# The Interplay of Teacher and Student Actions in the Teaching and Learning of Geometric Proof

- 570 Downloads
- 21 Citations

## Abstract

Proof and reasoning are fundamental aspects of mathematics. Yet, how to help students develop the skills they need to engage in this type of higher-order thinking remains elusive. In order to contribute to the dialogue on this subject, we share results from a classroom-based interpretive study of teaching and learning proof in geometry. The goal of this research was to identify factors that may be related to the development of proof understanding. In this paper, we identify and interpret students' actions, teacher's actions, and social aspects that are evident in a classroom in which students discuss mathematical conjectures, justification processes and student-generated proofs. We conclude that pedagogical choices made by the teacher, as manifested in the teacher's actions, are key to the type of classroom environment that is established and, hence, to students' opportunities to hone their proof and reasoning skills. More specifically, the teacher's choice to pose open-ended tasks (tasks which are not limited to one specific solution or solution strategy), engage in dialogue that places responsibility for reasoning on the students, analyze student arguments, and coach students as they reason, creates an environment in which participating students make conjectures, provide justifications, and build chains of reasoning. In this environment, students who actively participate in the classroom discourse are supported as they engage in proof development activities. By examining connections between teacher and student actions within a social context, we offer a first step in linking teachers' practice to students' understanding of proof.

## Keywords

classroom interaction geometry link between teaching and learning pedagogical choices proof and reasoning secondary school mathematics## Preview

Unable to display preview. Download preview PDF.

