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 its status as truth. By writing this paper, we are accepting Sfard's invitation to examine whether a useful theory of education can be built with this concept of reification as the central component. We are not denying the integrity of a construct of mathematical object; rather, we question the wisdom of making it the centerpiece of a theory to guide the teaching and learning of mathematics. We hope to demonstrate that the construct carries too much unintended baggage and marginalizes too many robust alternative approaches to make it more than an incremental approach to change in school or university practices.

In our discussion, we begin by proposing a set of components of the theory and critiquing the positions of a variety of theorists on these components. Later, focusing more specifically on Sfard's version of the theory and we distinguish her: 1) meta-theoretical stance; 2) theory of reification; and 3) reports of mathematicians' theorizing about mathematical objects, we offer an institutional, philosophical, and methodological critique. We then compare this approach briefly with the work of some French theorists, and finally offer the outlines of our own alternative model.

Components of the Theory

What unites the theorists described herein is their desire to explain advanced mathematical thinking. They all believe that a formalist account, from axiom to definition to theory to proof is an insufficient basis for either the development of mathematical ideas or their instruction. Recognizing that reform has been more extensive at the early grades, they seek to extend these analyses to include secondary and post-secondary content and to improve the chances of success of a greater proportion of the student body. Often these researchers' goals include acting as quality control agents to ensure that, as reform efforts extend up the grade levels, they do not diminish the preparation of adolescents to undertake advanced study. All of these are worthy goals.

The central approach of these theorists seems to be to elucidate what distinguishes higher level mathematics from more elementary mathematics and, in doing so, to find a satisfactory framework for talking about advanced mathematics. Their shared feature is a focus on a transformation of thought processes into a cognitive structure which then is referred to as an object. However, this transformation is cast as requiring only "action" and” operation," and seldom do we see any discussion of the purposes of the constructor, the determination of satisfaction of a goal, or the use of context of the construction beyond reference to other mathematical objects. According to the proponents, progress in mathematics depends on the successful acquisition of objects. Moreover, to our reading, there' is an acceptance that the primary motivation to learn mathematics is to acquire these objects, thus satisfying an aesthetic desire or permitting continued successful course-taking. We will argue that it is precisely this point of view that will continue to allow the disenfranchisement of the majority of the population from mathematics.

In what follows, we will elaborate on the individual components and then discuss and critique the assumptions inherent in these components:

1. Mathematics is strictly hierarchically ordered. We are using the term "strictly" here to refer to the fact that in their historical and classroom-based analysis, reification theorists rely over and over again on the typical sequence of content development and assert its necessary ordering. This assumption is explicit in nearly all treatments and discussions, such as in Sfard: "After all, mathematics is a hierarchical structure in which some strata cannot be built before another has been completed"??(Sfard & Linchevski, 1994, p. 195)t??. Sfard and Linchevski refer to this emphasis on hierarchy as a “logical analysis." In a later section, we offer an example of a classroom practice that demonstrates flexibility in sequencing and runs counter to all previous “logical analyses" of the multiplicative conceptual field, and yet was very successful with children.

2. The path to learning advanced mathematics is filled with cognitive obstacles. Drawing on the French tradition of examining the curriculum for epistemological obstacles ??(Bachelard, 1938)t ??, the reification theorists look for obstacles to progress in mathematics. As stated by Tall ??(1991c)t ??, "It is clear that the formal presentation of material to students in university mathematics courses--including mathematics majors, but even more for those who take mathematics as a service subject--involves conceptual obstacles that make the pathway very difficult for them to travel successfully" (ibid., p. 251). Later, he specifies that these difficulties are "due to the epistemological nature of mathematics and in part to misconceptions by mathematicians of how students learn" (ibid., p. 252).

While we support the view that analyzing content for epistemological obstacles is a useful didactical strategy, treating obstacles as” compulsory passages on the road to knowledge"=ult??(Artigue, 1992)t ?? presents some difficulties for us when this treatment signals the overcoming of an obstacle by eradication or logical necessity. Epistemological obstacles present important occasions for reconceptualization, but that reconceptualization: a) can be on the part of teacher, researcher, or student, and b) does not require eradication but a conscious decision to abandon or modify one approach in preference to another. In reification theories there is a strong tendency for these obstacles instead to be seen as schemes of thought to "act against" as opposed to being recognized as valuable indicators of fore-conceptions ??(Sierpinska, 1992, p. 28)t??. In science education a similar controversy has arisen as to whether to speak of misconceptions or alternative conceptions, or just simple concepts result??(Confrey, 1990)t ??.

3. There is a sequence from process- to object-based thinking that is the key to ascending the hierarchy and traversing the obstacles. These theorists tend to assume that failure to complete these sequences will drastically curtail one's mathematical pursuit at advanced levels. Sfard describes this by stating that "mathematics is a hierarchy in which what appears to be a process at one level must be transformed into a full-fledged abstract object at a higher level to become a building block of more advanced mathematical constructs"??(Sfard, 1992, p.64)t ??. In Tall's work in this arena, discussions of perception and conceptions resulted in his introduction of the term, procept, which stands "for a process which is symbolized by the same symbols as the product" ??(Tall, 1991c, p.254)t ??.

Sfard (1994) discusses her use of the term "object,” writing: "the metaphor of an ontological object, even though ostensibly only an option in mathematical thinking, is in fact indispensable for the kind of [mathematical] understanding people are prepared to call 'deep' or 'true'" (p. 54). Sfard and Linchevski ??(1994)t ?? explain their use of the distinction as follows:

The distinction between the two models of thinking, operational and structural, is delicate and not always easy to make. The ability to perceive mathematics in this dual way makes the universe of abstract ideas into the image of the material world: like in real life, the actions performed here have their 'raw materials' and their products in the form of entities that are treated as genuine, permanent objects. Unlike in real life, however, a closer look at these entities will reveal that they cannot be separated from the processes themselves as self-sustained beings. Such abstract objects like ?-1, -2, or the function 3(x+5)+ 1 are the result of a different way of looking on the procedures of extracting the square root from -1, of subtracting 2 and of mapping the real numbers onto themselves through a linear transformation, respectively. Thus, mathematical objects are an outcome of reification-- of our mind’s eye ability to envision the result of processes as permanent entities in their own right (p. 193-194). 2

In reification, Sfard and others choose the metaphor of object because it expresses a sense of permanence??(Sfard, 1994, p.53)t ??, of determinism and imposition (ibid., p. 52) and of physicality. Over and over, an image of a mental space filled with objects is communicated, yet the relationship between the knower and the known is exclusively about the knower's ability to move the objects successfully into the mental space, neglecting other rich types of relationships. These would include issues of purpose, of satisfaction (or dissatisfaction) of needs, and of negotiation of meaning. These issues, when they are discussed, are defined as being with the concerns of pure mathematics.

In the reification work there has been some acknowledgment of the constructivist alternative-- Sfard, for example,??(Sfard & Thompson, 1994, p. 23)t?? has recognized the value of "concreteness" as "not a property of an object but rather a property of a person's relationship with the object" (cited in reference to Wilensky, 1991, p. 198). While promising, these relationship issues are not yet fully developed within reification theory. Our concern is that the object language can deter such person-object discussions, especially within communities of mathematicians where the dominance of Platonism, even "practical Platonism" ??(Sfard, 1994, p.51)t ?? has made such conversations difficult and unacknowledged.

