga, Tallinn and Vilnius 1 st gymnasiums, is described. Keywords: argumentation in mathematics, mathematical contests, preparation process. Introduction The present day situation in teaching of mathematics at school has a clearly negative feature: proofs are paid a little attention to, and this becomes more and more widespread. So the students are not stimulated to make logical corollaries, nor they are urged to seek their own way to the solution. Mathematics is perceived by the majority as a discipline where a lot of formulas must be learned, some customary tricks and schemas must be acquired thus gaining the ability to solve any problem. Instead, students should learn the ways of thinking, practicise in creative approaches and seek for a solution also in the situations which has never occurred before. Unfortunately, the number of lessons is very limited and reduced, and this often urges the teachers to avoid creative problems, as they require a lot of time. Instead, the problems of the type available at the final examinations are considered with great care. As the result, the proper understanding of mathematics is disappearing among students. At this moment the problems of math olympiads are the best examples of proper mathematics generally available to high school students. It is a common situation that a student gets into contact with olympiad problems only 1 2 times during a year: before the annual school/ regional olympiad a problem set of previous year together with solutions is given to him by the teacher. The solutions are those provided by the central olympiad committee; they are written in a way to give the teacher conspective hints about possible approaches, right answers and typical expected flaws of the students. The teacher is expected to analyze the problems and their solutions with his students; it is very hard for even a good student to get through on his own, if he hasn t a good training in advance. Nevertheless, teachers expect their students to do this. As a result, a great part of students are sure that contest mathematics is too hard for them and only exceptional persons are able to manage it. 179

180 To escape from this situation, a regular training is needed. It is most welcome that the students could perform it on their own, because a large part of teachers is overloaded and underpaid and can hardly pay a lot of time for extra activities. Such a possibility can be achieved in a simple and effective way by creating teaching aids, containing characteristic problems from various contests, the solutions of which explore basic ideas and are written in a clear, detailed and precise manner. The olympiad of Riga, Vilnius and Tallinn 1 st gymnasiums As a part of the national awakening which resulted into the reestablishment of the independence of Baltic States, much closer contacts than earlier were established between Latvia, Lithuania and Estonia at the middle of 1980-ies. They were many-faced, the accent being put on culture, education, sport; the political collaboration came later. Since 1986 a competition between the students of Riga, Vilnius and Tallinn first gymnasiums is organized in turn in all three capitals. The students compete in mathematics, physics, chemistry, informatics and biology; each school is represented by two students in Grade 10, Grade 11 and Grade 12. The problems are composed by the annual host country. Riga 1st gymnasium is a mathematical school with 50 years long history and remarkable successes also on international scale. The 1st gymnasiums of Vilnius and Tallinn have other priorities, and they are strong in other areas. So in the years, when the competition is held in Riga, the problems are not chosen accordingly to, let s say, standards of IMO, which often require developed technic and additional knowledge far beyond the school curricula. Instead, the problems should be formulated in an interesting and non-traditional manner, admit short solutions with unexpected ideas, use only concepts from the regular curricula. So the book based on the materials of the abovementioned competition could be very appropriate for selfeducation of gifted and diligent high school students. Such a teaching aid was created at A. Liepa s Correspondence Mathematics School of University of Latvia. The problems and solutions The problems are usually chosen so that all five major areas of mathematics at school algebra, combinatorics, number theory, geometry and algorithmics are represented. Problems of type prove it!, is it true?, calculate!, decide whether etc. are included. We try also to have various levels of difficulty from easy exercises to quite hard questions. Some problems are particular cases of more general theorems or even unsolved questions; if so, it is mentioned in the solutions. 180

181 The solutions are written in a way to provide answer to any question that appears while studying it. More than one solution is often given, and readers are urged to seek for their own solutions too. One of the aims of the teaching aid, as well as of the whole education process, is to convince the students that in mathematics, like in life, there are various, sometimes very different ways how the only truth can be achieved. Some characteristic examples Problem in algorithmics (2001, Grade 10) A labyrinth consists of 11 rooms (see Fig. 1). A robber is hiding there. Fig.1 Each day police checks one of the rooms and goes home to have a rest during the night. Each night the robber goes to a room neighbouring to his former place. How can police catch the robber? Solution Let s denote the rooms in a row by numbers from 1 to 11. Police will catch the robber if they check the rooms in a sequence 2; 3; 4; ; 10; 2; 3; 4; ; 10. To prove this let s consider two possibilities. A. In fact the robber initially is in the room with an even number (but police doesn t know this). So, if police doesn t check this room at first day, then there is an odd number of rooms between police and robber. On the next day either police finds the robber, or there is again an odd number of rooms between them; besides, this number isn t greater than on the previous day. Moving into the room with a smaller number the robber gets closer to the police. If there is only one room between police and robber, then the only admissible move for the robber is to go to the room with greater number. Where police will come to the 9 th room, then (if the robber still isn t catched) there again is an odd number of rooms between police and robber; so the robber is in the 11 th room. Next night the robber will move into 10 th room, where he will be catched next day. B. In fact the robber initially is in the room with odd number (but police doesn t know this). In this case during the police s first passage from 2 nd room to 10 th room the robber is in a room with an odd number while police is checking the room with an even number, and vice versa. So, when police is checking the 2 nd room again, the robber is in a room with even number (the rooms with numbers 10 and 2 are checked in two consecutive days). We have already proved that police will now catch the robber checking the rooms in a row from the second to the tenth. 181

182 Comment. A nice problem for research is to find the minimal number of days which allow to catch the robber for sure. The next problems are given without solutions. Problem in algebra (2001, Grade 12) The graph of a function y = cos 2 x is constructed. By A the point (0;1) is denoted. For each point P on the graph let s denote by P 1 such a point that P A A is the midpoint of AP 1 (by definition, 1 ). The graph of which function is constituted by all such points P 1? (Answer for self-control: y = cosx.) Problem in combinatorics (2001, Grade 12) There are numbers 1; 2; 4; 8; ; 2 n-1 ; 2 n written on the blackboard. By one move any two numbers x and y on the blackboard can by replaced by one number x-y. This is repeated until only one number remains. What are the possible values of it? (Answer for self-control: any natural odd number not exceeding 2 n.) Problem in geometry (2001, Grade 11) Let A, B, C, D be consecutive vertices of a regular decagon inscribed in a circle with radius R. Prove that AD BC = R. (Hint: properties of parallel hords and parallelograms can be explored.) Problem in number theory (2001, Grade 12) Is it possible to choose 2001 consecutive natural numbers so that exactly 17 of them are primes? (Hint: consider the intervals [1; 2001] and [2002!+2; 2002!+2002] and rearrange one into another by successive shifts.) Conclusions There are several problems than are making a heavy influence on the teaching of mathematics: a lack of time and regularity in the plympiad problem solving, a students negative attitude to olympiad mathematics and insufficient collaboration between students and teachers in the preparation to Olympiads. Nevertheless, olympiad problems are developing not only mathematical way of thinking, but also general approaches to problem solving in real life. The teaching aid considered here is a step towards the improvement of the situation, but also a support from teachers is needed. 182

183 STUDENTS MATHEMATICS COMPETITIONS IN LITHUANIA Leonas Narkevičius, Kaunas University of Technology Gymnasium, Abstract. Various math competitions and preparation activities in Lithuania are considered. Keywords: mathematical competitions problems, preparation process. Types of competitions in mathematics Mathematics is the most popular discipline, in which competitions take place in order to find out who is better at problem solving. Various mathematics contests take place in Lithuania. The contests including lower grades ( the 1st 8th grade students) take place in separate regions. Few students are invited from all the schools of the district. The contests for students belonging to this age group are organized by Vilnius and Šiauliai Universities as well. Furthermore, open individual contests are organized for the senior students (9 th -12 th grade students) by different universities. In some regions team contests are organized by Vilnius University. In addition, this university organises the Republican team contest to win the Jonas Kubilius cup. That is probably the second most important contest in Lithuania. It takes place in autumn and according to the results in this contest a team from Lithuania is formed to compete in the team contest of the Baltic States, called Baltic Way. The main contest is the individual Republican mathematics contest of Lithuania for the 9th-12th grade students. It has three stages. The first takes place at school, the second in the city or region, and the third gathers participants from the whole country. According to the results of this contest, pupils are invited to the selective camp, where the strongest students are chosen to compete in the World contest. How are the high results achieved? The main competition between the most intelligent students and schools takes place in the Republican mathematics contest. By analyzing the results of the latest six Republican mathematics contests, it has been noticed that the most successful participants are concentrated at Kaunas University of Technology Gymnasium and Vilnius Lyceum. These two shools win the 183

184 majority of prizes. It is also important to mention another two localities, Kretinga and Visaginas, which also constantly win the prizes in Republican mathematics contests, although in smaller quantities year I place II place III place KTU gymnasium Vilnius Lyceum Kretinga Visaginas Other schools What are the reasons for the success? The domination of two particular schools mainly lies in their specifics. The Kaunas University of Technology Gymnasium and Vilnius Lyceum are sought after among the best students from Kaunas and Vilnius, as well as from other regions of Lithuania. Consequently, the general level of studies increases in these schools. Students consider both the team mathematics contest in autumn and the individual contest in spring being very important; therefore they put lots of effort into their improvement and try to do their best in order to surpass their rivals. It is pleasant to state that despite the intensive mathematical fight, the relationship between these two schools is very friendly. The students from Visaginas and Kretinga constantly achieve better results in these mathematical contests than the majority of those from much larger cities. Visaginas is the city of the Lithuanian atomic energy. It is inhabited by fairly educated specialists, the majority of which are Russian-speaking. It is obvious that the children of the educated parents are rather self-motivated and talented. There are teachers, who prepare pupils for the contests, working in two schools of this city, therefore the pupils achieve the high results. The high results of a small region of Kretinga prove that one enthusiastic teacher can achieve a lot. The teacher Vytautas Narmontas, who works in the city of Kretinga, is constantly looking for talented mathematicians and successfully preparing them for the contests. Two of his former pupils have participated and won prizes in the World mathematics contests. The complexity of mathematical problems It is very important to select the appropriate problems for the contests. If the problems are too simple, the majority of pupils achieve high results, therefore only some minor inaccuracies conditions the place. If the problems are rather complicated, then the most intelligent pupils achieve adequate results, although the weaker ones often do not manage to solve 184

185 anything at all. It is possible to notice that the problems in the latest contests organized by some universities have been rather simple. There has been the case that even 6 students have achieved the maximum score in their age group. As the contests organized by universities are one of the ways to enter the Republican contest, it is rather complicated to select the most intelligent pupils when the problems are too simple. It is a pity to state that there has been the time when the problems were too simple even in the Republican contests. Due to this fact, the results of the Republican contests were ignored while selecting students to the national team representing Lithuania in the World mathematics contest: the successful participants of the Republican contests had to prove their knowledge at the special selection camp. It is pleasant that the traditions are changing. The problems in this year s contest were rather complicated and appropriate to select the strongest students. How does the preparation for the contests go? We should talk about two age groups separately. Firstly, there are junior students, whose preparation is totally in the hands of their teachers, and secondly, there is the group of senior students that already have some achievements and need further development. The teachers hold the major responsibility while working with junior students. If a lower school student has a teacher who likes to prepare children for competitions, he will definitely have better possibilities than a child with whom nobody was working additionally. However, in many schools kids are taught additionally, they are taken to regional contests or to the contests organized by the universities. There is also a Junior mathematicians school in Kaunas, where students gather to solve various contest puzzles and problems. For senior students there is an extramural young mathematicians school organized by the University of Vilnius. All senior students who have a great interest in math can apply for it. Naturally, only the most talented and the most determined students stay to learn there. In this school students get their tasks via internet and, after solving them, they send them back by mail or via internet again. After graduating from this school students get their graduation diploma. The most talented mathematicians usually are invited to the preparation for the international contest s camp. The selection criteria for camps like that are the results on the regional contests. The National Student Academy (NSA) for senior students has been working in Lithuania for four years. The most talented students come there to study one of the seven subjects. The objective of the academy is not only 185

186 to help students to develop in the area they are interested in, but also to help them grow personally and to get the young talents used to communicate and collaborate with each other. The application procedure requires students to fill in the application form and to have some prior achievements in the study area they want to study in the academy. Firstly they receive the exercises via the internet and send the derivations to the head-teacher of their group. Then according to their activity in doing the exercises and also according to their achievements during the period they are invited to the sessions. There are three sessions within one year. The very best specialists of those subjects: professors, teachers who have raised a lot of contests winners, senior students of the universities prior international contests winners are invited to teach at the academy. The largest group of the NSA is the mathematicians. The mathematicians were the first to have a foreign teacher professor Agnis Andžāns from the University of Latvia. The majority of students became the laureates of the national math contest, many became the members of the national team going to the international math contest. Students with high achievements make the teachers, who have prepared them in the schools, happy. They also make happy the community of NSA. Everybody says that our student has won. It is impossible to measure who had the greatest impact for the results. The most important thing is that students are working in the study area they like and that they get good results. All these results are intended to achieve the same goal they raise the specialists needed to Lithuania. 186

187 MATHEMATICS AS AN ENVIRONMENT FOR DEVELOPING STUDENTS PERSONALITY Bohumil Novák, Faculty of Education, Palacký University in Olomouc, Abstract. The contribution reports on a non-traditional presentation of mathematical activities for pupils aged 11 15, which are being prepared as a part of grant focused on developing pupils interest in mathematics and change of their attitude to mathematics as a school subject. Solving non-standard tasks, competitions, games and manipulative activities provide pupils, teachers and parents with a chance to change their perception of school mathematics. Keywords: games, mathematical competitions, motivational activities, projects, teaching of mathematics. Introduction We discuss the Playful mathematics project conducted at Czech elementary schools, which included a performance of a set of activities aimed at presenting mathematics as an interesting subject to both pupils and their parents. Theoretical framework Mathematics for tomorrow s young children should become an environment for developing their personality. The idea of humanization, in which the school is a service to children and a tool in their development, the center of which are affective components of learning (Crowl et. al., 1997 ), is of key importance in this respect. Developing the personality of a child is seen as education in the broadest sense of the word. Children are not objects of lecturing but subjects of their own learning (Wittman, 1997). A teacher in school is to develop students know-how, their ability to reason as well as to encourage their creative thinking (Polya, 1966). This enables changing concepts, forms and methods of teaching mathematics so that the teachers could teach mathematics in a creative and interesting way and could become agents of a new and challenging class environment. At the same time, parents can learn at least part of what their children know to see the constructivist oriented teaching mathematics based on intersubject integration ( open classes ). They can see the challenges and experience of their children, however, on condition that they come and share the experience with their children (Kafoussi, 2006 ). 187