## References

- Balacheff, N.: 1991, ‘Treatment of refutations: Aspects of the complexity of a constructivist approach to mathematics learning’, in E. von Glasersfeld (ed.),
*Radical Constructivism in Mathematics Education*, Kluwer Academic Publishers, Dordrecht, Netherlands, pp. 89–110.Google Scholar - Ball, D.L. and Cohen, D.: 1999, ‘Developing practice, developing practitioners’, in L. Darling–Hammond and G. Sykes (eds.),
*Teaching as the Learning Profession: Handbook of Policy and Practice*, Jossey Bass Publishers, San Francisco, CA, pp. 3–32.Google Scholar - Boaler, J.: 2003, ‘Studying and capturing the complexity of practice: The case of the ‘dance of agency’, in N.A. Pateman, B.J. Dougherty, and J.T. Zilliox (eds.),
*Proceedings of the 27th Conference of the International Group for the Psychology of Mathematics Education Held Jointly with the 25th Conference of PME-NA*,Vol. 1, Honolulu, HI, pp. 3–16.Google Scholar - Brousseau, G.: 1984, ‘The crucial role of the didactical contract in the analysis and construction of situations in teaching and learning mathematics’, in H.G. Steiner (ed.),
*Theory of Mathematics Education*, Occasional paper 54, IDM, Bielefeld, Germany, pp. 110–119.Google Scholar - Brousseau, G.: 1997,
*Theory of Didactical Situations in Mathematics: Didactique des Mathématiques 1970–1990*in N. Balacheff, M. Cooper, R. Sutherland, and V. Warfield (trans. and eds.), Kluwer Academic Publishers, Dordrecht, Netherlands.Google Scholar - Brown, R.A.J. and Renshaw, P.D.: 2000, ‘Collective argumentation: A sociocultural approach to reframing classroom teaching and learning’, in H. Cowie and G. Van der Aalsvoort (eds.),
*Social Interaction in Learning and Instruction: The Meaning of Discourse for the Construction of Knowledge*, Pergamon, New York, pp. 52–66.Google Scholar - 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.CrossRefGoogle Scholar - Cobb, P.: 1999, ‘Individual and collective mathematical development: The case of statistical data analysis’,
*Mathematical Thinking and Learning*1(1), 5–43.Google Scholar - Cobb, P.: 2000, ‘Conducting teaching experiments in collaboration with teachers’, in A.E. Kelly and R.A. Lesh (eds.),
*Handbook of Research Design in Mathematics and Science Education*, Lawrence Erlbaum Associates, Mahwah, NJ, pp. 307–333.Google Scholar - Cobb, P. and Yackel, E.: 1996, ‘Constructivist, emergent, and sociocultural perspectives in the context of developmental research’,
*Educational Psychologist*31(3/4), 175–190.Google Scholar - De Villiers, M.: 1999,
*Rethinking proof with the Geometer's Sketchpad*, Key Curriculum Press, Emeryville, CA.Google Scholar - Hanna, G.: 2000, ‘Proof, explanation and exploration: An overview’,
*Educational Studies in Mathematics*44, 5–23.CrossRefGoogle Scholar - Harel, G. and Sowder, L.: 1998, ‘Students' proof schemes: Results from exploratory studies’, in A.H. Schoenfeld, J. Kaput, and E. Dubinsky (eds.),
*CBMS Issues in Mathematics Education*, Vol. 3, American Mathematical Society, Providence, RI, pp. 234–283.Google Scholar - Hart, E.W.: 1994, ‘A conceptual analysis of the proof-writing performance of expert and novice students in elementary group theory’, in J.J. Kaput and E. Dubinsky (eds.),
*Research Issues in Undergraduate Mathematics Learning: Preliminary Analyses and Results*, MAA Notes 33, Mathematical Association of America, Washington, DC, pp. 49–62.Google Scholar - Herbst, P.G.: 2002, ‘Engaging students in proving: A double bind on the teacher’,
*Journal for Research in Mathematics Education*33(3), 176–203.Google Scholar - Kumpulainen, K. and Mutanen, M.: 2000, ‘Mapping the dynamics of peer group interaction: A method of analysis of socially shared learning processes,’ in H. Cowie and G. Vander Aalsvoort (eds.),
*Social Interaction in Learning and Instruction: The Meaning of Discourse for the Construction of Knowledge*, Pergamon, NY, pp. 144–160.Google Scholar - Martin, T.S. and McCrone, S.S.: 2003, ‘Classroom factors related to geometric proof construction ability’,
*The Mathematics Educator*7(1), 18–31.Google Scholar - Martin, W.G. and Harel, G.: 1989, ‘Proof frames of preservice elementary teachers’,
*Journal for Research in Mathematics Education*20(1), 41–51.Google Scholar - McCrone, S.S. and Martin, T.S.: 2004, ‘Assessing high school students' understanding of geometric proof’,
*Canadian Journal for Science, Mathematics, and Technology Education*4(2), 223–242.Google Scholar - Middleton, J.A., Sawada, D., Judson, E., Bloom, I. and Turley, J.: 2002, ‘Relationships build reform: Treating classroom research as emergent systems’, in L.D. English (ed.),
*Handbook of International Research in Mathematics Education*, Lawrence Erlbaum Associates, Mahwah, NJ, pp. 409–431.Google Scholar - Miles, M.B. and Huberman, A.M.: 1994,
*Qualitative Data Analysis*, Sage Publications, Thousand Oaks, CA.Google Scholar - National Council of Teachers of Mathematics: 2000,
*Principles and Standards for School Mathematics*, National Council of Teachers of Mathematics, Reston, VA.Google Scholar - Senk, S.L.: 1985, ‘How well do students write geometry proofs?’,
*Mathematics Teacher*78(6), 448–456.Google Scholar - Sfard, A.: 2000, ‘On reform movement and the limits of mathematical discourse,’
*Mathematical Thinking and Learning*2(3), 157–189.CrossRefGoogle Scholar - Sfard, A.: 2001, ‘There is more to discourse than meets the ears: Looking at thinking as communicating to learn more about mathematical learning,’
*Educational Studies in Mathematics*46, 13–57.CrossRefGoogle Scholar - Vygotsky, L.: 1978,
*Mind in Society: The Development of Higher Psychological Processes*, Harvard University Press, Cambridge, MA.Google Scholar