Listening figures : listening to learners of mathematics in the secondary school and above

159 Pages·2009·2.76 MB·English

by  John Mason and Jenny Houssart (eds.)

Listening Figures cover 234x156mm_Lifelong Learning John Fi#6BF35 18/12/2012 14:50 Page 1
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listening to learners of mathematics listening to learners of mathematics
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at secondary school and above N at secondary school and above
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edited by John Mason and Jenny Houssart edited by John Mason and Jenny Houssart
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The underlying principle of this book is listening to people in secondary
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schools and beyond as they go about discovering, learning and doing
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mathematics. The contributors report on what is gained by listening to G
school students, undergraduates and adult learners. Sometimes the U
listening is informal; sometimes it is based on mathematics lessons or
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interviews. The accounts are used as models for discussing issues
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about mathematical understanding and about views of mathematics
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and approaches to it.
Listening Figuresis invaluable for secondary teachers, classroom
assistants, adult numeracy tutors and lecturers, and for people
preparing for taking such roles. Teachers of mathematics will be e
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encouraged by the examples here to consider the place of learners’ it
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voices in their own teaching. Many of the chapters draw on the d

contributor’s empirical research and will interest teachers who are b
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carrying out their own small scale research projects.
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John Mason is Professor of Mathematics Education at the Centre for h
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Mathematics Education at the Open University and senior research

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fellow at the University of Oxford. Jenny Houssart is a senior lecturer
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at the Institute of Education, London. s
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The contributors are Shafia Abdul-Rahman, Barbara Allen, Andy Begg, n

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Kenrick Cuffy, Helen Drury, Roger Duke, Hilary Evens, Alan Graham, n
Melissa Rodd, Peter Johnston-Wilder and Sue Johnston-Wilder. All are d

experienced in working with secondary and adult students and their Je
teachers and many have connections with the Centre for Mathematics n
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Education at the Open University. y

