Abstract
This chapter first analyses and compares mathematicians’ and mathematics educators’ different conceptualisations of proof and shows how these are formed by different professional backgrounds and research interests. This diversity of views makes it difficult to precisely explain what a proof is, especially to a novice at proving. In the second section, we examine teachers’, student teachers’ and pupils’ proof conceptions and beliefs as revealed by empirical research. We find that the teachers’ beliefs clearly revolve around the questions of what counts as proof in the classroom and whether the teaching of proof should focus on the product or on the process. The third section discusses which type of metaknowledge about proof educators should provide to teachers and thus to students, how they can do this and what the intrinsic difficulties of developing adequate metaknowledge are.
References
Alibert, D. (1988). Towards new customs in the classroom. For the Learning of Mathematics, 8(2), 31–35.
Arsac, G., Chapiron, G., Colonna, A., Germain, G., Guichard, Y., & Mante, M. (1992). Initiation au raisonnement déductif au collège. Lyon: Presses Universitaires de Lyon.
Balacheff, N. (1988). Aspects of proof in pupils’ practice of school mathematics. In D. Pimm (Ed.), Mathematics, teachers and children (pp. 216–238). London: Hodder & Stoughton.
Balacheff, N. (2009). Bridging knowing and proving in mathematics: A didactical perspective. In G. Hanna, H. N. Jahnke, & H. Pulte (Eds.), Explanation and proof in mathematics. Philosophical and educational perspectives (pp. 115–135). New York: Springer.
Ball, D. L. (1999). Crossing boundaries to examine the mathematics entailed in elementary teaching. In T. Lam (Ed.), Contemporary mathematics (pp. 15–36). Providence: American Mathematical Society.
Ball, D., & Bass, H. (2000). Making believe: The collective construction of public mathematical knowledge in the elementary classroom. In D. Phillips (Ed.), Constructivism in education (pp. 193–224). Chicago: University of Chicago Press.
Ball, D., & Bass, H. (2003). Making mathematics reasonable in school. In J. Kilpatrick, W. G. Martin, & D. Schifter (Eds.), A research companion to principles and standards for school mathematics (pp. 27–44). Reston: NCTM.
Barrier, T., Durand-Guerrier V., & Blossier T. (2009). Semantic and game-theoretical insight into argumentation and proof (Vol. 1, pp. 77–88).*
Bartolini Bussi, M. (2009). Proof and proving in primary school: An experimental approach (Vol. 1, pp. 53–58).*
Bell, A. (1976). A study of pupils’ proof-explanations in mathematical situations. Educational Studies in Mathematics, 7(23–40).
Biza, I., Nardi, E., & Zachariades, T. (2009). Teacher beliefs and the didactic contract in visualization. For the Learning of Mathematics, 29(3), 31–36.
Blum, W., & Kirsch, A. (1991). Preformal proving: Examples and reflections. Educational Studies in Mathematics, 22(2), 183–203.
Boero, P., Garuti, R., & Lemut, E. (2007). Approaching theorems in grade VIII. In P. Boero (Ed.), Theorems in schools: From history, epistemology and cognition to classroom practice (pp. 249–264). Rotterdam/Taipei: Sense.
Cabassut, R. (2005). Démonstration, raisonnement et validation dans l’enseignement secondaire des mathématiques en France et en Allemagne. IREM Université Paris 7. Downloadable on http://tel.ccsd.cnrs.fr/documents/archives0/00/00/97/16/index.html
Cabassut, R. (2009). The double transposition in proving (Vol. 1, pp. 112–117).*
Carpenter, T., Franke, M., & Levi, L. (2003). Thinking mathematically: Integrating arithmetic & algebra in elementary school. Portsmouth: Heinemann.
Chazan, D. (1993). High school geometry students’ justification for their views of empirical evidence and mathematical proof. Educational Studies in Mathematics, 24, 359–387.
Conner, A. (2007). Student teachers’ conceptions of proof and facilitation of argumentation in secondary mathematics classrooms (Doctoral dissertation, The Pennsylvania State University, 2007). Dissertations Abstracts International, 68/05, Nov. 2007 (UMI No. AAT 3266090).
Conner, A., & Kittleson, J. (2009). Epistemic understandings in mathematics and science: Implications for learning (Vol. 1, pp. 106–111).*
de Villiers, M. (1990). The role and function of proof in mathematics. Pythagoras, 24, 17–24.
Dormolen, Jv. (1977). Learning to understand what giving a proof really means. Educational Studies in Mathematics, 8, 27–34.
