CEPA eprint 2968

Information-processing psychology and mathematics education: A constructivist perspective

Cobb P. (1987) Information-processing psychology and mathematics education: A constructivist perspective. Journal of Mathematical Behavior 6(1): 3–40. Available at http://cepa.info/2968
Table of Contents
Computer Modelling
Formal Domains
Relevance, Common Sense, and Context
Sensory-Motor Action and Mathematical Meaning
Mathematical Cognition and Computer Simulation
Cognitive Behavior and Mathematical Experience
Social Life
Mathematical Practices
Instructional Implications
Possible Rapprochements
A distinction is made between weak and strong research programs in cognitive science, the latter being characterized by an emphasis on the development of runnable computer programs. The paper focuses on the strong research program and initially considers situations in which it claims to have advanced our understanding of mathematical activity. It is concluded that the program’s characterization of students as environmentally driven systems leads to: (a) a treatment of mathematical activity in isolated, narrow, formal domains; (b) a failure to deal with relevance, common sense, and context, and (c) a separation of conceptual thought from sensory-motor action. Taken together, these conclusions imply a failure to deal adequately with the issue of mathematical meaning. In general, the program’s primary focus appears to be on programmable mechanisms rather than fundamental problems of mathematical cognition. The purview of the discussion is then widened to consider the strong program’s difficulties in dealing with social interaction, intellectual communities, and the hidden curriculum. It is noted that instructional implications derived from this program typically involve the organization of mathematical stimuli that make explicit or salient the relevant properties of a propositional mathematical environment. Finally, it is argued that some members of the strong program have recently acknowledged that it has limitations. The possibility of a rapprochement in which the strong program is supplanted by a form of social constructivism is discussed.
The computer metaphor has increasingly become the dominant mode within which to characterize students’ mathematical cognitions in the United States in recent years. As Searle (1984) noted, the metaphorical use of technological advances to understand thought processes has a long history:
In my childhood we were always assured that the brain was a telephone switchboard (‘what else could it be?’). I was amused to see that Sherrington, the Great British neuroscientist, thought that the brain worked like a telegraph system. Freud often compared the brain to hydraulic and electromagnetic systems. Leibniz compared it to a mill (p. 44).
Johnson-Laird (1983) summarized the proposal for the latest technological metaphor when he claimed that:
… if both Turing’s thesis and functionalism are correct, any future theory of the mind will be completely expressible within computational terms. The computer is the last metaphor; it need never be supplanted (p. 10).
The central theme of this chapter will be to explore the value of this proposal for mathematics education. In doing so I will necessarily write as a member of the American mathematics education community. As van Oers (this issue) observed, the information- processing approach first emerged as a coherent, theoretically-ground research program in American psychology. Given the historically close relationship between psychology and mathematics education, it is not surprising that the influence of the computer metaphor is greater in the United States than in mathematics education communities in other countries. This influence extends beyond immediate implications such as writing computer programs to model mathematical cognition. New terms have become part of our vocabulary, others have shifted in meaning, and the metaphor has taken on an ontological significance. We are processors of information, teachers are decision-makers, students store information in their long term memories, memory is a warehouse for pieces of knowledge, thought is a process of manipulating symbols, and so forth. These initially tentative claims have almost become institutionalized as taken-for-granted facts that constrain the very ways we conceptualize the learning and teaching of mathematics.
Within the American scene, a crude dichotomy can be made between what might be called the weak and strong information-processing programs in mathematics education. This use of the terms “weak” and “strong” is borrowed from sociology of science. The terms refer to the force of the claims made about the value of the computer metaphor for understanding mathematical cognition. In this context, “weak” does not necessarily mean that a research program is feeble, unfit, or lacking in vigour. Rather, it implies that claims made are comparatively moderate and, I will argue, pragmatic. Conversely, “strong” implies that claims are extreme by comparison, not that the research program is necessarily healthy or progressive.
The weak program is perhaps most associated with the work of Minsky (1958), Lawler (1981), and Davis (1984). This work has both a phenomenological flavor in that the focus is on mathematical experience (Cobb, 1987a) and a pragmatic emphasis in that the computer metaphor is used as one possible source of explanatory analogies amongst others. The strong program is exemplified in the work of Anderson (1983), Van Lehn (1986), Brown and Burton (1978), Greeno (1987), Briars and Larkin (1984), and Riley, Greeno, and Heller (1983). In this program, the computer metaphor is the central if not the only source of analogies and, ideally, cognitive models should take the form of runnable computer programs. In contrasting the weak and strong programs, it can be argued that there is a trade-off between accounting for the subjective experience of doing mathematics and the precision inherent in expressing models in the syntax of computer formalisms (Cobb, 1987a). The weak program gives greater weight to mathematical experience and the strong program greater weight to precision.
A glance at recent publications in mathematics education that draw on cognitive psychology (e.g. Schoenfeld, 1987a; Silver, 1985) indicates that the strong program currently has greater influence in the United States. For this reason, my comments will focus most directly on this program. At the same time, several of the points to be made can be read as claims that the weak program will have to go beyond the computer metaphor if it is to cope with certain aspects of mathematical activity. My general conclusion will be that the strong program is inadequate with respect to the purposes of mathematics educators.
I will also draw on recent works of some of the adherents to the strong program to indicate that they tacitly acknowledge this inadequacy. These acknowledgements do not stem primarily from empirical results – experiments come to be seen as crucial only with hindsight (Lakatos, 1970). For example, Carpenter and Moser’s (1984) empirically based criticisms of Riley et al.’s (1983) and Briars and Larkin’s (1984) computational models of solutions to arithmetic word problems have been largely brushed aside. Instead, there seems to be a dawning awareness that the strong program cannot account for several phenomena that appear to obviously influence students’ mathematical activity. This leads me to speculate that the computer metaphor as manifested in the strong program’s production systems approach will be supplanted in the United States by a form of social constructivism. However, before speculating about the future, it is essential that we consider what is involved in computer modelling. Only then can we appreciate the constraints the modeller must accept to pursue this approach.
Computer Modelling
The computational approach to mathematical activity of the strong program hai its roots in the behaviorist tradition of American experimental psychology. In traditional psychological experiments, situations are created in which the subject’s action is strictly controlled and only a very limited number of aspects of the situation (i.e. variables) are considered relevant. Experiments of this type are conducted with the global identifying rules or laws that govern behavior. By capitalizing on the power of the computer, experimental psychologists can investigate the predictions of more detailed theories and empirically investigate processes that intervene between stimulus and response:
A program is a formal system that has some number of variables and that can be manipulated (run) to generate predictions about the behaviour (outputs) of some naturally occurring system that is intended to model [e.g. a mathematics student]. To the extent that the predicted behaviour corresponds to that observed, the theory is supported. The role of the computer is to enable the scientists to deal with more complex theories than those whose consequences could be determined by examination or hand computation (Winograd & Flores, 1986, p. 25).
One central tenet of the strong program, then, is the importance of achieving an input- output match between human subjects’ observed performance and the computer system’s (Anderson, 1983). In addition, researchers of the strong program accept constraints that are considered to be inherent in human thought when it is conceptualized as an information-processing system. The most obvious is the limited capacity of working or short-term memory. The claim that a program simulates and explains cognitive activity is based on conformity to such theoretically-dependent inferences about the characteristics of human thought and a demonstrated input-output match.
Gergen (1986) noted an immediate consequence of the strong program’s computer simulation approach that reflects its behaviorist origins:
Regardless of the character of the person’s behaviour, the mechanist theorist is virtually obliged to segment him [or hell from the environment, to view the environment in terms of stimulus or input elements, to view the person as reactive to and dependent on these input elements, to view the domain of the mental as structured (constituted of interacting elements). to segment behaviour into units that can be coordinated to the stimulus inputs, and so on (p. 146).
The separation of the individual from his or her environment is explicit in the strong program’s methodology. The epistemological and practical ramifications of this separation will be considered in subsequent sections of this paper. For the present, it suffices to observe that the programmer has first to analyze tasks and performance data to specify the potential set of procedures for the program. This requires that the programmer articulate an explicit account of what the system is intended to do and what the outcomes of its functioning will be. The tasks themselves must be precisely stated in terms of relevant objects and their properties (cf. van Oers, this issue). In the case of an arithmetical story problem such as “Jay has nine books; then he lost five books. Now how many does he have?,” relevant objects include books and sets. Relevant properties include the set membership of books and cardinality of sets. Relevant procedures including forming and enumerating sets of objects (Briars & Larkin, 1984; Riley et al., 1983).
Pursuing our example, distinctions are made between various types of one-step textbook addition and subtraction story problems in accord with the classification schemes developed by Carpenter and Moser (1982) and Riley et al. (1983). Each problem type is characterized by a constellation of the basic objects and properties. These constellations are sometimes called patterns of information (Greeno, 1987). The resulting task environment reflects the programmer’s interpretation of the situation in which the program will function. It is important to note that it is the programmer who decides what is and is not relevant. In effect, the programmer creates a propositional, context-free environment of objects and their properties. The elements of this environment reflect the goals and purposes for which the programmer carved them out – that of creating a computer model. As Dreyfus and Dreyfus (1988) noted:
When we try to find the ultimate context-free, purpose-free elements, as we must if we are going to find the primitive symbols to feed a computer, we are in effect trying to free aspects of our experience of just that pragmatic organization which makes it possible to use them intelligently in coping with everyday problems (pp. 26-27).
In effect, the programmer exercises crucial aspects of his or her own intelligence and thus avoids the issue of how to simulate them in terms of the context-free logic of a computer formalism.
