Paul Ernest

University of Exeter, Graduate School of Education, Faculty Member

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Um panorama da Filosofia da Educação Matemática

This paper focuses on the sub-field of study the philosophy of mathematics education from one per... more This paper focuses on the sub-field of study the philosophy of mathematics education from one perspective. The field is characterised in both narrow and broad terms, and from both bottom-up (questions and practices) and top-down perspectives (in terms of philosophy and its branches). From the bottom-up one can characterize the area in terms of questions, and I have asked: What are the aims and purposes of teaching and learning mathematics? What is mathematics? How does mathematics relate to society? What is learning mathematics? What is mathematics teaching? What is the status of mathematics education as knowledge field? I have characterized the sub-field using a ‘top down’ perspective using the branches of philosophy. Looking briefly into the contributions of ontology and metaphysics, aesthetics, epistemology and learning theory, social philosophy, ethics, and the research methodology of mathematics education reveals both how rich and deep the contributions of philosophy are to the theoretical foundations of our field of study. But even these two approaches leave many questions unanswered. For example: what are the responsibilities of mathematics and what is the responsibility of our own subfield, the philosophy of mathematics education? I conclude that the role of the philosophy of mathematics education is to analyse, question, challenge, and critique the claims of mathematics education practice, policy and research.

Critical issues in mathematics education

Information Age Pub. eBooks, 2009

... Gurion University of the Negev, Israel Lyn D. English, Queensland University of Technology, A... more

Mathematics Education and Philosophy

Routledge eBooks, Sep 2, 2003

Reconceptualizing the Philosophy of Mathematics: Fresh Winds in the Philosophy of Mathematics, Re... more 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.

The Philosophy of Mathematics Education

British Journal of Educational Studies, Aug 1, 1991

The use of general descriptive names, registered names, trademarks, service marks, etc. in this p... more The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, express or implied, with respect to the material contained herein or for any errors or omissions that may have been made.

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2007©The Montana Council of Teachers of Mathematics

This Article is brought to you for free and open access by ScholarWorks. It has been accepted for... more

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The Mathematical Attitudes, Beliefs and Ability of Students

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A Dialogue on the Ethics of Mathematics

The Mathematical Intelligencer, 2016

You have the unenviable task of upholding your view that mathematics is ethical. How will you bri... more You have the unenviable task of upholding your view that mathematics is ethical. How will you bridge the divide between ethics and pure mathematical knowledge? Consuela: Good morning, Absalom. I suppose it is harder to argue that mathematics is ethical instead of the standard view of mathematics as ethically free, because the 'received' absolutist view so permeates the culture of mathematics. However, the idea that mathematics is ethical goes back to Plato or earlier. Abs.: And will you use Plato's arguments to make your case? Con.: No. In the Republic Plato argues that mathematics is ethical because it teaches unity, concord, order, and proportion, and these are central features of ethics as he understands it. In modern times we do not see these as the core of ethics, even if they can be virtues when applied to some behaviours and social situations. 1 By the way, we notice that the name Absalom happens to begin with the same three letters as absolutism, the position that you will uphold. Abs.: Yes-just as your name begins the same as constructivism. 2 Con.: From my point of view, mathematics is ethical, that is, pure mathematical knowledge has ethical implications. My claim is that there are intrinsic ethical elements within mathematics itself. Abs.: Let's get this clear. I think it is important to distinguish: (1) the ethics of applications of mathematics, (2) the professional ethics of mathematicians, (3) the ethics of teaching mathematics, and (4) the ethics of pure mathematics itself, whatever that might be. We can clearly agree that ethical obligations are entered into within all professional and applicational work, as they are with all social practices. Any scientific application has an ethical dimension, there are professional codes of ethics for mathematicians, and clearly any teaching is bound by ethical codes. 3 Con.: That's right, Absalom, we can both agree on issues 1, 2, and 3. These are not controversial. What I am claiming is your point 4, that mathematics itself is intrinsically ethical. 1 Not everyone rejects Plato's (2000) position. Myles Burnyeat (2000), basing his arguments on Plato, proposes that mathematics is good for the soul by making it receptive to higher ideas including reasoning, truth, the good, and ethics. 2 Note that, as emerges in the dialogue, Consuela espouses social constructivism, the view that mathematical knowledge and objects are constructed socially. This should be distinguished from the foundationalist philosophy named constructivism, related to intuitionism, which proposes that all mathematical objects are constructed by finite procedures, and rejects completed infinities as well as the validity of proofs of existence by contradiction. Social constructivism has, in the past few decades, won a growing number of philosophers and mathematicians as adherents. For example, see the collections Gold and Simons (2008) and Tymoczko (1986). 3 A number of mathematicians have written on the need for ethics for applications of mathematics, including Chandler Davis (1988) and Philip Davis (2007); Reuben Hersh (1990, 2007) has written on mathematics and ethics and on ethics for mathematicians; and the American Mathematical Society (2005) has published Ethical Guidelines for mathematicians including research and teaching obligations. There is also an extensive literature on ethics in mathematics education, including Atweh (2013), Boylan (2013), and D'Ambrosio (1998).

