An analysis of the nature and function of mental computation in primary mathematics curricula

Arithmetic, Computation, Computational Estimation, Mathematics, Mathematics Curriculum, Mathematics Teaching, Mental Arithmetic, Mental Computation, Mental Strategies, Number, Number Sen

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

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

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

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

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

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 1 INTRODUCTION TO THE STUDY 1.1 Orientation of the Study 1

1.2 Context of the Study 4

1.2.1 Mental Computation: Overview 4

1.2.2 Mental Computation: Reasons For The Resurgence Of Interest 8

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 2 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

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

2.5.1 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

2.6 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 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

2.9 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

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: MENTAL COMPUTATION IN QUEENSLAND: 1860-1965 3.1 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

3.4.1 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

3.5 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

CHAPTER 4 MENTAL COMPUTATION IN QUEENSLAND: 1966-1997

4.1 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

CHAPTER 5 MENTAL COMPUTATION: A PROPOSED SYLLABUS

COMPONENT

5.1 Introduction 327

5.1.1 Context and Focus for Change 330

5.1.2 Framework for Syllabus Development 332

5.2 Mental, Calculator, and Written Computation 334

5.2.1 Traditional Sequence for Introducing Mental, Calculator, and Written Computation 334

5.2.2 A Sequential Framework for Mental, Calculator, and Written Computation 337

5.3 Mental Strategies: A syllabus Component 342

5.3.1 Background Issues 343

5.3.2 Developmental Issues 345

5.3.3 Mental Strategies for Addition, Subtraction, Multiplication, and Division 348

5.4 Concluding Points 354

CHAPTER 6: MENTAL COMPUTATION IN QUEENSLAND: CONCLUSIONS AND IMPLICATIONS 6.1 Restatement of Background and Purpose of Study 356

6.2 Mental Computation: Conclusions 358

6.2.1 The Emphasis Placed on Mental Computation 359

6.2.2 Roles of Mental Computation 361

6.2.3 The Nature of Mental Computation 364

6.2.4 Approaches to Teaching Mental Computation 366

6.3 Implications for Decision Making Concerning Syllabus Revision 369

6.3.1 Fostering Debate about Computation 369

6.4 Recommendations for Further Research 373

REFERENCES 376

APPENDIX A SUMMARY OF MENTAL ARITHMETIC IN QUEENSLAND

MATHEMATICS SCHEDULES AND SYLLABUSES (1860-1964)

A.1 1860 Schedule 419

A.2 1876 Schedule 419

A.3 1891 Schedule 420

A.4 1894 Schedule 422

A.5 1897 Schedule 423

A.6 1902 Schedule 425

A.7 1904 Schedule 426

A.8 1914 Syllabus 427

A.9 1930 Syllabus 428

A.10 1938 Amendments 432

A.11 1948 Amendments 434

A.12 1952 Syllabus 436

A.13 1964 Syllabus 441

APPENDIX B ADDITIONAL NOTES: CHAPTER 3 450

APPENDIX C SELF-COMPLETION QUESTIONNAIRE 456

Section 1 Beliefs About Mental Computation and How It Should Be Taught 457

Section 2 Current Teaching Practices 459

Section 3 Past Teaching Practices 461

Section 4 Background Information 464

APPENDIX D SURVEY CORRESPONDENCE D.1 Initial Letter: One-teacher Schools 466

D.2 Initial Letter: to All Schools Except One-teacher Schools 468

D.3 Letter to Contact Persons Accompanying Questionnaires 471

D.4 Initial Follow-up Letter to Principals of Schools Not Replying to Original Letter 472

D.5 Second Follow-up Letter to Schools Requesting Questionnaires From Which Completed Forms Had Not Been Received 473

APPENDIX E MEANS AND STANDARD DEVIATIONS OF SURVEY ITEMS IN FIGURES 4.1-4.6 474

LIST OF TABLES

2.1 Components of Mental Computation 65

2.2 Counting Strategies 86

2.3 Strategies Based Upon Instrumental Understanding 89

2.4 Heuristic Strategies Based Upon Relational Understanding 94

3.1 Queensland Mathematics Schedules and Syllabuses: 1860-1965 161

3.2 Selection of Textbooks Relevant to Mental Arithmetic Available to Queensland Teachers From the Mid-1920s 170

3.3 Extract From Recommended Mental Arithmetic Exercise for "Middle Standards" for Use by Teachers of Multiple Classes 227

3.4 Examples of Written Items from the 1925 Mathematics Scholarship Paper Given to Fifth Class Children as Mental 230

4.1 Queensland Mathematics Schedules and Syllabuses: 1965-1987 254

4.2 Sample of Schools by Band Within Educational Regions 263

4.3 Schools Returning Questionnaires 273

4.4 School Response Rate by Region and Band 274

4.5 Analysis of Number of Questionnaires Returned 275

4.6 Questionnaire Response Rate by Region and Band 276

4.7 Items for Which Significant Differences in Response were Observed, Based on Time of Receipt 278

4.8 Percentage of Responses Related to Beliefs About the Importance of Mental Computation 280

4.9 Percentage of Responses Related to Beliefs About the Nature of Mental Computation 281

4.10 Percentage of Responses Related to Beliefs About the General Approach to Teaching Mental Computation 283

4.11 Percentage of Responses Related to Beliefs About Issues Associated with

Developing the Ability to Calculate Exact Answers Mentally 285 4.12 Percentage of Responses Related to Current Teaching Practices for

Developing the Ability to Compute Mentally 288 4.13 Percentage of Responses Related to Past Beliefs About Mental Computation293 4.14 Percentage of Responses Concerning Past Teaching Practices Related to

Mental Computation 294 4.15 Percentage of Responses Related to the Importance of and Participation in

Inservice Sessions on Mental Computation 299 4.16 Source of Inservice on Mental Computation During Period 1991-1993 299 4.17 Categorisation of Resources Listed by Respondents in Sections 2.2 and 3.3

of the Survey Instrument 301 4.18 Textbooks Specific to Mental Computation Currently Used by Middle and

Upper School Teachers 301 4.19 Textbooks Specific to Mental Computation Used During the Period 1964-

