RESPONSE VERIFICATION
MATHEMATICS TEACHER EDUCATORS AS DEVELOPING PROFESSIONALS AN INTRODUCTION 1 PART 1 THEORIES AND CONCEPTUALISATIONS OF MATHEMATICS TEACHER EDUCATORS AND THEIR CHARACTERISTICS 13 1 HOW FAR IS THE HORIZON TEACHER EDUCATORS KNOWLEDGE AND SKILLS FOR TEACHING HIGH SCHOOL MATHEMATICS TEACHERS 15 2 DEVELOPING AS A MATHEMATICS TEACHER EDUCATOR LEARNING FROM THE OXFORD MSC EXPERIENCE 35 3 THEORETICAL PERSPECTIVES ON LEARNING AND DEVELOPMENT AS A MATHEMATICS TEACHER EDUCATOR 54 PART 2 MATHEMATICS TEACHER EDUCATORS LEARNING IN TRANSITIONS AND THROUGH COLLABORATIONS 80 4 THEORISING THEORISING ABOUT MATHEMATICS TEACHERS AND MATHEMATICS TEACHER EDUCATORS ENERGETIC LEARNING 82 5 MATHEMATICS TEACHER EDUCATOR COLLABORATIONS BUILDING A COMMUNITY OF PRACTICE WITH PROSPECTIVE TEACHERS 105 6 EDUCATING MATHEMATICS TEACHER EDUCATORS THE TRANSPOSITION OF DIDACTICAL RESEARCH AND THE DEVELOPMENT OF RESEARCHERS AND TEACHER EDUCATORS 133 7 MATHEMATICS TEACHER EDUCATORS LEARNING THROUGH SELF BASED METHODOLOGIES 160 CONTENTS PART 3 MATHEMATICS TEACHER EDUCATORS LEARNING FROM PRACTICE 191 8 CONCEPTUALIZATION AND ENACTMENT OF PEDAGOGICAL CONTENT KNOWLEDGE BY MATHEMATICS TEACHER EDUCATORS IN PROSPECTIVE TEACHERS MATHEMATICS CONTENT COURSES 193 9 LEARNING TO BE MATHEMATICS TEACHER EDUCATORS FROM PROFESSIONAL PRACTICE TO PERSONAL DEVELOPMENT 234 10 LEARNING WITH AND FROM TRU TEACHER EDUCATORS AND THE TEACHING FOR ROBUST UNDERSTANDING FRAMEWORK 274 11 MATHEMATICS TEACHER EDUCATORS LEARNING FROM EFFORTS TO FACILITATE THE LEARNING OF KEY MATHEMATICS CONCEPT WHILE MODELLING EVIDENCE BASED TEACHING PRACTICE 307 12 MATHEMATICS TEACHING DEVELOPMENT IN HIGHER EDUCATION 344 13 BECOMING A MATHEMATICS TEACHER EDUCATOR PERSPECTIVES FROM KAZAKHSTAN AND AUSTRALIA 369 PART 4 RESEARCHING MATHEMATICS TEACHER EDUCATORS 392 14 COMPETING PRESSURES ON MATHEMATICS TEACHER EDUCATORS 393 MATHEMATICS TEACHER EDUCATORS AS DEVELOPING PROFESSIONALS AN INTRODUCTION THIS CHAPTER PROVIDES AN INTRODUCTION TO VOLUME 4 IT BEGINS WITH AN OUTLINE OF DEVELOPMENTS IN RESEARCH ON MATHEMATICS TEACHER EDUCATORS OVER THE PERIOD SINCE THE FIRST EDITION OF THE HANDBOOK THE BULK OF THE CHAPTER IS STRUCTURED AROUND THE FOUR PARTS OF THE VOLUME THEORIES AND CONCEPTUALISATIONS OF MATHEMATICS TEACHER EDUCATORS AND THEIR CHARACTERISTICS MATHEMATICS TEACHER EDUCATORS LEARNING IN TRANSITIONS AND THROUGH COLLABORATIONS MATHEMATICS TEACHER EDUCATORS LEARNING FROM PRACTICE AND RESEARCHING MATHEMATICS TEACHER EDUCATORS THESE PARTS INCLUDE A BRIEF ACCOUNT OF EACH CHAPTER