Teaching with Tasks for Effective Mathematics Learning

MATHEMATICS TEACHER EDUCATION

VOLUME 9

SERIES EDITOR

Andrea Peter-Koop, University of Oldenburg, GermanyPatricia Wilson, University of Georgia, United States

EDITORIAL BOARD

Andy Begg, Auckland University of Technology, New ZealandChris Breen, University of Cape Town, South AfricaFrancis Lopez-Real, University of Hong Kong, ChinaJarmila Novotna, Charles University, CzechoslovakiaJeppe Skott, Danish University of Education, Copenhagen, DenmarkPeter Sullivan, Monash University, Melbourne, AustraliaDina Tirosh, Tel Aviv University, Israel

SCOPE

The Mathematics Teacher Education book series presents relevant research and innovative international developments with respect to the preparation and professional development of mathematics teachers. A better understanding of teachers’ cognitions as well as knowledge about effective models for preservice and inservice teacher education is fundamental for math-ematics education at the primary, secondary and tertiary level in the various contexts and cultures across the world. Therefore, considerable research is needed to understand what facil-itates and impedes mathematics teachers’ professional learning. The series aims to provide a signi fi cant resource for teachers, teacher educators and graduate students by introducing and critically re fl ecting new ideas, concepts and fi ndings of research in teacher education.

For further volumes:http://www.springer.com/series/6327

Peter Sullivan � Doug Clarke � Barbara Clarke

Teaching with Tasks for Effective Mathematics Learning

Peter Sullivan Faculty of Education Monash University Clayton , VIC , Australia

Barbara Clarke Faculty of Education Monash University Frankston , VIC, Australia

Doug Clarke Mathematics Teaching and Learning Research Centre Australian Catholic University Fitzroy , VIC , Australia

ISBN 978-1-4614-4680-4 ISBN 978-1-4614-4681-1 (eBook) DOI 10.1007/978-1-4614-4681-1 Springer New York Heidelberg Dordrecht London

Library of Congress Control Number: 2012945394

© Springer Science+Business Media New York 2013 This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part of the material is concerned, speci fi cally the rights of translation, reprinting, reuse of illustrations, recita-tion, broadcasting, reproduction on micro fi lms or in any other physical way, and transmission or infor-mation storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. Exempted from this legal reservation are brief excerpts in connection with reviews or scholarly analysis or material supplied speci fi cally for the purpose of being entered and executed on a computer system, for exclusive use by the purchaser of the work. Duplication of this publication or parts thereof is permitted only under the provisions of the Copyright Law of the Publisher’s location, in its current version, and permission for use must always be obtained from Springer. Permissions for use may be obtained through RightsLink at the Copyright Clearance Center. Violations are liable to prosecution under the respective Copyright Law. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publica-tion does not imply, even in the absence of a speci fi c statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. While the advice and information in this book are believed to be true and accurate at the date of publica-tion, neither the authors nor the editors nor the publisher can accept any legal responsibility for any errors or omissions that may be made. The publisher makes no warranty, express or implied, with respect to the material contained herein.

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v

Acknowledgements

This project was funded by an Australian Research Council Linkage Grant, and we are grateful for the support of the ARC. We also acknowledge the contribution of the Department of Education and Early Childhood Development (DEECD, Victoria, Australia) and the Catholic Education Of fi ce Melbourne (CEOM, Australia) who assisted with funding, provided links with schools, and offered helpful guidance throughout the project. In particular, we thank Nadia Walker and Helen Gist (DEECD) and Gerard Lewis and Janeane Anderson (CEOM) for their roles in this.

We thank the three clusters of schools and their cluster leaders, Merging Minds Cluster (Jan McCluskey): Armadale Primary School, Hawthorn Secondary College, Lloyd Street Primary School, Malvern Central School, and Toorak Primary School; Berwick South Cluster (Peter Sanders): Berwick Fields Primary School, Brentwood Park Primary School, Hallam Valley Primary School, Hillsmeade Primary School, Kambrya College, and Timbarra Primary School; and Geelong Cluster (Peter Caddy): Our Lady Star of the Sea Ocean Grove, St Aloysius Queenscliff, St Anthony’s Lara, St Ignatius Secondary College, St Margaret’s East Geelong, St Therese Torquay, and St Thomas Drysdale.

