CEPA eprint 2967

A constructivist alternative to the representational view of mind in mathematics education

Cobb P., Yackel E. & Wood T. (1992) A constructivist alternative to the representational view of mind in mathematics education. Journal for Research in Mathematics Education 23(1): 2–33. Available at http://cepa.info/2967
Table of Contents
Cognitive argument
Pragmatic considerations
Theoretical considerations
An anthropological argument
Pedagogical argument
The dualism
The social and cognitive aspects of mathematical learning
Pedagogical symbol systems
Students, teachers, and mathematics
Beyond the representational view
The representational view of mind in mathematics education is evidenced by theories that characterize learning as a process in which students modify their internal mental representations to construct mathematical relationships or structures that mirror those embodied in external instructional representations. It is argued that, psychologically. this view falls prey to the learning paradox. that. anthropologically, it fails to consider the social and cultural nature of mathematical activity and that. pedagogically, it reads to recommendations that are at odds with the espoused goal of encouraging learning with understanding. These difficulties are seen to arise from the dualism created between mathematics in students’ heads and mathematics in their environment. An alternative view is then outlined and illustrated that attempts to transcend this dualism by treating mathematics as both an individual, constructive activity and as a communal, social practice. It is suggested that such an approach might make i1 possible to explain how students construct mathematical meanings and practices that, historically, took several thousand years to evolve without attributing to students the ability to peek around their internal representations and glimpse a mathematically prestructured environment. In addition, it is argued that this approach might offer a way to go beyond the traditional tripartite scheme of the teacher, the student, and mathematics that has traditionally guided reform efforts in mathematics education.
Many of the current attempts to develop theories in mathematics education reelect the view that learning is a process of constructing internal mental representations. This work encompasses a diverse set of research efforts that differ in terms of underlying assumptions. For example. some constructivist researchers such as Confrey (1990a), Kaput (1987a, 1991). and Thompson (1989) have investigated the processes by which students modify their cognitive representations as they create external representations and use conventional symbols to express their thinking. This line of research rejects the view that mathematical meaning is inherent in external representations and instead proposes as a basic principle that the mathematical meanings given to these representations are the product of students’ interpretive activity. Consequently, these researchers argue that a pedagogical device such as “a graph window… becomes a ’representation’ only when a student uses it to express a conception” (Confrey, 1990a, p. 56).
This constructivist approach to representation stands in opposition to a philosophical position that Putnam (1988) called the representational view of mind. Rorty (1979) summarized the central tenet of this position as follows:
To know is to represent accurately what is outside the mind; so to understand the possibility and nature of knowledge is to understand the way in which the mind is able to construct such !internal] representations. (p. 3)
Given that mathematics educators almost universally accept that learning is a constructive process, it is doubtful if any take the representational view literally and believe that learning is a process of immaculate perception. However, as Ernest (1991a) observed, the term constructivism itself covers a panoply of theoretical positions. Some of these appear to be eclectic positions in which researchers attempt to combine the notion of learning as active construction with aspects of the representational view of mind. For example, it is sometimes argued that curriculum developers should devise materials that present mathematical meanings orstructure to students in a readily apprehendible or transparent form. One rationale for this recommendation is to ensure that the constructions that students make are correct as well as sensible. In such an approach, the mathematics to be learned is often said to be a salient property of external, instructional representations. Learning is then characterized as a process in which students gradually construct mental representations that accurately mirror the mathematical features of external representations.
Eclectic positions of this type in which students are presumed to build representations from the instructional materials presented to them clearly have a certain explanatory power. For example, our experience of mathematical truth and mathematical certainty need little explanation: we have learned to see the mathematical structures inherent in the physical and symbolic environment. Similarly, from the representational perspective, the reason why we are frequently able tocommunicate our mathematical thinking to each other is self-evident; we each come to see the same preexisting mathematical structures and relationships in the course of the communication process.
Our purpose in the first part of the paper is to probe the underlying assumptions of the representational view of mind and to consider their implications for mathematics education. In doing so, we make no claims about the current theoretical positions of particular researchers. Instead, our primary purpose is to encourage the reader to examine the consequences of some of the views that he or she might currently hold. To this end, we first question a basic metaphor that underlies the representational view of mind by summarizing arguments made from cognitive, anthropological, and pedagogical perspectives. At the same time, we attempt to demonstrate that this is not merely a matter of esoteric theoretical interest but that. on the contrary, the issue has significant pragmatic implications. In the remainder of the paper, we briefly outline and illustrate the implications of a philosophical orientation that attempts to dispense with all aspects of the representational view. Our central claim is that an approach that views mathematics as both an individual and a collective activity transcends the contradictions of the representational view and offers an account of truth, certainty, and intersubjectivity.
Cognitive argument
Pragmatic considerations
Three features of the eclectic position that we have outlined are highly compatible with the representational view of mind:
The overall goal of instruction is to help students construct mental representations that correctly or accurately mirror mathematical relationships located outside the mind in instructional representations.The method for achieving this instructional goal is to develop transparent instructional representations that make it possible for students to construct correct internal representations.External instructional materials presented to students are the primary basis from which they build their mathematical knowledge.
For the purposes of this paper, we will treat approaches that realize all three features in practice as instances of the instructional representation approach.
As an example, consider an approach to arithmetical computation that emphasizes a mapping between the steps of written algorithms and prescribed actions performed on Dienes blocks. Here, a transparent instructional representation of place value notation is assumed to be the primary basis from which students construct mental representations of preformed numerical relations and thus give meaning to the steps of the written algorithm. Holt (1982) described his reflections after he made similar assumptions about the instructional potential of particular manipulative materials.
Bill [a colleague] and I were excited about [Cuisenaire] rods because we could see strong connections between the world of rods and the world of numbers. We therefore assumed that children, looking at the rods and doing things with them, could see how the world of numbers and numerical operations worked. The trouble with this theory was that Bill and I already knew how the world of numbers worked. We could say, “Oh, the rods behave just the way the numbers do.” But if we had not known how numbers behaved, would looking at the rods have helped us to find out? (pp. 138-139)
Holt answered his own concern that instructional materials that were transparent to him might not he transparent to students when he discussed his experiences of using Dienes blocks in the classroom.
Children who already understood base and place value, even if only intuitively, could see the connections between written numerals and these Wracks…. But children who could not do these problems without the blocks didn’t have a clue about how to do them with the blocks….They found the blocks, as Edward had found the Cuisenaire rods, as abstract, as disconnected from reality, mysterious, arbitrary, and capricious as the numbers that these blocks were supposed to bring to life. (pp. 218-219)
Berieter (1985) called this concern raised by Holt the learning paradox.
If one tries to account for learning by means of mental actions carried out by the learner, then it is necessary to attribute to the learner a prior cognitive structure that is as advanced or complex as the one to be learned… The learning paradox does apply where – as in being introduced to rational numbers, for example – learners must grasp concepts or procedures more complex than those they already have available for application. (p. 202)
In other words, the assumption that students will inevitably construct the correct internal representation from the materials presented implies that their learning is triggered by the mathematical relationships they are to construct before they have constructed them (Cobb, 1987; Gravemeijer, 1991; von Glasersfeld, 1978). How then, if students can only make sense of their worlds in terms of their internal representations, is it possible for them to recognize mathematical relationships that are developmentally more advanced than their current internal representations? This would clearly seem to be an impossibility if students are simply left to their own devices to explore the materials without guidance. As a consequence. it would seem necessary to consider the teacher’s role in helping students construct replicas of the mathematical relationships presented to them in an easily apprehensible form. In the case of our example of mapping instruction, one possible role for the teacher is to spell out the correspondence between the two domains (i.e., blocks and numerals) in detail. In doing so, the teacher attempts to elicit a generalization across the two procedures by ensuring that the student realizes which components of each correspond to components of the other. However, as Steinbring (1989) noted, approaches in which the teacher becomes increasingly explicit about what it is that students are supposed to learn can lead to the excessive algorithmatization of mathematics and the disappearance of conceptual meaning. Brousseau (1984) made this point succinctly when he stated that
the more explicit I am about the behavior I wish my students to display, the more likely it is that they will display that behavior without recourse to the understanding which the behavior is meant to indicate: that is. the more likely they will take the form for the substance. [Quoted in Mason, 1989, p. 7]
The view of the teacher as one who accommodates to students’ activity by specifying predetermined mathematical relationships in increasing detail is consistent with the central features of the instructional representation approach. In particular, this image of teaching follows from the assumption that instructional representations are the primary source of students’ mathematical knowledge. As long as this assumption is taken for granted. the discussion of the teacher’s role centers on the explicitness with which the embodied mathematical relationships should be spelled out to students. We would note that this image of teaching as a process of imposition rather than negotiation (Bishop, 1985) clashes with that advanced in recent reform documents such as those of the National Council of Teachers of Mathematics (1989, 1991) and the National Research Council (1989).
