## Abstract

The emergence of a proof image is often an important stage in a learner’s construction of a proof. In this paper, we introduce, characterize, and exemplify the notion of proof image. We also investigate how proof images emerge. Our approach starts from the learner’s efforts to construct a justification without (or before) attempting any formal argument, and it focuses on the process by which a complete but not necessarily communicable image of that justification becomes available to the learner and provides explanation with certainty. We consider the interplay between the learner’s intuitive and logical thinking and, using the theoretical framework of Abstraction in Context, we trace the construction of knowledge that results from and enables progress of this interplay. The existence and identification of proof images and the nature of the processes by which they emerge constitute the theoretical contribution of this paper. Its practical value lies in the empirical analyses of these processes and in the potential to apply them to the design of tasks that support students in constructing their own proofs images and proofs. We believe that such processes are likely to considerably enrich students’ mathematical experience.

### Keywords

Proof image Justification Logical links Intuitive conviction Construction of knowledge Abstraction in Context### References

- Bingolbali, E., & Monaghan, J. (2008). Concept image revisited.
*Educational Studies in Mathematics, 68*, 19–35.CrossRefGoogle 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 school*(pp. 249–264). Rotterdam, The Netherlands: Sense Publishers.Google Scholar - Davydov, V. V. (1990).
*Soviet studies in mathematics education: Vol. 2. Types of generalization in instruction: Logical and psychological problems in the structuring of school curricula*(J. Kilpatrick, Ed., & J. Teller, Trans.). Reston, VA, USA: National Council of Teachers of Mathematics. (Original work published in 1972.)Google Scholar - Dreyfus, T. (1991). On the status of visual reasoning in mathematics and mathematics education. In F. Furinghetti (Ed.),
*Proceedings of the 15th conference of the International Group for the Psychology of Mathematics Education*(Vol. 1, pp. 33–48). Assisi, Italy: PME.Google Scholar - Dreyfus, T., Hershkowitz, R., & Schwarz, B. (2001). Abstraction in Context II: The case of peer interaction.
*Cognitive Science Quarterly, 1*, 307–368.Google Scholar - Dreyfus, T., & Kidron, I. (2014). From proof image to formal proof—A transformation. In S. Rezat, M. Hattermann, & A. Peter-Koop (Eds.),
*Transformation—A fundamental idea in mathematics education—Festschrift for Rudolf Strässer*(pp. 269–289). Berlin, Germany: Springer.CrossRefGoogle Scholar - Durand-Guerrier, V., Boero, P., Douek, N., Epp, S., & Tanguay, D. (2012). Argumentation and proof in the mathematics classroom. In G. Hanna & M. de Villiers (Eds.),
*Proof and proving in mathematics education—The 19th ICMI study*(New ICMI Study Series, Vol. 15, pp. 349–367). Dordrecht: Springer.CrossRefGoogle Scholar - Fischbein, E. (1994). The interaction between the formal, the algorithmic and the intuitive components in a mathematical activity. In R. Biehler, R. W. Scholz, R. Strässer, & B. Winkelmann (Eds.),
*Didactics of mathematics as a scientific discipline*(pp. 231–246). Dordrecht, The Netherlands: Kluwer.Google Scholar - 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. Gutiérrez (Eds.),
*Proceedings of the 20th conference of the International Group for the Psychology of Mathematics Education*(Vol. 2, pp. 113–120). Valencia, Spain: PME.Google Scholar - Giest, H. (2005). Zum Verhältnis von Konstruktivismus und Tätigkeitsansatz in der Pädagogik [On the relationship between constructivism and activity theory in education]. In F. Radis, M.-L. Braunsteiner & K. Klement (Eds.),
*Badener VorDrucke*(pp. 43–64). Baden/A.: Kompetenzzentrum für Forschung und Entwicklung (Schriftenreihe zur Bildungsforschung - Band 3).Google Scholar - Gray, E., Pitta, D., & Tall, D. (2000). Objects, actions and images: A perspective on early number development.
*Journal of Mathematical Behavior, 18*, 1–13.CrossRefGoogle Scholar - Hanna, G. (1990). Some pedagogical aspects of proof.
*Interchange, 21*, 6–13.CrossRefGoogle Scholar - Hanna, G., & de Villiers, M. (2012).
*Proof and proving in mathematics education—The 19th ICMI study*(New ICMI Study Series, Vol. 15). Dordrecht: Springer.CrossRefGoogle Scholar - Inglis, M., & Alcock, L. (2012). Expert and novice approaches to reading mathematical proofs.
*Journal for Research in Mathematics Education, 43*, 358–390.CrossRefGoogle Scholar - Kidron, I. (2011). Tacit models, treasured intuitions and the discrete–continuous interplay.
