Abstract
This chapter brings together two intersecting areas of research in mathematics education: teaching and learning with dynamic geometry environments (DGEs) and the teaching and learning of proof. We focus on developments in the literature since 2001 and, in particular, on (a) the evolution of the notion of “proof” in school mathematics and its impact on the kinds of research questions and studies undertaken over the past decade—including increasing use of DGEs at the primary school level; and (b) the epistemological and cognitive nature of dragging and measuring as they relate to proof.
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References
Arzarello, F., Micheletti, C., Olivero, F., Paola, D., & Robutti, O. (1998). A model for analysing the transition to formal proofs in geometry. In A. Olivier & K. Newstead (Eds.), Proceedings of the 22nd Conference of the International Group for the Psychology of Mathematics Education(Vol. 2, pp. 24–31). Stellenbosh, South Africa: International Group for the Psychology of Mathematics Education.
Arzarello, F., Olivero, F., Paola, D., & Robutti, O. (1999). I problemi di costruzione geometrica con l’aiuto di Cabri. L’insegnamento della matematica e delle scienze integrate, 22B(4), 309–338.
Arzarello, F., Olivero, F., Paola, D., & Robutti, O. (2002). A cognitive analysis of dragging practises in Cabri environments. ZDM—Zentralblatt fur Didaktik der Mathematik, 34(3), 66–72.
Arzarello, F., & Robutti, O. (2010). Multimodality in multi-representational environments. ZDM—The International Journal on Mathematics Education, 42(7), 715–731.
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. (1987). Processus de prevue et situations de validation. Educational Studies in Mathematics, 8, 147–176.
Bartolini Bussi, M. (2009). Proof and proving in primary school: An experimental approach. In F.-L. Lin, F.-J. Hsieh, G. Hanna, & M. De Villiers (Eds.), Proceedings of the ICMI Study 19 Conference: Proof and proving in mathematics education(Vol. 1, pp. 53–58). Taipei, Taiwan: National Taiwan Normal University.
Battista, M. T. (2007). The development of geometric and spatial thinking. In F. Lester (Ed.), Second handbook of research on mathematics teaching and learning(pp. 843–908). Reston, VA: National Council of Teachers of Mathematics.
Battista, M. T. (2008). Representations and cognitive objects in modern school geometry. In G. Blume & M. K. Heid (Eds.), Research on technology and the teaching and learning of mathematics: Cases and perspectives(Vol. 2, pp. 341–362). Charlotte, NC: Information Age Publishing.
Baulac, Y., Bellemain, F., & Laborde, J. M. (1988). Cabri-Géomètre, un logiciel d’aide à l’apprentissage de la géomètrie: Logiciel et manuel d’utilisation. Paris, France: Cedic-Nathan.
Bishop, A. J., Clements, M. A., Keitel, C., Kilpatrick, J., & Leung, F. K. S. (Eds.). (2003). Second international handbook of mathematics education. Dordrecht, The Netherlands: Kluwer.
Boero, P., Garuti, R., Lemut, E., & Mariotti, A. M. (1996). Challenging the traditional school approach to theorems: A hypothesis about the cognitive unity of theorems. In L. Puig & A. Gutiérrez (Eds.), Proceedings of the 20th Conference of the International Group for the Psychology of Mathematics Education(Vol. 2, pp. 113–120). Valencia, Spain: International Group for the Psychology of Mathematics Education.
Borba, M. C., & Villarreal, M. E. (2005). Humans-with-media and the reorganization of mathematical thinking. New York, NY: Springer.
Borwein, J., & Bailey, D. (2008). Mathematics by experiment: Plausible reasoning in the 21st century. Wellesley, MA: A. K. Peters/CRC Press.
Bruckheimer, M., & Arcavi, A. (2001). A Herrick among mathematicians or dynamic geometry as an aid to proof. International Journal of Computers in Mathematics Learning, 6, 113–126.
