Mathematics education : exploring the culture of learning

256 Pages·2004·2.23 MB·English

by  Barbara Allen, Sue Johnston-Wilder

Mathematics Education
Mathematics Education: exploring the culture of learning identifies some of the most
significant issues in mathematics education today. Pulling together relevant articles
fromauthorswellknownintheirfieldsofstudy,thebookaddressestopicalissuessuchas:
• Gender
(cid:127) Equity
(cid:127) Attitude
(cid:127) Teacher belief and knowledge
(cid:127) Community of practice
(cid:127) Autonomy and agency
(cid:127) Assessment
(cid:127) Technology
The subject is dealt with in three parts: culture of the mathematics classroom;
communication in mathematics classrooms; and pupils’ and teachers’ perceptions.
Studentsonpostgraduatecoursesinmathematicseducationwillfindthisbooka
valuable resource. Students on BEd and PGCE courses will also find this a useful
source of reference as will teachers of mathematics, mentors and advisers.
Barbara Allen is Director of the Centre for Mathematics Education at The Open
Universityandhaswrittenextensivelyonthesubjectofmathematicsteaching.
SueJohnston-WilderisaSeniorLectureratTheOpenUniversityandhasworked
formanyyearsdevelopingmaterialstopromoteinterestinmathematicsteachingand
learning.
Companion Volumes
The companion volumes in this series are:
Fundamental Constructs in Mathematics Education
Edited by: John Mason and Sue Johnston-Wilder
Researching Your Own Practice: the discipline of noticing
Author: John Mason
Allofthesebooksarepartofacourse:ResearchingMathematicsLearning,thatisitselfpartofTheOpen
UniversityMAprogrammeandpartofthePostgraduateDiplomainMathematicsEducationprogramme.
The Open University MA in Education
TheOpenUniversityMAinEducationisnowfirmlyestablishedasthemostpopularpostgraduate
degreeforeducationprofessionalsinEurope,withover3,500studentsregisteringeachyear.TheMA
in Education is designed particularly for those with experience of teaching, the advisory service,
educational administration or allied fields.
Structure of the MA
TheMAisamodulardegreeandstudentsarethereforefreetoselectfromarangeofoptionsinthe
programmewhichbestfitsinwiththeirinterestsandprofessionalgoals.Specialistlinesinmanagement
andprimaryeducationandlifelonglearningarealsoavailable.StudyinTheOpenUniversity’sAdvanced
Diploma can also be counted towards the MA and successful study in the MA programme entitles
studentstoapplyforentryintoTheOpenUniversityDoctorateinEducationprogramme.
OU Supported Open Learning
TheMAinEducationprogrammeprovidesgreatflexibility.Studentsstudyattheirownpace,intheir
own time, anywhere in the European Union. They receive specially prepared study materials
supported by tutorials, thus offering the chance to work with other students.
The Graduate Diploma in Mathematics Education
TheGraduateDiplomaisanewmodulardiplomadesignedtomeettheneedsofgraduateswhowish
todeveloptheirunderstandingofteachingandlearningmathematics.Itisaimedatprofessionalsin
educationwhohaveaninterestinmathematicsincludingprimaryandsecondaryteachers,classroom
assistants and parents who are providing home education.
The aims of the Graduate Diploma are to:
(cid:127) develop the mathematical thinking of students;
(cid:127) raise students’ awareness of ways people learn mathematics;
(cid:127) provideexperienceofdifferentteachingapproachesandthelearningopportunitiestheyafford;
(cid:127) develop students’ awareness of, and facility with, ICT in the learning and teaching of
mathematics; and
(cid:127) developstudents’knowledgeandunderstandingofthemathematicswhichunderpinsschool
mathematics.
How to apply
Ifyouwouldliketoregisterforoneoftheseprogrammes,orsimplytofindoutmoreinformation
about available courses, please request the Professional Development in Education prospectus by
writingtotheCourseReservationsCentre,POBox724,TheOpenUniversity,WaltonHall,Milton
KeynesMK76ZW,UKor,byphoning08709000304(fromtheUK)or+448709000304(from
outside the UK). Details can also be viewed on our web page www.open.ac.uk.
