DEVELOPING GEOMETRIC SKILLS THROUGH ACTIVITIES IN LESSONS OF MATHEMATICS

Acta Didactica Universitatis Comenianae Mathematics, Issue 14, 2014, pp. 71-86 DEVELOPING GEOMETRIC SKILLS THROUGH ACTIVITIES IN LESSONS OF MATHEMATICS LUCIA RUMANOVÁ, EDITA SMIEŠKOVÁ Abstract. The paper deals with problem of geometric constructions on secondary school level. There were 34 pupils who participated in our activity. It was focused on the practical use of constructions in school mathematics aiming to enhance pupils motivation and improve their geometrical skills in teaching geometry through construction of stone marks known from the past. Key words: fine arts, education, lesson, pupils, theory of didactic situation, solution 1 INTRODUCTION Geometry in the past was based on practical needs of life. Development of geometric knowledge most contributed to architectural activity, such as land surveying, construction of homes, forts etc. Similarly, geometric knowledge was required for orientations in the field - i.e. transport by sea or desert, manufacture of tools, weapons, and also in shipbuilding. Gradually, geometry became a science. Geometry has been well studied and developed because of these practical reasons, but also philosophical and theoretical reasons. (Surynková, 2010) The situation with arts was analogical. The arts have always accompanied mankind and it is a means of communication. For example, a full perception of a work of arts is unthinkable without vision. But it is not by any vision, but the conscious vision. Vision itself is intricate, as well as imaging what is seen. This is true even if it is only a view of the subject without any artistic ambitions. This fact is known from school practice. It must be taken into account how much work pupils have in order to sketch geometric objects into a picture correctly. (Šarounová, 1993) The study of geometry is demanding, therefore it is often circumvented at various levels of education, e. g. when there is not enough time left, geometry is

72 L. RUMANOVÁ, E. SMIEŠKOVÁ omitted. Geometry should be taught illustratively. Geometric constructions should never be taught as procedures under which pupils do not see anything in the picture or cannot imagine anything. Geometrical construction problems seem to be difficult for pupils. Those who have already encountered these constructions try to recall the algorithm, which in many cases they do imprecisely. Generally, pupils do not think about mathematical properties when doing geometrical constructions, they take only two approaches: remembering the algorithm, or drawing (not constructing) the required picture. They use many mathematical notions incorrectly. (Marchis, Molnár, 2009) One of the main aims of mathematical education as such is preparing the pupils for dealing effectively with the real-life situations. (Švecová et al., 2014) Geometry should be taught in an interesting and logical way, and also the use of geometry in practice should be emphasized. Geometry stems from practical activities. Use of geometry in practice is most evident in the architecture and arts. Creating a perfect piece of art, which is in harmony, it is possible to achieve use of some mathematical and geometric relationships. Architecture is primarily intended for practical use, but can also be "pleasing for eyes". Incorporating art in geometry inspires pupils to observe the world around them with the eyes of a mathematician. Pupils have an opportunity to demonstrate their knowledge of geometric shapes, as well as produce interesting works of art that can be displayed to show their understanding. Why arts? Activities for teaching geometry lesson for young pupils can often be bland. Adding an art component to lesson plans provides pupils with a hands-on medium for learning. Classroom teachers may want to coordinate their efforts with the art teacher; however, the following instructions can easily be accomplished without collaboration of the art teacher. (Neas, 2012) In modern educational theories necessity of pupils activity during lessons is highlighted. There is also demand for development of not only pupils knowledge but all key competencies. Therefore we need teaching methods which could meet these needs. (Vankúš, 2008) 2 THEORETICAL FRAMEWORK The activities presented in the article and the research brings together concepts and principles of the theory of didactical situations (TDS) and the analysis methods of problem solving processes. The core of TDS issued from this didactic school is analysis of problem in particular levels of didactic situations. A didactic situation according to the TDS consists of three main parts: devolution, a-didactical situation and institutionalization.