There are two major constraints that the programmer has to accommodate when analyzing the task environment. The first is that all the objects and properties deemed relevant must be represented by formal symbol structures. Further, operations that manipulate these symbols must be specified so as to derive new structures that represent the domain in such a way that an input-output match is achieved. This mapping and the representational function of the symbol is, of course, entirely in the mind of the programmer. “There is nothing in the design of the machine or the operation of the program that depends in any way on the fact that the symbol structures are viewed as representing anything at all” (Winograd & Flores, 1986, p. 96). Dreyfus and Dreyfus (1986) suggested that the symbols might better be called “squiggles” to avoid confusion over who or what is doing the representing. The second major constraint is that the formal representation must itself be set in correspondence with structures available on the computer. “The critical thing is that the embodiment remain faithful to the formal system – that the operations carried out by the computer system are consistent with the rules of the formal system” (Winograd & Flores, 1986, p. 96). As Orton (1988) put it, the programmer “must ‘serve two masters’: the machine that will ultimately simulate it and the child’s cognitive processes which the machine simulates” (p. 18).
Returning to our example of addition and subtraction word problems, the computational model developed by Riley et al. (1983) represents the task environment by means of schemas that correspond to the basic problem types. Each schema is, in effect, an elaborate production (Anderson, 1983) that consists of procedures for manipulating symbols and conditions that must be satisfied before the procedures can be carried out. Input to the system consists of a propositional representation of a story problem. Although we as observers can interpret this propositional representation in terms of books, sets, and their relations, the input does not represent anything from “the computer’s point of view.” When the input matches the conditions of a schema, parameters in the schema are specified by input data and the procedures associated with the schema are performed. In the case of the problem “Jay had nine books; then he lost five books. Now how many books does he have?,” the conditions of Riley et al.’s (1983) change schema are satisfied and the simulation can be interpreted as specifying the initial or start set (nine books) and the change set (five books).
A schema, thus characterized, is a sophisticated cognitive stimulus–response mechanism. The same can be said of all basic information-processing mechanisms that have been used to develop computer simulations of mathematical activity (e.g. productions, rules, and principles). As Lave (1988) put it, “relations between cognition and its ‘environments’ are still treated in terms of stimuli which evoke responses” (p. 90). These relations are explicit in Simon’s (1969) contention that:
… a man, viewed as a behaving system, is quite simple. The apparent complexity of his behaviour over time is largely a reflection of the complexity of the environment in which he finds himself (p. 25).
In fact, Anderson (1983, p. 6) characterized his production system models as “‘cognitive’ S–R theories.” This neo-behaviorist aspect of the strong program can be further illustrated by considering models developed to simulate the process of solving algebraic equations. The strategies mathematicians use “to categorize or transform algebraic equations can be identified by systematically analyzing the ‘preconditions’ that must be present before the classification processes or transformation procedures are applied” (Wenger, 1987, p. 237). Here again, it is assumed that mathematical activity consists of symbol manipulation procedures (i.e. responses) that are triggered by the input from a propositional environment.
The mechanisms discussed thus far all represent (in the mind of the programmer) the domain-specific knowledge sufficient to simulate the subject’s task performance. Some programs also simulate planning behavior by representing heuristics as cognitive condition–action pairs. In this case, the procedure or response is not a symbol manipulation procedure but the setting of parameters in other mechanisms. For the programmer, these parameters represent goals. Thus, “the computer can be given a rule which tells it that if certain facts are present, the situation should be organized in terms of a certain goal” (Dreyfus & Dreyfus, 1986, p. 65).
In summary, the computer simulation enterprise of the strong program as it is manifested in mathematics education involves developing propositional, context-free descriptions of task environments. In this approach, it is the programmer who decides what is and is not relevant or salient to task performance. The results of this analysis are embodied in the computer program as an organized set of cognitive stimulus – response mechanisms. “In setting up a program, the programmer has in mind a systematic correspondence by which the contents of certain storage cells represent objects and relationships within the subject domain” (Winograd & Flores, 1986, p. 84). In Riley et al.’s (1983) program, for example, the schemas correspond to types of arithmetical word problems. Each problem type is specified in terms of features that the programmer considers are essential.
The image of ourselves implied by the strong program is one in which we live in an objective propositional environment that is external to and independent of us. We perceive this environment to the extent that our perceptual systems allow us and use the information obtained to compute our behavior. The world “is viewed as containing information or ‘patterns’ existing prior to the organizing activity of the person” (Bidell, Wansart, & De Ruiter, 1986, p. 3). We are environmentally driven beings who depend on and are modified by our environment. For proponents of the computer simulation approach, this image makes possible the development of precise, rigorous computational models of human cognitive activity. One of the proposed strengths of this approach is that in the course of developing these models, cognitive psychologists “frequently discover performance requirements that were not evident when they merely thought about the task” (Greeno, 1987, p. 61). As a consequence, it is argued that tacit knowledge needed to perform the task can be made explicit.
As Lakatos (1976) observed:
It frequently happens in the history of thought that when a powerful new method emerges the study of those problems which can be dealt with by the new method advances rapidly and attracts the limelight, while the rest tends to be ignored or even forgotten, and its study despised (p. 1).
The problems that computer modellers can deal with are restricted by the constraints of their chosen methodology.
Formal Domains
As we have seen, the computer modeller must explicitly state what the system is intended to do and what the consequences of its procedures will be. In other words, it is essential that a set of goals and operators be specified that delimit what can and will be done. “This can be done for idealized ‘toy’ systems and for those with clearly circumscribed purposes (for example, programs that calculate mathematical formulas)” (Winograd & Flores, 1986, p. 53). The main emphasis of cognitive science is therefore “on providing detailed explicit models of cognitive development within a limited domain” (Brown & DeLoache, 1978, p. 9).
The computer modeller is obliged to select problems that involve the investigation of performance on a clearly bounded set of tasks. As a consequence, it is at least tacitly assumed that “‘knowledge’ consists of coherent islands whose boundaries and internal structure exist, putatively, independent of individuals” (Lave, 1988, p. 43). In effect, the researcher pursues the hunch that the activity of completing a set of mathematical tasks can be cut away from the rest of the people’s lives. This research strategy is typically justified by the argument that the researcher has to work in a narrow domain because he or she has yet to discover how to handle a wider domain. The implication is that this discovery will be made, allowing the development of more encompassing cognitive models. Work conducted within the hermeneutic and social anthropological traditions (e.g. Gadamer, 1986; Lave, 1988; Wittgenstein, 1982) explicitly questions whether this promise can be fulfilled even in principle. Regardless of how compelling one finds these arguments and counterarguments, we can observe that current computer modelling efforts must necessarily specify clearly bounded formal domains that are “characterized by a set of principles logically sufficient to solve problems in the domain” (Larkin, 1981, p. 311) and, presumably, do not open up into other areas of human activity. Examples of segments of the school mathematics curriculum that have been transformed into formal domains include addition and subtraction story problems (Briars & Larkin, 1984; Riley et al., 1983), addition and subtraction of multidigit numbers (Brown & Burton, 1978; Van Lehn, 1986), algebraic equations (Silver, 1984), and textbook geometry story ‘ problems (Anderson, Greeno, Kline, & Neves, 1981). The view of school mathematics that emerges from the work of the strong program is that of a subject composed of a collection of ideally isolated systems. Within each system, goals are achieved by making a series of moves that are constrained by formal rules.
Relevance, Common Sense, and Context
We have already noted that the programmer decides what is relevant when he or she analyzes the task environment. In relying on his or her own apparently intuitive ability to pick out what is relevant, the programmer sidesteps one of the more subtle aspects of cognition. Bereiter (1985) asked the following question:
Out of the infinitude 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).
In a similar vein, Minsky (1981, p. 214) concluded that “the problem of selecting relevance from excessive variety is a key issue.” This is not a problem for researchers who write their models using productions, rules, or principles because the models operate in pared down environments of preselected relevance. If it were the model rather than the programmer that did the representing, the model would be a literalist. Each symbol refers to a fixed object or property in the environment. The model would also be without contexts. The literal meanings of symbols do not shift or change depending on the situation in which they are used. In short, even if we attribute meaning to the model, we realize that we are dealing with a system that lacks common sense. In fact, renewed interest in the problem of common sense by the MIT group under the leadership of Minsky (1985) has been accompanied by the realization that available programming techniques are too limited for a broad theory. Fol. Minsky and other members of the weak program, the challenge of investigating some of the more subtle yet pervasive aspects of cognitive experience outweighs the desire for rigour and precision.
It might be argued that issues of relevance, context, and common sense are irrelevant to mathematical activity. However, DeCorte, Verschaffel, and DeWin (1985) found that even in the case of addition and subtraction story problems, children’s interpretations involve suppositions about both the intent of the tasks and ambiguities in the stories. In other words, children interpret and solve these types of tasks within a socially constituted context that constrains their sense of what is relevant. Findings with adults (Lave, Murtaugh, & de la Rocha, 1984) and children (Carraher, Carraher, & Schliemann, 1985, 1987; Cobb, 1987b; Saxe, 1988) also indicate that socially constituted contexts influence subjects’ interpretations of mathematical tasks. With regard to problem solving processes in general, Minsky (1975) contended that:
…at each moment one must work within a reasonably simple framework. I contend that any problem a person can solve at all is worked out at each moment in a small context and that the key operations in problem solving are all concerned with finding or constructing these working environments (p. 119).
In short, literal meanings are inadequate to account for mathematical activity beyond narrow sets of carefully selected tasks. And in these limited domains, the programmer carefully preselects what is relevant when he or she analyzes the task environment.
Sensory-Motor Action and Mathematical Meaning
I have previously drawn on the work of Davis and Hersh (1981), Hatano, Miyake, and Binks (1977), and Steffe, von Glaserfeld, Richards, and Cobb (1983) to support the contention that actual and represented sensory-motor action plays a crucial role in mathematical activity (Cobb, 1985, 1987a). Two further examples can be cited to support this position. One concerns young children and the other research mathematicians.