Durand-Guerrier, V. (2008). Truth versus validity in mathematical proof. ZDM – The International Journal on Mathematics Education, 40(3), 373–384.
Duval, R. (1991). Structure du raisonnement déductif et apprentissage de la démonstration. Educational Studies in Mathematics, 22(3), 233–262.
Fernandez, C. (2005). Lesson study: A means for elementary teachers to develop the knowledge of mathematics needed for reform minded teaching? Mathematical Thinking and Learning, 7(4), 265–289.
Fischbein, E. (1982). Intuition and proof. For the Learning of Mathematics, 3(2), 9–24.
Freudenthal, H. (1973). Mathematics as an educational task. Dordrecht: Reidel.
Fujita T., Jones K., & Kunimune S. (2009). The design of textbooks and their influence on students’ understanding of ‘proof’ in lower secondary school (Vol. 1, pp. 172–177).*
Furinghetti, F., & Morselli, F. (2007). For whom the frog jumps: The case of a good problem solver. For the Learning of Mathematics, 27(2), 22–27.
Furinghetti, F., & Morselli, F. (2009a). Leading beliefs in the teaching of proof. In W. Schlöglmann & J. Maaß (Eds.), Beliefs and attitudes in mathematics education: New research results (pp. 59–74). Rotterdam/Taipei: Sense.
Furinghetti, F., & Morselli, F. (2009b). Every unsuccessful solver is unsuccessful in his/her own way: Affective and cognitive factors in proving. Educational Studies in Mathematics, 70, 71–90.
Furinghetti, F., & Pehkonen, E. (2002). Rethinking characterizations of beliefs. In G. Leder, E. Pehkonen, & G. Toerner (Eds.), Beliefs: A hidden variable in mathematics education? (pp. 39–57). Dordrecht/Boston/London: Kluwer.
Galbraith, P. L. (1981). Aspects of proving: A clinical investigation of process. Educational Studies in Mathematics, 12, 1–29.
Hanna, G. (1983). Rigorous proof in mathematics education. Toronto: OISE Press.
Hanna, G., & Barbeau, E. (2009). Proofs as bearers of mathematical knowledge. In G. Hanna, H. N. Jahnke, & H. Pulte (Eds.), Explanation and proof in mathematics. Philosophical and educational perspectives (pp. 85–100). New York: Springer.
Harel, G., & Sowder, L. (1998). Students’ proof schemes: Results from exploratory studies. In A. H. Schoenfeld, J. Kaput, & E. Dubisnky (Eds.), Issues in mathematics education (Research in collegiate mathematics education III, Vol. 7, pp. 234–283). Providence: American Mathematical Society.
Harel, G., & Sowder, L. (2007). Toward a comprehensive perspective on proof. In F. Lester (Ed.), Second handbook of research on mathematics teaching and learning (pp. 805–842). Charlotte: NCTM/Information Age Publishing.
Healy, L., & Hoyles, C. (2000). A study of proof conceptions in algebra. Journal for Research in Mathematics Education, 31(4), 396–428.
Hemmi, K. (2008). Students’ encounter with proof: The condition of transparency. The International Journal on Mathematics Education, 40, 413–426.
Jahnke, H. N. (2007). Proofs and hypotheses. ZDM – The International Journal on Mathematics Education, 39(1–2), 79–86.
Jahnke, H. N. (2008). Theorems that admit exceptions, including a remark on Toulmin. ZDM – The International Journal on Mathematics Education, 40(3), 363–371.
Jahnke H. N. (2009a). Proof and the empirical sciences (Vol. 1, pp. 238–243).*
Jahnke, H. N. (2009b). The conjoint origin of proof and theoretical physics. In G. Hanna, H. N. Jahnke, & H. Pulte (Eds.), Explanation and proof in mathematics. Philosophical and educational perspectives (pp. 17–32). New York: Springer.
Jones, K. (2000). The student experience of mathematical proof at university level. International Journal of Mathematical Education, 31(1), 53–60.
Kirsch, A. (1979). Beispiele für prämathematische Beweise. In W. Dörfler & R. Fischer (Eds.), Beweisen im Mathematikunterricht (pp. 261–274). Hölder-Pichler-Temspsky: Klagenfurt.
Knuth, E. J. (2002a). Secondary school mathematics teachers’ conceptions of proof. Journal for Research in Mathematics Education, 33(5), 379–405.
Knuth, E. (2002b). Teachers’ conceptions of proof in the context of secondary school mathematics. Journal of Mathematics Teacher Education, 5(1), 61–88.
