## Abstract

This paper presents a theoretical framework for investigating the role of algorithms in mathematical thinking. In the study, a combined ontological-psychological outlook is applied. An analysis of different mathematical definitions and representations brings us to the conclusion that abstract notions, such as number or function, can be conceived in two fundamentally different ways: *structurally*-as objects, and *operationally*-as processes. These two approaches, although ostensibly incompatible, are in fact complementary. It will be shown that the processes of learning and of problem-solving consist in an intricate interplay between operational and structural conceptions of the same notions.

On the grounds of historical examples and in the light of cognitive schema theory we conjecture that the operational conception is, for most people, the first step in the acquisition of new mathematical notions. Thorough analysis of the stages in concept formation leads us to the conclusion that transition from computational operations to abstract objects is a long and inherently difficult process, accomplished in three steps: *interiorization, condensation*, and *reification*. In this paper, special attention is given to the complex phenomenon of reification, which seems inherently so difficult that at certain levels it may remain practically out of reach for certain students.

### Similar content being viewed by others

### Explore related subjects

Discover the latest articles, news and stories from top researchers in related subjects.## References

Anderson, J. R.: 1976,

*Language, Memory, and Thought*, Erlbaum, Hillsdale, N.J.Behr, M., Erlwanger, S., and Nichols, E.: 1976,

*How children view equality sentences*(PMDC Technical Report No. 3), Florida State University (ERIC Document Reproduction Service No. ED144802).Bishop, A. J.: 1988, ‘A review of research on visualisation in mathematics education’, in

*Proceedings of The Twelfth International Conference for the Psychology of Mathematics Education*, Hungary, Vol. 1, pp. 170–176.Brownell, W. A.: 1935, ‘Psychological considerations in learning and teaching arithmetic’, in

*The Teaching of Arithmetic: Tenth Yearbook of the NCTM*, Columbia University Press, New York.Cajori, F. A.: 1985,

*History of Mathematics*,*Fourth Edition*, Chelsea Publishing Company, New York.Carpenter, T. P., Corbitt, M. K., Kepner, H. S., Lindquist, M. M., and Reys, R.: 1980, ‘Results of the second NAEP mathematics assessment: Secondary school’,

*The Mathematics Teacher***73**(5), 329–338.Clements, K.: 1981, ‘Visual imagery and school mathematics’,

*For the Learning of Mathematics***2**(2), 33–38.Courant, R. and John, F.: 1965,

*Introduction to Calculus and Analysis*, Vol. I, Interscience Publishers, New York.Davis, R. B.: 1975, ‘Cognitive processes involved in solving simple algebraic equations’,

*Journal of Children's Mathematical Behavior***1**(3), 7–35.Davis, P. J. and Hersh, R.: 1983,

*The Mathematical Experience*, Penguin Books, London.Dörfler, W.: 1987, ‘Empirical investigation of the construction of cognitive schemata from actions’, in

*Proceedings of the Eleventh International Conference of PME*, Vol. III, pp. 3–9.Dörfler, W.: 1989, ‘Protocols of actions as a cognitive tool for knowledge construction’, in

*Proceedings of the Thirteenth International Conference of PME*, Paris, Vol. 1, pp. 212–9.Dubinsky, E. and Lewin, P.: 1986, ‘Reflective abstraction and mathematics education: The genetic decomposition of induction and compactness’,

*Journal of Mathematical Behavior***5**, 55–92.Eisenberg, T. and Dreyfus, T. (eds.): 1989,

*Visualization in the mathematics curriculum*, Special issue of*Focus on Learning Problems in Mathematics***11**(1&2).Frege, G.: 1970, ‘What is function’, in Geach, P. and Black, M. (eds.),

*Translations from the Philosophical Writings of Gottlob Frege*, Blackwell, Oxford.Hadamard, J. S.: 1949,

*The Psychology of Invention in the Mathematics Field*, Princeton University Press, NJ.Halmos, P. R.: 1985, ‘Pure thought is better yet...’,

*The College Mathematics Journal***16**, 14–16.Halmos, P. R.: 1985a,

*I Want to be a Mathematician, An Autobiography*, Springer, New York.Henrici, P.: 1974, ‘The influence of computing on mathematical research and education’, in

*Proceedings of Symposia in Applied Mathematics*, Vol. 20, American Mathematical Society, Providence.Hiebert, J. and Lefevre, P.: 1986, ‘Conceptual and procedural knowledge in mathematics: An introductory analysis’, in Hiebert, J. (ed.),