1987 302 5.1 Traditional Sequence for Introducing the Four Operations with Whole

Numbers as Presented in the Mathematics Sourcebooks for Queensland

Schools 336 5.2 Revised Sequential Framework for Introducing Mental, Calculator and

Written Procedures for Addition, Subtraction, Multiplication, and Division 340 5.3 Mental Strategies Component for Addition of Whole Numbers Beyond the

Basic Facts for Inclusion in the Number Strand of Future Mathematics

Syllabuses for Primary Schools 350 5.4 Mental Strategies Component for Subtraction of whole numbers Beyond the

basic facts for Inclusion in the Number Strand of Future Mathematics

Syllabuses for Primary Schools 351 5.5 Mental Strategies Component for Multiplication and Division of Whole

Numbers Beyond the Basic Facts for Inclusion in the Number Strand of

Future Mathematics Syllabuses for Primary Schools 353

LIST OF FIGURES

2.1 A model of the calculative process highlighting the central position of mental

calculation 39 2.2 Components of computational estimation 55 2.3 A view of memory processes for computing mentally 108 2.4 Traditional sequence for introducing computational procedures for each

operation 135 2.5 An alternative sequence for introducing computational procedures for each

operation 136

4.1 Position of means for items relating to the beliefs about the nature of mental

computation an a traditional-nontraditional continuum 282 4.2 Position of means for items relating to beliefs about the general approach to

teaching mental computation an a traditional-nontraditional continuum 284 4.3 Position of means for items relating to beliefs about specific issues

associated with developing mental computation skills an a

traditional-nontraditional continuum 287 4.4 Position of means for selected current teaching practices related to

developing mental computation skills on a traditional-nontraditional

continuum 291 4.5 Position of means for items relating to teaching practices used during the

periods 1964-1968, 1969-1974, 1975-1987 on traditional-nontraditional continua 297 4.6 Means for selected teaching practices and the beliefs which underpin them

for middle and upper school teachers 312

5.1 A conceptualisation of syllabus development to provide a focus on student

learning 336

AAMT Australian Association of Mathematics Teachers

AEC Australian Education Council

CDC Curriculum Development Centre

MSEB Mathematics Sciences Education Board

NCSM National Council of Supervisors of Mathematics

NCTM National Council of Teachers of Mathematics

NCMWG National Curriculum: Mathematics Working Group

NRC National Research Council

QSCO Queensland School Curriculum Office

STATEMENT OF ORIGINAL AUTHORSHIP

The work contained in this thesis has not been previously submitted for a degree or diploma at any other higher education institution To the best of my knowledge and belief, the thesis contains no material previously published or written by another person except where due reference is made

Signed: G R Morgan

Date: 12 January 1998

This study would not have been completed without the advice, support and operation of a number of people towards whom I wish to formally express my

co-appreciation Principal among these are:

• Dr Calvin Irons, Senior Lecturer, Queensland University of Technology, whose critical comments, advice and support as my supervisor were

invaluable at each stage of this study's development

• Associate Professor Tom Cooper, Head, Mathematics, Science and

Technology, Queensland University of Technology, who, as my assistant supervisor, provided constructive criticisms and direction at various stages

of the project

• My wife, Lena, and daughter, Fiona, whose understanding and support created an environment conducive to completing the task

Appreciation is also extended to:

• Mr Greg Logan, Ms Rosemary Mammino, and Mr Lex Brasher, History Unit, Queensland 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 Mr Barry Tainton, Research and Evaluation Unit, Department of Education, for their providing the information on which to form the sample of Queensland state primary schools

• The staff of the Centre for Mathematics and Science Education,

Queensland University of Technology, particularly for their assistance with the distribution of the questionnaires

• The teachers and administrators who returned completed questionnaires, and particularly to those staff members of the Lawnton State School who contributed to the questionnaire's development

CHAPTER 1

INTRODUCTION TO THE STUDY

1.1 Orientation of the Study

Several factors growth of technology, increased applications, impact of

computers, and expansion of mathematics itself have combined in the past quarter century to extend greatly both the scope and the application of the mathematical sciences Together, these forces have created a revolution in the nature and role of mathematics a revolution that must be reflected

in schools if students are to be well prepared for tomorrow's world (National Research Council [NRC], 1989, p 4)

In responding to this revolution in the nature and role of mathematics, the National Council of Supervisors of Mathematics (1989, pp 45-46) delineates twelve interrelated areas that it considers critical to the development of children's

mathematical competences essential for meeting the demands of the twenty-first century These are: problem solving, communicating mathematical ideas,

mathematical reasoning, applying mathematics to everyday situations, alertness to the reasonableness of results, estimation, appropriate computational skills (including mental, written, and technological procedures), algebraic thinking, measurement,

geometry, statistics, and probability In concert with these competences, A National

Statement on Mathematics for Australian Schools (Australian Education Council

[AEC], 1991, pp 11-13), suggests that the goals for learning mathematics involve students in (a) developing confidence and competence in dealing with commonly occurring situations, (b) developing positive attitudes towards their involvement in mathematics, (c) developing their capacity to use mathematics in solving problems individually and collaboratively, (d) learning to communicate mathematically, and (e) learning techniques and tools which reflect modern mathematics

These beliefs imply that computational skill per se can no longer be considered

an adequate measure of achievement in mathematics Nonetheless, computational competence remains an important goal of mathematics programs in primary

classrooms This goal, however, involves more than the routine application of memorised rules It involves children in developing:

• An expertise in problem solving and higher-order thinking

• A sound understanding of mathematical principles

• An ability to know when and how to use a variety of procedures for

calculating

(National Council of Supervisors of Mathematics [NCSM], 1989, p 44)

Such development is consistent with Willis’s (1995) advocacy for a curriculum that reflects the learning of mathematics which is significant and of value for an

individual's success in both private and professional endeavours

The ability to calculate exact as well as approximate answers mentally is essential to the repertoire of skills for computational competence in the 1990s and beyond (AEC, 1991, p 109) However, the development of an ability to arrive at exact answers mentally without the aid of external calculating or recording

devices mental computation (R E Reys, B J Reys, Nohda, & Emori, 1995, p 304) is one that has generally been neglected, or at least de-emphasised, in

classrooms during recent years, both in Australia and overseas (Koenker, 1961, p 295; McIntosh, 1990a, p 25; Shibata, 1994, p 17; Trafton, 1978, p 199; Wiebe,

