A Schema Theory

Next, for each category, Marshall devised a schema, i.e., an abstraction capturing its salient relational features. She then built an instructional computer program, incorporating these five schemas, to teach adults having limited problem-solving skills to use them in solving arithmetic story problems. She tested her program with college psychology students to see whether they could acquire the schema and solve story problems successfully. They could. The idea behind a schema theory is that people (not necessarily consciously) use abstract patterns, derived from regularities found in experience, to guide their actions, including problem solving. Naturally occurring schemas are hard to observe, but when Marshall could teach hers to subjects and they could solve story problems, this was evidence that they were using similar mental structures. [Cf. Schemas in Problem Solving, Cambridge University Press, 1995.]

Are there direct implications for mathematics education? Does one want to concentrate on training students to solve a specific collection of word problems, essentially procedurally, by providing a set of schemas? Is that what mathematics, even at the sixth-grade, ought to be about? Or, partly about?

Mathematics as Skills Acquisition

However, beginning in the 60's, American psychological research moved away from the sway of behaviorism, which valued mainly directly observable phenomena, and thus disparaged any mention of the mind or its contents as unscientific. There was a "cognitive revolution" in which the mind was often regarded metaphorically as a computer and was seen in information processing (IP) terms. Yet, despite having opened the mind to study, cognitive psychologists have continued to focus mainly on procedural knowledge, causing Pat Thompson, a mathematics education researcher, to ask, "In what way does a detailed understanding of how students perform tasks mindlessly help us improve mathematics education?" Indeed, Stellan Ohlsson, a cognitive psychologist at University of Pittsburgh, has also observed that current information processing theory is especially good at capturing skills acquisition (procedural knowledge), whereas schools, for the most part, aim to teach concepts and principles (conceptual knowledge). He feels that the IP-view of mind is limited, not false, that it has not yet risen to the real challenges of (conceptual) knowledge acquisition. [Cf. Ohlsson, "Cognitive Science and Instruction: Why the Revolution is Not Yet Here" in H. Mandl, et al Learning and instruction: European Research in an International Context, Vol. 2.1, Pergamon, 1990.]

In an attempt to address his own critique, Ohlsson, et al, designed two computational models of arithmetic, one simulating rote performance, the other simulating performance based on an understanding of place value. Was his attempt successful? Yes and no. In a published response to Ohlsson's work, Alan Schoenfeld, while appreciating the complexity of such models and acknowledging the intellectual work required to produce them, contends that Ohlsson, et al, finessed the crucial issue - "what it means to have a conceptual understanding of base 10 subtraction, as opposed to a procedural one." Furthermore, Ohlsson assumed rather ideal students - "point out an (arithmetic) error once, and it's fixed!" [Cf. "The Cognitive Complexity of Learning and Doing Arithmetic," JRME 23(2), 441-482.]

• Knowledge is acquired by construction, not by transmission alone. Compelling evidence for this is provided by the work on procedural bugs and misconceptions -- it is highly unlikely that students acquire them from direct teaching. For example, young children often make systematic subtraction errors, the most common of which is always subtracting the smaller digit from the larger, regardless of position, and many preservice elementary teachers believe "division (always) makes smaller." Surely, no one taught them this.

• Knowledge acquisition involves restructuring -- not only does the amount of a person's knowledge gradually increase, it gets reorganized. Children do not think like miniature or incomplete adults. For example, in attributing unknown properties to animate objects, Hatano found young children rely on similarity-based inference, whereas older children and adults use category-based inference. He finds studies of conceptual change, both in the history of science and in cognitive development, especially relevant because fundamental conceptual change is perhaps the most radical kind of (mental) restructuring. [Cf. Carey, "Conceptual differences between children and adults," Mind and Language 3, 1988; Kuhn, The Structure of Scientific Revolutions, University of Chicago Press, 1970.]

• The process of knowledge acquisition is constrained both internally, by what one already knows, and externally, by cultural artifacts such as shared language and notation. This explains, in part, why different individuals acquire similar, but not identical, knowledge.

• Knowledge is domain specific. This serves cognitive economy -- in problem solving, one need only access relevant knowledge. However, what is acquired in one domain can be transferred to another (e.g., through analogy) or generalized to a variety of domains (e.g., by abstracting structural commonalties).

