This study explores the way that belief systems, interactions with social or experimental environments, and skills at the “control” level in decision‐making shape people's behavior as they solve problems. It is argued that problem‐solvers' beliefs (not necessarily consciously held) about what is useful in mathematics may determine the set of “cognitive resources” at their disposal as they do mathematics. Such beliefs may, for example, render inaccessible to them large bodies of information that are stored in long‐term memory and that are (...) easily retrieved in other circumstances. In other cases, individuals' reactions to an experimental setting (fear of failure, or the desire to “look mathematical” while being videotaped) may induce behavior that is almost pathological—and at the same time, so consistent that it can be modeled. In general, such “environmental” factors establish the context within which individuals access and utilize the information potentially at their disposal.Protocols illustrating these points are presented and discussed. A model based on an axiomatization of students' beliefs about plane geometry is outlined, and is shown to correspond closely to their problem‐solving performance. A framework is offered for analyzing problem‐solving performance at three qualitatively different levels: access to cognitive resources stored in LTM, executive or control decision‐making, and belief systems. (shrink)

In this paper we look at some of the ingredients and processes involved in the understanding of mathematics. We analyze elements of mathematical knowledge, organize them in a coherent way and take note of certain classes of items that share noteworthy roles in understanding. We thus build a conceptual framework in which to talk about mathematical knowledge. We then use this representation to describe the acquisition of understanding. We also report on classroom experience with these ideas.