Mathematics Education
Exploring the culture of learning
Edited by Barbara Allen and
Sue Johnston-Wilder
First published 2004 by RoutledgeFalmer
11 New Fetter Lane, London EC4P 4EE
Simultaneously published in the USA and Canada
by RoutledgeFalmer
29 West 35th Street, New York, NY 10001
RoutledgeFalmer is an imprint of the Taylor & Francis Group
This edition published in the Taylor & Francis e-Library, 2004.
©2004 The Open University
All rights reserved. No part of this book may be reprinted or
reproduced or utilised in any form or by any electronic,
mechanical, or other means, now known or hereafter
invented, including photocopying and recording, or in any
information storage or retrieval system, without permission in
writing from the publishers.
British Library Cataloguing in Publication Data
A catalogue record for this book is available from the British Library
Libraty of Congress Cataloging in Publication Data
A catalog record has been requested
ISBN 0-203-46539-3 Master e-book ISBN
ISBN 0-203-47216-0 (Adobe eReader Format)
ISBN 0–415–32699–0 (hbk)
ISBN 0–415–32700–8 (pbk)
Contents
Listoffigures vii
Listoftables viii
Sources ix
Introduction:issuesinresearchingmathematicslearning 1
BARBARAALLENANDSUEJOHNSTON-WILDER
SECTION 1
Culture of the mathematics classroom – including equity
andsocialjustice 7
1 Imagesofmathematics,valuesandgender:aphilosophicalperspective 11
PAULERNEST
2 Towardsasociologyoflearninginprimaryschools 26
ANDREWPOLLARD
3 Learnersasauthorsinthemathematicsclassroom 43
HILARYPOVEYANDLEONEBURTONWITHCORINNEANGIER
ANDMARKBOYLAN
4 Paradigmatic conflicts in informal mathematics assessment as
sourcesofsocialinequity 57
ANNEWATSON
5 Constructing the ‘legitimate’ goal of a ‘realistic’ maths item:
acomparisonof10–11-and13–14-yearolds 69
BARRYCOOPERANDMÁIRÉADDUNNE
6 Establishing a community of practice in a secondary
mathematicsclassroom 91
MERRILYNGOOS,PETERGALBRAITHANDPETERRENSHAW
vi Contents
SECTION 2
Communicationinmathematicsclassrooms 117
7 Mathematics, social class and linguistic capital: an analysis of
mathematicsclassroominteractions 119
ROBYNZEVENBERGEN
8 What is the role of diagrams in communication of
mathematicalactivity? 134
CANDIAMORGAN
9 ‘Thewhisperers’:rivalclassroomdiscoursesandinquirymathematics 146
JENNYHOUSSART
10 Steeringbetweenskillsandcreativity:aroleforthecomputer? 159
CELIAHOYLES
SECTION 3
Pupils’andteachers’perceptions 173
11 The relationship of teachers’ conceptions of mathematics and
mathematicsteachingtoinstructionalpractice 175
ALBAGONZALEZTHOMPSON
12 Setting,socialclassandsurvivalofthequickest 195
JOBOALER
13 ‘I’ll be a nothing’: structure, agency and the construction of
identitythroughassessment 219
DIANEREAYANDDYLANWILIAM
14 Pupils’perspectivesonlearningmathematics 233
BARBARAALLEN
Index 243
Figures
1.1 The reproductive cycle of gender inequality in mathematics education 19
1.2 The simplified relations between personal philosophies of
mathematics, values, and classroom images of mathematics 21
2.1 The relationship betweenintra-individual, interpersonal and
socio-historical factors in learning 29
2.2 A model of classroom task processes 31
2.3 Individual, context and learning: an analytic formula 36
2.4 A social-constructivist model of the teaching/learning process 37
2.5 A model of learning and identity 38
4.1 Power relationships 61
5.1 Finding ‘n’: an ‘esoteric’ item 71
5.2 Tennis pairs: a ‘realistic’ item 71
5.3 Die/pin item and Charlie’s written response 80
6.1 The elastic problem 111
8.1 Richard’s ‘inner triangles’ 137
8.2 Craig’s response 139
8.3 Richard’s trapezium 140
8.4 Sally’s response to the ‘Topples’ task 142
10.1 Tim’s initial view of proof 162
10.2 Tim’s evaluation of a visual proof 163
10.3 A typicalExpressorscreen to explore the sum of three consecutive
