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The Use of Digital Technology to Estimate a Value of Pi: Teachers’ Solutions on Squaring the Circle in a Graduate Course in Brazil

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Abstract

In this research we investigate how mathematics teachers, as graduate students, estimate the value of π by exploring the problem of squaring the circle using digital technology. Initially, we mention some aspects of teaching and learning of calculus in the literature, emphasizing studies that use the notion of humans-with-media to highlight the role of technology in mathematical thinking and knowledge production. Insights on the history of mathematics in calculus are also discussed. We developed a qualitative study based on three different solutions created by groups of teachers using the software GeoGebra and Microsoft Excel. All the teachers’ solutions improved the approximation (8/9)2 \(\approx\) π/4, by determining p and q for (p/q)2 \(\approx\) π/4. The first two solutions with GeoGebra emphasized experimentation and visualization, improving the approximation from one to three decimal places. The third solution with Excel pointed out the elaboration of a formula and improved it up to six decimal places. We emphasize how media shaped the strategies and solutions of the groups. Based on these solutions, we explore an approach for cubing the sphere, discussing approximations for π, highlighting the role of media in enhancing conceptual complexity in the solution of mathematical problems. The nature of the strategies for solving a problem is discussed, especially regarding different ways of thinking-with-technology developed by collectives of teachers-with-media. Although we acknowledge an alternative design for the proposed task, the exploration of problems using aspects of the history of mathematics contributes to the state of the art concerning the studies in calculus developed by the Research Group on Technology, other Media, and Mathematics Education in Brazil.

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References

  • Araújo, J. L. (2002). Calculus, technology, and mathematical modeling: Students’ discussions. 173f. Dissertation (PhD in Mathematics Education), Sao Paulo State University: Rio Claro, Sao Paulo, Brazil.

  • Araújo, J. L., & Borba, M. C. (2004). Constructing research collaboratively in mathematics education. In M. C. Borba & J. L. Araújo (Eds.), Qualitative research in mathematics education (pp. 25–45). Belo Horizonte.

    Google Scholar 

  • Barbosa, M. A. (2009). Information and communication technology, Function composition, and chain rule. 199f. Dissertation (PhD in Mathematics Education), Sao Paulo State University: Rio Claro, Sao Paulo, Brazil.

  • Baron, M. E. (1969). The origins of the infinitesimal calculus. Pergamon Press.

    Google Scholar 

  • Baron, M. E. (1985). History of mathematics course: Origins and development of calculus. Editora Universidade de Brasília.

    Google Scholar 

  • Bezuidenhout, J. (2001). Limits and continuity: Some conceptions of first-year students. International Journal of Mathematical Education in Science and Technology, 32, 487–500. https://doi.org/10.1080/00207390010022590

    Article  Google Scholar 

  • Bicudo, M. A. V. (1993). Research in mathematics education. Pro-Posições (UNICAMP), 1(10), 8–23.

    Google Scholar 

  • Borba, M. C. (1993). Students understanding of transformations of functions using multirepresentational software. Faculty of graduate school of Cornell University.

    Google Scholar 

  • Borba, M. C. (2012). Humans-With-Media and continuing education for mathematics teachers in online environments. ZDM, 44(6), 1–14. https://doi.org/10.1007/s11858-012-0436-8

    Article  Google Scholar 

  • Borba, M. C., Almeida, H. R. F. L., & Gracias, T. A. S. (2018). Research on teaching and classroom: Different voices in an investigation. Autêntica.

    Google Scholar 

  • Borba, M. C., Scucuglia, R. R. S., & Gadanidis, G. (2014). Phases of digital technology in mathematics education: Classroom and internet in movement. Autêntica.

    Google Scholar 

  • Borba, M. C., & Villarreal, M. E. (2005). Humans-With-Media and the reorganization of mathematical thinking: information and communication technologies, modeling, experimentation and visualization. Springer.

    Google Scholar 

  • Byerley, C., & Thompson, P. W. (2017). Teachers’ meanings for measure, slope, and rate of change. Journal of Mathematical Behavior, 48, 168–193.

    Article  Google Scholar 

  • D’Ambrosio, U., & Borba, M. C. (2010). Dynamics of change of mathematics education in Brazil and a scenario of current research. ZDM Mathematics Education, 42(3–4), 271–279. http://hdl.handle.net/11449/71691. Accessed 5 Feb 2019.

