Abstract
In this article, we provide an empirical example of how digital technology; in this case, GeoGebra may assist students in uncovering—or whiteboxing—the content of a mathematical proof, in this case that of Proposition 41 from Euclid’s Elements. In the discussion of the example, we look into the impact of GeoGebra’s “dragging” functionality on students’ interactions and the possession and development of students’ proof schemes. The study and accompanying analysis illustrate that, despite the positive whiteboxing effects in relation to the mathematical content of the proposition, whiteboxing through dragging calls for caution in relation to students’ work with proof and proving—in particular, in relation to students seeing the necessity for formal proof. Moreover, caution must be paid, e.g., by teachers, so that students do not jump to conclusions and in the process develop inexpedient mathematical proof schemes upon which they may stumble in their future mathematical work.
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References
Aguilar, M., & Zavaleta, J. (2015). The difference as an analysis tool of the change of geometric magnitudes: The case of the circle. In E. Barbin, U. Jankvist & T. Kjeldsen (Eds), Proceedings of the seventh European summer university (pp. 391–399). Copenhagen, Denmark: The Danish School of Education, Aarhus University.
Artigue, M. (2002). Learning mathematics in a CAS environment: The genesis of a reflection about instrumentation and the dialectics between technical and conceptual work. International Journal of Computers for Mathematical Learning, 7(3), 245–274.
Artigue, M. (2010). The future of teaching and learning mathematics with digital technologies. In C. Hoyles & J.-B. Lagrange (Eds.), Mathematics education and technology: Rethinking the terrain. (pp. 463–475). Springer.
Arzarello, F., Olivero, F., Paola, D., & Robutti, O. (2002). A cognitive analysis of dragging practices in Cabri environments. ZDM: The International Journal on Mathematics Education, 34(3), 66–72.
Baki, A., & Guven, B. (2009). Khayyam with Cabri: Experiences of pre-service mathematics teachers with Khayyam’s solution of cubic equation in dynamic geometry environment. Teaching Mathematics and its Applications, 28(1), 1–9.
Balsløv, C. (2018). The mutual benefits of using CAS and original sources in the teaching of mathematics. Unpublished Master’s Thesis. Copenhagen, Denmark: The Danish School of Education, Aarhus University.
Barbin, E. (1997). Histoire et enseignement des mathématiques: Pourquoi? Comment? Bulletin de l’Association mathématique du Québec, 37(1), 20–25.
Brousseau, G. (1997). Theory of didactical situations in mathematics. . Kluwer Academic Publishers.
Buchberger, B. (1990). Should students learn integration rules? ACM SIGSAM Bulletin, 24(1), 10–17.
Buchberger, B. (2002). Computer algebra: The end of mathematics? ACM SIGSAM Bulletin, 36(1), 3–9.
Cedillo, T., & Kieran, C. (2003). Initiating students into algebra with symbol-manipulating calculators. In J. Fey, A. Cuoco, C. Kieran, L. McMullin, & R. Zbiek (Eds.), Computer algebra systems in secondary school mathematics education. (pp. 219–239). National Council of Teachers of Mathematics.
Chorlay, R. (2015). Making (more) sense of the derivative by combining historical sources and ICT. In E. Barbin, U. Jankvist, and T. Kjeldsen (Eds), Proceedings of the Seventh European Summer University on History and Epistemology in Mathematics Education (pp. 485–498). Copenhagen, Denmark: The Danish School of Education, Aarhus University.
Dreyfus, T. (1999). Why Johnny can’t prove. Educational Studies in Mathematics, 38(1–3), 85–109.
Dubinsky, E., Dautermann, J., Leron, U., & Zazkis, R. (1994). On learning fundamental concepts of group theory. Educational Studies in Mathematics, 27(3), 267–305.
Education Committee of the European Mathematical Society. (2011). Do theorems admit exceptions? Solid findings in mathematics education on empirical proof schemes. EMS Newsletter, 82, 50–53.
