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

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

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.

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

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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

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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

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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

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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

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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

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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

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

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ACKNOWLEDGMENTS

This study would not have been completed without the advice, support and co- operation 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.

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. 61- 62). 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 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. 32- 33). 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). 8

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 paper-and- 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). 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 9

[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 10

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 11

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 school- type problems, with little transfer to real-life problem situations (Carraher et al., 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 12

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, 13

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 14

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 year-old children and the 13 year- olds, a wide range of performances was recorded: Twenty percent correct for 36 - 9 to 52% correct for 64 + 20 (nine year-olds), and 32% correct for 60 ÷ 15 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 15

(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. 16

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 17

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 paper-and- pencil skills. Such skills should be ones that generally extend and support the use of mental strategies (McIntosh, 1990a, p. 37), and the development 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 19

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 20

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 22

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, socio- anthropological and psychological literature relevant to mental computation. 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? 23

2. What is the place of mental computation in the calculative process and particularly its relationships with computational estimation? 3. What is the nature of mental computation as perceived by mathematics educators, both contemporaneously and historically? 4. What are the roles currently perceived for mental computation within and beyond the classroom? 5. What are the affective and cognitive components, including commonly used mental strategies, that constitute skill in computing exact answers mentally? 6. What are the affective and cognitive characteristics exhibited by skilled mental calculators, including the role that memory plays in the process of calculating exact answers mentally? 7. What are the teaching approaches and sequence necessary for the development of mental computation skills?

Chapter 3: Mental Computation in Queensland: 1860-1965

1. What emphasis was given to mental computation in the various mathematics syllabuses for Queensland primary schools during the period 1860-1965? 2. What was the nature of mental computation as embodied in the various syllabuses and in the manner in which it was taught from 1860 to 1965? 3. What was the role of mental computation within the mathematics curricula from 1860 to 1965? 4. What was the nature of the teaching practices used to develop a child's ability to calculate exact answers mentally during the period 1860-1965? 5. What was the nature of the resources used to support the teaching of mental computation during the period 1860-1965?

Chapter Four: Mental Computation in Queensland: 1966-1997

1. What beliefs do teachers currently hold with respect to the nature and role of mental computation and how it should be taught? 24

2. What emphasis was given to mental computation in the period 1966-1987, with respect to both syllabus documents and teachers? 3. What emphasis is currently placed on developing the ability to compute mentally? 4. What are the characteristics of the teaching approaches currently used to develop the ability to calculate exact answers mentally? 5. What were the characteristics of the teaching approaches used to develop the ability to calculate exact answers mentally during the period 1966- 1987? 6. What need for inservice on mental computation is expressed by school personnel? 7. What was the nature of the resources used to support the teaching of mental computation during the period 1966-1987 and of those used currently? 8. What is the relevance to mental computation of recent initiatives in mathematics education in Queensland?

Chapter Five: Mental Computation: A Proposed Syllabus Component

1. How should the computation strand of the primary school mathematics syllabus be reorganised to incorporate a mental-written sequence for introducing computational procedures? 2. Which mental strategies for each operation are appropriate to recommend for direct teaching?

Chapter Six: Mental Computation: Conclusions and Implications

1. What conclusions about mental computation across the range of time periods can be drawn? 2. What implications do these conclusions hold for syllabus development? 3. What aspects of mental computation require further theoretical and empirical investigation? 27

