Sfard (1991)

On the Dual Nature of Mathematical Conceptions: Reflections on Processes and Objects as Different Sides of the Same Coin

• Author: Anna Sfard
• Journal: Educational Studies in Mathematics
• Year: 1991
• Source: http://link.springer.com/article/10.1007/BF00302715

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 incomplete, 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.

Outline of Headings

• Introduction
• 1. The Dual Nature of Mathematical Conceptions
• 2. The Role of Operational and Structural Conceptions in the Formation of Mathematical Concepts — Historical Outlook
• 3. The Role of Operational and Structural Conceptions in the Formation of Mathematical Concepts — Psychological Outlook
• 4. The Role of Operational and Structural Conceptions in Cognitive Processes
  • 4.1 Operational Approach: Certainly Necessary, Sometimes also Sufficient
  • 4.2 The Necessity of Structural Conceptions
  • 4.3 The Inherent Difficulty of Reification

Also

• Mendeley: http://www.mendeley.com/research/dual-nature-mathematical-conceptions-reflections-processes-objects-different-sides-same-coin/

APA

Sfard, A. (1991). On the dual nature of mathematical conceptions: Reflections on processes and objects as different sides of the same coin. Educational Studies in Mathematics, 22, 1–36. doi:10.1007/BF00302715

BibTeX

@article{Sfard1991,
author = {Sfard, Anna},
doi = {10.1007/BF00302715},
journal = {Educational Studies in Mathematics},
pages = {1--36},
title = {{On the dual nature of mathematical conceptions: Reflections on processes and objects as different sides of the same coin}},
url = {http://link.springer.com/article/10.1007/BF00302715},
volume = {22},
year = {1991}
}