In this paper we describe learners being asked to generate examples of new mathematical concepts, thus developing and exploring example spaces. First we elaborate the theoretical background for learner generated examples (LGEs) in learning new concepts. The data we then present provides evidence of the possibility of learning new concepts through…

Varying approaches to mathematics were raised in a Philosophy of Mathematics course. Students were asked which foundational schools they preferred and these preferences were compared with scores on hemispheric tests. The results indicate that preferences may be related to the brain's hemispheres. (MNS)

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)

Analysis of 116 students revealed those who preferred to process mathematical information by verbal-logical means outperformed more visual students on tests. Spatial ability and knowledge of spatial conventions had less influence on performance than expected. (MP)

It is argued that the view of mathematics as an activity emphasizing intuitive thinking is compatible with an axiomatic-deductive approach. (MP)

Features and tools to outline and shape the long-term learning process for ratio are presented, with examples. Ratio-anticipating activity and creating visual models and schemes are stressed, with the basis for a theory of teaching and learning ratio presented. (MNS)

Proposes a model for the growth of mathematical understanding based on the consideration of understanding as a whole, dynamic, leveled but nonlinear process. Illustrates the model using the concept of fractions. How to map the growth of understanding is explained in detail. (Contains 26 references.) (MKR)

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)

Ideas on a teaching-learning theory of ratio are presented. Views of children and of members of an international panel are presented, and five questions and their responses are compared with real classroom experience. (MNS)

A training experiment with 30 mentally handicapped children tested two models of mental-arithmetic development. Children's initial responses to a mental-addition task appeared to be the product of a mechanical prescription. Errors appeared to be more errors of method than of recall. (MNS)

Pupils from grades eight and nine were given problems, mostly referring to the idea of infinity. Scales measuring the level of confidence and degree of obviousness of answers were developed. (MP)

Some potential difficulties involved in using "real-life situations" in teaching problem solving are illustrated with a racing car problem. The importance of developing mental images and the ability to abstract in the solution process is indicated. (MP)

Describes a longitudinal constructivist teaching experiment analyzing children's ways of operating while solving fraction tasks. A case study of one third grader found that the evolution of her fraction schemes was the product of her cognitive activity and her collaborative interaction with the teacher. (42 references) (Author/MKR)

Mathematics educators have found sociocultural models of teaching and learning to be powerful in their ability to describe and support the pursuit of instruction based on recent standards documents (e.g., National Council of Teachers of Mathematics [NCTM], 1989, 2000). These models of instruction, however, have been criticized for their lack of…

Described is the Educational Leaders in Mathematics (ELM) Project, a four-stage inservice teacher program designed to stimulate teachers' development of a constructivist view of learning. The impact of the intervention was studied, and results indicated that teachers' beliefs were affected, producing changes in teaching strategies. (MDH)