Neuman (1987) conducted a longitudinal clinical investigation of young children’s developing conceptions of the first ten whole numbers. After analyzing her subjects’ solutions to a variety of tasks, she concluded that their physical movements and the physiological structure of their bodies profoundly influenced their conceptual development. In fact, she called one of the more primitive conceptualizations she identified “movements.” At a more sophisticated level, she accounted for children’s construction of relationships between the first ten whole numbers by observing that the children’s fingers:
…begin to form a gradually more and more precise ordinal/cardinal semi-decimal system, which can be used in a concrete way, and after that visualized. The numeral “name” of each finger gives ordinal qualities to the numbers in this system which in an indissociable way are related to the cardinal qualities inherent in the finger-patterns where a finger with a specific name is the last finger (pp. 179-180).
In other words, children’s relational knowledge of the first ten numbers is, as she put it, “bodily-anchored” (p. 206). This is because the knowledge derives from children’s sensory-motor activity of using their biological endowment of two collections of five fingers in arithmetical settings. Neuman’s findings are compatible with those reported independently by Steffe, Cobb, and von Glaserfeld (1988). In a related study, Cobb (1986) presented findings that indicate that children’s early learning of the doubles addition combinations (e.g. 3 + 3 = 6, 5 + 5 = 10) is related to the activity of making finger patterns that in turn reflects the bilateral symmetry of the central nervous system. These analyses contrast with those offered by information-processing psychologists of the strong program in which it assumed that children construct associations between stimulus (e.g. 3 + 5 = _.) and response (e.g. Ashcraft, 1982; Siegler & Robinson, 1982).
Turning now to consider more sophisticated forms of mathematical activity, the findings reported by McNeill (1985) indicate that mathematicians’ highly abstract concepts also have a sensory-motor aspect. He analyzed mathematicians’ verbal utterances and gestures as they engaged in discussions. In general, “the discussions are accompanied by a flow of gestures that show mathematical ideas in the form of actions” (p. 263). For example, in the case of the mathematical concept of limit, “the hands move towards some boundary marked by the other hand or a sudden stop; thus these gestures show an image of a mathematical concept in the action realm” (p. 264). McNeill also gave examples for the concepts of dual, inverse, and finiteness. In each case, “the form and manner of execution of the gesture depict a meaning relevant to the proposition being expressed verbally at the same time” (p. 265). Most significantly, mathematicians make the appropriate gesture even when they misspeak. This observation suggests that meaning as actual or represented action may be more primary than meaning as a proposition.
McNeill’s results should not be interpreted to imply that mathematicians cannot think if their hands are tied behind their backs. One only has to consider the work of Stephen Hawkings, the leading theoretical physicist who is almost completely paralyzed by amyotrophic lateral sclerosis, to realize that such a conclusion is false. Hawkings has said of his work, “Equations are just the boring part of mathematics. I attempt to see things in terms of geometry” (quoted by National Research Council, 1989, p. 35). And how else could Hawkings see things in terms of geometry other than by interpreting equations in terms of intuitive conceptual structures grounded in sensory motor experience? In this case, the sensory motor actions are represented rather than actual. Lakatos’s (1976) well- known reconstruction of the historical development of Euler’s theorem exemplifies the role that such bodily-grounded intuitions play in construction of mathematical knowledge. In Lakatos’s account, reflections on the re-presented sensory-motor activity of cutting and deforming various polyhedra were central to the development of both proofs and refutations. More recently, Fischbein (1987, this issue) has argued that intuitions of this type are essential to meaningful mathematical activity. He contends that the function of intuitions is to mediate between intellectually inaccessible formalisms and intellectually accessible and manipulable re-presentations. As he put it, “we think better with the perceptible, the practically manipulable, the familiar, the behaviorally controllable, the implicitly lawful, than with the abstract, the unrepresentable, the uncertain, the infinite” (1987, p. 123). This, I suggest, is exactly what Hawkings meant when he said that he “attempts to see things in terms of geometry” (emphasis added).
The argument that actual and re-presented sensory-motor action is central to mathematical meaning is compatible with the theory of meaning and reason developed by Johnson (1987). Image schemata are the pivotal constructs in Johnson’s analysis of cognition. These are “abstract patterns in our experience and understanding that are not propositional in any of the standard senses of the term, and yet they are central to the meaning and to the inferences we make” (p. 2). Johnson gives several mathematical examples including equality, symmetry, transitivity, and reflexity. He concludes, “in considering abstract mathematical properties, we sometimes forget the mundane bases in experience which are both necessary for comprehending those abstractions and from which the abstractions have developed” (p. 98).
The contention that the body and sensory-motor action are intimately involved in mathematical meaning and thought poses a challenge to computer modellers of the strong program. The computer formalisms used are propositional in nature and do not simulate proprioceptive and kinesthetic activity. However, even the development of such computer codes would not solve the problem. It would be necessary to “give the computer” our sensory-motor ways of knowing. We would have to simulate everything we know because we have a body. This is a tall order even if Johnson is wrong and image schemata are propositional in some sense. Proponents of the strong program emphasize the value of explicating tacit knowledge. It is therefore ironic that the constraints of their methodology preclude the simulation of the intuitive ways of knowing that seem crucial if we are to develop an adequate understanding of mathematical meaning.
These arguments concerning the importance of sensory-motor action are broadly compatible with the claim of van Oers (this issue) that cognitive processes and observable mathematical behavior are both based on actions. I would only add that, from the constructivist perspective, the mental actions or conceptual operations are constructed by internalizing and interiorizing sensory-motor activity (Furth, 1969; Piaget, 1971). The characterization of information-processing psychology by van Oers also complements my characterization of the strong program. In particular, his analysis of the strong program’s characterization of mathematics explores the consequence of ignoring sensory-motor action. For example, meaning of, say, place value numeration is reduced to a catalogue of mathematical principles to be followed (cf. Resnick & Omanson, 1987). The distinction between syntax and semantics then becomes in practice little more than that of labeling one set of if–then productions “symbol manipulation rules” and another set “mathematical principles.” Of course, this distinction is entirely in the researcher’s mind; a rule is a rule to the computer. One can legitimately ask members of the strong program, where’s the meaning?
Mathematical Cognition and Computer Simulation
In discussing the strong program thus far, I have focused on what proponents claim the methodology delivers – precise models of cognitive performance in limited domains. I have not considered the obviously tricky cases such as the process of generating conjectures and the role of metaphor and analogy in mathematical thought (Kieren & Pirie, 1988; Pimm, 1981, 1987). Further, I have ignored issues such as intentions, motivations, reasons for acting, and affects (McLeod, this issue). In addition, little mention has been made of the limitations inherent in proposed knowledge acquisition mechanisms (Beilin, 1981; Cobb, 1987a; Silver, 1987). I have taken a more conservative line of analysis and attempted to consider the implications of computer simulation in areas where proponents of the strong program claim they have succeeded. We have noted that the models simulate performance in isolated task domains. Buchanan (1982), a developer of expert systems, listed some of the characteristics of problems suitable for computer simulation. These include, “narrow domain of expertise; limited language for expressing facts and relations; limiting assumptions about problem and solution methods; little knowledge of own scope and limitations” (p. 283). We might have reservations about the strong program if we consider that mathematical activity is not characterized by these limitations.
We have also observed that meaning is dealt with in literalist terms and noted that as a consequence computer models do not operate in contexts, are unable to decide what is relevant, and generally do not simulate common sense. In addition, conceptual thought is divorced from sensory-motor action with the consequence that meaning is solely in the mind of the observer who interprets the computer program in terms of his or her own intuitively grounded mathematical knowledge. One ends up with an image of a mathematics student as a literalist who does exactly what he or she is told but does not think like the rest of us even in his or her literal way.
At times, one gets the impression that computer modellers would like to ignore the problem of meaning entirely if this were possible. For example, Van Lehn (1983) explained his decision to investigate children’s multidigit subtraction algorithms as follows:
Its main advantage, from a psychological point of view, is that it is a virtually meaningless procedure. Most elementary students have only a dim conception of the underlying semantics of subtraction, which are rooted in the base-ten representation of numbers …This isolation is the bane of teachers but a boon to the psychologist. It allows one to study a skill formally without bringing in a whole world’s worth of associations (p. 5, emphasis added).
Lave (1988) offered a further insight into why mathematical cognition might seem to be a suitable candidate for the computer simulation approach of the strong program. The findings of an extensive socio-anthropological study led her to conclude that in our culture:
…at this time, mathematics is a reified object as a career, academic discipline and body of knowledge. It is a subject in school and an object, “real math,” in folk belief …It indicates exactitude, rationality and ‘cold’ logic which stands in mutually exclusive relations with intuition, feeling, and expression (p. 125).
This folk belief is implicit in “the structuring of math lessons [in school] as specific, algorithmic prescriptions for a universally applicable set of procedures to be employed outside school” (p. 100). As Lave further noted, within this folk culture, it is self-evident that the formal scholastic structure of school mathematics can be taken as the basis for developing models of mathematical cognition. As we have seen, this is the structure that lends itself to the task analysis approach of the strong program.
In view of the apparent consistency between the folk belief about mathematics and the view of mathematics implicit in the strong program, we might want to scrutinize claims that computer simulations of mathematical understanding have been developed. “A major source of simple mindedness in AI programs is the use of mnemonics like ‘UNDERSTAND’ or ‘GOAL’ to refer to programs and data structures” (McDermott, 1981, p. 144):
Many instructive examples of wishful mnemonics by Al researchers come to mind once you see the point. Remember GPS (Ernst & Newell, 1969)? By now, “GPS” is a colourless term denoting a particularly stupid program to solve puzzles. But it originally meant “General Problem Solver,” which caused everybody a lot of needless excitement and distraction. It should have been called LFGNS – “LocalFeature-Guided Network Searcher.” …When you say (GOAL …), you can just feel the enormous power at your fingertips. It is, of course, an illusion (pp. 144-145).
As part of this scrutiny, we should not forget who is creating meaning, and doing the representing and understanding.