4. After reification, things get easier. As noted in Sfard and Linchevski ??(1994)t ??,

the transition from operational to structural algebra, although a significant step forward in the degree of abstraction and generality, results in facilitating the performance rather than in adding complexity. Although reification is difficult to achieve, once it happens, its benefits become immediately obvious. The decrease in difficulty and the increase in manipulability is immense." (p. 198)

Furthermore, although at times the operational-structural (process/object)discussion is treated in a dialectical and complementary relation (ibid., p. 193), at other times, the expert is portrayed as "ontologically blind” to the students' difficulties in reifying processes, suggesting that this is a one-way passage (ibid., p.225). With the exception of Dubinsky's =t??(1991)t ?? discussion of reversibility, a central tenet in Piaget, the implication is left open that flexibility and reversibility in returning to process may not be required for successful mathematical performance but only for instructional communication.

5. Mathematical development matures, through reification, to mathematical virtuosity including the formal construction of theorem and proof. The implications of this overall agenda for mathematics instruction include a number of assumptions: a) histories of geometry and number can be intellectually isolated from each other ??(Tall, 1995, p.62)t ??; b) algebraic symbolism captures function concepts most effectively??(Sfard & Linchevski, 1994, p.198)t??; and c) set- theoretical approaches, while still imperfect, represent our best thinking ??(Tall, 1995, p.62)t ??. In all of these cases, the "rational reconstruction" of history is done with thick modern lenses which obscure the rich controversies that did not always yield a clear winner ??(Shapin & Schaffer, 1985, p. 11)t ??. We comment further on these assumptions in a later section on the use of history.

6. Metaphor, action and operation are the psychological vehicles for the construction of mathematics, yet at the highest levels experiential knowledge and context are left behind. Sfard writes, "In advanced mathematics, at levels far removed from physical reality, it may well be that the immediate source of a basic metaphor is another, lower-level mathematical structure"??(Sfard, 1994, p.47)t ??.

Explaining further the demands of abstraction, Sfard writes: The frequent problem with new abstract ideas is that they have no counterpart in the physical world or, worse than that, that they may openly contradict our experiential knowledge. Obviously in the latter case no metaphor is available to support these abstractions. . . In fact, the very idea of reification contradicts our bodily experience; we are talking here about the creation of something out of nothing. Or about treating a process as its own product. There is nothing like that in the world of tangible entities where an object is an 'added value’ of an action, where processes and objects are separate, ontologically different entities which cannot be substituted one for the other. Our whole nature rebels against the ostensibly parallel idea of, say, regarding a recipe for a cake as the cake itself (ibid., p. 53). [Our emphasis added.]

Contrary to Sfard and Linchevski's claim of the uniqueness of this activity to mathematics, the movement from process to outcome is not uncommon in conversational English; for example, one often refers to a "haircut" as both the process and the product of the activity. The strength of structures does not necessarily depend on "objectification" but can rely on generality of application across settings. Thus, abstraction need not insist on the stripping away of context. It entails, rather, a recognition in apparently unlike things of a likeness, a commonality, which does not imply detachment but interconnections, and not just hierarchical ones.

In his work, Dubinsky relies on Piaget's construct of "reflective abstraction" as "present at the very earliest stages in the coordination of sensory-motor structures... and ... continues on up through higher mathematics ..." ??(Dubinsky, 1991, p. 95)t??. Dubinsky's use of process and object is also drawn from Piaget who said, The whole of mathematics may therefore be thought of in terms of the construction of structures. . . mathematical entities move from one level to another; an operation on such 'entities' becomes in its turn an object of the theory, and this process is repeated until we reach structures that are alternately structuring or being structured by 'stronger' structures (Piaget, 1972 p. 70),??(Tall, 1995)t ??.

In Dubinsky =t??(1991)t ??, as elsewhere in constructivism ??(Confrey, 1994b; Thompson,1994)t ??, "scheme" is used as a constructive building block, a means to describe when a person encounters a problematic situation established by purpose or goal, s/he acts and reflects on those acts, determining his or her success in resolving it. In this sense, the action sequence becomes a stable resource, a scheme. According to Dubinsky, a scheme is "a more or less coherent collection of objects and processes" ??(Dubinsky, 1991, p. 102)t??. He uses a variety of processes from Piaget--interiorization, coordination, encapsulation, generalization, and reversibility--in his theory. He talks very little about the situation, context or perturbation that spawns the construction other than to say that "a subject's tendency to invoke a schema in order to understand, deal with, organize, or make sense out of a perceived problem situation is her knowledge of an individual concept of mathematics" (ibid., p. 102). However, even this quote places the problem situation outside the schema which "begins with objects" to which "actions can be applied"(ibid., p. 104). Thus, although theories of reification personalize the construction of objects by recognizing the role of metaphor and action as they surface in process and are transformed into objects, the role of the personal and socio-cultural remains subdued and external to the course of intellectual development.

7. Fuller participation is obtained through a novice-expert approach. Educationally, many reification theorists have selected a novice-expert approach to student learning in which they study those successful in mathematics (including "able" students and mathematicians), create descriptive constructs to explain the retro-and introspective accounts of these experts, and then advocate increasing successful participation by inducing at-risk students to adopt these constructs. Tall ??(1991c)t ?? writes of "lower ability children" that instead of having a knowledge generator, they have an unencapsulated process which produces answers which are not manipulability objects. Thus, there grows a proceptual divide between the more able, using proceptual flexibility, and the less able, locked in process (p. 255).

And Sfard ??(1994)t ?? writes: The study of mathematicians' ways of thinking brings unimportant and probably quite universal message about the nature and conditions of understanding. . . the literature abounds in findings and arguments which support the claim that the natural tendency for structural thinking is typical not only of mathematicians but also of more able students. Thus, the immediate implication is that, as teachers, we should foster structural thinking and help 'novices' construct their structural metaphors (p. 54).

There is another trend in the literature, however, which questions the potential for novice-expert approaches to inform education. These critiques have concentrated on two weaknesses. First, novice-expert approaches typically rely on interview studies that tend to neglect the interactions among the community of practitioners as a significant force in promoting and defining expertise. Secondly, such approaches fail to examine the transformations and kinds of experiences and opportunities that permitted the expert to become expert.

Figure 1: A summary of components associated with reification theories

In this section, we have identified a set of components that represent seven common themes that occur across many of the papers on the theory of reification which we have arranged schematically in Figure 1. Of course, variation across theorists means that some are held more dear by one person than another. The viability of the theory depends on the acceptance and elaboration of these components. One could protest that these assumptions are prevalent in many treatments of mathematics education not just reification, and that, in reification, they are not deductively linked but only surround the hardcore of the theory. We would claim rather that they have a strong symbiotic relationship, serving each other and holding serious reform in check. While each component alone may appear insignificant, the aggregate of such thin strands can configure itself into a nearly impenetrable cage, an analogy created by feminist philosopher, Marilyn Frye??(1983)t?? to describe how oppression encloses the oppressed.