188 Research conducted at elementary schools in the Czech Republic as well as the PISA research outputs have shown that Czech pupils have very good knowledge of mathematics. A broadly organized research into popularity of respective school subjects and attitudes towards them (among pupils aged 11-18, 2006, Czech Republic) concluded that mathematics is not a popular subject and that pupils lack the ability to apply the obtained mathematical knowledge in real life situations (Grecmanová, Dopita 2007). Motivation plays definitely the most important role in this respect, as it is motivation that assigns subjective sense to the learning activities of the pupil (Skalková, 2007). Methodological starting points When preparing and conducting a research project aimed at making mathematics a more popular school subject, we follow the above mentioned attitudes. The National Programme of Research II project of the Ministry of Education, Youth and Sports Research on New Methods Competition of Use Creativity Focused on Motivation in Scientific Area, Especially in Mathematics, Physics and Chemistry is a chance to apply efficient instruments of motivation in teaching mathematics while following the basic constructivist principles. When solving one particular task of the project (preliminary name Playful mathematics, B. Novák) we were inspired by the above-mentioned ideas and tried to confront them with the actual elementary school practice. Our experiment focused on creation and support of and research into educational efficiency of a number of activities: school mathematical competitions, projects, events for parents and public. It is aimed at various elementary school target groups: mathematically talented pupils as well as average pupils (focus on raising and developing their interest in mathematics) or pupils with special needs. The events are prepared in order to give pupils (even the less mathematically talented ones) a chance to acquire new mathematical experience and especially to let them get to know mathematics as something else than a boring subject as an environment for personality development, interesting experimenting and discoveries. The teachers could see it as a chance to change their ways of teaching and forms and methods they use all in order to be able to teach mathematics in a creative and interesting way. Reflection of the participants view is very important in this respect the participants are welcome to subsequently give their comments on both the content and the form of the event. 188

189 Playful mathematics : Methods, ways of realization We aim at creating the basic framework of new competitions and other activities and advertising them. The activities include seminars on didactics for students and teachers, afternoon workshops on games, competitions, etc. When applying and performing the activities we rely on co-operation of the researchers (a team of 8) with elementary school teachers taking part in the project. Tutoring students and PhD students is another important part of the project. Between November 2006 and May 2007, 12 events took place at 8 schools with more than 1000 pupils as participants. Such activities offer possibilities to: pupils / students to be given space for interesting experiments and discoveries, activities connected with everyday experiences, projects, teachers to change the approach, forms and methods, to be able to teach mathematics in an inspiring and interesting way, to help in creating the new climate, challenges for both them and the pupils, parents to get to know part of what their children learn, find out that mathematics need not be boring and uninspiring formula training; however, they must come, see and share experience with their children. The events are preceded by a systematic preparation of pupils during classes. The pupils are trained in tasks interpretation and presentation, arguing skills when defending one s way of solving the task, communication skills when co-operating on solving the task (pair work, group work, class work) are promoted. This develops a number of competences: competence to learn, citizen competence, work competence, competence to solve problems, communicative, social and personal competences. Mathematical activities performed in the project could be categorised as follows: a) games, e.g.: sudoku, crosswords, board games, computer games, brain teasers; some examples of games include: surprising assembling geometrical jigsaws or assembling solids without gluing, polyomino, tangram type or Columbus egg type brain teasers, magic paper origami (water lily in blossom, box, dog, etc.), matches type brain teasers (move the matches so that... ), pyramid puzzles number pyramids, clusters and other number based tasks, funny tasks aimed at logical thinking development, 189

190 estimate the number of type tasks e.g. of beads, beans or other small objects in closed jars, words on a book page or on a handwritten sheet of paper. b) activities connected to everyday life (projects); some examples of such activities include: Building a town project, where pupils place themselves into a position of a citizen who wants to build a house. They have to consider financing (i.e. savings, loans, bank, jackpot, etc.), choosing the suitable plot (thus e.g. count the area), design, project, construction or buying the material (i.e. consider discounts, prices, etc.) Treasure hunt project, where pupils have to find the hidden treasure (such as sweets, small gift objects). The treasure can be found after solving a number of everyday life tasks including measuring, number and volume estimates (How much / how many? tasks), calculator calculations, navigating through the labyrinth, etc. c) unusual mathematical problems, e.g.: Kangaroo problems. Our experience Except for spontaneous reactions during and after the events a space is given to feedback from pupils, parents and teachers. A feedback questionnaire had been prepared, its evaluation will be performed before the project completion: a) Evaluate the following statements (use the ++, +, 0, -, -- range): 1. I enjoy mathematics. 2. I like solving non-traditional mathematical tasks. 3. I like solving brainteasers; I play chess or other board games. 4. I like working with a computer. 5. I found today s tasks simple. 6. I solved the tasks successfully. 7. What I was asked to do was new and unusual. 8. I enjoyed being a part of a team and helping each other. 9. On the whole, I liked the event. 10. I would welcome another event like this one. b) Choose and write the names of five activities, which you took part in, and sort them from the best one to the worst one: c) Write your own assessment of the event. 190

191 At the same time, both children and their parents responded on a board titled What I liked most / least. Conclusion We tried to show at least some possibilities offered by the National programme of research II project at Faculty of Science and Faculty of Education, Palacký University in Olomouc. We aim at increasing motivation of pupils for mathematics, improving perception of mathematics as an interesting school subject and making pupils learn mathematics. Our ambitions include utilizing non-traditional forms as a suitable tool for making mathematics more popular, forming positive attitude of pupils to mathematics and improving the overall class environment. Subsequent reflection and evaluation is important. We have learned that pupils as well as parents and general public like the events. We believe that this is caused by the fact that the events give everybody an unconventional view of mathematics, increases their interest in studying context and relations in problem solving. As far as motivation is concerned, this has enormous importance. We are happy to see pupils interest and enthusiasm, especially that fair play rules were never broken. This gives us the feeling of having done a good and meaningful thing. References 4. BROUSSEAU, G. Theory of didactical situations in mathematics. Dordrecht: Kluwer Academic Publisher, ISBN CROWL, T.K., KAMINSKI, S., PODELL, D.M. Educational Psychology. Windows on teaching. New York: Brown; Benchmark, GRECMANOVÁ, H., DOPITA, M. On the interest of pupils in natural sciences. (in Czech, an unpublished text, 2007). 7. KAFFOUSI, S. Parents and students interaction in mathematics: designed home mathematical activities. In: Proceedings of CIAEM 58, Srní, The Czech rep., NOVÁK, B. Reflection of a didactic game in teacher training. In: Proceedings international symposium elementary maths teaching. Praha: Prometheus, 1995, s POLYA, G. Mathematical discovery. New York: John Wiley et Sons, SKALKOVÁ, R. Playful mathematics as a method of motivation. Proceedings international symposium elementary maths teaching. Praha, UK 2007, s WITTMAN, E. CH. 10 Jahre Mathe Bilanz und Perspektiven. Dortmund: Universität Dortmund, Klett, Funded by the NPV II STM Morava, No. 2E06029 project. 191

192 DERIVATIVES USING PHP A SUITABLE TEACHING MATERIAL NOT ONLY FOR COMBINED OR DISTANCE STUDENTS Michal Novak, Brno University of Technology, Faculty of Electrical Engineering and Communication, novakm@feec.vutbr.cz Abstract. In the contribution I present a simple application, which could be used by university students as a supplementary material when revising the concept of differentiating. Since the application is based on a PHP technology, it can be used outside classes, which makes it suitable for uncontrolled practice of students of distance or combined forms of study. Keywords: derivative, distance learning, e-learning, PHP, teaching mathematics Introduction and background Understanding the concept of differentiating and the ability to find a derivative of a given function belong to basic skills of university mathematics. Naturally, the more practice the better. However, especially universities of technology at least in the Czech Republic have been experiencing a substantial reduction of classes of mathematics, which is unfortunately accompanied by a growth of subject matter students are required to absorb. This results in a necessity of passing the subject matter to students via a number of channels. Simultaneously, an increasing demand for tertiary education has resulted in growing numbers of students in other than attended form of study, i.e. students, whose contact with teachers is limited. The creation of the Basics of differential calculus: derivatives, which is presented in this contribution, was motivated by both of these phenomena: the application is one of a number of optional tools which students (most likely of the combined or distance forms of study) may use when learning one part of university mathematics. It is to be noted that the application has been intended as a supplementary material, not as a primary tool of teaching / learning how to differentiate functions. The aim and justification The application was programmed as a response to a typical situation: when practising the subject matter finding derivatives of functions in this particular case outside a class, the students usually have a collection of 192

193 exercises, i.e. tasks and results, at hand. They compute the tasks and compare the results obtained with the results recorded in the collection. Yet if the results are not equal, a process of identifying a mistake must take place. Naturally, identifying a mistake by students themselves is of a great didactic importance but the situation involved is uncontrolled practice. This type of practice is very often connected with time pressure e.g. when students prepare for exams. The mistake must therefore be identified swiftly, which is rather difficult if other than elementary tasks of differentiation are involved. As far as students of combined or distance form of study are concerned, their practice is almost exclusively an uncontrolled one. Uncontrolled, however, does not mean unguided. Yet the guidance is often missing. The swiftness in identifying the mistake made is even more important for these students as the time they can spend on learning is usually very limited. To sum up, the application Basics of differential calculus: derivatives was programmed in order to: give students such a tool for their uncontrolled practice, which would help them to identify mistakes made when differentiating functions give students of combined or distance forms of study a tool for practising in differentiating functions Additional aim of the application was to encourage experimenting such as finding derivatives of invented, i.e. increasingly complicated composite, functions. The requirements Given the aims of the situation discussed above, the application was to have the following functions and features: to find a derivative of any function (however complicated), which a student might expect to be asked to differentiate, not only to give a correct result, but, more importantly, to guide the user throughout the process of differentiation in order to help the user to identify the mistake. The issue of necessity A very relevant question arises of whether such an application must be programmed anew. In other words, whether there are not enough existing applications which could be used instead. The application Basics of differential calculus: derivatives was programmed primarily for students of our faculty. Students of the attended form of study attend practical computer-aided classes of a subject, in which 193

194 differential calculus is included. The numbers of computer-aided practical classes and numerical practical classes are roughly equal each type is allocated 2x50 minutes every other week throughout the 13week-long term. Students use Maple during the computer-aided practical classes. Given such a schedule (and the fact that parts of linear algebra and integral calculus are included in the same subject), it is obvious that students do not experience much controlled practice. As has been mentioned above, students in the combined form of study (distance form is not included at our faculty) have almost no controlled practice throughout their studies. Unlike the students in the attended form of study, they do not use Maple or any other mathematical software. Therefore, when left alone, the students of the attended form of study could use Maple during their uncontrolled practice (yet they would have to be present at a school computer with Maple installed) while students of other forms of study would be left at a loss with collections of exercises or their textbooks. In Maple an application similar to Basics of differential calculus: derivatives is included: it is the DiffTutor() command. However, this command launches a rather complicated English speaking window, which most of the students involved (i.e. either mostly 19- yearold students or middle-aged people; both with usually low level of general English) are unlikely to be able to work with. In any case a non-maple solution must be looked for, either an existing one or programmed anew. The form Given the background discussed above, the application should fulfill further requirements: availability: if it is to be used for uncontrolled practice (and / or by combined students) it must be independent of the school network. This restricts the solutions to Java applets, downloadable exe- files, web applications, etc. straightforwardness and simplicity of use: the application should be a single-purpose one not distracting students attention. This is especially necessary in the case of combined students whose time for learning and practice must be used as efficiently as possible. In order to maximize the availability of the application, it was programmed as a web application using the PHP technology. This means that the student can use the application once he/she has access to the Internet (which is assumed for students of all forms of study). No plug-ins must be installed (such as Java), which might be a problem at Internet cafes or computers or notebooks lent to combined students by their employers. 194

195 The form of a stand-alone application (programmed in Delphi, C## or Pascal) could have been chosen. Such application may even exist so that no programming is necessary. However, programming a differentiating application using a PHP technology implied solving an interesting and challenging problem. The obstacle a need for a new diff( ) function Even though some mathematical functions are included in the PHP specification, they can be used only to perform basic tasks such as basic elementary operations or evaluating elementary functions. Finding a derivative of a function, i.e. of an arbitrary composite function made up from an arbitrary number of arbitrary composite addends or members of a product or quotient, is beyond functionality of a single PHP function. A new function named diff(function, variable) had to be programmed. In order to do so, the task of finding a derivative was divided into three separate tasks: 1. reading the function: the function entered by the user is read as a string, i.e. sin(2*x) is decoded as a sine function, the argument of which is 2*x. For details cf. the application help files. 2. the differentiating algorithm based on the onion principle : I am going to demonstrate how the application works using sin(2*x+x^3)+x as an example: 1. Testing if the function is a sum, a product or a quotient of functions. The algorithm finds out that the function is a sum of functions. Therefore, it differentiates the function term by term. It starts with sin(2*x+x^3) and continues with x. 2. Testing if sin(2*x+x^3) is a sum, a product or a quotient of functions. It isn't, therefore the algorithm tests if sin(2*x+x^3) is an elementary sine function; the interior function 2*x+x^3 is recognized. The derivative of sine is cosine. The algorithm simultaneously finds out that 2*x+x^3 is a composite function. The application stores the derivative-so-far cos(2*x+x^3) and supplements it with the information that further differentiating is necessary. 3. In a similar way x is tested. It is neither a sum nor a product nor a quotient of functions. Therefore, the algorithm tests if x is an elementary function. A power function with exponent 1 is recognized. The term is differentiated accordingly and the 195

196 derivative-so-far together with the information that further differentiating of this term is not necessary is stored. 4. Differentiating 2*x+x^3. The algorithm knows that this is the interior function of sine. A sum of functions is recognized. Each term is therefore differentiated separately. 5. The term 2*x is a product of functions; the formula for differentiating products is therefore applied; 2 and x are differentiated and the derivative of a product is assembled. Simultaneously, the algorithm finds out that none of the functions is a composite function. The information that none of the functions requires further differentiation is stored. 6. The term x^3 is recognized as a power function; the respective formula is applied. Simultaneously, the information that its interior function doesn't require further differentiation is stored. 7. The derivative of 2*x+x^3 is placed at its position, i.e. after cos(2*x+x^3). 8. The algorithm finds out that there are no interior functions to differentiate. The end of the differentiating algorithm. 3. simplifying the result as various redundancies, such as zero members, multiple brackets, multiplications by one, etc. occur during the differentiating algorithm. For details cf. the application help files. Presentation of the application The application is located at The user enters the function using the common Maple / Matlab type of input, i.e. using +-*/^ characters and argument bracketing conventions, such as sin(x), exp(x). Multiple names are enabled, thus e.g. asin(x) and arcsin(x) both refer to the arcsine function. Then, the variable with respect to which the function is to be differentiated, is entered. Upon pressing the Differentiate button, the result is displayed. The user can then display the procedure of differentiating and is made aware of the limitations of the displayed text (see below). The application exists in two language versions Czech and English and is supplemented by extensive online help. The limitations The application Basics of differential calculus: derivatives has several minor limitations described in its help files. However, there is also a not-toneglected limitation regarding the notation of the procedure process 196

