Artificial Intelligence researchers and cognitive scientists commonly believe that thinking involves manipulating representations. Thinking involves search, inference, and making choices. This is how we model reasoning and what goes on in the brain is similar. Winograd and Flores present a radically different view, claiming that our knowledge is not represented in the brain at all, but rather consists of an unformalized shared background, from which we articulate representations in order to cope with new situations. In contrast, computer programs contain only pre-selected objects and properties, and there is no basis for moving beyond this initial formalization when breakdown occurs. Winograd and Flores provide convincing arguments with examples familiar to most artificial intelligence researchers. However, they significantly understate the role of representation in mediating intelligent behavior, specifically in the process of reflection, when representations are generated prior to physical action. Furthermore, they do not consider the practical benefits of expert systems and the extent of what can be accomplished. Nevertheless, the book is crisp and stimulating, and should make artificial intelligence researchers more cautious about what they are doing, more aware of the nature of formalization, and more open to alternative views.

Discusses the implications of information processing psychology for mathematics education, with a focus on the works of schema theorists such as D. E. Rumelhart and D. A. Norman and R. Glaser and production system theorists such as J. H. Larkin, J. G. Greeno, and J. R. Anderson. Learning is considered in terms of the actor’s and the observer’s perspective and the distinction between declarative and procedural knowledge. Comprehension and meaning in mathematics also are considered. The role of abstraction and generalization in the acquisition of mathematical knowledge is discussed, and the difference between helping children to “see, ” as opposed to construct abstract relationships is elucidated. The goal of teaching is to help students modify or restructure their existing schema in predetermined ways by finding instructional representations that enable students to construct their own expert representations.

The theoretical basis of the interviews reported about here is a Piagetian-like approach to the origin and genesis of cognitive schemata representing mathematical concepts. Such schemata are postulated to reflect the abstract and general structure of material, imagined or mental actions and of relations induced by these actions. The main cognitive tools for the mental construction of such schemata are seen to be: Actions, symbolic representations, prototypes of objects, reflection and abstraction, schematization, generalization. The interviews were devised such that the subjects were guided appropriately in their individual cognitive constructions. The mathematical topics treated are: Place value system, divisibility, word problems, geometric sequence, Riemann integral. In general the results support the view that the individual construction of cognitive schemata is possible and effective in the proposed way.

I propose a mechanism to model aspects of Piagetian development in infants. The mechanism combines a powerful empirical learning technique with an unusual facility for constructing novel elements of representation-- elements designating states that are not mere logical combinations of other represented states. I sketch how this mechanism might recapitulate the infant's gradual recognition that there exist physical objects that persist even when the infant does not perceive them. I also report results of a preliminary, partial implementation.