1987, p 57) French (1987) suggests that "one reason for the lack of interest [in mental computation] is the association that [this] has with the daily mental tests once used universally in schools, with their emphasis on recall of facts and speed" (p 39) This emphasis characterised the mental arithmetic programs that were regularly conducted in classrooms as precursors to the main focus of arithmetic lessons: the development of the standard written algorithms for the four basic

operations

Given that it is essential that the development of an ability to calculate exact answers mentally gains greater prominence in classroom mathematics programs (Gough, 1993, p.2) and that little research relevant to its development has been

undertaken (McIntosh, Bana, & Farrell, 1995, p 2; B J Reys, 1991, p 1; R E Reys

et al., 1995, p 324), the aims of this study were:

• To analyse key aspects of mental computation within primary school

curricula from past, contemporary and futures perspectives

• To formulate recommendations concerning mental computation to inform future revisions to the Number strand of the mathematics syllabus for Queensland primary schools

In planning for and guiding the implementation of suggested changes to the nature of school mathematics, cognisance needs to be given to the nature of past mathematics curricula, as well as those of the present As R E Reys et al (1995) suggest: "In order to get where we want to be, it is essential to know where we are" (p 324); integral to which is knowing where we have been (Skager & Weinberg,

1971, p 50) Hence, a significant aspect of this study is the analysis of the nature and function of mental computation in past syllabuses, particularly from a

Queensland perspective A focus such as this can assist mathematics educators to (a) understand educational movements (their "why" and "how,” their relevance to the period in which they received prominence, and their relevance to current problems); and (b) analyse suggested innovations to determine whether the proposals are likely

to be successful in meeting current and future needs (Best & Kahn, 1986, pp 62)

61-To complement the data from the analysis of past syllabuses (Chapter 3), a survey of Queensland state primary school teachers' and administrators' attitudes and teaching practices related to mental computation has been undertaken (Chapter 4) This has enabled the linking of the literature review (Chapter 2) and the

historical information to the present situation in Queensland primary classrooms, thus providing a comprehensive summary of the state of knowledge about mental computation, particularly from a Queensland perspective This summary has

provided the basis for the recommendations concerning the ways in which mental computation may be explicitly included in the Number strands of future mathematics syllabuses (Chapter 5)

1.2 Context of the Study

To provide an understanding of the context in which this investigation has occurred, it is necessary to give consideration to (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 programs; (d) the degree to which students show proficiency with calculating exact answers mentally; (e) the essential changes to the ways in which mental computation is viewed by teachers and students, changes regarded as critical for mental computation to fulfil the roles for which it is envisaged; and (f) issues related to mental computation in need of further clarification

1.2.1 Mental Computation: Overview

As intimated above, "the teaching of mathematics is shifting from a

preoccupation with inculcating routine skills to developing broad-based

mathematical power" (NRC, 1989, p 82) A key element in the repertoire of skills that underpins the development of mathematical power is the ability to compute exact answers mentally In endeavouring to create curricula and learning

environments conducive to ensuring that children gain power over the mathematics they use, an understanding of the historical context is critical to informed debate and the decision-making process

Little has been documented in relation to the nature and role of mental

computation in mathematics curricula in Australia (McIntosh et al., 1995, p 2) However, the importance placed on it by teachers and students is a function of factors which include the availability of particular tools for calculating, the prevailing psychological theory, and the objectives for teaching arithmetic during a particular period (Atweh, 1982, p 53)

With respect to the United States of America, the place of mental computation

in mathematics curricula has a "long and sporadic history" (B J Reys, 1985, p 43) The emphasis placed on mental computation has fluctuated with the prevailing psychological and pedagogical theories during any given period In contrast, mental computation has received a continuing emphasis in Soviet (Russian) elementary schools (Menchinskaya & Moro, 1975, p 73) and in Japanese schools, particularly

prior to the introduction of a new mathematics curriculum in 1989 (Shibata, 1994, p 17), albeit for different reasons In the Soviet Union mental computation has been viewed as a means for deepening mathematical knowledge (Menchinskaya & Moro,

1975, p 74), while in Japan the focus has been on the utility it provides for day calculations (Shibata, 1994, p 17)

day-to-Mental computation first gained prominence during the mid-nineteenth century

in formal mathematics curricula as part of a reaction to the perceived slowness with which students carried out written calculations Following Pestalozzi's work in

Europe, Warren Colburn, in the United States, encouraged an emphasis on oral

arithmetic in which problems were orally stated and computed mentally "as a protest against the intellectual sluggishness, lack of reasoning, and slowness of operation

of the old written arithmetic" (Smith, 1909, cited in Wolf, 1966, p 272)

The rationale for the inclusion of oral arithmetic in the curriculum was based on the tenets of Formal Discipline which held that the mind was a muscle and therefore

in need of exercise if it was to become strong (Kolesnik, 1958, p 4) Exercises in oral arithmetic were used as a form of drill to improve general mental discipline Speaking in 1830 of arithmetic in general, but relevant to the oral aspects, Colburn suggested that:

Arithmetic, when properly taught, is acknowledged by all to be very important

as discipline of the mind; so much so that even if it had no practical application which should render it valuable on its own account, it would still be well worth while to bestow a considerable portion of time on it for this purpose alone (Colburn, 1830, reprinted in Bidwell & Clason, 1970, p 24)

Nonetheless, despite the importance placed on the need to develop mental

discipline, Colburn (1830, reprinted in Bidwell & Clason, 1970, p 24) considered that it was secondary to the practical utility of arithmetic; a view that was to be echoed during the 1930s and 1940s with respect to mental computation, following its decreased emphasis early in the twentieth century (B J Reys, 1985, p 44) The near total neglect of mental methods of computation in mathematics curricula in the United States during the first quarter of the twentieth century was due to the Theory of Mental Discipline, and by association, Formal Discipline, falling

