Sfard (1991)

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Sfard (1991)

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

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. Yet 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 they 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]\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.

Sfard describes three stages in concept development: interiorization, condensation, and reification. During interiorization, learners become familiar with processes and operations on lower-level mathematical objects, and eventually gain skills with these operations. As examples, Sfard describes:

In the case of negative number, it is the stage when a person becomes skillful in performing subtractions. In the case of complex number, it is when the learner acquires high proficiency in using square roots. In the case of function, it is when the idea of variable is learned and the ability of using a formula to find values of the "dependent" variable is acquired. (pp. 18-19)

Condensation is the stage where learners "squeeze" sequences of operations and are more able to think of the process as a whole rather than a series of steps. At this stage students are able to combine processes, make comparisons, and generalize. Sfard continues her examples:

In the case of the negative numbers, condensation may be assessed through student's proficiency in combining the underlying processes with other computational operations; or, in other words, in his or her ability to perform such arithmetic manipulations as adding or multiplying negative and positive numbers. In the case of complex numbers, condensation is what helps the learner to realize that reversing the operation of squaring may be useful as part of lengthy calculations even if it would not, by itself, yield a legitimate mathematical object. The student may still treat such symbol as [math]5 + 2i[/math] as nothing but shorthand for a certain procedure, but at this stage it would not prevent him from skillfully using it as part of a complex algorithm. When function is considered, the more capable the person becomes of playing with mapping as a whole, without actually looking into its specific values, the more advanced in the process of condensation he or she should be regarded. Eventually, the learner can investigate functions, draw their graphs, combine couples of functions (e.g. by composition), even to find the inverse of a given function. (p. 19)

Condensation gives way to reification when a concept is seen as a fully fledged object. This is a leap in understanding, not a gradual progression, as the new object separates itself from the process that created it, and becomes a member of a category. Now the new object can be used as an input into new processes. Sfard's examples:

In the case of negative numbers, it is the learner's ability to treat them as a subset of the ring of integers (without necessarily being aware of the formal definition of ring) which can be viewed as a sign of reification. Complex number may be regarded as reified when the symbol [math]5 + 2i[/math] is interpreted as a name of a legitimate object — as an element in a certain well-defined set — and not only (or even not at all) as a prescription for certain manipulations. In the case of function, reification may be evidenced by proficiency in solving equations in which "unknowns" are functions (differential and functional equations, equations with parameters), by ability to talk about general properties of different processes performed on functions (such as composition or inversion), and by ultimate recognition that computability is not a necessary characteristic of the sets of ordered pairs which are to be regarded as functions. (p. 20)

Operational and Structural Conceptions in Cognitive Processes

Sfard suggests that mathematical conceptions only become fully formed when they are conceived both operationally and structurally, but considers what is lacking if only one or the other is realized. At some level, a solely operational approach seems necessary and perhaps sufficient, and many seem to take a view of mathematics as a collection of processes. Historically this has often appeared to be the case, but structural perspectives on mathematics in the 20th century reveal the need for structural conceptions and an understanding of the objects on which processes operate. Operational conceptions are required, but structural conceptions are equally necessary to ease the burden of endless process and organize it into more useful structures. Sfard paraphrases Henrici (1974) by saying, "the structural approach invites contemplation; the operational approach invites action; the structural approach generates insight; the operational approach generates result" (p. 28). Furthermore, as more processes are reified into objects, our cognitive capacity to do mathematics expands.

Sfard now faces the challenge of answering her initial question: Why is mathematical abstraction — now understood as a process of reification — so difficult? Why must it be such a significant leap in our understanding? Sfard explains with her three-stage model, where "reification of a process occurs simultaneously with the interiorization of higher-level processes" (pp. 30-31). In the punchline of the article, Sfard describes the reification of functions:

[In] order to see a function as an object, one must try to manipulate it as a whole: there is no reason to turn process into object unless we have some higher-level processes performed on this simpler process. But here is a vicious circle: on one hand, without an attempt at the higher-level interiorization, the reification will not occur; on the other hand, existence of objects on which the higher-level processes are performed seems indispensable for the interiorization — without such objects the processes must appear quite meaningless. In other words: the lower-level reification and the higher-level interiorization are prerequisite for each other! (p. 31)

Sfard closes her article contemplating the implications of her theories:

The thesis of the "vicious circle" implies that one ability cannot be fully developed without the other: on one hand, a person must be quite skillful at performing algorithms in order to attain a good idea of the "objects" involved in these algorithms; on the other hand, to gain full technical mastery, one must already have these objects, since without them the processes would seem meaningless and thus difficult to perform and remember. (p. 32)

The co-dependence of operational and structural conceptions, and the difficulty of reification, may explain why many people struggle so mightily to learn mathematics, even at the highest levels. Students may become discouraged when reification is slow to happen, and the challenge to mathematics educators is to find ways to "unravel the harmful tangle and to stimulate reification" (p. 34).

References

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://doi.org/10.1007/BF00302715},
volume = {22},
year = {1991}
}