DEVELOPING GEOMETRIC SKILLS THROUGH ACTIVITIES IN LESSONS 73 A framework to this approach is based on the works by Brousseau (1998), Chevallard (1992), Sierpinska, (2001). Brousseau (1998) defines didactic situation as a situation for which it is possible to describe the social intention of student s knowledge acquirement. This situation is realized in a system called the didactic system (didactic triangle) that is composed of three subsystems: learner (student), learning (teacher), information and relations between them. The relations the didactic contract represents results of intervention explicitly or implicitly defined relations between students or group of students. It is the environment and the educational system that prepare students to accept completed or nascent knowledge. They are exactly the rules of game to activate the student. The basic notion of TDS is the Didactic milieu, following Piaget s theory the milieu is source of contradictions and non-steady states of learner (subject) by process of adaptation (by Brousseau (1986) it is assimilation and accommodation). The environment is specific for every piece of knowledge. Different levels of milieu are embedded one inside another, a situation at one level becoming a milieu for a situation at a higher level action at an upper level presumes reflection on the previous level. (Regecová, Slavíčková, 2010) We can see this structure of milieu in Table 1. Table 1 Structure of the milieu in didactical and a-didactical situation (Földesiová, 2003) M 3 Constructional milieu P 3 Teacher - didactic S 3 Noosferic situation M 2 Project milieu P 2 Teacherconstructor S 2 Constructional situation M 1 Didactic milieu E 1 Reflective student P 1 Teacher- designer S 1 Project situation M 0 Milieu of learning E 0 student P 0 Teacher- designer S 0 Didactic situation M -1 Modelling milieu E -1 Cognizant intellect student S -1 Learning situation M -2 Objective milieu E -2 Active student S -2 Modelling situation M -3 Material milieu E -3 Objective student S -3 Objective situation

74 L. RUMANOVÁ, E. SMIEŠKOVÁ According to Brousseau (1998), the a-priori analysis is one of the tools that teacher can use in lesson planning. Analysis a-priori is necessary to do before finding a solution of a particular problem. It is useful for a teacher to prepare background for various possibilities that can be observed in lessons. A good preparation of a-priori analysis is a condition for successful devolution and a-didactical situation. Therefore it helps one prepare better a-didactical situation, a situation where children get the knowledge on their own. (Novotná et al., 2010) We apply the principles of this theory confirming the importance of the analysis a-priori and a-posteriori results. 2.1 DETAILS OF PRESENTED ACTIVITY AND RESEARCH The activity was realized in March 2014 with 16 17 year old pupils. They were 34 pupils of grammar school in Nitra. In school practice the ability of pupils to solve an open problem independently develops poorly. Pupils mechanically apply earlier acquired geometric skills and algorithms when solving these problems. The main aim of the submitted paper was to determine whether pupils can apply and use effectively their knowledge and expertise from different parts of mathematics in solving a particular geometric problem. Pupils looked for the beauty of arts, became acquainted with the rules of formation, so the arts was not only admired, but also understood. In the activity, including the research, the main focus was on the practical use of constructions in school mathematics, aiming to enhance pupils motivation and improve their geometrical skills in learning geometry through construction of stone marks known from the past. In some parts of mathematics it is difficult to invent such a game between the teacher and pupils which would meet qualified conditions of TDS so that it would be possible to assemble a set of problems and activities. Here definitely belongs geometry and certainly the constructional problems. Therefore, we have created problem tasks that are based on formerly acquired knowledge of geometry. The following activities, without naming the topics which we want to deal with, it is also possible to route of TDS. Before meeting the pupils we prepared a detailed a-priori analysis of activities later on realized with them, and we thought about what we expected from the pupils (Chapter 2.2). In the context of devolution, we thought about how pupils would explain important rules and then we would not interfere in the individual activity. When the pupils understood what to do, our help (work of the teacher) was terminated. Devolution was completed and followed by a- didactical situation. We also thought that during the activity pupils would go through all three phases of a-didactical situation (action, formulation, validation). Pupils would obtain new knowledge during these three phases. We expected that