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The unit of analysis in mathematics education: bridging the political-technical divide?

Educational Studies in Mathematics, 2016

Mathematics education is a complex, multi-disciplinary field of study which treats a wide range o... more Mathematics education is a complex, multi-disciplinary field of study which treats a wide range of diverse but interrelated areas. These include the nature of mathematics, the learning of mathematics, its teaching, and the social context surrounding both the discipline and applications of mathematics itself, as well as its teaching and learning. But research and researchers in mathematics education fall loosely into two camps: On the one hand there is technical research, drawing on mathematics, psychology and pedagogy, concerned with narrow questions about the teaching and learning of mathematics. On the other hand there is political and social research drawing on sociology and philosophy, addressing large scale problems of social consequence. These two camps tend to draw on different theoretical underpinnings as well as having different interests. Is it possible to find a shared theoretical element, a single unit of analysis for mathematics education which provides a unified approach to both analysing and explaining all of these diverse aspects? Can such a unit provide a bridge across the technical-political divide? Methodological and ontological senses of the term ‘unit of analysis’ are distinguished. Units of analysis in the ontological sense are proposed for each of the four listed subdomains of mathematics education. Drawing on Blunden’s (2009, 2010) interdisciplinary version of cultural historical activity theory a single over-arching unit of analysis, the collaborative project, is proposed for the whole field of mathematics education. This is applied to the case study of a single learner, illustrating the practical utility of this approach, as well as the way the technical-political divide might be bridged in our field.

Postmodern Mathematics

The Proceedings of the 12th International Congress on Mathematical Education, 2015

The goal was to elucidate the nature of postmodern mathematics and mathematics education, includi... more The goal was to elucidate the nature of postmodern mathematics and mathematics education, including the multiplicity of the subject of mathematics; and to explore and share ideas of how postmodern perspectives offer new ways of seeing mathematics, teachers, learners, mathematics education theories, research and practice. The two key themes originally used to organizer the sessions are as follows. Theme 1 What does postmodern mathematics mean? Part of this means exploring perspectives of mathematics as a plurality, as having multiple forms, identities, locations. Even the name mathematics is plural, although modern(ist) usage overlooks this anomaly in treating it/them as a single entity. Theme 2 What does postmodern mathematics education mean? This involves considering multiple-self perspectives of the human subject as teacher, learner or researcher. Organizers Co-chairs: Paul Ernest (UK), Regina Möller (Germany); Team Members: Ubiratan D'Ambrosio (Brazil), Allan Tarp (Denmark), Sencer Corlu (USA), Gelsa Knijnik (Brazil), Maria Nikolakaki (Greece); Liaison IPC Member: Michèle Artigue (France).

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The nature of mathematics: Towards a social constructivist account

Science and Education, 1992

Two dichotomies in the philosophy of mathematics are discussed: the prescriptive--descriptive dis... more Two dichotomies in the philosophy of mathematics are discussed: the prescriptive--descriptive distinction, and the process-product distinction. By focusing on prescriptive matters, and on mathematics as a product, standard philosophy of mathematics has overlooked legitimate and pedagogically rewarding questions that highlight mathematics as a process of knowing which has social dimensions. In contrast the social-constructivist view introduced here can affect the aims, content, teaching approaches, implicit values, and assessment of the mathematics curriculum, and above all else, the beliefs and practices of the mathematics teacher.

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The philosophy of mathematics and mathematics education

International Journal of Mathematical Education in Science and Technology, 1985

It is suggested that the philosophy ofmathematics isrelevant tomathematics education (1) because ... more

The Nature of Mathematics and Teaching

PERSPECTIVES-EXETER-, 1995

Rejection, Disagreement, Controversy and Acceptance in Mathematical Practice: Episodes in the Social Construction of Infinity

Deleted Journal, Jan 31, 2023

The concept of infinity has a long and troubled history. Thus it is a promising concept with whic... more The concept of infinity has a long and troubled history. Thus it is a promising concept with which to explore rejection, disagreement, controversy and acceptance in mathematical practice. This paper briefly considers four cases from the history of infinity, drawing on social constructionism as the background social theory. The unit of analysis of social constructionism is conversation. This is the social mechanism whereby new mathematical claims are proposed, scrutinised and critiqued. Minimally, conversation is based on the two roles of proponent and critic. The proponent puts forward a proposal, which is reacted to and evaluated by those in the role of critic. There is a continuum of contexts in which such conversations take place from inner conversations the mathematician has within themselves, and casual face-to face interactions between mathematicians at the chalkboard, all the way to the formal responses of referees and editors to submitted journal papers. Such responses vary from unconditional acceptance, partial acceptance through to outright rejection. There may be disagreements between proponents and critics, among those in the joint role of critic, and broader, community-wide disagreements and controversies, according to specific mathematical proposal and the critical judgements of it.