197 displayed to the user. It does not fully correspond to the usual differentiating done by a human, as it is the description of respective steps of the algorithm as described above. This seems to be the price for algorithmizing the task. The algorithm differentiates all level 1 outer functions, than all level 2 outer functions the outer functions of internal functions of level 1 outer functions, than all level 3 outer functions, etc. and keeps inserting derivatives of internal functions at their respective places. A human, on the other hand, works in a different way: every addend of the function is always differentiated until the end, and only than is the next addend differentiated. In the example above, sin(2*x+x^3)+x, the second addend, i.e. x, is treated as a level 1 outer function and differentiated immediately after sin(inner-something). A human, however, would differentiate it only after the whole first addend is differentiated until the end, i.e. until the whole of 2*x+x^3 is differentiated. Further applications based on the same technology The PHP technology can be used to treat a number of other problems relevant in teaching university mathematics. The examples include references [2] (currently in Czech only) or [4]. References 1. CASTAGNETTO, J. et. al. Programujeme PHP profesionálně. 2nd edition. Brno: Computer Press, pp. ISBN (PHP manual) 2. NOVÁK, M. Linear algebra: work with matrices. [online]. [cit ]. Available from 3. NOVÁK, M. Basic sof differential calculus: derivatives. [online]. [cit ]. Available from 4. NOVÁK, M., LANGEROVÁ, P. English/Czech Czech English dictionary of mathematical terminology. [online]. [cit ]. Available from 197

198 PARADOXICAL MATHEMATICAL PROBLEMS Raitis Ozols, University of Latvia, Abstract. This paper offers 12 mathematical problems with unexpected solutions. Keywords: mathematical problems, paradoxes. Problem 1. Prove that for any natural n 4 there exists a polygon with property: all it s sides and all it s diagonals are greater than 1 kilometer but area is smaller than 1 square milimeter. Solution. Such a polygon is shown in Fig. 1, all it s vertices are on a circle. The length of the base is L = 2 ( n 1) km. Then the length of any L of it s sides will be at least 2 n 1 = kilometers and length of any diagonal will be greater than 2 kilometers. Choosing h equal to milimeters, the area S will be not greater than Lh = 0, 2 mm n 1 Fig. 1 L Problem 2. Given is a squared rectangular paper sheet with size squares. In any square we can write no more than one decimal digit. Can we compute the value of number on this paper? In Fig. 2 it is showed how to add two numbers. Solution. In the upper right corner of the sheet we write number 11, and below it we write 10 times larger number 110 (see Fig. 3). Adding these numbers we obtain 121=11 2. Below 121 we write 1210 and, adding with 121, we obtain 1331=11 3, etc. Therefore in the 39 th row we will obtain the decimal expression of Remark. There is a problem in Ya.Perelman s book [1, p. 13] with the answer 11 11, however there is also a remark: I wonder if someone has patience to compute this number by repeated multiplication. As we see, it is not so difficult to compute that number. By the way, with more 198 h Fig = = = = = 11 5 Fig

199 ingenuity it is possible to compute also the numbers 11 n (n 25) easily. Problem 3. Do there exist such real positive irrational numbers a and b that both numbers a b and ab are rational? 1 Solution. Let us consider the function f ( x) = x. Since f < 0, 5 3 but f ( 1) > 0, 5, such x > 0 exists that f ( x) = 0, 5. Now suppose that x is rational. Then it can be expressed in the form q / p p relative primes, q>p. Then = 0, 5 q x = 1 x p q, where p and q are q p q 2 q is even. p = q Therefore p is also even. This is a contradiction with the fact that q p is a reduced fraction. Therefore x is irrational. Now let us consider number 1 y = which is also irrational. Then x y and xy will be rational numbers x y since x = 0, 5 and xy = 1. Problem 4. Suppose that plane figure is given with perimeter P and area S S. By α we denote the ratio 2. Is it possible to divide the square into n P parts so that α of every part is greater than α of the square if: a) n = 5; b) n = 2? Solution. We can assume that the length of the side is 1. Then a solution of a) is showed in Fig. 4 (h = 0,01). Solution of b) is showed in Fig. 5. Figure Q is a square from which an isosceles right triangle with legs 3 1 is cut off. Fig. 4. Fig

200 Remark. Author can prove that for any n 2 a square can be dissected into n parts such that α of every part is greater than α of a square (which is equal to 16 1 ). Problem 5. A horizontal table is given. Is it possible to make a stable construction consisting of 6 matches such that only 3 matches touch the table and the construction will collapse if any match will disappear? Each end of each match touches the table or is hanging in the air. Solution. Yes, it is possible, see Fig. 6. R 1 R 2 R 3 Fig. 6. Fig. 7. Problem 6. Does there exist such natural n that a square consisting of n n unit cells can be divided into N different rectangles with N > 2n? The cuts are made along grid lines. Solution. Yes, it is possible. This is impossible if n is small. Therefore we choose n = 150. First we divide the square into rectangles R 1, R 2, R 3 and R 4 (see Fig. 7.). After this we divide the rectangles R 1, R 2 and R 3 into 148, 72 and 52 different rectangles recpectively (see Fig. 8. and Fig. 9.). The shortest sides of rectangles which belongs to R i are equal to i, i {1;2;3}. Lengths of other sides are shown inside the rectangles. In Fig. 9 the division of R 4 is showed. The total number of obtained rectangles is = 301 > R Fig R 4 3 R

201 R 3 78 R Fig. 9. Problem 7. Is it possible to make two quadrangles using only four matches? Solution. Yes. The answer is showed in Fig. 10. k-th row row 1. row Fig. fig Fig. fig. 11. Problem 8. Suppose we can choose any square the side length of which is a natural number. Also an unbounded amount of equal circles with diameter 1 is given. Is it possible to place some circles in the square without overlapping so that they cover more than 90% of the area of square? Solution. First we use the fact that circles can be placed sufficiently dense if we use a hexagonal packing (see Fig. 11). However, if the side length n of a square will be small then the solution can t been obtained. Let us choose n = 98. It is easy to understand that the height of first k rows h k (k 1) in hexagonal packing satisfies h = 1 1 and 3 h i+ 1 hi =. 2 Therefore h k = 1+ ( k 1) If n = 98, then the maximal number of rows 2 (98 1) is k = 1 + = 113 and the total number of circles is 3 h k n 201

202 N = = (113 summands). Then the density will be ρ = π = 0, > 90,1% Problem 9. An arbitrary convex quadrangle is given. Is it possible to construct a net such that moving on it we can get from any vertex to any other and the total length of the net is less than the sum of the diagonals of the quadrangle? Solution. Let us first solve this problem for arbitrary rectangle with side lengths a and b, a b. We can make a net which is shown in Fig. 12; the marked angles are 120. The calculations are left to the reader. In Fig. 13 we see a solution for arbitrary quadrangle. The net is obtained choosing a rectangle whose all vertices are on diagonals. Inside of this rectangle we draw a net mentioned above. a b Fig. fig Fig. fig. 13. Problem 10. A rectangle with shortest side 1 and longest side smaller 3 than 1 + is given. Can it happen that there are 5 points in it such that the 2 distance between any two of them is larger than 1? 1,866 1 d Fig. 14. Solution. Yes, it can happen. Suppose that the largest side is 3 1,866 < 1+. We can chose A, B, C, D and O as shown in Fig. 14; B and 2 C are vertices of the rectangle, O its center, A and D are on the sides in a distance d from the nearest vertices. Choosing d = 0,05 we obtain CO = OB > AO = OD = ( 1, ,05) > 1, 01

203 and AC = BD = 1+ 0,05 2 > 1, 001. Problem 11. Does there exist a polyhedron whose all faces are regular triangles, regular 7-gons and regular 11-gons (at least one face of each type must be present)? All edges of the polyhedron must have the same length. Solution. Such a polyhedron exists. To construct it we first construct an antiprism with a regular 7-gon as a base. All the remaining faces are equilateral triangles. An antiprism with regular 11-gon as a base and all remaining faces being equilateral triangles is also made. Gluing these antiprisms so that they have a common triangular face we obtain the necessary polyhedron. Problem 12. There are n 2 disjoint polygons given in the plane. Can we draw a closed broken line which crosses each its segment exactly k times and which crosses also any plane curve joining any two points which belong to different polygons? These closed broked lines should not intersect any polygon. Solve this problem for: a) k = 1; b) k = 2. Solution. Solution of a) is showed in Fig. 15; solution of b) is showed in Fig. 16. In both cases n = 3. For any larger n the construction of the closed broken line is similar. k = 1 k = 2 Fig. fig. 15. Fig. fig. 16. References 1. J. Perelmanis. Saistošā algebra, Rīga, Latvijas Valsts izdevniecība, E. RiekstiĦš, A. Andžāns. Atrisini pats!, Rīga, Zvaigzne,

204 204 MATHEMATICAL TASKS CAUSING DIFFICULTY FOR PRIMARY SCHOOL STUDENTS Anu Palu, Eve Kikas, University of Tartu, Abstract. Several international studies regarding primary school students knowledge in mathematics and assessing that knowledge have been made worldwide. Estonia participated in the International Project on Mathematical Attainment initiated by CIMT (the Centre for Innovation in Mathematics Teaching). In a longitudinal project, the development of a group of students in mathematics was followed from early first grade until the end of the sixth grade. Estonian students were assessed in the first three grades (from 2002 to 2005). The aim of the present research was to describe the mathematical tasks that cause most difficulty for primary school students on the basis of the IPMA tests and analyse the reasons for these difficulties. Altogether 269 primary school students were tested four times within a three-year period. The study confirmed that in the case of most tasks the results improved with time. Yet, with some tasks the percentage of correct answers increased very little. The analysis of the solutions to the given tasks showed that the difficulties students had in solving mathematical tasks were connected with functional reading skills as well as not understanding the concepts properly. Keywords: difficulties in mathematics, errors, mathematical tasks and primary school student. Introduction Mathematical tasks are part of children s learning from the very beginning to the end of their studies at school. Learning often takes place through tasks. Therefore, the teacher has to choose the tasks carefully and with responsibility. In school mathematics, two types of mathematical tasks are solved: practice tasks and problem solving tasks. In the former, students know the algorithm for solving the task and the main aim is to practise the algorithm until it is acquired. In problem solving tasks, the algorithm is not transparent and the solution is not readily available. By means of these two types of tasks, children s knowledge in mathematics is assessed. The assessment of children s knowledge in mathematics may be characterized as being composed of two broad components: computation or operations and application (Thurber, Shinn & Smolkowski, 2002). Calculation tasks mostly test the result of practice. Application tasks demand applying acquired knowledge and thus we may describe them as problem solving tasks. Past practices of testing often assessed relatively trivial aspects of mathematics

205 like knowledge and skills. Problem solving and mathematical thinking are more difficult to assess trough tests (Zevenbergen, Dole & Wright, 2004). Teaching mathematics in primary school emphasizes the calculating skill and therefore calculating tasks are more frequently assessed than problem solving tasks. However, according to the national curriculum, a basic school student should, in addition to doing simple mathematical calculations, be able to generalise, discuss logically, describe real-world problems mathematically, analyse, solve problems and interpret results (Põhikooli ja gümnaasiumi riiklik õppekava, 2002). Such skills are mostly tested by means of problem solving tasks. In order to solve calculation tasks successfully, it is necessary to possess the respective conceptual and procedural knowledge. Conceptual knowledge can be defined as the understanding of principles that govern the domain. Procedural knowledge means the possession of strategies necessary for solving the task (Rittle-Johnson & Siegler, 1998). Fridman (1987) describes mathematical tasks as procedures. Simpler procedures (for example, writing and reading numbers in arithmetics, operations with single-digit numbers etc.) are components in more complicated procedures. Therefore it is inevitable that the student should be able to conduct simpler procedures automatically and without error. More complicated procedures, such as problem solving tasks, require the possession of all procedures necessary to apply knowledge and skills. Often, the student is accustomed to performing simpler procedures but unable to apply these procedures where necessary. In a problem task presented as a text, the student first has to read and then give a solution. Earlier studies have shown that in solving primary school mathematics tasks, an important role is played by the students reading skill (Geary, 1994; Mercer & Sams, 2006; Thurber, Shinn & Smolkowski, 2002; Zevenbergen, Dole & Wright, 2004). When a student makes an error, there are a number of levels at which the error could be made that do not necessarily represent a mathematical error. There are other sites for errors namely in reading and comprehending the task. The aim of the present research was to describe the mathematical tasks that cause most difficulty for primary school students on the basis of the IPMA tests. As Estonian class teachers consider doing calculations very important (Palu & Kikas, 2007), we assumed that primary school students have a good calculating skill but they remain weaker in solving problems. Secondly, we described the incorrect solutions to tasks that pose difficulties to students and analysed possible reasons for the errors made. 205

206 Methods Primary school students were tested four times within a 3-year period: at the beginning and the end of the first grade, at the end of the second grade and at the end of the third grade. Altogether 269 students from 20 schools participated in all the four tests, out of them 119 boys and 150 girls. The students were tested in writing during mathematics lessons. The tests were conducted by mathematics teachers. Tests were identical with the corresponding IPMA tests: Test 0, Test 1, Test 2, Test 3 (IPMA Tests, 1999). The first test comprised 10 tasks, with each following test having more but containing tasks from earlier tests. The last test comprised 60 mathematical tasks. According to the Estonian national curriculum, topics in mathematics are divided into 3 groups: number and operations, geometry and measurement. Of the tasks included in the IPMA tests, only one concerned measurement and one geometric figure. The remaining tasks were all from the realm of numbers: reading and writing numbers; mental addition, subtraction, multiplication and division; understanding the meaning of fractions; finding patterns in sequences of numbers; problem solving tasks. In the current paper, the answers to five problems are analysed. These are tasks, the solutions of which did not improve by the years or improved little. Results and conclusions Estonian students solved the majority of tasks well every year. Solving repeated tasks improved every year. But there were five tasks, the solutions of which did not improve by the years or improved little. The first task was: Circle all the odd numbers (12, 7, 15, 4, 1, 10, 18). Incorrect answers most often included single-digit numbers 7, 4 and 1 and even numbers 12, 4, 10 and 18. According to the national curriculum, the student should be familiar with the concept of odd numbers since Grade 1. That students do not develop a proper understanding of the concept may be due to the fact that even and odd numbers are introduced when learning numbers from 1 to 10. Later the term is not used systematically, which is why the students understanding of the concept remains incomplete. Some students confused odd numbers with even numbers. An analysis of incorrect answers showed that some students derived the concept on the basis of some other principle. When left to generalise on one s own, the student may develop a false concept. Thus, odd and even numbers were considered to be connected with the concept of single- and double-digit numbers: single-digit numbers were given as odd. Memorizing it once does not suffice for the acquisition of a concept, it has to be reminded and used more often. 206

