The Learning and Development of Mathematics Teacher Educators: International Perspectives and Challenges (Research in Mathematics Education)

Research in mathematics teacher education as a distinctive field of inquiry has grown substantially over the past 10-15 years. Within this field there is emerging interest in how mathematics teacher educators (MTEs) themselves learn and develop. Until recently there were few published studies on this topic, and the processes by which mathematics teacher educators learn, and the forms of knowledge they require for effective practice, had not been systematically investigated. However, researchers in mathematics education are now beginning to investigate the development of MTE expertise and associated issues. This volume draws on the latest research and thinking in this area is therefore timely to stimulate future development and directions. It will survey the emerging field of inquiry in mathematics education, combining the work of established scholars with perspectives of newcomers to the field, with the aim of influencing development of the field, invite cross-cultural comparisons in becoming a mathematics teacher educator by highlighting issues in the development of MTEs in different countries, and examine the roles of both mathematics educators and mathematicians in preparing future teachers of mathematics. The primary audience will be university-based mathematics teacher educators and MTE researchers, and postgraduate research students who are seeking academic careers as MTEs. Additional interest may come from teacher educators in disciplines other than mathematics, and education policy makers responsible for accreditation and quality control of initial teacher education programs. The Learning and Development of Mathematics Teacher Educators Acknowledgments Contents Contributors Authors Editors Editor and Author Biographies Untitled Chapter 1: Introduction: The Learning and Development of Mathematics Teacher Educators 1.1 Rationale 1.2 Who Is a Mathematics Teacher Educator? 1.3 Structure of the Book 1.3.1 Theme 1: The Nature of Mathematics Teacher Educator Expertise 1.3.1.1 Questions Addressed by Theme 1 Chapters 1.3.2 Theme 2: Learning and Developing as a Mathematics Teacher Educator 1.3.2.1 Questions Addressed by Theme 2 Chapters 1.3.3 Theme 3: Methodological Challenges in Researching Mathematics Teacher Educator Expertise, Learning, and Development 1.3.3.1 Questions Addressed by Theme 3 Chapters 1.3.4 Commentary Chapters 1.4 Contributions to Advancing the Field References Part I: The Nature of Mathematics Teacher Educator Expertise Chapter 2: What Do Mathematics Teacher Educators Need to Know? Reflections Emerging from the Content of Mathematics Teacher Education 2.1 Introduction 2.2 Mathematics Teacher Educator Knowledge 2.3 Mathematical Knowledge 2.4 Knowledge About Teachers’ PCK 2.5 Knowledge About Mathematics Teaching Practices and Skills 2.6 Knowledge About Professional Identity 2.7 Pedagogical Content Knowledge: What Does ‘Content’ Mean Here? 2.8 Knowledge of the Features of the Professional Development of Mathematics Teachers 2.9 Knowledge of Teaching the Content of Initial Mathematics Teacher Education Programmes 2.10 Knowledge of the Standards of Mathematics Teacher Education Programmes 2.11 Three Profiles of MTE 2.12 Concluding Remarks References Chapter 3: Applying the Knowledge Quartet to Mathematics Teacher Educators: A Case Study Undertaken in a Co-teaching Context 3.1 Introduction 3.2 Review of Literature 3.2.1 Mathematical Knowledge for Teaching 3.3 Theoretical Framework 3.3.1 The Knowledge Quartet 3.4 Methodology 3.5 Results and Discussion 3.5.1 Lesson Episode 1: Algebraic Thinking 3.5.1.1 Lesson Observations 3.5.1.2 Post-lesson Data 3.5.1.3 Post-lesson Reflections: Co-teachers 3.5.2 Lesson Episode 2: Measurement 3.5.2.1 Post-lesson Data 3.5.2.2 Links to the Knowledge Quartet 3.5.2.3 Foundation 3.5.2.4 Transformation 3.5.2.5 Connection 3.5.2.6 Contingency 3.6 Conclusions and Implications References Chapter 4: The Research Mathematicians in the Classroom: How Their Practice Has Potential to Foster Student Horizon 4.1 Undergraduate Studies in Mathematics and the