As a further point, our discussion of the teacher’s role acknowledges that social interaction plays an important role in students’ mathematical learning. Later in the paper, we will return to the issue of interaction and suggest that teachers might in practice challenge the assumptions that give rise to the learning paradox by attempting to see beyond their expert interpretations of instructional materials. They might then consider the various ways that students actively interpret the materials as they engage in genuine mathematical communication in the social context of the classroom. The materials would then no longer be used as a means of presenting readily apprehensible mathematical relationships but would instead be aspects of a setting in which the teacher and students explicitly negotiate their differing interpretations as they engage in mathematical activity.
Theoretical considerations
The discussion of pragmatic implications serves as a backdrop against which to consider four potential theoretical difficulties that arise when certain tenets of constructivism are combined with aspects of the representational view. The first difficulty concerns a tension in eclectic characterizations of mathematical learning. On the one hand, learning is described as a process in which students actively construct mathematical knowledge as they strive to make sense of their worlds. On the other hand. learning can. in practice, be treated as a process of apprehending or recognizing mathematical relationships presented in instructional representations. These two characterizations of mathematical learning reflect differences in the emphasis given to the students’ and to the teacher’s interpretations of instructional representations. The view of learning as active construction implies that students build on and modify their current mathematical ways of knowing. A detailed understanding of the various ways that students interpret particular situations is therefore crucial to both instructional development and teaching ( Steffe. 1987). In contrast, the view of learning as the correct recognition of mathematical relationships places the emphasis almost exclusively on the teacher’s expert interpretation of instructional materials. For example, it is not necessary to consider qualitative differences in students’ understandings of place value numeration when developing a rationale for the mapping instruction. In general, the tension between the two characterizations of mathematical learning indicates a difficulty in reconciling the belief that students actively construct their mathematical knowledge on the basis of what they currently know with the fact that we as educators have certain end points in mind for those constructive processes. In other words, within this theoretical scheme, the acknowledgement that students must necessarily make sense of their worlds conflicts with the realization that anything does not go, that any way of sense making is not as good as any other.
The second potential difficulty is closely related to the first and stems from an implicit appeal to two incommensurable semantic theories. The view that mathematical learning is a process of active construction locates the source of meaning in students’ purposeful. socially and culturally situated mathematical activity (Bartolini Bussi, 1991; Cobb, 1989: Sinclair, 1990: Steiner, 1989). By way of contrast, the characterization of mathematical learning as accurate apprehension rests on an objectivist theory of semantics directly derived from the representational view of mind. In this theory, meaning is analyzed in terms of fixed mappings between arbitrary symbols and objects or events in the world (Putnam, 1988). As Johnson (1987) noted, semantic theories of this type have been developed in a highly technical fashion by information-processing psychologists. In general, the tension between constructivist and objectivist semantics indicates that the representational view of mind as applied to mathematics education suffers from a deep-rooted conceptual anomaly. Further, the resolution of this difficulty would seem to require a major theoretical restructuring.
The third potential difficulty with the representational approach follows from the first two and concerns the dualism erected between mathematics in students’ heads and mathematics in the external environment. Presumably, the teacher, like the students, actively makes sense of his or her world by constructing internal representations. The crucial theoretical move that gives rise to the dualism is to project the teacher’s expert mathematical interpretations into the students’ environment, thereby treating them as mind-independent external representations. The learning situation then becomes one in which students are separated from fixed mathematical relationships contained in a prestructured environment. This separation is in fact institutionalized in the dual use of the term representation; internal representations are located in students’ heads and external representations are located in the environment (von Glasersfeld. 1987). It is this separation that underlies Berieter’s (1985) formulation of the learning paradox; because the concepts more complex than those that learners have available are located in a preformed environment, it is unclear how learners could ever grasp them. One way of attempting to resolve the paradox is to challenge the dualism that gives rise to it by reconsidering the relationship between mathematical knowledge and the knower.
The fourth potential difficulty with the assumption that specific mathematical meanings are inherent in external representations is that this idea conflicts with the empirical finding that mathematical meanings are socially and culturally situated (Billig, 1987; Bishop, 1988; Rogoff, 1990; Saxe, 1991a; Winograd & Flores, 1986). For example, Saxe’s (1991a) analysis of the computational algorithms developed by Brazilian child candy sellers indicates that the children’s constructions were closely linked to their participation in particular forms of economic exchange. More generally, analyses by Saxe and others illustrate the intimate relationship between mathematical cognition and cultural practices. As a consequence, their results directly challenge the assumption that fixed mathematical relationships are contained in a prestructured environment independent of individual and collective human activity. These findings are, however. compatible with the view that the activity of constructing mathematical knowledge in school is subject to explanation and justification as students participate in the intellectual practices of the classroom community. Thus, Greeno (1991) recently argued that
a person’s knowing of a conceptual domain is a set of abilities to understand, reason, and participate in discourse…. Any particular activities that a person engages in or learns to perform are embedded in a conceptual ecology that has been developed within a community of intellectual work ….Critical components of these sets of practice include the appreciation and use of explanatory ideals that are shared within the community and provide basic modes and goals of explanatory discourse. (p. 176)
Such a view acknowledges that knowing is a socially and culturally situated constructive process. We will consider some of the implications of this view when we discuss an alternative philosophical orientation. First, however, we continue our examination of the representational approach by presenting arguments framed in anthropological and pedagogical terms.
An anthropological argument
The apparent plausibility of the representational approach rests on the experienced transparency of external representations for the expert. We now take an anthropological stance to account for this intuition of direct access to mathematical relationships contained in a prestructured environment. As part of the argument. we question whether it is necessary to assume that we can peek around our internal representations and glimpse a pristine world unaffected by human interests, purposes, and concerns.
To avoid possible miscommunication, we should stress that the experience of a world structured by mathematical relationships is, in our view, a central aspect of meaningful mathematical activity (Davis & Hersh, 1981). The issue at hand arises when we step back from the activity of doing mathematics and attempt to explain how we come to have experiences of this sort. As a first step, we note that the experienced transparency of instructional representations reflects the expert’s own relatively sophisticated mathematical conceptions. However, this observation is not by itself sufficient to fully account for the expert’s belief that his or her interpretations of external representations are. in some sense. natural and self evident, Not only can adults individually interpret, say, Dienes blocks in terms of their conceptions of place value numeration, but each assumes without question that anyone else who can be said to understand place value will make a compatible interpretation (cf. Frank, 1979; Schutz, 1962). In short, we, asexperts,each make an interpretation that we take as shared with other mathematically acculturated members of society. The conviction that the blocks are transparent external representations of place value is then not merely the product of the sophistication of our individual mathematical constructions but also derives from the compatibility of our constructions with those of others who can be said to know mathematics. In particular, we have each experienced place value numeration as something public that can be pointed to and spoken about unproblematically while interacting with others. As a consequence. we experience place value numeration as an objective property of the blocks that exists between people independently of their individual and collective mathematical activity. Thus, in this account, the experienced transparency of instructional representations is a consequence of our own mathematical acculturation in the course of which we each actively constructed relatively sophisticated conceptions that make it possible for us to participate in the taken-as-shared mathematical practices of our society (cf. Ernest, 1991b).
