The Philosophy of Mathematics Education Today (ICME-13 Monographs)

Annotation This book offers an up-to-date overview of the research on philosophy of mathematics education, one of the most important and relevant areas of theory. The contributions analyse, question, challenge, and critique the claims of mathematics education practice, policy, theory and research, offering ways forward for new and better solutions. The book poses basic questions, including: What are our aims of teaching and learning mathematics? What is mathematics anyway? How is mathematics related to society in the 21st century? How do students learn mathematics? What have we learnt about mathematics teaching? Applied philosophy can help to answer these and other fundamental questions, and only through an in-depth analysis can the practice of the teaching and learning of mathematics be improved. The book addresses important themes, such as critical mathematics education, the traditional role of mathematics in schools during the current unprecedented political, social, and environmental crises, and the way in which the teaching and learning of mathematics can better serve social justice and make the world a better place for the future Contents Editor and Contributors 1 A Plea for a Critical Transformative Philosophy of Mathematics Education References Introduction to the Field 2 The Philosophy of Mathematics Education: An Overview Abstract Introduction: What Is the Philosophy of Mathematics Education? Question 1: What Is Mathematics? (The Basic Question of the Philosophy of Mathematics) Question 2. How Does Mathematics Relate to Society? (The Philosophy of the Milieu) Question 3: What is Learning and Learning Mathematics, in Particular? (The Philosophy of Learning) Question 4. What is Teaching and Teaching Mathematics, in Particular? (Pedagogical Philosophy for Mathematics) Question 5: What is the (Philosophical) Status of Mathematics Education as Knowledge Field? Applying Philosophy to Mathematics Education Ontology and Metaphysics Aesthetics Epistemology Learning Theory Social and Political Philosophy Ethics Methodology Conceptual Analysis Conclusion References The Nature of Mathematics 3 The Who and What of the Philosophy of Mathematical Practices Abstract Introduction A Short and Incomplete Historical Outline Lakatos as the Starting Point Kitcher as the Next Step A First Tension Is Introduced to Stay Enter the Sociologists (and a Second Tension), The Educationalists and the Ethnomathematicians Brain and Cognition Complete the Picture A Methodological Note Topics in the Philosophy of Mathematical Practices and What They Can Tell Us The View from Above: From Bird to Frog Could It Have Been Otherwise? The Rich Nature of ‘Real’ Proofs A Second Look at Foundational Studies Conclusion References 4 The Philosophy of Mathematical Education Between Platonism and the Computer Abstract Introduction Platonism 1 Platonism 2 General Ideas and Social Knowledge Knowledge as a Historical Process The Complementarity of Syntax and Semantics The Double Nature of Mathematics Mathematics as Problem Solving and as a Universal Language There is Another Side to Plato Episteme Versus Techne Mathematics: Theories and Algorithms Conclusion References 5 A Dialogical Conception of Explanation in Mathematical Proofs Abstract Introduction Explanatoriness in Mathematics A Dialogical Conception of Mathematical Proof Mathematical Proofs as Explanatory Fictive Dialogues Conclusion Acknowledgements References 6 The Amalgam of Faith and Reason: Euclid’s Elements and the Scientific Thinker Abstract Introduction Setting the Scene The Amalgamation of Faith and Science When Scholastics Met Euclid Sacred Mathematics and the Path to God The Discourses on Faith Acknowledgements References Critical Mathematics Education 7 Students’ Foregrounds and Politics of Meaning in Mathematics Education Abstract Meaning as a De-politicised Educational Issue Meaning as a De-politicised Philosophical Issue A Foreground-Interpretation of Meaning Polarised Foregrounds Destroyed Foregrounds Pointed Foregrounds Multiplied Foregrounds Politics of Meaning as Research Acknowledgements References 8 The Struggle Is Pedagogical: Learning to Teach Critical Mathematics Abstract Teaching Critical Mathematics Teaching Others to Teach Critical Mathematics From the Community to the Classroom References 9 Some Thoughts on a Mathematics Education for Environmental Sustainability Abstract The Planetary Context Environmental Sustainability as Post-normal Science A Critical Mathematics Education Perspective on Environmental Sustainability A Critical Mathematics Education Approach Reflective Knowing, Post-normal Science and Environmental Sustainability Mathematics Education and Environmental Sustainability: Some Concluding Suggestions Acknowledgements References 