Presents a refined form of a personal theory of learning that examines two distinct parts of the complex interaction between a learner's internal mental structures and his or her world of experience. (Author/MKR)

This article builds on two previous ones in which we presented the processes of construction and consolidation of one student's knowledge structures about comparisons of infinite sets, according to a recently proposed theory of abstraction. In the present article, we show that under slight variations of context, knowledge structures that have…

Differences in students' (aged 12-14) construction and analysis of mathematical algorithms may be explained by the differences between predicative and functional cognitive structures. Analyzes the central role of internal and external representations and the characteristics of each. Discusses prototypical research design and development of a…

Presents an episode in which a conflict arose as sixth-grade students attempted to create nonstandard units of measurement. Discusses the conflicting requirements of language and symbol use in mathematical communication in the activity. (MKR)

Argues that the organization of cognitive structures for technical domains can be visualized as a network of connected thinkable models. Describes a taxonomy of models that has been developed and discusses the issue of how representations relate to human modes of perception and action. Contains 25 references. (DDR)

Reviews recent research in the area of mathematical understanding and compares and contrasts it with a model formulated for the growth of understanding. Uses the analysis of a transcript from an interview with an eight-year-old boy to illustrate the power of the model to describe and map the growth of his mathematical understanding. (18…

Describes how an eight year old devised a part-whole schema during a school mathematics process involving the development of a geometric model to conceptualize fractions. Provides examples that utilize this schema in dealing with the relative size of fractions, as well as addition and subtraction, multiplication, and division of fractions.…

Discusses dilemmas teachers may face when shifting their concept of teaching or tutoring from explaining or telling to a constructivist perspective of facilitating the development of students' knowledge structures. Examines teachers' decisions during a 20-minute segment of dialogue between a student and teacher while studying linear functions. (14…

Discusses a new approach to mathematics education that moves away from methods that tell students what to do and toward methods that help students build mental representations of mathematical concepts. Contrasts the previous and emerging views and examines aspects of the new approach. (26 references) (MDH)

Reports results from a project designed to bring about student conceptual change by using multiple modeling and representation as well as links to other areas. Students developed rules as conveniences and meanings for symbols, were active in learning, made interpretations, and had confidence in their thinking. (MKR)