AN ANALYSIS OF THE NATURE AND FUNCTION OF MENTAL COMPUTATION IN PRIMARY MATHEMATICS CURRICULA by GEOFFREY ROBERT MORGAN Cert T., B.Ed.St., B.A., M.Ed (Primary Mathematics) A thesis submitted in fulfilment of the requirements for the degree of Doctor of Philosophy at the Centre for Mathematics and Science Education, Queensland University of Technology, Brisbane 1999 i KEYWORDS Arithmetic, Computation, Computational Estimation, Mathematics, Mathematics Curriculum, Mathematics Teaching, Mental Arithmetic, Mental Computation, Mental Strategies, Number, Number Sense, Queensland Educational History, Queensland Mathematics Syllabuses, Teacher Beliefs and Practices ii ABSTRACT This study was conducted to analyse aspects of mental computation within primary school mathematics curricula and to formulate recommendations to inform future revisions to the Number strand of mathematics syllabuses for primary schools The analyses were undertaken from past, contemporary, and futures perspectives Although this study had syllabus development in Queensland as a prime focus, its findings and recommendations have an international applicability Little has been documented in relation to the nature and role of mental computation in mathematics curricula in Australia (McIntosh, Bana, & Farrell, 1995, p 2), despite an international resurgence of interest by mathematics educators This resurgence has arisen from a recognition that computing mentally remains a viable computational alternative in a technological age, and that the development of mental procedures contributes to the formation of powerful mathematical thinking strategies (R E Reys, 1992, p 63) The emphasis needs to be placed upon the mental processes involved, and it is this which distinguishes mental computation from mental arithmetic, as defined in this study Traditionally, the latter has been concerned with speed and accuracy rather than with the mental strategies used to arrive at the correct answers In Australia, the place of mental computation in mathematics curricula is only beginning to be seriously considered Little attention has been given to teaching, as opposed to testing, mental computation Additionally, such attention has predominantly been confined to those calculations needed to be performed mentally to enable the efficient use of the conventional written algorithms Teachers are inclined to associate mental computation with isolated facts, most commonly the basic ones, rather than with the interrelationships between numbers and the methods used to calculate To enhance the use of mental computation and to achieve an improvement in performance levels, children need to be encouraged to value all methods of computation, and to place a priority on mental procedures This requires that teachers be encouraged to change the way in which they view iii mental computation An outcome of this study is to provide the background and recommendations for this to occur The mathematics education literature of relevance to mental computation was analysed, and its nature and function, together with the approaches to teaching, under each of the Queensland mathematics syllabuses from 1860 to 1997 were documented Three distinct time-periods were analysed: 1860-1965, 1966-1987, and post-1987 The first of these was characterised by syllabuses which included specific references to calculating mentally To provide insights into the current status of mental computation in Queensland primary schools, a survey of a representative sample of teachers and administrators was undertaken The statements in the postal, self-completion opinionnaire were based on data from the literature review This study, therefore, has significance for Queensland educational history, curriculum development, and pedagogy The review of mental computation research indicated that the development of flexible mental strategies is influenced by the order in which mental and written techniques are introduced Therefore, the traditional written-mental sequence needs to be reevaluated As a contribution to this reevaluation, this study presents a mental-written sequence for introducing each of the four operations However, findings from the survey of Queensland school personnel revealed that a majority disagreed with the proposition that an emphasis on written algorithms should be delayed to allow increased attention on mental computation Hence, for this sequence to be successfully introduced, much professional debate and