There was a group of teachers within the clusters who volunteered to develop and teach extended units of work as the research team studied their teaching. We are particularly grateful to these teachers. They were Shaye Bradbury, Elizabeth Campbell, Anthony Cavagna, Robyn Frigo, John Hassett, Alison Jordan, and Chris Powers.

We also thank Helen O’Shea who was a Senior Research Fellow on the project for the last two years and who made a major contribution to the data on students which are reported in this book. We acknowledge too the work of Anne Roche in data collection, professional development leadership, and research assistance. We also thank Orit Zaslavasky, Ron Tzur, and Irit Peled who contributed to the project when visiting Monash University. Meredith Begg and Leicha Bragg assisted with

vi Acknowledgements

the documentation of lessons and the development of the website, while Alison Jansz-Senn and Nike Prince undertook important administrative and research work in the earlier years of the project. Wendy May assisted us greatly in bringing this book to fruition with her management and editorial skills in the later stages, while Sheryl Sullivan assisted with re fi ning the tasks included in the fi nal chapter.

Sarah Ferguson and Richard O’Donovan completed PhDs which focused on aspects of the project involving low attaining students and the challenges teachers experience with using tasks effectively, respectively.

vii

Barbara Clarke is an Associate Professor in Mathematics Education at Monash University (Peninsula Campus) where she teaches primary pre-service teachers. Barbara has considerable experience in conducting and supporting research and directing or supervising major research projects and contracts. The major focus of her writing and research has been concerned with mathematics teachers, their prac-tice, and their professional development.

Doug Clarke is a Professor of Mathematics Education at the Australian Catholic University (Melbourne), where he directs the Mathematics Teaching and Learning Research Centre. In recent years, Doug has worked on four Australian Research Council grants, focusing on integrating mathematics and science, the role of tasks in mathematics learning, encouraging student persistence while working on chal-lenging tasks, and providing appropriate support for teachers implementing national curricula, respectively. Doug’s professional interests include young children’s math-ematical learning, using mathematics to explore current events and students’ inter-ests, the role of task-based assessment interviews with students, problem solving and investigations, manageable and meaningful assessment, and the professional growth of mathematics teachers.

Peter Sullivan is a Professor of Science, Mathematics, and Technology at Monash University. His main professional achievements are in the fi eld of research. His recent research includes four Australian Research Council grants funded projects. He is an author of the popular teacher resource Open-ended maths activities: Using good questions to enhance learning that is published in the US as Good questions for math teaching . Until recently he was chief editor of the Journal of Mathematics Teacher Education , was immediate past president of the Australian Association of Mathematics Teachers, was the author of the Shape paper that outlines the princi-ples for the development of the Australian Curriculum in Mathematics, and was the author of the 2011 Australian Education Review on research informed strategies for teaching mathematics.

About the Authors

ix

1 Researching Tasks in Mathematics Classrooms ................................... 1The Tasks Type and Mathematics Learning Project ................................. 1Using This Book ....................................................................................... 4

2 Perspectives on Mathematics, Learning, and Teaching ....................... 7The Mathematics It Is Intended that Students Learn ................................ 7Fostering a Breadth of Mathematical Actions .......................................... 8Considering Students’ Perspectives on Mathematical Tasks .................... 9Approaches to Teaching Mathematics ...................................................... 10Summary ................................................................................................... 12

3 Tasks and Mathematics Learning.......................................................... 13Introduction ............................................................................................... 13The Connection Between Tasks and Learning .......................................... 14Types of Tasks for Mathematics Teaching ................................................ 15The Role of Teacher Knowledge in Effective Task Use ........................... 15Teacher Beliefs, Attitudes, and Self-Goals ............................................... 18Constraints ................................................................................................ 19Teacher Intentions ..................................................................................... 20Summary ................................................................................................... 21

4 Using Purposeful Representational Tasks ............................................ 23A Rationale for Purposeful Representational Tasks ................................. 23Defining Purposeful Representational Tasks in the TTML Project .......... 25An Example of a Purposeful Representational Task: Colour in Fractions .................................................................................................... 25

Colour in Fractions ............................................................................... 25Some Additional Examples ................................................................... 28

Sourcing and Creating Purposeful Representational Tasks ...................... 29

Contents

x Contents

Some Reactions from Project Teachers to Purposeful Representational Tasks .............................................................................. 29

