Educational Studies in Mathematics

, Volume 24, Issue 2, pp 139–162 | Cite as

The theory of figural concepts

  • Efraim Fischbein


The main thesis of the present paper is that geometry deals with mental entities (the so-called geometrical figures) which possess simultaneously conceptual and figural characters. A geometrical sphere, for instance, is an abstract ideal, formally determinable entity, like every genuine concept. At the same time, it possesses figural properties, first of all a certain shape. The ideality, the absolute perfection of a geometrical sphere cannot be found in reality. In this symbiosis between concept and figure, as it is revealed in geometrical entities, it is the image component which stimulates new directions of thought, but there are the logical, conceptual constraints which control the formal rigour of the process. We have called the geometrical figuresfigural concepts because of their double nature. The paper analyzes the internal tensions which may appear in figural concepts because of this double nature, development aspects and didactical implications.


Geometrical Figure Figural Property Development Aspect Image Component Internal Tension 
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.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. Anderson, J. R.: 1978, ‘Arguments concerning representations for mental imagery’,Psychological Review, 249–277.Google Scholar
  2. Anderson, J. R.: 1990,Cognitive Psychology and Its Implications, Carnegie-Mellon University, Third Edition, W. H. Freeman and Co., New York.Google Scholar
  3. Anderson, J. R. and Bower, G. H.: 1973,Human Associative Memory, Hemisphere Press, Washington DC.Google Scholar
  4. Blanc-Garin, J.: 1974, ‘Recherches recentes sur les images mentales: Leur rôle dans les processus de traitment perceptif et cognitif’,Année Psychologique 74, 533–564.Google Scholar
  5. Denis, M. and Dubois, D.: 1976, ‘La représentation cognitive’,Année Psychologique 76, 541–562.Google Scholar
  6. Fischbein, E.: 1963,Conceptele Figurale (in Rumanian), Editura Academiei RPR, Bucuresti.Google Scholar
  7. Fischbein, E. and Kedem, I.: 1982, ‘Proof and certitude in the development of mathematical thinking’, in A. Vermandel (ed.),Proceedings of the Sixth International Conference for the Psychology of Mathematical Education, Universitaire Instelling, Antwerpen.Google Scholar
  8. Kline, M.: 1980,Mathematics. The Loss of Certainty, Oxford University Press, New York.Google Scholar
  9. Kosslyn, S. M.: 1980,Image and Mind, Harvard University Press, Cambridge, MA.Google Scholar
  10. Kosslyn, S. M.: 1983,Ghosts in the Mind's Machine. Creating and Using Images in the Brain, W. W. Norton & Company, New York.Google Scholar
  11. Mariotti, A.: 1992, ‘Imagini e concetti in geometria’,L'Insegnamento Della Matematica e Delle Scienze Integrata 15(9), 863–885.Google Scholar
  12. Paivio, A.: 1970, ‘On the functional significance of imagery’,Psychological Bulletin 73(6), 385–392.Google Scholar
  13. Paivio, A.: 1971,Imagery and Verbal Processes, Holt, Rinehart and Winston, New York.Google Scholar
  14. Panzycz, B.: 1988, ‘“Knowing” vs. “Seeing”. Problems of the plane representation of space geometry figures’,Educational Studies in Mathematics 19, 79–92.Google Scholar
  15. Piaget, J. and Inhelder, B.: 1966,LImage Mentale Chez l'Enfant. Etude sur le Développement des Représentations Imagées, PUF, Paris.Google Scholar
  16. Pylyshyn, Z. W.: 1973, ‘What the mind's eye tells the mind's brain: A critique of mental imagery’,Psychological Bulletin 80, 1–24.Google Scholar
  17. Piéron, H.: 1957,Vocabulaire de la Psychologie, PUF, Paris.Google Scholar
  18. Rohwer, W. D., Jr.: 1970, ‘Images and pictures in children's learning’,Psychological Bulletin 73(6), 393–403.Google Scholar
  19. Shepard, R. N.: 1978, ‘Externalization of mental images and the act of creation’, in B. S. Randhawa and W. E. Coffman (eds.),Visual Learning, Thinking and Communication, Academic Press, New York, pp. 133–189.Google Scholar
  20. Shepard, R. N. and Cooper, L. A.: 1982,Mental Images and Their Transformations, MIT Press, Cambridge, MA.Google Scholar
  21. Tall, D.: 1991, ‘The psychology of advanced mathematical thinking’, in D. Tall (ed.),Advanced Mathematical Thinking, Kluwer Academic Publishers, Dordrecht, pp. 4–21.Google Scholar

Copyright information

© Kluwer Academic Publishers 1993

Authors and Affiliations

  • Efraim Fischbein
    • 1
  1. 1.School of EducationTel-Aviv UniversityRamat AvivIsrael

Personalised recommendations