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
Summary
In her introduction, Sfard sets out to explore both the psychology and philosophy that makes the process of mathematical abstraction so difficult for so many:
There is probably much more to mathematics than just the rules of logic. It seems that to put our finger on the source of its ostensibly surprising difficulty, we must ask ourselves the most basic epistemological questions regarding the nature of mathematical knowledge. Indeed, since in its inaccessibility mathematics seems to surpass all the other scientific disciplines, there must be something really special and unique in the kind of thinking involved in constructing a mathematical universe. (p. 2)
The Dual Nature of Mathematical Conceptions: Structural versus Operational
Sfard distinguishes between mathematical concepts, the theoretical and formal constructs within "the formal universe of ideal knowledge," and conceptions, the "internal representations and associations evoked by the concept" (p. 3). Our conceptions have two types, structural and operational. A structural conception treats a mathematical concept as an abstract object, as if it is real. It can be recognized and operated with as a whole. An operational conception, on the other hand, sees mathematical concepts as processes, algorithms, and/or actions. Instead of seeing the concept as real, we see it as something that has potential, "which comes into existence upon request in a sequence of actions" (p. 4). Sfard emphasizes that there is a significant gap between operational and structural conceptions, even for the same mathematical concept, but that the two conceptions are complementary rather than incompatible. As examples, Sfard describes the concept of a function structurally, as a set of ordered pairs (Bourbaki, 1934), as well as operationally, as a computational process or defined method of getting from one system to another (Skemp, 1971). For rational number, a pair of integers may exist as a structural conception, while the result of dividing two integers would be an operational conception. Many visualizations help treat concepts structurally, while verbal descriptions often yield themselves to operational kinds of thinking.
Sfard recognizes that this kind of dichotomy is not unfamiliar in the research literature. Some have divided math into abstract and algorithmic (Halmos, 1985), declarative and procedural (Anderson, 1976), process and product (Kaput, 1979, Davis, 1975), as well as dialectic and algorithmic (Henrici, 1974). Piaget (1970) distinguished figurative from operative, while others have focused on conceptual and procedural (e.g., Lesh & Landau, 1983; Hiebert, 1985) or instrumental and relational (Skemp, 1976). Sfard does not mean to say these are all synonymous, but that "[the] classification suggested in this article puts us, therefore, in good company" (p. 8). What sets the structural and operational perspective apart, says Sfard, is the "combined ontological-psychological nature and its complementarity" (p. 8), which she likens to the complementary wave/particle view in physics that enables a rich explanation of observed phenomena. In this way, Sfard is "dealing here with duality rather than dichotomy" (p. 9).
Operational and Structural Conceptions in the Formation of Mathematical Concepts
In general, says Sfard, the operational conception comes before the structural conception, and the structural conception is viewed as more abstract and advanced. Exceptions may include figures from geometry, where the static object seems simpler than the algorithm or construction required to create it. Yes historically and developmentally, the operational tends to come first. Piaget (1952) observed that when children first learn to count, they respond to the question, "How many are there?" by recounting, not simply repeating the last number-name the used. Other researchers (Carpenter, Corbitt, Kepner, Lindquist, & Reys, 1980) noticed about half of 13-year-old students could not represent a division problem like 7 divided by 4 as a fraction. It took centuries for mathematicians to treat [math]\displaystyle{ \sqrt{-1} }[/math] as a legitimate object, even though the operation of finding roots was well-understood. In other words, it takes time for a process on an object to become reified into a new object in and of itself. This process of reification played a prominent role in the development of functions, which began as computational processes but eventually came to be used as objects.
Corrolary
- 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. http://doi.org/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}
}