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Forms of Proof and Proving in the Classroom

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Proof and Proving in Mathematics Education

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

Abstract

This chapter discusses forms of proof and proving in the learning and teaching of mathematics, including different representations used in proof production, different ways of arguing mathematically, different degrees of rigour in proving, and multiple proofs of the same statement. First, we focus on external forms of proof. We report research on students’ and teachers’ beliefs about visual aspects of proving and discuss the importance of visibility and transparency in mathematical arguments, particularly those using visualisation. We highlight the pedagogical potential of proving activities involving visualisation and reflect on its limitations. Next, we discuss the importance of various mathematical, pedagogical, and cognitive aspects of different forms of proof in multiple-proof tasks. We then examine which forms of proof might support students’ transition from empirical arguments to general proofs, using examples from the history of mathematics and discussing the roles of operative and generic proofs. We conclude by indicating potential future research agendas.

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References

  • Alsina, C., & Nelsen, R. B. (2006). Math made visual. Washington, DC: The Mathematical Association of America.

    Google Scholar 

  • Arzarello, F., Paola, D., & Sabena, C. (2009a). Proving in early calculus (Vol. 1, pp. 35–40).*

    Google Scholar 

  • Arzarello, F., Paola, D., & Sabena, C. (2009b). Logical and semiotic levels in argumentation (Vol. 1, pp. 41–46).*

    Google Scholar 

  • Australian Education Council. (1991). A national statement on mathematics for Australian schools. Melbourne: Curriculum Corporation.

    Google Scholar 

  • Balacheff, N. (1987). Processus de preuve et situations de validation [Proof processes and situations of validation]. Educational Studies in Mathematics, 18(2), 147–176.

    Google Scholar 

  • Biza, I., Nardi, E., & Zachariades, T. (2009a). Do images disprove but do not prove? Teachers’ beliefs about visualization (Vol. 1, pp. 59–64).*

    Google Scholar 

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

    Google Scholar 

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

    Google Scholar 

  • Borwein, J. (2011). Exploratory experimentation: Digitally-assisted discovery and proof. In G. Hanna & M. de Villiers (Eds.), Proof and proving in mathematics education (pp. XX–XX). Dordrecht: Springer.

    Google Scholar 

  • Bosch, M., & Chevallard, Y. (1999). Ostensifs et sensibilité aux ostensifs dans l’activité mathématique. Recherches en didactique des mathématiques, 19(1), 77–124.

    Google Scholar 

  • Brousseau, G. (1997). Theory of didactical situations in mathematics. Dordrecht/Boston/London: Kluwer Academic Publishers.

    Google Scholar 

  • Brown, J. (1997). Proofs and pictures. The British Journal for the Philosophy of Science, 48(2), 161–180.

    Google Scholar 

  • Buchbinder, O., & Zaslavsky, O. (2009). Uncertainty: a driving force in creating a need for proving. Technion IIT: Technical report available from the authors. [Paper accepted by but withdrawn from the ICMI 19 Study conference.]

    Google Scholar 

  • Byers, W. (2007). How mathematicians think: Using ambiguity, contradiction, and paradox to create mathematics. Princeton: Princeton University Press.

    Google Scholar 

  • Cabassut, R. (2009). The double transposition in proving (Vol. 1, pp. 112–117).*

    Google Scholar 

  • Chin, E-T., Liu, C-Y., & Lin, F-L. (2009). Taiwanese junior high school students’ proof conceptions in algebra (Vol. 1, pp. 118–123).*

    Google Scholar 

  • Cooney, J., Shealy, B. E., & Arvold, B. (1998). Conceptualizing belief structures of preservice secondary mathematics teachers. Journal for Research in Mathematics Education, 29(3), 306–333.

    Google Scholar 

  • Davis, P. J., & Hersh, R. (1981). The mathematical experience. Boston: Birkhäuser.

    Google Scholar 

  • Dreyfus, T. (1994). Imagery and reasoning in mathematics and mathematics education. In D. Robitaille, D. Wheeler, & C. Kieran (Eds.), Selected lectures from the 7th International Congress on Mathematical Education (pp. 107–122). Sainte-Foy: Les presses de l’université Laval.

    Google Scholar 

  • Dreyfus, T. (2000). Some views on proofs by teachers and mathematicians. In A. Gagatsis (Ed.), Proceedings of the 2nd Mediterranean Conference on Mathematics Education (Vol. I, pp. 11–25). Nicosia: The University of Cyprus.

