Paul ErnestUniversity of Exeter | UoE · The Graduate School of Education
Paul Ernest
BSc MSc PhD
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159
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Introduction
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September 1984 - present
Publications
Publications (159)
Mathematics education, especially the central activities of teaching and learning mathematics, is thoroughly entangled with issues of power, authority, necessity, compulsion and submission. Schooling is compulsory almost everywhere as is mathematics. What is the ethical basis of universal schooling and the imposition of mathematical study within it...
The widespread ‘ethical turn’ in philosophy, the humanities and social sciences has left the research field of mathematics education behind. Mathematics education, especially its central activities of teaching and learning mathematics, is thoroughly entangled with issues of ethics and justice. While there are many books on social justice and mathem...
A number of authors have claimed that Division by Zero and in particular the
Abstract
A number of authors have claimed that Division by Zero and in particular the Division of Zero by Zero (0/0) can be computed and has a definite value (Mwang 2018, Saitoh & Saitoh 2024). I refute these claims. This is trivial, but despite its elementary standing, some peripheral or recreational mathematicians make claims about 0/0 or k/0 hav...
This chapter addresses the ontological problems of mathematics and mathematics education. The ontological problem of mathematics is that of accounting for the nature of mathematical objects. The ontological problem of mathematics education concerns the chief entities in the domain, namely persons. What is the nature of persons, restricting the inqu...
This is a special issue of the Philosophy of Mathematics Education Journal on Self Based Methodologies (SBM) in mathematics teacher education. It explores the philosophical foundations of SBM, as well as 8 specific studies, making a valuable addition to the research literature.
Chapman et al. (2020) describe self-based methodologies (SBM) as “appr...
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 s...
This paper establishes two linked problems of necessity and authority in mathematics. First, in what way are the ideas, methods and results of mathematics taught in school necessary? Second, on what basis is the metaphysical necessity of mathematical solutions, results, and truths asserted, and is this justified? These primary problems raise a suit...
A traditional problem of ethics in mathematics is the denial of social responsibility. Pure mathematics is viewed as neutral and value free, and therefore free of ethical responsibility. Applications of mathematics are seen as employing a neutral set of tools which, of themselves, are free from social responsibility. However, mathematicians are con...
This is an exploration of the mathematical and philosophical significance of zero. It looks at the mathematics of zero, and the history of its development from three perspectives: syntactic, (signs) semantic (meanings) and pragmatic (social use and context). It draws a clear distinction between nothing and zero, which avoids various paradoxes and c...
I have now finished the draft of my paper, I have now finished the draft of my paper, Nought Matters: The Mathematical and Philosophical Significance of Zero. If you have an interest in looking at it,
any thoughts or critical feedback will be gratefully received. it is a bit long and a bit cheeky! But I am pleased with the breadth and depth of cont...
Os benefícios indiscutíveis e as virtudes intrínsecas da matemática não deveriam provocar uma cegueira quanto ao possível dano colateral provocado pela sua imensa, impiedosa e destrutiva força**, que é capaz de fazer a educação e a sociedade se renderem em direção a um futuro reformulado. Por causa do grande poder da matemática e da sua influência...
Two university academics Matthew and Sophia engage in a dialogue on the deep ethics of mathematics. Deep ethics refers not to mathematics being good or bad, virtuous or vicious, but to the social responsibility that it bears as a humanly invented technology. Mathematics is both an human practice and an technological artefact that has a deep effect...
This is a Dialogue on the Deep Ethics of Mathematics. What Deep Ethics means is clarified, as is the contention that humanity shares reponsibility for all of its products including mathematics. Sophia makes these claims. They are disputed in the dialogue by Matthew who defend Platonism and the ethics free nature of mathematics. Both views are expou...
A little paper unpicking the meaning of the deceptive mathematics behind the Covid alert levels and the spurious equation CAL=R+N in Boris Johnson's presentation on 10 May 2020
This paper explores the ethics of the mathematics teacher, starting from the ethical obligations that all human being and professionals share towards those in their care. Most notably this involves a duty of care for students, since teachers can be the most influential persons after their parents. The ethics of mathematics teaching is analysed as c...
A central problem for ethics in mathematics and its teaching is that the goodness of mathematics is taken for granted. It is widely and uncritically assumed not only to be wholly beneficial but also to be beyond any ethical responsibility. In this chapter I question this assumption and undertake an ethical audit of mathematics for education and soc...
A central problem for ethics in mathematics and its teaching is that the goodness of mathematics is taken for granted. It is widely and uncritically assumed not only to be wholly beneficial but also to be beyond any ethical responsibility. In this chapter I question this assumption and undertake an ethical audit of mathematics for education and soc...
Brief overview and list of contents of powerful new publication on education
PoME Journal 34 Frontpage Dec 2018 for pub.docx
The Philosophy of Mathematics Education Journal No. 34, 2018
This is the table of contents for the Philosophy of Mathematics Education Journal No. 34 (December 2018) which is freely available at http://socialsciences.exeter.ac.uk/education/research/centres/stem/publications/pmej/
This short paper explores ethics and the mathematics teacher, and the ethics of teaching mathematics.