Teaching Profession: Teachers’ Mathematical Horizon 4.2 Research Mathematicians’ Teaching Practices that Have Potential Implications on Teacher Education Programmes 4.3 Research Mathematicians’ Teaching Practices with the Potential to Foster Students’ Horizon 4.3.1 Methodology and Settings 4.3.2 Teaching Work on Fostering Student Horizon 4.4 Drawing on Examples 4.5 Connecting Mathematical Areas 4.6 Visualising 4.7 Simplifying 4.7.1 In a Nutshell 4.8 Implications for Mathematics Teacher Education References Chapter 5: Pedagogical Tasks Toward Extending Mathematical Knowledge: Notes on the Work of Teacher Educators 5.1 Introduction 5.2 Script-Writing in Mathematics Education 5.3 The Usage-Goal Framework 5.4 Context for the Examples 5.5 Example 1: Functions, Not Just Linear 5.5.1 The Scripting Task: Functions 5.5.2 Snapshots from the Scripts: Functions 5.5.2.1 On the Notion of Function 5.5.2.2 Polynomial Expressions 5.5.3 Follow-Up Activities: Functions 5.5.3.1 Function Definition 5.5.3.2 Fitting Polynomials 5.6 Example 2: Irrational Exponents, Not Just with a Calculator 5.6.1 The Scripting Task: Irrational Exponents 5.6.2 Snapshots from the Scripts: Irrational Exponents 5.6.2.1 Irrationals Can Only Be Approximated 5.6.2.2 Attempting to Make Sense of Irrational Exponents with the Use of Graphs 5.6.3 Follow-Up Activities: Irrational Exponents 5.6.3.1 Finding Irrational Numbers on the Number Line 5.6.3.2 Graphing Rational Exponents 5.7 Conclusion References Chapter 6: Characterisation of Mathematics Teacher Educators’ Knowledge in Terms of Teachers’ Professional Potential and Challenging Content for Mathematics Teachers 6.1 Introduction 6.2 Background 6.2.1 Students’ Mathematical Potential as Challenging Content for MTs 6.2.2 MTs’ and MTEs’ Proficiency as a Function of Varying Mathematical Challenge 6.3 Framing Challenging Content for MTs Using Mathematical Challenge and Mathematical Potential 6.4 MTEs’ Knowledge and Skills in Terms of MTs’ Professional Potential and Challenging Content for MTs References Chapter 7: Learning to Teach Mathematics: How Secondary Prospective Teachers Describe the Different Beliefs and Practices of Their Mathematics Teacher Educators 7.1 Beliefs About Mathematics and Mathematics Teaching 7.2 This Study 7.3 Survey Results and Discussion 7.3.1 Beliefs About Mathematics 7.3.2 Beliefs About Teaching Mathematics 7.4 Beliefs About Learning Mathematics 7.4.1 Differences Between the Beliefs of Subgroups of MTEs and Between MTEs and Prospective Teachers 7.5 Differences Related to MTEs’ Qualifications 7.6 Interviews with MTEs and Prospective Teachers 7.6.1 The Case of Ryan 7.6.2 The Case of Paul 7.6.3 The Case of Sam 7.6.4 Discussion of the MTE Cases 7.6.5 Prospective Teachers’ Views on Mathematics Teaching 7.7 Conclusions References Part II: Learning and Developing as a Mathematics Teacher Educator Chapter 8: Supporting Mathematics Teacher Educators’ Growth and Development Through Communities of Practice 8.1 Background 8.2 Forming the Community of Practice 8.3 Theoretical Framings 8.3.1 Reflection and Inquiry 8.3.2 Mathematical Knowledge for Teaching 8.4 Our CoP Processes 8.5 What Did We Learn? 8.5.1 Mathematics Content Knowledge 8.5.2 Working with Young Adult Learners 8.5.3 Thinking About Our Questioning 8.5.4 Learning from Our Community of Practice 8.6 Communities of Practice in the MTE Community 8.7 Conclusions References Chapter 9: Artifact-Enhanced Collegial Inquiry: Making Mathematics Teacher Educator Practice Visible 9.1 The Methods Course 9.1.1 General Information 9.1.2 Cycle of Enactment and Investigation 9.1.3 Contemplate then Calculate (CtC) 9.2 Theoretical Perspective 9.3 Artifact-Enhanced Collegial Inquiry (ACI) 9.4 Illustrating ACI 9.4.1 Phase 1: Proposing and Negotiating the Focus of Inquiry Within MTE Practice 9.4.2 Phase 2: Reconstructing and Enhancing the Focus of Inquiry with Artifacts 9.4.3 Phase 3: Consolidating and Projecting Forward from Focal Analysis to Future MTE Practice 9.4.4 Coda 9.5 Discussion References Chapter 10: Working with Awareness as Mathematics Teacher Educators: Experiences to Issues to Actions 10.1 Introduction 10.2 Background Ideas 10.2.1 Working with Awarenesses 10.2.2 Metacommunication 10.2.3 Second-Person Perspectives 10.3 A Way of Working: Experiences to Issues to Actions (Laurinda) 10.3.1 Story: Planning for the 4-Minute Workshop 10.3.1.1 Task 1: Limitations We Put on Ourselves 10.3.1.2 Task 2: What to Do When Students Have Finished? 