by Annie and John Selden
Different Questions and (often very) Different Views of Mathematics
Shared Underlying Principles of Knowledge Axquisition
Where is the Mind?
Some Concluding Reflections
Some two decades ago, about the time the International Group for the Psychology of Mathematics Education was being formed through the efforts of people like the mathematician, Hans Freudenthal, and the psychologist, Efraim Fischbein, one leading scholar observed, "It appears that mathematicians and psychologists have nothing to say to one another." Not everyone feels this way. For some time, accomplished mathematicians, such as Lebesgue, PoincarÃ©, Hadamard, and PÃ³lya, have commented on education and the psychological mechanisms underlying mathematical discoveries, but without doing systematic psychological research.
[Cf. Science and Method, Dover, 1914; Hadamard, An Essay on the Psychology of Invention in the Mathematical Field, Princeton University Press, 1945; Polya Mathematics and Plausible Reasoning, Princeton University Press, 1954.]
Mathematicians, mathematics educators, and psychologists often seem to be addressing different questions. While creativity may interest mathematicians and mathematics educators seek to understand mathematical learning with the aim of improving it, psychologists tend to use mathematical tasks to study aspects of general cognition such as problem solving. Even so, couldn't general learning theories tell us something about learning mathematics? Many mathematics educators would say "not much" -- knowledge acquisition is largely domain specific. Learning mathematics has features unlike learning, say, biology. Solving mathematics problems is not the same as employing a heuristic search to solve syntactic reasoning puzzles, like the Tower of Hanoi. Rather, what counts is a rich, organized set of connections between concepts, together with imagery and reasoning.
When cognitive psychologist Sandra Marshall wanted to lay out a theory of schema development, she selected the domain of arithmetic story problems. She combed sixth-grade, eighth-grade, and remedial college textbooks, as well as the standardized tests given to third-, sixth-, and eighth-grade California public school students. Altogether, she analyzed 3,027 story problems. She found that most (2,695) could be categorized according to the situations they depicted, despite having highly diverse surface features. She classified these as Change, Group, Compare, Restate, and Vary problems. A sample Restate problem is: At the pet store there are twice as many kittens as puppies in the store window. There are 8 kittens in the window. How many puppies are also in the window? Taken alone or in combination, these five categories proved sufficient to describe almost all situations found in common arithmetic story problems. Marshall then conducted four studies (with accelerated sixth-graders, less able sixth-graders, elementary and secondary teachers, and college psychology students) to determine whether subjects could learn to recognize these situations and sort them into the above categories. They could -- some even learned to do so with only 1/2 to 2 1/2 hours of instruction.
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?
In addition to testing their own theories of cognition, psychologists tend to work with tasks that mathematicians and mathematics educators often consider of lesser importance. For example, early in the 20th-century, when facts and rote skills were emphasized in the public schools, the psychologist, Thorndike, studied the psychology of arithmetic, arguing that bonds between stimuli and responses are strengthened through repetition and reward. Studies were done on which addition and multiplication facts were easiest to learn, with implications for classroom sequencing. Subsequently, notwithstanding Dewey's progressive influence and Brownell's emphasis on the meaningful learning of mathematics, behaviorist inquiry, with its emphasis on purely procedural knowledge, became the dominant American psychological research paradigm. Efforts to improve teaching proved disappointing.
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.]
Despite the above differences, cognitive psychologists and mathematics education researchers do agree on certain underlying principles of knowledge construction. In the Working Group on Theories of Learning Mathematics at the Seventh International Congress on Mathematical Education , the Japanese cognitive psychologist, Giyoo Hatano, gave the following five "characterizations" of long-term knowledge acquisition, with which, he felt, most cognitive psychologists would agree. And with which, most mathematics education researchers would agree.
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.
[Cf. Hatano, "A Conception of Knowledge Acquisition and Its Implications for Mathematics Education." In Steffe and Nesher (eds.) Theories of Mathematical Learning, Erlbaum, 1996, pp. 197-217.]
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.
One easy answer is "in the head." But despite the truly remarkable results concerning the brain being obtained by cognitive neuroscientists, there are cogent arguments for both individual cognition and a broader, societal perspective. Knowledge may be constructed by individuals, but it may also reside in the culture or in the language, with individuals acquiring it somehow. Or, one might want to consider knowledge as being "distributed" amongst systems, with a person and his/her tools, like a computer, in some kind of cooperating, symbiotic relationship.
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.
