Graphing functions and solving equations, inequalities and linear systems with pre-service teachers in Excel

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Graphing functions and solving equations, inequalities and linear systems with pre-service teachers in Excel Ján Beňačka, Soňa Čeretková To cite this version: Ján Beňačka, Soňa Čeretková.

CERME 9 - Ninth Congress of the European Society for Research in Mathematics Education, Feb 2015, Prague, Czech Republic. pp.

fr/hal-01289242 Submitted on 16 Mar 2016 HAL is a multi-disciplinary open access archive for the deposit and dissemination of scientific research documents, whether they are published or not.

1 Graphing functions and solving equations, inequalities and linear systems with pre-service teachers in Excel Ján Beňačka, Soňa Čeretková To cite this version: Ján Beňačka, Soňa Čeretková. Graphing functions and solving equations, inequalities and linear systems with pre-service teachers in Excel. Konrad Krainer; Naďa Vondrová. CERME 9 - Ninth Congress of the European Society for Research in Mathematics Education, Feb 2015, Prague, Czech Republic. pp , Proceedings of the Ninth Congress of the European Society for Research in Mathematics Education. <hal > HAL Id: hal Submitted on 16 Mar 2016 HAL is a multi-disciplinary open access archive for the deposit and dissemination of scientific research documents, whether they are published or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers. L archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

2 Graphing functions and solving equations, inequalities and linear systems with pre-service teachers in Excel Ján Beňačka and Soňa Čeretková Constantine the Philosopher University in Nitra, Nitra, Slovakia, The article presents the results of a survey on using Excel for graphing functions, solving equations and inequalities without and with a parameter, and solving systems of linear equations with pre-service mathematics teachers. The experimental group were 28 master students of teaching mathematics. The aim was to ascertain that the methodology can benefit teaching the topics and make mathematics more interesting. Keywords: Spreadsheets, modelling, constructivism. INTRODUCTION The spreadsheet is a tool that enables access to ideas and concepts through a computational experiment without any need for programming. It allows using inquiry and heuristic methods and the immediate feedback provokes into experimenting and discovering. The use of spreadsheets in mathematics education has been researched for decades, however, mainly on primary and lower secondary level (Healy & Sutherland, 1990; Rojano & Sutherland, 1993; Hošpesová, 2002ab; Haspekian, 2005, 2011, 2014; Ainley, Bills, & Wilson, 2005; O Reilly, 2006; Wilson, 2006; Tabach, Arcavi, & Hershkowitz, 2008; Tabach & Friedlander, 2008; Tabach, Hershkowitz, & Arcavi, 2008; Drake, Wake, & Noyes, 2012; González-Calero, Arnau, & Puig, 2013; Watson & Callingham, 2013; Geiger, Goos, & Dole, 2014). There is a considerably smaller number of articles written on the use of spreadsheets in mathematics education at upper secondary level (Molyneux-Hodgson, Rojano, Sutherland, & Ursini, 1999; Sivasubramaniama, 2000; Neurath & Stephens, 2006; Forster, 2007; Topcu, 2011; Benacka & Ceretkova, 2013). Research of Molyneux-Hodgson and colleagues (1999) with 16 to 18 years old students from England and Mexico showed that the mathematical culture in the country affected the choice of the means when the students solved tasks with spreadsheets. Research of Sivasubramaniam (2000) with student of age 14 and 15 showed that the use of spreadsheets had a positive impact on understanding Cartesian graphs. Forster (2007) researched a group of 17 and 18 years old girls on the use of technology when investigating the trend of a set of data from jewellery. Projecting graphs created in spreadsheets had a positive effect. Topcu s (2011) research with 16 years old boys showed that the boys who used spreadsheets to solve algebra tasks were considerably more confident as they were aware of the possibility of checking and correcting errors. Neurath and Stephens (2006) researched the effect of integrating Excel into teaching of algebra with 14 to 17 years old students. The students opinion of the lesson was surveyed by a questionnaire. The result was that 69% were interested in solving tasks with Excel and understood algebra better, and 77% of the students enjoyed the course because solving tasks with Excel made algebra more interesting. Just 8% disapproved in both cases. Benacka and Ceretkova (2013) gave account on an experiment in which 16 to 19 years old students developed Excel applications to graph functions, find extremes, solve systems of linear equations, calculate the area of a planar figure by the rectangle method and Monte Carlo method, and simulate motion of a projectile in a vacuum. The students opinion of the lesson was surveyed by a questionnaire. The result was that 100% of the students found the lessons interesting, 35% understood all and 65% understand most of the mathematics. This article presents the results of a survey on using Excel in mathematics teachers education. The experimental group were 28 first year master students of CERME9 (2015) TWG

Aplications developed by the first author were used.