207 Out of the tasks of Test 1, one task received considerably less correct solutions than other task by the end of the second and third years. The percentage of correct solutions even decreased by the end of the third year as compared to the second. The task was: Write the letter A on the third shape from the left and the letter B on the fourth shape from the right. Determining the third shape from the left and the fourth from the right in a row proved to be difficult for some students in all the three grades. A portion of students was unable to correctly determine the right or the left in all the grades. The increase of such children in Grade 3 is apparently connected with the fact that respective tasks are practiced more in Grades 1 and 2, but no longer in Grade 3. An analysis of the mistakes showed that students had difficulties in understanding the instructions. Often, it was an issue of careless reading: the letter was written on the triangle instead of the third shape; on the quadrangle instead of the fourth shape. Several students had trouble in fulfilling two given instructions at once: determining the right-hand or the left-hand side and counting. Students also found difficult to count from the right to the left, since we read texts from the left to the right. A seemingly simple task proved surprisingly difficult probably because solving it required the skills of reading as well as comprehending the text. In the present study, the fact that reading skills may influence assessment results of mathematics in primary school was partially confirmed. Out of the tasks for Grade 2, repeated bad solutions occurred in the case of three tasks. Finding the pattern in a sequence of numbers (Fill in the missing numbers 3, 9, 27, ) proved most difficult. Proposed solutions included numbers from 3 to 243. The most frequent incorrect answer proposed was 33 (28,6 per cent). Other popular incorrect answers included 30 and 45, the relative frequency of both was 5,6 per cent. The task designed to test the concept of the fraction (Colour in a quarter of the total number of circles) proved too difficult in Grade 2, but also in Grade 3. There are various ways to test if a concept has been acquired or not. Often, comprehension of concepts is tested by means of certain tasks, which means that a slightly modified task will pose a difficulty for the students. Whereas it is usually required to find a portion of a definite number or a shape, in this study, the task was to find a quarter of eight circles. It proved difficult for the students to consider the eight circles as a whole. Several students coloured a quarter of each small circle, thus presenting a correct answer. Colouring four circles (an incorrect answer) is probably derived from the denominator of the fraction fourth part. Previous studies have shown that 207

208 fractions can be taught already in the pre-school age (Mix, Levine & Huttenlocher, 1999) but the concept is acquired only if it is generalized. Most primary school students are not capable of this. Next poorly solved task was the problem solving task: Peter thinks of a number. He multiplies it by three, takes away 2 and gets 25. What was his number? It was not a typical primary school task. As the algorithm was not familiar, students conducted arithmetic operations just by manipulating symbols, operating with the numbers from the text. The most frequent incorrect answer, 27, was simply the sum of 25 and 2, which both appeared in the text. Subtracting 2 from 25 led to the other incorrect answer, 23. This means that students manipulated with numbers written in the text by means of digits, as the number three was represented by means of letters and was not used in the solution. Previous studies (such as Schoenfeld, 1991) have shown that older students similarly combine random operations from the numbers given in the text. This means that students do not consider it necessary to analyse the task or assess the result. The task in question gave a good opportunity for self-assessment, if the student had used the resulting number to make operations designated in the task. But the great proportion of incorrect answers indicates that students did not use that possibility. It was possible to assess a fairly small part of knowledge and skills in mathematics to be acquired in Grades 1 to 3 by means of the IPMA tests. The Estonian national curriculum presents 21 learning objectives in mathematics to be achieved by the end of third grade. The IPMA tests cover only certain areas of these learning objectives, mostly representing objectives related to calculation. Despite this, some conclusions can be drawn. Most of the tasks within the tests were simple calculation tasks, problem solving tasks were not so numerous. Therefore, no fundamental generalizations on the solubility of problem solving tasks can be made. Estonian students did solve the mental calculation tasks of the tests very well. In investigating problem solving tasks or application tasks, it could be noted that these pose greater difficulties to the students. Solving these tasks requires functional reading skills and an understanding of mathematical concepts. As these difficulties would not be aggravated in middle school, more attention should be paid to application tasks besides calculation tasks already in primary school. 208 References 1. Fridman L. (1987). Matemaatika õpetamise psühholoogilis-pedagoogilisi probleeme. Tallinn: Valgus. 2. Geary D. C. (1994). Children s mathematical development: research and practical applications (pp ).washington: APA. 3. IPMA Tests (1999).

209 4. Mercer N. & Sams C. (2006). Teaching Children How to Use Language to Solve Maths Problems. Language and education, 20 (3), Mix K.S., Levine S.C. & Huttenlocher J. (1999). Early Fraction Calculation Ability. Development Psychology, 35 (5), Palu A. & Kikas E. (2007). Primary school teachers beliefs about teaching mathematics. Nordic Studies in Mathematics Education, 12 (1), Põhikooli ja gümnaasiumi riiklik õppekava (2002). [National Curriculum for Basic Schools and Upper Secondary School]. Riigi teataja I osa 20, Tallinn: Riigi Teataja kirjastus. 8. Rittle-Johnson B. & Siegler R.S. (1998). The Relation Between Conceptual and Procedural Knowledge in Learning Mathematics: A Review. In C. Donlan (Ed.). The development of Mathematical Skills (pp ). Psychology Press Ltd. 9. Schoenfeld A. H. (1991). What's all the fuss about problem solving. Zentralblatt für Didaktik der Mathematik, 23 (1). 10. Thurber R.S., Shinn M.R. & Smolkowski K. (2002). What is Measured in Mathematics Tests? Construct Validity of Curriculum-Based Mathematics Measures. School Psychology Review, 31 (4), Zevenbergen R., Dole S. & Wright R.J. (2004). Teaching mathematics in primary schools. Crows Nest, NSW, Australia : Allen & Unwin. 209

210 ON THE HISTORY OF LITHUANIAN MATHEMATICAL TERMS Vidmantas Pekarskas, Kaunas University of Technology, Abstract. The paper presents the history of Lithuanian mathematical terms development, starting with the rise of the first mathematical terms and ending with their standardization. Contribution of the authors of Lithuanian mathematical textbooks and vocabularies of terms to the development of Lithuanian mathematical terminology is discussed. Difficulties involved in the Lithuanian terms development until they settled down and were universally accepted are disclosed. Keywords: history of mathematics, Lithuanian mathematical textbooks, mathematical terms. Introduction History of Lithuanian mathematical terms had its start at the end of the 19th century together with the national revival of the Lithuanian nation. At that time developers of the first mathematical terms found poor resources of Lithuanian language of science. On the basis of spoken Lithuanian they had either to coin absolutely new words or to apply common words to mathematical purposes. Development of Lithuanian mathematical terms involved difficulties and a number of different words were used to express a basic notion before the term settled down. Lithuanian mathematical terms history is being a little older than a centenary, it can still be divided into several periods. The first period covers the years from the publication of the first Lithuanian mathematical textbooks at the very end of the 19th century to 1915 when intensive standardization of terms gained strength. This is the period of search and development of the first terms. The second period can be characterized by standardization and establishment of standards in mathematics. During the period of two bi-lingual glossaries of arithmetic, algebra, geometry and trigonometry terms containing new, previously never used terms were published. In those term glossaries many international words were replaced by purely Lithuanian newly coined words. It would be wrong to assume that new terms had a preference in their usage. Quite on the contrary, fierce discussions were held among mathematicians until the disputes on their 210

211 usage calmed down little by little and new terms were accepted faster or slower. The third period set in with the Lithuanian University founded in Kaunas in During the two decades of the Inter-War period the first Lithuanian higher mathematics coursebooks were published whose authors improved the former and developed new terms, thus laying the foundations for the present terminology. Their contribution was immense, as the authors of the following textbooks could work in a well cultivated field of Lithuanian terminology. The post-war period is also of great importance to the terms development. The scientific activities of mathematicians were intensified, however, regrettably, Russian was introduced as the language of science bringing Russian mathematical terms which had to replace the well-formed Lithuanian terms. Whereupon a job of a basic consideration was taken and great efforts were made to resist against russification in the education. Likewise, as 50 years ago, the standardization of mathematical terms became an essential requisite in mathematics. That work was undertaken by the teams under prof. J. Kubilius and proceeded with some interruptions for several decades. It was crowned by the publication of a fundamental trilingual Dictionary of Mathematical Terms, edited by prof. Jonas Kubilius, History of Lithuanian arithmetic terms A book of arithmetic problems published by J.Gailutis in Tilsit, 1885 [1] may be considered as the first Lithuanian printed textbook of mathematics. Its author was a researcher of Lithuanian language, medical corps general Jonas Spudulis ( ). When the book was being compiled, J.Spudulis was a fifth year student in the Military Medicine Academy in Sanct Petersburg According to V.Biržiška [2], the first arithmetic textbook that should be considered is Arithmetic written in Samogitian language by Reverend Jeronimas Stanevičius (1793- after 1854) in The book was reviewed by mathematics teacher Adomas Dambrauskas from Kražiai Gymnasium, who considered himself to be Polish. That review criticized the textbook saying that introduction of Samogitian language to schools contradicted the welfare of peasants and their wish to teach their children in Polish and in Russian. That textbook was never published and its fate is unknown. In 1856 a similar textbook was prepared by teacher Motiejus Pranas Martynaitis (about after 1856). That information was found by V.Biržiška in a letter written to publisher A.Zavadzkis dating P.Martynaitis called the textbook Początki aritmetyki po litewsku. 211

212 Unfortunately, that handbook was also not published and its manuscript was never found. Notwithstanding the publication of the first Lithuanian arithmetic textbooks at the end of the 19th century, the first Lithuanian mathematical texts had been published in the much earlier ABCs, in the works of S.Daukantas, A.Juška, and other authors. There were the notions of numbers and figures which differ from the nowadays used ones, as the notion of a number was adopted in 1906, and that of a figure in In 1886 P.N ris published an arithmetic textbook in Tilsit [3]. In 1897 Arithmetic by S.Skačkauskas was published in Chicago [4]. P.N ris is a pseudonym of mathematician, engineer, and successful businessman Petras Vileišis ( ) while S.Skačkauskas remains unknown. In the years of , mathematics and English teacher Anupras Karalius ( , arrested by NKVD in Biržai, his fate unknown) published Arithmetic for learning by correspondence. Jonas Jablonskis ( ) put a final touch to the development of arithmetic terms by translating a Russian arithmetic textbook by A.Kiseliov and publishing it in Voronezh in 1917 [6]. The tables beneath illustrate the terms used by different authors which are compared with the terms given in Dictionary of Mathematical Terms (DMT) Term J.Gailutis (J.G.) terms, Table 1. Number and operation terms P. N ris S. Skačkauskas A.Karalius (P.N.) (A.K.) terms, (S.S.) terms, terms, J.Jablonskis (J.J.) terms, DMT number skaitlius skaitlius skaitlius numeris skaičius skaičius figure numeris rokuotin skaitlin skaitlin skaitmuo skaitmuo null niekis nulius ritin lis nolius nulius nulis (zero) (zero) unit vienatis vienyb vienut vienut vienetas vienetas fraction nuotrupa dalinis trupinys smulkmena trupmena trupmena skaitlius numeratodis rokuotojas skaitliaro- nominatorius skaitiklis skaitiklis denominatonitorius žim tojas pažym ti- denomina- vardiklis vardiklis operation veikm veikalas pradarbis veikm veiksmas veiksmas sum suma suma suma suma suma suma difference likius liekana atskirm liekana liekana skirtumas skirtumas product padaras vaisius sandaugm vaisius sandauga sandauga 212

213 This table containing several terms shows their different usage except the term sum. The description of operations differed, too. For example, subtraction = was written by J.G. in atimti 268 nuo 523; P.N. in 523 minus 268; S.S in iš 523 atimdami 268, gauname; A.K. in iš 523 atimti 268; J.J. in atimti 268 iš 523. At the end of his book J. Gailutis gave a list of terms calling it Some unusual words. In fact, it is the first Lithuanian mathematical terms trilingual vocabulary, as the meaning of a Lithuanian term was explained by means of Polish and German terms, e.g. fraction nuotrupos, Brüche, ułamki; unit vienatis, Einheit, jedność, etc. Only about 20 terms coined by J.Gailutis have survived. Even fewer terms, i.e. four or five, have reached our days from P.N ris glossary. The book by P.Neris was the first arithmetic textbook which presented theory. For that reason the author had to coin the words used in the theory of mathematics, such as definition, problem, excercise, conclusion, example, solution, etc. None of them was adopted, while the terms suggested by J.Jablonskis had the other fate and almost all, except some five-six of them, are used today. In the period of , several arithmetic textbooks were published, one among them is to be mentioned. Rev. Silvestras Gimžauskas ( ) published a book in Tilsit in 1888, however its title page carried 1862 as a publishing date. It was done deliberately for misinforming tzarist officials because it was the time of Press Prohibition in Lithuania (Lithuanian press prohibition started in 1864 and lasted for 40 years). In that book 19 pages were devoted to arithmetic. The above mentioned authors developed the terms linking them with spoken Lithuanian, while S.Gimžauskas used very strange terms which could not be related to any language and they seem to have had no influence on the terminology of mathematics. History of Lithuanian geometrical terms Consistent development of geometrical terms started in 1900 with the publication of the first Lithuanian textbook of geometry by P.N ris [8]. Prior to it some geometrical terms appeared in arithmetic textbooks. J.Gailutis used to write such words as cube, surface (meaning area), point. P.N ris used a pimple for a point. 213

214 Bishop and poet Antanas Baranauskas ( ) was also interested in mathematics. Prelate, Lithuanian University Doctor of Honour in Mathematics Adomas Jakštas-Aleksandras Dambrauskas ( ) had suggested to A.Baranauskas to coin Lithuanian geometry terms. The latter in his Polish letter to A.Jakštas, dating , said that in his Lithuanian correspondence with German H.Veber he had used some geometry terms. That letter revealed that A.Baranauskas had used the word rim for circle, wheel for disk, spoke for radius. That is the background of Lithuanian geometry terms. P. N ris textbook was the only Lithuanian teaching and learning source of geometry for almost two decades. It was in 1916 that A. Jakštas translated from Russian P.Mironov s textbook of geometry and complimented it with his own appendix [9]. In its preface he said he had trouble with the terminology of geometry as he could not accept all the terms suggested by P. N ris because, according to A.Jakštas, some of them were translated loanwords, the others- unskilful Lithuanian coinage. At the end of his book A.Jakštas gave a short bi-lingual glossary in geometry. For comparison, some of these terms are given in Table 2. Term Table 2. Geometry terms P.N ris A.Jakštas (P.N.), (A.J.), DMT volume ruima talpa tūris point taškas punktas taškas plane plokšt plokštis plokštuma calliper cirkelis skriestuvas skriestuvas angle kertis kampas kampas other side kat tas kat tas statinis hypotenuse gypotenuza hipotenūza įžambin circle ratlankis ratlankis apskritimas disk ratas ratas skritulys radius spinduolys stipinas spindulys diameter ratkirtis skersinis skersmuo cylinder v lan lis rulys ritinys sphere kamuolys skitulys rutulys P.N ris used some international words: pyramid, catete, hypotenuse, while other terms were taken from spoken Lithuanian e.g. for cylinder he used little shaft, for sphere he used ball.a. Jakštas preferred to use international words. He found suitable such words as: paralellogram, perpendicularity, bisectrix. Later neologisms were coined to all those words. Today some of P.N ris words are incomprehensible, however, both authors had put great efforts to maintain wholeness and continuity of the 214