Introduction of Critique

To begin with, let's consider the positive contributions of this work, of which there are many. Most notably, here are a set of authors who are risking their professional stature as mathematicians to wrestle mightily with educational issues. Not only do they take on these topics, but they are taking positions that may raise the eyebrows of many of their colleagues. They are making the claim that the failure of students to learn mathematics rests both on an inadequate analysis of advanced mathematical thinking and on immaturity in mathematicians' ability to de-center and to envision students' worlds.

Secondly, they are giving psychological, albeit essentially cognitive, interpretations to mathematical knowing. As a result, mathematical ability shifts from its portrayal as a genetically predetermined capacity to a learnable set of skills and ideas. Their clear commitment to improving the successful performance of students is in marked contrast to the views of some who tolerate and often even expect high failure rates as a necessary outcome of learning their demanding disciplines.

Thirdly, in all of this work there is a clear and unambiguous role for actions, operations and reflection in the processes. Most explicitly articulated in Dubinsky, a discussion of scheme based on the work of Piaget emphasizes the origins of operations in action. In Sfard's most recent discussion (1994), she draws on the work of the linguists Lakoff and Johnson, citing them as suggesting that metaphor is "a mental construction which plays a constitutive role, in structuring our experience and in shaping our imagination and reasoning" (p. 46). More explicitly, "Lakoff and Johnson's central claim is that abstract ideas inherit the structure of physical, bodily, perceptual, experience" (p. 47). This allows Sfard to interpret mathematicians' introspective accounts of their own thinking as drawing on physical actions to form mental constructs that are transformed metaphorically into ontological objects.

We would claim that these ideas have philosophical precedents--in externalization of the problem, the use of heuristics in mathematical discovery and, as previously mentioned, in the building of abstraction in constructivism--which support these positive aspects of the theories while concurrently leading more directly to the inclusion of socio-cultural relations. In Michael Polyani's Personal Knowledge ??(1958)t ??, he wrote of the way in which an unknown solution is treated as pre-existent, an act similar to the externalization of mathematical idea: When I am looking for my fountain pen, I know what I expect to find; I can name it and describe it... Though the solution of a [mathematical] problem is something we have never met before, yet in the heuristic process it plays a part similar to the mislaid fountain pen or the forgotten name which we know quite well. We are looking for it as if it were there, pre-existent (p. 125-6).

In addition, Lakatos's quasi-empiricism was explicitly developed to acknowledge the process of discovery, conjecture and refutation that contrasted with Euclidean formalism. This served to provide a psychological/social basis for the construction of mathematics that is less individualistic than what is promoted by reificationists. Lakatos used the idea of "heuristics in a quasi-empirical system" as a means of enriching the two processes of induction and deduction, as he writes "The basic rule of [quasi-empirical systems] is to search for bold, imaginative hypotheses with high explanatory and’ heuristic' power. Indeed it advocates a proliferation of alternative hypotheses to be weeded out by severe criticism" ??(Lakatos,1984, p. 34)t ??. In sum, Polyani and Lakatos have used problem-posing and problem-solving to describe mathematical activity. Does exchanging these for mathematical objects gain one any substantial advantage?

Polyani's and Lakatos's discussions, although they are psychological, lend themselves more readily to social interpretation, partly because they devote attention to human purpose and context. In her own work, Sfard clearly acknowledges the importance of human issues such as “utility" and "intersubjective agreement," and she has rejected objectivism in part due to its failure to acknowledge the role of psychology, controversy, social interaction and culture in the construction of theory, data, method and interpretation. Yet, paradoxically, these goals are undermined by her focus on objects as expressions of mathematical structure.

Mathematical objects hinder a social agenda of this sort primarily because the model, as we have elaborated, is overly individualistic. There is, however, an additional reason for limiting object- based metaphors for mathematical thinking. Object-based language may accurately describe mathematicians' sense of utility and their intersubjective experience within their limited professional communities; however, the dilemma is that it can simultaneously support a denial of their exclusive positioning in relation to the fuller culture. The question becomes: should one choose as a basis for reform a metaphor of description that permits this dilemma to endure? par

There are four possible reasons one might decide to such a dilemma intact:1) one wishes only to offer an explanation of an existing state of affairs(in which case reform is not an objective); 2) one advocates change but has deep faith in the reflectiveness of the group and in its ability to "see the light" without needing to change terminology; 3) one sees value in tolerating potential ambiguity and miscommunication which permits two contradictory agendas to coexist--often a form of survival or resistance; or 4) one sees no alternative way to discuss mathematics other than to adopt the language of mathematicians.

A Reexamination of the Adequacy of the Reification Approach

In this section, we discuss three questions: a) does Sfard's account keep a consistent distinction between her position as a non-Objectivist theorist and the frequently Platonist views expressed by mathematicians; b) can anon-Objectivist metatheoretical position be fruitfully juxtaposed with mathematicians' accounts in organizational, philosophical and methodological structure; and c) what are the choices facing mathematics educators and designers as regards the use of the metaphor of mathematical objects? The reasons for selecting these three questions is that the first allows us to clarify the position taken by Sfard, which arises in the writings of many other reificationists; the second examines the viability of the overall approach, particularly in the context of reform; and the third presents our view as an alternative to reification and invites the reader to decide whether object-based thinking promotes the goals of reform. a) Sfard's non-Objectivist Platonism. Sfard elaborates her metatheory in denying an Objectivist perspective and endorsing a non-Objectivist point of view. Although she does not make explicit her views of the relationship between a non-Objectivist point of view and a constructivist one, she describes her interpretation of the non-Objectivist position as: While Objectivism views understanding as somehow secondary to, and dependent on, predetermined meanings, non-Objectivism implies that it is our understanding which fills signs and notions with their particular meaning. While Objectivists regard meaning as a matter of relationship between symbols and a real world and thus as quite independent of the human mind, the non-Objectivist approach suggests that there is no meaning beyond that particular sense which is conferred on the symbols through our understanding ??(Sfard, 1994)t ??.

This metatheory commits Sfard to a theory of reification in which she denies mathematical objects' deriving their meaning from a reality that is independent of the human mind, while at the same time permitting her to argue for objects utility to and intersubjective meaning among mathematicians. That is, if she can show that "the particular sense conferred on the symbols through [their] understanding" is that of "mathematical object," she can attempt to articulate an ontology of mathematical objects as expressed by mathematicians, while asserting a non- Objectivist view for herself.

We believe that Sfard falters in her consistency of viewpoint by alternately speaking of Platonism or idealists' accounts as (i) in themselves valid and as (ii) only as metaphors for their users. For example, denying that reification bears any resemblance to Platonism, Sfard writes, "an abstract object is nothing 'real,' nothing that would exist even if we did not talk about it"??(Sfard & Thompson, 1994, p.14)t??. Yet, she ??(Sfard, 1994)t ?? seems unprepared to make this move, writing statements such as, "we spontaneously behave and feel like Platonists. . . Our mind will immediately go back to its 'natural' state--the state of a Platonic belief in the independent existence of mathematical objects, the nature and properties of which are not a matter of human decision" (p. 51). Although she holds that this state serves to construct a metaphor and is in that sense temporary, by claiming that it is natural she is left in a contradictory position: she produces anon-objectivist Platonism.

b) The Viability of the Reification Model for Reform. Even when one disregards these inconsistencies in perspective, a question of the viability of the conceptual framework needs to be evaluated. We propose to do so from three related perspectives: institutional, philosophical and methodological.