Centre de Recherche sur l’Ensignement des Mathématiques (CREM). (1995). Les mathématiques de la maternelle jusqu’à 18 ans. Essai d’élaboration d’un cadre global pour l’enseignement des mathématiques. In L. Grugnetti & V. Villani [Italian edition, La matematica dalla scuola materna all’Università]. Bologna, Italy: Pitagora Editrice.
Christou, C., Mousoulides, N., Pittalis, M., & Pitta-Pantazi, D. (2005). Proofs through exploration in dynamic geometry environments. International Journal of Science and Mathematics Education, 2, 339–352.
de Villiers, M. (1990). The role and function of proof in mathematics. Pythagoras, 24, 17–24.
de Villiers, M. (1997). The role of proof in investigative, computer-based geometry: Some personal reflections. In J. King & D. Schattschneider (Eds.), Geometry turned on(pp. 15–24). Washington, DC: Mathematical Association of America.
de Villiers, M. (1998). An alternative approach to proof in dynamic geometry? In R. Lehrer & D. Chazan (Eds.), Designing learning environments for developing understanding of geometry and space(pp. 369–393). Mahwah, NJ: Lawrence Erlbaum.
de Villiers, M. (1999). Rethinking proof with the Geometer’s Sketchpad. Emeryville, CA: Key Curriculum Press.
de Villiers, M. (2010). The role of experimentation in mathematics and mathematics education. In G. Hanna & H. N. Jahnke (Eds.), Explanation and proof in mathematics: Philosophical and educational perspectives(pp. 205–222). Basel, Switzerland: Springer Books.
Drijvers, P., Doorman, M., Boon, P., Reed, H., & Gravemeijer, K. (2010). The teacher and the tool: Instrumental orchestrations in the technology-rich mathematics classroom. Educational Studies in Mathematics, 75(2), 213–234.
Drijvers, P., Kieran, C., Mariotti, M., Ainley, J., Andresen, M., Chan, Y., Dana-Picard, T., Gueudet, G., Kidron, I., Leung, A., & Meagher, M. (2010). Integrating technology into mathematics education: Theoretical perspectives. In C. Hoyles & J. B. Lagrange (Eds.), Mathematics education and technology—Rethinking the terrain(pp. 89–132). New York, NY: Springer.
Duval, R. (2006). A cognitive analysis of problems of comprehension in the learning of mathematics. Educational Studies in Mathematics, 61, 103–131.
Erez, M. M., & Yerulshalmy, M. (2006). “If you can turn a rectangle into a square, you can turn a square into a rectangle …”: Young students experience the dragging tool. International Journal of Computers for Mathematical Learning, 11, 271–299.
Frant, J. B., & de Costra, R. M. (2000, July). Proofs in geometry: Different concepts build upon very different cognitive mechanisms.Paper presented at the ICME 9, TSG12: Proof and Proving in Mathematics Education, Tokyo, Japan.
Guala, E., & Boero, P. (1999). Time, complexity and learning. Annals of the New York Academy of Sciences, 879, 164–167.
Guven, B., Cekmez, E., & Maratas, I. (2010). Using empirical evidence in the process of proving: The case of dynamic geometry. Teaching Mathematics and Its Applications, 29(4), 193–207.
Hadas, N., Hershkowitz, R., & Schwartz, B. (2000). The role of contradiction and uncertainty in promoting the need to prove in dynamic geometry environments. Educational Studies in Mathematics, 44, 127–150.
Hanna, G., Jahnke, H. N., & Pulte, H. (Eds.). (2010). Explanation and proof in mathematics: Philosophical and educational perspectives. New York, NY: Springer.
Healy, L., & Hoyles, C. (2001). Software tools for geometrical problem solving: Potentials and pitfalls. International Journal of Computers for Mathematical Learning, 1(3), 235–256.
Herbst, P. (2002). Engaging students in proving: A double bind on the teacher. Journal for Research in Mathematics Education, 33(3), 176–203.
Hollebrands, K. F., Conner, A. M., & Smith, R. C. (2010). The nature of arguments provided by college geometry students with access to technology while solving problems. Journal for Research in Mathematics Education, 41(4), 324–350.