215 terms. The names of triangles which were derivatives of the names of their angles may be taken as an example. Standardization of Lithuanian mathematical terms Valuable contribution to the terminology of mathematics made by the authors of the first mathematical textbooks is evident when comparing or collating the present terms with the first ones. However, analysing the first term development period, it is obvious that promiscuity dominated among the terms as each author was frequently seeking to use his own coined word for it. It had an importand bearing on taking to standardization of terms. When the First World War broke out and schoolchildren were evacuated far into Russia, to Voronezh, the Lithuanian gymnasia for refugee boys and girls were established there in In those schools prominent pedagogues, mathematicians Pranas Mašiotas ( ), Marcelinas Šikšnys ( ), Zigmas Žemaitis ( ), were working and they devoted much time and work to the development of the terminology of mathematics. During the Inter-War period they all occupied high positions. P.Mašiotas was Vice Minister of Education, Headmaster,and made his mark as an author of books for children. M.Šikšnys was Headmaster of Lithuanian gymnasium in Vilnius occupied by the Poland. Z.Žemaitis worked as Dean of the faculty of Mathematics and Natural Sciences in Kaunas University, later he became its Rector and headed the Chair of Mathematical Analysis. Coming back to Voronezh, the then prominent Lithuanian linguist Jonas Jablonskis was teaching Lithuanian and Latin there, and he agreed to join the mathematicians who turned their efforts to develop the terminology. They all agreed on selection of acceptable terms and development of the missing ones. Z.Žemaitis wrote [10] that J.Jablonskis had fixed the time for them to come to him for discussions on terminology. That was the second period in the history of mathematical terminology. The discussions, as described by Z.Žemaitis, proceeded in the following way: mathematicians explained to J.Jablonskis the essence of the notion characterized by a new term, its origin and usage in other languages. J. Jablonskis either approved the terms or improved them. In those discussions the Lithuanian terms for number, unit, addition, subtraction, multiplication, division, term, nominator, denominator, product, circle, radius, plane and many others were coined and are still in use without any changes today. In developing the terms of trigonometry it was agreed to take international words as a basis and merely to add Lithuanian endings to them. Those discussions resulted in the dictionary of arithmetic and algebra terms edited by M.Šikšnys, published in 1919 [11], and the dictionary of 215

216 geometry and trigonometry terms edited by Z.Žemaitis, published in 1920[12]. The first one contained about 250 arithmetic and algebra terms, the second about 700 terms one third of whose were arithmetic and algebra terms which, according to the title, should not have been included. Both dictionaries consisted of two parts: Lithuanian Russian and Russian Lithuanian. Thus in mathematical terms two trends were formed: voronezh type with its prominent representatives M.Šikšnys and Z.Žemaitis, and vilniustype with its distinguished representative A.Jakštas who was the fiercest critic of voronezh-types. He published two articles in the magazine Draugija [13], [14], whose editor-in chief he was, in which he attacked M.Šikšnys and Z.Žemaitis because of their neologisms, especially those replacing the already used international words. Much to the surprise of nowadays schoolchildren and students would be that A.Jakštas had rejected the words of all four arithmetic operations, the term used for a fraction, he fiercely criticized neologisms which replaced such international words as parallelogram, diagonal, perpendicular, other side, hypotenuse, etc. A.Jakštas persistently stuck to his terms, and in his book published in 1931 [15] old vilnius-type terms were decisively used. A.Jakštas was a fierce polemisicist but his dispute was overdue. New terms were soon transferred from glossaries to textbooks and were universally adopted. The third period of the mathemetical terms development covering the years of was not so turbulent and full of disputes. Most of the authors were keeping to standardized terms. During the Inter-War period about thirty authors wrote and published over a hundred mathematical textbooks, among them a few coursebooks of higher mathematics. In all of them the terms of elementary mathematics were being improved and perfected, the terms of higher mathematics were being coined. They laid the foundations for the present Lithuanian terminology of mathematics. During the Post War period the development of Lithuanian terminology was debased due to the lack of original textbooks, none of them intended for universities were published in Lithuanian. Nevertheless, much work was done in accomplishing the terms of differential and integral calculation, analytic geometry, coining the terms for the theory of probabilities, mathematical statistics, theory of sets and other up-to-date branches of mathematics. 216 References 1. J.Gailutis. Užduotinas, tai ira Rankius užduocziu Aritmetikos arba Rokundos mokslo. Tilž, V.Biržiška. Aleksandrynas. Lietuvių rašytojai, 2 tomas. Čikaga, P.N ris. Keturi svarbiausieji veikalai Aritmetikos. Tilž, 1886.

217 4. S.Skaczkauskas. Aritmetika. Chicago, A.Karalius. Aritmetika. Lietuvių korespondencijin mokykla. Chicago, Aritmetika. Mokslo pradžia ir terminai (mokytojo reikalui). Voronežas, S.Gimžauskas. Pradžomokslis Lietuwiszko spaudraszczo ir rankraszczo. Warszowoje, P.N.Trumpa. G om trija. Tilž, P.M.Mironovas. Pradedamasis geometrijos vadov lis. Vilnius, Z.Žemaitis. Lietuviškos matematin s terminologijos istorijai. Lietuvių kalbotyros klausimai. 1966, 8, M.Šikšnys. Aritmetikos ir algebros terminų žodyn lis. Vilnius, Z.Žemaitis. Geometrijos ir trigonometrijos terminų rinkin lis. Kaunas, A. Jakštas. Mūsų techniškieji žodyn liai. Draugija. 1920,1-2, ( ), (59-66). 14. A.Jakštas. Z.Žemaitis. Geometrijos ir trigonometrijos terminų rinkin lis. Kaunas, 1920, 99 p. Draugija. 1920, 9-10 ( ), ( ). 15. A.Jakštas. Meno kūrybos problemos. Kaunas,

218 218 ON UNDERSTANDING PRINTED TEXTS IN THE STUDY OF MATHEMATICS Kaarin Riives-Kaagjärv, Estonian University of Life Sciences, Abstract. A positive experience relating to the development of the skill of understanding and meaningful use of a written text when learning classical higher mathematics is touched upon. Keywords: mathematical text: meaningful use of, understanding of. With surprising frequency in recent times, I have witnessed a problem which had remained hitherto almost unnoticed: there are university students who are unable to read information from existing printed texts. The question does not only pertain to mathematical texts, but seems to be a wider problem. In a particular case, a student had evidently failed to read an information sheet with sufficient attention in order to have learned all that was necessary to know about the arrangements for doing a task, or maybe he had not even seen the notice and acted according to oral information which had reached him. However, as one can gather from newsreports from time to time, such a regrettable situation is fairly widespread already in schools of general education, where many pupils do not acquire knowledge so much from textbooks as from what the teacher has said. They memorise and reproduce the material spoken by the teacher. Coming to mathematics, in situations described above, it is expected that typical exercises are solved during contact hours with the teacher, and the assignment and examination problems are dealt with on that basis. This approach is unfortunately also fostered in effect by the system of our state examinations because every school and every teacher is interested in the results turning out as good as possible. This would determine how the pupil could continue his or her education as well as subsequent life. Many problems are caused by the public ordering of schools according to success of their pupils in state examinations. True enough, education leaders on the highest levels have promised to abandon this practice, but the actual life has not changed and some schools continue to be overburdened by pupils while others can scarcely muster their contingents and have certainly to do with less able or less studious learners. This only intensifies the existing disproportionate picture. The way of teaching school mathematics based predominantly on the solving of model exercises leads the pupils to think that theoretical

219 aspects are of no particular importance. As teachers, we have all heard the statement: let me first get the problems solved, then I will study up the theory. In my work of teaching classical higher mathematics to engineering students, I have tried to impress on them just the opposite, namely, that without knowledge of theory it is impossible to solve problems, neither by hand nor by computer. The latter method would of course demand an accurate input of data and a substantial interpretation of the solution. This, in turn, requires comprehension of the concepts involved and often also of their properties and certainly their possibilities of application. If we are to agree to the idea that the role of theory is trifling in teaching mathematics, we would thereby fail to implement an important aspect of the very point of educating a young person mathematically. We would thus neglect the opportunity to mould the learner into a logical thinker, one who would be able to assess his or her chances and to behave optimally and ethically within their bounds. The development of the latter aspect is assisted, first and foremost, apart from one s family, by study courses in humanities and by the general social mentality. As it is not within the power of an individual to change the main features of the situation described above, I should like to touch upon a positive experience relating to the development of the skill to use, to understand and to interpret mathematical material in the course of mastering mathematical facts and their applications, whether the material is on paper or in Internet. The latter opportunity has created an additional problem. As it happens, there are materials circulating in Internet compiled by the students themselves, and they contain misleading errors as well as symbols handy in computer work but unused in traditional texts. To use these materials uncritically for the study towards examinations and for formalising examination answers would not lead to expected results. While the compiler of these materials may be able himself to explain the meaning of the symbols, it is a matter of experience that even those people who have solved practice problems on a reasonable level and who use extraneous materials in good faith to study theory, can almost never explain what the symbols they have written down denote and how these would look in traditional script. On the one hand, such a situation may point to dishonesty and to the use of a crib. On the other hand, it may mean that the written material has been mechanically memorised and completely ununderstood. Such a state at any stage of the educational process is to be condemned, and ought certainly not to be found in a university. It would contradict the very concept of higher education. Effort is needed for its prevention, but unfortunately it is difficult and often impossible to replace a mechanical study style that has developed in the course of time, and change it to a 219

220 meaningful approach. In many instances, the problem lies in the offhand attitude of young people towards study in general, and towards study of mathematics in particular. At the same time it is precisely the study of mathematics which provides one of the best opportunities to develop the skill of meaningful work for the solving of existing problems with the use of suitable helps. I can point to productive experience in the area of development of the skill of meaningful use of a written text. Work with students over a span of three semesters has led to notable results. If the method were used over a shorter time period, it would become difficult to assess whether a student had already previously learnt to utilise prepared texts meaningfully or he has gained this skill first in the course of our joint work. The availability of materials in Internet leads many university students to believe that lecture attendance is unimportant. Thus they deprive themselves of the chance to follow the logical trail of thought from setting up the problem through the introduction of relevant concepts to final arriving at the solution. As a result they have no model on which to support their own advance. In order, nevertheless, to give all students an opportunity to work through the relevant theoretical materials during practice periods in the course of solving practical problems, I have compiled a brief theoretical summary for each topic dealt with during the practice periods. They contain main definitions, properties of concepts, some schemes of solving multistage problems. These sheets are duplicated and laid on the desks of all participants and are also projected on the wall of the room. In this way, everybody could participate in working out the solutions of the problems given, ask relevant questions and search answers from the existing materials. Unfortunately, it has been impossible to engage all the students in this joint mental effort. Some think that when they have copied down all the solutions from the chalkboard without understanding the thought process, it suffices for subsequent successful work. If they would ever reach the stage of presenting themselves at an examination, their answers would betray that a great amount of effort had been expended to learn the material by heart while there is a complete lack of one s own logical thinking and comprehension. One can only sympathize with them because their further studies would not be easy and it would perhaps be difficult for them to work in positions of employment that demand creativity. Those who have successfully passed the course have testified that working through proposed summaries of theory have notably assisted in essential comprehension of the subject. 220

221 STUDY METHODICAL COMPLEX FOR MATHEMATICAL EQUALIZATION COURSE FOR THE FIRST YEARS STUDENTS Tatyana Shamshina, Ilona Zasimchuk, Transport and Telecommunication Institute, Today we have a situation, when our teacher s experience and the results of mathematical testing showed quite a low level of knowledge in mathematics. Besides traditional lecture in the process of higher education used to occupy one of the central positions and ut ot a present moemnt lecture courses constituted about 50% of studying time. Such system was characterized by clear-cut lectures when a lecturer simultaneously played a role both of organizer of a studying process and of a cordinator of student s activity. In this connection, it is necessary to organize additional classes on math for the first year students to help them to acquire the course of Higher mathematics in technical or economical institute. Transport and Telecommunication Institute of Riga (TTI) offers to include in study methodical complex (SMC) new instrument, which is called Electronic Navigator (EN). Especially famous is the usage of EN as a facility organization of students self preparation on equalization courses (or at home) on the most difficult chapter of higher mathematics, such as, differential and integral calculus. But at the beginning we must consider the problem of study methodical complex in connection with some common modern tendencies in teaching math. They are the following: 1) information technologies more and more activity are included in all spheres of education; 2) each year a number of contact hours in the course of Higher mathematics is limited; 3) and inversetly, a number of hours for individual study is added; 4) to our great regret students level knowledge in elementary math is quite law; 5) students inability to organize a sufficient self preparation is a serious problem for normal studies; 6) the necessity to supplement traditional studying of Higher Mathematics with different courses for equalization of students knowledge in this subject. 221

222 As you can see, some of these tendencies in teaching math are good, but some of them are bad. In this connection, modern study methodical complex on mathematics must include not only traditional normative documentation (study program, estimation criteria, testing system, etc.), but also some new study tools, for example, such as EN. (picture 1). The example of such SMC creation you can find on address: Later in more detail we will consider EN as additional instrument, which we offer to include in study methodical complex of math. Materials included in EN can be divided into two groups: 1) study-theoretical materials (basic notes for the lectures including necessary schemes and pictures); 2) study practical materials (set of tasks to be solved at practical lessons, texts of current and test home tasks, training patterns of individual works, demo version of final test, as well as questions for the test on each part of the course). Also EN has some special characteristics, such as: the simplicity and convenience of EN creation by means of standard programs MS Office (MS Power Point, HTML Help Workshop); EN is rather informative and organizational aid; EN is especially useful for the students with flexible scheme of attendance and for the part-time students. Where EN can be used? It can be used on the following lessons and courses: 1) practical lessons in a group with a teacher and individually; 2) preparational courses on elementary math; [1] 3) courses aimed at solving typical tasks from home assignment; [2] 4) equalization courses on differential and integral calculation [3], because this part of discipline Higher Mathematics is more difficult for a lot of first year students. At our institute the equalization courses is more popular among students. EN, included in SMC, for mathematical equalization courses performs the following main tasks: to organize the process of mathematical preparation and selfpreparation; to teach students necessary skills in taking notes; to form thinking ability of students in the given direction; to make for thoughtful and productive approach in solving typical tasks; 222