The Teachers’ Definitions of the Tasks ................................................. 29Some Examples of the Tasks that Teachers Valued .............................. 30The Advantages of Purposeful Representational Tasks as Seen by the Teachers ..................................................................................... 31The Constraints on the Use of Purposeful Representational Tasks as Seen by the Teachers ........................................................................ 32

The Challenges of Teaching Purposeful Representational Tasks: Learning to Use a Ratio Table................................................................... 32

Making Cordial: Taking Opportunities ................................................. 33Bottles Task: Accepting Different Strategies ........................................ 33Medals at the Shrine: Operationalising the Tool ................................... 36

Summary ................................................................................................... 36

5 Using Mathematical Tasks Arising from Contexts .............................. 39A Rationale for Tasks Built Around Practical Contexts ........................... 39The Contextualised Tasks That Were the Focus of Our Project ............... 42A Specific Example of a Contextualised Task .......................................... 43

Setting the Scene ................................................................................... 43Enabling Prompts .................................................................................. 43Students’ Strategies ............................................................................... 44Building upon Students’ Insights .......................................................... 45Pulling the Lesson Together .................................................................. 45Students as Problem Posers .................................................................. 46

Some Examples of Contextualised Tasks Which Teachers Valued .......... 46Some Reactions from Teachers to Contextualised Tasks .......................... 48

Teachers’ Views on Advantages and Difficulties in Using Contextualised Tasks ............................................................................. 49

Teacher Actions and Their Impact on Task Potential and Student Learning .................................................................................................... 50

Teacher A .............................................................................................. 50Teacher B .............................................................................................. 52Teacher C .............................................................................................. 53Some Reflections on the Three Lessons ............................................... 54

Sourcing and Creating Contextualised Tasks ............................................ 54Taking Prompts and Using Them to Develop Contextualised Tasks ........ 55Summary ................................................................................................... 56

And Where Was the Photograph Taken? .............................................. 56

6 Using Content-Specific Open-Ended Tasks .......................................... 57The Potential Contribution from Open-Ended Tasks to Student Learning .................................................................................................... 57Creating a Learning Experience Around a Content-Specific Open-Ended Task ...................................................................................... 59

xiContents

Insights into Related Teacher Actions Based on Observations During the Project ..................................................................................... 61

Finding Out What the Students Are Actually Doing Is Difficult .......... 62Sometimes Communicating the Precision of Mathematical Language Is Helpful .............................................................................. 63It Is Important that Questions Are Purposeful ...................................... 63There Is No Need to Talk All the Time ................................................. 64It Is Useful to Be Aware of Reluctant Contributors .............................. 64

Examples of Content-Specific Open-Ended Tasks ................................... 64Some Reactions from Project Teachers to Open-Ended Tasks ................. 65

The Teachers’ Definitions of the Tasks ................................................. 65Some Examples of the Tasks that Teachers Valued .............................. 66The Advantages of Open-Ended Tasks as Seen by the Teachers .......... 67The Constraints on the Use of Open-Ended Tasks as Seen by the Teachers ..................................................................................... 67

Sourcing and Creating Content-Specific Open-Ended Tasks ................... 68Method 1: Working Backwards ............................................................ 68Method 2: Adapting a Standard Question ............................................. 69

Summary ................................................................................................... 70

7 Moving from the Task to the Lesson: Pedagogical Practices and Other Issues ..................................................................... 71Looking for Three More: An Open-Ended Task to Challenge and Enhance Students’ Understanding of (Mean) Average ..................................................................................................... 72

Being Clear on the Mathematical Focus and the Goals of the Lesson for Students ..................................................................... 72Considering the Background Knowledge Which Student Are Likely to Bring to the Task, How to Establish This, and Likely Responses Students Will Make to the Tasks, Including the Difficulties They Might Experience ............................... 73Considering Ways in Which Students Who Have Difficulty Making a Start on the Problem and Students Who Solve the Problem Quickly Might Best Be Supported ......................................... 74Monitoring Students’ Responses to Tasks as They Work Individually or in Small Groups on the Tasks ....................................... 74Selecting Students Who Will Be Invited to Share During the Discussion Time ................................................................. 79Focusing on Connections, Generalisation, and Transfer ...................... 80Considering What the Next Lesson Might Look Like .......................... 81