3 Figure 1: Graph of a function with definition domain R (left) and not R (right) teaching mathematics. There were 6 men and 22 women in the group. The topics were Graphing functions, Solving inequalities, Solving equations with a parameter, and Solving systems of three linear equations with three variables. Aplications developed by the first author were used. The aim was to ascertain that the applications, the tasks and the presented teaching methodology can make teaching mathematics at upper secondary level attractive, benefit the teaching of the topics and contribute to the technological and pedagogical knowledge (TPACK) of the pre-service teachers. Each topic took a 90 minute lesson. A questionnaire survey with the following questions was carried out at the end of each lesson: A) The lesson was (1 = very; 2 = quite; 3 = little; 4 = not) interesting. B) I understood (1 = all; 2 = most; 3 = little; 4 = nothing) of the mathematics involved in the activity. C) The lesson contributed to my TPACK D) (1 = a lot; 2 = quite a lot; 3 = little; 4 = not at all). E) The applications benefit the teaching of the topic F) (1 = a lot; 2 = quite a lot; 3 = little; 4 = not at all). G) Developing the applications helps comprehend the core of the topic H) (1 = a lot; 2 = quite a lot; 3 = little; 4 = not at all). I) I am a man (1 = yes; 2 = no). Answers 1 and 2 in questions A E were considered positive. Altogether, 96 questionnaires were evaluated. The survey summary is in the last section. GRAPHING FUNCTIONS The application on the left side of Figure 1 graphs functions if the definition domain is R. The graph is produced from 100 points in range B16:C116. Cell C12 contains the formula =(C10-C9)/C11. Cell B16 contains =C9, Cell B17 contains the formula =B16+$C$12. Cell C16 contains the formula =$C$3*(B16-$C$5)^2+$C$6. The two formulas are copied down as far as row 116. The application on the right side of Figure 1 graphs functions if the definition domain is not R. The graph is made over 5000 points in range B16:B5016 and J16:J5016. Cell C16 contains the formula =$C$3*TAN($C$4*(B16-$C$5))+$C$6. Cell M17 contains the formula =IF(ISERROR(C16);NA();C16). The two formulas are copied down as far as row The points that are out of the definition domain are skipped by function NA(). Lesson and survey: There were 28 students in the lesson, 6 men and 22 women. The students downloaded the template (Figure 1 with empty white and grey cells) from a website and developed the application with the teacher s help. The shape of the graph, definition domain, range and symmetry of elementary functions y = x, y = x2, y = x3, y = 1/x, y = 1/x2, y = x, y = sin x, y = cos x, y = 2 x, y = log 2 x, y = tan x and y = cot x, was discussed and visualized by the applications. The effect of the sign and absolute value of the parameters on the orientation, steepness, period and shift of the graph were checked. The following algorithm for graphing functions y = af(a m) + n or trigonometric 2312

Figure 2: Relative frequency of the answers y = af(b(a m) )+ n by hand was deduced and exemplified by graphing the functions y = (2x 4)2 5 and y =2tan(2 2x) + 1 (Figure 1).

The result is: (A) 92% found the lesson interesting (71% very, 21% quite); (B) 100% understood the topic (96% all, 4% most); (C) 89% had the feeling that the lesson contributed to his/her TPACK (71%