223 to provide optimal preparation for passing a compulsory minimum on math. Full schema of new material with EN help is presented on Fig. 1. Theme 1 Theory Practice Knowledge for control Qestione 1 (date 1) Compulsory tasks for solving Home tasks Questions for self - testing Question 2 (date 2) Date 1 Date 1 Testing (0 variant) Question 3 (date 3) Date 2 Date 2 Examination (date) Fig. 1. Study Shema of New Material with EN help. You can see, that it is really good and simple organizer for students and teachers, which is able to answer on the set of the questions What? Where? And When? In conclusion we would like to note more positive and perspective sides of EN: 1) EN makes for clear understanding of course structure; 2) EN reflect the demands for students skills and abilities; 3) EN contains all tasks necessary for successful passing the test; 4) EN creates visible perspective of studying mathematics. All these make above mentioned factors SMC supplemented with EN is an integrated system of teaching math and makes for optimal organization of study process. 223

224 References 1. Shamshina T., Zasimchuc I. TTI Experience in Teaching of Elementary Mathematics Through the Systematization of Its Sections. Материалы межвузовской научно-практической конференции проф.-препод. состава ноября 2004 года, Мурманск: МГИ, 2004, с Shamshina T. Intranet Application in the Course of Higher Mathematics. Abstracts of Course of Higher Mathematics, VI International Conference: Teaching Mathematics: retrospective and perspectives, May 2005, Vilnius, p Gorbinko A. Application of the Electronic Navigator as a Means of Organizing Students Self-Training Based on Course Fundamentals of Higher Mathematics for the First Course Students of MHI Economic Faculty. Starptautiskā zinātniski praktiskā un mācību metodiskā konference Mūsdienu izglītības problēmas, gada februārī. Programma un tēzes. Rīga: TSI, 59.lpp. 224

225 MATHEMATICS TEACHING INNOVATION IN TECHNOLOGICAL EDUCATIONS Jaak Sikk, Tallinn University of Technology, Abstract. The rapid technological development is changing society and its attitudes towards education. This process is causing the urgent need to change the education environment, both at school and university. There is a gap between mathematics offered by technical universities and mathematics needed to educate a modern engineer. The shift of the teaching-learning paradigm is a necessity. Our challenge is to find synthesis of different attitudes to teaching mathematics and to move away from mathematics as an isolated subject. The dialogue between mathematicians and engineers is needed. In order to save all the rational in teaching mathematics we must investigate mathematical culture of engineering. We also know that in increasingly more countries the descent in the mathematical ability of new entrants to the university degree programmes is a major problem. A dialogue between mathematics teachers at school and university is needed. The study of educational innovations must lead to new didactics. Keywords: engineering education, horizontal mathematization, learning environment, syllabus, realistic mathematics education, vertical mathematization. Mathematics teaching and core mathematics for engineering The aim of current mathematical education at a technical university is not an efficient mathematical practice to support a technological innovation; rather, it is concerned with the transmission of basic ideas of existing traditional mathematical culture, considered by mathematicians as crucial. The value of such traditional approach to teaching of mathematics is more social-cultural than practical. Like all other social values, the value of a phenomenon named mathematical culture has a stable core, and this core is considered by mathematicians as a necessary instrument for interpretation of the surrounding world. It seems to be acute to reconsider this viewpoint. Any calculus course and teaching process must reflect the needs of new engineering disciplines and new technologies. An important social value for the community of engineers is an engineering mathematics basis certain amount of core mathematical knowledge and ideas considered by the engineering community as natural. Under the influence of constantly modernizing technologies the subject domain of this basis has been in a constant change. There exists a 225

226 difference between the mathematics taught at a technical university and mathematics needed by engineers in their work. According to the SEFI Mathematics Working Group [2] the mathematical topics of particular importance include: fluency and confidence with numbers; fluency and confidence with algebra; knowledge of trigonometric functions; understanding of basic calculus and its application to real-world situations; proficiency with the collection, management and interpretation of data. The SEFI Mathematics Working Group also stresses the importance of using the elements of mathematical modelling in calculus: 1. It is important that the exposition of the modelling process should be introduced as early in the curriculum as is reasonable. 2. The first models that are presented should be simple, so that the process is not obscured by the complexity of the problem, by concepts in engineering not yet encountered and by notation with which the student is unfamiliar. 3. The mathematics used in the first models should be straightforward. 4. The models must be realistic. The physical situation should be one to which the model has been applied in practice. It is of little value to apply a mathematical model to a situation to which it has never been applied, simply to make a pedagogic point. By using perfect teaching methods and having students with excellent mathematical background it would be possible to teach adequately all topics proposed by the SEFI core mathematics program. In theses circumstances the result would give students a good knowledge of traditional mathematical culture as well as understanding of mathematics, needed by engineers. Requirements for achieving these results are the perfect calculus teaching for excellent students. But the reality is somewhat different. A decline in the mathematical ability of new entrants to the university degree programmes is well-known in Europe. For instance, according to L. Mustoe [5], The decline of mathematical ability of undergraduate entrants to undergraduate engineering courses in the United Kingdom has been well documented. According to a report of the Swedish National Agency for Higher Education, the decline in the mathematical ability is an international problem. A group of Russian mathematicians have also pointed to the dangerous tendency of declining knowledge of mathematics at school [6]: today we are painfully aware of the deterioration of mathematical education in our society and decline of its mathematical culture Within the educational system itself it is mathematics that appears to be in an unfavorable situation as an academic subject which is not consistent with market economy. Global computerization doesn't diminish the role of mathematical education, but quite the reverse, it has set new aims to the science educational system. Further deterioration of mathematical literacy 226

227 and mathematical culture may turn a man from a master of computer into a slave of it. Clearly, it would be important to have the new students with better mathematical literacy. Working with newcomers for years, staff of mathematics at university has become confident about developments at school, especially about positive and negative tendencies in mathematics teaching. It would be productive to execute real cooperation between university lecturers and mathematics teachers at school. Of course, it would be impossible to start again to teach school mathematics in a way, as it was done earlier. But it seems reasonable to make a concise re-study of mathematics teaching methodology at school during the 1950-ies and 1960-ies. Theoretical bases of curriculum development process During a long period I have worked at Estonian Agricultural University (EAU). In EAU I started to investigate the theoretical aspects of an engineering calculus syllabus designing process (see [6-8]). The present paper is a continuation of this work. There is a need to investigate the effect of the use of computers in mathematics teaching. We are aware, that achieving positive results by teaching mathematics in network environments requires much more than simply providing access to hardware, software and net. The computer packages typically used in the teaching process are designed for professional use rather than for learning. It would be better to use packages explicitly designed for teaching. There is also a danger that the educational environment will become shaped by the technology, and enhancing student mathematics learning will become a secondary consideration. Our idea is to activate students by making the learning process more efficient. One of the purposes of our research is to reorganize the teaching process of calculus, changing also the substance of the subject. We are aware that using mathematical models in a computerized environment would help students to gain deeper understanding about mathematics. As a base of our considerations we follow some modern ideas of mathematics didactic theory. What do we want students to understand? There exist many understandings of mathematics: 1. Mathematics as a subject of study is considered as an obligatory part of the degree programme, to be studied via various teaching and learning techniques. 2. Mathematics as the basis for other subjects, both for study and in the world at large, is considered as something existing in its own right, something to be tackled (learned or understood) for future appropriate use. 227

228 3. Mathematics as a tool for analyzing problems that occur in the world at large and hence solving them is considered as something which co-exists with other areas of knowledge and supports the study and development of that knowledge. Our challenge is to find a synthesis of all these viewpoints and to develop a common approach, satisfying general aspects and needs for both mathematicians and engineers. Mathematics must be integrated into the curriculum of study and into the world it describes. To start this process, the dialogue between mathematicians and engineers is needed. In order to save all rational what we have in our mathematics syllabus, we must investigate, determine and estimate the existing mathematical culture of engineering. According to J.L.Schwartz the mathematics educators goal is to make mathematics a real part of the cultural heritage of humanity. She adds: In contrast to the community of mathematics educators, in many ways, the community of mathematics-using disciplines has already embraced the opportunities offered by the new technologies. It has done this by changing the ways in which mathematics is used within the various disciplines, often without waiting for the mathematics community to sanction the changes. There are already universities in United States which have abolished their mathematics departments and where the responsibility for mathematics instruction is being assumed by engineers and economists, physicists and physiologists. The teaching of mathematics has always been dependent upon technical facilities available for computation. A technical development of computation had caused a change of mathematics teaching methods and subject of mathematics. As a consequence, a certain evolutionary process had lead to an equilibrium situation between concepts of mathematics teaching and computation. Rapid advances in computation, linked with a development of mathematical software, have already started to destroy the existing equilibrium. The changes in computer technology make it impossible to estimate the sophistication of facilities which will be available to engineering students after five years. This phenomenon is playing a major role as a motivator in a calculus syllabus redesign. Having in mind a calculus renewal process by using modern mathematical software we must have answers to the following questions (see [5], p.7): What do we want students to understand? What do we want students to do with their understanding? What is the purpose of teaching? What are the goals of lectures for the students? What are the goals of the students? 228

229 One of the cornerstones of our research of the syllabus design is Realistic Mathematics Education (RME) (see [1,3]). The RME is a domain-specific instruction theory of mathematical education for secondary school students. It appears that it is possible to use the same ideas for calculus redesign. We will use in our investigation some basic standing points of the RME by using the work of Drijvers [1] and Freudenthal [3]. The Realistic Mathematics Education (RME) is a domain-specific instruction theory for mathematical education. One of the basic concepts of RME is an idea of mathematics as a human activity. This organizing activity is called in RMA mathematization. According to Freudenthal, it can improve the learning of mathematics, and mathematization is the core goal of mathematics education. Usually two types of mathematization are distinguished: horizontal mathematization and vertical mathematization. A second important characteristic of RME is a level principle. Students pass through different levels of understanding on which mathematization can take place. Within RME the term model is not taken in a very strict way, as in theory of mathematical models. It means that materials, visual sketches, situations, schemes, diagrams and even symbols can serve as models. These models must have the following properties: they have to be rooted in realistic, imaginable contexts, and they must be sufficiently flexible to be applied also on a more advanced, more general level. One can add the principle of anthropological approach in didactics to the principles of Realistic Mathematics Education. The anthropological approach and socio-cultural approache in the educational field share the vision that mathematics is a product of human activity. Mathematical production and thinking modes are thus seen as dependent on the social and cultural contexts where they develop. As a consequence, mathematical objects are not absolute objects, but are entities that arise from the practice of given institutions. Theory uses terms as pragmatic value, epistemic value, the routinisation of techniques etc. For obvious reasons of efficiency, the advance of knowledge requires the routinisation of some techniques. This routinisation is accompanied by a weakening of the associated theoretical discourse and by a naturalisation of associated knowledge which tends to become transparent, to be considered as natural. A technique that has become routine becomes de-mathematizised for the institution. It is important to be aware of this naturalisation process, because through this process techniques lose their nobility and become simple acts. The SEFI Core Curriculum [7] with its hierarchically structured and detailed list of topics provides us an excellent research material. By using principles of horizontal and vertical mathematization we can make 229

230 didactical considerations for calculus teaching. In Core Curriculum each of its three lower levels (0, 1; 2) are divided into many component parts in which the main topics and subtopics lie. Levels 0, 1, 2 are representing hierarchical progression from school towards to the first two stages of university education. This structure of knowledge, with its levels and sublevels, gives to us a possibility to analyze all topics of calculus by paradigm of mathematization theory. By this analysis we can determine how to approach different parts of calculus. The mathematical operations from lower level would be considered as purely algorithmic and executed by students by using the computers and software when used at higher level. The time saved would enable us to shift students attention to realistic problems related to the material considered, and to uses of simple mathematical models. Clearly, in these circumstances the mathematics software would support the creative development of the student. References 1. Drijvers P. What issues do we need to know more about: Questions for future educational research concerning CAS. The state of Computer algebra in Mathematics Education, Bromley, Chartwell-Brat, SEFI Mathematics Working Group Freudenthal H. Revisiting Mathematics Education. China Lectures, Kluwer Academic Publishers, Dordrecht, The Netherlands. 4. Lawson D. Notes on Round Table Discussions. 12 th SEFI Maths Working Group Seminar, Proceedings, Vienna University of Technology, 2004, pp Mustoe L. The Future of Mathematics in the United Kingdom. 12 th SEFI Maths Working Group Seminar, Proceedings, Vienna University of Technology, 2004, pp Sikk J. Calculus for Engineers: A conceptual Modelling Paradigm. 12 th SEFI Maths Working Group Seminar, Proceedings, Vienna University of Technology, 2004, pp Sikk J. Mathematics curriculum development generated by technologies. Proceedings of International conference Teaching Mathematics: Retrospective and Perspectives, Liepājas Pedagoăijas akadēmija, Liepāja, 2005, pp Sikk J. The development of the mathematics teaching environment through the networking paradigm. Proceedings of the Fourth Nordic-Baltic Agrometrics Conference, Ulf Olsson and Jaak Sikk (Editors), Uppsala, SLU, Department of Biometry and Informatics, Report 81, 2003, pp

231 NEW ENTRANCE MATHEMATICS TESTS AT ELS Jaak Sikk, Eve Aruvee, Estonian University of Life Sciences, Abstract. The correlations between students successes on state examinations, entrance tests and during the studies at agricultural university is investigated. Keywords: state exams, entrance tests, mathematics and statistics at university, correlation. Introduction Our chair of mathematics has had a long standing cooperation in mathematics and statistics with mathematicians of agricultural universities of the Baltic and Nordic countries. We have had five joint Nordic-Baltic agrometrics conferences. The term Agrometrics is used as a summary for mathematics and statistics in the agricultural sciences and education. One of the results of our cooperation is a common diagnostic test in mathematics for agricultural students. The test contains 15 elementary tasks of secondary school mathematics. The students had to find the correct answers from among many alternatives. The test was given in Sweden, Estonia, Latvia and Lithuania at the fall of The figure below shows a percentage of correct answers by different countries. The results of almost all countries are quite similar, especially those of Estonia and Lithuania. It is understandable as our backgrounds, basic teaching methods and ideas are similar. Only Sweden s results are a little bit different Sweden Lithaunia Estonia Fig. 1. The test results by different countries 231

232 Analysis In the present analysis we compare the test results by years for specialties of Land Engineering faculty. The specialties of Land Engineering faculty are: Real Estate Planning and Evaluation (kv), Geomatics (mm), Water Management (ve), Agricultural Buildings (eh). They are always given the same test. To enter the university there is a necessary condition for engineering specialty students to have a state exam in mathematics. We see (Table 1) that in year 2000 the average of all test scores was the lowest. The highest result on average was in the year 2003, at the same time the average of the examination mark on the state was the lowest. Evidently in year 2003 the tasks of state exam have been very difficult. In year 2000 nobody got the maximum (15) points, but in year 2003 even 4 students got the maximum of points and 3 students got 14 points. In each of years 2002, 2004 and students got the maximum of points. Table 1. Results of test and average of state exam in different years Tests Number Average Variance State mark test , , ,40 test , , ,25 test ,93 10, ,06 test , , ,11 test , , ,40 Which topics from school mathematics had caused more problems for new students? Analyzing the test results by task we see (Fig. 2) that in all years the results were very close, there was not any significant difference between them. In the test there were always 15 questions on the topics: 1 operations with fractions; 2 simplify; 3 general square root function; 4 raising to a power; 5 logarithmic function; 6 root of an equation; 7 solution of inequality; 8 solution of quadratic equation; 9 trigonometry; 10 equation of a line; 11 - square root; 12 - solution of inequality with an evaluation; 13 percentage; 14 simplify; 15 Pythagorean theorem. By tests it seems that the level of knowledge is quite the same during last five years. 232