experimentation needs to occur to demonstrate its efficacy to teachers Of significance to the development of efficient mental techniques is the way in which mental computation is taught R E Reys, B J Reys, Nohda, and Emori (1995, p 305) have suggested that there are two broad approaches to teaching mental computation a behaviourist approach and a constructivist approach The former views mental computation as a basic skill and is considered an essential prerequisite to written computation, with proficiency gained through direct teaching In contrast, the constructivist approach contends that mental computation is a process of higher-order thinking in which the act of generating and applying mental strategies is significant for an individual's mathematical development Nonetheless, this study has concluded that there may be a place for the direct teaching of selected mental strategies To support syllabus development, a sequence of mental iv strategies appropriate for focussed teaching for each of the four operations has been delineated The implications for teachers with respect to these recommendations are discussed Their implementation has the potential to severely threaten many teachers’ sense of efficacy To support the changed approach to developing competence with mental computation, aspects requiring further theoretical and empirical investigation are also outlined v TABLE OF CONTENTS KEYWORDS i ABSTRACT ii TABLE OF CONTENTS v LIST OF TABLES xii LIST OF FIGURES xiv ABBREVIATIONS xv STATEMENT OF ORIGINAL AUTHORSHIP xvi ACKNOWLEDGMENTS xvii CHAPTER INTRODUCTION TO THE STUDY 1.1 Orientation of the Study 1.2 Context of the Study 1.2.1 Mental Computation: Overview 1.2.2 Mental Computation: Reasons For The Resurgence Of Interest 1.2.3 Mental Computation: Place In Current Mathematics Curricula 12 1.2.4 Mental Computation: Student Performance 13 1.2.5 Mental Computation: Essential Changes In Outlook 16 1.2.6 Mental Computation: Needed Research 18 1.3 Purposes and Significance of the Study 19 1.4 Overview of the Study 21 1.4.1 Method And Justification 22 1.4.2 Chapter Guidelines 23 CHAPTER MENTAL COMPUTATION 2.1 Introduction 27 2.2 Research Questions 29 2.3 Recent Developments in Mathematics Education of Relevance to Mental Computation 30 2.3.1 Numeracy 31 2.3.2 Computation 32 vi 2.3.3 Number Sense 34 2.3.4 Learning Mathematics 35 2.4 The Calculative Process 37 2.5 The Nature of Mental Computation 40 2.6 2.5.1 Mental Computation Defined 41 2.5.2 Mental and Oral Arithmetic 42 2.5.3 Mental Computation and Folk Mathematics 44 2.5.4 Characteristics of Mental Procedures 48 Mental Computation and Computational Estimation 53 2.6.1 Components of Computational Estimation 54 2.6.2 Computational Estimation Processes 56 2.6.3 Comparison of Mental Computation and Computational Estimation 60 2.7 Components of Mental Computation 63 2.7.1 Affective Components 67 2.7.2 Conceptual Components 68 2.7.3 Related Concepts and Skills 69 2.7.4 Strategies for Computing Mentally 74 Models for Classifying Mental Strategies 77 Counting strategies 84 Strategies Based Upon Instrumental Understanding 88 Heuristic Strategies Based Upon Relational Understanding 92 2.7.5 Short-term and Long-term Memory Components of Mental Computation 107 2.8 2.9 Characteristics of Proficient Mental Calculators 112 2.8.1 Origins of the Ability to Compute Mentally 114 2.8.2 Memory for Numerical Equivalents 117 2.8.3 Memory for Interrupted Working 118 2.8.4 Memory for Calculative Method 120 Developing the Ability to Compute Mentally 123 2.9.1 Approaches to Developing Skill with Mental Computation 125 Traditional Approach 125 Alternative Approaches 127 2.9.2 General Pedagogical Issues 131 vii 2.9.3 Sequence for Introducing Computational Methods 134 2.9.4 Assessing Mental Computation 137 2.10 Summary and Implications for Mental Computation Curricula 139 2.11 Concluding Points 150 CHAPTER 3: 3.1 MENTAL COMPUTATION IN QUEENSLAND: 1860-1965 Introduction 152 3.1.1 Method 153 Sources of Evidence 153 Research Questions 155 Structure of Analysis 156 3.2 Selected Background Issues Related to Syllabus Development and Implementation 157 3.2.1 Focus of Syllabus Development and Implementation 158 3.2.2 Principles Underlying the Syllabuses from 1905 163 3.2.3 Syllabus Interpretation and Overloading 167 3.2.4 Summary of Background Issues 177 3.3 Terms Associated with the Calculation of Exact Answers Mentally 178 3.4 Roles Ascribed to Mental Arithmetic 183 3.5 3.4.1 Mental