DEVELOPING GEOMETRIC SKILLS THROUGH ACTIVITIES IN LESSONS 75 pupils would acquire knowledge of stone marks, principles of their constructing, history and arts. We would finish the didactic process with the last part of TDS - institutionalization. In our case the institutionalization would be a discussion with pupils, no matter how pupils would solve the problem. We would try to explain the solution of the problem and to affirm their acquired knowledge. We proceeded with the pupils as follows: - pupils filled in the initial test we wanted to determine their basic geometric knowledge needed to solve the problems; - we familiarized pupils with the activity which was then carried out with them; - pupils individually solved a given problem on the topic. After realization of activity we compared our projection from a-priori analysis of pupils solutions and made results in a-posteriori analysis. 2.2 A-PRIORI ANALYSIS OF PUPILS ACTIVITIES We formulated analysis a-priori before pupils solutions (the research), and the analysis includes descending analysis (analysis of the teacher s work) and ascending analysis (analysis of pupils work). Since we had not known the pupils in the experimental group before, we decided to check their geometric knowledge. The initial test was focused on the basic concepts related to the settlement of problems. We were interested if these pupils had necessary knowledge to comprehend the situation and to solve the given task. In accordance with the tenets theory of didactic situations frame: within the frame of the didactic situation S 3 (Noosferic didactic situation) we made an analysis of math textbooks for secondary schools and an analysis of various mathematical materials. The end of Noosferic situation would be the milieu for the following situation. Then we chose geometric notions that the pupils needed to follow solutions during the activities. The initial test is listed in Appendix 1. Relating to the pupils' knowledge, we formulated possible responses to the individual questions of the test. The teacher should attempt to read the pupils minds and get at the same level of their thinking. Similarly, we took into account the pupils when compiling tasks in other planned activities. We were prepared for the situation that the initial test would indicate insufficient pupils knowledge and then it would be needed to improve their knowledge through other tasks. In the context of problem tasks we solved the task with pupils in order to use the geometric constructions in creating the so-called stone marks. Activities with pupils were situated into the story of builders in the middle Ages, who signed those marks and sculpted them into the building. Relating to the knowledge of pupils, we compiled problems and tried to engage pupils into the

76 L. RUMANOVÁ, E. SMIEŠKOVÁ process of their solving. Problems and sequence of activities with pupils are listed in Appendix 2. Pupils would get acquainted with the problem and with the material milieu. Social component of material milieu would be minimal because other help would not be allowed in our experiment (self activity of pupils). Pupils should be able to solve the given tasks using drawing aids compasses or triangular rulers. Within problem solving pupils would become familiar with several types of stone marks and procedures for their constructions. They would have to know the construction principles of three basic keys to stone marks. Finally, we would want them to give the key construct its own mark. Pupils could then realize that they as stonemasons in the past would try the uniqueness and originality of their own signature. We expected that pupils would: - be able to work correctly with the drawing aids, such as compasses, triangular rulers; - know the geometric construction of basic geometric figures, such as a square and an equilateral triangle; - be able to find constructively the middle point of the diagonal (and sides) of the object, construct the perpendicular led by point to the side, construct an angle axis and the axis of side; - be able to construct inscribed and circumscribed circle of a triangle and a square; - know the construction of triangle height, medians and diagonal of a square. Finally, in the end of the activities pupils would solve the problem alone and they could choose from two options (Appendix 3). While addressing these tasks pupils could use the knowledge which we would also use. Pupils would receive a text of problem in which we put them in the position of medieval stonemasons. They would have to demonstrate their drawing skills and geometric skills. We assumed that the biggest problem would be with the accuracy of pupils construction. Therefore, if the mark of pupils would not be accurate enough, they would not get the correct final key. We would observe the proposed aims and also pupils solutions. It would be a situation where the analysis of teacher s work and analysis of pupils work meet, and this situation would be the result of the teaching process. In didactic situation the work of pupils would be affected by teachers and their advice in form of institutionalization, which could help the pupils to solve the given tasks, but teacher must first of all take into consideration pupils solution. The teacher could help with individual elements of construction marks.

DEVELOPING GEOMETRIC SKILLS THROUGH ACTIVITIES IN LESSONS 77 2.3 A-POSTERIORI ANALYSIS At the beginning of the lesson the pupils attended the initial test. We wanted to found out their knowledge before the problem task. As we expected within the ascending analysis (a-priori analysis) pupils have necessary knowledge to comprehend the situation and then to solve the given task. The initial test consisted of 8 questions. The most successfully answered question was question 6: A median of a triangle is... ; while the least successfully answered question was question 4: According to length ratio we divided triangles The results are summarized in Graph 1. Graph 1. Pupils results of initial test The most frequent incorrect answers to question 4 were: - isosceles triangle, equilateral triangle; - isosceles triangle, equilateral triangle, irregular triangle. Answers to question 3 Write how many axes of symmetry figures in the picture below have. were surprising; some of the pupils answers were: - a circle has 360 axes of symmetry; - a circle has 1 axis of symmetry; - a pentagon has 1 axis of symmetry; - a pentagon has 3 axes of symmetry. We did not consider the misunderstanding of terms in the test. We worked with pupils of grammar school, so according to the National Program of Education (2011) and also having in mind the type of school the pupils should have known these terms. We think that the incorrect and unusual answers of pupils could be the result of inattention or less knowledge of mathematics. After the initial test pupils were given the problem task which was inspired by the work of Vienna architect Franz Rizha who discovered the secret of the stone marks. He searched for the geometrical construction of keys of stone