    Google Scholar 

  • Dreyfus, T., & Hadas, N. (1996). Proof as answer to the question why. Zentralblatt für Didaktik der Mathematik, 28(1), 1–5.

    Google Scholar 

  • Ersoz, F. A. (2009). Proof in different mathematical domains (Vol. 1, pp. 160–165).*

    Google Scholar 

  • Fischbein, E. (1982). Intuition and proof. For the Learning of Mathematics, 3(2), 9–18.

    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. (2009). Teachers’ beliefs and the teaching of proof (Vol. 1, pp. 166–171).*

    Google Scholar 

  • Giaquinto, M. (2007). Visual thinking in mathematics. New York: Oxford University Press.

    Google Scholar 

  • Grabiner, J. (2011). Why proof? A historian’s perspective. In G. Hanna & M. de Villiers (Eds.), Proof and proving in mathematics education (pp. XX–XX). Dordrecht: Springer.

    Google Scholar 

  • Greer, B., de Bock, D., & van Dooren, W. (2009). The ISIS problem and pre-service teachers’ ideas about proof (Vol. 1, pp. 184–189).*

    Google Scholar 

  • Hadas, N., Hershkowitz, R., & Schwarz, B. (2000). The role of contradiction and uncertainty in promoting the need to prove in dynamic geometry environments. Educational Studies in Mathematics, 44(1–2), 127–150.

    Google Scholar 

  • Hanna, G. (1989). Proofs that prove and proofs that explain. In G. Vergnaud, J. Rogalski, & M. Artigue (Eds.), Proceedings of the 13th Conference of the International Group for the Psychology of Mathematics Education (Vol. 2, pp. 45–51). Paris: CNRS.

    Google Scholar 

  • Hanna, G., & Sidoli, N. (2007). Visualisation and proof: A brief survey of philosophical perspectives. ZDM – The International Journal on Mathematics Education, 39, 73–78.

    Google Scholar 

  • Hanna, G., de Villiers, M., Arzarello, F., Dreyfus, T., Durand-Guerrier, V., Jahnke, H. N., Lin, F.-L., Selden, A., Tall, D., & Yevdokimov, O. (2009). ICMI study 19: Proof and proving in mathematics education: Discussion document (Vol. 1, pp. xix-xxx).*

    Google Scholar 

  • Harel, G., & Sowder, L. (1998). Students’ proof schemes: Results from exploratory studies. In A. H. Schoenfeld, J. Kaput, & E. Dubinsky (Eds.), Research in collegiate mathematics education III (CBMS: Issues in mathematics education, 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: Information Age Publishing.

    Google Scholar 

  • Harel, R., & Dreyfus, T. (2009). Visual proofs: High school students’ point of view. 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, p. 386). Thessaloniki: PME.

    Google Scholar 

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

    Google Scholar 

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

    Google Scholar 

  • Hemmi, K., & Jaworski, B. (2009). Transparency in a tutor-student.interaction concerning the converse of Lagrange’s theorem (Vol. 1, pp. 202–207).*

    Google Scholar 

  • Iannone, P. (2009). Concept usage in proof production: Mathematicians’ perspectives (Vol. 1, pp. 220–225).*

    Google Scholar 

  • Israeli Ministry of Education. (1994). Tomorrow 98. Jerusalem: Ministry of Education [in Hebrew].

    Google Scholar 

  • Kidron, I., & Dreyfus, T. (2009). Justification, enlightenment and the explanatory nature of proof (Vol. 1, pp. 244–249).*

    Google Scholar 

  • Kidron, I., Bikner-Ahsbahs, A., Cramer, J., Dreyfus, T., & Gilboa, N. (2010). Construction of knowledge: Need and interest. 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. 3, pp. 169–176). Belo Horizonte: PME.

    Google Scholar 

  • Kondratieva, M. (2009). Geometrical sophisms and understanding of mathematical proofs (Vol. 2, pp. 3–8).*

    Google Scholar 

  • Kwon, O. N., Park, J. S., & Park, J. H. (2006). Cultivating divergent thinking in mathematics through an open-ended approach. Asia Pacific Education Review, 7, 51–61.

    Google Scholar 

  • Lave, J., & Wenger, E. (1991). Situated learning: Legitimate peripheral participation. Cambridge: Cambridge University Press.