Hollebrands, K., Laborde, C., & Sträßer, R. (2008). Technology and the learning of geometry at the secondary level. In K. Heid & G. Blume (Eds.), Research in technology and the teaching and learning of mathematics: Research syntheses(Vol. 1, pp. 155–203). Charlotte, NC: Information Age Publishing.
Hoyles, C., & Noss, R. (2003). What can digital technologies take from and bring to research in mathematics education? In A. J. Bishop, M. A. Clements, C. Keitel, J. Kilpatrick, & F. Leung (Eds.), Second international handbook of mathematics education(pp. 323–349). Dordrecht, The Netherlands: Kluwer Academic Publishers.
Jackiw, N. (1991, 2001). The Geometer’s Sketchpad[Computer Program]. Emeryville, CA: Key Curriculum Press.
Jackiw, N. (2006). Mechanism and magic in the psychology of dynamic geometry. In N. Sinclair, D. Pimm, & W. Higginson (Eds.), Mathematics and the aesthetics: New approaches to an ancient affinity(pp. 145–159). New York, NY: Springer.
Jahn, A. P. (2002). “Locus” and “Trace” in Cabri-géomètre: Relationships between geometric and functional aspects in a study of transformations. ZDM—Zentralblatt fur Didaktik der Mathematik, 34(3), 78–84.
Japanese Society of Mathematics Education. (2000). Mathematics programme in Japan. Tokyo, Japan: Author.
Jones, K. (2000). Providing a foundation for deductive reasoning: Students’ interpretations when using dynamic geometry software and their evolving mathematical explanations. Educational Studies in Mathematics, 44, 55–85.
Jones, K., Mackrell, K., & Stevenson, I. (2010). Designing digital technologies and learning activities for different geometries. In C. Hoyles & J.-B. Lagrange (Eds.), Mathematics education and technology—Rethinking the terrain New ICMI Study Series(Vol. 13, pp. 47–60). Dordrecht, The Netherlands: Springer.
Laborde, C. (1992). Solving problems in computer based geometry environments: The influence of the features of the software. ZDM—Zentrablatt für Didactik des Mathematik, 92(4), 128–135.
Laborde, C. (1998). Relationship between the spatial and theoretical in geometry: The role of computer dynamic representations in problem solving. In J. D. Tinsley & D. C. Johnson (Eds.), Information and communications technologies in school mathematics(pp. 183–195). London, UK: Chapman & Hall.
Laborde, C. (2000). Dynamical geometry environments as a source of rich learning contexts for the complex activity of proving. Educational Studies in Mathematics, 44(1–2), 151–161.
Laborde, C. (2001). Integration of technology in the design of geometry tasks with Cabri-Géomètre. International Journal of Computers for Mathematical Learning, 6, 283–317.
Laborde, C. (2004). The hidden role of diagrams in pupils’ construction of meaning in geometry. In J. Kilpatrick, C. Hoyles, & O. Skovsmose (Eds.), Meaning in mathematics education(pp. 1–21). Dordrecht, The Netherlands: Kluwer Academic Publishers.
Laborde, C., Kynigos, C., Hollebrands, K., & Sträßer, R. (2006). Teaching and learning geometry with technology. In A. Gutiérrez & P. Boero (Eds.), Handbook of research on the psychology of mathematics education: Past, present and future(pp. 275–304). Rotterdam, The Netherlands: Sense Publishers.
Lassak, M. (2009). Using dynamic graphs to reveal student reasoning. International Journal of Mathematical Education in Science and Technology, 40(5), 690–696.
Lehrer, R., Jenkins, M., & Osana, H. (1998). Longitudinal study of children’s reasoning about space and geometry. In R. Lehrer & D. Chazan (Eds.), Designing learning environments for developing understanding of geometry and space(pp. 137–167). Mahwah, NJ: Lawrence Erlbaum.
Leung, A. (2003). Dynamic geometry and the theory of variation. In N. A. Pateman, B. J. Doughherty, & J. T. Zillox (Eds.), Proceedings of PME 27: Psychology of Mathematics Education 27th International ConferenceVolume 3 (pp. 197–204). Honolulu: University of Hawaii.
