The purpose of this paper is to develop a means to illustrate and analyse thecognitive paths taken by students in solving problems. The approach is builtupon the notion of cognitive unit (small enough to be consciouslymanipulated). Our interest is in the nature of the student’s cognitive units andthe connections between them. We find that a student may have an overallstrategy and even formulate goals to achieve all or part of a solution.However, if conceptual structures are too diffuse, the student may concentrateon procedures that occupy most of the focus of attention. This may cause themto lose touch with the ultimate goal and be faced with sequences of activitythat are longer, more detailed, and more likely to break down.IntroductionWhy is it that some students find algebra so essentially simple, yet others struggle sobadly that they fail in school and need to take remedial algebra courses in college? Theliterature abounds in distinctions between the conceptual thinking of some students andthe procedural thinking of others (e.g. Hiebert & Lefevre, 1986). But why does thisoccur? What is the nature of procedural thinking that makes it the default position for somany? Hiebert and Carpenter (1992) suggest two metaphors for cognitive structures, asvertical hierarchies or as webs:
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