CHAPTER 2

MENTAL COMPUTATION

2.1 Introduction

Although "many of the fundamental ideas which underpin the school mathematics curriculum of ‘old'...are not necessarily ‘old-fashioned' or ‘out-of-date'" (Willis, 1990, p. 12), the ways in which mathematics is produced and applied are undergoing rapid change through the availability of technological devices for calculating. This has resulted in reanalysing what should constitute mathematics for primary school children. In the reappraisements to date, significant changes are suggested for the number strand of school mathematics curricula (AEC, 1991; NCTM, 1989; NCMWG, 1988). In outlining principles for change, the Mathematical Sciences Education Board and National Research Council (1990, p. 36) suggest that mathematics education must focus on the development of mathematical power (Principle 1). Mathematical power enables individuals to (a) understand mathematical concepts and procedures, (b) perceive the relationships between mathematical ideas embodied in a variety of situations, (c) reason logically and to solve both routine and nonroutine problems, and (d) use mathematical ideas intelligently in all aspects of their lives. With respect to an understanding of number concepts, mathematical power flows from the development of a strong sense of number and from an ability to confidently select from, and apply, a range of computational procedures appropriate to particular contexts. As Plunkett (1979, p. 5) suggests, calculators are ideal tools for difficult and laborious computations. Their acceptability and use within classrooms are, however, yet far from ideal, necessitating a reassessment of teaching practices (Australian Association of Mathematics Teachers [AAMT], 1996a, p. 5). For most computational needs, the Australian Education Council (1991, p. 109) believes that mental methods, combined with informal written procedures to

provide memory support, are the most appropriate. Elevating the importance of mental computation (and calculator use) in school mathematics is a recognition of the central role of such calculation outside the classroom. As Cockcroft (1982) comments: "In almost all jobs the ability to carry out some calculations mentally is of value and lack of ability to do this is a frequent cause of complaint by employers" (p. 20, para 71). Therefore, there is a need for positive attitudes towards mental computation to be fostered in classrooms (J. P. Jones, 1988, p. 42), a view supported by the performance data referred to in Section 1.2.4. The importance of a focus on mental computation does not merely centre on its role as a computational procedure per se. Barbara Reys et al. (1993, p. 314) suggest that mental computation is a vehicle for developing each of the curriculum standards with respect to problem solving, reasoning, number sense and communication. These standards, which are in accord with recommendations contained in A National Statement on Mathematics for Australian Schools (AEC, 1991), were proposed by the National Council of Teachers of Mathematics (1989, p. 2) as a means for indicating educational goals, ensuring quality of teaching, and promoting change in education. Mental computation, as now perceived, is considered to be a more creative activity than that embodied in what has been traditionally called mental arithmetic (Curriculum Programmes Branch, 1989, p. 26), where the focus was primarily on the correctness of the answer rather than on the mental methods employed. Joy Jones (1988, p. 44) suggests that the real value of emphasising mental procedures is the development of the soundness of number that comes from considering a variety of methods. Indeed, mental computation constitutes the process most readily available to assist in the understanding of how numbers operate (McIntosh, 1990a, p. 37). Additionally, the thinking that is involved facilitates the growth of a sense for computational routines (B. J. Reys, 1985, p. 43). Number sense and the ability to calculate exact and approximate answers mentally evolve concurrently, with each supporting the development of the other. Children who are adept at mental computation can flexibly use a rich variety of reasoning strategies (Carraher et al., 1987, p. 96). This ability is firmly grounded in their sense of number, their drive and capacity to formulate and apply logical connections between new and previously acquired information (B. J. Reys, 1992, p. 94). It follows, therefore, that effective mental methods cannot be acquired by rote learning (French, 1987, p. 39). Indeed, mental computation helps prevent a reliance

on acquiring facts and algorithms through meaningless verbal memorisation (Atweh, 1982, p. 53). Several benefits for both students and teachers arise from including mental computation in the mathematics curriculum. Mental computation can be a means for individualising learning. As Barbara Reys (1985, p. 45) suggests, through a focus on personal strategies that make sense to the child, the brightest students can be challenged while allowing all students to use their knowledge of basic mathematical concepts and relationships to solve particular problems. Obtaining answers mentally is a problem solving process. In many situations mental algorithms do not exist for the child, or are forgotten (Atweh, 1982, p. 18). Students are allowed to become cognitive apprentices (Collins, Brown, & Newman, 1989, p. 457). Such a situation arises where relevant conceptual and factual knowledge are exemplified and situated in the contexts of their use, and where learning experiences are structured to "allow skill to build up bit by bit, yet permit participation even for the relatively unskilled, often as a result of the social sharing of tasks" (Resnick, 1989a, p. 13). Insights are therefore provided into the ways children think and understand as they solve problems mentally (R. E. Reys, 1992, p. 65).