At times, advocates of computer simulation talk as though precision is the hallmark of science. Champagne and Rogalska-Saz (1984) argued, for example, that:
Formal cognitive models are verified by working computer programs whose output mimics significant aspects of observed human performance. Thus, in the scientific study of cognition, productions are written in programming languages (p. 8).
Marr (1981), who did some of the pioneering work in developing computational theories of perceptual processes, seemed to believe that there is more to science than precision:
Studies – particularly of natural language understanding, problem-solving, or the structure of memory – can easily degenerate into the writing of programs that do no more than mimic in an unenlightened way some small aspect of human performance …[One] is left in the end with unlikely looking mechanisms whose only recommendation is that they cannot do something we cannot do. Production systems seem to fit this description quite well. Even taken on their own terms as mechanisms, they leave a lot to be desired. As a programming language they are poorly designed and hard to use, and! cannot believe that the human brain could possibly be burdened with such prior implementation decisions at sobasic a level …Studying our performance in close relation to production systems seems to me a waste of time, because it amounts to studying a mechanism, not a problem (pp. 139-140).
Marr went on to say that the same criticism applies to schema theories because there is a fascination with a mechanism that “is so simple and general as to be devoid of theoretical substance” (p. 141).
Marr, McDermott, Minsky, and other members of the weak program seem to imply that explanation should outweigh precision in the scientific enterprise (cf. Toulmin, 1963). In assessing this contention, we should remember that there is no neutral algorithm of theory choice (Kuhn, 1970). It is a matter of exercising judgment and deliberation to weigh up values that frequently conflict. Mathematics educators will have to decide whether precision is worth the price the strong program has to pay for it.
Cognitive Behavior and Mathematical Experience
We have seen that the computer models of the strong program are environmentally driven systems that execute a sequence of symbol manipulating procedures in a step-by- step manner in response to input. The goal of achieving an input – output match between the model and the human subject is a manifestation of the Turing test, itself unashamedly behaviorist. From this perspective, the goal is to model cognitive behavior. A subject’s mathematical solutions are treated as objects at the investigator’s disposal to be taken apart to see how they work. This can be contrasted with an alternative view in which a solution is thought of as an expression of the subject’s coping rationality. Here it is assumed that the subject’s activity is rational given his or her understanding, interests, purposes, obligations, and responsibilities. To be adequate, an analysis must specify the subject’s reasons for acting and account for the reasonableness of behavior that at first blush might seem bizarre. From this perspective, which is common to constructivism and the weak program, the goal is to infer and account for mathematical experience rather than to simulate cognitive behavior. As a caveat, it should be noted that this emphasis on experience does not mean that any action or interpretation is as good as any other. We can attempt to understand someone’s reasons for acting without necessarily condoning his or her actions.
The contrast between explanations that focus on cognitive behavior and mathematical experience can be exemplified by considering two analyses of the same phenomenon. Siegler and Shrager (1984) reported that a substantial proportion of their four- and fiveyear-old subjects solved symbolic addition sentences by giving the immediate successor of the second addend as their answer (e.g. 3 + 3 = 4, 3 + 5 = 6). Siegler and Shrager interpreted answers of this kind by first noting that most children know the counting-string up to ten long before they begin to add and subtract. They concluded that “the last addend in an addition problem may always activate its immediate successor as a potential answer” (p. 265). In other words, the second addend (i.e. stimulus) triggers a rule-following response in the environmentally driven child. This explanation is adequate when the focus is on cognitive behavior because it specifies a rule that relates the stimulus to the observed behavior.
Neuman (1987) reported similar observations but suggested an alternative explanation. She analyzed her subjects’ solutions to other tasks and inferred that they have a numerical concept that she termed “names”:
Its hallmark is that the numerals have become namesof objects. Theseseem to constitute an imagined row of named objects, or simply a rowof digits, each of them being a symbol for the object thought of (p. 104).
Consequently, with regard to solutions such as 3 + 5 = 6:
There is a big difference between the action of choosing one counting-word following another in the counting-word sequence [as proposed by Siegler and Shrager] and that of adding one imagined object, which is represented by a digit [“3” in 3 + 5 = – ] to the larger set, which is represented by a row of digits [of which the last is “5” in 3 + 5 = _] This is the difference between an operation which has quantitative meaning, and an answer given without any relation to quantitative problem-solving (p. 114).
Neuman’s explanation focuses on children’s mathematical experience. It is an account that specifies how children might have interpreted the task. In contrast, Siegler and Shrager focused on the effect of this stimulus on the child. For Neuman, children actively construct meanings whereas for Siegler and Shrager children respond to stimuli that cause them to behave. By reading Neuman’s explanation, we begin to understand why children give these answers. An initially strange response of “4” to 3 + 3 = now seems reasonable given the child’s numerical concepts. We have an idea of what things might be like from the children’s point of view. Siegler and Shrager identified a regularity in children’s behavior, but with regard to understanding why children respond in this way, we are none the wiser. Neuman took it as self-evident that children act on the basis of their experientially grounded mathematical understandings whereas Siegler and Shrager assumed that children act by following rules in their heads. In the trade-off between experience and precision, Neuman seems to value the former more, and Siegler and Shrager the latter.
Social Life
Thus far we have tacitly accepted the strong program’s characterization of students as solo learners who do not know of the existence of other beings. We will now broaden our perspective by viewing cognitive development as the process by which students grow into the mathematical practices of a cultural community. This expansion of our preview is essential in that students’ developing cognitions have “clout” only to the extent that they enable them to increasingly participate in communal mathematical practices. Conversely, opportunities to learn arise in the course of social interactions with other members of the community (Bruner & Haste, 1987). Students’ developing mathematical cognitions are inevitably constrained by the need to coordinate their mathematical actions with those of other members of the community. As an example, we have already noted that even young children’s successful solutions to arithmetical word problems involve culturally appropriate suppositions.
For ease of explication, it is possible to distinguish two processes that constrain students’ cognitive development in the course of their mathematical acculturation in school. The first concerns the negotiation of classroom social norms and the second, the negotiation of mathematical meanings and practices (Cobb, Wood, & Yackel, in press, a, b). As Balacheff (1986) noted, classroom social norms constrain both what is an acceptable mathematical problem and what is an acceptable mathematical solution. It is by initiating and guiding the continual renegotiation of classroom norms that the teacher influences students’ beliefs about their own role, the teacher’s role, and the general nature of mathematical activity (Cobb, Yackel, & Wood, in press; Nicholls, Cobb, Wood, Yackel, & Patashnik, in press). In effect, the teacher inducts students into the interpretive standpoint he or she takes to mathematical activity. The learning that occurs in the process of negotiating the classroom social norms is typically called “the hidden curriculum.” It is hidden in that the obligations and expectations negotiated to make possible the relatively smooth flow of classroom social interactions are, for the most part, outside the teacher’s and students’ awareness (Voigt, 1985). This tacit knowledge, which makes it possible for students to just know what is appropriate in particular classroom situations, appears to profoundly influence their construction of mathematical knowledge and yet is beyond the scope of the strong program. This is a serious weakness, for as Bernstein (1983) observed:
An adequate analysis of inquiry must take account of the norms embedded in intersubjective communication, norms which serve as regulative and critical ideals of such inquiry (and which themselves can be subject to further interpretation and criticism) (p. 78).
The task of constructing computer models that participate in the negotiation of social norms that regulate mathematical activity would seem to be beyond the scope of the strong program. The programmer would have to specify all relevant operations and their results when the objects being operated on are other people. This would involve reducing our entire repertoire of interpersonal knowledge to a set of rules. This seems a daunting task in view of the fact that discourse analysts such as Sachs, Schegloff, and Jefferson (1974) have succeeded in doing so only for highly ritualized interactions such as greeting someone when answering the telephone. Not surprisingly, all attempts to develop a formalizable theory of dialogue have been conducted without notable progress (Dreyfus & Dreyfus, 1986).
A more modest goal for the strong program might be to ignore the seemingly intractable problem of social interaction and instead simply consider the beliefs of solo students. In a sense, this has already been done. Computer models of the strong program implicitly reflect the belief that mathematics is an algorithmic activity. Even meaning is a matter of following rules rather than of experientially-based intuitions. In effect, the program is a model of a student whose activity reflects the folk belief about mathematics described by Lave (1988).
Mathematical Practices
The difficulties the strong program has in coping with the negotiation of classroom social norms applies equally to mathematical meaning’, and practices. Mathematics educators who believe that learning involves the negotiation of meaning (Bauersfeld, 1980; Bishop, 1985), the resolution of conflicting p( nts of view (Perret-Clermont, 1980), the distancing of the self from on-going activity to give or make sense of an explanation (Levina, 1981; Sigel, 1981), and, more generally, the construction of consensual domains (Barnes & Todd, 1977), will have to look elsewhere for a psychological theory that fits their purposes.
The difficulty in dealing with all but, perhaps, the most highly ritualized of social interactions accounts in large measure for the strong program’s treatment of mathematics as a propositional environment that students perceive according to the adequacy of their conceptual apparatus. Mathematics as a cultural practice is, in effect, dumped into the environment and reduced to a carefully organized collection of stimuli. In the absence of social interaction and negotiation, it is these stimuli that shape mathematical thought. This view of mathematics is incompatible with historical reconstructions of the development of mathematics (Bloor, 1976, 1983; Lakatos 1976), recent developments in the philosophy of mathematics (Tymoczko, 1986), and the finding that self-evident mathematical practices vary across communities (Carraher & Carraher, 1987; D’Ambrosio, 1985; Lave et al., 1984; Saxe, 1988). An alternative to this characterization of mathematics takes seriously the view of mathematics as a cultural practice (Wittgenstein, 1964). From this perspective, mathematics is not absolute but is instead continually negotiated and institutionalized by members of intellectual communities (Schutz, 1962). As Peirce (1935) put it, the “very origin of the conception of reality [including mathematical reality] shows that this conception involves the notion of community” (p. 186).
