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From visual reasoning to logical necessity through argumentative design

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Abstract

Our main goal in this study is to exemplify that a meticulous design can lead pre-service teachers to engage in productive unguided peer argumentation. By productivity, we mean here a shift from reasoning based on intuitions to reasoning moved by logical necessity. As a subsidiary goal, we aimed at identifying the kinds of reasoning processes (visual, inquiry-based, and deductive) pre-service teacher's students adopt, and how these reasoning processes are interwoven in peer-unguided argumentation. We report on a case study in which one dyad participating in a pre-service teachers program solved a mathematical task. We relied on three principles to design an activity: (a) creating a situation of conflict, (b) creating a collaborative situation, and (c) providing a device for checking hypotheses/conjectures. We show how the design afforded productive argumentation. We show that the design of the task entailed argumentation which first relied on intuition, then intertwined the activities of conjecturing and checking conjectures by means of various hypotheses-testing devices (measurement, manipulations, and dynamic change of figures with Dynamic Geometry software), leading to a conflict between conjectures and the outcome of the manipulation of DG software. Peer argumentation then shifted to abductive and deductive considerations towards the solution of the mathematical task. These beneficial outcomes resulted from collaborative rather than adversarial interactions as the students tried to accommodate their divergent views through the co-elaboration of new explanations.

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Notes

  1. According to Reid (2003), abduction is a kind of logical inference described by Charles Sanders Peirce as “guessing.” The term refers to the process of arriving at an explanatory hypothesis. Peirce said that to abduce a hypothetical explanation a from an observed surprising circumstance b is to surmise that a may be true because then b would be a matter of course. Thus, to abduce a from b involves determining that a is sufficient (or nearly sufficient), but not necessary…the ability to reason abductively, what Peirce calls the “guessing instinct” and the degree of which such “instinct” might be natural or derived from our cultures.

References

  • Andriessen, J. E. B., & Schwarz, B. B. (2009). Argumentative design. In N. Muller-Mirza & A. N. Perret-Clermont (Eds.), Argumentation and education: Theoretical foundations and practices (pp. 145–174). New York: Springer.

    Google Scholar 

  • Asterhan, C. S. C., & Schwarz, B. B. (2009). Argumentation and explanation in conceptual change: Indications from protocol analyses of peer-to-peer dialogue. Cognitive Science, 33, 374–400.

    Article  Google Scholar 

  • Arzarello, F. (2008). The proof in the 20th century. In P. Boero (Ed.), Theorems in schools: From history epistemology and cognition to classroom practices (pp. 43–64). Rotterdam: Sense Publishers.

    Google Scholar 

  • Arzarello, F., Olivero, F., Paola, D., & Robutti, O. (2008). The transition to formal proof in geometry. In P. Boero (Ed.), Theorems in schools: From history, epistemology and cognition to classroom practices (pp. 307–424). Rotterdam: Sense Publishers.

    Google Scholar 

  • Baker, M. (2003). Computer-mediated argumentative interactions for the co-elaboration of scientific notions. In J. Andriessen, M. Baker, & D. Suthers (Eds.), Arguing to learn: Confronting cognitions in computer-supported collaborative learning environments (pp. 47–70). Dordrecht: Kluwer Academic.

    Google Scholar 

  • Balacheff, N. (1991). The benefits and limits of social interaction: The case of mathematical proof in mathematics knowledge: Its growth through teaching. Boston: Kluwer Academic.

    Google Scholar 

  • Barron, B. (2003). When smart groups fail. Journal of the Learning Sciences, 12(3), 307–359.

    Article  Google Scholar 

  • Bartolini Bussi, M., Boero, P., Ferri, F., Garuti, R., & Mariotti, M. A. (1997). Approaching geometry theorems in context. In E. Pehkonen (Ed.), Proceedings of 21st Conference of the International Group for the Psychology of Mathematics Education (App, Vol. 1, pp. 180–195). Lahti: University of Helsinki.

    Google Scholar 

  • Birdsell, D., & Groarke, L. (1996). Toward a theory of visual argument. Argumentation and Advocacy, 33, 1–10.

    Google Scholar 

  • Boero, P. (2007). Theorems in school: An introduction. In P. Boero (Ed.), Theorems in schools: From history, epistemology and cognition to classroom practices (pp. 3–8). Rotterdam: Sense Publishers.