    Google Scholar 

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

    Google Scholar 

  • Leder, G. C., Pehkonen, E., & Törner, G. (Eds.). (2002). Beliefs: A hidden variable in mathematics education? Dordrecht: Kluwer.

    Google Scholar 

  • Leikin, R. (2007). Habits of mind associated with advanced mathematical thinking and solution spaces of mathematical tasks. In D. Pitta-Pantazi & G. Philippou (Eds.), Proceedings of the Fifth Conference of the European Society for Research in Mathematics Education – CERME-5 (pp. 2330–2339) (CD-ROM). Retrieved March 7, 2011, from http://www.erme.unito.it/CERME5b/

  • Leikin, R. (2009a). Multiple proof tasks: Teacher practice and teacher education (Vol. 2, pp. 31–36).*

    Google Scholar 

  • Leikin, R. (2009b). Exploring mathematical creativity using multiple solution tasks. In R. Leikin, A. Berman, & B. Koichu (Eds.), Creativity in mathematics and the education of gifted students (pp. 129–145). Rotterdam: Sense Publishers.

    Google Scholar 

  • Leikin, R., & Levav-Waynberg, A. (2007). Exploring mathematics teacher knowledge to explain the gap between theory-based recommendations and school practice in the use of connecting tasks. Educational Studies in Mathematics, 66, 349–371.

    Google Scholar 

  • Leikin, R., & Levav-Waynberg, A. (2009). Development of teachers’ conceptions through learning and teaching: Meaning and potential of multiple-solution tasks. Canadian Journal of Science, Mathematics, and Technology Education, 9(4), 203–223.

    Google Scholar 

  • Leron, U., & Zaslavsky, O. (2009). Generic proving: Reflections on scope and method (Vol. 2, pp. 53–58).*

    Google Scholar 

  • Leung, A. (2009). Written proof in dynamic geometry environment: Inspiration from a student’s work (Vol. 2, pp. 15–20).*

    Google Scholar 

  • Lin, C. C. (2009). How can the game of Hex be used to inspire students in learning mathematical reasoning? (Vol. 2, pp. 37–40).*

    Google Scholar 

  • Lin, F.-L., Hsieh, F.-J., Hanna, G., & de Villiers, M. (Eds.). (2009). Proceedings of the ICMI Study 19 Conference: Proof and Proving in Mathematics Education (2 vols.). Taipei: Department of Mathematics, National Taiwan Normal University.

    Google Scholar 

  • Lithner, J. (2008). A research framework for creative and imitative reasoning. Educational Studies in Mathematics, 67, 255–276.

    Google Scholar 

  • Malek, A., & Movshovitz-Hadar, N. (2009). The art of constructing a transparent p-proof (Vol. 2, pp. 70–75).*

    Google Scholar 

  • Malek, A., & Movshovitz-Hadar, N. (2011). The effect of using transparent pseudo-proofs in linear algebra. Research in Mathematics Education, 13(1), 33–58.

    Google Scholar 

  • Mamona-Downs, J., & Downs, M. (2009). Proof status from a perspective of articulation (Vol. 2, pp. 94–99).*

    Google Scholar 

  • Mancosu, P., Jorgensen, K. F., & Pedersen, S. A. (Eds.). (2005). Visualization, explanation and reasoning styles in mathematics. Dordrecht: Springer.

    Google Scholar 

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

    Google Scholar 

  • Mason, J., & Pimm, D. (1984). Generic examples: Seeing the general in the particular. Educational Studies in Mathematics, 15, 277–289.

    Google Scholar 

  • Mejía-Ramos, J. P., & Inglis, M. (2009). Argumentative and proving activities in mathematics education research (Vol. 2, pp. 88–93).*

    Google Scholar 

  • Moore, R. C. (1994). Making the transition to formal proof. Educational Studies in Mathematics, 27(3), 249–266.

    Google Scholar 

  • Morselli, F. (2007). Sui fattori culturali nei processi di congettura e dimostrazione. Unpublished Ph.D. dissertation. Università degli Studi di Torino, Turin.

    Google Scholar 

  • Morselli, F., & Boero, P. (2009). Habermas’ construct of rational behaviour as a comprehensive frame for research on the teaching and learning of proof (Vol. 2, pp. 100–105).*

    Google Scholar 

  • Nardi, E. (2008). Amongst mathematicians: Teaching and learning mathematics at university level. New York: Springer.

