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

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 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


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
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


ix

CHAPTER 5

MENTAL COMPUTATION: A PROPOSED SYLLABUS
COMPONENT

5.1

5.2

Introduction ................................................................................................... 327
5.1.1

Context and Focus for Change ....................................................... 330

5.1.2

Framework for Syllabus Development ............................................ 332

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

6.3

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

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 ASUMMARY OF MENTAL ARITHMETIC IN QUEENSLAND
MATHEMATICS SCHEDULES AND SYLLABUSES (1860-1964)


x

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 BADDITIONAL 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 EMEANS AND STANDARD DEVIATIONS OF
SURVEY ITEMS IN FIGURES 4.1-4.6 ....................................... 474

xi

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


xii

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 19641987............................................................................................................ 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


xiii

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 traditionalnontraditional 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


xiv

ABBREVIATIONS
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


xv

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


xvi

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


1

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.

2

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


3

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. 6162).
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).


4

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


5

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-today calculations (Shibata, 1994, p. 17).
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


6

form of mental activity, including mental arithmetic" (Reys & Barger, 1994, pp. 3233). This was despite such beliefs as those of Suzzallo who, in 1911, contended
that:

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


7

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).