Against the background of this contention that mathematical learning is both a process of cognitive construction and of acculturation, consider a second aspect of the learning paradox discussed by Berieter (1985):
Out of the multitude of correspondences that might be noticed between one event and another, how does it happen that children notice just those that make for simple theories about how the world works – and that. furthermore. different children, with a consistency far beyond chance, tend to notice the same correspondences? (p. 204)
One could cite the literature on students’ conceptions to challenge the uniformity and consistency that Berieter attributes to students’ constructive efforts (Confrey, 1990b). However, for our purposes, it suffices to take the paradox at face value and merely add that the correspondences that children apparently notice are presumably those experienced as natural and self-evident by acculturated members of society. Berieter’s observation that a multitude of possible correspondences can be constructed leads one to question whether this experienced naturalness of one particular interpretation can be accounted for in terms of a direct contact with a prestructured environment. Why. for example. should the correspondence between pieces of wood scored with lines and the numerals “1,” “10,” “100,” and “1000” seem self- evident? This is indeed a conundrum if one subscribes to the view that external representations present one particular set of relationships to students in transparent form. Thus, as Dearden (1967) put it,
when a teacher presents a child with some apparatus or materials…he [or she] typically has in mind some one particular conception of what he [or she] presents in this way. But then the incredible assumption seems to be made that the teacher’s conception of the situation somehow confers a special uniqueness on it such that the children must also quite inevitably conceive of it in this way too. (pp. 145-146)
In making this comment, Dearden is questioning the plausibility of an approach that attempts to place learners in situations where the mathematical constructions that they naturally and inevitably make from the materials presented are correct as well as sensible.
The problem of explaining how students make constructions compatible with those that the expert has in mind seems intractable as long as we fail to make our self- evident interpretations of external representations an object of an analysis. We experience mathematical relationships as being readily apprehensible in external representations precisely because we assume that our interpretation of the materials is shared with everyone else who knows mathematics. In other words, the experienced transparency of instructional representations is a consequence of taken-asshared interpretations that constitute a basis for communication (cf. Bauersfeld, 1990). As long we continue to assume that these interpretations are self-evident, we do not consider the possibility that they might he but one of a variety of alternatives or that students might not see what we see. Further, if we assume without question that the relationships we have in mind are in the students’ environment waiting to be perceived, our only recourse when our initial attempts to bring the relationships to their attention are unsuccessful is to be increasingly explicit and spell it out for them. In doing so, we open ourselves to the possibility that the students will take form for substance and merely learn to behave in ways that convince us that they see what we consider self-evident (cf. Brousseau. 1984).
The paradox seems less ominous once we broaden our focus and acknowledge that the naturalness of certain interpretations is a product of our own acculturation into the mathematical practices of our society. We might then find some merit in Putnam’s (1981) contention that
signs do not intrinsically correspond to objects, independently of how those signs are employed and by whom. But a sign that is actually employed in a particular way by a particular community of users can correspond to particular objects within the conceptual scheme of those users…. Since the objects and the signs are alike internal to the conceptual scheme of description, it is possible to say what matches what. (p. 49)
In other words. the assumed pregiven, fixed correspondences between arbitrary symbols and the world that underpin objectivist semantic theory are constructed within a taken-as-shared interpretive framework that constitutes a basis of communication for members of a community (cf. Kaput. 1991). As Johnson (1987) put it.
some statements will correspond to the world more accurately, for our purposes. than others – some of them will be obviously true, others will be clearly false, and many will be problematic borderline cases. But in every case, this “correspondence” will always be relative to our understanding of our world for present situation) and of the words we use to describe it. (p. 203)
The anthropological stance that Putnam and Johnson take to account for the apparent naturalness of certain correspondences is compatible with several other analysts’ contention that mathematical activity is a social as well as a cognitive phenomenon (e.g., Lave. 1988a; Minick, 1989; Solomon, 1989; van Oers, 1990; Walkerdine, 1988). It is also compatible with recent developments in the philosophy of mathematics (e.g., Bloor, 1983; Kitcher, 1983; Tyrnoczko I 986a, 1986b) and in the philosophy and sociology of science (cf. Cobb. Wood, & Yackel. 1991). In addition, it sits well with the empirical finding that taken-for-granted mathematical practices can differ from one social group to another (e.g.. Carraher. Carraher, & Schliemann, 1987; D’Ambrosio, 1985; Saxe, 1989, 1991a). From this point of view, we would have no reason to assume a priori that the correspondences that acculturated members of society consider natural should also he self-evident to the uninitiated. Instead, we would acknowledge that the experienced naturalness of certain mathematical interpretations is relative to the taken-as-shared conceptual schemes that we have each actively constructed in the course of our mathematical acculturation. As a consequence. we would not characterize teaching as an activity in which we attempt to focus students’ attention on things we see in their environment in increasingly explicit ways. Instead, we would view it as an activity in which we guide students’ constructive efforts, thereby initiating them into taken-as-shared mathematical ways of knowing. Concomitantly, learning would be viewed as an active. constructive process in which students attempt to resolve problems that arise as they participate in the mathematical practices of the classroom. Such a view emphasizes that the learning-teaching process is interactive in nature and involves the implicit and explicit negotiation of mathematical meanings. In the course of these negotiations, the teacher and students elaborate the taken-asshared mathematical reality that constitutes the basis for their ongoing communication (Bauersfeld, Krummheuer. & Voigt. 1988).
In this characterization of the learning-teaching process. teachers as well as students modify their interpretations in light of their developing understandings of each others’ mathematical activity (Bauersfeld, 1980; Cobb, Wood, & Yackel, in press a: Voigt, 1985). This is not to deny that the teacher is necessarily an authority in the classroom in that only he or she can assess the potential fruitfulness of students’ individual and collective activity for their future learning. The teacher’s role involves. in part, making inferences about what he or she and the students can take-as-shared for the purposes at hand. His or herpotentially revisable assumptions about both consensual understandings and individual students’ conceptions constitute the background against which the teacher selects or develops instructional activities and initiates and guides discussions. In doing so, the teacher reformulates and thereby implicitly legitimizes selected aspects of students’ mathematical contributions, a process that Leont’ev (1981) and Newman. Griffin, and Cole (1989) call appropriation. In addition, the teacher might frame conflicting interpretations or solutions as a topic for discussion, thus encouraging students to explicitly negotiate mathematical meanings by engaging in mathematical argumentation. More generally, it is by capitalizing on students’ mathematical activity that the teacher initiates and guides the classroom community’s development of taken-as-shared ways of mathematical knowing that are compatible with those of the wider community. The paradox that Berieter raised then disappears. It is only a paradox if one separates knowledge from the knower and considers that learning is a process of apprehending mathematical relationships that are self-evident only to the initiated.
Pedagogical argument
Thus far, we have suggested that two general problems arise if the process of learning and teaching mathematics is approached from the representational perspective. First, we noted that it is difficult to explain how students can apprehend the mathematical relationships presented to them in instructional representations unless we attribute to them what it is that they are supposed to learn. Second. we argued that it is not at all clear how students manage to pick out just the relationships that are self-evident to the initiated from amongst the multitude of alternatives. We have also attempted to demonstrate that this is not a matter of purely theoretical interest. The instructional strategy of being increasingly explicit and spelling out what students are supposed to apprehend brings with it the danger that mathematics will become excessively algorithmatized at the expense of conceptual meaning. Microsociological analyses conducted by Bauersfeld (1980) and by Voigt (1985, 1989) document the manner in which this occurs in the course of ongoing classroom interactions when the teacher’s goal of ensuring that students make specific predetermined constructions is inflexible.
We now consider a third potential difficulty inherent in the representational approach, namely, that students might make a separation between their mathematical activity in school and in other settings. This issue becomes relevant once we note that in the representational approach, the expert mathematical concepts or proce‑dures that are to be the end point of the learning process are taken as a starting point. Typically, formal structural analyses of the expert’s mathematical ways of knowing are used to develop physical materials or diagrams that embody these formal structures in an apparently transparent way. As Treffers (1987) observed, most of these analyses “are satisfied with rather simple networks of knowledge structures, where the structural element is more conspicuous than the phenomenological aspect” (p. 291). Dienes blocks were, for example, developed as a means of presenting the formal structure of place value numeration in a readily apprehensible way. Following Fischbein (1987), we can observe that materials of this type are symbolic – they symbolize taken-as-shared concepts or procedures for mathematically acculturated members of the community. Consequently, it is not surprising that representational theorists consider that their expert interpretations of external representations are natural; the materials were specifically developed as expressions of these interpretations.
