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A Critique of the Selection of "Mathematical Objects” As a Central Metaphor for Advanced Mathematical Thinking Jere Confr

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A Critique of the Selection of A Critique of the Selection of "Mathematical Objects” as a Central Metaphor for Advanced Mathematical Thinking Jere Confrey Shelley Costa Cornell University Introduction Reform in mathematics education at the secondary and post-secondary level is proceeding very slowly and, at least in the United States, the drop-out and failure rates for students continue to be intolerably high. Beginning in grade nine, fifty percent of the students in the United States enrolled each year drop mathematics the following year. Even at major research universities with selective student populations, failure rates in introductory calculus can range from 17-22%.This is astounding considering that students are permitted to withdraw from the courses up to four weeks from their termination, suggesting that even this statistic under estimates the seriousness of the situation. Although reform approaches are widely lauded in elementary and middle school education, at the secondary and post-secondary level, faculties are often unmotivated to change. Two contributing factors to this conservatism are that secondary teachers feel constrained by college entrance requirements and exams and also take their lead from university and college faculty who themselves tend to resist reform. Secondly, although the teachers find research on student learning interesting, they view many reform methods (such as use of small groups, materials, tools and manipulatives, projects, and of multiple forms of assessment) as age- dependent. In their communities, using such materials and approaches at the upper grades constitutes providing students with a crutch, thus diluting the content. Thus there is a real need for a widely respected theory of advanced mathematical thinking that could: 1) account for the breadth of topics in advanced mathematics; 2) address learning problems as they are experienced in the classrooms; and 3) provide a robust vision and guidance towards the kind of reform that is needed. In the last tenures, largely due to the Working Group on Advanced Mathematical Thinking of the International Group for the Psychology of Mathematics Education, a body of work has emerged that sets as its goal the description of advanced mathematical thinking. Central to this work has been a discussion of the relationship between "process" and "object" in learning mathematics. In this paper, we wish to discuss the intellectual gains and costs of selecting mathematical objects as the central metaphor for a theory of advanced mathematical thinking. Although there are differences in the treatments of this term by members of this group, the similarities outweigh them, and in this paper, we are classifying them all as a whole under the label, theories of reification.1 Anna Sfard has published most prolifically on this topic, so we will rely heavily on her thoughtful articulation of the theory, citing additional fundamental work as well. One of the most difficult parts of the task of analyzing and critiquing the theory of reification is that it is changing at a rapid pace. Its proponents have delivered numerous invited lectures??(Sfard & Thompson, 1994; Tall,1995)t ?? written articles ??(Dorfler, 1993; Dubinsky & Harel, 1992; Sfard, 1992; Tall,1991b)t ?? and edited volumes ??(Harel & Dubinsky, 1992; Tall,1991a)t ??. The theory is a candidate for assignment to the Lakatosian category of a progressive research paradigm ??(Lakatos & Musgrave,1970)t ??. Yet to our knowledge, no reviews, no analyses, and little critique has been written. Hence a major contribution of this article is to articulate and clarify the components of the model from our perspective. In this paper, we will attempt to summarize and critique the theory. We will argue that while the theory's presentations have been interesting and its goals admirable, selecting mathematical objects as the critical metaphor in the theory may be retrogressive for the field of mathematics education. Despite the fact that its proponents offer psychological explanations for mathematicians' use of the term "object," (claiming that it is metaphorical and/or action-based), those explanations frequently serve only to recast Platonism. In a sense, the work can be viewed as a modern "mathematician's apology." We will also show how the over-reliance on a single intellectual mechanism, a kind of universal description of abstraction in these theories, fails to legitimate and take into account the richness and diversity of mathematical practice, inside and out of mathematics departments. For instance, this leads to a neglect of the development and use of mathematical tools. In this paper we offer some historical perspectives on the critical role tools have always played in the development of mathematical thought. We further discuss how the use of appropriate technologies and tools with children further their mathematical progress. A reification approach tends to neglect the use of tools making mathematical objects a sufficient basis for learning rather than relying on cognitive technologies. Finally, we strive to show that the theory of reification may serve to reinforce a narrow perspective of the mathematics community rather than engendering reflective or critical discussion. Furthermore, by minimizing the role of human interaction in mathematical practice, theories of reification separate mathematical thinking from its origins in social contexts, and weaken the bridge to the examination of school practices. As a result, its proponents run the risk of de-tracking the educational debate more than contributing to it. It is our position that other formulations with more critical components will prove more progressive and generative for the field. The relevance of this critique to the audiences of this journal lies in the close relationships among the mathematics education community and those developing quantitatively related software. Much of the language of reification has migrated back and forth between these communities but with a great deal of variation in meaning and objective. There is a relatively frequent use of the terminology of mathematical object in software design, such as claims made that the dynamical objects of the computational medium eliminate a gap between concrete and abstract thinking ??(Hoyles, 1993; Hoyles & Noss, 1993; Papert, 1980; Wilensky,1991)t ??. Our hope is that this paper will promote discussion and will lead to clarification of the issues for both communities. The Theory of Reification The theory of reification explains the value and use of the term “mathematical object" in describing mathematical activity and makes the acquisition of mathematical objects central to the pursuit of advanced mathematical thinking. "Mathematical objects," Sfard writes, "are theoretical constructs expected to help in making sense of things we see when observing people engaged in mathematical activity"??(Sfard & Thompson, 1994, p. 14)t??. They are, "a kind of link (a glue if you wish) we add to the observables in order to make the latter hang together as a coherent structure" (ibid., p. 14). Thinking with objects, in her terminology, is "structural thinking” which she contrasts with "operational thinking," writing, "The advantage of the [structural] type of schema over the [operational] is that it is more integrative, more economical, and manipulable" (p. 53). As an example of this, she argues that embracing the notion of function as abstract object allows her to see her students' faulty behavior as different symptoms of basically the same malady: students' inability to think in structural terms, or, as she puts it, "learners' 'blindness' to the abstract objects called functions" (ibid., p. 15). We will show through further discussion that the idea of structure in reification is valuable, because it can emphasize the importance of relational understanding which then could include the use of context and community. However, we fear the primacy of "objects" language as evidence of structure tends to signal a priori discovery, which then encourages a dismissal of alternative, challenging student-generated approaches. Before delving more deeply into the components of the theories, let us reflect first on what kind of theories these are. Sfard makes it clear that for her, "mathematical object" is a theoretical construct. She carefully advises, ". .the notion of a 'mathematical object' can only function as a theoretical construct, and it should only be used as such if a good theory may be built around it"(ibid., p. 9). She argues that the tentative status of such ideas is widely established, noting: To a great extent, it is the growing abandonment of the Objectivist epistemology that made us more daring than ever in our theorizing about the human mind and about its functioning. Indeed, we came a long way since the times when 'mind' itself sounded somewhat dirty. Nowadays, people are no longer concerned with the objectivity of knowledge--with the question how well a given scientific theory reflects the 'real' state of affairs. There is no belief anymore in the 'God's eye view' of reality. The concern about truthfulness of our representation of the pre-given world has been replaced with the pragmatic questions of usefulness (Lyotard, 1992) and of 'intersubjective agreement' (Rorty, 1991). (ibid., p. 10). Such a declaration belies the current perspective of many, if not most practicing mathematicians and scientists and ignores the fact that many of the authors decrying objectivism make allowances and exceptions that permit mathematics to maintain
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