WITH ATTENTION DRAWN TO SOME OF THE CONNECTIONS AMONG THEM THE CHAPTER CONCLUDES WITH A SUMMARY OF THE VARIOUS WAYS IN WHICH THERE HAS BEEN PROGRESS IN OUR UNDERSTANDING OF MATHEMATICS TEACHER EDUCATORS ALONG WITH A REMINDER OF THE POLITICAL IMPORTANCE OF OUR WORK SINCE THE FIRST EDITION OF THE INTERNATIONAL HANDBOOK OF MATHEMATICS TEACHER EDUCATION THERE HAS A BEEN A BURGEONING OF RESEARCH INTEREST IN THOSE WHOSE RESPONSIBILITIES INCLUDE THE EDUCATION OF PROSPECTIVE AND PRACTISING TEACHERS OF MATHEMATICS THERE HAVE FOR EXAMPLE BEEN DISCUSSION GROUPS AND WORKING SESSIONS AT AN INTERNATIONAL CONGRESS ON MATHEMATICS EDUCATION BESWICK CHAPMAN GOOS ZASLAVSKY 2015 ANNUAL CONFERENCES OF THE INTERNATIONAL GROUP FOR THE PSYCHOLOGY OF MATHEMATICS EDUCATION E G BESWICK CHAPMAN 2013 BESWICK GOOS CHAPMAN 2014 AND A SPECIAL ISSUE OF THE JOURNAL OF MATHEMATICS TEACHER EDUCATION BESWICK GOOS 2018 ALL OF THESE HAVE FOCUSSED ON THE KNOWLEDGE OF MATHEMATICS TEACHER EDUCATORS THE NATURE OF THAT KNOWLEDGE AND ITS ACQUISITION AND DEVELOPMENT IN ADDITION A SPECIAL ISSUE OF THE JOURNAL MATHEMATICS TEACHER EDUCATION AND DEVELOPMENT FOCUSSED ON MATHEMATICS TEACHER EDUCATORS INQUIRIES INTO THEIR PRACTICES AND THEIR IMPACTS IN PROSPECTIVE TEACHERS MUIR BRAGG LIVY 2018 THE CHAPTERS IN THIS BOOK DEVELOP THE THEMES CONCERNING MATHEMATICS TEACHER EDUCATORS KNOWLEDGE AND THE WAYS IN WHICH THEIR WORK AND THEIR REFLECTION ON IT CONTRIBUTE TO THEIR DEVELOPMENT 1 ONE SEEMINGLY SUPERFICIAL DIFFERENCE BETWEEN THIS VOLUME AND THE CORRESPONDING VOLUME IN THE FIRST EDITION OF THE HANDBOOK IS THE MUCH GREATER NUMBER OF AUTHORS INVOLVED WHEREAS THERE WERE 23 CONTRIBUTORS TO THE FIRST EDITION VOLUME THERE ARE 54 AUTHORS INVOLVED IN THIS VOLUME AND THIS IN SPITE OF THE NUMBER OF CHAPTERS BEING FEWER 14 COMPARED WITH 17 EVEN IF WE DISCOUNT 22 OF THE 23 AUTHORS OF THE SCHOENFELD ET AL CHAPTER THERE ARE 32 CONTRIBUTORS THIS TIME THE EXTENT OF COLLABORATION EVIDENT IN THE AUTHORSHIP OF CHAPTERS IN THIS VOLUME IS CONSISTENT WITH A BROADENING OF THE CONVERSATION ABOUT MATHEMATICS TEACHER EDUCATORS THAT SPANS MANY COUNTRIES AND A MOVEMENT AWAY FROM INDIVIDUAL MATHEMATICS TEACHER EDUCATORS REFLECTING ALONE ON THEIR LEARNING TO UNDERTAKING THIS REFLECTIVE WORK AS A COLLABORATIVE ENTERPRISE IT SUGGESTS THAT THE WORK OF MATHEMATICS TEACHER EDUCATORS MAY BE ENTERING THE MAINSTREAM OF MATHEMATICS EDUCATION RESEARCH THE REFLECTIONS AND THEORETICAL CONTRIBUTIONS OF INDIVIDUALS ARE OF COURSE STILL VITAL AND THESE ALSO FEATURE IN THE CURRENT VOLUME DEVELOPMENT