    Google Scholar 

  • Boero, P., Garuti, R., & Lemut, E. (2007). Approaching theorems in grade VIII. Some mental processes underlying producing and proving conjectures, and conditions suitable to enhance them. In P. Boero (Ed.), Theorems in schools: From history, epistemology and cognition to classroom practices (pp. 251–266). Rotterdam: Sense Publishers.

    Google Scholar 

  • Brown, A., & Palincsar, A. (1989). Guided, cooperative learning and individual knowledge acquisition. In L. B. Resnick (Ed.), Knowing, learning and instruction: Essays in honor of Robert Glaser (pp. 393–451). Hillsdale, NJ: Lawrence Erlbaum.

    Google Scholar 

  • Cazden, C. (2001). Classroom discourse: The language of teaching and learning. Portsmouth: Heinemann.

    Google Scholar 

  • Chi, M. T. H., Bassok, M., Lewis, M. W., Reimann, P., & Glaser, R. (1989). Self-explanations: How students study and use examples in learning to solve problems. Cognitive Science, 13, 145–182.

    Article  Google Scholar 

  • Clements, D. H., & Batista, M. T. (1992). Geometry and spatial reasoning. In D. A. Grouws (Ed.), Handbook of research on mathematics teaching and learning (pp. 420–464). New York: Macmillan.

    Google Scholar 

  • Cobb, P., & Bauersfeld, H. (Eds.). (1995). The emergence of mathematical meaning: Interaction in classroom culture. Hillsdale, NJ: Lawrence Erlbaum Associates.

    Google Scholar 

  • Davis, P. J., & Anderson, J. A. (1979). Nonanalytic aspects of mathematics and their implication for research and education. SIAM Review, 21(1), 112–127.

    Article  Google Scholar 

  • Dawes, L., Mercer, N., & Wegerif, R. (2000). Thinking together: A program of activities for developing speaking, listening and thinking skills for children aged 8–11. Birmingham: Imaginative Minds Ltd.

    Google Scholar 

  • Dillenbourg, P. (1999). What do you mean by collaborative learning? In P. Dillenbourg (Ed.), Collaborative-learning: Cognitive and computational approaches (pp. 1–19). Oxford: Elsevier.

    Google Scholar 

  • Douek, N. (1999). Argumentation and conceptualisation in context: A case study on sun shadows in primary school. Educational Studies in Mathematics, 39, 89–110.

    Article  Google Scholar 

  • Dreyfus, T. (2004). What counts as proof in the mathematics classroom? In M. Kourkoulos, G. Troulis, & C. Tzanakis (Eds.), Proceedings of the 3rd Colloquium on the Didactics of Mathematics (pp. 114–132). Rethymnon, Crete: University of Crete.

    Google Scholar 

  • Duval, R. (1998). Geometry from a cognitive point of view. In C. Mammana & V. Villani (Eds.), Perspectives on the teaching of geometry for the 21st century. An ICMI Study (pp. 37–62). Dordrecht, The Netherlands: Kluwer Academic.

    Google Scholar 

  • Garuti, R., Boero, P., & Lemut, E. (1998). Cognitive unity of theorems and difficulties of proof. In_A. Olivier, & K. Newstead (Eds.), Proceedings of 22nd Conference of the International Group for the Psychology of Mathematics Education. Vol. 2, (pp. 345–352). Stellenbosch: Stellenbosch University.

  • Garuti, R., Boero, P., Lemut, E., & Mariotti, M. A. (1996). Challenging the traditional school approach to theorems: A hypothesis about the cognitive unity of theorems. In L. Puig, & A. Gutierrez (Eds.), Proceedings of 20th Conference of the International Group for the Psychology of Mathematics Education. (Vol. 2, pp.113–120). Valencia: University of Valencia.

  • Gutierrez, A., Jaime, A., & Fortuny, J. M. (1991). An alternative paradigm to evaluate the acquisition of the van Hiele levels. Journal For Research In Mathematics Education, 22(3), 237–251.

    Article  Google Scholar 

  • Hadas, N., Hershkowitz, R., & Schwarz, B. B. (2001). The role of surprise and uncertainty in promoting the need to prove in computerized environment. Educational Studies in Mathematics, 44(1&2), 127–150.

    Google Scholar 

  • Hadas, N., Hershkowitz, R., & Schwarz, B. B. (2002). Between task design and students’ explanations in geometrical activities. Canadian Journal of Research in Mathematics Education, 2, 529–551.

    Google Scholar 

  • Hanna, G. (1990). Some pedagogical aspects of proof. Interchange, 21, 6–13.