"into such disrepute that it was difficult to maintain a place in the curriculum for any

form of mental activity, including mental arithmetic" (Reys & Barger, 1994, pp 33) This was despite such beliefs as those of Suzzallo who, in 1911, contended that:

32-It is altogether probable that many simple calculations or analyses can be done

"silently" from the beginning; that others require visual demonstrations, but once mastered can thereafter be done without visual aids; that still others will always be performed, partially at least, with some written work It is purely a matter for concrete judgment in each special case, but the existing practice scarcely recognises this truth The result is that many problems are arbitrarily done in one way, and it is too frequently the uneconomical and inefficient way that is used (Suzzallo, 1911, p 78, cited in Reys & Barger, 1994, p 33)

Hall (1954, p 349) observed that it was unfortunate that mental arithmetic should also have been discredited However, the Theory of Mental Discipline's promise of transfer of knowledge through exercising each general faculty was

questioned when it was shown that learning arithmetic (and Latin) did not facilitate learning other subjects Its demise was accentuated by "the rise of associationism

as a dominant psychological account of mental functioning" (Resnick, 1989a, p 8) Similar concerns to those of the mid-nineteenth century began to be expressed during the 1930s in the United States, with respect to the perceived

overdependence on written methods of calculation The rationale for a renewed emphasis on mental (oral) methods was one of social utility, namely, that mental arithmetic was more useful outside the classroom than were paper-and-pencil procedures (B J Reys, 1985, p 44) This advocacy for an emphasis on mental computation coincided with attempts to improve instruction in mathematics, such as

Brownell's (1935) promotion of the meaning theory of arithmetic instruction

Reflected in these recommendations were the beginnings of a shift in the

philosophic orientation in teaching mathematics, away from drill and practice

towards discovery learning and independent inquiry (Reys & Barger, 1994, p 34) During the 1940s and early 1950s there was an increased emphasis on mental computation until the concern for developing an understanding of mathematical

structure gained prominence during the New Mathematics era in the late 1950s to

mid-1970s During this period the issue of paper-and-pencil versus mental

computation was virtually ignored Nevertheless, most proponents of mental

computation have always advocated that a focus on mental computation supports a deeper understanding of numbers and the use of structural relationships when calculating (Menchinskaya & Moro, 1975, p 74; R E Reys, 1984, p 549) As Hall (1954) points out, the renewed emphasis on mental arithmetic, in the 1940s in the United States, was geared to:

(a) Functional problem situations, including those requiring approximate and exact answers; (b) an [increased] understanding of place value and of ten as the foundation of our number system; (c) an awareness of number

relationships in the discovery of acceptable short cuts, once the conventional procedure [was] understood; and (d) recreational exercises to motivate and enrich number experiences (p 349)

The revival of interest in mental computation since the late 1970s initially coincided with, and was strengthened by, a reevaluation of what constitutes school

mathematics as a reaction by the mathematics education community to the Back to

Basics movement, principally in the United States (NCSM, 1977; National Council of

Teachers of Mathematics [NCTM], 1980) Together with this reevaluation was "a growing realization that many students apply written algorithms mechanically, with little sense as to why, how, or what they are doing" (B J Reys, 1985, p 44), an echo of previous calls for a renewed interest in mental computation However, the relative importance and the nature of mental computation as now proposed differ markedly from the oral arithmetic of the past that emphasised oral drill "mental gymnastics" in Koenker's (1961, pp 295-296) view rather than exploration and discussion

1.2.2 Mental Computation: Reasons for the Resurgence of Interest

The current resurgence of interest in mental computation stems from the recognition that computing mentally remains a viable computational alternative in the calculator age and that the development of mental procedures contributes to the formation of powerful mathematical thinking strategies (R E Reys, 1992, p 63)

Encouraging children to develop idiosyncratic cognitive methods for carrying out computations is compatible with the constructivist approach to learning, which asserts that "human beings acquire knowledge by building it from the inside instead

of internalizing it directly from the environment" (Kamii, 1990, p 22)

Except for the implied belief that mental computation skills should only be developed "once the conventional procedure is understood," the goals outlined by Hall (1954, p 349) are ones echoed in the current advocacy for mental computation

to be given prominence in the mathematics curriculum Robert Reys (1984, p 549) suggests that an emphasis on mental computation contributes to the development of:

• A deeper understanding of the structure of numbers and their properties

• Creative and independent thinking

• Ingenious ways of manipulating numbers

• Skills and strategies associated with problem solving and computational estimation, the latter being an essential skill in the efficient use of electronic calculating devices

Despite the similarities in the goals for developing mental computational skills,

as expressed by Hall (1954) and Robert Reys (1984), there are marked differences

in the nature of mental computation as now envisaged and in the ways that such skills should be developed The current thrust, which had its beginnings in the late 1970s, has a broader focus than earlier movements Besides highlighting a

recognition of the applicability of the constructivist theory of learning to the

development of mathematical abilities, it also emphasises the belief that pencil skills, particularly the traditional written algorithms, should receive decreased attention (B J Reys, 1991, p 7) Additionally, it also gives recognition to 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 the society of the 1990s (Masingila, Davidenko, Prus-Wisniowska, & Agwu, 1994, p 3)

paper-and-The present interest in mental computation coincides with a need to redefine the way in which calculations are performed, particularly as a consequence of the availability of calculators and computers As McIntosh (1990a) points out, "none of

