Applications and Misapplications of Cognitive Psychology to Mathematics Education

There is a frequent misperception that the move from behaviorism to cognitivism implied an abandonment of the possibilities of decomposing knowledge into its elements for purposes of study and decontextualizing these elements for purposes of instruction. We show that cognitivism does not imply outright rejection of decomposition and decontextualization. We critically analyze two movements which are based in part on this rejection--situated learning and constructivism. Situated learning commonly advocates practices that lead to overly specific learning outcomes while constructivism advocates very inefficient learning and assessment procedures. The modern information-processing approach in cognitive psychology would recommend careful analysis of the goals of instruction and thorough empirical study of the efficacy of instructional approaches. Following on the so-called "cognitive revolution" in psychology that began in the 1960s, education, and particularly mathematics and science education, has been acquiring new insights from psychology, and new approaches and instructional techniques based on these insights. At the same time, cognitive psychologists have being paying increasing attention to education as an area of application of psychological knowledge and as a source of important research problems. There is every reason to believe that as research in cognitive psychology progresses and increasingly addresses itself to educational issues, even closer and more productive links can be formed between psychology and mathematics education. However, there is a tendency now to present all manner of educational opinion as bearing a stamp of approval from cognitive psychology. For instance, Lamon and Lesh (1992) write in the introduction to a recent book they edited: "Behavioral psychology (based on factual and procedural rules) has given way to cognitive psychology (based on models for making sense of real-life experiences), and technology-based tools have radically expanded the kinds of situations in which mathematics is useful, while simultaneously increasing the kinds of mathematics that are useful and the kinds of people who use mathematics on a daily basis. In response to these trends, professional and governmental organizations have reached an unprecedented, theoretically sound, and future-oriented new consensus about the foundations of mathematics in an age of information." (p. 18-19) In fact, as in many recent publications in mathematics education, much of what is described in that book reflects two movements, "situated learning" and "constructivism", which have been gaining influence on thinking about education and educational research. In our view, some of the central educational recommendations of these movements have questionable psychological foundations. We wish to compare these recommendations with current empirical knowledge about effective and ineffective ways to facilitate learning in mathematics and to reach some conclusions about what are the effective ways. A number of the claims that have been advanced as insights from cognitive psychology are at best highly controversial and at worst directly contradict known research findings. As a consequence, some of the prescriptions for educational reform based on these claims are bound to lead to inferior educational outcomes and to block alternative methods for improvement that are superior. These two schools, of situated learning and constructivism, are not identical: situated learning emphasizes that knowledge is maintained in the external, social world; constructivism argues that knowledge resides in an individual's internal state, perhaps unknowable to anyone else. However, both schools share the general philosophical positions that knowledge cannot be decomposed or "decontextualized" for purposes of either research or instruction, and each group often appeals to the writings of the other for support. Since rejection of decomposition and decontextualization seems to be the core common ground of this "new look" in mathematics education, we will first examine the degree to which modern cognitive psychology lends support to that rejection. Decomposition and Decontextualization In an influential educational paper, Resnick and Resnick (1992) provide a succinct statement of a common theoretical understanding in cognitive psychology called the information-processing approach: "Information-processing theories of cognition (Anderson, 1983; Newell and Simon 1972), for example, analyze cognitive performances into complexes of rules, but performances critically depend on interactions among those rules. Each rule can be thought of as a component of the total skill, but the rules are not defined independently of one another. The `competence' of a problem-solving system thus depends on how the complex of rules acts together." (p. 43) A number of educational researchers (e.g., Shepard, 1991) have cited Resnick and Resnick as reporting that cognitive psychology has shown that cognition cannot be analyzed into components. On the contrary, what the above quote states (and what the cognitive literature they allude to says) is quite the opposite. This literature, incorporating extensive empirical evidence, deals both with the "rules" (components or processes) to which Resnick and Resnick refer, and also, emphatically, with the interactions among these processes: the interaction between these processes and sensory stimuli (the organism's awareness of its current environment), and the interaction of processes with information (other components of knowledge) that has been assembled in memory through previous engagement with the environment. The whole purpose of modeling cognition with computer programs--a central tool in information-processing approaches--is to develop a full picture of these interactions among components of knowledge. Unlike earlier behaviorist theories, information-processing theories do not posit a simple one-toone mapping between individual rules or knowledge components and individual bits of behavior. They deny this precisely because continual interaction can be observed among components of knowledge and behavior. Information-processing psychology has advanced rapidly by developing methods both for identifying the components and for studying them in their interactions with their entire contexts. This is the meaning of the "unified theories of cognition" (e.g., Newell, 1991) which has guided so much of the recent research and theory-building. Thus, componential analysis is very much alive and well in modern cognitive psychology. The information-processing approach tries both to deepen our understanding of the components and to understand the relations among them and with their environments. Examples of these methods of componential analysis are the use of think-aloud protocols as data (Ericsson and Simon, 1993) and the use of models that simulate the interactions of perceptual, memory, learning and thinking processes over a wide range of cognitive tasks (e.g., Anderson, 1993; Feigenbaum and Simon, 1984; Newell, 1991). With respect to decomposition, the correct principle is: Assessing learning and improving learning methods requires careful task analysis at the level of component skills, intimately combined with study of the interaction of these skills in the context of broader tasks and environments. o much for decomposition; what about decontextualization? Because components interact with one another, it might prove impossible to invoke and study them outside certain contexts. To cite a simple example, processes for carrying out multi-column addition will only be evoked in the context of a problem large enough to require carrying; they cannot be studied by posing problems of adding 3+4 or 5+2. While some context will often be required to assess a component, there are always bounds on how complex such a context need be. It is a well-documented fact of human cognition that large tasks decompose into nearly independent subtasks (Simon, 1981, Chapter 7; Card, Moran & Newell, 1983), so that only the context of the appropriate subtask is needed to study its components. For instance, there is no need to teach or assess the ability to perform multi-column addition in the context of calculating income taxes. The process of adding tax deduction items is the same as the process of taking sums in other tasks. And whether one does the sum by hand or by calculator is unlikely to affect the rest of the tax calculation procedures. Thus, the larger procedure is independent of the summing procedure, just as the summing procedure is independent of the larger procedure. The addition procedures might become tied into the tax calculation procedures--for example, ignoring cents in calculating the sums. Such specialized subprocedures are especially frequent at high levels of expertise. However, this just means that the expert's procedure involves a structure of different subtasks than the novice's, not that it cannot be analyzed into components nor that these components cannot still be assessed in subtasks of the original task. Thus, with respect to decontextualization, while it may be difficult to get behavioral measures of individual components; these components organize themselves into subtasks to achieve subgoals, and these subgoals can have independent, assessable, behavioral realizations. It does not require recondite research to demonstrate the near-decomposability of human tasks. Every page of a good cookbook contains examples of assumed component procedures (e.g., sauté, parboil) as do the how-to books in domains like carpentry, plumbing or car repair. Moreover, one can apply these procedures in new contexts such as when a chemistry lab requires us to boil water. Fortunately for us human beings, with our very limited short-term memories, the workings of each component can be understood without simultaneous awareness of the details of all the other components. With r