In this account, the external representation can be seen to serve as the medium through which the expert attempts to transmit his or her mathematical ways of knowing to students. As long as we fail to consider the potential relevance of communicative interactions in which the teacher initiates and guides the classroom community’s negotiation of mathematical meanings and practices, we will continue to interpret the instructional problem as that of developing new and improved ways to express and transmit mathematical relationships that are self-evident to the expert. We then concern ourselves with the development of instructional situations that are not directly related to the out-of-school settings in which students engage in mathematical activity. It is only after mathematical structures have been presented to students in finished form that are they taught to apply this knowledge to
situations that bear a closer resemblance to those that they might encounter outside the mathematics classroom.
As Resnick (1987) observed, this issue is not merely one of theoretical speculation.
It seems that children treat arithmetic class as a setting in which to learn rules, but are somehow discouraged from bringing to school their informally acquired knowledge about number… .The process of schooling seems to encourage the idea that… there is not supposed to be much continuity between what one knows outside school and what one learns in school. (p. 15)
Saxe (1991b) addressed this issue by considering how analyses of the solutions that children develop as they participate in out-of-school mathematical practices might inform mathematics instruction in the classroom. In his view, it is important “to engage children in a classroom practice that has the properties of the daily practices involving mathematics in which children show sustained engagement” (p. 22). The practice-linked mathematics that emerges might then serve as a starting point in the learning-teaching process. This approach is consistent with the observation that. historically, pragmatic, informal mathematical problem solving constituted the basis from which formal, codified, academic mathematics evolved (Ekeland, 1988; Tymoczko, 1986a). The approach that Saxe proposed therefore avoids what Freudenthal called the anti-didactical inversion wherein applications follow an attempt to teach formal mathematical structures (Gravemeijer, 1990). More generally, it is highly compatible with Freudenthal’s (1981) contention that the first instructional phase should involve an extensive phenomenological exploration in a range of specific situations that are experientially real for students. In contrasting this and the representational approach, Treffers (1987) argued that
a vast phenomenological exploration is not a luxury one can dispense with but, on the contrary, a sheer necessity. This necessity is shunned by the formal. logical approach…leading to a kind of mathematics detached from and isolated from reality. (p. 257)
The reality to which Treffers refers is the students’ phenomenological or experiential reality rather than an environment that contains preformed mathematical structures. In his view, the primary features that distinguish the two instructional approaches are
’embeddedness’ versus ’embodiment.; the naturally organizable versus the artificially organized matter: eliciting structuring activity in an everyday life or physical or imagined reality versus debasing mathematical structures and forcing them into an artificially created environment. (p. 275)
For us, these arguments imply that we should not presume to devise artificial settings that express our expert mathematical conceptions in transparent form. Instead, we should attempt to develop instructional situations in which the teacher can draw on students’ prior experiences to guide the negotiation of initial conventions of interpretation (cf. Hiebert & Carpenter, in press). It should be stressed that our goal here is not to advocate “everyday” mathematics in school as an end in itself, for the ways these instructional activities are interpreted in the social setting of the classroom must necessarily differ from everyday interpretations in which people typically invent ways to reduce or eliminate the need to calculate (cf. Lave, 1988a. b: Walkerdine. 1988). Instead, the initial taken-as-shared interpretationsestablished in the classroom constitute an occasion for students to progressively mathematize their experiences with the teacher’s initiation and guidance. The frequent appeals to constructivism that one finds in the literature then become “more than a credo. a nebulous thought, an elusive principle” (Treffers. 1987, p. 249).
In cognitive terms, instructional situations of the sort discussed above are of educational value to the extent that students’ informal activity constitutes an experiential basis that they can progressively mathematize by internalizing and interiorizing that activity (cf. Plage!, 1970). In anthropological terms. these situations are of value to the extent that students’ informal mathematical activities constitute a starting point from which the teacher can guide their problem-solving efforts and thus facilitate their acculturation into the mathematical ways of knowing of wider society. In general, the learning-teaching process in such an approach commences from students’ initial, informal mathematizations of hypothetical situations described in instructional activities. The possible end points are conceived of as the potential consequences of students’ constructive activities as the teacher guides the classroom community’s elaboration of a taken-as-shared mathematical reality. Instructional situations in which students are traditionally expected to apply previously acquired knowledge then come to be seen as settings in which they engage in constructive mathematical activity (Streefland, 1990). The possibility that students might separate their mathematical activity in school from that in other settings is therefore addressed directly by challenging the traditional separation between acquisition and application.
The dualism
We have critiqued the representational view of the learning-teaching process from cognitive, anthropological, and pedagogical perspectives to argue that it has deep-rooted theoretical difficulties and is at odds with current reform efforts in mathematics education. It is important that this discussion not be interpreted as a critique of the intentions and values of mathematics educators who implicitly or explicitly have adopted aspects of this theoretical perspective. In many cases, the educational goals they propose arc highly compatible with those advocated in reform documents. Thus. we should acknowledge that external representations are typically developed in an attempt to help students construct conceptual meaning. The thrust of our argument has been that rigid adherence to the representational view is at odds with many of these goals. including that of encouraging the development of conceptual meaning. In particular, these implications lead to both an excessive algorithmatization of mathematics and to the possibility that many students will continue to separate their mathematical activity in school from that in other settings.
We have suggested that the theoretical and practical difficulties of the representational view stem from the dualistic nature of the basic underlying, metaphor. At the outset, mathematics in students’ heads (internal representations) is separated from mathematics in their environment (external representations that arc transparent for the expert). The basic problem is then to find ways of bringing the two hack into contact. The history of western philosophy can be seen as a series of unsuccessful attempts to offer a solution to this problemin more general terms (von Glasersfeld, 1984). In our century, Dewey, Mead. Wittgenstein, and, more recently, European hermeneuticists (e.g., Gadamer, 1986; Habermas, 1979) and American neo-pragmatists (e.g., Bernstein. 1983, 1986; Putnam, 1987; Rorty, 1979, 1982) have all questioned whether this philosophical problem bequeathed to us by Descartes is worth posing. They each scrutinized the underlying assumption that the mind represents nature independently of history, culture, and human purpose and developed arguments to demonstrate that this assumption is unwarranted. As a consequence. they each rejectedany representational theory of knowledge. Dewey (1973) in fact called representational theories spectator theories of knowledge to emphasize that they ignore individual and collective human activity.
More generally, critics of the representational view of mind have suggested that the traditional epistemological concern with the relationship between knowledge and a world independent of our experience is misguided. in their view, the only world we will ever know is that in which we individually and collectively live our lives and go about our daily business. As Searle (1983) put it,
my commitment to “realism” is exhibited by the fact that I live the way that I do, drive my car, drink my beer, write my articles give my lectures, and ski in my mountains, Now in addition to all these activities…there isn’t a further “hypothesis” that the real world exists. (pp. 158-159)
Searle and other critics of the representational view would brand as neurotic philosophers’ traditional worries about an unknowable and therefore fictional world that is claimed to exist apart from the reality in which we act and interact. Thus, in Dewey’s view, philosophy will not recover itself until “it ceases to be a device for dealing with the problems of philosophers and becomes a method. cultivated by philosophers, for dealing with the problems of men [and women)” (1973, p. 73). From this perspective, our investigations should focus on the only reality we will ever know, that in which problems and dilemmas arise for people in the course of their ongoing activities. These investigations would treat people in general and mathematics teachers and students in particular as active constructors of their ways of knowing and as participants in social practices rather than as mirrors of a world independent of experience. history, and culture. Knowing would then be seen as a matter of being able to participate in mathematical practices in the course of which one can appropriately explain and justifying one’s actions (Cobb, 1990a; Greeno, 1991: Much & Shweder, 1978). From this perspective, the goal when analyzing certain mathematical acts of knowing would be to account for students’ developing ability to engage in and justify these acts. In doing so, we would have no reason to characterize mathematical knowing in terms of the degree of accuracy of internal representations.
We will shortly begin to outline a few initial aspects of an alternative to the representational view of mind in mathematics education. First, however. we address possible misunderstandings by briefly attempting to dispel what Bernstein (1986) called the Cartesian Anxiety: “Either there is some support for our being. a fixed foundation for our knowledge, or we cannot escape the forces of darkness that envelop us with madness, with intellectual and moral chaos” (p. 11)
To say that truth and knowledge can only be judged by the standards of the inquirers of our own day is not to say that human knowledge is less noble or Important. or more “cut off from the world” than we have thought. It merely says that nothing counts as justification unless by reference to what we already accept. and that there is no way to get outside our beliefs and our language so as to find some test other than coherence. (Finny. 1979. p. 178)
This is not a skeptical position. Epistemological skepticism feeds on the desire to find an ahistorical foundation for knowledge and gains its force from the very metaphor of the mind as a representing mirror that Dewey, Kuhn, Bernstein, Rorty, and others have questioned. Skepticism becomes as irrelevant as representationalism once we give up this metaphor and transcend the Cartesian either/or dichotomy. The only claim being made is that we should question all claims to have escaped from culture and experience by discovering a permanent, neutral framework that tells us what is true and what is false.
