This paper reports on data from investigations on learners' understanding of inclusion relations of quadrilaterals, building on the ideas from our earlier study (Fujita & Jones, 2007). By synthesising past and current theories in the teaching of geometry (van Hiele's model, figural concepts, prototype phenomenon, etc.), we propose a theoretical…

Presents results of a project with second and third graders to establish concepts of equality that led to the realization that teachers often help students too much and may push them algorithmically beyond their ability without the development of proper referents. (Author/MKR)

How physiological disciplines can contribute to the study of how people learn mathematics is considered. Manipulative and experiential learning, sequential versus hierarchical organization, declarative versus procedural knowledge, and short-term versus long-term memory are among the points discussed. (MNS)

Describes the components of, and theoretical/research bases for, learning cycle pedagogy, which is especially appropriate for the special needs of the middle-school student. Three stages of the learning cycle are exploration, conceptual invention, and expansion. (Author/MKR)

Discusses why improvements in teaching mathematics are needed, including discussion of students, human development, knowledge and learning, structure of schools and instructional programs, and mathematics. Also discusses a need for two theories: (1) how young people form self-concepts and (2) classification of approaches to curricula and…

How multiplication is usually taught in school and how it could be taught are discussed. Development of understanding is illustrated through children's words and work. (MNS)

Based on the cognitive psychological theories of Vygotsky and Davydov, discusses the establishment of connections between mathematical elements, and the algorithmic rules that govern them, and children's spontaneous mathematical concepts. Presents examples that establish connections involving addition and subtraction, comparing numerical…

The discussions of the theme group on Technology and Cognitive Development at the Fifth International Congress on Mathematical Education are summarized. How computers can be used to engage students actively, to promote problem-solving skills, and to achieve better understanding is discussed. (MNS)

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…

The second in a series of three articles, some themes of an undercurrent of educational objectives that move beyond the evident content goals are outlined. The themes relate the computer software materials analyzed in Part I to the cognitive experiences that students should have in learning about subject matter. (MDH)

A year-long study in 40 eighth grade mathematics classrooms compared the effect on student achievement of teaching with meaning with algorithmic-practice teaching. In general, teaching with meaning was found to increase student achievement. However, differences in response to the methods was seen in the top, middle, and lower thirds of the…

Discusses the first session and events from other sessions of a mathematics circle consisting of the author and four preschool children. Presents discussions that ensued with the children when asked to solve problems related to Piaget's stages of cognitive development. (MDH)

Using case studies that are functionalist in orientation and computational in technique, the role of control knowledge in developing constructive thinking is illustrated. Further, the integration of related knowledge structures, emanating from diverse sensory modes and pertaining to both place value in addition and angle relationships in geometry,…

Responses of students in grades K-10 (42 interviewed; 98 written) to 4 common fractions problems were analyzed for iconic mode processes in relationship to concrete symbolic mode development. Concludes that progress in the latter, without development of complementary iconic support, limits both understanding and flexibility in solving problems.…

This study explored how one eighth grade girl learned about the structure of recursive programs in BOXER programing language. The method of impasse resolution was presented in detail and used both to understand how she constructed her understanding and as an empirical method to analyze her learning. (MDH)