    Google Scholar 

  • Nardi, E. (2009). ‘Because this is how mathematicians work!’ ‘Pictures’ and the creative fuzziness of the didactical contract at university level (Vol. 2, pp. 112–117).*

    Google Scholar 

  • Nardi, E., & Iannone, P. (2006). Conceptualising formal mathematical reasoning and the necessity for proof. In E. Nardi & P. Iannone (Eds.), How to prove it: A brief guide for teaching proof to year 1 mathematics undergraduates (pp. 5–16). Norwich: Higher Education Academy.

    Google Scholar 

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

    Google Scholar 

  • Nelsen, R. B. (1993). Proofs without words. Washington, DC: The Mathematical Association of America.

    Google Scholar 

  • Nelsen, R. B. (2000). Proofs without words II. Washington, DC: The Mathematical Association of America.

    Google Scholar 

  • Nogueira de Lima, R., & Tall, D. (2006). The concept of equation: What have students met before? In J. Novotná, H. Moraová, M. Krátká, & N. Stehlíková (Eds.), Proceedings of the 30th Conference of the International Group for the Psychology of Mathematics Education (Vol. 4, pp. 233–241). Prague: PME.

    Google Scholar 

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

    Google Scholar 

  • Perry, P., Camargo, L., Samper, C. Echeverry, A., & Molina, Ó. (2009a). Assigning mathematics tasks versus providing pre-fabricated mathematics in order to support learning to prove (Vol. 2, pp. 130–135).*

    Google Scholar 

  • Perry, P., Samper, C., Camargo, L., Molina, Ó., & Echeverry, A. (2009b). Learning to prove: Enculturation or…? (Vol. 2, pp. 124–129).*

    Google Scholar 

  • Pólya, G. (1945/1973). How to solve it. Princeton: Princeton University.

    Google Scholar 

  • Presmeg, N. C. (2006). Research on visualization in learning and teaching mathematics: Emergence from psychology. In A. Gutierrez & P. Boero (Eds.), Handbook of research on the psychology of mathematics education (pp. 205–235). Dordrecht: Sense Publishers.

    Google Scholar 

  • Raman, M., Sandefur, J., Birky, G., Campbell, C., & Somers, K. (2009). “Is that a proof?”: Using video to teach and learn how to prove at the university level (Vol. 2, pp. 154–159).*

    Google Scholar 

  • Rowland, T. (1998). Conviction, explanation and generic examples. In A. Olivier & K. Newstead (Eds.), Proceedings of the 22nd Conference of the International Group for the Psychology of Mathematics Education (Vol. 4, pp. 65–72). Stellenbosch: PME.

    Google Scholar 

  • Rowland, T. (2001). Generic proof in number theory. In S. Campbell & R. Zazkis (Eds.), Learning and teaching number theory: Research in cognition and instruction. Westport: Ablex.

    Google Scholar 

  • Schoenfeld, A. H. (1985). Mathematical problem solving. New York: Academic.

    Google Scholar 

  • Schwarz, B., & Kaiser, G. (2009). Professional competence of future mathematics teachers on argumentation and proof and how to evaluate it (Vol. 2, pp. 190–195).*

    Google Scholar 

  • Selden, A. (2005). New developments and trends in tertiary mathematics education: Or, more of the same? International Journal of Mathematical Education in Science and Technology, 36(2/3), 131–147.

    Google Scholar 

  • Silver, E. A. (1997). Fostering creativity through instruction rich in mathematical problem solving and problem posing. ZDM – The International Journal on Mathematics Education, 3, 75–80.

    Google Scholar 

  • Siu, M-K. (2009). The algorithmic and dialectic aspects in proof and proving (Vol. 2, pp. 160–165).*

    Google Scholar 

  • Sriraman, B. (2003). Mathematical giftedness, problem solving, and the ability to formulate generalizations. Journal of Secondary Gifted Education, 14, 151–165.

    Google Scholar 

  • Stenning, K., & Lemon, O. (2001). Aligning logical and psychological perspectives on diagrammatic reasoning. Artificial Intelligence Review, 15(1–2), 29–62.

    Google Scholar 

  • Stevenson, I. (2009). Dynamic geometry and proof: The cases of mechanics and non-Euclidean space (Vol. 2, pp. 184–189).*

    Google Scholar 

  • Stylianides, G. J., & Stylianides, A. J. (2009). Ability to construct proofs and evaluate one’s own constructions (Vol. 2, pp. 166–171).*

    Google Scholar 

  • Sun, X. H. (2007). Spiral variation (bianshi) curriculum design in mathematics: Theory and practice. Unpublished doctoral dissertation, The Chinese University of Hong Kong, Hong Kong.