[the previous] pendulum shifts [has] suggested other than that mental computation [be] the bridesmaid of written computation" (p 36) This was despite such contrary views for their time as those of Branford (cited by McIntosh, 1990a) who suggested,

in 1908, that "mental arithmetic should come first and form the solid food: written arithmetic should be the luxury, given where and when it can be appreciated" (p 36), a view now promoted, given the influence of technological calculating devices,

as well as for pedagogical reasons

With respect to the latter, the Mathematical Sciences Education Board and the National Research Council (1990, p 19) believe that there is now sufficient

evidence to suggest that an overemphasis on paper-and-pencil skills may hinder a child's effective use of mental techniques for calculating Mental computation is increasingly being considered as an essential prerequisite to the successful

development of written algorithms (R E Reys, 1984, p 549) This is in marked contrast to the traditional view of the place of mental computation within the

mathematics curriculum

A "novel facet of [the current] revival is the interest in using mental computation

as a vehicle for promoting thinking, conjecturing and generalizing based on

conceptual understanding rather than as a set of skills which serve as an end of

instruction" (Reys & Barger, 1994, p 31) By focusing on conceptual understanding rather than on the memorisation of rules, the manipulation of quantities rather than symbols, in Reed and Laves' (1981, p 442) terms, is involved Rathmell and

Trafton (1990) assert that:

The varied and thoughtful ways children manipulate quantities when doing mental computation promote number sense as well as mathematical thinking This kind of activity brings a dynamic quality to learning mathematics because

children are actually doing mathematics rather than learning to repeat

conventional procedures (p 157)

During this process, children construct their own mathematical knowledge that not only enhances learning but also encourages them to view mathematics as meaningful, rather than as a collection of arbitrarily derived rules Children are compelled to seek novel ways to use numbers and number relations, methods that are likely to increase an understanding of the structure of the number system

(Sowder, 1992, p 15) Ironically, although the development of an understanding of

mathematical structure was an important goal of the new maths movement, mental computation was virtually ignored in the syllabuses of the 1960s and 1970s, despite such beliefs as those espoused, in the United States, by Beberman (1959, cited by Josephina, 1960) He asserted that:

Mental arithmetic is one of the best ways of helping children become

independent of techniques which are usually learned by strict

memorisation Moreover, mental arithmetic encourages children to

discover computational short cuts and thus to gain deeper insight into the number system (p 199)

Undertaking mental calculations is not only the simplest method of performing many arithmetical procedures, it is also the main form of calculation used in

everyday life Wandt and Brown (1957, pp 152-153) found that during a 24 hour period 75% of nonvocational uses of mathematics by college students were mental

in nature, rather than ones involving paper-and-pencil Forty-eight percent entailed mental computation to provide exact answers, while 27% involved computational estimation The ability to compute mentally is therefore a valuable skill for everyday living (NCSM, 1989, p 45)

That mental computation is not emphasised in classrooms is seen by Cockcroft (Cockcroft, 1982, p 75, para 255) as representing a failure to recognise the central place that mental procedures occupy throughout mathematics Flournoy, in 1957, contended:

Because activities of everyday life require competence in mental arithmetic, schools must provide pupils with [opportunities] to learn to think without paper and pencil in solving problems involving simple computation, making

approximations, and interpreting quantitative data, terms and statements (p 147)

However, those who are proficient at mathematics in daily life, and in the workplace, seldom make use of the standard written algorithms during mental calculations Rather, idiosyncratic methods are used or else the written algorithms are adapted in unique ways (Cockcroft, 1982, p 75, para 256) People are involved

with, what Maier (1980, pp 21-23) calls, folk mathematics or, what Howson and Wilson (1986, p 21) term, ethnomathematics The methods used differ with the

situation in which an arithmetical problem is to be solved (Carraher, Carraher, & Schliemann, 1987, p 83) Further, Lave (1985, p 173) suggests that the

organisation of arithmetic varies qualitatively from one situation to another It

appears that algorithms taught in schools are only likely to be used to solve type problems, with little transfer to real-life problem situations (Carraher et al.,

school-1987, p 95) Conversely, self-taught strategies used in every-day situations are unlikely to be used by children in the classroom without specific encouragement by teachers (Gracey, 1994, p 75)

While school mathematics remains largely oriented towards paper-and-pencil algorithms, folk mathematics predominantly involves mental calculations and

algorithms that lend themselves to mental use Calculators and computers are used for the more difficult and cumbersome calculations, with paper-and-pencil

procedures considered as final choices (Maier, 1980, p 22) Therefore for school mathematics to become more meaningful and useful in non-classroom situations an emphasis needs to be placed on encouraging children "to develop personal mental computational strategies, to experiment with and compare strategies used by others, and to choose from amongst their available strategies to suit their own strengths and the particular context" (AEC, 1991, p 109)

1.2.3 Mental Computation: Place in Current Mathematics Curricula

Despite the on-going advocacy for increased attention to mental computation, this has not yet been translated significantly into mathematics classrooms in the United States (Coburn, 1989, p 47; Koenker, 1967, p 295; Resnick & Omanson,

1987, p 65; Reys & Barger, 1994, p 46; R E Reys, 1992, p 69; Sachar, 1978, p 233) However, some progress is beginning to be made with attempts by education

authorities to implement the Curriculum and Evaluation Standards for School

Mathematics (NCTM, 1989); for example, the Ohio Department of Education with its Model Competency-Based Mathematics Program in which Strand 7 focuses

explicitly on mental computation and estimation (Ohio Department of Education, n.d., pp 89-99)

In the United Kingdom, Cockcroft (1982, pp 74-75, para 254) had also

expressed concern at the decline in attention given to skills associated with

calculating mentally (Cockcroft, 1982, pp 74-75, para 254) One reason for this is that the nature of mental computation, when viewed as a higher-order thinking process, does not facilitate the delineation of a fixed scope and sequence (R E Reys, 1992, p 69) The encouragement of mental computation through textbooks, therefore, becomes an even more difficult task Strategies for computing mentally are naturally developed through discussion and exploration (R E Reys, 1992, p 70) This implies that a less structured approach to lesson design than that

traditionally used to develop the written algorithms is imperative, a requirement that necessitates changes to the teaching practices of many teachers

In Australia, the place of mental computation in mathematics curricula is only beginning to be seriously considered (AEC, 1991, pp 106-134) As McIntosh

(1990a, p 25) has observed, little attention has been given to teaching mental computation, with such attention predominantly 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, usually the basic ones, rather than "with networks of relationships between numbers or with methods of computation" (McIntosh, 1990a, p 25)

