USING MENTAL CALCULATION TO PERFORM THE FOUR MATHEMATICAL OPERATIONS: EXPERIENCES OF GRADE 2 TEACHERS WITH ADDITION AND SUBTRACTION IN SELECTED SCHOOLS IN LUSAKA, ZAMBIA By Muhau Tabakamulamu A thesis submitted to the University of Zambia in fulfillment of the requirements for the degree of Doctor of Philosophy in Mathematics Education THE UNIVERSITY OF ZAMBIA LUSAKA 2010 i COPYRIGHT Copyright © 2010 by Muhau Tabakamulamu. All rights reserved. No part of this Thesis may be reproduced or stored in any form without the written permission of the author or the University of Zambia ii DECLARATION I, MUHAU TABAKAMULAMU, declare that this Thesis (a) Represents my own work; (b) Has not previously been submitted for a degree of either this or another university; and (c) Does not incorporate any published work or material from another thesis. Signed:……………………………………………………………………………………. Date:………………………………………………………………………………………. iii APPROVAL This Thesis of MUHAU TABAKAMULAMU has been approved as fulfilling the requirements for the award of the Degree of Doctor of Philosophy in Mathematics Education by the University of Zambia Signature Date …………………………………. …………………………………………… …………………………………. …………………………………………… …………………………………. …………………………………………… …………………………………. …………………………………………… iv ABSTRACT Since the 1980s, mathematics educators around the world have accepted the view that to develop numeracy effectively, young children need initially to use strategies for mental calculation rather than to memorise textbook procedures. The term mental calculation refers to informal, often untaught calculation methods used by children to solve a variety of arithmetic problems. This primarily qualitative study used a quasi-experimental design to assess the extent to which teachers in early primary mathematics in Zambia could adopt the use of strategies for mental calculation for double-digit addition and subtraction. The study also sought to determine the impact of teachers’ use of such strategies on pupils’ performance in mathematics. Participants were 10 Grade 2 teachers (5 from control schools and 5 from experimental schools) and 311 Grade 2 pupils. The Grade 2 pupils, 167 of whom were from experimental schools and 144 from control schools, were evenly distributed with regard to sex, age, intellectual ability and socioeconomic background. In experimental schools the study had two phases: Grade 2 teachers attended a staff development workshop to prepare them to implement the study; followed by the main study when the teachers implemented in the classroom the ideas discussed during the workshop. In control schools the same set of mathematics topics were covered, but Grade 2 teachers there used the usual textbook procedures. Data were collected through lesson observations, interviews with teachers, teachers’ journal entries, and document analysis; while pupils’ performance was assessed by means of two numeracy tests. One numeracy test was administered at the beginning of the main study, and the other at the end of the study. Results of the study suggested that over the ten-week implementation period teachers in experimental schools changed their existing beliefs about mathematics teaching and learning and to some extent their classroom practices as well, to support the use of strategies for mental calculation for double-digit addition and subtraction. Their pupils performed significantly better in the post-test compared to pupils in control schools, and developed more positive attitudes towards learning mathematics. Based on these findings, it was recommended among other things that, to show pupils that their informal solution strategies were valued, teaching in early primary mathematics should build on them. At the same time teachers should demonstrate that standard textbook procedures resulted from pupils’ informal strategies and were more efficient. v DEDICATION This Thesis is dedicated to my parents, my late father Pastor Moses Tabakamulamu Singulwani and my mother Namatama Muyunda Tabakamulamu, who together struggled with little resources to ensure that I had a good education, although my father did not live long enough to see the final