Institutional. Unlike an ethnographer who can work in unfamiliar settings or cultures, gain a form of entry, participate to achieve an understanding of this "other" culture and depart, mathematics educators, especially those working on learning in advanced mathematics, typically reside in mathematical communities. Educators concerned with advanced mathematics depend on these mathematics communities to implement their perspectives and often to confer them with professional status and opportunity. It is almost always an asymmetric relationship whose asymmetry tends to decrease as the mathematics educator's theories coincide more closely with those of the dominant culture of mathematicians. This institutional structure typically leads to a situation where communities of mathematicians either trivialize the distinctiveness of an approach, assimilate it to their own agendas, or dismiss it outright. This situation reaffirms the privileged status of mathematicians over mathematics educators in determining directions for reform.

Philosophical. Let us ignore for a moment these institutional factors and assert the freedom of theorists to hold any philosophical position they choose. Even then, the reificationists' conceptual framework proves epistemologically challenging. We model Sfard's version of this framework in Figure 2 but we believe that the issues we will raise apply more broadly.

Figure 2: Sfard's Conceptual Framework and its Relationship to Educational Reform

This model is philosophically consistent but it suffers from being overly self-referential. It is consistent because, through the lens of Lakoff and Johnson's metaphorical analysis, any belief mathematicians have about their thinking can be incorporated. Thus the question is, from what sources can refutation or validation of the theory arise? If no outside validation can be drawn on, the theory becomes purely interpretative, and it is not clear what educational implications can be drawn. For instance, does the theory require one to encourage students to explicitly recognize the role of metaphor in their use of objects? Since many successful mathematicians express their own lack of such awareness, preferring instead Platonic viewpoints, it seems clear that either awareness or unawareness is tenable under this model.

Because in her model, educational practice derives from mathematical practice, the only obligatory educational implication is that teachers should be aware of the object-based thinking mechanism in their teaching of math. Yet Sfard herself documents the limited efficacy of this approach: "Studies have shown that even the most sincere efforts to bring about the appropriate metaphor will often be rewarded with only limited success" ??(Sfard, 1994, p.54)t ??. Perhaps its key problem is that its central goal is to reproduce expert mathematicians, a result that may support only incremental and modest reform. An alternative position would be to see the arrow in the model to education as bidirectional and thus be able to argue that changes in mathematicians' practices can flow from an examination of educational practices ??(Tymoczko, 1979)t ??.

Methodological. Methodologically, these philosophical tangles are exacerbated. First, Sfard's informants are typically those mathematicians who succeeded in the current institutional and intellectual arrangements, which can be a serious obstacle to engaging in vigorous reform. Sfard does recognize the thorniness of her methodology, stating that "I knew that what I would find in my subjects would not necessarily be generally true" (p. 45). Further, since her goal is the examination of “inner sensations" rather than "visible behaviors", she is caught in a second methodological bind (p. 45): descriptions of one's own inner sensations are chosen from culturally embedded descriptors. To help alleviate the first, however, she could have chosen also to interview other mathematically competent users, such as engineers, physicists, chemists, architects or others outside the academy. For the second, mathematical activity and discourse could have been pursued in public and communal settings, where visible interactions can shed light on how a community's values influence the self-descriptions of an individual member. c.) The Need for a Decision. A fair question is, can one avoid self-referentiality? The French tradition of didactical engineering offers one alternative. In the French work, we see similar constructs to those in reification, but there are critical differences. Although we cannot do this work justice in such a brief space, we wish to highlight its alternative assumptions. The French tradition combines attention to epistemological obstacles ??(Bachelard, 1938; Brousseau, 1983; Cornu,1991)t ?? with analysis of teaching and learning in institutional settings. Asstated by Artigue, "the epistemological work in question here is only a tool of the didactic work itself: although it is a useful guide, it cannot be a substitute." They explicitly develop views of didactical transposition to connect the two??(Chevallard,1991)t ??. By separating these elements, these theorists have constructed a framework that, while linking these two essential elements, maintaining them as independent sources of their theories. In doing so, they lessen the problem of self-referentiality.

The work of Douady ??(1991)t ??, another French theorist, offers a different view from that of the reification theorists. She explicitly and consistently argues for a dialectic of tool and object, thus building in reversibility at all levels. Her descriptions of history often acknowledge the values of "changing of settings" (pp. 115-117) which allow mathematical concept development to thrive due to placement in multiple forms of representation. Further, attention to the role of "socio-cultural dimensions" (p. 117) recognizes the possibility of different portrayals of mathematical content.

On the other hand, Douady, like the reificationists, restricts her consideration of context to tools only. She explicitly defines objects as “considered in a cultural dimension as a piece of knowledge independent of any context, of any person which has a place in the body of the socially recognized scientific knowledge. An object is "mathematically defined" (p. 116). In contrast, "tools are contextualized and personalized" (p. 124). The cadence of instruction is then described as a sequence from a) implicit use of old tools to b) explicit use of old tools to c) the construction of objects and d) a final stage where objects are converted to tools to be used again in the construction process. Although Douady's language involves tools, context, and culture, the stripping away of these elements is still her objective. As a result, although the conceptual framework of the French school is less self-referential, its choices of mathematical and educational perspectives are unlikely to support as extensive a reform agenda as we see necessary to alter participation patterns and failure rates in the United States.

In the next two sections we wish to provide fertile territory for rethinking the choice of theoretical orientation for a discussion of advanced mathematical thinking in a reform setting. We discuss views of the history of mathematics and of classroom interactions that challenge many of the assumptions listed above and point toward the characteristics one might expect to find in an alternative theory to reification. The outlines of such a theory are offered in Confrey??(1994a; 1995;1995)t ?? and discussed briefly in the concluding section.

Reification's Use of History and Suggested Alternative Approaches

Theorists of reification (e.g., Sfard & Linchevski, 1994, and Tall, 1995) have used examples from the history of mathematics in various ways to support their theoretical positions. However, in corresponding artificially closely to certain components of their theory(such as mathematics as a hierarchy), their choices in this regard have painted a non-representative picture of history. Tall, for example (1995) gives a historical discussion to illustrate the large-scale development of a "versatile approach" to mathematical conceptualization which incorporates both of the modes of thinking ("visuo-spatial" and "manipulative symbolic") defined by him as separate but complementary: In the advanced stages of such a development, certain subtle difficulties occur which mean that advanced mathematical thinking must expunge itself of possible hidden assumptions that occur when visual ideas are verbalized. In the nineteenth century a number of flaws became apparent in Euclidean geometry and theoretical developments in algebra (such as non-commutative quaternions) were over-stretching simple beliefs in the manipulation of symbols. Research mathematics took a new direction using set-theoretic definition and logical deduction. Theorems inspired by geometric perception and symbolic manipulation were reformulated to give a new axiomatic approach to mathematics that led on to greater generality(p. 62).