OF THE FIELD IS REFLECTED IN MANY CHAPTERS INCLUDED IN THIS VOLUME TWO OF THESE BY THORNTON BEAUMONT LEWIS AND PENFOLD CHAPTER 2 AND ABBOUD ROBERT AND ROGALSKI CHAPTER 6 ARISE FROM EXPERIENCES OF TEACHING OR PARTICIPATING IN EDUCATION PROGRAMS FOR PROSPECTIVE MATHEMATICS TEACHER EDUCATORS SUCH PROGRAMS ARE RECOGNISED ELSEWHERE IN THE VOLUME AS NEEDED APPOVA CHAPTER 8 BUT RARE WU YAO CAI CHAPTER 9 THORNTON ET AL AND ABBOUD ET AL FOCUS ON THE LEARNING OF THE AUTHORS FROM THEIR EXPERIENCES IN THESE CONTEXTS AND ALTHOUGH ABBOUD ET AL PROVIDE A QUITE DETAILED DESCRIPTION OF THEIR PROGRAM THIS IS NOT THE MAIN FOCUS OF THESE CHAPTERS THE WAYS IN WHICH THE DESIGN OF PROGRAMS FOR MATHEMATICS TEACHER EDUCATORS RELATES TO CONCEPTUALISATIONS OF MATHEMATICS TEACHER EDUCATOR KNOWLEDGE AND VARIES WITH CONTEXT LARGELY REMAINS TO BE EXPLORED ALTHOUGH THESE CHAPTERS AND THAT OF APPOVA MAKE A CONTRIBUTION TO THE DISCUSSION OF WHAT A CURRICULUM FOR MATHEMATICS TEACHER EDUCATORS MIGHT COMPRISE WHAT APPROPRIATE AIMS FOR SUCH PROGRAMS MIGHT BE AND THE EXTENT TO WHICH STANDARDISATION ACROSS PROGRAMS MIGHT BE DESIRABLE GOODCHILD CHAPTER 12 EXTENDS THE RANGE OF MATHEMATICS TEACHERS WITH WHOM MATHEMATICS TEACHER EDUCATORS WORK IN HIS REFLECTIONS ON HIS LEARNING FROM WORKING WITH TEACHERS OF UNIVERSITY MATHEMATICS WHO IN MOST CASES HAVE NO CONNECTION WITH TEACHER EDUCATION EXTENSION IN A DIFFERENT WAY IS OFFERED BY WU ET AL CHAPTER 9 WHO ARGUE THAT THE TEACHING PRACTICE OF MATHEMATICS TEACHER EDUCATORS INVOLVES CONSIDERABLY MORE THAN THEIR WORK DIRECTLY WITH PROSPECTIVE OR PRACTISING MATHEMATICS TEACHERS IN HER INTRODUCTION TO THE CORRESPONDING VOLUME OF THE FIRST EDITION OF THE HANDBOOK JAWORSKI 2008 POINTED TO A SHIFT IN THE THEORETICAL ORIENTATION OF MATHEMATICS EDUCATION RESEARCH FROM A LARGELY INDIVIDUAL PSYCHOLOGICAL FOCUS TOWARDS ACCOUNTING FOR THE SOCIAL CONTEXTS IN WHICH TEACHING AND LEARNING OCCUR SHE NOTED THE INCREASED USE OF SOCIOCULTURAL THEORIES AND GREATER ATTENTION TO POLITICAL AND POLICY ISSUES THESE TRENDS HAVE CONTINUED AND ARE IN EVIDENCE IN THE CHAPTERS OF THIS VOLUME TWO OF THE CHAPTERS THOSE BY GOODCHILD CHAPTER 12 AND GOOS RESPONSE VERIFICATION BOOK 2 2 CHAPTER 3 PROVIDE UPDATES ON THESE AUTHORS CONTRIBUTIONS TO THE FIRST EDITION THE PROGRESSION IN THEIR RESEARCH REFLECTS DEVELOPMENT IN THE FIELD THIS VOLUME IS ORGANISED IN FOUR PARTS THEORIES AND CONCEPTUALISATIONS OF MATHEMATICS TEACHER EDUCATORS AND THEIR CHARACTERISTICS MATHEMATICS TEACHER EDUCATORS LEARNING IN TRANSITIONS AND THROUGH COLLABORATIONS