In this chapter I challenge the idea that mathematics is an unqualified force for good. Instead I show the harm that learning mathematics can inadvertently cause unless it is taught and applied carefully. I acknowledge that mathematics is a widespread force for good but make the novel case that there is significant collateral damage caused by learn...
This chapter offers an overview of the philosophy of mathematics education. This sub-field is characterised in both narrow and broad terms, concerning the aims of mathematics education and all philosophical aspects of research in mathematics education, respectively. The sub-field is also explored in terms of its questions and practices, which can b...
THE ETHICS OF MATHEMATICS: IS MATHEMATICS HARMFUL?
Paul Ernest
ABSTRACT
In this paper I challenge the idea that mathematics is an unqualified force for good. Instead I show the harm that learning mathematics can inadvertently cause unless it is taught and applied carefully. I acknowledge that mathematics is a widespread force for good but make th...
There is a paradox in exploring space-time and the limits of human understanding from the perspective of mathematics. For mathematics provides the language in which theories of space-time are formulated and yet is understood to be outside of space and time itself. Thus geometry is the science of space, and yet is itself beyond space.
This is a working paper I am giving on The Changing Role of the University and the Academic in the Knowledge Market: Inner and Outer Roles
This paper argues that mathematics is imbued with values reflecting its production from human imagination and dialogue. Epistemological, ontological, aesthetic and ethical values are specified, both overt and covert. Within the culture of mathematics, the overt values of truth, beauty, purity, universalism, objectivism, rationalism and utility are...
This survey provides a brief and selective overview of research in the philosophy of mathematics education. It asks what makes up the philosophy of mathematics education, what it means, what questions it asks and answers, and what is its overall importance and use? It provides overviews of critical mathematics education, and the most relevant moder...
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. Bu...
There is a widespread perception that mathematics is objective and value-free and that values only enter into mathematics subjectively in the personal preferences of mathematicians. In this paper I argue that, on the contrary, mathematics itself is value-laden; imbued with epistemological, ontological, aesthetic and ethical values. I identify a ran...
This survey provides a brief and selective overview of research in the philosophy of mathematics education. It asks what makes up the philosophy of mathematics education, what it means, what questions it asks and answers, and what is its overall importance and use? It provides overviews of critical mathematics education, and the most relevant moder...
Two questions about certainty in mathematics are asked. First, is mathematical knowledge known with certainty? Second, why is the belief in the certainty of mathematical knowledge so widespread and where does it come from? This question is little addressed in the literature. In explaining the reasons for these beliefs, both cultural-historical and...
This chapter addresses two issues. First, the status and the nature of the philosophy of mathematics education as a subfield of research in mathematics education. Second, Stephen Lerman's contributions to this subfield. A narrow interpretation of the philosophy of mathematics education concerns the aims and rationale of the practices of teaching an...
This session is also available as an audiorecording.
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 original...
In this paper, we look at what engaging with psychoanalysis, through psychosocial accounts of subjectivity, has contributed to our struggles for legitimacy and security within our ways of knowing. The psychosocial, with its insistence on the unconscious and the irrational, features as both a source of security and of insecurity. We use three exampl...
Three variants of constructivist learning theory for mathematics are contrasted. These are information processing, radical, and social constructivism. Each position is sketched, and the underlying metaphors of mind and models of the world (with their links to epistemology and ontology) are examined critically. The positions are compared with regard...
Although many agree that all teaching rests on a theory of knowledge, this is an in-depth exploration of the philosophy of mathematics for education, building on the work of Lakatos and Wittgenstein.
This journal is devoted to contributions concerning the philosophy of mathematics accessible to researchers in mathematics education, to philosophical aspects of mathematics education, theoretically interesting research studies in mathematics education, and any other papers relevant to this conflux of ideas.
Available freely at
http://www.people...
It is argued that “problem solving” has multiple meanings, according to the teacher's mathematics-related belief-system or
perspective. A model of teacher belief-systems is proposed. It is argued that the central feature is a personal philosophy
or conception of mathematics, and that this determines the teacher's understanding of the nature of prob...
The central focus of the paper is the nature of the mental representations of the meaning of the linguistic expressions of mathematics. An information-processing model for the construction of mental representations is presented. The model provides syntactical tree structures as meaning representations. It is argued that the major function of writte...
Commentaire et presentation de l'ouvrage de Solomon Y., The Practice of Mathematics, montrant l'interet de reinserer l'apprentissage des mathematiques et la genese des nombres dans le contexte epistemologique piagetien
One of the great clashes of ideas of our times is that between psychology and sociology. Sociologists accuse psychologists
of being narrowly technical, supporters of what the critical theorists term instrumental rationality. Psychologists have also
been stereotyped as apolitical and closed minded about social and political issues and social/politic...
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...
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 mathematic...
This chapter discusses some of the universalizing forces at work in the globalization and internationalization of mathematics
education. Wikipedia is used a both a definitional source for the concepts of globalization and internationalization, as well
as exemplifying the Anglophone and eurocentric domination of the knowledge economy worldwide. This...
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 h...
