In this paper the mathematical structures of Bourbaki, the Piagetian developmental structures, and Polya's heuristic structures are compared. (SD)

A theory that there is a correspondence between Piagetian conservation operations and groups of symmetry transformations, and that these symmetry transformations may be used in explaining human problem solving behaviors, is developed in this paper. Current research in artificial intelligence is briefly reviewed, then details of the symmetry…

We explore the nature and consequences of teachers' problem solving through an example of a teacher's mathematical problem solving as it was occasioned by a student's mathematics. This illustration demonstrates the value of an interpretive framework that points to the mathematics of the classroom as a collective. Arising from this exploration is…

For meaningful learning of mathematics, a learning set is required which demands that all things accepted as true should be demonstrable in terms of a paradigm appropriate to the child's cognitive development: preparatory, concrete-particular, concrete-general, formal-abstract. Future teachers should experience all paradigms to become aware that…

A sample of 24 subjects was presented with 800 addition problems to test a general model of simple and complex addition. According to this model, an addition problem is encoded and verified by column, with specific addition strategies determined by the size of the problem. Two hundred problems each of the following four types were presented: (1)…

This research has as its basic premise the belief that physical and mental skills are highly similar, enough so in fact that computation paradigms such as the ones used in Artificial Intelligence research about predominantly mental skills can be usefully extended to include physical skills. This thesis is pursued experimentally by the…

After considering the mathematical system common to various types of measure, the author argues that instruction needs to reflect the common aspects of measure systems as well as the distinctive attributes of different systems. (SD)

A concrete and a symbolic approach to learning multiplication in the second grade are studied. The results, indicating that children using the symbolic model performed at a higher level than those using a concrete model, are discussed. (DT)

Two approaches to teaching number concepts and operations are discussed. Their relationships to the foundations of mathematics as well as to the characteristics of the learner are explored. On the basis of this discussion the implicit approach is advocated over the definitial approach. (SD)

The author argues that teaching students to program a computer enhances their logical reasoning, problem solving skills, understanding of models, and ability to examine one's work critically. (SD)

This paper notes three objections to the use of hand-held calculators in schools: they would (1) block reasoning, (2) make individuals machine-dependent, and (3) broaden the gap between developed and underdeveloped nations. Each is addressed, with specific examples used to refute them. The belief is strongly expressed that calculators can aid in…