Arithmetic as a Pedagogical Tool 185 3.4.2 The Social Usefulness of Mental Arithmetic 190 3.4.3 Mental Discipline and Mental Arithmetic 192 The Nature of Mental Arithmetic 198 3.5.1 Interpretations of Mental Arithmetic 199 3.5.2 The Syllabuses and Mental Arithmetic 202 3.5.3 Mental Arithmetic as Implemented 216 3.6 Recommended Approaches to Teaching Mental Arithmetic 231 3.7 Conclusions and Summary 249 viii CHAPTER 4.1 MENTAL COMPUTATION IN QUEENSLAND: 1966-1997 Introduction 250 4.1.1 Background to Research Strategy 251 4.1.2 Research Focus 251 4.2 The Syllabuses and Mental Computation in Queensland: 1966-1987 252 4.3 Survey of Queensland Primary School Personnel 256 4.3.1 Survey Method 258 Research Questions 258 Instrument Used 259 Sample 262 Research Procedure 265 Methods of Analysis 267 4.3.2 Survey Results 272 Response Rate 272 Analysis of Nonresponse 275 Beliefs About Mental Computation and How It Should Be Taught 279 Current Teaching Practices 287 Past Teaching Practices 291 Inservice on Mental Computation 398 Textbooks Used to Develop Skill with Mental Computation 299 4.3.3 Discussion 302 Limitations of Findings 303 Conclusions 303 Concluding Points 317 4.4 Mental Computation in Queensland: Recent Initiatives 317 4.4.1 Student Performance Standards and Mental Computation 321 4.4.2 Number Development Continuum and Mental Computation 325 4.4.3 Implications for Mental Computation Curricula 325 461 SECTION PAST TEACHING PRACTICES If you were not a class teacher at any time during the period 1964 - 1987, please go to Section on Page SECTION 3.1 If you taught during a particular period, please circle a number to show the level of importance placed on mental computation during each period If you are unsure, please place a tick on the line provided No Little Some Great Importance Importance Importance Importance Unsure 39 40 For each period, how important did the syllabus consider the ability to calculate exact answers mentally to be? (a) 1964 - 1968 (1964 Syllabus) _ (b) 1969 - 1974 (PIM: 1st ed) _ (c) 1975 - 1987 (PIM: 2nd ed) _ (a) 1964 - 1968 _ (b) 1969 - 1974 _ (c) 1975 - 1987 _ For each period, how important did you consider the ability to calculate exact answers mentally to be? SECTION 3.2 If you taught during a particular period, please circle a number on each line to show how frequently you used each teaching technique to develop the ability to mentally calculate exact answers beyond the basic facts If you are unsure, please place a tick on the line provided Never 41 Seldom Sometimes Often Unsure Allowed children to decide the method to be used to arrive at an exact answer mentally (a) 1964 - 1968 _ (b) 1969 - 1974 _ (c) 1975 - 1987 _ 462 SECTION (PAST PRACTICES) cont Never 42 43 44 45 46 Seldom Sometimes Often Unsure Allowed children to explain and discuss their mental strategies for solving a problem: (a) 1964 - 1968 _ (b) 1969 - 1974 _ (c) 1975 - 1987 _ (a) 1964 - 1968 _ (b) 1969 - 1974 _ (c) 1975 - 1987 _ (a) 1964 - 1968 _ (b) 1969 - 1974 _ (c) 1975 - 1987 _ (a) 1964 - 1968 _ (b) 1969 - 1974 _ (c) 1975 - 1987 _ (a) 1964 - 1968 _ (b) 1969 - 1974 _ (c) 1975 - 1987 _ Gave several one-step questions and simply marked the answers as correct or incorrect: Emphasised speed when calculating exact answers mentally Insisted that children use the procedures for the written algorithms when calculating exact answers mentally Placed an emphasis on teaching rules for calculating exact answers mentally (e.g divide by ten by removing a zero) 463 SECTION (PAST PRACTICES) cont SECTION 3.3 Resources: Please list any resources (e.g texts) that were used to develop the ability to calculate exact answers mentally If any were used for a limited period only, please indicate Comments: Please write below any comments you may care to make: Please continue to Section over page 464 SECTION BACKGROUND INFORMATION For each question, please place a tick (√) in the appropriate box 47 In which Educational Region is your school located? Sunshine Coast Metropolitan West South West Wide Bay North West Metropolitan East Darling Downs Capricornia Peninsula Northern South Coast 48 What is the size of your school? Band: 49 10 How many years teaching experience have you had? < yr 1-5 yrs 6-10 yrs 11-15 yrs 16-20 yrs 21-25 yrs 26-30 yrs 30+ yrs 50 Are you a: Class Teacher Teaching Principal? Principal Deputy Principal? (Go to next question) (Go to Question 53) 51 If a class teacher, which year level/s are you currently teaching? None 52 