Leung, A. (2008). Dragging in a dynamic geometry environment through the lens of variation. International Journal of Computers in Mathematics Learning, 13, 135–157.
Leung, A., & Or, C. M. (2007). From construction to proof: Explanations in dynamic geometry environments. In J. H. Woo, H. C. Lew, K. S. Park, & D. Y. Seo (Eds.), Proceedings of the 31st Conference of the International Group for the Psychology of Mathematics Education(Vol. 3, pp. 177–184). Seoul, Korea: International Group for the Psychology of Mathematics Education.
Mariotti, M. A. (2000). Introduction to proof: The mediation of dynamical software environment. Educational Studies in Mathematics, 44(1–2), 25–53.
Mariotti, M. A. (2002). Influence of technologies advances on students’ math learning. In L. English (Ed.), Handbook of international research in mathematics education(pp. 695–723). Mahwah, NJ: Lawrence Erbaum.
Mariotti, M. A. (2006). Proof and proving in mathematics education. In A. Guttiérrez & P. Boero (Eds.), Handbook of research on the psychology of mathematics education: Past, present and future(pp. 173–204). Rotterdam, The Netherlands: Sense Publishing.
Mariotti, M. A. (2010). Proofs, semiotics and artefacts of information technologies. In G. Hanna, H. N. Jahnke, & H. Pulte (Eds.), Explanation and proof in mathematics: Philosophical and educational perspectives(pp. 169–190). Dordrecht, The Netherlands: Springer.
Marrades, R., & Gutierrez, A. (2000). Proofs produced by secondary school students learning geometry in a dynamic computer environment. Educational Studies in Mathematics, 44, 87–125.
Marton, F., & Booth, S. (1997). Learning and awareness. Mahwah, NJ: Lawrence Erlbaum.
Ministry of Education, People’s Republic of China. (2001). Full-time obligatory education mathematics curriculum standards (experimental version)[in Chinese]. Beijing, China: Beijing University Press.
National Council of Teachers of Mathematics. (2000). Principles and standards for school mathematics.Reston, VA: Author.
Noss, R., & Hoyles, C. (Eds.). (1996). Windows on mathematical meanings—Learning cultures and computers. Dordrecht, The Netherlands: Kluwer Academic Publishers.
Olivero, F. (2002). The proving process within a dynamic geometry environment(Doctoral thesis). University of Bristol, UK
Olivero, F. (2006). Students’ constructions of dynamic geometry. In C. Hoyles, J.-B. Lagrange, L.-H. Son, & N. Sinclair (Eds.), Proceedings of the 17th International Congress on Mathematical Instruction Study Conference “Technology Revisited”(pp. 433–442). Hanoi, Vietnam: Hanoi Institute of Technology and Didirem University.
Olivero, F., & Robutti, O. (2001). Measuring in Cabri as a bridge between perception and theory. In M. van den Heuvel-Panhuizen (Ed.), Proceedings of the 25th Conference of the International Group for the Psychology of Mathematics Education(Vol. 4, pp. 9–16). Utrecht, The Netherlands: International Group for the Psychology of Mathematics Education.
Olivero, F., & Robutti, O. (2002). An exploratory study of students’ measurement activity in a dynamic geometry environment. In J. Novotńa (Ed.), Proceedings of CERME 2(Vol. 1, pp. 215–226). Prague, Czech Republic: Charles University.
Olivero, F., & Robutti, O. (2007). Measuring in dynamic geometry environments as a tool for conjecturing and proving. International Journal of Computers for Mathematical Learning, 12(2), 135–156.
Oner, D. (2008). A comparative analysis of high school geometry curricula: What do technology-intensive, standards-based, and traditional curricula have to offer in terms mathematical proof and reasoning? Journal of Computers in Mathematics and Science Teaching, 27(4), 467–497.
Oner, D. (2009). The role of dynamic geometry software in high school geometry curricula: An analysis of proof tasks. International Journal for Technology in Mathematics Education, 16(3), 109–121.