4 Figure 2: Relative frequency of the answers y = af(b(a m) )+ n by hand was deduced and exemplified by graphing the functions y = (2x 4)2 5 and y =2tan(2 2x) + 1 (Figure 1). At the end of the lesson, the students answered the questionnaire. The result is graphed in Figure 2. The result is: (A) 92% found the lesson interesting (71% very, 21% quite); (B) 100% understood the topic (96% all, 4% most); (C) 89% had the feeling that the lesson contributed to his/her TPACK (71% a lot, 18% quite a lot); (D) 97% found the applications to be benefitting (86% a lot, 11 % quite a lot). (E) 93% were of the opinion that the developing helped comprehend the core (54% a lot, 39% quite a lot). The women found the lesson more interesting and benefitting than the men. SOLVING INEQUALITIES Solving an inequality analytically may be an intricate process that takes many minutes. With Excel, the solution is quick, transparent and credible. The inequality 3 5x x 2 x 2 is resolved by using the application in Figure 3 and 4. The solution is clear from the position of the graphs. Lesson and survey: There were 21 students, 5 men and 16 women. The analytical solution to the inequality was quickly gone through. Then the students developed the application from the application on the right side of Figure 1 and resolved the inequality in two ways. First, the intersection point was found by iteration (Figure 4). The maximum and minimum of axis x (cells C9:C10) were set up to be close to the intersection point from both sides. Then, the x coordinate of the intersection was found in the first column of range B16:D5016 in the row in which the values in the second and third column were equal (cell B3090). The other way of solving was by using Goal Seek (Figure 3, range K12:N13). The solution is x [ 0.19, 1.14] (2, ). The exact solution is x [ 5 29 x 2, x 2 ] (2, ); however, the bounds have to be converted into decimal numbers for practical use, which is the form shown above. At the end, the students answered the questionnaire. The result is graphed in Figure 5. The result is: (A) 96% found the lesson interesting (67% very, 29% quite); (B) 100% understood (90% all, 10% most); (C) 96% had the feeling that the lesson contributed to his/her TPACK (67% a lot, 29% quite a lot); (D) 100% found the applications to be benefitting (71% a lot, 29% quite a lot). (E) 100% were of the opinion that the developing helped comprehend the core (81% a lot, Figure 3: Solution with Goal Seek Figure 4: Solution by iteration 2313

5 Figure 5: Relative frequency of the answers 19% quite a lot). The men found the lesson more interesting while the women found it more benefitting. SOLVING EQUATIONS WITH A PARAMETER BY USING ANIMATION Solving an equation with a parameter is a hard task at upper secondary level. The graphical interpretation helps. If the parameter is changed quickly, the relation between the parameter and the solution can be easily revealed. The left side of the equation (p + 1)x2 + px + p = 0 is graphed in the application in Figure 6. Parameter p is controlled by a spinbutton. Clicking it and holding down animates the graph. The positions at other values of parameter p are depicted in Figure 7. The solution is: p (, 1) ( 1, 0.8): p = 1: x = 1 p = 0.8: x = 2 p = 0: x = 0 p ( 0.8, 0) (0, ): x { } x = p ± p2 4(p + 1) p 2(p + 1) Lesson and survey: There were 23 students, 4 men and 19 women. The analytical solution to the equation was quickly gone through. The students developed the application from the application on the left side of Figure 1. They found that the graph is in a special position if p = 0, 0.8 or 1. The solution at these boundary values was obtained by substitution. The solution for the values inside the intervals given by the boundary values was obtained by applying the quadratic equation. At the end, they answered the questionnaire. The result is graphed in Figure 8. The result is: (A) 87% found the lesson interesting (61% very, 26% quite); (B) 96% understood the topic (83% all, 13% most); (C) 83% had the feeling that the lesson contributed to his/her TPACK (57% a lot, 26% quite a lot); (D) 83% found the applications to be benefitting (65% a lot, 18% quite a lot). (E) 87% were of the opinion that the developing helped comprehend the core (65% a lot, 22% quite a lot). The women found the lesson more interesting and benefitting than the men. INTERACTIVE SOLUTION TO A QUADRATIC EQUATION AND TO A SYSTEM OF THREE LINEAR EQUATIONS WITH THREE VARIABLES Solving quadratic equations and systems of three linear equations with three variables are skills that are often applied in solving tasks at upper secondary level. The following task requires applying both skills: A projectile was fired in a vacuum. Three points of the trajectory were detected by radar. The trajectory is a parabola. Find the points of shot and impact. Figure 6: Position of the graph at p = 2 Substitution of the coordinates of the points in the equation y = ax2 + bx + c yields a system of three linear equations with variables a, b, and c. In Figure 9, the system is solved by using the Gaussian elimination method (GEM). 2314

The system always has a unique solution, which stems from the physics of the task. The discriminant of equation ax2 + bx + c = 0 is in cell L10. It is always positive.the roots are in cells L13 a L14.

