Skip to main content
SpringerLink
Log in
SpringerLink
Log in

Conceptions of Proof – In Research and Teaching

  • Chapter
  • First Online:
Download book PDF
Download book EPUB
Proof and Proving in Mathematics Education

Part of the book series: New ICMI Study Series ((NISS,volume 15))

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.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution

Institutional subscriptions

References

  • Alibert, D. (1988). Towards new customs in the classroom. For the Learning of Mathematics, 8(2), 31–35.

    Google Scholar 

  • Arsac, G., Chapiron, G., Colonna, A., Germain, G., Guichard, Y., & Mante, M. (1992). Initiation au raisonnement déductif au collège. Lyon: Presses Universitaires de Lyon.

    Google Scholar 

  • 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.

    Google Scholar 

  • 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.

    Google Scholar 

  • 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.

    Google Scholar 

  • 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.

    Google Scholar 

  • 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.

    Google Scholar 

  • Barrier, T., Durand-Guerrier V., & Blossier T. (2009). Semantic and game-theoretical insight into argumentation and proof (Vol. 1, pp. 77–88).*

    Google Scholar 

  • Bartolini Bussi, M. (2009). Proof and proving in primary school: An experimental approach (Vol. 1, pp. 53–58).*

    Google Scholar 

  • Bell, A. (1976). A study of pupils’ proof-explanations in mathematical situations. Educational Studies in Mathematics, 7(23–40).

    Google Scholar 

  • Biza, I., Nardi, E., & Zachariades, T. (2009). Teacher beliefs and the didactic contract in visualization. For the Learning of Mathematics, 29(3), 31–36.

    Google Scholar 

  • Blum, W., & Kirsch, A. (1991). Preformal proving: Examples and reflections. Educational Studies in Mathematics, 22(2), 183–203.

    Google Scholar 

  • 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.

    Google Scholar 

  • 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).*

    Google Scholar 

  • Carpenter, T., Franke, M., & Levi, L. (2003). Thinking mathematically: Integrating arithmetic & algebra in elementary school. Portsmouth: Heinemann.

    Google Scholar 

  • Chazan, D. (1993). High school geometry students’ justification for their views of empirical evidence and mathematical proof. Educational Studies in Mathematics, 24, 359–387.

    Google Scholar 

  • 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).

    Google Scholar 

  • Conner, A., & Kittleson, J. (2009). Epistemic understandings in mathematics and science: Implications for learning (Vol. 1, pp. 106–111).*

    Google Scholar 

  • de Villiers, M. (1990). The role and function of proof in mathematics. Pythagoras, 24, 17–24.

    Google Scholar 

  • Dormolen, Jv. (1977). Learning to understand what giving a proof really means. Educational Studies in Mathematics, 8, 27–34.

    Google Scholar 

  • Durand-Guerrier, V. (2008). Truth versus validity in mathematical proof. ZDM – The International Journal on Mathematics Education, 40(3), 373–384.

    Google Scholar 

  • Duval, R. (1991). Structure du raisonnement déductif et apprentissage de la démonstration. Educational Studies in Mathematics, 22(3), 233–262.

    Google Scholar 

  • 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.

    Google Scholar 

  • Fischbein, E. (1982). Intuition and proof. For the Learning of Mathematics, 3(2), 9–24.

    Google Scholar 

  • Freudenthal, H. (1973). Mathematics as an educational task. Dordrecht: Reidel.

    Google Scholar 

  • 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).*

    Google Scholar 

  • 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.

    Google Scholar 

  • 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.

    Google Scholar 

  • 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.

    Google Scholar 

  • 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.

    Google Scholar 

  • Galbraith, P. L. (1981). Aspects of proving: A clinical investigation of process. Educational Studies in Mathematics, 12, 1–29.

    Google Scholar 

  • Hanna, G. (1983). Rigorous proof in mathematics education. Toronto: OISE Press.

    Google Scholar 

  • 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.

    Google Scholar 

  • 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.

    Google Scholar 

  • 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.

    Google Scholar 

  • Healy, L., & Hoyles, C. (2000). A study of proof conceptions in algebra. Journal for Research in Mathematics Education, 31(4), 396–428.

    Google Scholar 

  • Hemmi, K. (2008). Students’ encounter with proof: The condition of transparency. The International Journal on Mathematics Education, 40, 413–426.

    Google Scholar 

  • Jahnke, H. N. (2007). Proofs and hypotheses. ZDM – The International Journal on Mathematics Education, 39(1–2), 79–86.

    Google Scholar 

  • Jahnke, H. N. (2008). Theorems that admit exceptions, including a remark on Toulmin. ZDM – The International Journal on Mathematics Education, 40(3), 363–371.

    Google Scholar 

  • Jahnke H. N. (2009a). Proof and the empirical sciences (Vol. 1, pp. 238–243).*

    Google Scholar 

  • 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.

    Google Scholar 

  • Jones, K. (2000). The student experience of mathematical proof at university level. International Journal of Mathematical Education, 31(1), 53–60.