  • de Sousa, G. C., & de Gomes, A. B. A. (2020). Support for the promotion of historic-with-technology activities. Research, Society and Development, 9(5), e86953206. https://doi.org/10.33448/rsd-v9i5.3206

    Article  Google Scholar 

  • Domingues, A. R., & Scucuglia, R. R. S. (2019). Experiência matemática estética: O uso da história da matemática e do software geogebra no estudo da geometria espacial [Aesthetic mathematical experience: The use of the history of mathematics and the software geogebra in the study of spatial geometry]. In Anais do VII Congresso Brasileiro de Educação [Proceedings of the VII Brazilian Congress of Education] (pp. 1–15). Bauru: UNESP.

  • Easterday, K., & Smith, T. (1991). A Monte Carlo application to approximate pi. Mathematics Teacher, 84(5), 387–390.

    Article  Google Scholar 

  • Edge, O., & Friedberg, S. (2015). Factors affecting achievement in the first course in calculus. The Journal of Experimental Education, 52, 136–140. https://doi.org/10.1080/00220973.1984.11011882

    Article  Google Scholar 

  • Eichler, A., & Erens, R. (2014). Teachers’ beliefs towards teaching calculus. ZDM, 46(4), 647–659. https://doi.org/10.1007/s11858-014-0606-y

    Article  Google Scholar 

  • Engelbrecht, J., Llinares, S., & Borba, M. C. (2020). Transformation of the mathematics classroom with the internet. ZDM Mathematics Education, 52, 825–841. https://doi.org/10.1007/s11858-020-01176-4.

    Article  Google Scholar 

  • Hegedus, S. & Kaput, J. (2004). An introduction to the profound potential of connected algebra activities: Issues of representation, engagement and pedagogy. In Proceedings of the 28th conference of the international group for the psychology of mathematics education (Vol. 3, pp. 129–136). Bergen, Norway: Conference Committee.

  • Hegedus, S. J., & Moreno-Armella, L. (2009). Introduction: The transformative nature of “dynamic” educational technology. ZDM, 41, 397–398. https://doi.org/10.1007/s11858-009-0201-9

    Article  Google Scholar 

  • Hoffkamp, A. (2011). The use of interactive visualizations to foster the understanding of concepts of calculus: Design principles and empirical results. ZDM, 43, 359–372. https://doi.org/10.1007/s11858-011-0322-9

    Article  Google Scholar 

  • Hohenwarter, M., Hohenwarter, J., Kreis, Y., & Lavicza, Z. (2008). Teaching and learning calculus with free dynamic mathematics software GeoGebra. In Paper presented at the 11th international congress on mathematical education. Monterrey, Nuevo Leon, Mexico.

  • Honorato, V. S., & Scucuglia, R. R. S. (2019). Geometric construction in space: Discussing the design of a task. Boletim GEPEM, 74, 72–87.

    Google Scholar 

  • Javaroni, S. L. (2007). Geometric approach: Possibilities for the teaching and learning of Introduction to Ordinary Differential equations. 230f. Dissertation (PhD in Mathematics Education), Sao Paulo State University: Rio Claro, Sao Paulo, Brazil.

  • Kaput, J. J., & Thompson, P. W. (1994). Technology in mathematics education research: The first 25 years in JRME. Journal for Research in Mathematics Education, 25(6), 676–684.

    Article  Google Scholar 

  • Katz, V. J. (1993). Using the history of calculus to teach calculus. Science & Education, 2, 243–249. https://doi.org/10.1007/BF00490066

    Article  Google Scholar 

  • Lévy, P. (1993). Technologies of intelligence: The future of thinking in the informatics era (p. 34). Editora.

    Google Scholar 

  • Marshall, M. N. (1996). Sampling for qualitative research. Family Practice, 13, 522–525. https://doi.org/10.1093/fampra/13.6.522

    Article  Google Scholar 

  • Moreno-Armella, L. (2014). An essential tension in mathematics education. ZDM, 46, 621–633. https://doi.org/10.1007/s11858-014-0580-4

    Article  Google Scholar 

  • Ponte, J. P. (2006). Case study in mathematics education. Bolema, 25, 105–132.

    Google Scholar 

  • Powell, A. B., Francisco, J. M., & Maher, C. A. (2004). An approach to video data analysis to study the development of learners’ mathematical ideas and reasoning. Bolema, 17(21), 81–140.

    Google Scholar 

  • Rabardel, P., & Bourmaud, G. (2003). From computer to instrument system. Interacting with computers, 15, 665–691.

    Article  Google Scholar 

  • Rasmussen, C., Marrongelle, K., & Borba, M. C. (2014). Research on calculus: What do we know and where do we need to go? ZDM, 46, 507–515. https://doi.org/10.1007/s11858-014-0615-x

    Article  Google Scholar 

  • Rochowicz, J. A. (1996). The impact of using computers and calculators on calculus instruction: Various perceptions. Journal of computers in Mathematics and Science teaching, 15(4), 423–435.