6 Figure 7: Positions at p = 0; -0.5; -0.8; -0.9; -1; -1.1 (left to right by rows) Figure 8: Relative frequency of the answers The matrix is in range N5:Q7. The upper triangular matrix is obtained in four steps over ranges N9:Q11, N13:Q15, N17:Q19 and N21:Q23. Variables a, b and c are calculated in range C3:C5. The system always has a unique solution, which stems from the physics of the task. The discriminant of equation ax2 + bx + c = 0 is in cell L10. It is always positive.the roots are in cells L13 a L14. The vertex coordinates are in cells K18 a L18. The application is interactive if the coordinates of the points in range K5:L7 change, the point of shot and impact (L13, L14) are recalculated. Lesson and survey: There were 24 students, 4 men and 20 women. The solution to the systems of linear equations by GEM was gone through. The students developed the application in Fig 9 from the application on the left side of Figure 1 and resolved the task. At the end, they answered the questions. The result is in Figure 10. The result is: (A) 92% found the lesson interesting (54% very, 38% quite); (B) 100% understood the topic (83% all, 17% most); (C) 92% had the feeling that the lesson contributed to his/her TPACK (54% a lot, 38% quite a lot); (D) 92% found the applications benefitting (63% a lot, 29% quite a lot); (E) 96% were of the opinion that the developing helped comprehend the core (50% a lot, 46% quite a lot). The men found the lesson more interesting while the women found it more benefitting. SURVEY SUMMARY Altogether 96 questionnaires were answered, 19 by men and 77 by women. The result is graphed in Figure

Figure 9: Trajectory of a projectile in a vacuum Figure 10: Relative frequency of the answers Figure 11: Relative frequency of the answers in total The result is: (A) 92% found the lesson interesting

7 Figure 9: Trajectory of a projectile in a vacuum Figure 10: Relative frequency of the answers Figure 11: Relative frequency of the answers in total The result is: (A) 92% found the lesson interesting (64% very, 28% quite); (B) 99% understood the topic (89% all, 10% most); (C) 90% had the feeling that the lesson contributed to his/her TPACK (63% a lot, 27% quite a lot); (D) 93% found the applications to be benefitting (72% a lot, 21% quite a lot). (E) 94% were of the opinion that the developing helped comprehend the core (62% a lot, 32% quite a lot). The outcome for all, men and women was: A) The lesson was interesting: 92%, 84 %, 94 % B) I understood of the mathematics involved: 99%, 100 %, 99 % C) The lesson contributed to my TPACK: 90%, 69 %, 95 % D) The applications benefit the teaching of the topic: 93%, 79 %, 96 % E) Developing the applications helps comprehend the core: 94%, 90 %, 95 % F) I am a man: 20% 2316