Lay S. R. (2009) Good proofs depend on good definitions: Examples and counterexamples in arithmetic (Vol. 2, pp. 27–30).*
Leatham, K. (2006). Viewing mathematics teachers’ beliefs as sensible systems. Journal of Mathematics Teacher Education, 9, 91–102.
Lee K., & Smith, J. P. III (2009). Cognitive and linguistic challenges in understanding proving (Vol. 2, pp. 21–26).*
Lee, K., & Smith, J. P. III. (2008). Exploring the student’s conception of mathematical truth in mathematical reasoning. Paper presented at the Eleventh Conference on Research in Undergraduate Mathematics Education, San Diego.
Ma, L. (1999). Knowing and teaching elementary mathematics. Mahwah: Erlbaum Associates.
Manin, Y. (1977). A course in mathematical logic. New York: Springer.
Martin, W. G., & Harel, G. (1989). Proof frames of preservice elementary teachers. Journal for Research in Mathematics Education, 20(1), 41–51.
Mingus, T., & Grassl, R. (1999). Preservice teacher beliefs about proofs. School Science and Mathematics, 99, 438–444.
Miyazaki, M., & Yumoto, T. (2009). Teaching and learning a proof as an object in lower secondary school mathematics of Japan (Vol. 2, pp. 76–81).*
Moore, R. C. (1994). Making the transition to formal proof. Educational Studies in Mathematics, 27, 249–266.
Morselli, F. (2006). Use of examples in conjecturing and proving: An exploratory study. In J. Novotná, H. Moarová, M. Krátká, & N. Stelíchová (Eds.), Proceedings of PME 30 (Vol. 4, pp. 185–192). Prague: Charles University.
National Council of Teachers of Mathematics (NCTM). (2000). Principles and standards for school mathematics. Reston: NCTM.
Neubrand, M. (1989). Remarks on the acceptance of proofs: The case of some recently tackled major theorems. For the Learning of mathematics, 9(3), 2–6.
Ouvrier-Buffet, C. (2004). Construction of mathematical definitions: An epistemological and didactical study. In Proceedings of the 28th Conference of the International Group for the Psychology of Mathematics Education (Vol. 3, pp. 473–480), Bergen, Norway.
Ouvrier-Buffet, C. (2006). Exploring mathematical definition construction processes. Educational Studies in Mathematics, 63, 259–282.
Philipp, R. A. (2007). Mathematics teachers’ beliefs and affect. In F. K. Lester (Ed.), Second handbook of research on mathematics teaching and learning (pp. 257–315). Charlotte: NCTM/Information Age Publishing.
Porteous, K. (1990). What do children really believe? Educational Studies in Mathematics, 21, 589–598.
Rav, Y. (1999). Why do we prove theorems? Philosophia Mathematica, 7(3), 5–41.
Semadeni, Z. (n.d.). The concept of premathematics as a theoretical background for primary mathematics teaching. Warsaw: Polish Academy of Sciences.
Simon, M., & Blume, G. (1996). Justification in the mathematics classroom: A study of prospective elementary teachers. The Journal of Mathematical Behavior, 15, 3–31.
Smith, J. C. (2006). A sense-making approach to proof: Strategies of students in traditional and problem-based number theory courses. The Journal of Mathematical Behavior, 25(1), 73–90.
Sørensen, H. K. (2005). Exceptions and counterexamples: Understanding Abel’s comment on Cauchy’s Theorem. Historia Mathematica, 32, 453–480.
Sowder, L., & Harel, G. (2003). Case studies of mathematics majors’ proof understanding, production, and appreciation. Canadian Journal of Science, Mathematics, and Technology Education, 3, 251–267.
Wittmann, E. C. (2009). Operative proof in elementary mathematics (Vol. 2, pp. 251–256).*
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NB: References marked with * are in F. L. Lin, F. J. Hsieh, G. Hanna, & M. de Villiers (Eds.) (2009). ICMI Study 19: Proof and proving in mathematics education. Taipei, Taiwan: The Department of Mathematics, National Taiwan Normal University.
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Cabassut, R., Conner, A., İşçimen, F.A., Furinghetti, F., Jahnke, H.N., Morselli, F. (2012). Conceptions of Proof – In Research and Teaching. In: Hanna, G., de Villiers, M. (eds) Proof and Proving in Mathematics Education. New ICMI Study Series, vol 15. Springer, Dordrecht. https://doi.org/10.1007/978-94-007-2129-6_7
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