Rabardel, P. (1995). Les hommes et les technologies—Approche cognitive des instruments contemporains. Paris, France: A. Colin.
Radford, L. (2010). The anthropological turn in mathematics education and its implication on the meaning of mathematical activity and classroom practice. Acta Didactica Universitatis Comenianae, 10, 103–120.
Rav, Y. (1999). Why do we prove theorems? Philosophia Mathematica III, 7, 5–41.
Sinclair, N. (2002). The kissing triangles: The aesthetics of mathematical discovery. International Journal of Computers for Mathematical Learning, 7(1), 45–63.
Sinclair, M. P. (2003). Some implications of the results of a case study for the design of pre-constructed dynamic geometry sketches and accompanying materials. Educational Studies in Mathematics, 52(3), 289–317.
Sinclair, N. (2008). The history of the geometry curriculum in the United States. Charlotte, NC: Information Age.
Sinclair, N., Arzarello, F., Gaisman, M. T., & Lozano, M. D. (2009). Implementing digital technologies at a national scale. In C. Hoyles & J.-B. Lagrange (Eds.), Digital technologies and mathematics teaching and learning: Rethinking the terrain(Vol. 13, pp. 61–78). New York, NY: Springer.
Sinclair, N., Moss, J., & Jones, K. (2010). Developing geometric discourse using DGS in K-3. In M. M. F. Pinto & T. F. Kawasaki (Eds.), Proceedings of the 34th Conference of the International Group for the Psychology of Mathematics Education(Vol. 4, pp. 185–192). Belo Horizonte, Brazil: International Group for the Psychology of Mathematics Education.
Sinclair, N., & Yurita, V. (2008). To be or to become: How dynamic geometry changes discourse. Research in Mathematics Education, 10(2), 135–150.
Sträßer, R. (2009). Instruments for learning and teaching mathematics. An attempt to theorise about the role of textbooks, computers and other artefacts to teach and learn mathematics. In M. Tzekaki, M. Kaldrimidou, & H. Sakonidis (Eds.), Proceedings of the 33rd Conference of the International Group for the Psychology of Mathematics Education(Vol. 1, pp. 67–81). Thessaloniki, Greece: International Group for the Psychology of Mathematics Education.
Stylianides, A. J. (2007). The notion of proof in the context of elementary school mathematics. Educational Studies in Mathematics, 65, 1–20.
Stylianou, D., Knuth, E., & Blanton, M. (Eds.). (2009). Teaching and learning proof across the grades: A K-16 perspective. New York, NY: Routledge.
Trouche, L. (2004). Managing complexity of human machine interactions in computerized learning environments: Guiding student’s command process through instrumental orchestrations. International Journal of Computers for Mathematical Learning, 9(3), 281–307.
Unione Matematica Italiana. (2004). G. Anichini, F. Arzarello, L. Ciarrapico & O. Robutti (Eds.), Matematica 2003. La matematica per il cittadino. Attività didattiche e prove di verifica per un nuovo curricolo di Matematica (ciclo secondario).Lucca, Italy: Matteoni Stampatore.
Vadcard, L. (1999). La validation en géomètrie au collège avec Cabri-Géomètre: Mesures exploratoire et mesures probatoires. Petit X, 50, 5–21.
Verillion, P., & Rabardel, P. (1995). Cognition and artifacts: A contribution to the study of thought in relation to instrumented activity. European Journal of Psychology of Education, 10(1), 77–101.
Vygotsky, L. S. (1978). Mind in society. The development of higher psychological processes. Cambridge, MA: Harvard University Press.
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Sinclair, N., Robutti, O. (2012). Technology and the Role of Proof: The Case of Dynamic Geometry. In: Clements, M., Bishop, A., Keitel, C., Kilpatrick, J., Leung, F. (eds) Third International Handbook of Mathematics Education. Springer International Handbooks of Education, vol 27. Springer, New York, NY. https://doi.org/10.1007/978-1-4614-4684-2_19
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