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Is Beauty the First Value? The Case of Mathematics1

Is beauty the first value of the classical triad of values comprising beauty, truth and the good?... more Is beauty the first value of the classical triad of values comprising beauty, truth and the good? Usually, the truth is placed before it. Science seeks true models of the world. But unlike science, because mathematical concerns a range of possible models, logical necessity and proof come before truth. Furthermore, creative mathematics is both about necessity, following rules, but also about choice. Choice over which problems, method, concepts, axioms and theories to investigate. The choice criterion is aesthetic. As in any pure art form, it is mostly beauty but also interest and other criteria that drives the artist. Mathematics depends on aesthetics in its making, research practices, and its appreciation. Thus, within mathematics beauty or rather aesthetics is claimed to be the first value, or at the very least, equal in importance to the other values of the triad.

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

Forms of Mathematical Knowledge

New forms of mathematical knowledge are growing in importance for mathematics and education, incl... more 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. Skemp's 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 student's 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.

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The Philosophy of Mathematics, Values and Keralese Mathematics

The Mathematics Enthusiast

This paper explores the philosophical significance of the Keralese and Indian subcontinent contri... more This paper explores the philosophical significance of the Keralese and Indian subcontinent contribution to history of mathematics. Identifying the most accurate genesis and trajectory of mathematical ideas in history that current knowledge allows should be the goal of every history of mathematics, and is consistent with any philosophy of mathematics. I argue for the need of a broader conceptualization of philosophy of than the traditional emphasis on scholastic enquiries into epistemology and ontology. For such an emphasis has been associated, though I add need not necessarily be so, with an ideological position that devalues non-European contributions to history of mathematics. The philosophy of mathematics needs to be broad enough to recognise the salient features of the discipline it reflects upon, namely mathematics.

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Varieties of Constructivism: A Framework for Comparison

Theories of Mathematical Learning

Postmodernity and Social Research in Mathematics Education

Mathematics Education Library

Postmodernism is a brash new style, which, in the mouths of some authors, claims to supersede wha... more Postmodernism is a brash new style, which, in the mouths of some authors, claims to supersede what came before. In my view much of its claimed originality is overhyped, and as a cultural-as opposed to philosophical-discourse it celebrates the hybrid, pastiche, and ironic. These exhibit double-or more-codings as multiple meanings collide. Thus hypocrisy, insincerity and double standards are all potential dangers for postmodernism. If I adopt, for the moment, the standpoint of a cynic, postmodernity can be read as an excuse for inauthenticity and ethical expediency. However, simultaneously, the philosophical discourse of postmodernity, much of which I support, celebrates authenticity, being-here-now (Heidegger's Dasein). These are attitudes that acknowledge our multiple existence in the linked but disparate worlds of our experience: the bodily, mundane, discursive, political, professional, institutional and cultural realms. Thus the irony is that the postmodern celebrates the transgressive-crossing the traditional boundaries of knowledge, values, economics-as well as those of good taste, decency, and morality. And this leaves me feeling split-both a supporter and a critic of the various stances attributed to post-modernism. This is all heady stuff. But it takes me away from the more down to earth and dare I say it-serious-aims of this chapter. Habermas (1990) distinguishes the philosophical and cultural discourses of postmodernity-and modernity. Most of the above musings are to be found in the cultural discourse. In the present context it is the philosophical discourse I shall draw upon. For the purposes of this chapter I will pick and choose those elements of postmodernity relevant to my project, namely developing the conceptualisation of socio-political and socio-cultural research in mathematics education. In doing so I will not take the space to offer a definitive account of postmodernism-not that I could necessarily supply this. Rather I will devote my space to three themes which, although varying in support from the pantheon of postmodernist and poststructuralist authors cited here, are undoubtedly important in mathematics education. The three elements or themes I have chosen to focus on are the rejection of metanarratives, a focus on conversation and language, and the knowledge / power / economics nexus.

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Topic Study Group No. 53: Philosophy of Mathematics Education

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Nominalism and Conventionalism in Social Constructivism

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