2.2 Research Questions

In Robert Reys' (1992, pp. 65-66) view, a child's ability to reason mathematically and to use a range of flexible thinking skills is enhanced by the process of computing mentally. This process (a) requires a considered inspection of the problem situation before computing, (b) promotes and encourages the use of basic mathematical properties, (c) rewards adaptable approaches to manipulating numbers, and (d) facilitates the use of visual thinking skills. These issues are considered in later sections of this chapter, the purpose of which is to delve more deeply into specific aspects of the nature and role of mental computation in primary classrooms. Specifically, the following questions are addressed:

1. What are the recent developments in mathematics education of relevance to mental computation?

2. What is the place of mental computation within the calculative process and particularly its relationships with computational estimation? 3. What is the nature of mental computation as perceived by mathematics educators, both contemporaneously and historically? 4. What are the roles currently perceived for mental computation within and beyond the classroom? 5. What are the affective and cognitive components, including commonly used mental strategies, that constitute skill in computing exact answers mentally? 6. What are the affective and cognitive characteristics exhibited by skilled mental calculators, including the role that memory plays in the process of calculating exact answers mentally? 7. What are the teaching approaches and sequence necessary for the development of mental computation skills?

2.3 Recent Developments in Mathematics Education of Relevance to Mental Computation

In proposing what was effectively Australia's first national statement on basic mathematical skills and concepts for children to effectively participate in Australian society, the Australian Mathematics Education Project (1982, p. 4) highlighted that any such statement needed to be time-dependent. Contrary to the belief that mathematics is the least dynamic of school subjects, the fundamental essential learnings cannot be considered as an absolute. This being the case, most countries in the western world are currently developing recommendations for school mathematics to meet the needs of future adults in the Information Age, an age in which "the world of work in the twenty-first century will be less manual but more mental; less mechanical but more electronic; less routine but more verbal; and less static but more varied" (NRC, 1989, p. 11). The exact nature and method of delivering the recommendations are dependent upon the unique socio-political environment in which mathematics education in a particular country is to occur (AAMT, 1996b, p. 6). Australia's A National Statement on Mathematics for Australian Schools constitutes "a framework

around which systems and schools may build their mathematics curriculum....It is descriptive rather than prescriptive" (AEC, 1991, p. 1). This is in contrast to England's national curriculum, which has legislative status for the teaching and learning of mathematics (NCMWG, 1988). Whatever the nature of the statements for the future of mathematics education, given the close contact between mathematics educators throughout the world, together with the increasing interdependence of nations, there are many issues of agreement and common concern, issues that centre on the concept of numeracy, computation, number sense and learning mathematics.

2.3.1 Numeracy

The development of numeracy (or mathematical literacy) is commonly held as a key purpose for studying mathematics in school. Willis (1990, p. 9) considers such development to be a service to students. It should equip them with skills and understandings necessary for successfully dealing with other aspects of their lives─in the home, in the workforce and across the curriculum. As the technological demands in relation to work and social interactions within a society increase, the way in which numeracy is conceptualised also needs modification (NRC, 1989, p. 8). This suggests that a wide definition needs to be formulated. In common with recommendations of the Cockcroft Report (1982, pp. 10-11, paras 35-39) and the position paper by the National Council of Supervisors of Mathematics (1989), the Queensland Years 1 to 10 Mathematics Syllabus (Department of Education, 1987a, p. 2) suggests that although there is a continuing need for students to develop competence and confidence in computational skills, they also need to develop a much broader competency, one that includes an ability to apply understandings of number, space and measurement in realistic situations, both familiar and unfamiliar, to be able to use a range of technological aids, and to recognise the reasonableness of results.