*Educational Studies in Mathematics, 78*, 109–126.CrossRefGoogle Scholar - Kidron, I., & Dreyfus, T. (2009). Justification, enlightenment and the explanatory nature of proof. In F.-L. Lin, F.-J. Hsieh, G. Hanna, & M. de Villiers (Eds.),
*Mathematics Education*(Vol. 1, pp. 244–249). Taipei, Taiwan: National Taiwan Normal University, Department of Mathematics.Google Scholar - Kidron, I., & Dreyfus, T. (2010). Justification enlightenment and combining constructions of knowledge.
*Educational Studies in Mathematics, 74*, 75–93.CrossRefGoogle Scholar - Liljedahl, P. G. (2005). Mathematical discovery and affect: The effect of AHA! experiences on undergraduate mathematics students.
*International Journal of Mathematical Education in Science and Technology, 36*, 219–235.CrossRefGoogle Scholar - Mariotti, M. A. (2006). Proof and proving in mathematics education. In A. Gutiérrez & P. Boero (Eds.),
*Handbook of research on the psychology of mathematics education—Past, present and future*(pp. 173–204). Rotterdam, The Netherlands: Sense Publishers.Google Scholar - Nardi, E. (2006). Mathematicians and conceptual frameworks in mathematics education…or: Why do mathematicians’ eyes glint at the sight of concept image/concept definition? In A. Simpson (Ed.),
*Retirement as process and concept—A festschrift for Eddie Gray and David Tall*(pp. 181–189). Prague, Czech Republic: Charles University.Google Scholar - Nasar, S. (1998).
*A beautiful mind*. London: Faber & Faber.Google Scholar - Pedemonte, B., & Buchbinder, O. (2011). Examining the role of examples in proving processes through a cognitive lens: The case of triangular numbers.
*ZDM—The International Journal on Mathematics Education, 43*, 257–267.CrossRefGoogle Scholar - Raman, M. (2003). Key ideas: What are they and how can they help us understand how people view proof?
*Educational Studies in Mathematics, 52*, 319–325.CrossRefGoogle Scholar - Rav, Y. (1999). Why do we prove theorems?
*Philosophia Mathematica, 7*, 5–41.CrossRefGoogle Scholar - Rota, G.-C. (1997).
*Indiscrete thoughts*(pp. 131–135). Boston, MA: Birkhäuser.CrossRefGoogle Scholar - Sandefur, J., Mason, J., Stylianides, G. J., & Watson, A. (2013). Generating and using examples in the proving process.
*Educational Studies in Mathematics, 83*, 323–340.CrossRefGoogle Scholar - Scataglini-Belghitar, G., & Mason, J. (2011). Establishing appropriate conditions: Students learning to apply a theorem.
*International Journal of Science and Mathematics Education, 10*, 927–953.CrossRefGoogle Scholar - Schwarz, B. B., Dreyfus, T., & Hershkowitz, R. (2009). The nested epistemic actions model for abstraction in context. In B. B. Schwarz, T. Dreyfus, & R. Hershkowitz (Eds.),
*Transformation of knowledge through classroom interaction*(pp. 11–41). London, UK: Routledge.Google Scholar - Tall, D. (1995). Cognitive development, representations and proof. In
*Proceedings of Justifying and Proving in School Mathematics*. London: Institute of Education.Google Scholar - Tall, D., & Vinner, S. (1981). Concept image and concept definition in mathematics with particular reference to limit and continuity.
*Educational Studies in Mathematics, 12*, 151–169.CrossRefGoogle Scholar - Treffers, A., & Goffree, F. (1985). Rational analysis of realistic mathematics education—The Wiskobas program. In L. Streefland (Ed.),
*Proceedings of the 9th conference of the International Group for the Psychology of Mathematics Education*(Vol. 2, pp. 97–121). Utrecht, The Netherlands: OW&OC.Google Scholar - Van der Waerden, B. L. (1954).
*Einfall und Überlegung: Drei kleine Beiträge zur Psychologie des mathematischen Denkens*[Idea and reflection: Three small contributions to the psychology of mathematical thinking]. Basel, Switzerland: Birkhäuser.Google Scholar - Vinner, S., & Hershkowitz, R. (1980). Concept images and common cognitive paths in the development of some simple geometrical concepts. In R. Karplus (Ed.),
*Proceedings of the 4th conference of the International Group for the Psychology of Mathematics Education*(pp. 177–184). Berkeley, CA: PME.Google Scholar - Weber, K., & Alcock, L. (2004). Semantic and syntactic proof productions.
*Educational Studies in Mathematics, 56*, 209–234.CrossRefGoogle Scholar - Weber, K., & Alcock, L. (2009). Proof in advanced mathematics classes: Semantic and syntactic reasoning in the representation system of proof. In D. A. Stylianou, M. L. Blanton, & E. J. Knuth (Eds.),
*Teaching and learning proof across the grades*(pp. 323–338). New York, NY: Routledge, Studies in Mathematical Thinking and Learning.Google Scholar - Wilkerson-Jerde, M. H., & Wilensky, U. J. (2011). How do mathematicians learn math?: Resources and acts for constructing and understanding mathematics.
*Educational Studies in Mathematics, 78*, 21–43.CrossRefGoogle Scholar