10.3.1.3 Task 3: What’s the Purpose of the Activity? 10.4 Current Stories and Discussions of Planning 10.4.1 Alf: Session on Using ICT 10.4.2 Tracy: Session on “Algebra” 10.4.3 Julian: Session on “Assessment” 10.5 Reflecting on Similarities and Differences in the Learning of Prospective Teachers and MTEs 10.6 Layers of Awareness References Chapter 11: Mapping the Territory: Using Second-Person Interviewing Techniques to Narratively Explore the Lived Experience of Becoming a Mathematics Teacher Educator 11.1 Introduction 11.2 Theoretical Underpinnings 11.2.1 Being an Enactivist 11.2.2 What Is Learning? 11.2.3 Second-Person Interviewing 11.3 Methodology and Methods 11.3.1 Using the Protocol for Second-Person Interviewing 11.3.2 Stabilising Attention 11.3.3 Turning the Attention from What to How? 11.3.4 Moving from a General Representation to a Singular Experience 11.3.5 Getting to New Basic-Category Labels 11.4 Case Study Written by Alistair: Becoming a Mathematics Teacher Educator 11.4.1 Narrative for Strapline: Setting Up the Culture 11.5 Discussion of Case Study 11.6 Multiple Perspectives 11.6.1 Strapline: Setting Up the Culture 11.6.2 Thoughts on Similarities and Differences for Setting Up the Culture 11.6.3 Strapline: Listening and Listening for 11.6.4 Thoughts on Similarities and Differences for Listening and Listening for 11.7 Final Discussion References Chapter 12: From Researcher in Pure Mathematics to Primary School Mathematics Teacher Educator 12.1 Introduction 12.2 Teacher Education in Norway 12.3 Literature on Becoming a Mathematics Teacher Educator 12.4 Methodology: Inner Research and Self-Study 12.5 Investigation of MTE Learning Within a Four-Dimensional Framework 12.5.1 Knowledge and Learning 12.5.2 Inquiry and Reflection 12.5.3 Insider and Outsider 12.5.4 Individual and Community 12.6 Conclusion References Chapter 13: Shaping our Collective Identity as Mathematics Teacher Educators 13.1 Introduction 13.2 Methodology 13.2.1 Methodological Framework 13.2.2 Actualising the Methodology 13.3 Analysis and Discussion 13.3.1 Collective Identity: The Ingredients 13.3.1.1 Common Interests 13.3.1.2 Common Experiences 13.3.1.3 Solidarity 13.3.2 Collective Identity as a Gestalt 13.3.3 Collective Identity as Partially Enabled by the Project 13.3.4 Collective Identity as Multi-layered 13.3.5 Effects of Disciplinary Boundaries 13.3.6 Transitions Between Layers of Collective Identity 13.4 Conclusions References Chapter 14: The Influence of and Interactions Between Different Contexts in the Learning and Development of Mathematics Teacher Educators 14.1 Introduction 14.2 Teacher Education Policy and Practice in England 14.3 Analytic Framework 14.4 Specific Professional Contexts and Their Influence on the Change Environment 14.5 The Master’s Programmes: Operating as External Sources of Stimulus and as Influences on the MTEs’ Change Environment 14.6 Analysis of Each of the Instances of Change Reported by the MTEs 14.6.1 Clare 14.6.2 Colin 14.6.3 Jen 14.7 Reflections on the Interactions Between Different Contexts and Their Impact on Individual MTEs’ Learning References Chapter 15: Mathematics Teacher Educators’ Learning in Supporting Teachers to Link Mathematics and Workplace Situations in Classroom Teaching 15.1 Introduction 15.2 Literature Review and Theoretical Background 15.3 Methodology 15.3.1 The Context of the Study 15.3.2 The Group of MTEs 15.3.3 Data Analysis Process 15.4 Results 15.4.1 MTEs’ Concerns 15.4.1.1 Making Sense of How Workplace Situations and IBL Can Be Linked to Mathematics Teaching 15.4.1.2 Enacting Workplace Situations and IBL in PD Meetings 15.4.2 Tensions and Attempts to Deal with Them 15.4.2.1 Tension: Authenticity of Workplace Situations Versus Classroom Teaching 15.4.2.2 Tension: High Versus Low Degree of Teachers’ Autonomy 15.5 Conclusion References Chapter 16: Mathematics Teacher Educators Learn from Dilemmas and Tensions in