2.3.2 Computation

In the late 1970s Girling (1977) provocatively, for that time, defined basic numeracy as "the ability to use a four-function electronic calculator sensibly" (p. 4). Whereas the accepted view of numeracy is now much broader than this, as delineated above, Girling's statement draws attention to the profound effect that the availability of calculators (with computers and other electronic calculating devices) has had on society and therefore on the features of school mathematics relevant to the present technological age. Australia's national statements on the use of calculators in schools (AAMT, 1996a; Australian Association of Mathematics Teachers & Curriculum Development Centre [AAMT & CDC], 1987) reflect the commonly held belief that school mathematics needs to capitalise upon the power of the calculator. The statement not only recommends that all students at all year levels (P-12) should use calculators, but that they should be used both as an instructional aid and as a computational tool during the learning process across the curriculum. The impact of electronic calculating devices on the nature of calculation within present western societies requires the number strands of mathematics curricula to be reassessed. This applies particularly to their goals and to the computational procedures with which children are expected to become proficient. Rathmell and Trafton (1990) assert that because "most complex computation is now done by calculators and computers...paper-and-pencil procedures can no longer be the focus of computation in the curriculum" (p. 54). Such a view is not new. Before the calculator (and personal computer) age, Biggs (1969) cautioned that "we must be quite clear...about out [sic] purpose in continuing to include written computational practice as a part of the primary school curriculum, as this will determine the nature and extent of this aspect of the work" (p. 25). To the extent that written algorithms continue to be taught they need to be ones that are relatively easy to learn and be ones that assist in the development of concepts and processes (Lindquist, 1984, pp. 602-603). With the implementation of the Years 1 to 10 Mathematics Syllabus (Department of Education, 1987a) in Queensland in 1988 a greater emphasis is given to the use of calculators, at least at the level, in Howson and Wilson's (1986, p. 37) terms, of the intended curriculum. Some written algorithms have been deleted from the primary school curriculum─multiplication of common fractions─whereas others have been delayed─division by two-digit numbers.

However, mastery of the traditional paper-and-pencil algorithms for the four operations remains the chief goal of the number strand. In contrast, Willis (1992, p. 4) points out that Australia's national statement takes the view that children need to be helped to develop sensible methods for calculating, many of which may be idiosyncratic with respect to the child or to the particular task. Further, many of these methods are likely to be mental rather than written. The statement asserts that "students should regard mental arithmetic as a first resort in many situations where a calculation is needed" (AEC, 1991, p. 109), a view supported by the United Kingdom's National Curriculum: Mathematics for Ages 5 to 16 (NCMWG, 1988, p. 9). In reviewing and restructuring the number strands of mathematics curricula, a shift in emphasis is required to ensure a realistic balance among the various forms of computation (B. J. Reys & R. E. Reys, 1986, p. 4). Trafton and Suydam (1975, p. 531) have suggested that the study of computation should promote broad, long- range goals of learning. Children should develop confidence in their ability to learn and perform mathematics, and gain insights into number ideas and relationships, through their ability to discover patterns and form generalisations, the culmination of which is the development of an ability to reason mathematically to solve problems of significance to the individual.

2.3.3 Number Sense

McIntosh (1990a) stresses that the importance of being able to handle numbers and to compute is not diminished by the availability of calculating devices:

What we all need to become, are thinking calculators with an ability to adapt and improvise methods and to test quickly the reliability of results produced by machines. And we need, now more than ever in this calculator age, a well- developed and flexible sense of number. (McIntosh, 1990a, p. 31)

Although number sense cannot be defined precisely (Hope, 1989, p. 12), and is dependent upon the number system being used (Sowder, 1992, p. 6), Howden (1989) suggests that it can be described "as good intuition about numbers and their

relationships" (p. 11). It may be elusive and difficult to pin down (Greeno, 1991, p. 36), but there are identifiable characteristics in the behaviour of those who have a well developed feel for number. The National Council of Teachers of Mathematics (1989, pp. 39-40) believes that children with good number sense (a) have well- understood number meanings, (b) have developed multiple relationships among numbers, (c) recognise the relative magnitudes of numbers, (d) know the relative effects of operating on numbers, and (e) have developed referents for measures of common objects and situations within their environment. Sowder (1992, pp. 5-6) and Resnick (1989b, p. 36) extend this list to include the ability to (a) perform mental computations with nonstandard strategies that take advantage of the ability to compose and decompose numbers, (b) use numbers flexibly to estimate numerical answers to computations and to realise when as estimate is appropriate, and (c) judge the reasonableness of solutions obtained, dependent upon their belief that mathematics makes sense and that they are capable of finding sense in a numerical situation. Whereas the development of number sense is something that has always occurred in many classrooms, the pivotal role that it plays in the ability of individuals to respond flexibly and creatively to number situations requires its development to be viewed as a major goal of primary school mathematics (AEC, 1991, p. 107; MSEB & NRC, 1990, p. 46). Markovits (1989, p. 78) suggests that when students with number sense are given a mathematical task, they are expected to have in mind that there is not always one answer, that there is not always one algorithm, that mathematics and real life are related, and that decisions and judgements are expected. Further, students with number sense tend to analyse the whole problem first, rather than immediately applying a standard algorithm. They look for relationships among the numbers, and with the operations and contexts involved. The computational procedure chosen or invented takes advantage of these observed relationships. At each step in the solution process individuals with number sense seem to be aware of the mathematical reasonableness of what is being done and of the answers obtained (Markovits, 1989, p. 79). Hence, Carroll (1996, p. 3) suggests that an individual’s number sense can be assessed through an analysis of the mental strategies used when calculating.