    Article  Google Scholar 

  • Hanna, G. (2007). The ongoing value of proof. In P. Boero (Ed.), Theorems in schools: From history, epistemology and cognition to classroom practices (pp. 11–24). Rotterdam: Sense Publishers.

    Google Scholar 

  • Heath, S. B. (1983). Ways with words: Language, life and work in communities and classrooms. Cambridge: Cambridge University Press.

    Google Scholar 

  • Hershkowitz, R. (1990). Psychological aspects of learning geometry. In P. Nesher & J. Kilpatrick (Eds.), Mathematics and cognition (pp. 70–95). Cambridge: Cambridge University Press.

    Chapter  Google Scholar 

  • Hershkowitz, R., Parzysz, B., & van Dormolen, J. (1996). Space and shape. In A. J. Bishop, K. Clements, C. Keitel, J. Kilpatrick, & C. Laborde (Eds.), International handbook of mathematics education (pp. 161–204). Dordrecht: Kluwer Academic Press.

    Google Scholar 

  • Hershkowitz, R., & Schwarz, B. B. (1999). Reflective processes in a technology-based mathematics classroom. Cognition and Instruction, 17, 66–91.

    Article  Google Scholar 

  • Howe, C., Toulmie, A., Duchak-Tanner, V., & Rattay, C. (2000). Hypothesis-testing in science: Group consensus and acquisition of conceptual and procedural knowledge. Learning and Instruction, 10, 361–391.

    Article  Google Scholar 

  • Inglis, M., Mejia-Ramos, J. P., & Simpson, A. (2007). Modeling mathematical argumentation: The importance of qualification. Educational Studies in Mathematics, 66(1), 3–21.

    Article  Google Scholar 

  • Johnsson, D. W., & Johnsson, R. T. (2009). Energizing learning: The instructional power of conflict. Educational Research, 38(1), 37–51.

    Article  Google Scholar 

  • King, A., & Rosenshine, B. (1993). Effects of guided cooperative questioning on children’s knowledge construction. Journal of Experimental Education, 61(2), 127–148.

    Article  Google Scholar 

  • Krummheuer, G. (1995). The ethnography of argumentation. In P. Cobb & H. Bauersfeld (Eds.), The emergence of mathematical meaning: Interaction in classroom cultures (pp. 229–269). Hillsdale, NJ: Erlbaum.

    Google Scholar 

  • Laborde, C. (1995). Designing tasks for learning geometry in computer-based environment, the case of Cabri-géomètre. In L. Burton & B. Jaworski (Eds.), Technology in mathematics teaching—A bridge between teaching and learning (pp. 35–68). London: Chartwell-Bratt.

    Google Scholar 

  • Lavy, I. (2006). A case study of different types of arguments emerging from exploration in an interactive computerized environment. Journal of Mathematical Behavior, 25, 153–169.

    Article  Google Scholar 

  • Limón, M. (2001). On the cognitive conflict as an instructional strategy for conceptual change: A critical appraisal. Learning & Instruction, 11, 357–380.

    Article  Google Scholar 

  • Mariotti, M. A. (2001). Introduction to proof: The mediation of a dynamic software environment. Educational Studies in Mathematics, 44(1&2), 25–53.

    Google Scholar 

  • Means, M. L., & Voss, J. F. (1996). Who reasons well? Two studies of informal reasoning among children of different grade, ability, and knowledge level. Cognition and Instruction, 14, 139–179.

    Article  Google Scholar 

  • Mercer, N. (1995). The guided construction of knowledge: Talk amongst teachers and learners. Clevedon: Multilingual Matters.

    Google Scholar 

  • Mercer, N. (2000). Words and minds: How we use language to think together. New York: Routledge.

    Book  Google Scholar 

  • Mercer, N., & Littleton, K. (2007). Dialogue and the development of children’s thinking: A sociocultural approach. London: Routledge.

    Google Scholar 

  • Mugny, G., & Doise, W. (1978). Socio-cognitive conflict and structure of individual and collective performances. European Journal of Social Psychology, 8, 181–192.

    Article  Google Scholar 

  • Pedemonte, B. (2007). How can the relationship between argumentation and proof be analyzed? Educational Studies in Mathematics, 66, 23–41.

    Article  Google Scholar 

  • Prusak, N. (2007). The Analysis of dialogues of dyads of pre-service teachers operating in an activity designed to foster argumentation—The case of dynamic geometry. Unpublished Master’s thesis. The Hebrew University: Jerusalem. (In Hebrew).

