It is argued that the philosophical and epistemological beliefs about the nature of mathematics have a significant influence on the way mathematics is taught at school. In this paper, the philosophy of mathematics of the NCTM's Standards is investigated by examining is explicit assumptions regarding the teaching and learning of school mathematics. The main conceptual tool used for this purpose is the model of two dichotomous philosophies of mathematics-absolutist versus- fallibilist and their relation to mathematics pedagogy. The main conclusion is (...) that a fallibilist view of mathematics is assumed in the Standards and that most of its pedagogical assumptions and approaches are based on this philosophy. (shrink)

This paper describes an attempt to develop a program for teaching history and philosophy of mathematics to inservice mathematics teachers. I argue briefly for the view that philosophical positions and epistemological accounts related to mathematics have a significant influence and a powerful impact on the way mathematics is taught. But since philosophy of mathematics without history of mathematics does not exist, both philosophy and history of mathematics are necessary components of programs for the training of preservice as well as inservice (...) mathematics teachers. (shrink)

A visible role in the theoretical discourses on education has been played in the last couple of decades by the constructivist epistemologies, which have questioned the basic assumptions of realist epistemologies. The increased popularity of interpretative approaches especially has put the realist epistemologies on the defensive. Basing itself on critical realism, this article discusses the ontological and epistemological commitments of educational research and its consequences for text interpretation. The article defends ontological realism and the semantic conception of truth against radical (...) constructivist ontology and the epistemic conceptions of truth. (shrink)

The study is focused on the relation between theology and mathematics in the situation of increasing secularization. My main concern in the second part of this paper is the early-twentieth-century foundational crisis of mathematics. The hypothesis that pure mathematics partially fulfilled the functions of theology at that time is tested on the views of the leading figures of the three main foundationalist programs: Russell, Hilbert and Brouwer.

The issue of proper functioning of operative computing and the utility of program verification, both in general and of specific methods, has been discussed a lot. In many of those discussions, attempts have been made to take mathematics as a model of knowledge and certitude achieving, and accordingly infer about the suitable ways to handle computing. I shortly review three approaches to the subject, and then take a stance by considering social factors which affect the epistemic status of both mathematics (...) and computing. I use the analogy between mathematics and computing in reverse—that is to say, I consider operative computing as a form of making mathematics, and so attempt to learn from computing to mathematics in general. I conclude that mathematics engineering is a field to be both developed for practical improvement of doing mathematics and taken into consideration while philosophizing about mathematics as well. (shrink)

The common sense notion of learning as the all-pervasive acquisition of new behaviour and knowledge, made vivid by experience, is an incomplete characterisation, because it assumes that the learning of behaviour and the learning of knowledge are indistinguishable, and that acquisition constitutes learning without reference to transfer. A psychological level of analysis is used to argue that conceptual learning should have priority in higher education.

Kitcher and Aspray distinguish a mainstream tradition in the philosophy of mathematics concerned with foundationalist epistemology, and a ‘maverick’ or naturalistic tradition, originating with Lakatos. My claim is that if the consequences of Lakatos's contribution are fully worked out, no less than a radical reconceptualization of the philosophy of mathematics is necessitated, including history, methodology and a fallibilist epistemology as central to the field. In the paper an interpretation of Lakatos's philosophy of mathematics is offered, followed by some critical discussion, (...) and an extension to a social constructivist position (which might well have been unacceptable to Lakatos). (shrink)

The article deals with the way in which authority was established and interpreted by teachers and students in two Flemish sixth-grade mathematics classrooms. Problem-solving lessons during a seven-month observation period were analysed regarding three aspects of teacher?student interactions that explicitly or implicitly reflect who bears mathematical authority: (1) to whom were students allowed to ask for help; (2) who was allowed to answer students? mathematics-related questions; and (3) who was allowed to evaluate students? responses. For each of these aspects, we (...) were interested whether authority was ascribed only to the teacher or whether authority was distributed more broadly, with students being considered accountable for their own and other?s learning process. Interviews with teachers and students were conducted to investigate the meaning that participants gave to the practices in which they were involved. (shrink)

In this article, modern standards of early years mathematics education are criticized and a proposal for change is presented. Today's early years mathematics education standards rest on a view of mathematics that became obsolete already at the end of the 19th century while the spirit of children's mathematics is precisely the spirit of modern mathematics. The proposal for change is not a return to the “new mathematics” movement, but something different.