Fitzpatrick, R. (2008). Euclid’s elements of geometry. (http://farside.ph.utexas.edu/Books/Euclid/Elements.pdf) [The Greek text of J.L. Heiberg (1883–1885) from Euclidis Elementa, edidit et Latine interpretatus est I.L. Heiberg, in aedibus B.G. Teubneri, 1883–1885 edited, and provided with a modern English translation, by Richard Fitzpatrick.]
Furinghetti, F., Jahnke, H., & van Maanen, J. (2006). Mini-workshop on studying original sources in mathematics education. Oberwolfach Reports, 3(2), 1285–1318.
Guven, B. (2008). Using dynamic geometry software to gain insight into a proof. International Journal of Computers for Mathematical Learning, 13(3), 251–262.
Harel, G., & Sowder, L. (1998). Students’ proof schemes: Results from exploratory studies. In A. Schoenfeld, J. Kaput, & E. Dubinsky (Eds.), Research in Collegiate Mathematics Education III. (pp. 234–283). The American Mathematics Society.
Harel, G., & Sowder, L. (2007). Toward comprehensive perspectives on the learning and teaching of proof F. In Lester (Ed.), Second handbook of research on mathematics teaching and learning. (pp. 805–842). Information Age Publishing.
Healy, L., & Hoyles, C. (2002). Software tools for geometrical problem solving: Potentials and pitfalls. International Journal of Computers for Mathematical Learning, 6(3), 235–256.
Isoda, M. (2004). Why we use historical tools and computer software in mathematics education: mathematics activity as a human endeavor project for secondary school. In F. Furinghetti, S. Kaijser & C. Tzanakis (Eds), Proceedings of HPM2004 & ESU4, revised edition (pp. 229–236). Uppsala University.
Jahnke, H., Arcavi, A., Barbin, E., Bekken, O., Furinghetti, F., El Idrissi, A., Silva da Silva, C., & Weeks, C. (2000). The use of original sources in the mathematics classroom. In J. Fauvel & J. van Maanen (Eds.), History in Mathematics Education. (pp. 291–328). Kluwer Academic Publishers.
Jankvist, U. (2009). A categorization of the ‘whys’ and ‘hows’ of using history in mathematics education. Educational Studies in Mathematics, 71(3), 235–261.
Jankvist, U., & Geraniou, E. (2019). Digital technologies as a way of making original sources more accessible to students. In E. Barbin, U. Jankvist, T. Kjelsen, B. Smestad & C. Tzanakis (Eds), Proceedings of the Eighth European Summer University on History and Epistemology in Mathematics Education (pp. 107–130). Oslo Metropolitan University.
Jankvist, U., & Misfeldt, M. (2015). CAS-induced difficulties in learning mathematics? For the Learning of Mathematics, 35(1), 15–20.
Jankvist, U., & Misfeldt, M. (2019). CAS-assisted proofs in upper secondary school mathematics textbooks. REDIMAT Journal of Research in Mathematics Education, 8(3), 232–266.
Jankvist, U., Misfeldt, M., & Aguilar, M. (2019a). Tschirnhaus’ transformation: Mathematical proof, history and CAS. In E. Barbin, U. Jankvist, T. Kjelsen, B. Smestad & C. Tzanakis (Eds), Proceedings of the Eighth European Summer University on History and Epistemology in Mathematics Education (pp. 319–330). Oslo Metropolitan University.
Jankvist, U., Misfeldt, M., & Aguilar, M. (2019). What happens when CAS procedures are objectified? The case of “solve” and “desolve.” Educational Studies in Mathematics, 101(1), 67–81.
Kieran, C., & Drijvers, P. (2006). The co-emergence of machine techniques, paper-and-pencil techniques, and theoretical reflection: A study of CAS use in secondary school algebra. International Journal of Computers for Mathematical Learning, 11(2), 205–263.
Kjeldsen, T., & Blomhøj, M. (2012). Beyond motivation: History as a method for learning meta-discursive rules in mathematics. Educational Studies in Mathematics, 80(3), 327–349.
Laborde, C. (2000). Dynamic geometry environment as a source of rich learning context for the complex activity of proving. Educational Studies in Mathematics, 44(1–2), 151–161.
Lagrange, J. (2005). Using symbolic calculators to study mathematics: The case of tasks and techniques. In D. Guin, K. Ruthven, & L. Trouche (Eds.), The didactical challenge of symbolic calculators: Turning a computational device into a mathematical instrument. (pp. 113–135). Springer.