results of his efforts; and to my wife Kubecte and my children Namatama and Sepo, who now understand why I had to work so hard to complete this task. vi ACKNOWLEDGEMENTS A monumental work of this kind cannot be accomplished alone. In view of this, I would like to express my sincere thanks and gratitude to the following: • In Zambia my supervisor, Professor Vasyl S. Kostyuk, who never gave up on me even though my research project seemed to go on forever; colleagues in the Department of Mathematics and Science Education who kept asking, “When are you completing your doctoral programme?”; Professor Michael J. Kelly who though retired agreed to read a draft of the work; and the University of Zambia which funded my study programme; • In the UK Professor Mike Askew and Professor Margaret Brown of King’s College, University of London, who supervised my work while I was there and allowed me to modify and use in Zambia research materials used in the Leverhulme Project (UK); and their colleagues Tamara Bibby and Jeremy Hodgens who, not long from completing their own doctoral programmes, showed me that it could be done; and • Others too numerous to mention, whether in Zambia or in the UK, who made my work lighter by what they said or what they did. vii TABLE OF CONTENTS Page Copyright declaration ii Declaration iii Certificate of Approval iv Abstract v Dedication vi Acknowledgements vii List of Tables xvi List of Figures xvii List of Appendices xviii List of Abbreviations and Acronyms xix CHAPTER 1: INTRODUCTION 1 1.0 Introduction 1 1.1Background to the Study 1 1.1.1 Zambia: Brief Facts and Figures 1 1.1.2 The Education System in Zambia 6 1.1.3 Medium of Instruction in Education 8 1.1.4 Learner Performance in Primary Mathematics 13 1.2 Statement of the Problem 17 1.3 Purpose of the Study 17 1.4 Objectives of the Study 17 1.5 Research Questions 18 1.6 Significance of the Study 19 1.7 Definition of Terms 19 1.8 Limitations and Delimitations of the Study 20 1.9 Layout of the Thesis 21 viii CHAPTER 2: REVIEW OF RELATED LITERATURE 23 2.0 Introduction 23 2.1 Influences for Change in Numeracy Teaching Practices 23 2.1.1 Realistic Mathematics Education in the Netherlands 24 2.1.2 The National Numeracy Strategy in the United Kingdom 27 2.1.3 Standards-Based Mathematics Curricula in the United States 28 2.2 Mental Calculation Strategies Used by Children 30 2.2.1 Children’s Strategies for Single-digit Addition and Subtraction 30 2.2.2 Children’s Strategies for Double-digit Addition and Subtraction 31 2.3 Teaching Mental Calculation Strategies 34 2.4 Theories of Learning Reflected in Current Developments in Early Primary Mathematics Teaching and Learning 35 2.4.1 Radical Constructivism 36 2.4.2 Sociocultural Theories of Learning 41 2.4.3 Social constructivism 48 2.4.4 Implications of Social Constructivism for the Mathematics Classroom 51 2.5 Teacher Change and Developments in Early Primary Numeracy Teaching Practices 53 2.5.1 Characteristics of Change 55 2.5.2 Local Characteristics 58 2.5.2.1 Changing Teacher Beliefs 59 2.5.2.2 Teacher Knowledge and Educational Change 61 2.5.2.3 Changing Teaching Practices 67 2.5.3 External Factors 71 2.6 Evaluating Mathematics Innovations 71 2.7 Summary of the Literature Review 72 ix CHAPTER 3: RESEARCH DESIGN AND METHODOLOGY 75 3.0 Introduction 75 3.1 Broad Methodological Approach 75 3.2 Research Design 79 3.2.1 The Mixed Method Strategy Used in this Study 79 3.2.2 Quasi-experimental Design 80 3.2.3 Case Studies 82 3.2.4 Staff Development Workshop 82 3.3 Sample 83 3.3.1 Sample and Sample Selection 83 3.3.2 Description of Research Sites 85 3.3.2.1 Control Schools 86 5.3.2.2 Experimental Schools 89 3.4 Methods 92 3.4.1 Lesson Observation 92 3.4.2 Interviews with Teachers 94 3.4.2.1 Group Interviews 95 3.4.2.2 Post-observation Interviews 95 3.4.2.3 Post-treatment Interviews 96 3.4.3 Teachers’ Journal Entries 96 3.4.4 Numeracy Assessment Tests 96 3.4.5 Document Analysis 98 3.4.6 Procedures for Data Collection 99 3.4.6.1 Negotiating Access to Research Sites 99 3.4.6.2 Data Collection 100 3.4.7 Data Analysis 102 3.4.7.1 Analysis of Quantitative Data 102 3.4.7.2 Analysis of Qualitative Data 104 x