A semiotic perspective on mathematical activity provides a way of conceptualizing the teaching and learning of mathematics
that transcends and encompasses both psychological perspectives focussing exclusively on mental structures and functions,
and performance-focussed perspectives concerned only with student' behaviours. Instead it considers the p...
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 'implicati...
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 'implicati...
Semiotics provides a way of conceptualising the teaching and learning of mathematics driven by a primary focus on signs and
sign use. It considers patterns of sign use and production, and the con-texts and social rules underlying sign use. It attends
to agency in the learner’s personal appropriation of signs and the meaning structures embodying the...
son 1983, Schofield 1981) The problem of poor attitudes can be particularly acute in the case of primary or elementary school teachers (particularly those who are not mathematics specialists). For teachers of younger students tend to have poorer attitudes than those teaching older or intermediate aged students (Early 1970, Raines 1971). "The result...
this paper I wish to indicate how this metaphor can be developed to describe mathematics itself, mind and the teaching and learning of mathematics
this paper I explore the meaning of empowerment in the teaching and learning of mathematics. The main part of the paper is devoted to distinguishing three different but complementary meanings of empowerment concerning mathematics: mathematical, social and epistemological empowerment. Mathematical empowerment concerns gaining the power to use mathem...
In Principia Mathematica Whitehead and Russell erected one of the cornerstones of modernist philosophy of mathematics. However, as an epistemological project seeking to secure the foundations of mathematics, this attempt failed, as Whitehead acknowledged. His later work in process philosophy represented a new direction. Although not explicitly a ph...
This paper discusses some of the main findings of a survey study that explores the range of images, beliefs and attitudes towards mathematics as held by the adult public, and their possible sources. Interpretative approaches employing both quantitative and qualitative methods were used. Five main categories of response emerged from the analysis. Th...
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...
The Culture of the Mathematics Classroom is becoming an increasingly salient topic of discussion in mathematics education. Studying and changing what happens in the classroom allows researchers and educators to recognize the social character of mathematical pedagogy and the relationship between the classroom and culture at large. The volume is divi...
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...
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...
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...
This paper seeks to explore values that are explicitly and implicitly documented in the Malaysian school mathematics curriculum and to compare them with mathematics teachers' perceptions of what values are appropriate to be taught through mathematics. The study finds that what is planned only partially matches what is espoused by the teachers in th...
"[Note: This abstract applies to all articles within this issue."] The chapters in this monograph describe qualitative research methods used to investigate students' and teachers' interactions with school mathematics. Each contributing author uses data from his or her own research to illustrate a particular technique or aspect of research design. T...
Discusses three misapplications of progressive thinking that have damaged the quality of mathematics learning in British schools: individualized learning, spiral curriculum, and applications of mathematics. (MKR)
Questions
Questions (3)
Epistemology is concerned with justifcation for knowledge claims. Why is 2+2=4 true, is my ring true gold (cf Heidegger). Blockchain validtes the genuineness of virtual currencies, it can also validate ownership, and there is even discussion of Blockchain and identity. Has anyone explored the philosophical implications of Blockchain??
Introduction
Mathematics is a very rich and powerful subject, with broad and varied footprints across education, science, culture and indeed all of human history. Both academia and society in the large accord mathematics a very high status as an art and as the queen of the sciences (Bell 1952). Mathematics has a uniquely privileged status in education as the only subject that is taught universally and to all ages in schools. Behind this elevated status is the hidden ethical assumption that mathematics is an unqualified force for good. Is this assumption correct? Does nothing but good flow from mathematics? In this chapter I argue that mathematics does harm as well as good. My claim is that mathematics in school has unintended outcomes in leaving some students feeling inhibited, belittled or rejected by mathematics. In sorting and labelling learners and citizens in modern society, mathematics reduces the life chances of those labelled as mathematical failures or rejects. In addition, even for those successful in mathematics, mathematics serves as a training that shapes thinking in an ethics-free and amoral way. Thus mathematics supports instrumentalism and ethics-free governance. This is exploited in warfare, psychopathic corporations, the misuse of humans and the environment, and in all acts that treats persons as objects rather than moral beings always entitled to respect and dignity. I conclude by suggesting solutions. To avoid or remedy the negative effects in schooling we need to attend more closely to the causes of success and failure, and become fully aware of how these have far-reaching impacts on learners. Further, in order to forestall the harmful effects of mathematics in society we need to teach the social responsibility of mathematics, through including philosophy and especially the ethics of mathematics alongside mathematics itself. All students of mathematics and fully fledged mathematicians should be able to view the uses and applications of mathematics critically, seeing the mathematics in play and understanding the ethical implications of the issues involved, including both benefits and harm.
As a mathematician myself, someone who has devoted his professional life to furthering the teaching of mathematics in school and university, I might be expected to be among the last to question the value of mathematics. However, I believe that both the place of mathematics in the world and the benefits it brings are strengthened through looking at mathematics as a critical friend. This entails not only lauding and feeling pride at the benefits mathematics brings but also recognising the harm it can do, and not shying away from it. Acknowledging that there are negative outcomes opens the doors to solutions, to possible means of ameliorating and rectifying the damage brought about through mathematics.
Is Mathematics an Untramelled Good?
