Paul Ernest

The Knowledge, Beliefs and Attitudes of the Mathematics Teacher: a model

Abstract Official pressure for reforms in the teaching of mathematics overlooks a key factor: the psychological foundations of the practice of teaching mathematics, including the teachers knowledge, beliefs and attitudes. Research on teaching and teacher education also under‐emphasises this area, which Shulman terms the ‘missing program’ in research on teacher cognitions. The present paper addresses this lack by proposing an analytic model of the different types of knowledge, beliefs and attitudes of the mathematics teacher, and their relationship with practice. Of particular note is the importance accorded to the teachers practical knowledge of the teaching of mathematics (both pedagogical and curricular knowledge), knowledge of classroom organisation, and knowledge of the school context. Also notable is the importance ascribed to the teachers beliefs concerning the nature of mathematics, and concerning the processes of teaching and learning mathematics. The model has implications for teacher educatio...

Mathematics, education and philosophy : an international perspective

Reconceptualizing the Philosophy of Mathematics: Fresh Winds in the Philosophy of Mathematics, Reuben Hersh What can the Sociologist of Knowledge Say About 2+2=4?, DavidBloor. Post-Modernist and Post- Structuralist Approaches: Reasoning in a Post-Modern Age, Valerie Walkerdine Mathematical Writing, Thinking, and Virtual Reality, Brian Rotman. The Human Face of Mathematics: Mathematics and Art: Cold Calipers against Warm Flesh?, Philip J. Davis Different Ways of Knowing: Contrasting Styles of Argument in Indian and Greek. The Social Context of Mathematics and Education: The Social Life of Mathematics, Sal Restive.

Forms of Knowledge in Mathematics and Mathematics Education: Philosophical and Rhetorical Perspectives

New forms of mathematical knowledge are growing in importance for mathematics and education, including tacit knowledge; knowledge of particulars, language and rhetoric in mathematics. These developments also include a recognition of the philosophical import of the social context of mathematics, and are part of the diminished domination of the field by absolutist philosophies. From an epistemological perspective, all knowledge must have a warrant and it is argued in the paper that tacit knowledge is validated by public performance and demonstration. This enables a parallel to be drawn between the justification of knowledge, and the assessment of learning. An important factor in the warranting of knowledge is the means of communicating it convincingly in written form, i.e., the rhetoric of mathematics. Skemps concept of ‘logical understanding’ anticipates the significance of tacit rhetorical knowledge in school mathematics. School mathematics has a range of rhetorical styles, and when one is used appropriately it indicates to the teacher the level of a students understanding. The paper highlights the import of attending to rhetoric and the range of rhetorical styles in school mathematics, and the need for explicit instruction in the area.

Reflections on theories of learning

Four philosophies of learning are contrasted, namely ‘simple’ constructivism, radical constructivism, enactivism and social constructivism. Their underlying explanatory metaphors and some of their strengths and weaknesses are contrasted, as well as their implications for teaching and research. However, it is made clear that none of these ‘implications’ is incompatible with any of the learning philosophies, even if they sit more comfortably with one of them.

Values, gender and images of mathematics: a philosophical perspective

This paper describes the widespread public image of mathematics as cold, abstract, and inhuman, and relates it to absolutist philosophies of mathematics. It is argued that this image is consistent with ‘separated’ values which help to make mathematics a ‘critical filter’ denying access to many areas of study and to fulfilling professional occupations, especially for women in Anglophone western countries. In contrast, an opposing humanized image of mathematics, consistent with ‘connected’ values, finds academic support in recent fallibilist philosophies of mathematics. It is argued that although these two philosophical positions have a major impact on the ethos of mathematics classrooms, there is no direct logicalconnection. It is concluded instead that the values realized in the classroom are probably the dominant factor in determining the learners image and appreciation of mathematics (and hence, indirectly, that of society).

Constructivism, the Psychology of Learning, and the Nature of Mathematics: Some Critical Issues.

Constructivism is one of the central philosophies of research in the psychology of mathematics education. However, there is a danger in the ambiguous and at times uncritical references to it. This paper critically reviews the constructivism of Piaget and Glasersfeld, and attempts to distinguish some of the the psychological, educational and epistemological consequences of their theories, including their implications for the philosophy of mathematics. Finally, the notion of ‘cognizing subject’ and its relation to the social context is examined critically.

Mathematically gifted and talented learners: Theory and practice

There is growing recognition of the special needs of mathematically gifted learners. This article reviews policy developments and current research and theory on giftedness in mathematics. It includes a discussion of the nature of mathematical ability as well as the factors that make up giftedness in mathematics. The article is set in the context of current developments in Mathematics Education and Gifted Education in the UK and their implications for Science and Technology. It argues that early identification and appropriate provision for younger mathematically promising pupils capitalizes on an intellectual resource which could provide future mathematicans as well as specialists in Science or Technology. Drawing on a Vygotskian framework, it is suggested that the mathematically gifted require appropriate cognitive challenges as well as attitudinally and motivationally enhancing experiences. In the second half of this article we report on an initiative in which we worked with teachers to identify mathematically gifted pupils and to provide effective enrichment support for them, in a number of London Local Authorities. A number of significant issues are raised relating to the identification of mathematical talent, enrichment provision for students and teachers’ professional development.

A Postmodern Perspective on Research in Mathematics Education

Over the past quarter century mathematics education has become established world-wide as a major independent area of knowledge and research. The field now has numerous dedicated journals, book series and conferences serving national and international communities of scholars. Many countries offer specialist master’s and doctoral programs of study in mathematics education, and new entrants often receive their postgraduate education within the field itself. Given this coming of age, as we approach the 21st century it is appropriate to engage in a period of critical reflection and self-scrutiny. Thus the present volume embodying the ICMI-sponsored inquiry into the nature of research in mathematics education and its products provides a welcome opportunity for our field to take stock of itself, and its outcomes and effects.

The number line as a teaching aid

The number line is often recommended as an aid for the teaching of whole number addition and subtraction. The possible stages in young childrens use of the aid are described, as is their ultimate outgrowing of this aid.Number line diagrams are also used in diagnostic and achievement test items to assess the understanding of whole number addition. This practice is discussed and criticised with the aid of the empirical findings of the second NAEP. Finally the wider role of number line diagrams in mathematics teaching is outlined.

The philosophy of mathematics and mathematics education

It is suggested that the philosophy of mathematics is relevant to mathematics education (1) because the philosophical schools of thought have a direct bearing on educational issues and (2) because new entrants to teaching may bring with them undiluted theoretical views on the nature of mathematics. The views of the following five schools of thought are sketched: Logicism, Formalism, Constructivism, Platonism and Falliblism. Subsequently the relationship between these views and certain current issues in mathematics education is discussed. It is concluded that each of the views provides insights as to the nature of mathematics, but that Falliblism is perhaps the only viewpoint compatible with humane mathematics education.