MATHEMATICS TEACHER EDUCATORS LEARNING FROM PRACTICE RESEARCHING MATHEMATICS TEACHER EDUCATORS THESE ARE DISCUSSED IN TURN IN THE FOLLOWING SECTIONS THEORIES AND CONCEPTUALISATIONS OF MATHEMATICS TEACHER EDUCATORS AND THEIR CHARACTERISTICS THE CHAPTERS IN THIS PART ILLUSTRATE THE GROWING USE OF SOCIOCULTURAL THEORIES AND COMMUNITY OF PRACTICE IDEAS IN MATHEMATICS EDUCATION RESEARCH JAWORSKI 2008 WITH SOCIOCULTURAL THEORIES AS WELL AS PSYCHOLOGICAL VIEWS IN EVIDENCE MERRILYN GOOS FOR EXAMPLE PROVIDES AN ACCOUNT OF HOW ZONE THEORY VALSINER 1997 CAN BE APPLIED NOT ONLY TO THE DEVELOPMENT OF MATHEMATICS TEACHERS GOOS 2008 BUT ALSO TO THE DEVELOPMENT OF MATHEMATICS TEACHER EDUCATORS AND HOW MATHEMATICS TEACHER EDUCATORS WITH DIFFERING BACKGROUNDS E G MATHEMATICIANS AND MATHEMATICS EDUCATION RESEARCHERS CAN BE VIEWED AS CONSTITUTING COMMUNITIES OF PRACTICE WENGER 1998 SHE ARGUES THAT ENCOUNTERS AT THE BOUNDARIES OF THESE COMMUNITIES ESPECIALLY IF SUSTAINED OVER TIME OFFER POTENTIAL FOR LEARNING BY MATHEMATICS TEACHER EDUCATORS ON BOTH SIDES OF THE BOUNDARY ROZA LEIKIN ON THE OTHER HAND BUILDS HER MODEL OF MATHEMATICS TEACHER EDUCATORS KNOWLEDGE AND SKILLS ON CONCEPTIONS OF TEACHER KNOWLEDGE FOUNDED IN CONSTRUCTIVIST VIEWS OF LEARNING THAT ALTHOUGH ACKNOWLEDGING THE ROLE OF THE SOCIAL CONTEXT IN LEARNING PORTRAY KNOWLEDGE AS AN ATTRIBUTE OF THE INDIVIDUAL IN HER CHAPTER LEIKIN DESCRIBES SEVERAL CONCEPTUALISATIONS OF MATHEMATICS TEACHER EDUCATOR KNOWLEDGE INCLUDING THAT OF GOOS 2009 USING ZONE THEORY THAT HAVE IN COMMON A HIERARCHICAL RELATIONSHIP BETWEEN MATHEMATICS TEACHER EDUCATORS KNOWLEDGE AND THE KNOWLEDGE OF SCHOOL MATHEMATICS TEACHERS SHE MAKES THE IMPORTANT POINT THAT THE PRECISE NATURE OF THE RELATIONSHIP BETWEEN LAYERS OF THESE HIERARCHIES DEPENDS UPON THE SPECIFIC GOALS AND ACTIVITIES OF THE MATHEMATICS TEACHER EDUCATORS CONCERNED AS WELL AS THE CHARACTERISTICS OF THE MATHEMATICS TEACHERS WITH WHOM THEY ARE WORKING MATHEMATICS TEACHER EDUCATORS MAY BE UNIVERSITY BASED MATHEMATICIANS MATHEMATICS EDUCATION RESEARCHERS OR EXPERT SCHOOL MATHEMATICS TEACHERS DESPITE THEIR SHARED OBJECTIVE OF DEVELOPING MATHEMATICS TEACHERS KNOWLEDGE AND SKILLS FOR TEACHING MATHEMATICS IN SCHOOLS THE SPECIFIC GOALS THAT MATHEMATICS TEACHER EDUCATORS FROM THESE GROUPS HAVE FOR THEIR INTERACTIONS WITH MATHEMATICS TEACHERS PROSPECTIVE OR PRACTISING AND THE ACTIVITIES IN WHICH THEY ENGAGE VARY TO THE EXTENT THAT THERE MAY BE MINIMAL OVERLAP BETWEEN THE KNOWLEDGE AN AFFECTIVE CHARACTERISTICS RESPONSE VERIFICATION BOOK 2 3