This passage implies that the history of mathematics is a metaphor for straightforward, purifying progress. While Tall softens this slightly by adding that "the existence of a systematic body of formal mathematical knowledge is not the final quest in mathematics, although it does offer a vital foundation upon which even more sophisticated ideas can be built" (ibid., p. 63), this additional comment retains a strong sense of mathematics as progressing hierarchically. Moreover, his dismissal of quaternions as an inferior product of the dissonance between two modes of thought ignores the productive influence the concept had on the development of nineteenth-century mathematical physics. As is the case with many core assumptions of the theory of reification, Tall's history disregards important scientific context in its unidirectional march "forward."

Similarly, in focusing on the concept of function as arising exclusively from symbolic algebra, Sfard & Linchevski ??(1994)t ?? omit important historical perspectives. "Function's turbulent biography," Sfard has written elsewhere, "can be viewed as a three-centuries long struggle for reification" ??(Sfard, 1992,p. 62))t ?? --a notion corroborated by Sfard and Linchevski in their more recent treatment. They begin(p. 196) by describing the early "rhetoric algebra" as conceptualized in bulky, procedural terms prior to the sixteenth century. They herald the ease of manipulation that accompanied the introduction of symbolism: "algebraic symbolism is unrivaled in its power to squeeze the operationally conceived ideas into compact chunks and thus in its potential to make the information easier to comprehend and manipulate" (ibid., p 198). From this, they tell us, a formal algebraic calculus was created, after which “the new invention was transferred (mainly by Descartes and Fermat) to geometry to serve as an alternative to the standard graphic representations and then applied in science (by Galileo, Newton, and Leibniz, among others) to represent natural phenomena"(ibid., p. 200).

The thrust of Sfard and Linchevski's historical argument is to use the concept of function to depict a global process of reification. Because of a need for logical foundations of algebra, they claim, mathematicians abandon[ed] the idea of defining the variable as such and by offering instead an interpretation for algebraic formula in its totality. Function, a new kind of abstract mathematical object, was created to serve as a referent for such expressions as 3(x+ 5) +1 or x2 + 2y + 5"(ibid., p. 200).

Overall, Sfard's view of history is that "structural conceptions usually develop out of operational, or in other words, abstract objects emerge from certain computational processes"??(Sfard, 1994, p.61)t ??.

Notice that in this version of history, all developments have collaborated to produce modern notation and a formal approach, which, while only a temporary vantage point on the way to greater things, is cast as the pinnacle of past work. Confrey ??(1980)t ??, referred to such an approach as "progressive absolutism" and argued that it diminishes the recognition that mathematical approaches may be abandoned or not pursued but still have tremendous potential contributions to our understanding. The course of mathematics is for more like a meandering stream than a direct path. An alternative interpretation of history does not attempt to compress it into the repeated application of a single process, but points to controversy and diversity in values and interests. More like the use of history suggested by Jahnke ??(1994)t ?? or Dennis ??(1995)t ?? and Costa (1995)t ??, this history is used to provoke and value multiple historical trends or critique modern approaches.

An alternative historical approach would embed the development of function in its original, problem-centered contexts: scientific investigations of motion; analyses of images resulting from curve-drawing devices; use of tables to aid in computational accuracy for navigation or for economics. We see Oresme in the 14th century and Galileo in the sixteenth seeking to transform static geometry diagrams into expressions of distance, time and velocity. A need to move from seeing these as varying magnitudes that relate both length and area to numerical expressions made the later inventions of Viete??(1591)t ?? promising candidates for mediating symbolism. These sixteenth-century events are, summarized in Sfard and Linchevski as the application of algebra to science. Our description attempts to reverse their order, as theirs is one that permits the interpretation that mathematics has its own trajectory and these other activities are relegated to application.

In Sfard, Descartes transferred algebraic notation to geometry. Descartes, however, contrary to the assumptions of many, did not simply apply algebraic techniques to geometric figures by using his invention of the Cartesian plane ??(Dennis, Smith, &Confrey, 1992)t ??. Rather, he was interested in analyzing the curves produced by curve-drawing devices. Such devices had close connections to the tools of the time; in fact, the study of geometry at this time was, for the most part, a study of civil and mechanical engineering. Descartes did not impose axes and then place letters on geometric points to create "analytic geometry;" he analyzed how things moved and what properties of the curves could be expressed by means of descriptions of magnitudes and their relationships. For a detailed discussion of his work see Dennis ??(1995)t ??, Dennis and Confrey ??(1995)t ?? and van Schooten =t??(1657)t ??.

In addition, Wallis (for example) used tables extensively to analyze the behavior of curves--an approach which gets no mention in most mathematical histories, including that of the reificationists. Wallis's work on tables demonstrates the historical productiveness of moving between and among mathematical forms. In addition to the need for accurate tables for economic and navigational purposes, there was a keen interest in improved logarithmic values and for alternative methods of approximation for irrational values. In Dennis and Confrey ??(in press)??, we trace this history, showing that tabular and geometric approaches played central roles in the work not only of Wallis, but of Pascal and Euler, as well as in the development of calculus by both Newton and Leibniz.

Why has this part of the historical development been traditionally absent from accounts of the development of function? One answer is that since education in the twentieth century has avoided the use of tables (until spreadsheets) by emphasizing symbolism while devoting only minimal effort to tabular display, historians of mathematics have not seen this part of the history as essential to the development of the concept. This reminds us that all historical analysis is deeply theory-laden, and one’s theoretical commitments determine what is reported and what is suppressed. It raises serious questions about a strictly hierarchical view of mathematics.

The mathematician and historian Arnol'd tells a remarkable story related to this unfortunately commonplace absence of historical reference to context and tools. According to Arnol'd (1990)t??, Newton received a letter from Hooke discussing the law of universal gravitation, seemingly arrived at independently by Hooke. Newton did not wish to offer any credit to Hooke and when it was insisted upon by members of the Royal Society, he responded bitterly and sarcastically. In addition, as Curator of the Society, Hooke was required to design and produce apparatus for weekly scientific demonstrations. Newton, who had witnessed a number of these demonstrations, later destroyed Hooke's extensive drawings and the physical apparatus, as well as all extant portraits of Hooke (p. 51). Arnol'd's tale is much more than one of professional envy. It illustrates a disturbing force in the history of mathematics: the obscuring of the role of tools and physical experiment in the development of mathematics.

Michael Otte ??(1993)t?? discusses this trend further in his work as he contrasts the role of geometry and mechanics on the evolution of mathematics. Otte locates the social roots of modern science during the Renaissance in interactions among engineers, artists and medical practitioners. For artists, the geometry guided construction. For engineers, carpenters, architects, farmers, and sailors, mechanics formed the basis of the enterprise. Most importantly, Otte identifies a critical period in the nineteenth century when a real positivism and empirically-oriented methodologies developed only when the problem of the application of scientific knowledge turned into asocial and epistemological question of great relevance and pure mathematics arose in the course of increased specialization and division of labor. In the course of this change. . . 18th century algebraic analysis in the sense of Euler and Lagrange was replaced by arithmetized analysis and function theory in the sense of Fourier and Cauchy. It was only then that people turned to a closer study of empirical phenomena of all kinds by means of arithmetized mathematics(ibid., p. 285, emphasis added).