This inaugural handbook documents the distinctive research field that utilizes history and philosophy in investigation of theoretical, curricular and pedagogical issues in the teaching of science and mathematics. It is contributed to by 130 researchers from 30 countries; it provides a logically structured, fully referenced guide to the ways in which science and mathematics education is, informed by the history and philosophy of these disciplines, as well as by the philosophy of education more generally. The first handbook to cover the (...) field, it lays down a much-needed marker of progress to date and provides a platform for informed and coherent future analysis and research of the subject. -/- The publication comes at a time of heightened worldwide concern over the standard of science and mathematics education, attended by fierce debate over how best to reform curricula and enliven student engagement in the subjects There is a growing recognition among educators and policy makers that the learning of science must dovetail with learning about science; this handbook is uniquely positioned as a locus for the discussion. -/- The handbook features sections on pedagogical, theoretical, national, and biographical research, setting the literature of each tradition in its historical context. Each chapter engages in an assessment of the strengths and weakness of the research addressed, and suggests potentially fruitful avenues of future research. A key element of the handbook’s broader analytical framework is its identification and examination of unnoticed philosophical assumptions in science and mathematics research. It reminds readers at a crucial juncture that there has been a long and rich tradition of historical and philosophical engagements with science and mathematics teaching, and that lessons can be learnt from these engagements for the resolution of current theoretical, curricular and pedagogical questions that face teachers and administrators. (shrink)

This study focuses on undergraduate students' ability to unpack informally written mathematical statements into the language of predicate calculus. Data were collected between 1989 and 1993 from 61students in six small sections of a “bridge" course designed to introduce proofs and mathematical reasoning. We discuss this data from a perspective that extends the notion of concept image to that of statement image and introduces the notion of proof framework to indicate the top-level logical structure of a proof. For simplified informal (...) calculus statements, just 8.5% of unpacking attempts were successful; for actual statements from calculus texts, this dropped to 5%. We infer that these students would be unable to reliably relate informally stated theorems with the top-level logical structure of their proofs and hence could not be expected to construct proofs or evaluate their validity. (shrink)

Many students experience mathematics as ahistorical and acultural. We review the philosophical roots of this experience and pose alternatives. We argue that there is evidence that the inclusion of a historical dimension into the teaching of mathematics courses at all levels, combined with an ‘active’ approach to learning, will improve motivation and achievement.

This chapter examines the relationship between the two fields of science education and philosophy of education to inquire how philosophy could better contribute to improving science curriculum, teaching, and learning, especially science teacher education. An inspection of respective research journals exhibits an almost complete neglect of each field for the other (barring exceptions).While it can be admitted that philosophy has been an area of limited and scattered interest for science education researchers for some time, the subfield of philosophy of education (...) has been little canvassed and remains an underdeveloped area. To help bring science education closer into the fold of educational philosophy and theorizing, the historical development of science education and philosophy of education are sketched to reveal their common roots, interests, and concerns. Thereto, the contours of a new philosophy of science education are presented (as an integration of three academic fields). Arguments are provided which seek to illustrate why philosophy in general and philosophy of education in particular can make positive contributions to teacher education and the research field together with suggesting future directions and possible reform contributions (scientific literacy, educational aims, educational theory, pedagogical content knowledge, science teacher, and curricular epistemologies). (shrink)

Although much creativity research has suggested that creativity is influenced by cultural and social factors, these have been minimally explored in the context of mathematics and mathematics learning. This problematically limits who is seen as mathematically creative and who can enter the discipline of mathematics. This paper proposes a framework of creativity that is based in what it means to know or do mathematics and accepts that creativity is something that can be nurtured in all students. Prominent mathematical epistemologies held (...) since the beginning of the twentieth century in the Western mathematics tradition have different implications for promoting creativity in the mathematics classroom, with fallibilist and social constructivist perspectives arguably being most conducive for conceiving of creativity as a type of action for all students. Thus, this paper proposes a framework of creative mathematical action that is based in these epistemologies and explains key aspects of the framework by drawing connections between it and research in the field of creativity. (shrink)

This paper claims that the awareness of crucial philosophical questions and controversies, which have arisen during the historical evolution of fundamental concepts, ideas and processes in mathematics, should be an essential component of the professional knowledge of student teachers who intend to teach children mathematics.

The purpose of this paper is to critically analyse and discuss the views of constructivism, on the teaching and learning of mathematics. I provide a background to the learning of mathematics as constructing and reconstructing knowledge in the form of new conceptual networks; the nature, role and possibilities of constructivism as a learning theoretical framework in Mathematics Education. I look at the major criticisms and conclude that it passes the test of a learning theoretical framework but there is still a (...) gap between theory and mathematics classroom practice. (shrink)