10 Epistemological Questions About School Mathematics Abstract Introduction Traditions of ‘Knowing Mathematically’ An Ethics of the Self Moving Forward References 11 The Concept of Culture in Critical Mathematics Education Abstract Ethnomathematics and Its Discontents Culture, Again The Interest of Mathematics Education in Culture References 12 The Ethics of Mathematics: Is Mathematics Harmful? Abstract Introduction Is Mathematics an Untrammelled Good? The Intrinsic Value of Mathematics The Extrinsic and Social Value of Mathematics Features and Characteristics of Mathematics Mathematical Thinking as Detached Instrumental and Calculative Reasoning Qualifying the Critique of Instrumental Thinking The Social Impacts of Mathematics and Its Application The Social Impact of the Image of Mathematics Summary and Provisional Solutions Teaching the Philosophy of Mathematics Teaching the Ethics and Social Responsibility of Mathematics Conclusion References Philosophical Theory in Mathematics Education Research 13 On the Need for Theory of Mathematics Learning and the Promise of ‘Commognition’ Abstract What Is Theory and Why Do We Need It? The Commognitive Keywords and Their Use (Basic Concepts) Commognitive Mediators (Data) Commognitive Routines (Method of Analysis) Commognitive Stories of Mathematics Learning (Theory) References 14 On the Roles of Language in Mathematics Education Abstract Introduction—Language and Change A Glimpse into History—The Layers of Reification in Geometry Language as a Tool of Epistemological Analysis Acknowledgements References 15 The Separation of Mathematics from Reality in Scientific and Educational Discourse Abstract Introduction The Separation of Mathematics from Reality Reality in Mathematics Classrooms Final Remarks References 16 Mathematics Education Actualized in the Cyberspace: A Philosophical Essay Abstract The Ontological Aspects of Cyberspace Understanding Cyberspace as Real The Interweaving of Anthropological and Epistemological Aspects The Modes of Dialogue Present in the Humans-with-Computers Mathematics Education Realized in the Cyberspace Concluding... References Philosophy of/in Teaching, Learning and Doing Mathematics 17 Making Distinctions: A Phenomenological Exploration in Mathematics Education Abstract Introduction Methods Two Phenomena Immediate Comment Extended Comment Enquiring Two Task-Exercises Distinguishing Forms of Attention Associated Actions and Predictions Extended Commentary Theoretical Considerations Classifying Noticing and Acting An Epistemological Stance A Phenomenological Stance Reflection References 18 Using Rules for Elaborating Mathematical Concepts Abstract Introduction Inferential Use of Words in Language Games Methodology Generating Mathematical Rules for Expressing Meaning Final Remarks Appendix References 19 Towards a Wider Perspective: Opening a Philosophical Space in the Mathematics Curriculum Abstract Introduction Widening the Perspective: Bringing an Outsider View to the Mathematical World-Vision Philosophical Inquiry in the Classroom Community of Philosophical Inquiry: Methodology and Mathematical Practice Fragments from a Philosophical Text for Middle School Students Conclusion References 20 Creativity Research in Mathematics Education Simplified: Using the Concept of Bisociation as Ockham’s Razor Abstract Introduction State of the Creativity Research—Brief Summary Koestler Theory of Creativity as Ockham Razor Mathematics Education: Bisociation, Schema Accommodation and Reflective Abstraction The Bisociative Edge of OR as a Heuristic Guide to Question, Challenge and Critique Certain Claims in Mathematics Education Research Cognitive-Affective Duality of Aha! Moment Bisociation and Simultaneity of Attention Bisociation and Teaching—Research References 21 Teaching of Velocity in Mathematics Classes—Chances for Philosophical Ideas Abstract Introduction Didactical Orientated Analysis Historical and Philosophical Aspects Some Mathematical Aspects of the Velocity Concept Some Philosophical Aspects in Mathematics Class Conclusions References 22 Time for Work: Finding Worth-While-Ness in Making Mathematics Abstract Introduction Mathematics in the Making (MiMa) Background to the Intervention Theoretical Underpinnings for MiMa The Intervention Reflections 1: Time and Worth-Whiling Reflections 2: Worth-While Work Conclusion References 23 Hades—The Invisible Side of Mathematical Thinking Abstract Introduction Hermeneutics Analytics Dialectics Experience/Phenomenology Speculation Outlook References 24 Developing Rules Due to the Use of Family Resemblances in Classroom Communication Abstract Family Resemblances Methodology and Theoretical Framework Empirical Examples Description and Interpretation of the Scene The Use of (Family) Resemblances in the Scene Conclusions and Outlook References