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ISBN: 978-1-85856-449-4 o
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Trentham Books
www.trentham-books.co.uk
listening figures book 3/6/09 2:39 pm Page i
Listening Figures
listening to learners of mathematics
in the secondary school and above
listening figures book 3/6/09 2:39 pm Page ii
listening figures book 3/6/09 2:39 pm Page iii
Listening Figures
listening to learners of mathematics
in the secondary school and above
Edited by John Mason
and Jenny Houssart
Trentham Books
Stoke on Trent, UK and Sterling, USA
listening figures book 3/6/09 2:39 pm Page iv
Trentham Books Limited
Westview House 22883 Quicksilver Drive
734 London Road Sterling
Oakhill VA 20166-2012
Stoke on Trent USA
Staffordshire
England ST4 5NP
©2009 John Mason and Jenny Houssart
All rights reserved.No part of this publication may be reproduced or
transmitted in any form or by any means, electronic or mechanical
including photocopying, recording or any information storage or
retrieval system, without prior permission in writing from the
publishers.
First published 2009
British Library Cataloguing-in-Publication Data
A catalogue record for this book is available from the British Library
ISBN:978 1 85856 449 4
Designed and typeset by Trentham Print Design Ltd, Chester and
printed in Great Britain by Page Bros (Norwich) Ltd, Norfolk.
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Contents
Introduction• vii
John Mason and Jenny Houssart
Chapter 1
What Pupils Want:A Friendly Mathematics Classroom • 1
Barbara Allen
Chapter 2
Learning to Listen • 13
Helen Drury
Chapter 3
Matchbox Algebra• 25
Alan Graham and Roger Duke
Chapter 4
Listening to Learners Working with Robots• 39
Sue Johnston-Wilder
Chapter 5
‘If you don’t know math, people can rob you’: secondary Students
discuss why they learn mathematics • 49
Kenrick Cuffy and Jenny Houssart
Chapter 6
Random Listenings• 65
Peter Johnston-Wilder
Chapter 7
Awareness and Understanding Revealed• 77
Shafia Abdul Rahman
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LISTENING FIGURES
Chapter 8
Listening as a Relation(ship)• 93
Andy Begg
Chapter 9
Listening to Mathematics Undergraduates:
what are they telling us? • 107
Melissa Rodd
Chapter 10
Listening to Mathematics Students:
is our Language the same?• 117
Hilary Evens
Chapter 11
Learning from Listening to yourself• 127
John Mason
Notes on Contributors • 143
Index• 145
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Introduction
John Mason and Jenny Houssart
Introduction
W
hat can be learned by listening to learners of mathematics in
secondary, post-secondary and pre-service teacher educa-
tion? The contributors to this book offer the insights they
obtained through attending closely to what learners of mathematics do
and say. All the authors were at one time associated with the Centre for
Mathematics Education at the Open University, and all retain the prac-
tical yet theoretically well-founded stance typical of the Centre.
Looking back over the Centre’s twenty-five years of trying to support
people teaching children and adults mathematics, it is evident that one
of our cornerstones was that listening figures, both because listening to
learners makes sense, and because listening is an ongoing source of in-
sight into learning that can inform teaching. Under listening we include
attending to what learners do, to what they say about what they are
doing, and to what they say about how they feel about what they are do-
ing. The chapters offer a wide range of ways of listening in a wide variety
of contexts.
Listening
It is one thing to appear to listen to what learners say and do. It is quite
another to hear what they are saying and make use of this to inform
teaching. We are not suggesting that what learners say is always taken at
face value, but being listened to can have a significant effect on learners
as well as on the listeners.
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LISTENING FIGURES
Listening to learners is a popular practice in teacher education. Pre-
service teachers are usually asked to shadow a single learner over an ex-
tended period or else to work closely with a few learners, listening and
making observations. Listening is something that most teachers would
claim to do, or try to do, at least to some extent. However the pressures
of curriculum and assessment, behaviour management and lesson pre-
paration tend to displace intensive listening and watching. Researchers
have the advantage of having the time to listen carefully and at length
and not having to step in and direct learner attention in order to correct
misapprehensions and confusions.
When teachers are occasionally granted sufficient time to listen compre-
hensively, their appreciation of what learners are capable of is amplified
and transformed. For example, Michael Armstrong (1980) describes his
revelations when taking a year out simply to sit with and observe his
learners while someone else taught them; Philip Jackson (1992) reports
insights obtained when he observed secondary classrooms at length;
Jean Piaget and Hans Freudenthal both built their theories on extensive
conversations with their young relatives; a host of researchers have
contributed to the mathematics education literature through reporting
astonishing observations and perceptions of learners that might other-
wise have gone unnoticed. Most notable is the paper by Stanley Erl-
wanger (1973) reprinted in 2004 as a classic in mathematics education.
The distinction between listeningand hearingis evidenced by whether
or not anything happens as a result. McLeod (2008) points out that the
literature on listening to children includes writers who define listening
in terms of attitude and those who take it to imply action. The second
type of listening appears to be supported by a recent report on primary
mathematics (Williams, 2008) which stresses the importance of listening
as part of discussion in primary mathematics classrooms, and draws on
a recent review of literature (Kyriacou and Issit, 2008) advocating trans-
formative listening, when teachers listen to their pupils’ contributions in
a way that suggests a genuine meeting of minds and the teacher being
willing to change their thinking in the light of what the pupils have said.
In our definition, listening draws on a range of information and is fol-
lowed by action. However, although action may sometimes appear to
be immediate, there is often an intermediate step of interpretation,
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INTRODUCTION
which can be far from straightforward (Wallach and Even, 2005). This
leads us to a three step model of listening involving (a) openness to
information, verbal and otherwise, from learners, (b) consideration of
possible interpretations of this information and (c) action based on this
information. This was the model we had in mind when putting together
this collection. Our contributors were encouraged to write data rich
chapters which were based around accounts or quotes obtained by
listening or paying attention to learners of mathematics. We asked our
authors to consider what might be learned from their accounts, and
finally, what can be done with what is learned. Our main emphasis
throughout is on teaching and learning mathematics. The information-
interpretation-action cycle usually directly concerns mathematical
understanding, though it also sometimes concerns issues that impact
on this understanding such as preferred ways of learning.
This volume brings out different kinds and degrees of insight. Some-
times learners display unexpectedly deep insight into mathematics,
and this raises questions about the effect on pupils when their teachers,
parent-guardians, schools, and the education system reveal low expec-
tations of them. Sometimes learners display an unexpected lack of
awareness or understanding of what they can apparently do quite com-
petently. This raises questions about whether students’ understanding
is actually assessed, or whether what is assessed is more often about
their trained behaviour than their understanding.
Sometimes learners display insight into the ways in which they go
about studying; sometimes they display remarkable ignorance about
how they go about learning and what is actually expected of them. This
raises questions about the strategies and tactics used for teaching.
Benson Snyder (1970) formulated the notion of a hidden curriculum
consisting of requirements for success which learners are expected to
pick up for themselves and he used it to explain learner dissatisfaction
and frustration. David Bloor (1976) and more recent authors who adopt
an explicitly sociocultural perspective, are quick to draw attention to
how some students, particularly those with behavioural or socio-
economic disadvantage, are often slower than their more advantaged
peers to work out what behaviour is required of them at school and
often more reluctant to conform. So assessment based simply on a form
of listening is frequently impoverished and incomplete.
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