78 L. RUMANOVÁ, E. SMIEŠKOVÁ marks. In accord to the TDS Objective situation was the same in every strategy. Pupils acquaint with the problem task (see Appendix 3) and material milieu (basic writing tools, drawing aids compasses or triangular rulers, context of the task...). Social part of material milieu was minimalized because pupils solved problem task separately. In the Modelling situation is the pupil active and he try to solve the given problem task with material milieu. In the next Learning situation is pupil in the position of solver of the problem task, he start to formulate to the own initial findings and conclusions we (as teacher) did not helped pupils. So realization of research do not assume the interruption by teacher to the solution of pupils, pupils do not reach Didactic situation. Now we give a few specific observations and the results of the pupils' solutions. Pupils were provided with the text by which they were put into the role of journeymen in the middle Ages. The middle Ages journeymen had to show their competencies in geometry. Pupils who could not solve the problem task often made mistakes at the beginning; therefore we think it was caused by inattentive reading of the text. Other mistake was inaccurate and confusing drawing which resulted in impossibility to find out the secret of the key of their stone mark. We were interested in the connection between pupils' knowledge and their ability to solve the problem task. Pupils could obtain 1 point for each correct answer in the initial test. The following graph depicts the frequency of pupils awarded 0 8 points in the initial test, and correct or incorrect solution of the problem task (one of the variants A or B). Graph 2. Frequency of pupils according to solution the problem task A and B

DEVELOPING GEOMETRIC SKILLS THROUGH ACTIVITIES IN LESSONS 79 The contingent table for group of 34 pupils was made. Pupils' results of the initial test were divided into two groups. Pupils who had 0 4 correct answers were included in the first group; pupils who had 5 8 correct answers were included in the second group. Then the solutions of the problem task were marked by 1 if they were correct, by 0 if they were incorrect. Table 2 shows that pupils who did better in the initial test were also more successful when solving the problem task. On the other hand pupils who are in the first group of the initial test were not so successful with the problem task. Table 2 The contingent table of pupils' results Tasks A or B Questionnaire 1. group 2. group 0 15 5 20 1 4 10 14 Total 19 15 34 Total A solution sample of pupils' who solved the problem task correctly: A solution sample of pupils' who solved the problem task incorrectly:

80 L. RUMANOVÁ, E. SMIEŠKOVÁ 3 CONCLUSION The education process attempts to teach pupils reading comprehension, so that pupils would be able to follow the instructions, and/or should accept mathematics as a part of human culture as an important society tool through cross-curricular relations. (NPE ISCED 3A, 2011) Basing on the reactions of pupils in the questionnaire on the activities we write about in the article, we conclude that pupils: - are interested in such activities; - learned something new or deepened their geometric knowledge; - by drawing more complicated units (how pupils named it) they learned patience and precision, the lesson was enjoyable for them and they felt free to apply their creativity (often in mathematics they do not feel like that); - learned something from the history and discovered connections between well-known facts which they had not been aware of before; - described the activities as very interesting and the procedure of drawing was clear and understandable for them; - had time for individual work and moreover a good deal of freedom in solving the problem tasks. Therefore, we think that including such kind of tasks in teaching process is useful, although we are aware of the fact that their preparation takes a lot of time and space. We believe that giving pupils this type of tasks remains an open problem. That is why teachers diagnose the formal knowledge of their pupils by these tasks, encourage and develop their geometric skills. This is the reason why the teacher should not forget that geometry should be taught from the very basic level. In the future, this problem will be a subject for next research. REFERENCES Brousseau, G. (1986). Fondaments et methods de la didactique des mathématiques. Reserches en Didactique des Mathematiques. Grenoble, La Pensée sauvage. Brousseau, G. (1998). Théorie des situations didactiques. Grenoble, La Pensée sauvage. Chevallard, Y. (1992). Concepts foundamentaux de la didactique: perspectives apportées par une approache antropologique. Recherches en Didactique des Mathématiques, Vol. 12/1. Grenoble, La Pensée sauvage. Földesiová, L. (2003). Sequence analytical and vector geometry at teaching of solid geometry at secondary school. In: Quaderni di Ricerca in Didattica, Number 13, Palermo, 2003, ISSN 1592-4424, p. 33-42 Kadeřávek, F. (1935). Geometrie v uměni v dobách minulých. Praha: Jan Štenc, (1935), pp. 43-48