Is your school a trial school for Student Performance Standards in mathematics? Yes No (If NO, go to Question 55) 465 SECTION (BACKGROUND INFORMATION) cont 53 If your school is a trial school and you are currently a class teacher, are you trialing the standards in your classroom? Yes No 54 Do you consider it important that inservice sessions on mental Computation be made available to teachers? Yes No 55 Have you attended, during the last three years, inservice sessions in which Mental Computation was a specific topic for discussion? Yes No (If NO, skip the next question) 56 If you have attended inservice on Mental Computation, who conducted the inservice? Teaching colleague Administrator Tertiary Lecturer Other Mathematics Adviser If Other, please specify: Comments: Please write any further comments you may care to make: THANK YOU for completing this survey Please return it, in the envelope provided, to the contact person in your school 466 APPENDIX D SURVEY CORRESPONDENCE D.1 Initial Letter: One-teacher Schools Dear Colleague As detailed in the attached "Memorandum to Principals" from the Executive Director, Review and Evaluation, approval has been granted for me (Geoff Morgan, Deputy Principal, Lawnton State School) to approach you to invite your participation in a survey of beliefs and teaching practices related to Mental Computation (Mental Arithmetic) Your school has been randomly selected as part of a sample of 115 State Primary Schools from all Education Regions and bands of schools The attached Research Outline indicates that the survey is designed as a culmination to historical research currently being undertaken The aim of the project is to document the nature and place of Mental Computation in Queensland primary classrooms from 1860 to the present The research is part of a doctoral program supervised by Dr Calvin Irons of the Queensland University of Technology's School of Mathematics, Science and Technology Rather than simply send a copy of the questionnaire to you in this mailing, the purpose of this letter is to determine whether you would be prepared to participate in the survey The questionnaire should take about 15-20 minutes to complete, with not all respondents having to complete all sections The information to be gathered concerns: Current beliefs about Mental Computation and how it should be taught; Present teaching practices; and Past teaching practices related to the 1964 Syllabus, and to the 1966-68 and 1975 "Programs in Mathematics" I am also interested in identifying the resources used by teachers to support the development of Mental Computation skills While some personal background information is asked for, all information obtained will be treated confidentially In fact, you are asked not to identify yourself or your school on the questionnaire itself as this information does not have any valid use during data analysis Some questions relate to recent inservice opportunities with regard to Mental Computation Besides being able to document the status of Mental Computation, it is also hoped to be able to provide recommendations to the Department on any inservice needs which may be identified /2 467 Your participation in this survey will be greatly appreciated Please tear off the form below and return in the envelope provided, preferably by Monday, 18 October A questionnaire will then be dispatched should a preparedness to be involved be expressed Thanking you, Yours faithfully MENTAL COMPUTATION SURVEY Please complete the appropriate section below and return in the envelope provided (by Monday, 18 October) to: Geoff Morgan Lawnton State School P.O Box LAWNTON 4501 Phone: Fax: (07) 285 2968 (07) 285 6506 Please SEND the questionnaire on Mental Computation Name of school: _ State School Your name (optional): _ (For addressing envelope to send questionnaire.) OR Please DO NOT SEND the questionnaire on Mental Computation 468 D.2 Initial Letter: to All Schools Except One-teacher Schools Dear Colleague As detailed in the attached "Memorandum to Principals" from the Executive Director, Review and Evaluation, approval has been granted for me (Geoff Morgan, Deputy Principal, Lawnton State School) to approach you to invite your participation, and that of your staff, in a survey of beliefs and teaching practices related to Mental Computation (Mental Arithmetic) Your school has been randomly selected as part of a sample of 115 State Primary Schools from all Education Regions and bands of