It should also be noted that this is not a position of absolute relativism (Barnes, 1982; Bernstein, 1983; Brannigan, 1981; Rorty, 1983). We are not denigrating the notions of truth, objectivity, or standards for concluding that some arguments and interpretations are better than others for particular purposes. The only contention being made is that claims of truth are justified “by appealing to those social practices which have been hammered out in the course of human history and are the forms of inquiry within which we distinguish what is true and false, what is objective and idiosyncratic” (Bernstein, 1986, p. 41, emphasis in the original). In short, such a view most emphatically does not mean that anything goes, that any interpretation is as good as any other, or that we are each free to construct our own private truths without constraint. In the case of mathematics teaching. forexample, the goal is still that students eventually construct correct or true mathematical understandings with the teacher’s guidance. We are merely questioning the viability of correspondence theories of truth and the soundness of explanations in which it is claimed that students do so by putting aside their internal representations for a moment to glimpse ahistorical, culture-free mathematical truths presented to them in external representations. Instead, we suggest that it is potentially more fruitful for our purposes as mathematics educators to view students as actively constructing mathematical ways of knowing that make it possible for them to participate increasingly in taken-as-shared mathematical practices. From this perspective, mathematical truth is accounted for in terms of the taken-as-shared mathematical interpretations, meanings, and practices institutionalized by wider society. The notion of mathematical truth is therefore dealt with pragmatically. We do not dwell on the traditional question of whether or not something is true in an ahistorical, culture-free sense but instead treat mathematical interpretations or solutions that are considered to be true by members of a community as being practically true in particular situations. In the case of a classroom community, a central issue of interest for us is how students each actively learn as they participate in the community’s establishment of mathematical truths.
The social and cognitive aspects of mathematical learning
We have argued that the representational view leads to the intractable problem of bridging the separation made between mathematics in students’ heads and mathematics in students’ environment. The situation becomes more tenable when we view both experts and students as active interpreters who attempt to make sense of their worlds. We are then in position to reconsider whether it is reasonable for pedagogical purposes to assume that experts’ interpretations of instructional representations reveal fixed mathematical structures located in the students’ environment in the form of external representations. In addressing this issue, we would agree with Blumer (1969) that
the contention that people act on the basis of the meaning of their objects has profound methodological implications. It signifies immediately that if the scholar wishes to understand the actions of people it is necessary for him to see their objects as they see them. Failure to see their objects as they see them or a substitution of his [or her] meanings of the objects for their meanings, is the gravest kind of error that the social scientist can commit. It leads to the setting up of a fictitious world. Simply put, people act towards things on the basis of the meaning that these things have for them, not on the basis of the meaning that these things have for the outside scholar. (pp. 50-51)
Such objects include instructional materials that are described as external representations. If we accept Blumer’s argument, then the distinction is not between students’ internal representations and mathematical relations contained in external representations but between the students’ interpretations of instructional materials and the taken-as-shared interpretations of mathematically acculturated members of the wider community. This latter distinction is nondualistic in that we do not have to explain how knowledge projected into the environment by an expert interpreter somehow gets inside students’ heads. The challenge is to explain how students construct their mathematical ways of knowing as they interact with others in the course of their mathematical acculturation (Bruner & Haste, 1987; Rogoff, 1990). In addressing this issue, we need not appeal to a process of glimpsing a mathematically prestructured environment but can instead analyze how teachers and students mutually influence and adapt to each others’ mathematical activity as they interact in instructional situations. Such an analysis would emphasize that mathematics is both a collective human activity (what DeMillo, Lipton, and Perlis, 1986, call a community project) and an individual constructive activity. It is in fact because we want to bring to the fore the individual as well as the collective aspect of mathematical activity that we have consistently spoken of taken-as-shared rather than of shared mathematical meanings and practices. On the one hand, the teacher and students implicitly and explicitly negotiate intersubjective interpretations of their mathematical activity in particular situations, some of which might involve the use of instructional representations. On the other hand, when we focus on the teacher’s or individual students’ interpretations it becomes apparent that they do not have direct access to each others’ mathematical experiences and consequently have no way of knowing whether their individual interpretations of a situation actually correspond to those of others (von Glasersfeld, 1990).We therefore view the process of attaining intersubjectivity as one in which the teacher and students each construct individual interpretations that they take as being shared with the others. In accounting for the experience of intersubjectivity in this way, we are suggesting that successful communication requires only that individual interpretations are compatible or fit for the purposes at hand in that possible discrepancies do not become apparent i n the course of ongoing social interactions (Mehan & Wood, 1975; Schutz, 1962; von Glasersfeld, 1983).
It should be noted that in advancing in this account, we are treating intersubjectivity as a reflexive phenomenon. On the one hand, speakers typically formulate their communicative acts in order to be understood as they intend – they act on the assumption that intersubjectivity will be attained. On the other hand, it is by making this assumption and attempting to communicate that discrepancies in individual interpretations become apparent. Previously unquestioned aspects of the taken-as-shared basis for communication can then become the object of explicit negotiation, in the course of which the participants modify their individual interpretations as they attempt to achieve intersubjectivity. Thus, in this view, intersubjectivity has to be assumed in order to be attained (Mehan, 1979; Rommetveit, 1986).
As a further point, the distinctions we have made between fit and match and between taken-as-shared and shared mathematical meanings and practices are more than philosophical quibbles. On the one hand, the establishment of intersubjective mathematical meanings and practices is crucial for communication in the classroom. It is therefore essential to initiate the explicit negotiation of conventions of interpretation when instructional representations are introduced (Solomon, 1989), and this is what most teachers do despite the suggestion by some that the materials are transparent. On the other hand, the tensions between the mathematical interpretations made by different members of the classroom community. whether explicit or not, are a crucial source of learning opportunities in all instructional situations including those that involve the use of instructional representations (Yackel, Cobb, & Wood, 1991). In other words, both the explicit problems and conflicts that arise in the course of social interactions and the generally unnoticed mutual appropriations of meanings that occur in any communicative interaction serve as occasions for individual students’ constructive activities. This being the case, an account of a student’s mathematical learning in the classroom should consider the development of both the taken-as-shared, communal meanings and practices and the individual student’s personal meanings and practices. It should, for example, then he possible to explain how learning occurs in instructional situations when, as frequently happens, students working together make differing interpretations and yet each assumes that the others each share his or her personal interpretation (Cobb, Wood, & Yackel, in press b). Such an explanation would not attribute to the students the ability to peek around their internal representations and glimpse pregiven mathematical relationships. Instead, it would focus on the processes by which students learn as they reciprocally adapt to each others’ and the teacher’s mathematical activity, thereby establishing taken-as-shared mathematical meanings.
Thus far, we have emphasized that learning opportunities arise for individual students as they participate in classroom social interactions. This focus on interaction as a catalyst for individual mathematical development does not, however, do justice to the full extent of the social aspect of mathematical learning. If we consider the collective as well as the individual nature of mathematical activity, we can note with Solomon (1989) that the mathematical problems students attempt to solve and the solutions they construct in the course of their development have a social as well as a cognitive aspect. As an illustration, we consider how one particular manipulative is used in second-grade classrooms where we collaborate with teachers to develop alternatives to traditional American textbook instruction.
The instructional materials of interest are collections of 100 Unifix cubes stored in bars of ten. The students in these classrooms are expected to decide when the use of these materials might help them as they attempt to complete a variety of arithmetical tasks while working in pairs (Cobb. Wood, Yackel, Nicholls, Wheatley, Trigatti, & Perlwitz, 1991). The decision to make these materials available to students is justifiable because almost all the children can create and count abstract units of one when they solve a wide range of arithmetical tasks at the beginning of the school year. The construction of units of this type is indicated by solutions that carry the significance of double counting in that they imply that the children experience units of one as entities that can themselves be counted (Steffe, von Glasersfeld, Richards, & Cobb, 1983). The construction of ten as a conceptual unit composed of individual units of one is therefore a development possibility for them (Steffe, Cobb, & von Glasersfeld, 1988).