Lopez-Real, F., & Leung, A. (2006). Dragging as a conceptual tool in dynamic geometry environments. International Journal of Mathematical Education in Science and Technology, 37(6), 665–679.
Mariotti, M. (2002). Justifying and proving in the Cabri environment. International Journal of Computers for Mathematical Learning, 6(3), 257–281.
Mariotti, M. (2006). Proof and proving in mathematics education. In A. Gutiérrez & P. Boero (Eds.), Handbook of research on the psychology of mathematics education: Past, present and future. (pp. 173–204). Sense Publishers.
Mason, J. (1991). Questions about geometry. In D. Pimm & E. Love (Eds.), Teaching and learning school mathematics: A reader. (pp. 77–90). Hodder and Stoughton.
Misfeldt, M. & Jankvist, U. (2018). Instrumental genesis and proof: Understanding the use of computer algebra systems in proofs in textbook. In. L. Ball, P. Drijvers, S. Ladel, H.-S. Siller, M. Tabach & C. Vale (Eds), Uses of technology in K–12 mathematics education: Tools, topics and trends (pp. 375–385). Springer-Verlag.
Monaghan, J., Trouche, L., & Borwein, J. (2016). Tools and mathematics: Instruments for learning. . Springer.
Nabb, K. (2010). CAS as a restructuring tool in mathematics education. In P. Bogacki (Ed.), Electronic proceedings of the 22nd international conference on technology in collegiate mathematics (pp. 247–259). (http://archives.math.utk.edu/ICTCM/VOL22/R007/paper.pdf)
Olsen, I., & Thomsen, M. (2017). History of mathematics and ICT in mathematics education in primary education. Unpublished Master’s Thesis. Copenhagen, Denmark: The Danish School of Education, Aarhus University.
Papadopoulos, I. (2014). How Archimedes helped students to unravel the mystery of the magical number pi. Science & Education, 23(1), 61–77.
Pengelley, D. (2011). Teaching with primary historical sources: Should it go mainstream? Can it? In V. Katz & C. Tzanakis (Eds.), Recent developments on introducing a historical dimension in mathematics education. (pp. 1–8). The Mathematical Association of America.
Rabardel, P., & Bourmaud, G. (2003). From computer to instrument system: A developmental perspective. Interacting with Computers, 15(5), 665–691.
Sfard, A. (2008). Thinking as communicating: Human development, the growth of discourses, and mathematizing.Cambridge University Press.
Thomsen, M. (2021). Working with Euclid’s geometry in GeoGebra: Experiencing embedded discourses. Paper presented at the Ninth Nordic Conference on Mathematics Education. Oslo, Norway.
Thomsen, M. & Jankvist, U. (2020). Reasoning with digital technologies: Counteracting students’ techno-authoritarian proof schemes. In A. Donevska-Todorova, E. Faggiano, J. Trgalova, Z. Lavicza, R. Weinhandl, A. Clark-Wilson & H.-G. Weigand (Eds), Proceedings of the Tenth ERME Topic Study Conference on Mathematics Education in the Digital Age (pp. 483–490). Johannes Kepler University.
Trouche, L. (2005). Calculators in mathematics education: A rapid evolution of tools, with differential effects. In D. Guin, K. Ruthven, & L. Trouche (Eds.), The didactical challenge of symbolic calculators: Turning a computational device into a mathematical instrument. (pp. 11–40). Springer.
Vygotsky, L. (1978). Mind in society: The development of higher psychological processes. . Harvard University Press.
Zengin, Y. (2018). Incorporating the dynamic mathematics software GeoGebra into a history of mathematics course. International Journal of Mathematical Education in Science and Technology, 49(7), 1083–1098.
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Jankvist, U.T., Geraniou, E. “Whiteboxing” the Content of a Formal Mathematical Text in a Dynamic Geometry Environment. Digit Exp Math Educ 7, 222–246 (2021). https://doi.org/10.1007/s40751-021-00088-6
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DOI: https://doi.org/10.1007/s40751-021-00088-6