2.3.4 Learning Mathematics

Current developments related to the teaching and learning of mathematics, including the development of mental computation skills, are fairly consistent across developed countries (Literacy and Numeracy Diagnostic Assessment Project, 1991, p. 7). These are based on beliefs which hold that:

• Learning is enhanced where children are placed in problem solving situations. • Learning requires opportunities for active involvement and reflection. • Learning is enhanced where recognition is given to what the child already knows. • Learning is highly tuned to the situation in which it occurs.

Such beliefs place the child at the centre of the learning process where personally meaningful solutions can be developed. Mathematical power is gained. This requires classroom environments to be "cultures of sense-making" (MSEB & NRC, 1990, p. 32), environments in which the mathematics presented is seen to be predictable, purposeful and personally relevant. That this view of mathematics was not an outcome of the curricula of the new maths era was one factor in their failure to meet the goals set, particularly those with respect to developing positive attitudes towards mathematics and to being able to think purposively and effectively in mathematical situations. This child-centred focus is consistent with the constructivist approach to learning that has its contemporary genesis in the work of Piaget who believed that children learn through the assimilation and accommodation of new with existing knowledge. Learning mathematics is an active, problem solving process in which social interaction plays an important role. In the view of Yackel, Cobb, Wood, Wheatley, and Merkel (1990, pp. 12-13), the problems that arise when attempting to communicate are equally valuable learning experiences as the problem tasks themselves. Further, providing for reflection on experiences allows children time to link new to existing knowledge, an essential process for the expansion and refinement of current understandings (AEC, 1991, p. 17).

Learning as acquiring situated knowledge is interrelated with the beliefs that learning is knowledge-dependent and a process of knowledge construction (Resnick, 1989a, p. 1). Lave (1985) suggests that people engage in mathematical tasks in richly varied ways in different situations. When engaged in arithmetical situations:

People changed problems, decomposed and recomposed them in ways that reflected the organisation of the activity at hand as well as the structure of the number system, and often turned the social and physical environment into a calculating device. (p. 173)

This contrasts with school mathematics in which the procedures traditionally taught are designed to be context free and therefore universally applicable. Howson and Wilson (1986, p. 21) suggest that there are three types of mathematics. These are: (a) ethnomathematics, the idiosyncratic ways in which people in particular socio-cultural groups think about and engage in mathematical tasks; (b) school mathematics; and (c) higher (pure) mathematics. If mathematics is to be meaningful to the child, school mathematics needs to embody clear links among the three types. Only then will children gain mathematical power and therefore learn "that mathematics can help in the solution of their problems and in their own decision making" (Howson & Wilson, 1986, p. 22). The syllabi of the 1960s attempted to create links between school mathematics and higher mathematics by transplanting the aims, methods and structures of the latter onto the former with the concomitant ignoring of the needs of the student. A task for the 1990s is to develop ways in which key features of the ethnomathematical domain can be incorporated into school mathematics. Fostering classroom environments in which, for example, idiosyncratic computational strategies are accepted and valued would be a step towards this goal.

