Cognitive Development of Proof

  • David TallEmail author
  • Oleksiy Yevdokimov
  • Boris Koichu
  • Walter Whiteley
  • Margo Kondratieva
  • Ying-Hao Cheng
Part of the New ICMI Study Series book series (NISS, volume 15)


This chapter traces the long-term cognitive development of mathematical proof from the young child to the frontiers of research. It uses a framework building from perception and action, through proof by embodied actions and classifications, geometric proof and operational proof in arithmetic and algebra, to the formal set-theoretic definition and formal deduction. In each context, proof develops over the long-term from the recognition and description of observed properties and the links between them, the selection of specific properties that can be used as definitions from which other properties may be deduced, to the construction of ‘crystalline concepts’ whose properties are a consequence of the context. These include Platonic objects in geometry, symbols having relationships in arithmetic and algebra and formal axiomatic systems whose properties are determined by their definitions.


Knowledge Structure Euclidean Geometry Formal Proof Mathematical Thinking Mathematical Proof 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.



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Copyright information

© Springer Science+Business Media B.V. 2011

Authors and Affiliations

  • David Tall
    • 1
    Email author
  • Oleksiy Yevdokimov
    • 2
  • Boris Koichu
    • 3
  • Walter Whiteley
    • 4
  • Margo Kondratieva
    • 5
  • Ying-Hao Cheng
    • 6
  1. 1.Mathematics Education Research CentreUniversity of WarwickCoventryUK
  2. 2.Department of Mathematics & ComputingUniversity of Southern QueenslandToowoombaAustralia
  3. 3.Department of Education in Technology and ScienceTechnion – Israel Institute of TechnologyHaifaIsrael
  4. 4.Department of Mathematics and StatisticsYork UniversityTorontoCanada
  5. 5.Faculty of Education and Department of Mathematics and StatisticsMemorial University of NewfoundlandSt. John’sCanada
  6. 6.Department of MathematicsTaipei Municipal University of EducationTaipeiTaiwan

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