Teaching About/Through Culturally Relevant Pedagogy 16.1 Introduction 16.2 Literature Review 16.2.1 Mathematics Teacher Educator Learning 16.2.2 Teacher Education 16.2.3 Culture, Mathematics, and CRP 16.3 Lindsay’s Narrative: Who and What Is “Relevant” in CRP? 16.3.1 Kathy’s Response to Lindsay’s Narrative 16.4 Kathy’s Narrative: How Do Students Respond to CRP? 16.4.1 Push 16.4.2 Patience 16.4.3 Pressure 16.4.4 Lindsay’s Response to Kathy’s Narrative 16.5 Concluding Thoughts References Chapter 17: Supporting Secondary Mathematics Teacher Educators in China: Challenges and Opportunities 17.1 Introduction 17.2 Secondary MTEs in China: Composition, Responsibilities, and Developmental Trajectories 17.2.1 A Brief Overview of the Teaching Research System and Teacher Promotion System in China 17.2.2 School-Based Mathematics Mentor Teachers 17.2.3 Mathematics Teaching Researchers (MTRs) 17.2.4 University-Based MTEs 17.2.5 Preparing Secondary MTEs in China 17.3 University-Based Secondary MTEs: Challenges and Responses 17.3.1 Methodology 17.3.2 Results 17.3.3 Summary 17.4 Discussion References Part III: Methodological Challenges in Researching Mathematics Teacher Educator Expertise, Learning and Development Chapter 18: What Influences Mathematics Teacher Educators’ Decisions in Course Design: Activity Theory and Professional Capital as an Investigative Approach 18.1 Introduction and Impetus for the Study 18.2 Researching the Work of Mathematics Teacher Educators 18.3 Theoretical Perspectives 18.3.1 Activity Theory 18.3.2 Professional Capital 18.4 The Course-Review Process 18.5 Results and Analysis of Meetings and Interviews 18.5.1 Beliefs and Values 18.5.1.1 Perceptions of Mathematics 18.5.1.2 Pre-service Teachers’ Productive Dispositions 18.5.1.3 Deepening Pre-service Teachers’ Content Knowledge 18.5.1.4 What Is Important for Pre-service Teachers to Learn Regarding Pedagogy in Mathematics? 18.5.2 Agency and Autonomy 18.5.3 PCK for Pre-service Teachers 18.5.4 Alignment 18.5.5 Summary of Analysis 18.6 Reflection on the Themes 18.6.1 Reflection on Themes in Relation to Activity Theory 18.6.2 Reflection on Themes in Relation to Professional Capital 18.7 Implications References Chapter 19: Researching Modelling by Mathematics Teacher Educators: Shifting the Focus onto Teaching Practices 19.1 Introduction 19.2 Concept of Modelling 19.3 Methodological Challenges in the Study of Modelling 19.3.1 Modelling as a Two-Sided Practice 19.3.2 Modelling as a Situated Practice 19.4 The Next Step in Researching Modelling with a Focus on Teaching Practices Inside the Classroom References Chapter 20: Mathematics Teacher Educators Within the New Technological Environments: Changing the Perspective 20.1 Introduction 20.2 Meta-Didactical Transposition 20.3 MOOC-MDT: The MDT Framework Adapted to the MOOC 20.4 MOOC’s Zone Theory: Networking Between MOOC-MDT and Zone Theory 20.5 The Math MOOC UniTo Project 20.6 Analysing the MTEs Involved in Our MOOCs 20.6.1 Target 20.6.2 Theme 20.7 Discussion and Conclusion References Part IV: Commentaries Chapter 21: Mathematics Teacher Educator Knowledge for Teaching Teachers 21.1 Mathematics Teacher Knowledge: A Basis for MTE Knowledge 21.2 MTE Knowledge for Teaching Mathematics Teachers 21.2.1 MTE Knowledge as MTK 21.2.2 MTE Knowledge as KMTEd 21.2.3 MTE Knowledge as Beliefs 21.3 Reflection on Conceptualization of MTE Knowledge 21.4 Conclusion References Chapter 22: Who Are We as MTEs: And How Do We Learn and Develop? 22.1 Background and Introduction 22.2 Theme 2: Learning and Development as an MTE 22.2.1 Reflection and Voice 22.2.2 Collaboration and Inquiry 22.2.3 Theoretical Underpinnings 22.2.4 Methodology 22.2.5 Learning from the Literature 22.2.6 Another Chapter 22.3 Theme 3: Methodological Challenges in Researching MTE Expertise, Learning and Development 22.4 In Conclusion References Correction to: Pedagogical Tasks Toward Extending Mathematical Knowledge: Notes on the Work of Teacher Educators Correction to: Chapter 5 in: M. Goos, K. Beswick (eds.), The Learning and Development of Mathematics Teacher Educators, Research in Mathematics Education, https://doi.org/10.1007/978-3-030-62408-8_5 Author Index Subject Index