DEVELOPING GEOMETRIC SKILLS THROUGH ACTIVITIES IN LESSONS 81 Marchis, J. Molnár, A. É. (2009). Research on how secondary school pupils do geometrical constructions. In: Acta Didactica Napocensia, Volume 2, Number 3, Romania, 2009, ISSN 2065-1430, p. 119-126 National Institute for Education (2011). National Program of Education Mathematics ISCED 3A. Retrieved April 4, 2014, from http://www.statpedu.sk/sk/statny-vzdelavaci-program/statnyvzdelavaci-program-pre-gymnaziaisced-3a/matematika-a-praca-s-informaciami.alej Neas, L. M. R. (2012). Using Geometry in Art Class. Retrieved April 24, 2014, from http://www.brighthubeducation.com/lesson-plans-grades-3-5/63018-teaching-geometry-artand-shapes/ Novotná, J. et al. (2010). Devolution as a motivating factor in teaching mathematics. In Motivation via natural differentiation in mathematics. Rzeszów: Wydawnictwo Uniwersytetu Rzeszowskiego, 2010, pp. 38-46. Regecová, M. Slavíčková, M. (2010). Financial literacy of Graduated students. In: Acta Didactica Universitatis Comenianae Mathematics, Issue 10, Bratislava, 2010, ISSN 1338-5186, p. 121-147 Sierpinska, A. (2001). Théorie des situations didactiques. Retrieved February 18, 2014, from http://www-didactique.imag.fr Struhár, A. (1977). Geometrická harmónia historickej architektúry na Slovensku. Bratislava: Pallas, (1977), pp. 68-72 Surynková, P. (2010). Geometrie, architektura a umění. Retrieved April 26, 2014, from http://www.surynkova.info/dokumenty/ja/prezentace/geometrie_brno_2010.pdf Šarounová, A. (1993). Geometrie a malířství. In: Historie matematiky. I. Seminář pro vyučující na středních školách. Brno: Jednota českých matematiků a fyziků, 1993. pp. 190-219 Švecová, V. - Pavlovičová, G.- Rumanová, L. (2014). Support of Pupil's Creative Thinking in Mathematical Education. In: Procedia-Social and Behavioral Sciences: 5 th World Conference on Educational Sciences - WCES 2013, 2014. - ISSN 1877-0428, Vol. 116 (2014), p. 1715-1719 Vankúš, P. (2008). Games based learning in teaching of mathematics at lower secondary school. In: Acta Didactica Universitatis Comenianae Mathematics, Issue 8, Bratislava, 2008, ISSN 1338-5186, p. 103-120 LUCIA RUMANOVÁ, Department of Mathematics, Faculty of Natural Sciences, Constantine the Philosopher University in Nitra, 949 74 Nitra, Slovakia E-mail: lrumanova@ukf.sk EDITA SMIEŠKOVÁ, Department of Mathematics, Faculty of Natural Sciences, Constantine the Philosopher University in Nitra, 949 74 Nitra, Slovakia E-mail: edita.smieskova@ukf.sk

82 L. RUMANOVÁ, E. SMIEŠKOVÁ Appendix 1 INITIAL TEST FOR PUPILS 1. Which dimensional figure could be hide under ground? 2. Mark, which the pair of figures in the picture are axially symmetrical? a) b) c) d) 3. Write, how many axis of symmetry have figures in the picture below? cccc 4. According to length ratio we divided triangles 5. A height of triangle is... 6. A median of triangle is... 7. The center of circumcircle of a triangle is (mark only one answer) a. the intersection of the angle bisectors, b. the intersection of the axis of sides, c. the intersection of the medians of triangle and we called it the centroid, d. the intersection of heights of triangle and we called it the orthocenter.

