Different types of hierarchies in the learning of mathematics and criteria by which they might be judged are examined. Current works on mathematical learning hierarchies are illustrated and in particular the work of the mathematics team of the research project "Concepts in Secondary School Mathematics and Science" is examined. (MP)

The concept of equivalence is shown to be an elusive one that provides difficulty for preschoolers through college students. The idea persists that the equal sign is a "do something," or operator symbol, rather than a symbol for an equivalence relation. (MP)

The author takes exception to Piaget and Inhelder's conclusion that a child's first conceptualization of space is topological. (JP)

The hypothesis investigated is that understanding of the long division algorithm requires a higher cognitive level than understanding of fundamental division concepts. Sixth-grade children were tested on performance and understanding of a given algorithm and concepts of division. (MP)

Presents a theory of the development of algebraic abstraction that extends Sfard's and Mason's ideas on learners' progress from operational or process-oriented thinking to the abstract or structural perspective. Analyzes secondary school students' approaches to classifying a set of complex equations. (Author/MKR)

Task performance experiments concerning properties of mathematical groups with 10 and 11 year-old children are discussed including transfer effects, effects of structural relations, individual differences, and memory (both short and long term). (JG)

A review of work related to the topological primacy thesis and a critique of some aspects of it are presented. On the basis of the psychological research, it is concluded that topological concepts ought to be taught. (MP)

Experiments by Russian educators on teaching algebra in elementary grades are summarized by the author. The algebra taught consisted of generalized solutions to concrete problems that are usually presented with fixed numerical values. (JP)

Presented is a theory of how competence with written mathematical symbols develops, tracing a succession of cognitive processes that cumulate to yield competence. Arguments supporting the theory are drawn from the history, philosophy, and psychology of mathematics. (MNS)

Research conducted in several countries has shown consistent patterns of performance on "change,""combine," and "compare" word problems involving addition and subtraction. These findings are interpreted within a theoretical framework which emphasizes development of levels of word problem-solving ability related to…

Several critical questions concerning the topological primacy thesis were raised in an extensive literature survey (SE 531 428). Three points related to this criticism are discussed and reinforced, including a reexamination of Laurendeau and Pinard's data (showing that they do not support the hypothesis of topological primacy in children's…

This study considers two problems related to cognitive development: "Is development hierarchical?" and "If so, what are the mechanisms involved in the process of development?" Analysis of the results of an experiment lead to differentiation of stages of development, and problem-solving strategies at each level are discussed.…

A multidisciplinary team research project to document the relationship between environmental and cultural features of Papua New Guinea, cognitive development, and mathematics learning is described. (MP)

Analyzed are the different stages in the conceptual evolution of infinity as developed historically through the work of Bolzano and Cantor. Results of a study of 18 to 20 year old's concept of infinity prior to instruction produced aspects of the passage between conceptual levels of infinity. (MDH)

Profiled interviews of (n=41) kindergarten and first-grade children's numerical development over a school year. Significant mismatches between children's numerical development and typical curricula are described, and changes to current practice are recommended. Appendix includes sample interview tasks. (Contains 21 references.) (MKR)