With this perspective, we can see pure mathematics’ origins in, on the one hand, specialization and division of labor, and on the other from the movement from geometry and mechanics to new areas of empirical reality.

Otte goes on to discuss the important roles of axiomization and of arithmetized mathematics in the development of mathematics. For the former, he cites Israel (1981) who has argued that "the axiomatic approach stresses that theories are realities sui generis that refer to a reality different from that of mere empirical phenomena." "Arithmetized mathematics," Otte writes of the latter, "served the consolidation of pure mathematics as defining a separate group of professionals as well as setting the training requirements for membership in such a group" (ibid., p. 285). These two forces from pure mathematics exert considerable influence on the development of mathematics and its relationship to technology. As elegantly stated by Otte, "the purpose for a social history of mathematics is to account realistically for the seemingly aprioristic necessity of mathematics." It is this "seeming" necessity which Otte wishes to examine that reificationists often take uncritically as their starting point.

Otte argues that arithmetized mathematics is an expression of "mentalism." Mentalism means the notion that mental activity is autonomous and independent of any context, being characterized by a certain substance of our mind or will. He claims that one result of mentalism is "held by pure mathematicians [who] emphasize mainly the individualist element, the cult of genius and creationism, abominating real applications of mathematics, particularly because of the necessity for the permanent social cooperation and the continued exchange of experience involved with it" (ibid., p. 295).

These historical episodes present a different historical picture than the one that emerges from the theorists on reification. These stories reveal first and foremost that the choice of historical lens exerts significant influence on the reconstruction of the mathematical enterprise. This challenges the view that mathematics is strictly hierarchical, and makes this belief a product of modern lenses rather than of historical necessity. The retelling of the history of function can uncover the importance of geometry and science to a fuller understanding of function while questioning the emphasis on algebraic symbolism as the ultimate achievement of mathematics.

The stories suggest that the history of mathematics also includes an attempt to suppress its relationship to tools, to everyday thought and to controversy ??(Costa, 1995)t ??. It also raises a serious challenge to the theorists in reification. It suggests that reducing the analysis of history to identify a narrow set of psychological processes reinforces the lens of isolated invention. One's choice of theories for advanced mathematics determines one's participation in this activity.

Voice and Perspective

The educational choices of theorists of reification also bear closer examination.

The theorists' proposals to elevate object-based thinking into the central position of mathematics learning express a strong Piagetian influence. Their interpretation of Piaget accepts both his description of the construction of schemes (micro-level) and his stage theory(macro-level) which assumes that development universally follows successive hierarchical stages from sensori-motor to abstraction. In contrast, we take the position that while Piaget's description of equilibration, accommodation and assimilation are useful in modeling student thought, his strictly hierarchical approach to development is limiting. To our minds, his most valuable contribution to educational theory is his recognition that student conceptions may not mirror--or be partial versions of--an adult perspective.

By listening closely to children, one can gain insight into fresh perspectives on mathematics. The question is: What does one do with these perspectives? Are they assumed always to be arbitrary locations on the establish path to mathematical sophistication, or can they be seen as indicators of potential alternatives? To allow for this interpretation, the dialectic of "voice and perspective" result??(Confrey, 1995)t ?? was developed. According to this approach, a student voice can be used as an opportunity to reconsider curricular decisions and to rethink one's own perspectives on mathematical thinking.

Since we have established the position that development is often cast as historical recapitulation, and history is often cast as progressive absolutism, we have the basis for a fundamental reexamination of student voice. Just as alternative approaches to history can counter the suppression of mathematical context, controversy, and diversity, a focus on voice counters the tendency to devalue students' need for personally and socially vigorous constructions of knowledge in favor of pushing them toward a mathematical "competence" which is too often narrowly defined as proficiency in self-referential symbolism and language.

Recently, our research group completed a three-year teaching experiment with twenty 11-year- olds. Beginning when the students were in third grade, we approached multiplication, division and ratio as a co-defining trio of operations. This contrasted with the traditionally ordered curriculum, a disparate sequence of increasingly difficult operations that starts with multiplication as repeated addition, introduces division (usually as repeated subtraction) only later, and addresses ratio and proportion still later. In fourth grade, we introduced fractions as a subset of ratio. This approach was the result of the "splitting conjecture" based in an examination of young children's intuitive use of sharing, splitting and similarity ??(Confrey, 1994a)t ??. In her conjecture, Confrey argued that repeated addition was an insufficient basis for multiplication and that multiplication/division and ratio must be developed earlier in the curriculum and with stronger ties to two-dimensional graphing, geometry, and the use of tables.

According to reification theory, such a reversal is hardly advisable, for in these reification theories, logical analysis would reveal that fractions contain the concrete referents necessary for young children and ratio and proportion introduces harder symbolic and conceptual demands. However, our data support that: 1) the students performed at exceptionally high levels on standardized tests; 2) the differential in the performance of students with differing preparation levels declined; and 3) the evidence of "standard misconceptions" (the use of additive strategies) decreased??(Confrey & Scarano,1995)t ??.

The results of the experiment can be traced to three elements: the use of projects and tools, the careful use and sequencing of a variety of representational forms, and the type of classroom interactions. The children engaged in projects such as the design of a handicapped-access ramp, the development of a tourist's guide for a foreign weight system, and the gathering of data for a school-wide poll. The representational forms included the use of tables, graphs on the two- dimensional plane, dot drawings, ratio boxes, and common fractional forms. Classroom interactions were of two major categories: whole-class discussion, where problems were introduced, progress reported and discussed, and closure and assessment solicited; and small- group interactions for projects.

Clearly, data are open to interpretation, and complex data such as long-term classroom studies offer no singular or assured solutions. However, we would suggest that the theory of reification, as it currently stands, is simply not a rich enough framework for interpretation of such events and interactions. A classically hierarchical interpretation would imply that our approach, despite its successes, was simply inappropriately ordered. In fact, in Confrey??(1995)t ??, I argue that it was the tendency to import axiomization and arithmetization into our treatment of elementary mathematics that squeezed geometry from the curriculum. Due to our enculturated preference to view the real numbers in relation to a number-line correspondence, we reduced the rational numbers to points on a one-dimensional line. This necessitates a definition of addition of rationals which requires a common denominator. In turn, this delegitimizes the addition of ratios by numerator and denominator, a common activity in everyday life such as when adding a ratio of odds (e.g. combining "yes"es to "no"es in two samples).Moving to the two dimensional plane with vectors as ratios permits this additive operation, but requires student to know in which frame of reference they are operating.

Conclusions and Future Directions

The type of approach we have advocated contrasts with a reification approach. It makes all of mathematics a tool for modeling, although it does not rule out a decision by some pure mathematicians to focus on the development, elaboration and improvement of those tools ??(Doerr & Confrey,1994)t ??. It does not assign mathematical objects a separate reality, but recognizes that making distinctions and forming constructs is an essential part of modeling. Selecting the language of tools over the language of objects signals a clearer relationship between mathematics and human activity. Using this modeling or project-based approach, one is forced to reconsider the question of sequencing. Instead of relying on a purely logical analysis to decide how to develop mathematics, one must consider rather what activities students are interested in and familiar with and how these activities relate to other possibilities. Although the classroom data reported here derives from young children, our studies indicate that this is an approach that can be used as a higher grade level ??(Doerr, 1995; Rizzuti, 1991)t??.