• Knowledge acquisition is "situated," i.e., it reflects how it was originally acquired and has been used -- it consists not only of abstract rules, laws, and formulas, but also of personal experiences. Becoming an expert, say in mathematics or physics, may be a process of "desituating" one's knowledge to make it less context-bound, less tied to surface features.

Despite the above common principles, one can find quite different answers to some basic questions among those who work in cognitive psychology and mathematics education.

All these views occur in psychology and mathematics education (although not always widely) and can be fruitfully considered. However, we will focus on a recent debate in Educational Researcher between the traditional information processing (or computational) view in cognitive psychology and the newer, situated learning view espoused by some, but not all, cognitive scientists and (mathematics) education researchers. One might think that such differing perspectives were a purely academic matter, having little practical significance. This is not the case. They can influence what one sees as appropriate teaching methods, as well as the kinds of research questions asked.

Cognition as Information Processing

John Anderson and Herbert Simon are psychologists who have been in the forefront of the cognitive revolution since its beginnings in the 60's. They helped prevail over the stimulus-response ideas of behaviorism and legitimized the idea that the mind works on internal representations (production rules), albeit in a complex way. Anderson's ACT* was the first well-developed cognitive architecture. [Cf. M. I. Posner (ed.), Foundations of Cognitive Science, MIT Press, pp. 109-119]. Simon pioneered the use of think-aloud protocols to gain insight into human problem solving and developed, with Allen Newell, the General Problem Solver, whose basic heuristic was means-ends analysis -- a general method for narrowing the "distance" between the problem solver's current state and his/her goal state. More recently, such work has begun to focus on the importance of domain knowledge and has inspired practical programs for medical diagnosis and electronic trouble-shooting. Anderson has also designed a geometry tutor, GPTutor, which has been implemented in the classroom [AERJ 31(3), 579-618]. And now, after decades of effort, just as their work is beginning to address itself effectively to educational issues, they see a threat to this potential for real progress coming from situated learning and constructivism -- two quite different perspectives found in education research, which tend to ignore, if not reject, an IP-view of mind.

Situated Cognition

The situated view is advocated by such researchers as Jean Lave of Berkeley, who co-authored Situated Learning: Legitimate Peripheral Participation, and Jim Greeno of Stanford. They study how individuals learn to act within complex social situations, for example, as apprentices might. Such studies often have an anthropological flavor, e.g., Carraher et al's work on Brazilian street sellers who can calculate the cost of three items costing fifty centavos, but not the cost of fifty items costing three centavos, or Lave's study of U.S. homemakers who did well when making supermarket best-buy calculations, but much worse on equivalent paper-and-pencil problems. (Is workplace mathematics, e.g., for engineers, significantly different from their university mathematics? If so, in what ways?) Within mathematics, an apprentice-like situation can sometimes be found in Moore method courses and in dissertation supervision. Furthermore, examiners of Ph.D. orals sometimes concentrate on whether a candidate "acts like" and "sounds like" a mathematician, suggesting they are (implicitly) taking a situated perspective.

The Debate Between Information Processing and Situated Cognition

>From the situative perspective, one is less likely to speak of knowledge and tasks than of improved participation. Whether transfer occurs depends on how a situation is transformed. Whether it is difficult or easy for the learner depends on how the learner was "attuned to the constraints and affordances" in the initial learning activity. For example, when students are given instruction about refraction prior to shooting targets under water, they are more likely to become attuned to the angular disparity of a projectile's trajectory before and after entering the water, and hence, perform better. Also, Greeno distinguishes between generality and abstraction using an example from mathematics. If students learn correct rules for manipulating symbols without learning that mathematical expressions represent concepts and relationships, what they learn may be abstract, but it is not general.

While it is certainly true that all knowledge is learned in specific contexts, information-processing psychologists like Anderson, et al, tend to think in terms of acquiring abstract rules, which are subsequently applied in specific situations, whereas situated cognition adherents, such as Lave and Greeno, focus on how individuals learn to participate within communities of practice and how their development is shaped by the activities they engage in. In fact, they tend to avoid speaking in terms of abstract knowledge.