A representational analysis of the second graders’ developing ability to count the bars of Unifix cubes by ten would presume that each bar is a transparent external representation of ten as a numerical unit and argue that children recognize the correspondence between these embodied units and the number words ten, twenty, thirty, etc. In contrast to this account, which projects the expert’s interpretation of the bars into the children’s environment, an alternative analysis focuses on the inferred quality of the children’s experiences as they count the cubes that compose the bars. First, it can be observed that the rhythmic nature of the children’s counting activity is momentarily interrupted as they complete the count of one bar and begin to count the next. In other words, the instructional materials made available by the teacher give rise to a brief experiential pause when the children reach ten, twenty, thirty, etc. as they count by ones. In addition, the children have previously constructed the number word sequence (but not necessarily the numerical counting sequence) “ten, twenty, thirty,…” by abstracting regularities from their forward number sequence in the course of their prior mathematical acculturation. It is therefore possible that the pauses that arise when counting the bars by one might become significant within the experiential worlds of the children. In particular, they might experience the unintentioned production of the number word sequence of “ten, twenty, thirty…” as a surprise that could lead them to reflect on their counting activity (cf. von Glasersfeld, 1987). They might then realize that this familiar number word sequence signifies the result of counting the cubes that compose each of the bars. in coming to this realization, they would in effect have interpreted each bar as signifying a composite unit of ten (Steffe. Cobb, & von Glasersfeld. 1988). This conceptual construction would involve the internalization of the activity of counting by one and would be indicated by a curtailment of that activity as the children begin to count the bars “ten, twenty, thirty….” when they use them to solve a variety of arithmetical tasks.
For ease of explication, we have spoken thus far as though the children act in isolation from each other and the teacher. The need to complement this account by locating the children’s mathematical activity in the social context of classroom life is indicated by our tentativeness when we described their possible constructions. We would also note that the construction of ten as composite unit is but one of several alternative possibilities. For example. we have observed several children who begin tocount the cubes that compose the bars by twos rather than tens, indicating that they have constructed two rather than ten as a composite unit. (Such observations are problematic from the representational perspective given its emphasis on the transparency of instructional materials).
Despite our tentativeness and our acknowledgment of alternative developmental possibilities, all the children in the classrooms in which we work eventually come to routinely count the bars by ten. To account for this observation, we focus on the social interactions that occur during the whole-class discussions that follow smal I- group work. Some of the children routinely count the bars by ten as soon as they are made available and the second-grade teachers typically appropriate this aspect of the students’ solutions when they redescribe students’ explanations and justifications in terms that they infer other students might understand (Cobb, Yackel, Wood, & McNeal, in press; Wood, Cobb, & Yackel, 1991). Metaphorically, it is as though the teachers give a running commentary on the students’ contributions from the perspective of those who can judge which aspects of students’ mathematical activity might be significant for their future mathematical learning and acculturation. As a consequence. the act of creating units of ten becomes an increasingly explicit topic of conversation with the teachers’ guidance. Learning opportunities therefore arise for children who initially count by ones as they attempt to make sense of others’ mathematical activity and participate in interactions in which others appropriate their activity.
The contention that learning opportunities of this type arise in the course of whole-class discussions is supported by observed differences between children’s solutions during small-group work and those that they subsequently explain in whole-class discussions (Yackel, 1991). More generally, the whole-class discussions in the second-grade classrooms are as much about constructing novel solutions, often while interacting with the teacher or peers, as they are about reporting previously completed solutions (Krummheuer. 1991). Consequently, it is reasonable to conclude that the teachers facilitate the development of individual children’s interpretations of and actions with the bars when they make interventions of the sort outlined above. For their part, the children contribute to the evolution of their classroom community’s basis for mathematical communication when they make these conceptual constructions. Thus, when we focus on the classroom community rather than individual children, the whole-class discussions are at least in part situations for the institutionalization of mathematical ways of knowing compatible with those of wider society (cf, Balacheff, 1990a). It is as a consequence of this process of institutionalization that the interpretation of the bars as signifying composite units of ten becomes established as a mathematical truth for the classroom community. As far as the students are concerned, this taken-as-shared interpretation is then beyond question; in the world of their experience, the bars simply are tens. Thus, like adherents to the representational view, they take it for granted that the mathematical relationships they see are self-evident properties of the instructional materials. As we have already noted, this ontological commitment is an essential aspect of meaningful mathematical activity. We do, however, question the viability of explanations that account for this commitment in terms of the ability to glimpse pregiven mathematical objects.
The account we have given of the process by which the second graders eventually come to interpret the bars in ways compatible with the expert acknowledges that their accomplishment involved internalizations. These are not, however, internal‑izations of the mathematical relationships that are a salient property of the bars for the expert Instead, the students internalize and interiorize their mathematical activity as they attempt to resolve situations that they find problematic while participating in classroom mathematical practices such as counting. In doing so, they simultaneously contribute to the classroom community’s establishment of a mathematical truth. The analysis we have presented also illustrates that what counts as a problem and as a conceptual advance has a social aspect even at the most elementary levels of mathematics. In effect, mathematical problems arise for students against the background of the ways of knowing that they have actively constructed in the course of their prior mathematical acculturation. To the extent that they make conceptual advances while resolving these problems, they participate in the elaboration of taken-as-shared mathematical understandings. and these advances constitute the background against which further problems arise. This analysis takes on added significance in the current climate of reform that characterizes mathematical learning as in large part a communicative, problem-solving process (cf. Confrey, 1990c; National Council of Teachers of Mathematics, 1989; Thompson, 1985; von Glasersfeld, 1987).
More generally, our discussion of the cognitive and social aspects of mathematical knowing has involved an attempt to coordinate three points of reference – individual students’ personal mathematical ways of knowing, the taken-as-shared mathematical practices of the classroom community, and the taken-as-shared mathematical practices of wider society. It is the problem of coordinating these complementary perspectives rather than that of reconciling the poles of the Cartesian dualism that faces us if we reject the representational view of mind (Balacheff, 1990h; Cobb, 1989, 1990b; Maturana, 1980; Smith & Confrey, 1991). In analyzing the illustrative example, we focused on the relationship between students’ interpretations and classroom community’s taken-as-shared mathematical practices. It is therefore as well to add that, in the new scheme of things, the taken-as-shared mathematical practices of wider society are privileged in the sense that they delimit the possible long-term cognitive goals that might be formulated for students’ mathematics education. Ideally, teachers’ knowledge of these potentially revisable goals and their understanding of their students’ developmental possibilities inform their practices as they plan for instruction and interact with students in the classroom (Lampert. 1990). For example, the second-grade teachers with whom we work typically draw on these understandings as they attempt to capitalize on students’ contributions in a flexible, highly situated manner. In doing so, they both facilitate individual students’ acculturation into the mathematical practices of wider society and guided the classroom community’s establishment of taken-as-shared mathematical ways of knowing that are compatible with those practices.
Pedagogical symbol systems
As the discussion in the previous section indicates, we see value in some instructional materials that have the characteristics of what are traditionally called instructional or external representations. From the constructivist perspective we have outlined, the role of these materials is very different from the traditional attempt to present mathematical relationships to students in a readily apprehensible form. To further clarify our position. we again observe that so-called instructional representations are symbolic – they are typically developed to symbolize the taken-as-shared mathematical interpretations of wider society. In discussing the possible educational value of such materials, we will therefore view them as possible means that students might use to symbolize their developing mathematical activity. Further, we will call them pedagogical symbol systems rather than instructional representations to emphasize their symbolizing role in individual and collective mathematical activity.