To complement its reification of mathematics as a propositional environment, the strong program treats long-term memory as the place where cultural acquisitions are stored:
The main difficulty with this view is that the nexus of cognition/culture relations is never constructed in the present, but always assumed to have an existence because of events which took place in the past... [Culture is defined] as “what people have acquired and carry around in their heads,” rather than as an immediate relation between individuals and the sociocultural order within which they live their lives (Lave, 1988, p. 91).
In other words, mathematical ways of knowing that make possible participation in communal practices are reduced to a well-indexed set of atomistic facts and their relations (cf. van Oers, this issue).
A second difficulty, this time epistemological in nature, also arises from the strong program’s difficulty in dealing with mathematical practices. It will be recalled that the methodology of the strong program requires the separation of the student from the propositional mathematical environment. This is the classic Cartesian dualism of object and representation. A distinction is made between the real, objective task and the student’s internalized version of it. The latter is termed the student’s representation. In general, mathematical problem solving is characterized “as a process of building successively richer and more refined problem representations” (Silver, 1987, p. 44). The difficulty is that:
…the Cartesian self is acquainted with the material [and mathematical] world only via its ideas [or representations] and only it is directly acquainted with those ideas. Thus, each Cartesian self lives in an entirely self-contained world. It is as if we each inhabit our own private picture show (Bakhurst, 1988, p. 34).
In short, the separation of object and representation leads straight to solipsism.
The strong program avoids this conclusion by suggesting that the student can come into direct contact with the propositional mathematical environment – where else could the contents of long-term memory come from? This process of direct contact is typically characterized as the taking in of information (Briars, 1982). As Bidell et al. (1986) noted, this proposal seems to involve an undue reliance on “the so-called doctrine of ‘immaculate perception’ “ (p. 3). We therefore have an unpalatable choice between solipsism and a form of direct realism that would seem to be untenable in view of severe critiques (e.g. Putnam, 1988; Rorty, 1982; Glasersfeld, 1984).
One way out of this dilemma is to transcend the very Cartesian dualism that is built into the strong program’s methodology. In such a view, “it is impossible to distinguish and thus contrast the interpretation of a thing from the thing itself …because the interpretation of the thing is the thing” (Mehan & Wood, 1975, p. 69). From this perspective, the experience of an external, objective mathematical reality is not any way denied but is instead taken as a phenomenon in need of explanation. Such an explanation will involve a theory of reality construction that specifies processes by which mathematical reality is accomplished. The work of both social interactionists (e.g. Blumer, 1969; Meade, 1934; Schutz, 1962) and ethnomethodologists (e.g. Mehan & Wood, 1975) suggests that these reality-constructing processes are interactional and cultural as well as cognitive. A mathematical truth is true because a community of knowers makes it so. Ethnomethodologists refer to this as the reflexivity of knowledge. In this view, mathematics is a human social activity – a community project (De Millo, Lipton, & Perlis, 1986). It is the dialectical interplay of many minds that determines whether a theorem is both interesting and true:
[After enough] transformation, enough generalization, enough use, and enough connexions, the mathematical community eventually decides that the central concepts of the original theorem, now perhaps greatly changed, have an ultimate stability. If the various proofs feel right and the results are examined from c gh angles, then the truth of the theorem is eventually established. The theorem is thought to be true in the classical sense – that is, in the sense that it could be demonstrated by formal deductive logic, although for almost all theorems no such deduction ever took place or ever will (De Millo et al., 1986, p. 273).
It is through this social process of negotiation and institutionalization that a theorem becomes an incorrigible proposition (Gasking, 1955) and acquires the status of a truth for the community that made it.
This characterization of the development of mathematical truth as a communal activity can be extended to the classroom by viewing the teacher and students as constituting an intellectual community (Cobb, in press). The processes by which mathematical meanings and practices are negotiated and institutionalized can then be investigated. Cobb, Yackel, and Wood (1988) took this perspective when analyzing life in a second grade classroom during mathematics instruction. They argued that the practice of conceptualizing whole numbers as composed of units of ten and of one emerged as a mathematical truth for the students in the course of their on-going interactions with each other and the teacher. A point was reached during the school year after which solutions that involved the construction of units of ten and of one were beyond question and justification. The students and teacher had together constituted a fact about mathematical reality. In an analysis of this type, students, as members of an intellectual community, are seen as active constructors of their mathematical realities. We can acknowledge that mathematics is practically real (Goodman, 1986) without accepting solipsism or appealing to philosophical realism. If we fail to take the notion of community seriously, we are faced with a choice between these two undesirable options and can only characterize students either as environmentally driven systems or as inhabitants of their own private picture shows.
Instructional Implications
Given the strong program’s characterization of students as environmentally driven systems, one would predict that the resulting instructional recommendations will focus on the development and manipulation of mathematical stimuli. This, for the most part, is what we find. For example, Resnick (1983) contended that “effective instruction must aim to place learners in situations where the constructions they naturally and inevitably make as they try to make sense of their worlds are correct as well as sensible ones” (pp. 30-31). The instructor should therefore find instructional representations that should “be ‘transparent’ to the learner (i.e. represent relationships in an easily apprehended form or decompose procedures into manageable units)” (p. 32). In other words, the instructor should devise stimulus materials in which the relevant mathematical properties are particularly salient and thus make it possible for students to perceive immaculately. Similarly, Wenger (1987) stated that “our primary leverage for instruction lies in making …the careful selection and design of the examples and the sets of practice tasks that students will work” (p. 221). This care is necessary because the tacit knowledge identified by computer modellers “‘exists between the lines’ …in examples that are presented in texts or by teachers; and if students are to acquire it, they have to ‘read between the lines’ and do so by a strongly abductive process” (Greeno, 1987, p. 67). It is therefore necessary to be more explicit about the objects or properties students are to pick out and the procedures they are to use. This will “help students to acquire the knowledge and skill necessary for success” (Greeno, 1987, p. 70). For Wenger, this implies that the purpose should be “to help students learn to perceive and use relevant patterns in algebraic expressions to create strategies and recall procedures for carrying them out” (p. 238). It is, of course, Wenger who has decided which patterns are relevant. He wants students to attend to the objects or properties he has picked out in a propositional mathematical environment. In his view, computer models are useful in accomplishing this instructional goal because the objects and properties that trigger a procedure (i.e. the preconditions of a procedure) “often make explicit the kinds of ‘understanding’ that mathematics educators hope their students will infer from practice tasks” (p. 238). Production rule descriptions can therefore be “exploited to develop text (and computer-based) materials that support students’ attempts to infer salient patterns and use their planning strategies and recalling relevant procedures for carrying them out” (p. 238). In this discussion, “understand” means knowing the conditions that should trigger a particular response. Recalling McDermott’s (1981) comments, we might question whether this has much to do with what we typically mean by mathematical understanding. Further, although the programmer specifies the features of the stimulus the student should attend to, the model does not simulate how the student is to accomplish this because it is the programmer who has decided what is relevant.
Whereas Wenger is concerned with the sequence of exercises and practice tasks, Greeno (1987), like Resnick (1983), argues that we should attempt to find “ways to present explicit representations of the information patterns that students need to recognize” (p. 69). It will be recalled that these information patterns are the stimulus conditions for different schemas. The rationale for this approach is that “providing explicit expressions of tacit knowledge could demystify the tasks significantly and thus enable teachers and texts to communicate more completely with students about what is to be learned” (p. 62). In this case, communication seems to mean perceiving the specific attributes of stimuli that an authority has previously picked out. Taking this recommendation at face value, there does not seem to be much room for genuine dialogue or the negotiation of meanings. Greeno went on to note that:
…this approach expands upon a principle of instructional design that is familiar in behavioural task analyses and has a tradition that extends at least back to Thorndike (e.g. 1922) and that has recently been developed in more detail by Gagn6 and his associates …The idea is that instruction should be given for the components of a complex skill, as well as for the integrated skill …Cognitive theory extends this idea by providing analyses of a different set of components – cognitive processes and knowledge structures – in addition to the behavioural components that are considered in earlier theories (pp. 62-63).
All this is, of course, compatible with the strong program’s focus on cognitive behavior rather than lived mathematical experience.
In general, instructional recommendations derived from the strong program all involve the development of mathematical stimuli that will hopefully cause students to learn to make the correct perceptions. This approach is consistent with the view of students as environmentally driven systems. Leaving aside the details of specific recommendations, we can note that the strongprogram leads us to functionalist cultural transmission theories of education (Kohlberg & Mayer, 1972). The mathematical meanings and practices negotiated and institutionalized by past generations are taken as constituting a pool of information to be transmitted from one generation to the next accurately and with verisimilitude (Lave, 1988, p. 7). In contrast to Thorndike and Gagne, computer modellers can give more details about the information to be transmitted, and address cognitive as well as external behaviors. However, as a trade-off, they can only do so for relatively small domains. In line with the functionalist cultural transmission tradition, educational goals are defined “in terms of standards of knowledge and behavior learned in school” (Kohlberg & Mayer, 1972, p. 150; Lave 1988). The goal might, for example, be proficient use of the standard multidigit subtraction algorithm, successful solution of addition and subtraction word problems, or successful solution of algebraic expressions. In general, the educational goal is defined in terms of the conformity of behavior to particular standards – ideally an input–output match with the computer model. The models thus come to serve a normative role because they are used as the criteria by which to distinguish between correct and incorrect responses (Lave, 1988). There is little thought of attempting to arouse interest and challenge or of viewing educational change as the active reorganization of cognitions made possible by experiential problem solving (Kohlberg & Mayer, 1972). Further, the focus is on mastery in limited domains rather than the influence of instruction on later developments.