Further, in Atweh's (1982, p 57) view, teachers often discourage mental calculation

by insisting that children write down their solutions so that each individual step may

be detailed This expectation may stem from the belief that children are not

productively engaged in mathematics unless they are writing, or, it may be a

classroom management procedure designed to enable the teacher to retain control over the pace of a lesson

In the Queensland context, the need for an emphasis on mental computation is not inconsistent with many recommended teaching practices Associated with the

implementation of the Years 1 to 10 Mathematics Syllabus (Department of

Education, 1987a) emphasis is being given to issues relevant to the development of the ability to calculate exact answers mentally It is believed that learning is

enhanced by (a) teaching through problem solving, (b) encouraging children to explore and discuss mathematical ideas with their peers and teachers, and (c) accepting a range of solutions as well as various strategies for arriving at a

particular solution (Department of Education, 1987b, pp 3-4)

Further, the need to develop a range of strategies for arriving at approximate answers is being emphasised Strategies for calculating exact answers mentally,

different to those traditionally used for paper-and-pencil calculations are given consideration, albeit a limited one, in the year-level sourcebooks for Years 4, 6 and

7, and particularly in that for Year 5 (Department of Education, 1988, pp 51-57) However, due to the unavailability of research data, the degree to which these recommendations are being implemented in classrooms is unknown Nonetheless, given the inadequacies of teacher inservice associated with the implementation of the current mathematics syllabus in Years One to Seven and the abandonment, in

1996, of the implementation of Student Performance Standards in Mathematics for

Queensland Schools (1994), which supported an explicit focus on mental

computation, the extent to which these recommendations may be observed in classrooms could be limited This issue is a focus of the survey of Queensland state primary school teachers and administrators reported in Chapter 4

1.2.4 Mental Computation: Student Performance

Data related to performance levels on mental computation are limited, with none available for large samples of Australian students However, data provided by McIntosh et al (1995) do permit some insight into the abilities of West Australian children in Years 3, 5, 7 and 9 Robert Reys (1985, p 14) suggests that, within the standardised achievement testing programs in the United States, the lack of

information is primarily due to the unique conditions required for testing mental computation, namely, that the examples given need to be paced and that answers need to be open-ended to permit children to record their individual responses, conditions that depart significantly from those normally associated with standardised achievement tests Another factor is that there is insufficient information from research to assist in the designing of performance outcomes for mental computation (Coburn, 1989, p 47)

The data that are available, primarily from the United States and in common with that available from Australia (McIntosh et al., 1995) and Japan (R E Reys et al., 1995), suggest that children perform rather poorly on tasks designed to measure mental computation abilities and that children generally prefer not to calculate mentally even where the items lend themselves to mental calculation (Carpenter, Matthews, Lindquist, & Silver, 1984, p 487; B J Reys, R E Reys, & Hope, 1993,

p 314; R E Reys et al., 1995, p 323) However, two Australian studies, although

limited in scope, suggest that Australian children may not be as biased against mental computation, even though little instruction is undertaken Gracey (1994, p 113) found that mental computation was the most commonly preferred option for calculating one-step multiplication items, 945 x 100 for example, by a class of Year

6 children Similar findings are reported for all four basic operations by McIntosh et

al (1995, p 12) For children who did not prefer to calculate such items as 100 x 35 mentally, it was concluded that this was due to a lack of conceptual understanding rather than a lack of computational skill (McIntosh et al., 1995, p 11-12)

Results from the United States’ Third National Mathematics Assessment reveal that the ability of nine year-old children to compute mentally is only beginning to emerge, with performance levels on such items as 6 + 47, 36 + 9 and 90 x 3 being below 50% correct (R E Reys, 1985, p 15) Barbara Reys (1991, p 3) suggests that this may be because children of this age are primarily concerned, in present curricula, with developing the written algorithms and that this emphasis may

interfere with their ability to compute mentally, which requires an abstract and

flexible manipulation of numbers For the nine old children and the 13 olds, a wide range of performances was recorded: Twenty percent correct for 36 - 9

year-to 52% correct for 64 + 20 (nine year-olds), and 32% correct for 60 ÷ 15 year-to 92% correct for 700 - 600 (13 year-olds) Addition was the easiest operation, with

division being the most difficult (R E Reys, 1985, p 15)

In contrast, results from Periodiek Peilings Onderzoek in The Netherlands

indicate 70% to 90% accuracy on items similar to those of the Third National

Mathematics Assessment (Treffers, 1991, p 336) Dutch 13 year-olds scored approximately 90% correct on 480 ÷ 6 and 7 x 90, and on 600 ÷ 300 and 20 x 2400 about 70% While these scores are superior to those of the United States, Treffers (1991) concluded that "rather than being proud of our students' achievement, [Dutch educators] should draw the general conclusion that mental arithmetic badly needs improvement" (p 336)

Barbara Reys et al (1993) report similar results to those of the National

Assessment from research undertaken with a sample of children in the second, fifth and seventh grades in Texas and Saskatchewan For each of these year-levels, performances on applied problems were also quite low For six of the 10 items, Grade 2 children scored below 20% correct This may reflect their relative lack of experience with problem solving and particularly with tests in which items are paced

(B J Reys et al., 1993, p 309) Although a narrower range of performances was evident for children in Grades 5 and 7, for most items a success rate of

approximately 10% was achieved The highest percentage correct for Grade 5 was 32% for the item: "Chuck's family lives 100km from Chicago They stop after driving 65km How much farther do they have to go?" (B J Reys et al., 1993, p 311) Chaining of addition and subtraction was difficult for both Grade 2 and Grade 5 children who also dropped in performance where chaining of multiplication was involved (B J Reys et al., 1993, p 310) Examples such as 75 + 85 + 25 + 2000 (fifth grade) require partial values to be mentally retained for later computation, a skill that is not developed during the traditional focus on paper-and-pencil skills For performance levels in mental computation to rise, children need to be encouraged to calculate mentally and to develop a range of strategies for carrying out such calculations They also must develop an appreciation of when it is

appropriate to calculate mentally, in context with their abilities, and in so doing develop the confidence to use mental procedures Such confidence appears to be seriously lacking at this time R E Reys et al (1995, p 323) conclude that this is a theme that appears to be common across various school systems Flournoy (1959,

p 134), following a survey of children's use of mathematics across a 7 day period, concluded that greater use of mental arithmetic would be made if they felt confident

in solving number situations without paper-and-pencil Of relevance to this finding is that of Case and Sowder (1990) with respect to computational estimation They concluded that the lack of confidence children exhibit in their mathematical abilities results from the "split between the understanding of number that children glean from their everyday quantitative activity and the school-based algorithms they learn

to execute" (Case & Sowder, 1990, p 100) This conclusion has implications for how mental computation is defined and for how it is to be taught