Many mathematicians may protest that an over-reliance on contextual activities reduces the purity of the enterprise. However, by working with tools as a metaphor, one does not deny the value of focusing on the tools themselves in order to improve their performance or develop newer and better methods of use. In seeking to understand the relationship between the function and the form of tools, we place the issue of structure centrally. Here, however, structure is not an a priori set of arrangements, but is the result of the kinds of activities and actions a tool permits when used in certain ways. Thus, a tool-based approach would share a central feature with reification, asserting the importance of structure. However, rather than demanding departure from activity, the act of seeing similarities in structure across different contexts would be the basis for abstraction.

Another distinction of this approach is that it places interactions at the center of the construction of meaning. Rather than relying on expertise alone and at a distance to define the goals of instruction, it focuses on the negotiation of meaning through classroom interactions. An expert teacher, in contrast with an expert mathematician, will enable students' interactions to generate discussion, demonstration and argument which with careful support can evolve into definition, axiomatization and proof. This need not be seen as an exclusive territory, but one in which all participants' views would contribute to the development of the ideas. Independent work would not be eschewed, but placed within the context of group progress. This approach does not diminish the importance of expert knowledge on the part of the teacher, but in fact requires a deeper, more flexible mathematical understanding.

If such an approach were taken, a fair question is whether it would produce fewer exceptional mathematicians or more of them. Whereas reification theorists might fear that with this approach, no student would receive adequate training to produce theoretically sound results, we would contend that by avoiding the exclusion of such a large and diverse pool of students, the quality of mathematicians would be at least as strong as it is now. We would further predict that a less splintered preparation, rich in connections, would constitute stronger mathematics. In addition, a wider segment of the public might be more supportive of mathematics and value it more thoroughly.

Although any detailed treatment of the use of object-based language in computers is beyond the scope of this work, some initial comments might lead to further discussions. The computer presents us with a flexible medium into which we can embed our metaphors, express our constructs, and impose our values. The use of the graphics on the screen invites a description of objects, and the process-object discussion should therefore have serious appeal. Languages, like Logo, in which processes become named procedures, can lead us to point out objects on the screen beyond the initial turtle. This resembles reification to some extent, but it does not necessarily entail the other components: these microworlds and simulations do not assume strict hierarchies nor are there presumed cognitive obstacles to be overcome. Furthermore, in most cases, no complete departure from everyday experience is desired. Even if a microworld or simulation allows one to tamper with realities, the assumption is that this leads to a deeper understanding of everyday experience, not its replacement.

Furthermore, the role of the computer as a communication device also lessens its object-based image. For many children, the attraction of the video game with its characters and drama (and violence) is being challenged by the appeal of electronic mail, access to conversation groups and the like. A fundamental question is how mathematics instruction will be treated in this dichotomous environment. Can we imagine and develop strong uses of interactions around mathematics in computer-based environments?

Our critique of reification theories suggests that it is important to avoid using the computer to teach mathematics as if it were a set of rigidly ordered constructs. In multi-media, one immediately encounters the possibility, even the necessity, of flexible sequencing. This environment also makes it easy to draw connections among mathematical topics and cultural contexts in geography, history, biology, and discussions across communities is an easily accomplished enhancement. An outstanding question is how the question of access will be treated. Will the computer become as ubiquitous as the telephone, or will its acquisition further stratify our population?

We began this article decrying the high failure and drop-out rates in mathematics. We end by joining our colleagues in asserting that computers can contribute significantly to changing this unacceptable state of affairs. However, in order to do so, both the software environments and the ways in which mathematics is communicated within them need to be reconceptualized in light of the arguments raised in this paper. Equity in access, as one important example, will have to before vigorously pursued. Our goal in presenting this paper has been to inspire debate and what kinds of theories can move reform actively to the secondary and post-secondary level. Ideally, such debate will include responses from the theorists on reification as well as responses by people in educational technology that also have a stake in the decisions about language and description.

References

??Arnol'd, V. I. (1990). Huygens& Barrow and Newton & Hooke (E. J. F. Primose, Trans.). Basel:Bierkh?user Verlag.

Artigue, M. (1992). The importance and limits of epistemological work in didactics. In W. Geeslin & K. Graham (Eds.), Sixteenth Psychology of Mathematics Education, 3 (pp. 195-216). Durham, NH:

Bachelard, G. (1938). La formation de l'esprit scientifique (reprinted 1983 ed.). Paris: J. Vrin.

Brousseau, G. (1983). Les obstacles epistemolgiques et les problemes en mathematiques. Recherche in Didactiques des Matematiques, 4(2),164-198.

Chevallard, Y. (1991). Concepts fondamentaux de didactique: Perspectives apportees par une perspective anthropologique. In Lecture at the Vieme Ecole d'Ete de didactique des mathematiques, . Plestin les Greves,France:

Confrey, J. (1980). Conceptual change, number concepts and the introduction to calculus. doctorial dissertation, Cornell University, Ithaca, NY.

Confrey, J. (1990). A review of the research on student conceptions in mathematics, science and programming. In C. Cazden (Ed.), Review of Research in Education (pp. 3-56). Washington, DC: American Education Research Association.

Confrey, J. (1994a). Splitting, similarity, and rate of change: New approaches to multiplication and exponential functions. In G. Harel & J. Confrey (Eds.), The Development of Multiplicative Reasoning in the Learning of Mathematics (pp. 293-332). Albany, NY: State University of New York Press.

Confrey, J. (1994b). A theory of intellectual development, Part I. For the Learning of Mathematics, 14(3, November), 2-8.

Confrey, J. (1995). Student voice in examining "splitting" as an approach to ratio, proportion, and fractions. In L. Meira & D. Carraher (Eds.), Proceedings of the Nineteenth International Conference for the Psychology of Mathematics Education, 1 (pp. 3-29). Recife, Brazil: Universidade Federal de Pernambuco.

Confrey, J., & Scarano, G. H. (1995). Splitting reexamined: Results from a three-year longitudinal study of children in grades three to five. In D. T. Owens, M. K. Reed, & G. M. Millsaps (Eds.), Proceedings of the Seventeenth Annual Meeting for the Psychology of Mathematics Education-NA, 1 (pp. 421-426). Columbus, OH: ERIC Clearinghouse for Science, Mathematics, and Environmental Education.

Cornu, B. (1991). Limits. In D. Tall (Ed.), Advanced Mathematical Thinking (pp. 153-166). Dordrecht: Kluwer Academic Publishers.

Costa, S. (1995). "Our" notation from their quarrel: The Leibniz-Newton controversy as embodied in calculus textbooks. In Paper presented at the History of Science Society Annual Meeting, October . Minneapolis, MN:

Dennis, D. (1995). Historical perspectives for the reform of mathematics curriculum: Geometric curve drawing devices and their role in the transition to an algebraic description of functions. doctoral, Cornell University, Ithaca, NY.