    Google Scholar 

  • Kirsch, A. (1979). Beispiele für prämathematische Beweise. In W. Dörfler & R. Fischer (Eds.), Beweisen im Mathematikunterricht (pp. 261–274). Hölder-Pichler-Temspsky: Klagenfurt.

    Google Scholar 

  • Knuth, E. J. (2002a). Secondary school mathematics teachers’ conceptions of proof. Journal for Research in Mathematics Education, 33(5), 379–405.

    Google Scholar 

  • Knuth, E. (2002b). Teachers’ conceptions of proof in the context of secondary school mathematics. Journal of Mathematics Teacher Education, 5(1), 61–88.

    Google Scholar 

  • Lay S. R. (2009) Good proofs depend on good definitions: Examples and counterexamples in arithmetic (Vol. 2, pp. 27–30).*

    Google Scholar 

  • Leatham, K. (2006). Viewing mathematics teachers’ beliefs as sensible systems. Journal of Mathematics Teacher Education, 9, 91–102.

    Google Scholar 

  • Lee K., & Smith, J. P. III (2009). Cognitive and linguistic challenges in understanding proving (Vol. 2, pp. 21–26).*

    Google Scholar 

  • 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.

    Google Scholar 

  • Ma, L. (1999). Knowing and teaching elementary mathematics. Mahwah: Erlbaum Associates.

    Google Scholar 

  • Manin, Y. (1977). A course in mathematical logic. New York: Springer.

    Google Scholar 

  • Martin, W. G., & Harel, G. (1989). Proof frames of preservice elementary teachers. Journal for Research in Mathematics Education, 20(1), 41–51.

    Google Scholar 

  • Mingus, T., & Grassl, R. (1999). Preservice teacher beliefs about proofs. School Science and Mathematics, 99, 438–444.

    Google Scholar 

  • 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).*

    Google Scholar 

  • Moore, R. C. (1994). Making the transition to formal proof. Educational Studies in Mathematics, 27, 249–266.

    Google Scholar 

  • 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.

    Google Scholar 

  • National Council of Teachers of Mathematics (NCTM). (2000). Principles and standards for school mathematics. Reston: NCTM.

    Google Scholar 

  • 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.

    Google Scholar 

  • 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.

    Google Scholar 

  • Ouvrier-Buffet, C. (2006). Exploring mathematical definition construction processes. Educational Studies in Mathematics, 63, 259–282.

    Google Scholar 

  • 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.

    Google Scholar 

  • Porteous, K. (1990). What do children really believe? Educational Studies in Mathematics, 21, 589–598.

    Google Scholar 

  • Rav, Y. (1999). Why do we prove theorems? Philosophia Mathematica, 7(3), 5–41.

    Google Scholar 

  • Semadeni, Z. (n.d.). The concept of premathematics as a theoretical background for primary mathematics teaching. Warsaw: Polish Academy of Sciences.

    Google Scholar 

  • Simon, M., & Blume, G. (1996). Justification in the mathematics classroom: A study of prospective elementary teachers. The Journal of Mathematical Behavior, 15, 3–31.

    Google Scholar 

  • 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.

    Google Scholar 

  • Sørensen, H. K. (2005). Exceptions and counterexamples: Understanding Abel’s comment on Cauchy’s Theorem. Historia Mathematica, 32, 453–480.

    Google Scholar 

  • 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.

    Google Scholar 

  • Wittmann, E. C. (2009). Operative proof in elementary mathematics (Vol. 2, pp. 251–256).*

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. LDAR Laboratoire de Didactique André Revuz, Paris 7 University, Paris, France

    Richard Cabassut

  2. IUFM Institut Universitaire de Formation des Maîtres, Strasbourg University, Strasbourg, France

    Richard Cabassut

  3. Department of Mathematics & Science Education, University of Georgia, Athens, GA, USA

    AnnaMarie Conner

  4. Department of Mathematics and Statistics, Kennesaw State University, Kennesaw, GA, USA

    Filyet Aslı İşçimen

  5. Dipartimento di Matematica, Università di Genova, Genoa, Italy

    Fulvia Furinghetti & Francesca Morselli

  6. Fakultät für Mathematik, Universität Duisburg-Essen, Essen, Germany

    Hans Niels Jahnke

Authors
  1. Richard Cabassut
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. AnnaMarie Conner
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Filyet Aslı İşçimen
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Fulvia Furinghetti
    View author publications

    You can also search for this author in PubMed Google Scholar

  5. Hans Niels Jahnke
    View author publications

    You can also search for this author in PubMed Google Scholar

  6. Francesca Morselli
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Hans Niels Jahnke .

Editor information

Editors and Affiliations

  1. Ontario Institute for Studies in Education (OISE), University of Toronto, Toronto, ON, Canada

    Gila Hanna

  2. Faculty of Education, University of KwaZulu-Natal, Edgewood Campus, Ashwood, South Africa

    Michael de Villiers

Additional information

 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.

Rights and permissions

Open Access This chapter is licensed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence and indicate if changes were made.

The images or other third party material in this chapter are included in the chapter's Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the chapter's Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder.

Reprints and permissions

Copyright information

© 2012 The Author(s)

About this chapter

Cite this chapter

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

Download citation

Publish with us

Policies and ethics

Search

Navigation