One sentence answer to the question in the title is that the ability to prove depends on forms of knowledge to which most student are rarely, if ever, exposed. Presents more detailed analysis, drawing on research in mathematics education and classroom experiences. (Contains 44 references.) (Author/ASK)

Answers Burn's question related to a previous study on the nature of knowledge about abstract algebra. Claims that a previous paper presents research that attempts to contribute to knowledge of how students' understanding of certain group concepts may develop instead of teaching abstract algebra with the computer software ISETL. (ASK)

Illustrates distinctions among kinds of knowing. Distinguishes knowing-to from other forms of knowing, and explores implications of that distinction for teaching and learning mathematics. Proposes that knowing-to act in the moment depends on the structure of attention at the moment, and on what one is aware of. (Contains 75 references.)…

Examines the mathematical proof schemes of students starting their studies at the University of Cordoba and relates these schemes to the meanings of mathematical proof in different institutional contexts. Concludes that deductive mathematical proof is difficult for these students. Suggests that the different institutional meanings of proof might…

Views of (n=70) 16- and 17-year olds on mathematical knowledge, activity, and learning were analyzed using factorial techniques. Findings suggest there was no simple systematic relationship between beliefs about the nature of mathematical knowledge and the teaching and learning of mathematics. (19 references) (Author/MKR)

To understand an individual student's learning in the complexity of the mathematics classroom, it is necessary to examine the events before, during, and after learning. To illustrate, the process by which two children each construct new mathematical meanings is examined from these perspectives. (Author/MKR)

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)

How the computer as a metaphor affects our understanding of the processes of learning and teaching is explored. After describing reflection and recursion in mathematics and their roles in thinking and learning, models of the mind and teaching effects are discussed, with self-awareness as a theme. (MNS)

In two parallel one-year studies, solution strategies for subtraction number facts and achievement patterns of matched groups of first graders in two different instructional programs were examined. Significant differences between groups were found favoring the strategy approach. (Author/CW)

Addresses issues related to the ways teachers learn mathematics and the teaching of mathematics, and the relevance of those ways to their professional development. Mathematical experiences need to be congruous with the kind of teaching expected of the reflexive, adaptive teacher. Presents both practical and theoretical considerations of how these…

Discusses what important ideas about forms of knowing mathematics should be included in mathematics methods courses for preservice teachers. Proposes ideas related to Shulman's framework of teacher knowledge. Provides a brief discussion of the implications each idea holds for teaching mathematics, and makes some suggestions about experiences that…

Describes the characteristics of progressive schematization with regard to column multiplication and column division. Contrasts this with column arithmetic based on progressive complexity. Presents a summary of research data concerning column arithmetic. (TW)

Presents examples and reflections hinting at the role of formal thought in the process of knowledge growth. Discusses proof, language, and mathematics; knowledge as form versus development; the difference between concept and object; how mathematical texts can be improved; and geometry and the idea of diagrammatic reasoning. (Contains 36…

Reported is a naturalistic study of the role of mathematical paradoxes in the preservice education of high school mathematics teachers. Findings indicate that the model of resolving paradoxes as applied in this study has relevance to such aspects of mathematics education as cognitive conflicts, motivation, misconceptions, and constructive…

The main thesis of the paper is that geometry deals with mental entities (the so-called geometrical figures) which possess simultaneously conceptual and figural characters. The paper analyzes the internal tensions which may arise in figural concepts because of their double nature, developmental aspects, and didactical implications. (Author/MDH)