# From Proof Image to Formal Proof—A Transformation

## Abstract

We propose the notion of proof image as an intermediate stage in a learner’s production of a proof. A proof image consists of the cognitive structure in the learner’s mind that is associated with the given proof. It consists of previous constructs that the learner has selected for potential use in the proof to be constructed and of the links between these previous constructs, links that the learner expects to play a role in the proof. In this chapter, we focus on the transition of a proof from proof image to formal proof. We do this within the theoretical framework of Abstraction in Context, leaning on Davydov’s notion of abstracting, according to which abstraction proceeds from an unrefined and vague form to a final coherent construct. We exemplify this transition by means of the story of K, who constructs a proof for a theorem in analysis from his proof image. We discuss in more detail the notion of proof image by means of the story of L, another learner, this one being related to bifurcations in dynamical systems.

## Keywords

Proof Justification Proof image Inevitability Formal proof Transition to formal proof Davydov’s view of abstraction Abstraction in context Constructs Previous constructs and links between them Dynamic and static proof images Concept image and concept definition## Notes

### Acknowledgment

This research was supported by the Israel Science Foundation under grant number 843/09.

## References

- 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., & Kidron, I. (2006). Interacting parallel constructions: A solitary learner and the bifurcation diagram.
*Recherches en Didactique des Mathématiques*,*26*(3), 295–336.Google Scholar - Fischbein, E. (1982). Intuition and proof.
*For the Learning of Mathematics, 3*(2), 9–18.Google Scholar - Giest, H. (2005). Zum Verhältnis von Konstruktivismus und Tätigkeitsansatz in der Pädagogik. 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 - Kidron, I., & Dreyfus, T. (2010a). Justification enlightenment and combining constructions of knowledge.
*Educational Studies in Mathematics*,*74*(1), 75–93.CrossRefGoogle Scholar - Kidron, I., & Dreyfus, T. (2010b). Interacting parallel constructions of knowledge in a CAS context.
*International Journal of Computers for Mathematical Learning, 15*(2), 129–149.CrossRefGoogle Scholar - Nasar, S. (1998).
*A beautiful mind*. London: Faber & Faber.Google Scholar - Ozmantar, M. F., & Monaghan, J. (2007). A dialectical approach to the formation of mathematical abstractions.
*Mathematics Education Research Journal, 19*(2), 89–112.CrossRefGoogle Scholar - Rota, G.-C. (1997).
*Indiscrete thoughts*. Boston: Birkhäuser.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.Google 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: Routledge.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 - Thurston, W. P. (1994). On proof and progress in mathematics.
*Bulletin of the American Mathematical Society*,*30*(2), 161–177.CrossRefGoogle Scholar - Van Oers, B. (2001). Contextualisation for abstraction.
*Cognitive Science Quarterly, 1*(3/4), 279–305.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 Annual Meeting for the Psychology of Mathematics Education*(pp. 177–184). Berkeley: PME.Google Scholar