numbers 164
10.4 Tim’s proof that the sum of four consecutive numbers is not divisible
by four 165
10.5 Tim’sinductiveproofthatthesumoffiveconsecutivenumbersis
divisiblebyfive 165
10.6 Tim’s two explanations 166
10.7 Susie’s rule for consecutive numbers 167
12.1 Relationship between mathematics GCSE marks and NFER entry
scores at (a) Amber Hill and (b) Phoenix Park 210
Tables
5.1 Response strategy on the tennis item (interview) by class (10–11 years) 74
5.2 Response strategy on the tennis item (interview) by sex (10–11 years) 74
5.3 Marks achieved (one mark available) on the tennis item in the
interview context: initial response (10–11 years) 75
5.4 Marks achieved (one mark available) on the tennis item in the
interview context after cued response (10–11 years) 77
5.5 Response strategy on the tennis item (interview) by class (13–14 years) 77
5.6 Responsestrategyonthetennisitem(interview)bysex(13–14years) 77
5.7 Marks achieved (one mark available) on the tennis item in the
interview context: initial response (13–14 years) 77
5.8 Marks achieved (one mark available) on tennis item in the interview
context: after cued response (13–14 years) 78
6.1 Assumptions about teaching and learning mathematics implicit in
teacher–student interactions 99
6.2 Year 11 maths lesson 1: Finding the inverse of a 2 × 2 matrix 101
6.3 Year 11 maths lesson 2: Inverse and determinant of a 2 × 2 matrix 102
9.1 Comparison of cultures and domains of discourse 151
9.2 Outcome when whisperer’s discourse is audible 156
12.1 Means and standard deviations (SD) of GCSE marks and
NFER scores 211
12.2 Amber Hill overachievers 212
12.3 Amber Hill underachievers 212
12.4 Phoenix Park overachievers 212
12.5 Phoenix Park underachievers 213
12.6 GCSE mathematics results shown as percentages of students in
each year group 214
Sources
Chapter1 Reproduced,withkindpermissionoftheauthor,fromachapteroriginally
published in Keitel, C. (ed.), Social Justice and Mathematics Education, pp. 45–58,
Taylor & Francis (1998).
Chapter 2 Reproduced from an article originally published in British Journal of
Sociology of Education,11(3) pp. 241–56, Taylor & Francis (1990).
Chapter3 ReproducedfromachapteroriginallypublishedinBurton,L.(ed.),Learning
Mathematics:fromhierarchiestonetworks,pp.232–45,FalmerPress(1999).
Chapter4 ReproducedfromanarticleoriginallypublishedinEducationalReview,
52(2) pp. 105–15, Taylor & Francis (1999).
Chapter5 ReproducedfromachapteroriginallypublishedinFiler,A.(ed.),Assessment–
SocialPracticeandSocialProduct,pp.87–109,RoutledgeFalmer(2000).
Chapter6 ReproducedfromachapteroriginallypublishedinBurton,L.(ed.),Learning
Mathematics:fromhierarchiestonetworks,pp.36–61,FalmerPress(1999).
Chapter 7 Reproduced from a chapter originally published in Atweh, B. and
Forgasz, H. (eds), Socio-cultural Aspects of Mathematics Education: An International
Perspective, pp. 201–15, Lawrence Erlbaum (2000).
Chapter8 ReproducedfromanarticleoriginallypublishedinProceedingsoftheBritish
SocietyforResearchinMathematicsLearning,pp.80–92,InstituteofEducation(1994).
Chapter 9 Reproduced from an article originally published in For the Learning of
Mathematics,21(3) pp. 2–8, FLM Publishing Association (2001).
Chapter10 ReproducedfromanarticleoriginallypublishedinFortheLearningof
Mathematics,21(1) pp. 33–9, FLM Publishing Association (2001).
Chapter11 ReproducedfromanarticleoriginallypublishedinEducationalStudies
in Mathematics,15(2) pp. 105–27, Taylor and Francis (1984).
Chapter12 ReproducedfromanarticleoriginallypublishedinBritishEducational
Research Journal, 23(5) pp. 575–95, Taylor & Francis (1997).
Chapter13 ReproducedfromanarticleoriginallypublishedinBritishEducational
Research Journal, 25(3) pp. 343–54, Taylor & Francis (1999).