Differences Between Task Types in Relation to the Process of Turning a Task into a Lesson ................................................................ 82Summary ................................................................................................... 83

xii Contents

8 Constructing a Sequence of Lessons ..................................................... 85Introduction ............................................................................................... 85A Sequence of Lessons on Presenting and Interpreting Data ................... 86An Investigative Project as an Overarching Theme .................................. 87Lesson One: Most Popular Letter ............................................................. 88Lesson Two: Introducing Graphical Representations ............................... 89

Features of Graphical Representations ................................................. 89Matching Graphical Representations .................................................... 89

Lesson Three: Conducting and Representing Survey Results .................. 90Lesson Four: Making Decisions on Representations ................................ 91

Finding Similar Graphs ......................................................................... 91Rock, Paper, Scissors ............................................................................ 92

Lesson Five: Two-Way Tables .................................................................. 92Lesson Six: Average Height of Class ........................................................ 93Lesson Seven: Mean Average ................................................................... 94

A Sentence with Five Words ................................................................. 94Average Height of the School ............................................................... 95

Lesson Eight: Comparing and Contrasting Measures of Centre and Spread ................................................................................. 95

Seven People Went Fishing ................................................................... 95Clues on Cards ...................................................................................... 96Post-sequence Assessment and Evaluation ........................................... 96

The Lesson Observations, Including Observations of Lesson Five ........................................................................................... 97Summary ................................................................................................... 98

9 Students’ Preferences for Different Types of Mathematics Tasks ............................................................................. 99Seeking Students’ Opinions About Tasks ................................................. 99Responses of Students to Pre-determined Prompts About Tasks and Pedagogies................................................................................ 100Students’ Attitudes .................................................................................... 101

In Summary ........................................................................................... 106Tasks Preferences Within a Lesson Sequence .......................................... 107Summary ................................................................................................... 109

10 Students’ Perceptions of Characteristics of Desired Mathematics Lessons .............................................................................. 111Responses from the Overall Survey on Desired Lesson Characteristics .............................................................................. 111Students’ Essays on Their “Ideal Maths Class” ........................................ 115Some Themes in the Data ......................................................................... 117

Classroom Grouping ............................................................................. 117Interaction Between Teacher and Students ........................................... 119

Summary ................................................................................................... 120

xiiiContents

11 Contrasting Types of Tasks: A Story of Three Lessons ....................... 121The Tasks and the Lessons ........................................................................ 122

Inside and Outside the Square ............................................................... 122Dido’s Problem ..................................................................................... 122What the L? ........................................................................................... 123

Results: Teacher Data ............................................................................... 126Order of Teaching ................................................................................. 126Teaching and Learning Challenges ....................................................... 127Teacher Preferences .............................................................................. 128One Teacher’s Contrast Between Types of Tasks ................................. 130

Student Responses .................................................................................... 131Student Preferences ............................................................................... 131Advice on Improving Lessons .............................................................. 132Summary Comments from the Student Data ........................................ 133Relationship Between the Student Preferences and the Teacher Preferences ........................................................................ 133

Summary ................................................................................................... 134

12 Conclusion ............................................................................................... 135What We Learnt About the Different Types of Tasks ............................... 136

Purposeful Representational Tasks ....................................................... 136Contextual Tasks ................................................................................... 137Open-Ended Tasks ................................................................................ 137Other Types of Tasks ............................................................................. 138

Planning for Task Implementation (or Turning the Task into a Lesson) ................................................................................... 138

Differences in Task Implementation ..................................................... 139Opportunities and Constraints in Task Use ........................................... 141

Students and Mathematics Tasks .............................................................. 143

13 A Selection of Mathematical Tasks ........................................................ 145Purposeful Representational Tasks ........................................................... 146

Chocolate Blocks .................................................................................. 147Clues on Cards ...................................................................................... 149Decimal Maze ....................................................................................... 152Smartie Predictions ............................................................................... 155Matching Graphical Representations .................................................... 159

Five Contextualised Tasks ......................................................................... 161Music Cards .......................................................................................... 162Mike and His Numbers ......................................................................... 165Block of Land ....................................................................................... 168Comparing Coins from Different Countries ......................................... 173

xiv Contents

Content-Specific Open-Ended Tasks ........................................................ 176Wrap the Present ................................................................................... 177Painting a Room .................................................................................... 180Writing a Sentence ................................................................................ 183Different Ways to Represent Data......................................................... 186Money Measurement ............................................................................ 190

References ........................................................................................................ 195

Index ................................................................................................................. 201