schools The attached Research Outline indicates that the survey is designed as a culmination to historical research currently being undertaken The aim of the project is to document the nature and place of Mental Computation in Queensland primary classrooms from 1860 to the present The research is part of a doctoral program supervised by Dr Calvin Irons of the Queensland University of Technology's School of Mathematics, Science and Technology Rather than simply send copies of the questionnaire to you in this mailing, the purpose of this letter is to determine whether you and/or members of your staff would be prepared to participate in the survey The questionnaire should take about 15-20 minutes to complete, with not all respondents having to complete all sections The information to be gathered concerns: Current beliefs about Mental Computation and how it should be taught; Present teaching practices; and Past teaching practices related to the 1964 Syllabus, and to the 1966-68 and 1975 "Programs in Mathematics" I am also interested in identifying the resources used by teachers to support the development of Mental Computation skills While some personal background information is asked for, all information obtained will be treated confidentially In fact, respondents are asked not to identify themselves or their school on the questionnaire as this information does not have any valid use during data analysis Some questions relate to recent inservice opportunities with regard to Mental Computation Besides being able to document the status of Mental Computation, it is also hoped to be able to provide recommendations to the Department on any inservice needs which may be identified /2 469 Your school's participation in this survey will be greatly appreciated To facilitate the distribution of the questionnaire to staff members, I am wondering if you (or one of your staff) would be prepared to act as Contact Person for the receipt of the questionnaires, their distribution to other staff members, and their return after completion for which a postage-paid envelope will be provided It is hoped that at least one teacher from each year level, as well as school administrators, will be prepared to complete the questionnaire Could the form attached please be returned in the envelope provided, preferably by Monday, 18 October Questionnaires will then be dispatched should a preparedness to be involved be expressed Thanking you, Yours faithfully 470 MENTAL COMPUTATION SURVEY Please complete the appropriate section below and return in the envelope provided (by Monday, 18 October) to: Geoff Morgan Lawnton State School P.O Box LAWNTON 4501 Phone: Fax: (07) 285 2968 (07) 285 6506 Please SEND the questionnaire on Mental Computation (a) Name of School: State School (b) Name of Contact Person: _ (c) Position in school: (d) Number of Questionnaires required: _ (e) Information to assist data analysis: Number of class teachers on staff: _ Number of Deputy Principals: _ OR Please DO NOT SEND the questionnaire on Mental Computation 471 D.3 Letter to Contact Persons Accompanying Questionnaires Dear < Contact Person/Principal > Thank you for your offer to act as Contact Person for the survey on Mental Computation As requested, I have enclosed < number > copies of the questionnaire Also enclosed is a Reply Paid envelope in which the completed questionnaires may be returned Hopefully the survey will not take too long to complete For those who have been teaching since the mid-1960s, to which all sections may apply, it is anticipated that it should take around 15-20 minutes of their time As indicated in my original letter, all information received will be treated confidentially It will constitute a valuable contribution to the knowledge about the teaching of Mental Computation in Queensland during the last quarter century Your assistance, and that of the other members of your school's staff who complete the survey, is greatly appreciated Could the questionnaires please be returned by Friday, < Date > November Thanking you Yours sincerely 472 D.4 Initial Follow-up Letter to Principals of Schools Not Replying to Original Letter Dear Colleague I am writing to you with respect to the survey of Queensland state primary school teachers and administrators on Mental Computation that I am currently undertaking, to which documentation sent earlier this term referred A couple of schools have indicated that they wish to participate in the survey, but have