The beliefs of preservice teachers about mathematics teaching and learning.
Preservice teachers have had a twelve year "apprenticeship of observation" in the practice of teaching and as a consequence have internalised a set of beliefs about Mathematics teaching and learning. These beliefs are usually implicit but nevertheless influence the teaching practice of the preservice teachers to the extent that they "teach as they were taught." A preservice Mathematics Education course, falling as it does between the prospective teachers' experiences as scholars, and their future teaching experience provides an ideal opportunity for preservice teachers to review their personal beliefs prior to carrying them over to teaching practice. In order to facilitate this review, a series of activities was designed as part of a Mathematics Education course for preservice secondary phase Mathematics teachers. These activities provided opportunities for student teachers to examine their beliefs, to discuss and write about these beliefs, to read about the beliefs of others, and finally to decide whether they wished to retain or modify their personal beliefs. Data on the personal theories of the preservice teachers was obtained from the written responses to various critical incidents, from metaphors for the teaching and learning of Mathematics drawn and described by the preservice teachers, and from interviews with selected participants. These theories were classified into qualitatively different categories. After completing several developmental activities and a five week period of classroom teaching, the preservice teachers were invited to reconsider their personal theories and amend their metaphors. This provided evidence of reflection and development in their thinking. It is contended that the personal theories of preservice teachers are not only reflected in their classroom practice but also function as barriers to impede acceptance of novel ideas and innovations. The findings of this study contribute to the understanding of the thinking of preservice teachers and inform the development of a curriculum for the Mathematics Education component of a Professional Studies course.