Those taking an information processing view see knowledge as residing in individual minds. They commonly consider it decomposable into small units and analyze "cognitive performances into complexes of rules," with each rule thought of as a component of the total skill. They emphasize careful task analyses: "It is a well-documented fact of human cognition that large tasks decompose into nearly independent subtasks, so that only the context of the appropriate subtask is needed to study its components." Rules can be combined in rather complex ways, with the overall organization sometimes referred to as a "cognitive architecture," which can be modeled theoretically or as a runnable computer program. If the output of a model duplicates that of humans, this is taken as evidence for the model's correctness, i.e., that human minds work like the model.
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.
Those taking a situated cognition point of view appear not to treat knowledge as entirely in one's head. Their main interest is in the way individuals interact with, or function in, various situations, often social situations. Taking such interactions (between individuals and situations) as a principle unit of analysis means it is not very enlightening to look at what is in an individual's mind separately from the situation. In mathematics, this would be something like trying to understand a function by looking at just its domain and range. One can do so, but one looses key information about the function, namely the relationship between the elements of the domain and range.
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.
In a recent article, Anderson, et al, argued against what they see as the four central claims of situated cognition: (1) Action is grounded in the concrete situation in which it occurs. (2) Knowledge does not transfer between tasks. (3) Training in abstraction is of little use. (4) Instruction must be done in complex, social environments. In each case, they provide objections based on psychological findings and suggest that situation cognition doesn't make much sense. Subsequently, Greeno responded, noting that the above four points are not really claims of situated cognition. It seems that, in attempting to capture the essence of situated cognition using the concepts and viewpoint of IP, Anderson, et al, produced a caricature.
>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.
Additional commentary on this debate can be found in an addendum.
[Cf. Anderson, Reder, and Simon, "Situated Learning and Education," Educational Researcher, May 1996; Greeno, "On Claims that Answer the Wrong Questions," Educational Researcher, January/February 1997].
Teaching abounds with practical questions. For example, how will I get these thirty, just barely attentive, students to appreciate -- understand, use in any way -- the Chain Rule? A few years back, one might have looked for answers in general psychological principles involving, say, the way (untrained) people naturally reason. However, in order to obtain reliable results, psychologists are likely to work in laboratory settings, on clearly defined and relatively simple, though not easy, research questions, quite different from the messy one's found in day-to-day teaching. They also tend to view mathematics as a collection of facts and standard algorithms and to focus on students' actions, rather than on their thought processes.
Perspectives from Math Ed Research
More recently, the emphasis has moved towards looking for answers that are domain specific. Hence, the rise of specialists doing mathematics education research. They try to answer questions like: How do students come to understand the concept of function? Such questions are not quite the messy ones of actual teaching, but they are closer than those asked by most cognitive psychologists. This, together with a shift in interest away from procedural knowledge towards conceptual knowledge, has left much solid work in cognitive psychology unused.
There are perhaps four schools of thought that influence mathematics education research today: the situated view, the sociocultural perspective, and the moderate and radical constructivist views. There are both conflicts and consistencies among these, but they all agree with cognitive psychology that what happens in people's minds can be profitably studied, in contrast to the earlier behaviorist view. In addition, they all agree that the mathematics teaching in schools and colleges could be greatly improved. They might even agree that for various reasons, not necessarily under teachers' control, mathematics is now often learned in small, isolated bits, which tend to be computational or procedural, devoid of conceptual understanding, and largely useless in applications requiring much originality.
What we Might Learn from Each Other
In this situation, cognitive psychology might have much to say about the efficiency of breaking tasks into subtasks that could be taught separately. This applies especially well to procedural knowledge, but might also be applied to the solving of somewhat familiar problems. On the other hand, anyone who has asked a student to prove a new theorem, or make an unusual application, will be interested in the solution of truly novel problems and the necessary conceptual understanding. Here the situated cognition adherents might study apprenticeships and the socioculturalists might examine ways conceptual understanding arises from social interaction, e.g., by discussing a proof with someone more knowledgeable. The constructivists would be more likely to examine how individual reflection leads to concept construction. A cognitive psychologists might worry about the reliability of this kind of research because ideas like "understanding" are hard to pin down precisely, while the others may worry more that dividing tasks into subtasks sounds a lot like the kind of teaching they hope to improve.
Although mathematicians, mathematics educators, and cognitive psychologists have their differences, perhaps we should be listening to one another more, lest there be a balkanization "of an area of intellectual activity that deserves better." [Cf. R. Davis' review, "One Very Complete View (Though Only One) of How Children Learn Mathematics," JRME 27(1), of a recent American Psychological Association volume on children's mathematical development.] Indeed, there are interesting psychological results on reasoning and short- and long-term memory, which may prove helpful in examining the learning of more advanced mathematics, e.g., how students learn to check the correctness of proofs. Furthermore, it takes only a little familiarity with conditioning to understand why a conscientious preservice teacher of ours developed a distaste for mathematics -- in her elementary school, working math problems was meted out as punishment.