Abstract
The paper describes two versions of an inquiry-based activity in geometry, designed as a game between two players. The game is inspired by Hintikka’s semantic game, which is a familiar tool in the field of logic to define truth. The activity is designed in a dynamic geometry environment (DGE). The inquiry is initially guided by the game itself and later by a questionnaire that helps students discover the geometry theorem behind the game. The activity is emblematic of describing a geometry-based inquiry that can be implemented with various Euclidean geometry theorems. The analysis of the first “student vs. student” version associates the example space produced by the students with their dialogue, to identify the different functions of variation. Based on the results of this version, we designed a “student vs. computer” version and created filters for the automatic analysis of the players’ moves. Our findings show that students who participated in the activity developed forms of strategic reasoning that helped them discover the winning configuration, formulate if-then statements, and validate or refute conjectures. Automation of the analysis creates new research opportunities for analyzing and assessing students’ inquiry processes and makes possible extensive experimentation on inquiry-based knowledge acquisition.
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Notes
We thank one of the reviewers for pointing out this interpretation.
References
Antonini, S. (2003). Non-examples and proof by contradiction. In N. Pateman, B. J. Dougherty, & J. Zilliox (Eds.), Proceedings of the 27 th annual meeting of the International Group for Psychology in Mathematics Education (Vol. 2, pp. 49–55). Honolulu: University of Hawaii.
Arbib, M. A. (1990). A Piagetian perspective on mathematical construction. Synthese, 84(1), 43–58.
Arzarello, F., Olivero, F., Paola, D., & Robutti, O. (2002). A cognitive analysis of dragging practises in Cabri environments. ZDM, 34(3), 66–72.
Arzarello, F., & Sabena, C. (2011). Semiotic and theoretic control in argumentation and proof activities. Educational Studies in Mathematics, 77(2–3), 189–206.
Baccaglini-Frank, A., & Mariotti, M. A. (2010). Generating conjectures in dynamic geometry: The maintaining dragging model. International Journal of Computers for Mathematical Learning, 15(3), 225–253.
Balacheff, N. (1988). Aspects of proof in pupils’ practice of school mathematics. In D. Pimm (Ed.), Mathematics, teachers and children (pp. 216–235). London: Holdder & Stoughton.
Balacheff, N. (1999). Is argumentation an obstacle? Invitation to a debate. Preuve International Newsletter on the Teaching and Learning of Mathematical Proof, 1–7.
Brousseau, G. (1997). Theory of didactical situations in mathematics. Dordrecht, The Netherlands: Kluwer Academic Publishers.
Chazan, D. (1993). High school geometry students’ justification for their views of empirical evidence and mathematical proof. Educational Studies in Mathematics, 24(4), 359–387.
Cobb, P., Jackson, K., & Dunlap, C. (2015). Design research: An analysis and critique. In L. English & D. Kirshner (Eds.), Handbook of international research in mathematics education (3rd ed., pp. 481–503). New York: Routledge.
de Villiers, M. D. (2010). Experimentation and proof in mathematics. In G. Hanna, H. N. Jahnke, & H. Pulte (Eds.), Explanation and proof in mathematics: Philosophical and educational perspectives (pp. 205–221). New York: Springer.
Gómez Chacón, I. M. (1992). Desarrollo de diversos juegos de estrategia para su utilización en el aula. Epsilon, 22, 77–88.
Harel, G. (2001). The development of mathematical induction as a proof scheme: A model for DNR-based instruction. In S. Campbell & R. Zazkis (Eds.), The learning and theaching of number theory (pp. 185–212). Dordrecht, The Netherlands: Kluwer.
Healy, L. (2000). Identifying and explaining geometrical relationship: Interactions with robust and soft Cabri constructions. In T. Nakahara & M. Koyama (Eds.), Proceedings of the 24 th conference of the International Group for the Psychology of Mathematics Education (PME) (Vol. 1, pp. 103–117). Hiroshima, Japan: PME.
Hintikka, J. (1998). The principles of mathematics revisited. Cambridge: Cambridge University Press.
Hintikka, J. (1999). Is logic the key to all good reasoning?. In Inquiry as inquiry: A logic of scientific discovery. Jaakko Hintikka Selected Papers (Vol. 5). Dordrecht: Springer.
Hoyles, C., & Jones, K. (1998). Proof in dynamic geometry contexts. In C. Mammana & V. Villani (Eds.), Perspective on the teaching of geometry for the 21st century (pp. 121–128). Dordrecht: Kluwer Academic Publishers.
Laborde, C. (1993). The computer as part of the learning environment: The case of geometry. In C. Keitel & K. Ruthven (Eds.), Learning from computers: Mathematics education and technology (pp. 48–67). Berlin: Springer-Verlag.
Laborde, C. (2005). The hidden role of diagrams in students’ construction of meaning in geometry. In J. Kilpatrick, C. Hoyles, & O. Skovsmose (Eds.), Meaning in mathematics education. Mathematics education library (Vol. 37). Boston, MA: Springer.
Leung, A. (2008). Dragging in a dynamic geometry environment through the lens of variation. International Journal of Computers for Mathematical Learning, 13(2), 135–157.
Luz, Y., & Yerushalmy, M. (2015). E-assessment of geometrical proofs using interactive diagrams. In K. Krainer & N. Vondrová (Eds.), Proceedings of the ninth conference of the European Society for Research in Mathematics Education (CERME9) (pp. 149–155). Prague, Czech Republic: Charles University.
Maheux, J.-F., & Proulx, J. (2015). Doing|mathematics: Analysing data with/in an enactivist-inspired approach. ZDM—The International Journal on Mathematics Education, 47(2), 211–221.
Marton, F., Runesson, U., & Tsui, A. B. M. (2004). The space of learning. In F. Marton & A. B. M. Tsui (Eds.), Classroom discourse and the space of learning (pp. 3–40). Mahwah, NJ: L. Erlbaum Associates.
Mason, J., & Pimm, D. (1984). Generic examples: Seeing the general in the particular. Educational Studies in Mathematics, 15(3), 277–289.
Prediger, S., Gravemeijer, K., & Confrey, J. (2015). Design research with a focus on learning processes: An overview on achievements and challenges. ZDM Mathematics Education, 47(6), 877–891.
Soldano, C., & Arzarello, F. (2016). Learning with touchscreen devices: A game approach as strategies to improve geometric thinking. Mathematics Education Research Journal, 28, 9–30.
Weber, K. (2001). Student difficulty in constructing proofs: The need for strategic knowledge. Educational Studies in Mathematics, 48(1), 101–119.
Yerushalmy, M., & Chazan, D. (1992). Guided inquiry and geometry. Some aspects of teaching with technology. Zentralblatt Fur Didaktik Der Mathematik (ZDM), 24, 172–177.
Yerushalmy, M., Chazan, D., & Gordon, M. (1990). Mathematical problem posing: Implications for facilitating student inquiry in classrooms. Instructional Science, 19(3), 219–245.
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Soldano, C., Luz, Y., Arzarello, F. et al. Technology-based inquiry in geometry: semantic games through the lens of variation. Educ Stud Math 100, 7–23 (2019). https://doi.org/10.1007/s10649-018-9841-4
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DOI: https://doi.org/10.1007/s10649-018-9841-4