Children need to make appropriate choices between applying paper-and-pencil procedures, mental calculations and the use of a calculator Data from the Third National Mathematics Assessment indicate that on items considered appropriate for mental computation most 13 year-old students preferred to use either paper-and-pencil or a calculator For example, for 4 x 99, although 44% indicated that they would "Do it in their heads,” 39% would use paper-and-pencil and 16% would use a calculator (R E Reys, 1985, p 15) A similar pattern of preferred methods for calculating items whose structure encourages mental calculation is reported by B J

Reys et al (1993, p 310) for children in Grades 5 and 7, with the majority preferring

to employ paper-and-pencil for all except 1000 x 945, a fifth grade example The preference for written methods rather than calculator use is also reported by Gracey (1994, p 113) and reflects the emphasis on written procedures in the classroom

1.2.5 Mental Computation: Essential Changes in Outlook

To enhance the use of mental computation and to achieve an improvement in performance levels, Rathmell and Trafton (1990, p 156) believe that children should

be encouraged to value all methods of computation This requires that teachers come to recognise that the development of mental skills is a legitimate goal for computational programs in classrooms (B J Reys et al., 1993, p 312), a goal supported by recent curriculum reform documents in mathematics education (AEC,

1991, pp 114-115, 120-121, 126-127; NCTM, 1989, pp 44-49, 94-97; National Curriculum: Mathematics Working Group [NCMWG], 1988, pp 20-21)

These documents recommend that the emphasis on paper-and-pencil skills in current curricula should be reduced, particularly with respect to the standard written algorithms Representative of these recommendations, is the belief of the

Australian Education Council (1991) that "less emphasis should be given to

standard paper-and-pencil algorithms and, to the extent that they continue to be taught, they should be taught at later stages in schooling" (p 109) Terezinha Carraher et al (1987, p 83) suggest that school-learned algorithms may not be the preferred ways for solving numerical problems outside the classroom Further, by continuing the emphasis on the written algorithms, the commonly-held but

erroneous view of arithmetic as necessarily involving linear, precise and complete calculations is maintained Such a view restricts the development of a mental agility

with numbers number sense which is an essential ingredient for the development

of a range of flexible strategies for calculating, whether using paper-and-pencil, a calculator or the mind

The procedures involved in performing the traditional written algorithms for addition, subtraction, and multiplication, in particular, with their emphasis on right-to-left processing, conflict with the procedures commonly employed when calculating mentally Cooper, Haralampou, and Irons (1992) suggest that "if the pen-and-paper algorithms interfere, deter, impede, replace or stop the development of mental

strategies there may be a need to consider resequencing the teaching of

computation for each operation throughout the primary school" (p 101) Except in relation to the development of the basic facts, a focus on mental calculation, albeit a limited one, has traditionally occurred after the written algorithm has been

introduced for a particular operation

Children need to be given opportunities to explore mathematical relationships and to invent idiosyncratic strategies for computing mentally, unencumbered by patterns of thought developed through a premature focus on the written algorithms, either idiosyncratic or standard Mental computation, and computational estimation, should receive an ongoing emphasis throughout all computational experiences in the classroom (NCTM, 1989, p 45) Rathmell and Trafton (1990, p 156) advocate

an early and ongoing emphasis on mental computation, estimation, and an

appropriate use of calculators to provide a framework for developing pencil skills Such skills should be ones that generally extend and support the use

paper-and-of mental strategies (McIntosh, 1990a, p 37), and the development paper-and-of number sense

1.2.6 Mental Computation: Needed Research

Although much has been written about mental computation during the past century concerning its place and importance in the classroom, little research is available to guide either curriculum or instruction (B J Reys, 1991, p 7), an

observation that is particularly relevant to the Australian context Additionally, there

is only a limited body of knowledge, albeit one that is gradually expanding,

concerning how children think when they compute exact answers mentally This applies particularly to mental strategies for multiplication and division beyond the basic facts (McIntosh, 1990b, p 18)

Robert Reys and Nohda (1994, p 5) suggest that the basic question of "What

is mental computation?" is one that continues to be analysed and debated, a

situation suggestive of the paucity of definitive knowledge about mental

computation Essential to a comprehensive understanding of mental computation are a number of issues identified in the research literature as needing clarification These include:

• The nature of the interface between self-generated strategies and formally taught written algorithms

• The extent to which children of varying abilities develop efficient mental strategies

• The effects of direct and indirect teaching methods to encourage the

development of self-generated strategies for computing mentally

• The relationship between the development of strategies for mental

computation and those for computational estimation

(Reys & Barger, 1994, p 45)

• The interrelationship between mental computation and number sense

• The methods for and timing of the introduction of alternative computational procedures, namely, mental computation, computational estimation, paper-and-pencil procedures and calculator use

(Reys & Nohda, 1994, p 5)

It is suggested that for children to successfully implement the range of

computational alternatives available to them, further knowledge, some of which may

be country specific, is required on each of these issues (Reys & Nohda, 1994, p 6)