233 1,00 0,80 0,60 0,40 0,20 0, Fig.2. Correct answers percentage by task in years 2000, 2002, 2003, 2004 and 2005 The Pythagorean theorem was the best answered task, good results were achieved also in tasks 1, 3, 6 and 12. More than half of the students did not find the right answers for the tasks 5, 7, 8, 9, 11 and 14. Trigonometry was a big problem: students were not able to decide which angle is positive. They also could not extract the root, could not use auxiliary formulas in simplification tasks, did not know general formula for quadratic equation. Here are the texts of the most difficult tasks 7, 9 and Which of the following assertions are correct? A. 0, = 1 6 B. 0, > 0, D. 0, > E. None of the above C Which of the following values is positive? 3π A. cos( 4 ) D. tan( 3π 4 ) 3π B. sin( 4 ) C. sin( 3π 4 ) E. None of the above ( a ) = A. 5 + a B. -5 +a C. 5 a D. -5 a E. None of the above The students who started university studies in year 2003, completed the learning of mathematics (8 points) and statistics (3 points) in spring of During their studies the students of engineering specialties must pass three exams in mathematics and one exam in statistics. As we can see from 233

234 Table 2 the average marks of state exam were in all specialties close to test and exam marks at university. The Geomatics (mm) specialty students had passed the tests on a lower level. They also passed through exams in mathematics and statistics worse than other students. Table 2. Average of different marks Statemark statistics specialty m1 m2 m3 test kv 2,3 3,4 3,4 3,8 4,2 3,5 mm 2,4 1,7 2,1 3,0 3,7 3,0 ve 3,3 2,6 2,8 3,6 3,6 3,1 eh 2,7 3,3 3,3 3,8 3,5 3,1 What is the reason? Maybe students have low motivation to work or they have bad group climate. The students maybe were interrupted by adapting a new learning system; they started not to work regularly. It seems that all student who had problems with mathematics at school, have also difficulties to acquire the university mathematics course. Specialties Real Estate Planning (kv) and Agricultural Buildings (eh) have better results at the end of mathematics course than the other two specialties (Fig. 3). Specialty Geomatics (mm) had made all exams of averages different from other and that difference is statistically significant. 4,0 3,5 3,0 2,5 2,0 1,5 1,0 m1 m2 m3 test mark statistics kv mm ve eh Fig. 3. Average marks 4,0 3,5 3,0 2,5 2,0 1,5 1,0 Next we are going to study a correlative dependence between variables sex, state-exam mark, test mark and marks in mathematics and statistics. 234

235 Table 3. Correlation matrix sex m1 m2 m3 Test State-exam Statistics m m m Test State-exam Statistics Specialty From correlation matrix we find dependence between test and stateexam. Also dependence between test and mathematics marks and dependence between mathematics marks can be identified. You can see dependence between third mathematics mark and test result (Fig. 4) test mark m3 Fig. 4. Correlative dependence between third math mark and test mark We can compose the regression equations to predict different mathematics marks. The regression equation for first mathematics mark is m1 = sex test The variables sex and test describe about 32 % of the first math mark. We can predict math mark when test result is 10 and sex is female; then math mark is 2.4. If sex is male then math mark is 2.9. That means, boys got better marks than girls. The regression equation for second mathematics mark is m2 = m test The variables first math mark and test result describe about 51% of the second math mark. 235

236 We can predict second math mark when test result is 10 and first mark is 3; then the mark is 3.2. If test result is 14 then the mark is 4. The third math mark can be found from the regression equation m3 = m m2 The variables first and second math marks describe about 77% of the third mark. Summary The mathematics tests we had used since year 2000 show that certain topics of mathematics are difficult for students to acquire (tasks 7, 9, 11). These topics and other topics related to them are very important for future engineers. The cooperation is needed between technical university and secondary school institutions to improve teaching of these items which would be important later. References 1. Edlund T. Basic Mathematical Knowledge Possessed by our Students. Second Nordic-Baltic Argometrics Conference, Karaski, Estonia, September 23 25, Aruvee E. Correlative analysis of mathematics tests and examinations data at EAU. Third Nordic Baltic Argometrics Conference, Jelgava, Latvia, May 24-26,

237 METHODOLOGY FOR TEACHING POLYNOMIAL APPROXIMATION OF FUNCTIONS IN CAS ENVIROMENT Regina Dalia Sileikiene, Vilija Dabrisiene, Kaunas University of Technology, Abstract. The paper attempts to reveal the methodology of teaching function approximation in a computer class and indicates the advantages and disadvantages of this method in comparison with the traditional one. The objectives of the research are to indicate general didactic attitudes of the method suggested by the authors; to present several examples of the application of the suggested method; to analyze and discuss the results of empirical study (student survey) that reveal the efficiency of the proposed method. Keywords: approximation of functions, methodology of teaching mathematics in a computer class. A number of changes have taken place in recent years, which have profoundly affected and conditioned the teaching and learning of mathematics at the tertiary level. The International Commission on Mathematical Instruction indicates the increasing difference between secondary and tertiary education as one of the most significant as it includes different educational aims, didactic attitudes and assessment systems. Besides this, another tendency clearly demonstrates that with an increase of the total number of students, there are students with low basic knowledge as well as motivation for studying. The third change, influencing the teaching and learning of mathematics, is a rapid development of technologies, enabling to seek for new possibilities in the training process. Teaching methods of mathematics in a university tend to remain conservative [1]. The content is oriented towards learning the theory of mathematics with low emphasis on practical applications and modelling of the theory. Therefore, modern students consider such practice-adrift training content, conveyed only in verbal or written form, to be unattractive and useless. On the other hand, as it was already mentioned before, the basic student s knowledge and especially skills on performing standard mathematical operations are often poor. The problems described above were faced while teaching mathematics in the Faculty of Mechanical Engineering and Mechatronics in Kaunas University of Technology. During the last three years more than one third of students failed in the course of mathematics at the faculty. Therefore, a need 237

238 to develop such methods for teaching mathematics that would form a positive attitude of students to knowledge and studying process, would emphasise the importance and applicability of mathematical knowledge, would be based on active and independent student performance, appeared. The need to develop the methods, enumerated above, coincided with the establishment of computer classes for mathematical training in the University, where during workshops students could use software applications (for example, MathCad). Preparing teaching methods for the theory of function approximation in the computer lab, it was necessary to analyse both traditional teaching methods and possibilities provided by the computer. The aim of the paper is to reveal the methodology of teaching function approximation in a computer class and indicates the advantages and disadvantages of this method in comparison with the traditional one. The objectives of the research are: to indicate general didactic attitudes of the method suggested by the authors; to present several examples of the application of the suggested method; to analyze and discuss the results of empirical study (student survey) that reveal the efficiency of the proposed method. Didactic attitudes Planning. While preparing exercise copybooks, it was attempted to change a traditional linear model theory problem solving, used for teaching mathematics in a tertiary level, in such a way that the basic theoretical issues would be thoroughly analyzed and evaluated. It was decided to follow an assumption that the first stage of problem solving defined by comprehension of the task and planning how to solve the problem, requires skills on information analysis and search. However, during this stage a student superficially gets acquainted with the theory. The second stage, when the prepared plan for problem solution has to be applied, requires a lot of skills to perform standard mathematical operations. Here poor skills of students are outweighed by the software, but mistakes are still inevitable. When performing operations, drawing pictures by a computer, mistakes are noticed immediately and must be corrected at once. Thus, a student is forced to get a deeper knowledge of the theory. The third stage involves checking of the result obtained, analysis and conclusion making. The stage allows forming an overall approach and notice possibilities of theoretical knowledge application and, thus, becomes 238

239 extremely significant due to a possibility to reflect on theoretical knowledge once again and get convinced in the efficiency of the method applied. Applying the basis of the stages, described before, it was attempted to compile such tasks, where the activity of students would follow the scheme, presented below : Information search. Analysis of theoretical facts. Creation of a solution algorithm. Deeper theoretical analysis Computer-aided solution algorithm. Analysis and correction of the mistakes observed. Finding of the solution Solution analysis. Checking of theoretical facts. Analysis of method reliability. Fig. 1. Student s activity Organization. Classes and practical works are organized in a traditional way where a student has to prepare for practical work, to fill in his exercise book, record theoretical facts and formulas. Students are able to use their records during practical works that are organized in two ways. One part of practical works is devoted to a common activity with a help of a teacher while cooperating together and another part includes an individual practical work of a student when he/she receives individual tasks that are completed and carried out using personal records and experience, gained during common practical work. Evaluation. Students can get 4 points for practical works and 6 points for a test. The area of student s activity. Theoretical course and practical trainings in a computer class for students of all specialities in the Faculty of Mechanical Engineering and Mechatronics of Kaunas University of Technology are delivered. The subject in the third semester is called Applied mathematics. The area of student s activity is based on the scheme in Fig. 2. Examples on method application Seeking to demonstrate the methodology suggested several examples are presented that show how graphs could be used to illustrate problem solving and theoretical ideas as well. 239

240 Number series Power series, Taylor s and McLaurent series MathCad working environment: Calculus; Variables and functions; Operations with matrix and vectors; Symbolical calculus; Graphics. Fourier series Polynomial of Lagrange and Newton Non-linear approximation Solving differential equations applying Euler s method Solving equations of oscillation and thermal spread, using a grid method Fig. 2. The area of student s activity The first example presents a function, approximated by interpolative polynomials of Lagrange and Newton. The diagram allows a student to notice if all interpolation polynomials are recorded correctly. Making a view larger, students could observe interpolation formulas that approximate the function more precisely. While analyzing differences in the graphs of interpolation functions it is possible to illustrate theoretical statements on approximation errors (Fig. 3, Fig. 4). y 4 L1( x) N1( x) N2( x) X, x, x, x Fig. 3. Interpolation of a function 4 240

241 0.002 f ( x ) N1 ( x ) π 6 x 1 π f ( x ) N2 ( x ) π 6 Fig. 4. Indication of approximation errors graphically. x 1 π It is believed that it is rather important to formulate new knowledge while modelling and making experiments. This possibility could be supported by a simple example. Analyzing convergence of number series, a series 1 is used for comparison. After the introduction of the p n = 1 n definition on series convergence, it could be suggested to perform such a task, using a computer: Try to experiment while drawing the graphs of the series 1 of p n=1 n partial sums with various values of p. Try to indicate what values of p cause the convergence of the series. Carrying out the task, students quickly notice that if p < 1 and the number of terms increase the sum of the series increases infinitely. However, if p> 1 and the number of terms extremely increase, the sum of the series approximates some particular value (Fig. 5) n = 1 n n 1 n n = n 1 1 n 1 10 n = 1 n 2 n = 1 n 5 n n Fig. 5. Modelling tasks, allowing to form a hypothesis

242 The results of the method applied. Monitoring the work of students during computer-based workshops it became clear that the suggested method has justified itself. It was also proved by the results of students questionnaire survey (n=112). The view of students on the studying process appeared to be very positive. About 90 percent of the students stated that workshops organised in such a way were very interesting and useful, 72 percent considered that during computeraided workshops they learned much more than during usual workshops. In addition to this, about 96 percent of students thought that workshops, organised in such a way, promote thinking, encourage analysing processes and comparison making as well as conclusion drawing. The students stated that it was more interesting and easier to learn during computer aided workshops (about 90 percent), more is learned in a class and less has to be done at home (about 78 percent). The attitude of the students to the knowledge obtained was favourable as well. About 90 percent of the students thought that the understanding of function approximation would be useful for their further studies or work activities. About 50 percent of the students stated that even during the course they already applied the knowledge obtained while studying other subjects. Conclusions Computer-aided workshops help to form a positive attitude of students to the training process and the knowledge obtained as well as forms a deeper comprehension of theoretical models. Computer-aided workshops enable students, who possess weaker skills of mathematical operations, to assimilate a new course successfully, to understand theoretical models and to learn how to solve different problems practically. The usage of applied mathematical programs in teaching mathematics enables to prepare such methods that focus not on details but on a holistic approach to the subject taught, as well as flexible analysis and practical testing of theoretical models. References 1. On the Teaching and Learning of Mathematics at University Level. ICMI study Kidron I. Polynomial Approximation of Functions: Historical Perspective and New Tools

243 ON PROOFS IN TEACHING MATHEMATICS Eugenijus Stankus, Vilnius University, Abstract. The proofs of mathematical propositions are very different subject to students knowledge, to their ability, to definition of original concepts, to the moment (100 or 1000 years ago, or today) of lecture.the concept of mathematical proof is analyzed. Keywords: courses, examples, interpretations, mathematics, methods, proof, teaching. The role of proofs in mathematics is particularly important, so it is in teaching mathematics. Concept of proof in mathematics is exceptional from others. In the proof we operate with mathematical concepts and relations between them. A proof is a demonstration that, assuming certain axioms, some statement is necessarily true; proof is a logical argument (Wikipedia). But this definition of proof became indeterminate considering the growth of mathematics in twentieth century. In [2] Brian Davies wrote: we will argue that developments of the classical Greek view of mathematics do not adequately represent current trends in the subject. It proved remarkably successful for many centuries, but three crises in the twentieth century force us to reconsider the status of an increasing amount of mathematical research. The all three crises are pertinent to the concept of the proof in mathematics. First crisis (1930s) Kurt Gödel demonstrated that within any sufficiently rich axiomatic system there must exist certain statements that cannot be proved or disproved. [2]. Mathematicians and nonmathematicians assert that the proofs are too long and complex (second crisis, 1970s). Discussion about proof in mathematics The Nature of Mathematical Proof is inconclusive third crisis, 2004s (see, e.g., [1]). Deep comprehending of mathematical concepts is the first and main purpose of everybody studying and learning mathematics. The whole bag of tricks is good in order to achieve this purpose, for example, discussions on the historical or philosophical nature of concepts, geometrical interpretations, true-life examples, etc. [3]. The concept of proof is especially complicated. A student may wonder why obvious facts are being proved at the geometry lessons. Therefore the question emerges what needs to be proved for students and what doesn t. Of course, we could consider some propositions to be true without proof. However, there would be many of them, and the history of mathematics 243