Dennis, D., & Confrey, J. (1995). Functions of a curve: Leibniz's original notion of functions and its meaning for the parabola. The College Mathematics Journal, 26(3, May), 124-131.

Dennis, D., & Confrey, J. (in press). The creation of continuous exponents: A study of the methods and epistemology of Alhazen and Wallis. In J. Kaput& E. Dubinsky (Eds.), Research in Collegiate Mathematics Education II Providence, RI: American Mathematical Society.

Dennis, D., Smith, E., & Confrey, J. (1992). Rethinking functions: Cartesian constructions. In S. Hills (Ed.), Second International Conference on the History and Philosophy of Science in Science Education, 2 (pp. 449-466). Queens University: Kingston Ontario: The Mathematics, Science, Technology and Teacher Education Group.

Doerr, H. (1995). An integrated approach to mathematical modeling: A classroom study. In Annual Meeting of the American Educational Research= Association, April . San Francisco, CA:

Doerr, H., & Confrey, J. (1994). A modeling approach to understanding the trigonometry of forces: A classroom study. In A paper presented at the Eighteenth Annual Meeting of Psychology of Mathematics Education, .Lisbon, Portugal:

Dorfler, W. (1993). Computer use and views of the mind. In C. Keitel & K. Ruthven (Eds.), Learning from Computers: Mathematics Education and Technology (pp. 159-186). New York: Springer-Verlag.

Douady, R. (1991). Tool, object, setting, window: Elements for analysing and constructing didactical situations in mathematics. In A. J. Bishop & S. Mellon-Olsen (Eds.), Mathematical Knowledge: Its Growth Through Teaching Dordrecht: Kluwer Academic Publishers.

Dubinsky, E. (1991). Reflective abstraction in advanced mathematical thinking. In D. Tall (Ed.), Advanced Mathematical Thinking (pp.95-126). Dordrecht: Kluwer Academic Publishers.

Dubinsky, E., & Harel, G. (1992). The nature of the process conception of function. In G. Harel & E. Dubinsky (Eds.), The Concept of Function, Washington, DC: Mathematical Association of America.

Frye, M. (1983). The Politics of Reality: Essays in Feminist Theory. Trumansburg, NY: Trumansburg Press.

Harel, G., & Dubinsky, E. (Eds.). (1992). The Concept of Function. Washington, DC: Mathematical Association of America.

Hoyles, C. (1993). Microworlds/schoolworlds: The transformation of an innovation. In C. Keitel & K. Ruthven (Eds.), Learning from Computers: Mathematics Education and Technology New York: Springer-Verlag.

Hoyles, C., & Noss, R. (1993). Out of the cul-de-sac? In J. R. Becker & B.J. Pence (Eds.), Proceedings of the Fifteenth Annual Meeting of the Psychology of Mathematics Education-NA, 1 (pp. 83-90). Pacific Grove, CA: Center for Mathematics & Computer Science Education, San Jose State University.

Jahnke, H. N. (1994). The historical dimension of mathematical understanding--objectifying the subjective. In J. P. daPonte & J. F. Matos(Eds.), Proceedings of the Eighteenth International Conference of the Psychology of Mathematics Education, 1 (pp. 139-156). Lisbon, Portugal: University of Lisbon.

Lakatos, I. (1984). A renaissance of Empiricism in the recent philosophy of mathematics? In T. Tymockzko (Ed.), New Directions on the Philosophy of Mathematics (pp. 29-48).

Lakatos, I., & Musgrave, A. (Eds.). (1970). Criticism and the Growth of Knowledge. London: Cambridge University Press.

Otte, M. (1993). Towards a social theory of mathematical knowledge. In C. Keitel & K. Ruthven (Eds.), Learning from Computers: Mathematics, Education, and Technology (pp. 280- 327). New York: Springer-Verlag.

Papert, S. (1980). Mindstorms. New York: Basic Books, Inc.

Polanyi, M. (1958). Personal Knowledge: Towards a Post-Critical Philosophy. Chicago: University of Chicago Press.

Rizzuti, J. (1991). Students' Conceptualizations of MathematicalFunctions: The Effects of a Pedagogical Approach Involving Multiple Representations. doctoral dissertation, Cornell University, Ithaca, NY.

Sfard, A. (1992). Operational origins of mathematical objects and the quandary of reification: The case of function. In G. Harel & E. Dubinsky(Eds.), The Concept of Function (pp. 59-84). Washington, DC: Mathematical Association of America.

Sfard, A. (1994). Reification as the birth of metaphor. For the Learning of Mathematics, 14(1), 44-55.

Sfard, A., & Linchevski, L. (1994). The gains and the pitfalls of reification: The case of algebra. Educational Studies in Mathematics,26, 191-228.

Sfard, A., & Thompson, P. W. (1994). Problems of reification: Representations and mathematical objects. In D. Kirshner (Ed.), Proceedings of the Sixteenth Annual Conference of the Psychology of Mathematics Education-NA, 1 (pp. 3-32). Baton Rouge, LA: Louisiana State University.

Shapin, S., & Schaffer, S. (1985). Leviathan and the Air-pump: Hobbes, Boyle, and the Experimental Life. Princeton, NJ: Princeton University Press.

Sierpinska, A. (1992). On understanding the notion of function. In G. Harel& E. Dubinsky (Eds.), The Concept of Function (pp. 25-58).Washington, DC: Mathematical Association of America.

Tall, D. (Ed.). (1991a). Advanced Mathematical Thinking. Dordrecht: Kluwer Academic Publishers.

Tall, D. (1991b). The psychology of advanced mathematical thinking. In D. Tall (Ed.), Advanced Mathematical Thinking (pp. 3-21). Dordrecht: Kluwer Academic Publishers.

Tall, D. (1991c). Reflections. In D. Tall (Ed.), Advanced Mathematical Thinking (pp. 251- 260). Dordrecht: Kluwer Academic Publishers.

Tall, D. (1995). Cognitive growth in elementary and advanced mathematical thinking. In L. Meira & D. Carraher (Eds.), Proceedings of the Nineteenth International Conference for the Psychology of Mathematics Education, 1 (pp. 61-76). Recife, Brazil: Graduate Program in Cognitive Psychology, Universidade Federal de Pernambuco.

Thompson, P. W. (1994). The development of the concept of speed and its relationship to concepts of rate. In G. Harel & J. Confrey (Eds.), The Development of Multiplicative Reasoning in the Learning of Mathematics (pp. 181-236). Albany, NY: State University of New York Press.

Tymoczko, T. (1979). The four-color problem and its philosophical significance. Journal of Philosophy, 76(2), 57-82.

van Schooten, F. (1657). Exercitationum Mathematicorum, Liber IV, Organica Coniccarum Sectionum in Plano Descriptione. (original edition in the rare books collection of Cornell University, Ithaca, NY).

Viete (1591). The Analytz Avt (Avs Analytica).

Wilensky, U. (1991). Abstract meditations on the concrete and concrete implications for mathematics education. In I. Harel & S. Papert (Eds.),Constructionism Norwood, NJ: Ablex Publishing Corp.t ??