1.3 Purposes and Significance of the Study

Focussing on the development of personal strategies for calculating exact answers mentally places the child at the centre of the learning process This

highlights that the advocacy for an increased emphasis on mental computation goes well beyond simply focussing on it as a computational method per se While it is important for mental computation, and the use of calculators, to receive greater emphasis in the calculative process, with a concomitant de-emphasis on the

standard written algorithms, a focus on the relationships between the development

of idiosyncratic thinking strategies and the development of number sense and

numeracy is of equal importance

By undertaking (a) an analysis of past and present syllabuses, from a mental computation perspective, and (b) a survey of Queensland school personnel, issues related to these recommendations are able to be placed in context with beliefs and

teaching practices, both past and current This should lead to an enriched context

in which these issues can be debated from a Queensland perspective, issues which encompass aspects of the areas of needed research identified by Barbara Reys and Barger (1994, p 45), and Robert Reys and Nohda (1994, p 5) Consequently, the principal purposes of this study, in accordance with the aims delineated in Section 1.1, were:

1 To formulate a mental computation strand, encapsulating key elements of the research data, for inclusion as a core element in future mathematics syllabuses

2 To draw implications for decision making about mental computation, from past, contemporary and futures perspectives These implications may be essential to any discussion for change within the Number strand of the Queensland primary school mathematics syllabus

3 To provide an in-depth analysis of key psychological, socio-anthropological and pedagogical issues related to mental computation

4 To document the nature and role of mental computation, and associated pedagogical practices, under each of the Queensland mathematics

syllabuses

5 To investigate the beliefs about mental computation currently held by Queensland primary school teachers and administrators

6 To gain an insight into the status of mental computation in current

mathematics programs, and to identify current pedagogical practices

related to mental computation in Queensland primary school classrooms

7 To compare and contrast past and current beliefs and practices with those recommended as essential for children to gain mastery of the calculative process

Achieving these purposes has provided a comprehensive summary of the state

of knowledge about mental computation, both from theoretical and Queensland perspectives This has significance for the following areas:

• Psychological and pedagogical significance: The analysis of the nature of mental computation, together with the identification of recommended

teaching practices, should not only be significant as a comprehensive summary of the knowledge about mental computation, but it also should provide guidance in selecting and implementing teaching strategies that will enhance the development of a student's ability and confidence to calculate exact answers mentally Additionally, where mental strategies are to be formally taught, suggestions as to those considered appropriate for each operation have been highlighted

• Mathematical significance: Students' attitudes towards, and performances

on, tasks involving mental computation should be improved by the

implementation of teaching approaches and sequences based on data relevant to the Queensland context This may lead students to view mental computation as the method of first resort rather than one that is to be avoided for all but the simplest calculations

• Significance for curriculum development: This study should provide a comprehensive source of data to support the decision making processes of curriculum developers during any future reshaping of the Queensland mathematics syllabus for the primary school This applies particularly to the degree to which mental computation should be emphasised viz-a-viz paper-and-pencil and calculator use, as embodied in the proposed mental computation strand

• Significance for Queensland educational history: The analysis of the nature and role of mental computation in Queensland state primary school

classrooms since 1860 should extend and deepen the available knowledge about the mathematics taught, the teaching methods used, and issues which were of importance to their development and implementation

1.4 Overview of the Study

This study has been conceived as comprising three major investigations, which, although necessarily interrelated, are essentially distinct Consequently, a more detailed analysis of methodological issues is presented in the respective chapters The following provides an overview of the research procedures employed

1.4.1 Method and Justification

In a paper prepared, in the mid-1960s, by three District Inspectors for the Queensland Department of Education's Syllabus Committee, it was suggested that

"one of the major functions of a study of [the past] is to enable a community to avoid future errors, and to assist them to solve problems which may arise in the future, by

a consideration of events of the past" (Schildt, Reithmuller, & Searle, n.d., p 1) With respect to curricula, the preoccupation in such analyses should principally be with the gathering of data which will assist in understanding contemporary

curriculum issues (Goodson, 1985, p 126; Fox, 1969, p 45), in this instance, aspects of mental computation

The preparation of the paper by Schildt et al (n.d.) occurred during a period when the mathematics syllabus for Queensland state primary schools was

undergoing major revisions due to administrative changes and altered beliefs about

how children learn mathematics While the Years 1 to 10 Mathematics Syllabus

(Department of Education, 1987a), which was introduced into Queensland schools

in 1988, is not at present scheduled for revision, there is an increasing emphasis by mathematics educators on the need to determine an appropriate balance between

mental, written and technological calculations within the mathematics curriculum A

National Statement on Mathematics for Australian Schools (AEC, 1991) suggests

that mental computation should be the method of "first resort" (p 109) in many calculative situations This implies that the primary focus on written computation embodied in the current Queensland syllabus will need to be revised For this to occur successfully, the two revolutions delineated by McIntosh (1992, pp 131-134) will need to occur The first revolution, in particular, namely the need for change in the way mental computation is viewed by teachers, should be supported by the data gathered in this study These provide an understanding of how mental computation has been viewed under each of the mathematics syllabuses in Queensland from

1860, an outcome of which should be the fostering of an enriched context for

debate

Consequently, this study is characterised by four approaches to gathering and analysing data, and drawing conclusions, relevant to the nature and role of mental computation within the primary school, both from a theoretical perspective and from

one that relates directly to its teaching in Queensland state schools These

approaches entailed undertaking:

1 A literature review, the aim of which was to analyse the pedagogical, anthropological and psychological literature relevant to mental computation

socio-2 An analysis of Queensland syllabuses in relation to the nature and function

of mental computation within Queensland primary school curricula from

1860, with particular emphasis on the period 1860-1965 during which Queensland mathematics syllabuses made specific references to the mental calculation of exact answers

3 A survey, using a postal, self-completion questionnaire, of Queensland state primary school teachers and administrators, from a random sample of Queensland state schools, to ascertain their beliefs and teaching practices pertaining to mental computation

4 A synthesis of the data from the first three approaches, the aim of which

was to highlight similarities and differences in beliefs and practices

concerning the nature and function of, and teaching methods related to, mental computation In so doing it was aimed to provide recommendations for future action with respect to syllabus revision in Queensland, the key element of which is the proposed mental computation strand for future mathematics syllabuses

1.4.2 Chapter Guidelines

Developed from the study's purposes and from key theoretical and practical aspects of mental computation, each of the research strategies was guided by a series of questions for which answers were sought These questions are presented below

Chapter 2: Mental Computation

1 What are the recent developments in mathematics education of relevance

to mental computation?