244 shows that it s been endeavoured to have as few obvious (primary) concepts and propositions (axioms) as possible. On the other hand, a mathematical proposition usually has more than one way to prove it. There are at least 300 known proofs of Pythagorean Theorem. Teaching any mathematical course we have to decide which facts we are going to prove and what way of proof we choose. The following considerations come from experience in teaching the course Selected Chapters of Number Theory within master study programme Mathematics and Informatics teaching at Vilnius University. People have been interested in the distribution of prime numbers for many centuries till nowadays. Euclid in IV-III BC proved that the set of prime numbers is infinite. It is interesting that even contemporary methods and powerful computers are powerless to solve the problem of twine primes (the set of pairs of primes of form (p, p+2) is infinite). This shows the significance of mathematical proof. It is extremely important to demonstrate such facts to the students - future teachers and lecturers. By elementary methods we may prove some results on distribution of 1 prime numbers. For ( ) 1 1 P x = and S( x) = we have that p x p p x p P ( x) > ln x, 1 S ( x) > ln ln x. From these inequalities the divergence 2 1 of series p is deduced and herewith the infinity of the set of prime 1 numbers (Euler, ). Consequently, the infinity of the set of prime numbers was reproved in XVIII century using other methods. Let π (x) be the number of primes not exceeding x. If prime numbers are distributed quite randomly, maybe the behaviour of function π (x) is more understandable? Using elementary method we prove the theorem of Euler, that π ( x ) = o( x), and other results, e.g., x x c1 π ( x) c2, c2 > c1 > 0, of Chebyshev ( ). But these ln x ln x x methods are unsatisfactory to prove that π ( x ) = ( 1 + o(1) ). ln x The new era in number theory started with the creation of analytical method. This method is based on using the Riemann ( ) zeta function s ς ( s) = n, where s = σ + it, σ > 1. We may prove some = n 1 244

245 properties of Riemann zeta function and demonstrate the possibilities of analytical method. However, not everything is known about Riemann zeta function up until now nobody could prove or deny the Riemann hypothesis (RH): the real part of any non-trivial zero of the Riemann zeta 1 function isσ =. But if the Riemann hypothesis was proved, many results 2 of number theory would become more exact, among them we would find dt prime number theorem: π ( x ) = Li x + O( x ln x), where Li x = ln. t 2 Number theory is fascinating because it is usually simple to formulate its tasks but in many cases difficult to solve them. It allows demonstrating different mathematical methods and develops thinking. One of main concepts in algebra is the concept of determinant. In traditional courses for mathematicians the determinant of order n is defined after studying permutations and substitutions. Then we teach the properties of determinants, define the concepts of minor and adjunct and then prove the Laplace ( ) theorem. Such order of presentation is good for students of mathematics. But is there any use to prove the Laplace theorem for non-mathematicians? Maybe after definitions of minor and adjunct we can formulate the rule of calculation of determinant based on non-proved Laplace theorem? We use such approach in Vilnius Management Academy (see [4]). Mathematical proof itself is a philosophical concept. When we approach some mathematical truth then presenting it to the listeners is a very subtle issue and depends on the audience, on the purposes of the course, etc. However, by no means we are going to manage without proofs at all. References 1. Alan Budy, Mateja Jamnik, Andrew Fugard. What is a proof? Phil. Trans. R. Soc A, October 15, 2005, vol. 363, no.1835, pp Brian Davies. Whither Mathematics? Notices of the AMS. 2005, vol.52, no.11, pp Eugenijus Stankus. On the problems of teaching mathematics to philosophy students. VII International Conference, May, 2006 Teaching Mathematics: Retrospective and perspectives, Proceedings of the Conference, Tartu, 2006, pp A. Apynis, E. Stankus. Applied mathematics. VVK publishing house, Vilnius, 2000 (in Lithuanian). x 245

246 246 ON THE EVOLUTION OF MATHEMATICAL CONTESTS AT JUNIOR LEVEL Zane Škuškovnika, Agnis Andžāns, University of Latvia, Abstract. The changes in the tasks and content of mathematical contests caused by the shifts in general educational policy are considered. Some recommendations to the preparation process are given. Keywords: advanced education, mathematical contests, problem set composition, proof. Introduction As we can learn from many written and oral sources the number of lessons devoted to exact disciplines in school is decreasing in (almost?) all countries. Of course, in some sense this can be compensated by introducing new technologies into education. Nevertheless, today not all teachers are ready to explore their advances. So the official curriculum in mathematics today is far from the level of 1980-ies. Of course, advanced topics were first to go, and advanced mathematical education in schools met danger to become occasional and disintegrated. In this situation mathematical olympiads appeared to be very strong consolidating factor. Olympiad curricula wasn t changed; it was developed further in an essential way. The standards that were elaborated in olympiad movement during many years in some sense became the unofficial standards for advanced education in mathematics. There is a lot of topics that are not included in any official school program but nevertheless are discussed regularly with all students interested in mathematics because it is clear that they can very well occur in some of mathematical olympiads (preparatory, regional, open). The other positive feature of mathematical olympiads is their stability. Teachers are aware that the olympiads will be held, and they can organize their activities and encourage their students to work additionally for a clear and inspiring aim. Although the general principles of organization and evaluation remain unchanged, the mathematical content has to follow the evolution of the changing world. Some particular topics have been well-acquired by the pedagogical community during years and there is no need to focus on them still today. Some new classes of problems have emerged, and they are worth attention to be included into contests. As some material isn t longer

247 considered at school, the mathematical ideas behind it should be redressed to remain, etc. Positive changes: general observations The spectrum of the mathematics covered by olympiads and other contests gradually becomes wider. This is caused by the fact that the traditional areas are more or less exhausted, and fresh ideas are needed to preserve the element of unexpectedness and creativity. In many countries a new generation of olympiad organizers has appeared, consisting of actively working young scientists who themselves have been long-term participants of various contests during their school years; they bring the ideas from their research areas into olympiads. The ICT are providing rich opportunities to study the experience of various countries worldwide, so original findings are introduced broadly and with a little delay. Negative changes: general observations As the general curricula at schools become simpler, the level of difficulty of contest problems should follow these changes if we don t want to make the contests accessible only to elitarian public. Besides, as it was mentioned, olympiads now have much larger educational significance than earlier; also this reduces the mathematical level of them. From the other side, there is a socially significant part of students who s mathematical background becomes stronger (due to math circles, correspondence courses, summer camps etc.); for them simple olympiads are of little interest. In order not to allow the olympiad movement disintegrate the organizers are forced to compose very nonhomogenous problem sets, the simplest problems gradually becoming even more simple and the hardest ones even more hard. This brings us to the threat that the participants are split into two groups those who are able to solve only few problems (a majority) and olympiad professionals, with a very little transition group between these major parts. As teaching of mathematics at schools becomes more and more practically oriented, it becomes hard to follow the idea of equal representation of all principal parts of mathematics in an olympiad problem set. In ideal, algebra, geometry, arithmetics/ number theory, combinatorics and algorithmics should be included. The underrepresentation of geometry and especially reducing the role of the proof at school cause serious deviations from this setting. 247

248 Characteristic negative examples A. Multiple-choice problems are often considered as low-level ones, not requiring any proof or even explanation. Nevertheless, during last years some of them appear in contest problem sets quite regularly. Giving the opportunity to weaker students, they nevertheless can cause confusion about the necessity of argumentation. B. In geometry the problems are more and more often formulated for the situation given in the figure, thus avoiding the analysis of possible different configurations. C. The topics of impossibility have almost disappeared from the junior olympiad curricula in many countries. Particularly problems based on the idea of invariant have become rare, although they should be very thoughtprovoking. D. Problems from previous years contests are repeated more and more often, confirming that there is an acute need for new ideas suitable at junior level. E. Algorithmical problems are clearly underrepresented in junior contests, with few exceptions. Characteristic positive examples A. Problems of combined nature (e.g., combinatorics in algebra, algebra in arithmetics) appear more often than earlier. B. Some ideas that earlier were used only on elder grades are represented now also on junior level, redressed into appropriate form. C. Problem books with solutions, containing also the classification of problem by content/ method of solution, are available. D. Teaching aids are developed especially for junior students, devoted to general approaches to problem solving. Some recommendations A structured system of appropriate teaching aids on various levels of difficulty should be developed by the international community. The experience of long-lasting junior contests (Sankt-Petersburg, Moscow, Bulgaria, Latvia etc.) should be made internationally available. A new type of general textbooks with advanced topics integrated in them should be developed and introduced. Contests of various scale and level are welcome; nevertheless, one of them in each country should be regarded as the main one, and the high mathematical level should be preserved by any means. 248

249 Literature 1. K.Kokhas e.a. Problems of Sankt-Petersburg Mathematical Olympiads (in Russian). SPb, S.Genkin, I.Itenberg, D.Fomin. Leningrad Mathematical Circles (in Russian). Kirov, A.Leman. Problem Book of Moscow Mathematical Olympiads (in Russian). Moscow, Andžāns, I. BērziĦa, B. Johannessons. Profesora CipariĦa kluba uzdevumi un atrisinājumi gados. LU, Rīga, Andžāns, D. Bonka, Z. Kaibe, L. Rācene, B. Johannessons. Matemātikas sacensības klasēm. Rīga: Mācību grāmata, A. Andžāns, Z. Škuškovnika, B. Johannessons. Latvijas Atklātās Matemātikas Olimpiādes klases. Biznesa augstskola Turība, Rīga, A. Andžāns, B. Johannesson. Dirichlet Principle. Part I. Mācību grāmata, Rīga, A. Andžāns, B. Johannesson. Dirichlet Principle. Part II. Mācību grāmata, Rīga, A.Reihenova, A.Andžāns, L.Ramāna, B.Johannessons. Invariantu metodas. TEV,Vilnius, R. Kašuba. What to do when You don t Know What to do? Mācību grāmata, Rīga, R. Kašuba. What to do when You don t Know What to do? Part II. Mācību grāmata, Rīga, E.Barabanov, I.Voronovich e.a. Problems of Minsk mathematical olympiad for junior students (in Russian). Minsk, I.Tonov, K.Bankov, D.Rakovska, T.Vitanov. Selected Problems from math Competitions, Grades 4-7 (in Bulgarian). Yambol, I.BērziĦa, D.Bonka, G.Lāce. The Mathematical Content of Junior Contests: Latvian Approach. Mathematics Competitions, vol. 20, no 1., pp Квант (Russia), Математика+ (Bulgaria),

250 250 THE MEAN VALUE METHOD IN MATHEMATICAL GAMES Ingrida Veilande, LMA, Abstract. Mathematical games provide an attractive way to raise the pupil s interest in search for innovative methods of problem solving. They can solve puzzles and play mathematical games for to observe the regularities to identify the winning strategy of the game. On the other hand, games have considerable mathematical appeal. In spite of the simplicity of some games we can study complex problem solving methodologies. It is important to recognize the application of significant mathematical and general thinking methods. Varieties of the mean value method are usable in investigation of some particular problems to determine the winning strategies of the games. Keywords: Dirichlet s box principle, mathematical game, mean value method, winning strategy. Introduction The main goal of mathematical circles is to draw a deeper interest in mathematics. Students can become more proficient in solving special tasks as well as in combinatorial regularities by solving mathematical puzzles and analysing brainteasers. There are some puzzles that require special game sets, but others could be done with the pencil and paper only. Mathematical laws could be discerned even in the simplest games like Tic-tac-toe, Dots and boxes, Nim and other [4, 5], which are carried on by younger students. New complicated mathematical theorems are heuristically uncovered while progressively improving skills. Various methods of thinking could be developed with the help of more complex games. Organizers of a competition named after M.Lomonosov are sure in the utility of mathematical games, which are included into students multisubject contest together with math, physics and other exact sciences and humanities as an individual discipline [9]. Mathematical games Considering games and particularly looking for a winning strategy is a task of special branch of mathematics combinatorial game theory. Within this theory also two-person games with full information are considered: Players move in turns. Moves are not taken randomly. Game always ends after finite number of moves.

251 Final position defines the outcome of the game one s win or a draw. We must distinquish between: impartial games with identical next move option (Nim, Hackenbush, Dots and boxes) and partisan games, where each has its own pieces and permitted moves (Tic-tac-toe, Draughts, Hex). Solitaires also are subject to our investigation. Mean Value Method Mean value method (MVM) is one of the universal methods of reasoning [2]. Its essence is to estimate individual element in comparison with any of the mean values of the given group of elements. This is a well-known method in combinatorics, graph theory, number theory and other branches of mathematics. The manifestations of the mean value method can be found on different stages of mathematical game analysis: examining the structure of the given elements or their layout at the initial position of the game; examining substructures of elements and composing algorithms to achieve them ; seeking some regularities which can be generalized to achieve the final outcome of the game. Rules of the game and initial position There are two possible situations in beginning of any game playing board is either empty or contains a special displacement of the objects. If the game starts on an empty board, demonstration the MVM in some cases depends on the type or properties of the board. By some features of the initial positions the outcome of the game could be predetermined with the help of MVM. Examples 1) Simple game Tic-tac-toe on 3x3 or 5x5 board with both players optimal strategy without fail ends in a draw [5]. Summarizing the 3D 3x3x3 board game, where cell is one of unit squares, first player can create a fork on the third move if the set is started by crosses, then noughts are not able to block double crosses, which appear at once in several directions. The fork is losing position for opponent. This position expresses the Dirichlet s box principle (DP) [2] one of MVM s varieties - in a simple form. Hales-Jewett s theorem shows the significant generalization of the Tic-tac-toe game; it is one of Ramsey theory s results of presence of onecolour straight line in a multidimensional square with multicoloured cells [5, 7]. The theorem implies that with sufficient dimensionality a draw is not possible. 251

252 2) There is a wide class of graph colouring games. They could be divided into edge, vertex or area colouring games. For example, the SIM (by Gustav Simmons [8]) game is played on a full six vertex graph C 6, colouring in turn the edges in red or blue colour. The one who first makes a monochromatic triangle is considered as a loser. Ramsey s theorem for full graphs (which is the generalization of the DP) guaranties the unavoidable loss for second player (by correct play of the first player). Ramsey s number R(3,3,2) = 6 means that there will be one-colour red or blue triangle in any bichromatic colouring of C 6. 3) Let us consider an ancient Japanese brainteaser [6]: black and white pearls are scattered on squared paper one by one in some squares. The necklace should be made, i.e. a closed broken line, consisting of horizontal and vertical segments, should be drawn according to two rules: a) The broken line rotates on 90º degrees in each black square, but there is no turn before or after the black one. b) Going through the white square the turn should be made right before or after crossing the square. Any of given situations could be shown as a graph with coloured vertices and edges. Then the solution of such problem is equivalent to finding a Hamilton cycle. In some particular cases the impossibility of Hamilton cycle existence could be proved with a help of DP. For example, according to b), it is impossible to draw a straight line through more than two white squares, when the pearls are scattered in more than three blackand-white lines. 4) Diamond-shaped playing board of Hex game (by Piet Hein) consists of regular hexagons (size of the original 11x11) and has two white and two black opposite sides. Players in turn put their pieces on free cells. The first player has black pieces, the second player white ones. The winner is one who reaches the opposite side of the board through a continous chain of appropriate pieces of his colour (see Fig. 1). 252 Fig. 1. In Hex game the winner has black pieces. Relying on the fact that on totally filled board there will be a chain of black pieces that connects black sides or a chain of white pieces that