  • Rav, Y. (1999). Why do we prove theorems? Philosophia Mathematica, 7, 5–41.

    Google Scholar 

  • Rasmussen, C., Zandieh, M., King, K., & Teppo, A. (2005). Advancing mathematical activity: A practice-oriented view of mathematical thinking. Mathematical Thinking and Learning, 7(1), 51–73.

    Article  Google Scholar 

  • Rasmussen, C., & Stephan, M. (2008). A methodology for documenting collective activity. In A. E. Kelly, R. A. Lesh, & J. Y. Baek (Eds.), Handbook of design research methods in education (pp. 195–215). New York: Routledge.

    Google Scholar 

  • Reid, D. (2003). Forms and uses of abduction. In M. Mariotti (Ed.), Proceedings of the third ESRME conference Bellaria Italy. (www.dm.unipi.it/~didattica/CERME3/proceedings/Groups/TG4/TG4_list.html).

  • Rogoff, B. (1990). Apprenticeship in thinking: Cognitive development in social context. New York: Oxford University Press.

    Google Scholar 

  • Rogoff, B. (1995). Observing sociocultural activity on three planes: Participatory appropriation, guided participation and apprenticeship. In J. W. Wertsch, P. del Rio, & A. Alvarez (Eds.), Sociocultural studies of mind (pp. 139–164). Cambridge: Cambridge University Press.

    Google Scholar 

  • Schwarz, B. B. (2009). Argumentation and learning. In N. Muller-Mirza & A.-N. Perret-Clermont (Eds.), Argumentation and education: Theoretical foundations and practices (pp. 91–126). New York: Springer.

    Google Scholar 

  • Schwarz, B. B., & Linchevski, L. (2007). The role of task design and of argumentation in cognitive development during peer interaction. The case of proportional reasoning. Learning and Instruction, 17, 310–331.

    Article  Google Scholar 

  • Schwarz, B. B., & Asterhan, C. S. C. (2010). Argumentation and reasoning. In K. Littleton, C. Wood, & J. Kleine Staarman (Eds.), International handbook of psychology in education (pp. 137–176). Bradford: Emerald Group.

    Google Scholar 

  • Schwarz, B. B., Hershkowitz, R., & Prusak, N. (2010). Argumentation and mathematics. In C. Howe & K. Littleton (Eds.), Educational dialogues: understanding and promoting productive interaction (pp. 115–141). London: Routledge.

    Google Scholar 

  • Stephan, M., & Rasmussen, C. (2002). Classroom mathematical practices in differential equations. Journal of Mathematical Behavior, 21, 459–490.

    Article  Google Scholar 

  • Toulmin, S. (1958). The uses of argument. Cambridge: Cambridge University Press.

    Google Scholar 

  • Vygotsky, L. S. (1981). The genesis of higher mental functions. In J. V. Wertsch (Ed.), The concept of activity in Soviet psychology (pp. 144–188). New York: Sharpe.

    Google Scholar 

  • Webb, N. M. (2009). The teacher’s role in promoting collaborative dialogue in the classroom. The British Journal of Educational Psychology, 79, 1–28.

    Article  Google Scholar 

  • Wells, G. (1992). The meaning makers: Children learning language and using language to learn. London: Hodder and Stoughton.

    Google Scholar 

  • Yackel, E. (2002). What we can learn from analyzing the teacher’s role in collective argumentation. Journal of Mathematical Behavior, 21, 423–440.

    Article  Google Scholar 

  • Yerushalmy, M., & Houde, R. A. (1986). The Geometric Supposer: Promoting thinking and learning. Mathematics Teacher, 79, 418–422.

    Google Scholar 

  • Zeilberger, D. (1993). Theorems for a price: Tomorrow’s semi-rigorous mathematical culture. Notices of the American Mathematical Society, 40(8), 978–981.

    Google Scholar 

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Authors and Affiliations

  1. The Hebrew University, Jerusalem, Israel

    Naomi Prusak & Baruch B. Schwarz

  2. The Weizmann institute, Rehovot, Israel

    Rina Hershkowitz

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  1. Naomi Prusak
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  2. Rina Hershkowitz
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  3. Baruch B. Schwarz
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Correspondence to Baruch B. Schwarz.

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Prusak, N., Hershkowitz, R. & Schwarz, B.B. From visual reasoning to logical necessity through argumentative design. Educ Stud Math 79, 19–40 (2012). https://doi.org/10.1007/s10649-011-9335-0

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