We have suggested that initial instructional activities that constitute starting points for students’ mathematical constructions should satisfy two constraints. On the one hand, they should make it possible for the teacher to draw on students’ prior experiences when guiding the negotiation of initial conventions of interpretation. Activities that are traditionally viewed as applications problems might he appropriate in this regard. On the other hand, students’ interpretations of and actions in these situations should constitute highly situated, intuitive bases from which they might abstract as they construct increasingly sophisticated mathematical conceptions. These points can be illustrated by outlining a sequence of instructional activities called “The Candy Factory” developed in the course of a third-grade classroom teaching experiment conducted in collaboration with Koeno Gravemeijer. Our intent was to facilitate students’ construction of increasingly sophisticated conceptions of place-value numeration and increasingly efficient multidigit addition and subtractional algorithms. To this end. the teacher first described the scenario of a candy factory in which candies were packed into rolls, rolls into boxes. and boxes into cases. Classroom observations indicated that this scenario was experientially real for students and that the compatibility of their interpretations constituted an initial basis for communication. The first issue posed by the teacher in the whole- class setting was to decide how many candies to put in a roll, how many rolls to put in a box, and soon, with the constraint that the people in the factory wanted to figure out easily how many candies were in the factory storeroom. This constraint was experienced as reasonable by the students, and they made several alternative suggestions including that of repeatedly packing by tens. The teacher acknowledged that the various proposals were justifiable and explained that the people in the factory decided to pack the candies in tens. In doing so. she did not funnel the students to the solution she had in mind. but instead introduced this way of packing as a convention in the factory. In explicitly negotiating these initial conventions of interpretation, the teacher and students further elaborated their basis for communication.
The teacher next gave pairs of students bags of Unifix cubes and asked them to pretend that they worked in the factory. The task was to pack the “candies.” In this situation, the individual cubes and the bars the students made served a dual signifying function. On the one hand, they signified candies and rolls in the experientially real scenario of the candy factory. On the other hand. they signified units of one and of ten for the students as a consequence of their problem-solving activity in second grade. Consequently, these interpretations of the packing activity were immediately taken as shared. and the students spontaneously counted by tens and ones to find out how many candies they had packed.
With the teacher’s guidance. the pairs next pooled their “candies” to make boxes and cases. The teacher then raised a variety of issues including the number of candies in a box, the number of rolls in a case, and so on. These tasks gave rise to genuine mathematical problems for many students, and the mathematical arguments that they gave to explain and justify their responses reflected a range of qualitatively distinct conceptual interpretations of the situation. For example, some gave purely numerical explanations that indicated they could conceptually coordinate abstract arithmetical units of different ranks. Others gave explanations that indicated they coordinated arithmetical units by re-presenting packing activity. Still others were yet to internalize their activity in this way and actually packed or unpacked candies in order to find the total number of candies in rolls, boxes, etc. The teacher for her part appropriated the students’ explanations as she described purely numerical explanations in terms of packing activity and vice versa as guided by her knowledge of her students’ conceptual understandings. In doing so, she facilitated their individual conceptual development and initiated the classroom community’s gradual establishment of increasingly sophisticated taken-as-shared interpretations.
Our reason for describing these initial instructional activities is not to claim that they constitute an exemplary solution to place-value instruction. We do. however. contend that students’ differing interpretations of and actions in these situations constituted personally meaningful contexts from which they subsequently abstracted and constructed increasingly sophisticated arithmetical conceptions and algorithms. This is not to say that the situations were presented ready-made to students. Instead. students’ interpretations and actions contributed to the co- construction of the situations in which they learned. We would also note that the initial activities make sense mathematically when we take as our point of reference the taken-as-shared meanings and practices of wider society. Given our observations of students’ mathematical activity and our inferences about their differing interpretations, the pedagogical challenge for us was then to facilitate their reflective abstraction from, and progressive mathematization of. their initially situated activity. Here again. pedagogical symbol systems proved to be of value when analyzed in both cognitive and social terms.
For example, in one activity, students were given an array of small circles that signified the candies in the storeroom and were asked to make a drawing to show how they would pack the candies. In this activity, which was routine for all students. the act of circling a collection of candies signified the conceptual act of creating a unit of ten or one hundred. Once the students’ solutions had been discussed. the teacher explained that a shop had ordered a specified number of candies and asked the students to show how they would repack the candies in the storeroom so that they could fill the order. The array and the number of candies to be sent had been chosen so that the students would have to develop a means of symbolizing the act of unpacking candies (or of decomposing composite units). This activity was genuinely problematic for many students and the teacher appropriated explanations that reflected differing conceptual understandings in whole class discussions of students’ interpretations and solutions.
In subsequent instructional activities. the classroom community developed a taken-as-shared way of making freehand drawings to symbolize transactions in which candies were moved into the storeroom from the factory or sent out to a shop. It should be stressed that the students were free to decide whether to make a drawing as they completed instructional activities. Thus, the making of drawings was not practiced as an end in itself, but instead served as a possible way to make sense of events in the factory. Later, an inventory form was introduced as the means by which the factory manager decided to record transactions into and out of the storeroom. Conventions for symbolizing these transactions and the composition or decomposition of numerical units were explicitly discussed, and students eventually constructed and used several different efficient addition and subtraction algorithms.
Both the classroom observations and individual interviews suggested that students gave their overt acts of manipulating numerals on the inventory form a variety of different meanings. In other words, the interpretations they gave to their own and others’ mathematical activity did not necessarily match, but were instead compatible for the purposes at hand (Glasersfeld, 1990). It is therefore reasonable to say that the evolving basis for mathematical communication was taken as shared rather than shared. For example, some students’ symbol-manipulation acts signified purely numerical transformations, indicating that they had interiorized the acts of creating numerical units by packing or unpacking candies. Others seemed to give meaning to their manipulation of numerals by re-presenting a packing and unpacking activity. It would therefore seem that these children had internalized but not interiorized the acts of creating experientially real numerical units in the situation of the candy factory. Partly for this reason. another scenario that involved pedagogical symbol systems was developed and coordinated with activity in the candy factory setting to further facilitate the students’ conceptual constructions.
As we step back from the discussion of the candy factory scenario, we can first note that our treatment of pedagogical symbol systems is at least partially compatible with several other analyses of learning in instructional situations that emphasize the experiential aspects of mathematical knowing. These include Fischbein’s (1987) discussion of intuitive models, Tail’s (1989) illustrations of the role of generic organizers, Pimm’s (1981, 1987) analysis of metaphors, and Clement’s (1983) findings concerning the role of analogies in scientific and mathematical problem solving. The discussion of the candy factory scenario also illustrates the more general point that pedagogical symbol systems can be viewed as ways in which students might choose to express their mathematical thinking as they complete instructional activities. In cognitive terms, students who make these ways of symbolizing their own can create physical or iconic expressions of their mathematical interpretations, which they can then act on to solve tasks that would otherwise have been beyond their capabilities. In so doing, they simultaneously create records of the results of their mathematical activity. The crucial point for our purposes is that these expressions and records can support students’ re-presentation of and reflection on their activity and thus their construction of increasingly sophisticated mathematical conceptions (Laborde, 1990; Steinbring, 1989; Treffers, 1987).
In social terms, the development of taken-as-shared ways of symbolizing mathematical activity by the classroom community clearly facilitates mathematical communication. For example, students in the third-grade classroom manipulated physical materials or drew figures and diagrams as they explained and justified their mathematical interpretations and solutions. More generally, pedagogical symbol systems can be viewed as aspects of didactical situations in which the teacher initiates and guides the classroom community’s negotiation of mathematical meanings and practices. These negotiations might have as their theme ways of interpreting the materials, ways of acting with the materials, or ways of interpreting those symbolic actions. In the latter case, the teacher and students in effect create ways of symbolizing their symbolic actions and thus progressively mathematize those actions. This is what happened repeatedly in the candy factory scenario. In Walkerdine’s (1988) sociolinguistic terms, one can in fact speak of the classroom community’s establishment of taken-as-shared chains of signification. Further, if one focuses on the products of students’ constructive activities, then our analysis of the candy factory scenario can be seen to illustrate Kaput’s (1987b) observation that much of mathematics involves representing one mathematical structure by another.
In summary, we might say that materials typically characterized as instructional representations are of educational value to the extent that they facilitate students’ individual and collective constructive activities and thus their increasing participation in the mathematical practices of wider society. In this view, correctness does not mean conforming to the dictates of an authority who spells out his or her own interpretation. Instead. it means making mathematical constructions that have clout in that they enable students to increasingly participate in socio-historically evolving mathematical practices (Bruner, 1986). Moreover, the notion of instructional materials as a means of delivering mathematical knowledge to students is displaced by the view of teachers initiating and guiding emerging systems of mathematical meanings and practices in their classrooms.