8 The lessons contributed considerably more to the TPACK of the women than men (difference of 26%). The women found the applications more benefitting the teaching of the topics and more interesting than the men (difference of 17% and 9%). The women were stronger of the opinion that developing the applications helps comprehend the core of the topic (difference of 5%). The result implies that the presented method of teaching mathematics through developing spreadsheet applications that model and solve tasks is of interest to pre-service teachers, benefits the teaching of the topics and contributes to the technological and pedagogical knowledge of the teachers, mainly women. The outcomes correspond well with the result of the authors research with high school students (Benacka and Ceretkova, 2013). That suggests that learning by doing with spreadsheets has a potential in promoting mathematics at upper secondary level. ACKNOWLEDGEMENT The authors are members of the team of LLP Comenius project LLP AT-COMENIUS-CAM. REFERENCES Ainley, J., Bills, L., & Wilson, K. (2005). Designing spreadsheet-based tasks for purposeful algebra. International Journal of Computers for Mathematical Learning, 10(3), Benacka, J., & Ceretkova, S. (2013). Excel modelling in upper secondary mathematics a few tips for learning functions and calculus. In B. Ubuz, C. Haser, & M. A. Mariotti (Eds.), Proceedings of CERME 8 (pp ). Ankara, Turkey: Middle East Technical University and ERME. Drake, P., Wake, G., & Noyes, A. (2012). Assessing functionality in school mathematics examinations: what does being human have to do with it? Research in Mathematics Education, 14(3), Forster, P. A. (2007). Technologies for teaching and learning trend in bivariate data. International Journal of Mathematical Education in Science and Technology, 38(2), Geiger, V., Goos, M., & Dole, S. (2014). The role of digital technologies in numeracy teaching and learning. International Journal of Science and Mathematics Education. DOI /s Gonzalez-Calero, J. A., Arnau, D., & Puig, L. (2013). Solving word problems algebraically in a spreadsheet environment in a primary schol. Research in Mathematics Education, 15(3), Haspekian, M. (2005). An instrumental approach to study the integration of a computer tool into mathematics teaching: the case of spreadsheets. International Journal of Computers for Mathematical Learning, 10(2), Haspekian, M. (2011). The co-construction of a mathematical and a didactical instrument. In M. Pytlak, T. Rowland, & E. Swoboda (Eds.), Proceedings of CERME 7 (pp ). Rzeszów, Poland: University of Rzeszów. Haspekian, M. (2014). Teachers instrumental geneses when integrating spreadsheet software. In A. Clark-Wilson, O. Robutti, & N. Sinclair (Eds.), The Mathematics Teacher in the Digital Era: An International Perspective on Technology Focused Professional Development (pp ). London, UK: Springer. Healy, L., & Sutherland, R. (1990). The use of spreadsheets within the mathematics classroom. International Journal of Mathematical Education in Science and Technology, 21(6), Hošpesová, A. (2002a). What brings use of spreadsheets in the classroom of 11-years olds? In J. Novotná (Ed.), Proceedings of CERME 2 (pp ). Prague, Czech Republic: Charles University, Faculty of Education. Hošpesová, A. (2002b). Are spreadsheets worthwhile for all? In L. Bazzini & W. C. Inchley (Eds.), Proceedings of CIEAEM 53 (pp ). Milano, Italy: Ghisetti e Corvii Editori. Molyneux-Hodgson, S., Rojano, T., Sutherland, R., & Ursini, S. (1999). Mathematical modelling: the interaction of culture and practice. Educational Studies in Mathematics, 39(1 3), Neurath, R. A., & Stephens, L. J. (2006). The effect of using Microsoft Excel in a high school algebra class. International Journal of Mathematical Education in Science and Technology, 37(6), O Reilly, D. (2006). Learning together: student teachers, children and spreadsheets. Research in Mathematics Education, 8(1), Rojano, T., & Sutherland, R. (1993). Towards an algebraic approach: the role of spreadsheets. In Proceedings of the 17th International Conference for the Psychology of Mathematics Education (pp ). Tsukuba, Japan: University of Tsukuba. Sivasubramaniama, P. (2000). Distributed cognition, computers and the interpretation of graphs. Research in Mathematics Education, 2(1), Tabach, M., Arcavi, A., & Hershkowitz, R. (2008). Transitions among different symbolic generalizations by algebra beginners in a computer intensive environment. Educational Studies in Mathematics, 69(1),

9 Tabach, M., & Friedlander, A. (2008). Understanding equivalence of symbolic expressions in a spreadsheet-based environment. International Journal of Computers for Mathematical Learning, 13(1), Tabach, M., Hershkowitz, R., & Arcavi, A. (2008). Learning beginning algebra with spreadsheets in a computer intensive environment. Journal of Mathematical Behavior, 27(1), Topcu, A. (2011). Effects of using spreadsheets on secondary school students self-efficacy for algebra. International Journal of Mathematical Education in Science and Technology, 42(5), Watson, J., & Callingham, R. (2013). Likelihood and sample size: The understandings of students and their teachers. Journal of Mathematical Behavior, 32(3), Wilson, K. (2006). Naming a column on a spreadsheet. Research in Mathematics Education, 8(1),

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Students concept images of inverse functions Sinéad Breen, Niclas Larson, Ann O Shea, Kerstin Pettersson To cite this version: Sinéad Breen, Niclas Larson, Ann O Shea, Kerstin Pettersson. Students concept