Like all functionalist theories of cultural transmission, recommendations deriving from computer models support the status quo. The current educational values and practices institutionalized in schools are almost invariably taken, unquestioningly, as the starting point. We have noted, for example, that the folk belief about the nature of mathematical activity currently institutionalized in school culture is reflected in the work of the strong program. This program is not the place to look if one believes that this culture must itself be scrutinized. Similarly, with regard to content issues the question of whether students should be trained to used standard addition and subtraction algorithms or to manipulate algebraic expressions does not arise. Similarly, questions concerning the value of and role that word problems can play in students’ mathematical education are not addressed by the strong program. The goal of teaching students to master tasks of this sort is taken as a self- evident fact. In general, the strong program gives its allegiance to the way things are and implies an education that contributes to the reproduction of the status quo.
Possible Rapprochements
We have contrasted the view of students as isolated systems driven by a propositional mathematical environment with that of students as active reorganizers of their mathematical experiences. Development is, from one perspective, the acquisition of domain-specific mathematical facts and rules and, from another, growing participation in the evolving mathematical practices of a community. Adherents to and admirers of the strong program have recently begun to acknowledge the importance of both the hidden curriculum and mathematics as a cultural practice. For example, Silver (1987, p. 57) argued that “our students may realize greater educational benefits from our attention to the hidden curriculum of beliefs about the attitudes towards mathematics than from any improvements we could make in the ‘transparent’ curriculum of mathematics facts, procedures, and concepts.” Schoenfeld (1987b) provided an example of instruction that gives attention to the hidden curriculum:
With hindsight, I realise that what I succeeded in doing in the most recent versions of my problem-solving course was to create a microcosm of mathematical culture. Mathematics was the medium of exchange. We talked about mathematics, explained it to each other, shared false starts, enjoyed the interaction of personalities. In short, we became mathematical people …[By] virtue of this cultural immersion, the students experienced mathematics in a way that made sense, in a way similar to the way mathematicians live it (p. 213).
The interesting thing about Silver’s and Schoenfeld’s comments is that they appear in a book about the relationship between the strong program of cognitive science and mathematics education. Silver’s strongest instructional recommendation and Schoenfeld’s insights into the success of his course are in fact inconsistent with the implications of the strong program. They made these observations because their first hand knowledge of teaching and learning allowed them to transcend the constraints of the program. Schoenfeld, for example, talked of the way mathematicians live mathematics. His conceptualization of his problem-solving course calls into question the assumption that solo performance on a circumscribed set of tasks can be separated from other aspects of the student’s life. Schoenfeld’s example is particularly intriguing in that:
…early versions of the courses focused on the thinking tools I thought students needed for competent problem-solving performance. This meant giving training in particular heuristic strategies, teaching a prescriptive “managerial strategy” for self-regulation, and pointing directly to problems (or domains) where the “wrong” beliefs caused difficulty (p. 213).
This instructional approach, which was presumably found wanting, is precisely the type of recommendation that can be derived from the strong program.
More generally, Greeno (1988) recently argued that:
some information and skills. Some aspects of this view arehighlightcd if we think of mathematical learning as a kind of cognitive apprenticeship …The view is also informed by consid cring mathematical knowledge as it is embodied in the practice of the mathematical community …rather than as it is encoded in mathematical writings (p. 481).
Greeno also suggests that “we need to treat the concept of knowing relativistically... That is, the properties of someone’s knowing cannot be specified independently of the situation that provides a frame of reference for the person’s cognitive activity” (p. 482). Brown, Collins, and Duguid (1989) made similar points about the situated nature of knowledge, learning as enculturation, and cognitive apprenticeship. In a complete volte face with respect to Brown’s previous work (e.g. Brown & Burton, 1978; Brown & Van Lehn, 1982), they go on to suggest that concepts are in some ways analogous to a set of tools. Pursuing this analogy:
…learning how to use a tool involves far more than can be accounted for in any set of explicit rules. The occasions and conditions for use arise directly out of the context of activities of each community that uses the tool …Conceptual tools similarly reflect the cumulative wisdom of the culture in which they are used and the insights and experience of individuals. Their meaning is not invariant but a product of negotiation within the community (p. 33, emphasis added).
Greeno’s and Brown et al.’s comments suggest that we may be witnessing the first phase of a profound conceptual revolution that calls into question the basic tenets of the strong program. The radical reconceptualizations of knowledge, learning, and education proposed by these key members of the strong program indicate that a rapprochement with both social constructivism and action psychology (van Oers, this issue) might be possible. We can explore what such a rapprochement will involve by considering the implications of Greeno’s and Brown’s new position.
As we have noted, the strong program treats students as isolated environmentally driven systems. The new position views students as participants in cultural practices. This position, if it is to be more than mere rhetoric, implies that the relationship between the individual and the mathematical culture of the community to which he or she belongs is dialectical in nature. It is dialectical in that neither the individual nor the cultural is primary (Cobb, et al., in press a). Instead, the coordinated mathematical activities of individuals interactively constitute the communal mathematical practices that constrain the individuals’ mathematical activities. Conversely, the communal mathematical practices constrain the mathematical actions of the individuals who jointly constitute those practices (Cobb et al., in press b). From this perspective:
…culture is constantly in the process of being recreated as it is interpreted and renegotiated by its members. In this view, a culture is as much a forum for negotiating and renegotiating meaning and for explicating action as it is a set of rules …It follows from this view of culture as a forum that induction into the culture through education, if it is to prepare the young for life as lived, should also partake of the spirit of a forum, of negotiation, of the recreation of meaning. But this conclusion runs counter to traditions of pedagogy that derive from another time, another interpretation of culture, another conception of authority – one that looked at the process of education as transmission of knowledge and values by those who know more to those who knew less and knew it less expertly (Bruner, 1986, p. 123).
The tradition that Bruner’s conclusion runs counter to is that embodied in the strong program. In the new position articulated by Greeno and Brown et al., education as cognitive apprenticeship is the process of facilitating cognitive developments that enable students to increasingly participate in cultural mathematical practices. And opportunities to make these cognitive constructions arise for students as they are acculturated to those practices.
The teacher’s role in this process is to express his or her institutionalized authority by initiating and guiding the classroom community’s continual renegotiation of both social norms and mathematical meanings and practices. A case study of how one second grade teacher made possible the mutual construction of social norms more akin to those of mathematical communities than of traditional mathematics classrooms can be found in Cobb et al. (in press). It is important to note that these norms were jointly constructed by the teacher and students together. The teacher initiated these joint constructions and guided children’s participation in the process. This characterization of the teacher’s role should be distinguished from the view of the teacher as a social engineer who puts together social environments for students. In the latter view, students are not active participants in the construction of the classroom social context but are instead driven by their social environments.
The new position also implies that the teacher should initiate and guide the renegotiation of mathematical meanings and practices in the classroom. This can be contrasted with the strong program’s characterization of the teacher as one who organizes mathematical stimuli for environmentally driven students. The admonition to develop instructional representations that cause students to make correct constructions is displaced by a focus on the negotiation and institutionalization of mathematical meanings and practices in the classrooms. This does not rule out the use of manipulatives, diagrams and graphics in mathematics instruction. In fact, they would appear to play an essential role in helping students construct intuitive representations that make the abstract comprehensible. The crucial point is that the meanings of these instructional’ materials have to be negotiated by the teacher and students. In effect, the teacher as a participant in the mathematical practices of the wider community has to initiate the students into the interpretive stance he or she takes with regard to the materials. This, of course, is what good teachers do without thinking about it – they simply take the necessity of negotiating interpretations with their students for granted. The so-called instructional representations can then be seen as essential aspects of settings in which the teacher initiates and guides the classroom community’s negotiation of mathematical meanings. The students do not acquire mathematical knowledge by internalizing it from instructional representations in which the relevant mathematical relationships are said to be salient. Instead, they reorganize and elaborate their mathematical interpretations in the course of the negotiation process and thus come to participate increasingly in the mathematical practices of the wider community. Materials typically characterized as instructional representations are of value to the extent that they facilitate this increasing participation by making possible the negotiation of mathematical meanings and thus individual students’ construction of mathematical knowledge. In this view, correctness does not mean conforming to the dictates of an authority, be it a teacher, textbook, or the output of a computer model. Instead, it means making mathematical constructions that have “clout” in that they enable students to increasingly participate in socio-historically evolving mathematical practices of the community. More generally, the notion of mathematicg instruction as a delivery system is displaced by the view of teachers initiating and guiding emerging systems of mathematical meanings and practices.
I have been highly critical of the strong program in information-processing psychology with respect to my interests and purposes as a mathematics educator. In doing so, I have drawn on the work of members of the weak program to support my arguments. We have noted that the weak program acknowledges the importance of the common sense knowledge problem. Further, Minsky’s (1975) introduction of the frame as a data structure resulted in a step forward “from a passive model of information processing to one that tries to take account of the interactions between a knower and the world” (Dreyfus & Dreyfus, 1988, p. 28). It should therefore be clear that the points made in this chapter do not rule out the computer metaphor as one of many sources of analogies. The value of specific analogies will, of course, have to be judged on a case-by-case basis that is beyond the scope of this chapter. Admirers of the weak program can decide to what extent arguments concerning the importance of sensory-motor action, social interaction, and communal practices apply in various analogies. My primary concern has been to explore the consequences when we forget that the computer metaphor is just that – a metaphor.
I concluded by suggesting that the strong program’s attempt to equate cognitive processes with runnable programs may soon become a historical curiosity. The initial signs that this may already be happening do not reflect responses to empirical anomalies. Rather, they seem to stem from an acknowledgement that an alternative conceptualization of cognition is more attractive. This alternative views mathematical activity as culturally situated practice (e.g. Lave, 1988; Rogoff & Lave, 1984). With regard to the apparent attractiveness of this view, I would speculate that experientially based accounts of classroom life have had a powerful influence (e.g. Lampert, 1986; Schoenfeld, 1985). In addition, ethical values may well have played a central role. As Rorty (1983) contentiously stated:
The only sorts of policy makers who would be receptive to most of what presently passes for “behavioural science” would be rulers of the Gulag, or of Huxley’s Brave New World, or a conspiracy of those who personify Foucault’s “forces of domination” (pp. 161-162).