Students, teachers, and mathematics
The discussion of the candy factory scenario indicates that our critique of the representational view of mind is not merely a theoretical matter but that, on the contrary, it has significant practical implications. We now turn to consider a further issue that has both theoretical and pragmatic relevance for the mathematics education community, that of integrating research on learning and research on teaching mathematics (e.g., Cobb, Yackel, & Wood, 1991; Ferinema, Carpenter, & Larrion, 1991). The current interest in this topic can be viewed as an attempt to transcend what Bauersfeld (1988) called the classical didactical triad of the teacher, the student, and mathematics. As Bauersfeld observed, reform efforts in mathematics education have typically concentrated on one of the legs of this triad at the expense of the other two. He further noted that this tendency and, more generally, the didactical triad itself are consistent with a perspective that separates mathematical knowledge from the knower. Together with Bauersfeld, we contend that we can go beyond the usual limits of the classical triad only if we reject the metaphor of the mind as a mirror that reflects a mathematically prestructured environment unaffected by individual and collective human activity. As an alternative to the assumption that mathematical learning is a matter of somehow acquiring accurate internal representations of such an environment, we have suggested that students actively construct their mathematical ways of knowing as they are initiated into the taken-as-shared mathematical practices of wider society by the teacher. More specifically, we have contrasted the notion that students apprehend mathematical relationships contained in external representations with the view that the teachers and students negotiate taken-as-shared ways of interpreting and acting with pedagogical symbol systems. In this latter characterization, the activities of teaching and learning are intrinsically linked in that the teacher and students mutually influence each others’ activities (Bauersfeld, 1980). For example, we had to consider the teacher’s interpretations and actions when we discussed the second graders’ learning as they used the bars of Unifix cubes. More generally, an account of students’ activity involves inferences about their interpretations of others’ actions. As the discussion of the candy factory scenario illustrates, we would also want to identify the taken-as-shared meanings and practices that make mathematical communication possible and to isolate differences in personal meanings that give rise to explicit and implicit negotiations. In short. one would necessarily attend to students’ learning even if’ the primary objective is to explain the teacher’s activity (and vice versa) in that neither is seen to exist without the other once we give up the representational view. Consequently, analyses that focus on selected aspects of either the teacher’s or students’ behavior in isolation from the classroom social context that they co-construct in the course of their interactions are considered to be of limited value from the perspective we are advancing.
The remaining leg of the classic triad. mathematics. is sometimes viewed as content contained in the curriculum. Forexample, external representations as items in a curriculum are said to contain readily apprehensible mathematical relationships. As an alternative that unites mathematics and the knower, we have suggested that it is useful for educational purposes to characterize mathematics as both an individual and a collective human activity. For example, in the case of the candy factory scenario, we did not project our expert place value interpretation of candies, rolls, boxes, and cases into the instructional materials and assume that it was a salient property waiting to be apprehended by students. Instead, we focused on the ways that students interpreted and acted with pedagogical symbol systems as they negotiated mathematical meanings with others in the classroom. In doing so, we attempted to illustrate that active construction and the process of acculturation into the taken-as-shared practices of wider society are but different sides of the same coin. As a consequence, what is traditionally called the content could be seen to emerge in the course of the teacher’s and students’ interactions as the teacher guided both the individual student’s constructive activities and the evolution of the classroom community’s taken-as-shared meanings and practices. Such an account sidesteps the intractable problem of explaining how mathematics in students’ heads and mathematics located in their environment can possibly come into contact with each other.
Beyond the representational view
Throughout this paper, we have attempted to demonstrate that mathematical knowing has a social as well as cognitive aspect in that to know is to be able to participate in a social practice. It is against this background that we consider one final issue, the apparently common belief that constructivism as a theoretical position implies that mathematical learning should be a process of spontaneous. unguided, independent invention. In this interpretation of constructivism, a maxim that applies to the learning process, namely. that students must necessarily construct their own mathematical ways of knowing. is treated as a direct instructional recommendation. Constructivist theory is then interpreted to imply that students’ learning should be natural and that teachers should not tell them anything as they attempt to make sense of their worlds. Typically, this characterization of constructivism is contrasted with an alternative in which the teacher attempts to specify mathematical relationships for students, often by using instructional representations.
It should be clear from the arguments we have made that we do not believe that mathematical learning can ever be natural. if by natural we mean the unconstrained organic growth of mathematical knowledge independent of social and cultural circumstance. Further, the conclusion that teachers should not attempt to influence students’ constructive efforts seem indefensible, given our contention that mathematics can be viewed as a social practice or a community project. From our perspective, the suggestion that students can he left to theirown devices toconstruct the mathematical ways of knowing compatible with those of wider society is a contradiction of terms. A teacher who actually took this caricature of constructivism seriously would be abrogating his or her obligations to students, to the school as a social institution, and to wider society. Given the absurdity of this caricature from our perspective, it is of interest to understand why it might seem reasonable, given the assumptions that underlie the representational view of mind.
Recall that, from the representational perspective, learning is a process of acquiring accurate mental representations of fixed mathematical structures, relationships, and the like that exist independently of individual and collective activity. Within this frame of reference, the teacher is in a privileged position because he or she has apprehended these external mathematical structures, whereas the students are yet to do so. If we accept these assumptions, our instructional options are relatively limited as we attempt to make the mathematical structures apprehensible to students. We can of course develop external representations to make the mathematical structures as transparent as possible. The only other obvious variable is the extent to which we might spell out the mathematical relationships that students are to apprehend. At one extreme, we might be as explicit as possible in a manner compatible with behaviorism, whereas at the other extreme we might be radically noninterventionist in a manner compatible with romanticism. Given that the only instructional options that come to mind are located somewhere on this continuum as long as we adhere rigidly to the representational view, it seems plausible toequale the noninterventionist extreme with constructivism. To construct then means to learn by spontaneously apprehending fixed mathematical relationships without the teacher’s guidance. Again, we would emphasize that this can be interpreted as a plausible learning process only from a perspective that considers that mathematical structures exist independently of human activity, culture, and history.
In the view that we are advancing, serious consideration of what life in mathematics classrooms could be like requires that we go beyond the continuum of instructional options that follows from the assumptions of the representational view. Drawing on the work of others, but most notably that of Bauersfeld, Krummheuer, and Voigt (1988) and von Glasersfeld (1987), we have proposed the metaphor of mathematics as an evolving social practice that is constituted by, and does not exist apart from, the constructive activities of individuals as an alternative to the metaphor of mind as a mirror. We have further argued that this alternative metaphor allows us to go beyond the Cartesian dualism, and thus beyond the relatively narrow continuum of instructional options that follows from representational theory. Once we make this break with the representational view, we do not distinguish between instructional approaches on the basis of whether or not they allow students to construct mathematical knowledge. Instead, we contend that students must necessarily construct their mathematical ways of knowing in any instructional setting whatsoever. including that of traditional direct instruction. In the latter case. the teacher might, for example, tell the students the mathematical rules or procedures they are to follow and demonstrate how the rules are to be used. Even here, the students have to make sense of their worlds as they actively interpret what the teacher says and does, and they can only do so in terms of their current ways of knowing. The central issue is not whether students are constructing, but the nature or quality of those constructions.
Once we break with the representational view, we can observe that when we talk of students’ constructive activities we are emphasizing the cognitive aspect of mathematical learning. It then becomes apparent that we need to complement the discussion by noting that learning is also a process of acculturation. Once we do so, we begin to understand how it is possible for students to construct for themselves the mathematical practices that, historically, took several thousand years to evolve. In particular, we concur with the following comment of Freudenthal.
The young learner recapitulates the learning process of [humankind]. though in a modified way. He [or she] repeats history not as it actually happened but as it would have happened if people in the past would have known somewhat like we do now. It is a revised and improved version of the historical learning process that the young learner recapitulates…. ’ Ought to recapitulate’ we should say. In fact we have not understood the past well enough to give them this chance to recapitulate. [Quoted by Steiner, 1989. p. 281
The research reported in this paper was supported by the National Science Foundation under grant No. MDR 885-0560 and by the Spencer Foundation. The opinions expressed do not necessarily reflect the views of the foundations.
Several notions central to this paper were elaborated in the course of discussions with Heinrich Bauersfeld, Götz Krummheuer, and Jörg Voigt at the University of Bielefeld, Germany. The authors are also grateful to Jere Confrey and Guershon Harel for numerous insightful comments made on previous drafts.
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