    Google Scholar 

  • Sun, X. (2009). Renew the proving experiences: An experiment for enhancement trapezoid area formula proof constructions of student teachers by “one problem multiple solutions” (Vol. 2, pp. 178–183).*

    Google Scholar 

  • Sun, X., & Chan, K. (2009). Regenerate the proving experiences: An attempt for improvement original theorem proof constructions of student teachers by using spiral variation curriculum (Vol. 2, pp. 172–177).*

    Google Scholar 

  • Tabach, M., Levenson, E., Barkai, R., Tsamir, P., Tirosh, D., & Dreyfus, T. (2009). Teachers’ knowledge of students’ correct and incorrect proof constructions (Vol. 2, pp. 214–219).*

    Google Scholar 

  • Tall, D. (1979). Cognitive aspects of proof, with special reference to the irrationality of √2. In D. Tall (Ed.), Proceedings of the 3rd Conference of the International Group for the Psychology of Mathematics Education (pp. 206–207). Warwick: PME.

    Google Scholar 

  • Tall, D., Yevdokimov, O., Koichu, B., Whiteley, W., Kondratieva, M., & Cheng, Y. (2011). Cognitive Development of Proof. In G. Hanna & M. de Villiers (Eds.), Proof and proving in mathematics education (pp. XX–XX). Dordrecht: Springer.

    Google Scholar 

  • Tsamir, P., Tirosh, D., Dreyfus, T., Tabach, M., & Barkai, R. (2009). Is this verbal justification a proof? (Vol. 2, pp. 208–213).*

    Google Scholar 

  • Tymoczko, T. (1985). New directions in the philosophy of mathematics. Basel: Birkhäuser.

    Google Scholar 

  • Weber, K. (2001). Student difficulty in constructing proofs: The need for strategic knowledge. Educational Studies in Mathematics, 48, 101–119.

    Google Scholar 

  • Weber, K., & Alcock, L. (2004). Semantic and syntactic proof productions. Educational Studies in Mathematics, 56, 209–234.

    Google Scholar 

  • Whiteley, W. (2004). To see like a mathematician. In G. Malcolm (Ed.), Multidisciplinary approaches to visual representations and interpretations (Vol. 2, pp. 279–291). London: Elsevier.

    Google Scholar 

  • Whiteley, W. (2009). Refutations: The role of counter-examples in developing proof (Vol. 2, pp. 257–262).*

    Google Scholar 

  • Winicki-Landman, G. (2009). Mathematical games as proving seminal activities for elementary school teachers (Vol. 2, pp. 245–250).*

    Google Scholar 

  • Wittmann, E. Ch. (1998). Operative proof in elementary school. Mathematics in School, 27(5).

    Google Scholar 

  • Wittmann, E. Ch. (2005). The alpha and the omega of teacher education: Organizing mathematical activities. In D. Holton (Ed.), The teaching and learning of mathematics at university level (pp. 539–552). Dordrecht: Kluwer Academic Publishers.

    Google Scholar 

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

    Google Scholar 

  • Wong, N. Y. (2007). Confucian heritage cultural learner’s phenomenon: From “exploring the middle zone” to “constructing a bridge”. Regular lecture, presented at the Fourth ICMI-East Asia Regional Conference on Mathematical Education. Penang.

    Google Scholar 

  • Zehavi, N., & Mann, G. (2009). Proof and experimentation: Integrating elements of DGS and CAS (Vol. 2, pp. 286–291).*

    Google Scholar 

Download references

Acknowledgements

This research was partly supported by the Israel Science Foundation under grants 843/09 and 891/03, as well as by an EU Erasmus Staff Mobility Bilateral Agreement between the University of East Anglia in the UK and the University of Athens in Greece.

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Authors and Affiliations

  1. Department of Mathematics, Science and Technology Education, School of Education, Tel Aviv University, Tel Aviv, Israel

    Tommy Dreyfus

  2. School of Education, University of East Anglia, Norwich, UK

    Elena Nardi

  3. Department of Mathematics Education, Faculty of Education, University of Haifa, Haifa, Israel

    Roza Leikin

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  1. Tommy Dreyfus
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Correspondence to Tommy Dreyfus .

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  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

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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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Dreyfus, T., Nardi, E., Leikin, R. (2012). Forms of Proof and Proving in the Classroom. 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_8

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