DEVELOPING GEOMETRIC SKILLS THROUGH ACTIVITIES IN LESSONS 83 8. The center of inscribed circle of a triangle is (mark only one answer) a. the intersection of the angle bisectors, b. the intersection of the axis of sides, c. the intersection of the medians of triangle and we called it the centroid, d. the intersection of heights of triangle and we called it the orthocenter.

84 L. RUMANOVÁ, E. SMIEŠKOVÁ Appendix 2 Introduction of the activity motivation speech: Nowadays, the architect or designer is signed for his project. How was it in the past? In the time of Greece and Rome the builder was signed by a stone mark that engraves directly to masonry construction and architects made it in the same way in the middle Ages. As the builders were many, as well as stone marks were numerous and each mark was a different. We showed to pupils how stone marks look like and gave them examples where they might see them. The construction of the key to stone marks: "Every stone mark made up to a certain key which the building company strictly guarded. All marks should be able to put into such key. Since now it is known 14 keys, which we call the basic root of the mark. After then we showed 12 roots of the marks to pupils. We then showed 12 pupils roots marks: The use of the stone mark by journeyman: If the journeyman received his own stone mark, he should signed by it. The condition of use was that he had to know its details so that he was able to put it into the key. Unauthorized use was punishable.

DEVELOPING GEOMETRIC SKILLS THROUGH ACTIVITIES IN LESSONS 85 We offered pupils some examples of stone marks from different period of history: The organization of building companies: Builders worked together in the communities which we called building companies. Builders had their privileges and their master who took care about apprentices and journeymen. Each of them got his own mark with the key of construction. The key was a pattern created by repeating of simple root of the building company or developing the root to network of parallel lines. After promotion of the apprentice to journeyman, the mark was more developed. The mark of journeymen contained of lines, which at least one intersect other. The intersection was at right angel. If the journeyman was older and experienced, he got a new mark. This mark consisted of intersection of diagonal lines. The mark of master consisted of whole circle. The legend by which our activities were inspired: The journeyman had to travel a lot and so that to expanded their experiences and education. Supposedly it was so, if they came to other building company, they had to three times knock on the door and answered on three questions. And then the door of the company was opened, but they were not still received. They had to make an exam of the geometry of which part was the construction of their stone mark and the explanation of the construction. If they were successful in this exam the building company received them. The construction of three roots of stone marks:

86 L. RUMANOVÁ, E. SMIEŠKOVÁ Appendix 3 The problem for pupils Imagine that you are a journeyman of some Slovak building company from Middle Age. This building company granted you your own building mark. Your building company sent you to disseminate knowledge and education to the other parts of world. After few days of walking you stopped before a gate of Prague building company. How says a legend, you three times knocked on the door and gave a right answer on three questions. Gate of the building company is opened, but you are not still received. There is an exam from geometry. Demonstrate yourself by your building mark and explain the principles how was your building mark constructed. Choose one variant A or B and write a reason why you have decided so. Variant A You are experienced and skilled journeyman and you know the principles of many geometrical constructions, however the way to the Prague building company was long. Only one thing you remember about the construction your building mark is that the construction of the mark s key begin with the construction of two equilateral triangles. Their shapes created a six-pointed star and lie on the circle to which is your building mark inscribed. The longest mark s line is height of one that equilateral triangle. More than you know, that you have to construct some different equilateral triangles and their heights. Parts of your building mark lie on these triangles and heights. How to do so? Discover the secret of mark s key and prove to Prague building company that you are worthy to become their new journeyman. Variant B You are experienced and skilled journeyman and you know the principles of many geometrical constructions, however the way to the Prague building company was long. Only one thing you remember about the construction your building mark is that the construction of the mark s key begin with the construction of two equilateral triangles. Their shapes created a six-pointed star and lie on the circle to which is your building mark inscribed. More than you know, that you must inscribe six the same smaller circles which are touched from inside of the biggest circle. These circles intersect each other and some parts created the line of your building mark. How to do so? Discover the secret of mark s key and prove to Prague building company that you are worthy to become their new journeyman.