The view of students as environmentally driven systems and of teachers as people who manipulate students’ environments in order to control their behavior is rather dehumanizing. By comparison, a characterization of students as active participants in classroom communities and of teachers as initiators and guides of negotiation processes might seem positively enlightened. Perhaps the time is ripe for American cognitive science and mathematics education to question and transcend its behaviorist origins.
Anderson J. R. (1983) The architecture of cognition. Cambridge MA: Harvard University Press.
Anderson J. R., Greeno J. G., Kline P. J. & Neves D. M. (1981) Acquisition of problem-solving skill. In: J. R. Anderson (ed.) Cognitive skills and their acquisition (pp. 191–230) Hillsdale NJ: Erlbaum.
Ashcraft M. H. (1982) The development of mental arithmetic: A chronometric approach. Developmental Review 2: 213–236.
Bakhurst D. (1988) Activity, consciousness, and communication. Quarterly Newsletter of the Laboratory of Comparative Human Cognition 10: 31–39.
Balacheff N. (1986) Cognitive versus situational analysis of problem-solving behavior. For the Learning of Mathematics, 6 (3): 10–12.
Barnes D. & Todd F. (1977) Communicating and learning in small groups. London: Routledge & Kegan Paul.
Bauersfeld H. (1980) Hidden dimensions in the so-called reality of a mathematics classroom. Educational Studies in Mathematics 11: 23–41.
Beilin H. (1981, June) Raga and the new functionalism. Paper presented at the Eleventh Symposium of the Jean Piaget Society, Philadelphia.
Bereiter. C. (1985) Towards it solution of the learning paradox. Review of Educational Research 55: 201–226.
Bernstein R. J. (1983) Beyond objectivism and relativism: Science, hermeneutics, and praxis. Philadelphia: University of Pennsylvania Press.
Bidell T. R., Wansart W. & De Ruiter J. A. (1986, April) Model building: Towards a constructivist theor y of loaning merdttrmsnts. Pallor presented at the annual meeting of the American Educational Research Association, Chicago.
Bishop A. (1985) The social construction of meaning – a significant development for mathematics education? For the Learning of Mathematics, 5 (1): 24–28.
Bloor D. (1976) Knowledge and social imagery. London: Routledge and Kegan Paul.
Bloor D. (1983) Wittgenstein: A social theory of knowledge. New York: Columbia University Press.
Blumer II. (1969) Symbolic interactionism: Perspectives and method. Englewood Cliffs NJ: Prentice Hall.
Briars D. J. (1982) Implications from information-processing psychology for research on mathematics learning and problem solving. In: F. K. Lester & J. Garofalo (eds.) Mathematical problem solving: Issues in research. Philadelphia: Franklin Institute Press.
Briars D. J. & Larkin J. H. (1984) An integrated model of skill in solving elementary word problems. Cognition and Instruction 1: 245–297.
Brown J. S. & Burton R. B. (1978) Diagnostic models f or procedural bugs in mathematical skills. Cognitive Science 2: 155–192.
Brown J. S., Collins A. & Duguid P. (1989) Situated cognition and the culture of learning. Educational Researcher, 18 (1): 32–42.
Brown A. L. & DeLoache J. S. (1978) Skills, plans, and self-regulation. In: R. S. Siegler (ed.) Children’s thinking: What develops (pp. 3–35) Hillsdale NJ: Erlbaum.
Brown J. S. & Van Lehn K. (1982) Towards a generative theory of “bugs.” In T. R. Carpenter J. M. Moser, & T. A. Romberg (eds.) Addition and subtraction: A cognitive perspective (pp. 117–135) Hillsdale NJ: Erlbaum.
Bruner J. (1986) Actual minds, possible worlds. Cambridge MA: Harvard University Press.
Bruner J. & Haste H. (1987) Introduction. In: J. Bruner & H. Haste, (eds.) Making sense: The child’s construction o f the world (pp. 1–25) New York: Methuen.
Buchanan B. (1982) New research on expert systems. In: J. E. Hayes D. Michie, & Y.-H. Pao (eds.) Machine intelligence (pp. 269–299) Chichester, England: Ellis Horwood.
Carpenter T. P. & Moser J. M. (1982) The development of addition and subtraction problem solving skills. In: T. P. Carpenter J. M. Moser, & T. A. Romberg (eds.) Addition and subtraction: A cognitive perspective (pp. 9–24) Hillsdale NJ: Erlbaum.
Carpenter T. P. & Moser J. M. (1984) The acquisition of addition and subtraction concepts in grades one through three. Journal for Research in Mathematics Education 15: 179–202.
Carraher T. N. & Carraher D. W. (1987) Mathematics as personal and social activity. Paper presented at the International Conference on Success or Failure? The Child’s Development at School, Poitiers, France.
Carraher T. N., Carraher W. C. & Schliemann A. D. (1985) Mathematics in streets and schools. British Journal of Developmental Psychology 3: 21–29.
Carraher T. N., Carraher D. W. & Schliemann A. D. (1987) Written and oral mathematics. Journal for Research in Mathematics Education 18: 83–97.
Champagne A. B. & Rogalska-Saz J. (1984, April) Learning to borrow: A theoretical analysis of traditional subtraction instruction. Paper presented at the annual meeting of the American Educational Research Association, New Orleans.
Cobb P. (1985) Mathematical actions, mathematical objects, and mathematical symbols. Journal of Mathematical Behavior 4: 127–134.
Cobb P. (1986) An investigation into the sensory-motor and conceptual origins of the basic addition facts. In: L. Burton & C. Hoyles (eds.) Proceedings of the Tenth Conference of the International Group for the Psychology of Mathematics Education (pp. 141–146) London: University of London Institute of Education.
Cobb P. (1987a) Information-processing psychology and mathematics education – A constructivist perspective. Journal of Mathematical Behavior 6: 3–40.
Cobb P. (1987b) An investigation of young children’s academic arithmetic contexts. Educational Studies in Mathematics 18: 109–124.
Cobb P. (in press) Multiple perspectives. In: L. P. Steffe & T. Woods (eds.) Transforming early childhood mathematics education. Hillsdale NJ: Erlbaum.
Cobb P., Wood T. & Yackel E. (in press, a) A constructivist approach to second grade mathematics. In: E. von Glasersfeld (ed.) Constructivism in mathematics education. Dordrecht: Reidel.
Cobb P., Wood T. & Yackel E. (in press, b) Analogies from the philosophy and sociology of science for understanding classroom life. International Journal of Science Education.
Cobb P., Yackel E. & Wood T. (1988) Curriculum and teacher development: Psychological and anthropological perspectives. In: E. Fennema T. P. Carpenter, & S. J. Lamon (eds.) Integrating research on teaching and learning mathematics (pp. 92–131) Madison: Wisconsin Center for Educational Research, University of Wisconsin, Madison.
Cobb P., Yackel E. & Wood T. (in press) Young children’s emotional acts while doing mathematical problem solving. In: D. B. McLeod & V. M. Adams (eds.) Affect and mathematical problem solving: A new perspective. New York: Springer-Verlag.
Davis R. B. (1984) Learning mathematics: The cognitive science approach to mathematics education. Norwood NJ: Ablex.
Davis P. J. & Hersh R. (1981) The mathematical experience. Boston: Houghton Mifflin.
De Millo R., Lipton R. & Perlis A. (1986) Social processes and proofs of theorems and programs. In: T. Tymoczko (ed.) New directions in the philosophy of mathentatirs (pp. 2h7–285) Boston: Birkhauser.
DeCorte E., Verschaffel L. & DeWin L. (1985) The influence of rewording verbal problems on children’s problem representations and solulithis. Journal of Ethical lanai Psychology 77: 460–470.
Dreyfus H. L. & Dreyfus S. E. (1986) Mind over machine. New York: Free Press.
Dreyfus H. L. & Dreyfus S. E. (1988) Making a mind versus modeling the brain: Artificial intelligence back at a branchpoint. In: S. R. Graubard (ed.) The artificial intelligence debate: False starts, real foundations (pp. 15–43) Cambridge MA: MIT Press.
D’Ambrosio U. (1985) Ethnomathematics and its place in the history and pedagogy of mathematics. For the Learning of Mathematics, 5 (1): 44–48.
Ernst G. W. & Newell A. (1969) CPS: A case study in generality and problem solving. New York: Academic Press.
Fischbein E. (1987) Intuition in science and mathematics. Dordrecht, Holland: Reidel.
Furth H. G. (1969) Piaget and knowledge. Englewood Cliffs NJ: Prentice-Hall.
Gadamer, H-G. (1986) Truth and method. New York: Crossroad.
Gasking D. (1955) Mathematics and the world. In: A. Flew (ed.) Logic and language (pp. 412–454) Garden City NY: Anchor Books.
Gergen K. J. (1986) Correspondence versus autonomy in the language of understanding human action. In: D. W. Fiske & R. A. Shweder (eds.) Metatheory in social science (pp. 136–162) Chicago: University of Chicago Press.
Glasersfeld E. von (1984) An introduction to radical constructivism. In: P. Watzlawick (ed.) The invented reality (pp. 17–40) New York: Norton. http://cepa.info/1279
Goodman N. (1986) Mathematics as an objective science. In: T. Tymoczko (ed.) New directions in the philosophy o f mathematics (pp. 79–94) Boston: Birkhauser.
Greeno J. G. (1987) Instructional representationsbased on research about understanding. In: A. H. Schoenfeld (ed.) Cognitive science and mathematics education (pp. 61–88) Hillsdale NJ: Erlbaum.