    Google Scholar 

  • Scucuglia, R. R. S. (2006). The investigation of the Fundamental Theorem of Calculus with graphing calculators. 145f. Thesis (Masters in Mathematics Education), Sao Paulo State University: Rio Claro, Sao Paulo, Brazil.

  • Sierpinska, A., Bobos, G., & Knipping, C. (2008). Sources of students’ frustration in pre-university level, prerequisite mathematics courses. Instructional Science, 36(4), 289–320. https://doi.org/10.1007/s11251-007-9033-6

    Article  Google Scholar 

  • Soares, D. S. (2012). A pedagogic approach based on the analysis of models for biology majors: What is the role of the software. 341f. Dissertation (PhD in Mathematics Education), Sao Paulo State University: Rio Claro, Sao Paulo, Brazil.

  • Swidan, O., & Yerushalmy, M. (2014). Learning the indefinite integral in a dynamic and interactive technological environment. ZDM, 46, 517–531. https://doi.org/10.1007/s11858-014-0583-1

    Article  Google Scholar 

  • Tall, D. (1991). The psychology of advanced mathematical thinking. In D. Tall (Ed.), advanced mathematical thinking (pp. 3–21). Dordrecht: Kluwer.

    Chapter  Google Scholar 

  • Tall, D. (1993). Students’ difficulties in calculus. In Proceedings of Working Group 3 on Students’ Difficulties in Calculus, ICME7, Québec, Canada, pp. 13–28.

  • Tall, D., & Katz, M. (2014). A cognitive analysis of Cauchy’s conceptions of function, continuity, limit, and infinitesimal, with implications for teaching the calculus. Educational Studies in Mathematics, 86(1), 97–124. https://doi.org/10.1007/s10649-014-9531-9

    Article  Google Scholar 

  • Thompson, P. W. (1994). Images of rate and operational understanding of the Fundamental Theorem of Calculus. Educational Studies in Mathematics, 26(2–3), 229–274.

    Article  Google Scholar 

  • Thompson, P. W. (2016). Researching mathematical meanings for teaching. In L. English & D. Kirshner (Eds.), Handbook of international research in mathematics education (pp. 435–461). Taylor and Francis.

    Google Scholar 

  • Thompson, P. W., & Ashbrook, M. (2019). Calculus: Newton, Leibniz, and Robinson meet technology. http://patthompson.net/ThompsonCalc/

  • Thompson, P., Byerley, C., & Hatfield, N. (2013). A conceptual approach to calculus made possible by technology. Computers in the Schools. https://doi.org/10.1080/07380569.2013.768941

    Article  Google Scholar 

  • Tikhomirov, O. K. (1981). The psychological consequences of computarization. In J. V. Wertsch & M. E. Sharpe (Eds.), The concept of activity in Soviet Psychology (pp. 256–278). M. E. Sharpe.

    Google Scholar 

  • Villarreal. M. E. (1999). The mathematical thinking of university students on calculus and technology. 402f. Dissertation (PhD in Mathematics Education), Sao Paulo State University: Rio Claro, Sao Paulo, Brazil.

  • Villarreal, M. E., & Borba, M. C. (2010). Collectives of humans-with-media in mathematics education: Notebooks, blackboards, calculators, computers and … notebooks throughout 100 years of ICMI. ZDM, 42(1), 49–62. https://doi.org/10.1007/s11858-009-0207-3

    Article  Google Scholar 

  • Williamson, T. (2013). Calculating pi using the Monte Carlo method. The Physics Teacher, 51(8), 468–469. https://doi.org/10.1119/1.4824938

    Article  Google Scholar 

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Acknowledgements

CNPq—Brazilian Council for Scientific and Technological Development (428323/2018/-9).

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Authors and Affiliations

  1. São Paulo State University, UNESP, São José do Rio Preto, Brazil

    Ricardo Scucuglia Rodrigues da Silva

  2. São Paulo State University, UNESP, Rio Claro, Brazil

    Lara Martins Barbosa & Marcelo C. Borba

  3. Federal University of Pelotas, UFPEL, Pelotas, Brazil

    André Luis Andrejew Ferreira

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  1. Ricardo Scucuglia Rodrigues da Silva
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  2. Lara Martins Barbosa
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  3. Marcelo C. Borba
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  4. André Luis Andrejew Ferreira
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Correspondence to Ricardo Scucuglia Rodrigues da Silva.

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da Silva, R.S.R., Barbosa, L.M., Borba, M.C. et al. The Use of Digital Technology to Estimate a Value of Pi: Teachers’ Solutions on Squaring the Circle in a Graduate Course in Brazil. ZDM Mathematics Education 53, 605–619 (2021). https://doi.org/10.1007/s11858-021-01246-1

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