*Conceptual and Procedural Knowledge: The Case of Mathematics*, Erlbaum, Hillsdale, NJ.Jourdain, P. E. B.: 1956, ‘The nature of mathematics’, in Newman, J. R. (ed.),

*The World of Mathematics*, Simon and Schuster, New York.Kaput, J. J.: 1979, ‘Mathematics and learning: Roots of epistemological status’, in Lochhead, J. and Clement, J. (eds.),

*Cognitive Process Instruction*, Franklin Institute Press.Kieran, C.: 1981, ‘Concepts associated with the equality symbol’,

*Educational Studies in Mathematics***12**(3), 317–326.Kilpatrick, J.: 1988, ‘Editorial’,

*Journal for Research in Mathematics Education***19**(4).Kleiner, I.: 1988, ‘Evolution of the function concept: A brief survey’,

*College Mathematics Journal***20**(4), 282–300.Lesh, R. and Landau, M. (eds.): 1983,

*Acquisition of Mathematics Concepts and Processes*, Academic Press, New York.Markovits, Z., Eylon, B., and Bruckheimer, M.: 1986, ‘Functions today and yesterday’,

*For the Learning of Mathematics***6**(2), 18–24.Maurer, S. B.: 1985, ‘The algorithmic way of life is best’,

*College Mathematics Journal***16**, 2–5.Miller, G. A.: 1956, ‘The magic number seven plus minus two: Some limits on our capacity for processing information’,

*Psychological Review***63**, 81–96.Otte, M.: 1984, ‘Komplementarität’,

*Dialektik***8**, 60–75.Paivio, A.: 1971,

*Imagery and Verbal Processes*, Holt, Rinehart, and Winston, New York.Penrose, R.: 1989,

*The Emperor's New Mind*, Oxford University Press, Oxford.Piaget, J.: 1952,

*The Child's Conception of Number*, Routledge and Kegan, London.Piaget, J.: 1970,

*Genetic Epistemology*, W. W. Norton, New York.Poincaré, H.: 1952,

*Science and Method*, Dover Publications, New York.Sfard, A.: 1987, ‘Two conceptions of mathematical notions: operational and structural’, in

*Proceedings of the Eleventh International Conference of PME*, Montreal, Vol. 3, pp. 162–9.Sfard, A.: 1988, ‘Operational vs structural method of teaching mathematics: A case study’, in

*Proceedings of the Twelfth International Conference of PME*, Hungary, pp. 560–7.Sfard, A.: 1989, ‘Transition from operational to structural conception: the notion of function revisited’, in

*Proceedings of the Thirteenth International Conference of PME*, Paris, Vol. 3, pp. 151–8.Sinclair, H. and Sinclair, A.: 1986, ‘Children's mastery of written numerals and the construction of basic number concepts’, in Hiebert, J. (ed.),

*Conceptual and Procedural Knowledge: The Case of Mathematics*, Erlbaum, Hillsdale, N.J.Skemp, R. R.: 1971,

*The Psychology of Learning Mathematics*, Penguin Books, Harmondsworth, England.Skemp, R. R.: 1976, ‘Relational understanding and instrumental understanding’,

*Mathematics Teacher***77**, 20–26.Stein, S. K.: 1988, ‘Gersham's law: Algorithm drives out thought’,

*Journal of Mathematical Behavior***7**, 79–84.Steiner, H.-G.: 1985, ‘Theory of mathematics education: An introduction’,

*For the Learning of Mathematics***5**(2), 11–17.Tahta, D. (ed.): 1972,

*A Boolean Anthology*, Derby, ATM.Thompson, P. W.: 1985, ‘Experience, problem solving, and learning mathematics: considerations in developing mathematical curricula’, in E. A.Silver (ed.),

*Teaching and Learning Mathematical Problem Solving: Multiple Research Perspectives*, Erlbaum, Hillsdale, N.J.Vinner, S. and Dreyfus, T.: 1989, ‘Images and definitions for the concept of function’,

*Journal for Research in Mathematics Education***20**(5), 356–66.

## Author information

### Authors and Affiliations

## Rights and permissions

## About this article

### Cite this article

Sfard, A. On the dual nature of mathematical conceptions: Reflections on processes and objects as different sides of the same coin.
*Educ Stud Math* **22**, 1–36 (1991). https://doi.org/10.1007/BF00302715

Issue Date:

DOI: https://doi.org/10.1007/BF00302715