mislaid my initial letter On the off-chance that your school may be in a similar situation, I have enclosed copies of the questionnaire If you, or any of your staff members, are prepared to complete the questionnaire it would be greatly appreciated I have enclosed a Reply Paid envelope in which any completed questionnaires may be returned Hopefully the survey will not take too long to complete For those who have been teaching since the mid-1960s, to which all sections may apply, it is anticipated that it should take around 15-20 minutes of their time As indicated in my original letter, all information received will be treated confidentially It will constitute a valuable contribution to the knowledge about the teaching of Mental Computation in Queensland during the last quartercentury Once again, your assistance, and that of the other members of your school's staff who complete the survey, is greatly appreciated Could any completed questionnaires please be returned by Friday, 26 November Thanking you, Yours faithfully Geoff Morgan P.S Should you have already posted the form, which was included with my previous letter, indicating that your school did not want to receive any questionnaires, please disregard this mailing 473 D.5 Second Follow-up Letter to Schools Requesting Questionnaires From Which Completed Forms Had Not Been Received Dear < Contact Person/Principal > With this school term fast drawing to a close, I'm writing to once again express my thanks to you for agreeing to act as contact person for the mental computation survey which I'm currently undertaking Could you please pass on my thanks to those at your school who completed the questionnaire A preliminary look at the information so far received suggests that it will prove very useful in mapping the status of mental computation in Queensland primary classrooms I hope to begin analysing the data during the forthcoming vacation To this end, it would be appreciated if the questionnaires that have been completed could be sent to me at your earliest convenience Recognising how busy we all have been during this term, any that have been completed will be gratefully accepted Thanks once again for your assistance in conducting this survey Wishing you an enjoyable vacation Yours sincerely Geoff Morgan 474 APPENDIX E MEANS AND STANDARD DEVIATIONS FOR SURVEY ITEMS IN FIGURES 4.1 - 4.6 Figure 4.1: Beliefs about the nature of mental computation Item 3a 5a 8a Mean 2.89 3.17 2.63 3.17 2.89 3.24 Std Dev .72 56 74 59 65 55 Figure 4.2: Beliefs about the general approach to teaching mental computation Item 10 13 14 Mean 2.32 3.18 3.24 Std Dev .67 50 55 Figure 4.3: Beliefs about specific issues associated with developing mental computation skills Item 11 15a 16a 17a 19 20a 21a 22 Mean 3.31 2.16 2.26 2.45 3.50 3.06 2.92 3.37 Std Dev .55 68 65 76 50 66 70 51 Figure 4.4: Selected teaching practices of middle- and upper-school teachers Item 25 26 29a 30a 31 33a 34a 36a Mean 3.55 3.50 2.30 1.79 3.32 2.13 3.18 1.60 Std Dev .55 60 75 61 61 70 77 69 Figure 4.5: Teaching practices used during periods 1964-1968, 1969-1974, and 1975-1987 Item 41a 42a 43aa 44aa 45aa 46aa Mean 2.78 2.78 1.64 1.58 2.62 1.47 Std Dev 41b 42b 43ba 44ba 45ba 46ba 3.14 3.09 1.81 1.64 2.65 1.53 76 81 73 65 88 55 41c 42c 43ca 44ca 45ca 46ca 3.31 3.27 1.89 1.86 2.69 1.73 72 77 70 72 95 66 1.08 1.02 87 80 1.05 62 475 Figure 4.6: Selected teaching practices and beliefs of middle- and upperschool teachers Current Teaching Practices Item Mean Std Dev 25 3.55 3.50 26 29 a 31 34 a 36 a Note a Current Beliefs Item Mean Std Dev .55 14 3.27 52 60 19 3.46 50 2.98 63 2.30 75 20 3.32 61 22 3.18 1.60 77 69 a 3.26 50 a 2.87 67 a 2.63 70 Items representing traditional beliefs and teaching practices recoded to reflect a nontraditional orientation to facilitate placement of item means on traditional-nontraditional continua ... Department of Education, for their guidance and assistance in gathering the sources of primary data for the analysis of mental computation in Queensland mathematics curricula • Dr Shirley O'Neill and. .. the gulf between learning and practising school mathematics and learning and practising the mathematics used outside the classroom, a focus that centres on the utility of school mathematics in. .. (a) the nature and role of mental computation in past mathematics curricula; (b) the reasons for the contemporary resurgence of interest in mental computation; (c) its place within current mathematics