Transformation  A Fundamental Idea of Mathematics Education
pp 269289
From Proof Image to Formal Proof—A Transformation
 Tommy DreyfusAffiliated withSchool of Education, Tel Aviv University Email author
 , Ivy KidronAffiliated withDepartment of Applied Mathematics, Jerusalem College of Technology
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 Title
 From Proof Image to Formal Proof—A Transformation
 Book Title
 Transformation  A Fundamental Idea of Mathematics Education
 Book Part
 Part III
 Pages
 pp 269289
 Copyright
 2014
 DOI
 10.1007/9781461434894_13
 Print ISBN
 9781461434887
 Online ISBN
 9781461434894
 Publisher
 Springer New York
 Copyright Holder
 Springer Science+Business Media, LLC
 Additional Links
 Topics
 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
 Industry Sectors
 eBook Packages
 Editors

 Sebastian Rezat ^{(1)}
 Mathias Hattermann ^{(2)}
 Andrea PeterKoop ^{(3)}
 Editor Affiliations

 1. IEIM  Institut für Mathematik, Universität Paderborn
 2. Fakultät für Mathematik  IDM, Universität Bielefeld
 3. Fakultät für Mathematik  IDM, Universität Bielefeld
 Authors

 Tommy Dreyfus ^{(4)}
 Ivy Kidron ^{(5)}
 Author Affiliations

 4. School of Education, Tel Aviv University, Ramat Aviv, P.O. Box 39040, 69978, Tel Aviv, Israel
 5. Department of Applied Mathematics, Jerusalem College of Technology, Havaad Haleumi Str. 21, P.O. Box 16031, 91160, Jerusalem, Israel
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