Greeno J. G. (1988) The situated activities of learning and knowing mathematics. In: M. J. Behr C. B. Lacampagne, & M. M. Wheeler (eds.) Proceedings of the Tenth Annual Meeting of the North American Chapter of the International Group for the Psychology of Mathematics Education (pp. 481–521) DeKalb IL: Northern Illinois University.
Hatano G., Miyake Y. & Binks M. G. (1977) Performance of expertabacus operations. Cognition 5: 57–7 1.
Johnson M. (1987) The body in the mind: The bodily basis of reason and imagination. Chicago: University of Chicago Press.
Johnson-Laird P. N. (1983) Mental models: Towards a cognitive science of language, inference, and consciousness. Cambridge MA: Harvard University Press.
Kieren T. E. & Pirie S. E. B. (1988, February) Recursion and mathematical experience. Paper presented at the Conference on the Epistemological Foundations of Mathematical Experience, Athens GA.
Kohlberg L. & Mayer R. (1972) Development as the aim of education. Harvard Educational Review 42: 449–496.
Kuhn T. S. (1970) The structure of scientific revolutions (2nd ed.) Chicago: University of Chicago Press.
Lakatos I. (1970) Falsification and the methodology of scientific research programmes. In: I. Lakatos & A. Musgrave (eds.) Criticism and the growth of knowledge (pp. 91–196) Cambridge: Cambridge University Press.
Lakatos I. (1976) Proofs and refutations. Cambridge: Cambridge University Press.
Lampert M. (1986) Knowing, doing, and teaching multiplication. Cognition and Instruction 3: 305–342.
Larkin J. H. (1981) Enriching formal knowledge: A model for learning to solve textbook physics problems. In: J. R. Anderson (ed.) Cognitive skills and their acquisition (pp. 311–334) Hillsdale NJ: Erlbaum.
Lave J. (1988) Cognition in practice: Mind, mathematics and culture in everyday life. Cambridge: Cambridge University Press.
Lave J., Murtaugh M. & de la Rocha O. (1984) The dialectic of arithmetic in grocery shopping. In: B. Rogoff & J. Lave (eds.) Everyday cognition: Its development in social context (pp. 67–94) Cambridge MA: Harvard University Press.
Lawler R. W. (1981) The progressive construction of mind. Cognitive Science 5: 1–30.
Levina R. E. (1981) L. S. Vygotsky’s ideas about the planning function of speech in children. In: J. V. Wertsch (ed.) The concept of activity in Soviet psychology (pp. 279–299) Armonk NY: Sharpe.
Man D. (1981) Artificial intelligence -a personal view. In: J. Haugeland (ed.) Mind design (pp. 129–142) Cambridge MA: Bradford Books.
McDermott D. (1981) Artificial intelligence meets natural stupidity. In: J. Haugeland (ed.) Mind design (pp. 143–169) Cambridge MA: Bradford Books.
McNeill D. (1985) Language viewed as action. In: J. V. Wertsch (lid.) Culture. communication, and cognition (pp. 258–270) Cambridge: Cambridge Universii y Press.
Meade G. H. (1934) Mind, self; and society. Chicago: University of Chicago Press.
Mehan H. & Wood H. (1975) The reality of ethnomethodology. New York: Wiley.
Minsky M. (1975) A framework for representing knowledge. In: I’. II. Winston (ed.) The psychology of computer vision. (pp. 211–277) New York: McGraw Hill.
Minsky M. (1981) A framework for representing knowledge. In: J. Haugeland (ed.) Mind design (pp. 95–128) Cambridge MA: Bradford Books.
Minsky M. (1985) The society of mind. New York: Simon and Schuster.
National Research Council (1989) Everybody counts: A report to the nation on the future of mathematics education. Washington DC: National Academy Press.
Neuman D. (1987) The origin of arithmetic skills: A phenomenographic approach. Goteborg, Sweden: Acta Universitatis Gothoburgensis. (Goteborg Studies in Educational Sciences, 62.)
Nicholls J., Cobb P., Wood T., Yackel E. & Patashnik M. (in press) Dimensions of success in mathematics: Individual and classroom differences. Journal for Research in Mathematics Education.
Orton R. E. (1988, April) Constructivist and information processing views of representation in mathematics education. Paper presented at the annual meeting of the American Educational Research Association, New Orleans.
Peirce C. S. (1935) Collected papers of Charles Sanders Peirce, Vol. 5 (C. Hartshone & P. Weiss, Eds.) Cambridge MA: Harvard University Press.
Perret-Clermont A. N. (1980) Social interaction and cognitive development in children. New York: Academic Press.
Piaget J. (1971) Biology and knowledge. Chicago: University of Chicago Press.
Pimm D. (1981) Metaphor and analogy in mathematics. For the Learning of Mathematics, 1 (3): 47–50.
Pimm D. (1987) Speaking mathematically: Communication in mathematics classrooms. London: Routledge and Kegan Paul.
Putnam H. (1988) Representation and reality. Cambridge MA: Bradford Books.
Resnick L. B. (1983) Towards a cognitive theory ofinstruction. In: S. G. Paris G. M. Olson, & W. H. Stevenson (eds.) Learning and motivation in the classroom (pp. 5–38) Hillsdale NJ: Erlbaum.
Resnick L. B. & Omanson S. F. (1987) Learning to understand arithmetic. In: R. Glaser (ed.) Advances in instructional psychology (Vol. 3: 41–95) Hillsdale NJ: Erlbaum.
Riley M. S., Greeno J. G. & Heller J. 1. (1983) Development of children’s problem-solving ability in arithmetic. In: H. P. Ginsburg (ed.) The development of mathematical thinking (pp. 153–196) New York: Academic Press.
Rogoff B. & Lave J. (eds.) (1984) Everyday cognition: Its development in social context. Cambridge MA: Harvard University Press.
Rorty R. (1982) Consequences of pragmatism. Minneapolis: University of Minnesota Press.
Rorty R. (1983) Method and morality. In: N. Haan R. N. Bellah, & P. Robinson (eds.) Social science as moral inquiry (pp 155–176) New York: Columbia University Press.
Sachs H., Schegloff E. & Jefferson, G, (1974) A simplest systematics for the analysis of turn taking in conversation. Language 50: 696–735.
Saxe G. B. (1988, January) The interplay between children’s learning formal and informal social contexts. Paper presented at the conference on the Scientific Practice of Science Education, Berkeley, California.
Schoenfeld A. H. (1985) Mathematical problem solving. New York: Academic Press.
Schoenfeld A. H. (ed.) (1987a) Cognitive science and mathematics education. Hillsdale NJ: Erlbaum.
Schoenfeld A. H. (1987b) What’s all the fuss about metacognition. In: A. H. Schoenfeld (ed.) Cognitive science and mathematics education (pp. 189–216) Hillsdale NJ: Erlbaum.
Schutz A. (1962) The problem of social reality. The Hague: Martinus Nijhoff.
Searle J. (1984) Minds, brains, and science. Cambridge MA: Harvard University Press.
Siegler R. S. & Robinson M. (1982) The development of numerical understandings. In: H. W. Reese & L. P. Lipsitt (eds.) Advances in child development and behavior (Vol. 1: 241–312) New York: Academic Press.
Siegler R. S. & Shrager J. (1984) Strategy choice in addition: How do children know what to do. In: C. Sophian (ed.) Origins of cognitive skills (pp. 229–293) Hillsdale NJ: Erlbaum.
Sigel I. E. (1981) Social experience in the development of representational thought: Distancing theory. In: I. E. Sigel D. M. Brodzinsky, & R. M. Golinkoff (eds.) New directions in Piagetian theory and practice (pp. 203217) Hillsdale NJ: Erlbaum.
Silver B. (1984) Using meta-level inferences to constrain search and to learn strategies in equation solving. Unpublished doctoral dissertation. University of Edinburgh, Department of Artificial Intelligence.
Silver E. A. (ed.) (1985) Teaching and learning mathematical problem solving: Multiple research perspectives. Hillsdale NJ: Erlbaum.
Silver E. A. (1987) Foundations of cognitive theory and research for mathematics problem solving. In: A. H. Schoenfeld (ed.) Cognitive science and mathematics education (pp. 33–60) Hillsdale NJ: Erlbaum.
Simon H. A. (1969) The sciences of the artificial. Cambridge MA: MIT Press.
Steffe L. P., Cobb P. & Glasersfeld E. von (1988) Construction of arithmetical meanings and strategies. New York: Springer-Verlag.
Steffe L. P., Glasersfeld E. von, Richards, J & Cobb P. (1983) Children’s counting types: Philosophy, theory, and applications. New York: Praeger Scientific.
Thorndike E. L. (1922) The psychology of arithmetic. New York: MacMillan.
Toulmin S. (1963) Foresight and understanding. New York: Harper Torchbooks.
Tymoczko T. (ed.) (1986) New directions in the philosophy o f mathematics. Boston: Birkhauser.
Van Lehn K. (1983) On the representation of procedures in repair theory. In: H. P. Ginsburg (ed.) The development of mathematical thinking (pp. 201–252) New York: Academic Press.
Van Lehn K. (1986) Arithmetic procedures are induced from examples. In: J. Hiebert (ed.) Conceptual and procedural knowledge: The case o f mathematics (pp. 133–180) Hillsdale NJ: Erlbaum.
Voigt J. (1985) Patterns and routines in classroom interaction. Recherches en Didactique des Mathematiques 6: 69–118.
Wenger R. H. (1987) Cognitive science and algebra learning. In: A. H. Schoenfeld (ed.) Cognitive science and mathematics education (pp. 217–252) Hillsdale NJ: Erlbaum.
Winograd T. & Flores F. (1986) Understanding computers and cognition: A new foundation for design. Norwood NJ: Ablex.
Wittgenstein L. (1964) Remarks on the foundations of mathematics. Oxford: Blackwell.
Wittgenstein L. (1982) Last writings on the philosophy of psychology (Vol. 1) Chicago: University of Chicago Press.
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