2.4 The Calculative Process

Although the demands of the technological age require that a broad view of basic skills be taken, the development of appropriate skills for calculating provides

students with useful tools for everyday life, for the workplace, and for learning other topics where calculative proficiency is required. The aim, therefore, is not to develop calculative skill per se, but to provide children with skills useful for solving problems, making applications and exploring new knowledge in mathematics and other subject areas (Coburn, 1989, p. 47). With the ready availability of hand-held calculators, as well as other electronic calculating devices, together with the need to be able to flexibly adopt and improvise methods of calculation, "the thrust of curriculum reform...is not to reduce the importance of computation but rather to broaden the definition of computation and to elevate the importance of problem solving" (Coburn, 1989, p. 47), in context with the development of number sense. Paper-and-pencil procedures continue to be appropriate for situations requiring a written record or in which the numbers are too complicated for mental calculation but not so unwieldy as to require a calculator (Rathmell & Trafton, 1990, p. 155). However, A National Statement on Mathematics for Australian Schools (AEC, 1991, p. 109) presents the view that mental calculation should be the method of first resort, particularly for less complex calculations where the numbers are easy to work with, and where there is no need for recording partial answers. Further, mental calculation is involved when computing with paper-and-pencil and with a calculator: known facts need to be recalled, and estimates need to be mentally calculated as a check on the reasonableness of the solutions obtained. Hence mental calculation is central to the calculative process, and closely linked to paper-and-pencil and technological calculation. This position is encapsulated in Figure 2.1. This model of the calculative process, while based on that in the Curriculum and Evaluation Standards for School Mathematics (NCTM, 1989, p.9) and on that devised by Rathmell and Trafton (1990, p. 154), recognises that, for any individual, the computational process is firmly embedded in their sense of number, computation being considered as a dimension of number sense (Trafton, 1992, p. 9). Whatever the method of calculation─mental, paper-and-pencil, or technological─its proficient use is likely to involve the decomposition and recomposition of numbers, and is associated with the characteristics of number sense delineated in Section 2.3.3 (NCTM, 1989, pp. 39- 40; Resnick, 1989b, p. 36; Sowder, 1992, pp. 5-6). The need for computation most commonly arises from problem situations (NCTM, 1989, p. 9). In such situations an individual needs to be able to (a)

recognise that computation is required, (b) formulate the calculation by deciding what operations to use, (c) choose an appropriate method of calculation, (d) carry out the calculation, and (c) interpret the solution obtained in terms of its reasonableness to the characteristics of the situation. The latter may result in changes to the nature of the answer required or to the method of calculation. Additionally, the model (Figure 2.1) recognises that the method chosen to carry out the calculation is dependent upon a range of factors that includes:

• The nature of the operation to be performed: the specific operation, its degree of complexity, the type and size of the numbers involved.

Situation Requiring Calculation

Calculative tools Operation required available

Selection of Appropriate Method of Calculation

Confidence felt Degree of precision required

Emotional state

Exact Answer Approximate Answer

PAPER-AND-PENCIL MENTAL CALCULATION TECHNOLOGICAL CALCULATION CALCULATION

Precision - Critical Precision - Noncritical

MENTAL COMPUTATIONAL COMPUTATION ESTIMATION

Exact Answer Approximate Answer

Reasonableness of Answer

Figure 2.1. A model of the calculativeNUMBER process SENSE highlighting the central position of mental calculation (Adapted from: NCTM, 1989, p. 9; Rathmell & Trafton, 1990, p. 154) • The degree of precision required: whether an approximate or exact answer is more appropriate.

• The availability of particular calculative tools: the human brain, writing materials, electronic calculating devices. • The degree of confidence felt with respect to the mathematics embedded in the situation and to the use of the particular computational tools available. The level of confidence with which a calculative task is approached is influenced by the emotional state of the individual.

Irrespective of the approach taken─mental calculation, paper-and-pencil, technological─the solution may be an exact or approximate answer, dependent upon the needs of the calculative situation. With respect to mental calculation, Figure 2.1 suggests that mental computation and computational estimation are interrelated. The latter involves calculating exact answers using approximate numbers, derived from the numbers embedded in the problem task environment (Silver, 1987, p. 41), to provide a reasonable estimate of the true answer to the problem. Hence, during computational estimation a limited range of numbers are manipulated as compared to that for mental computation. This and other features of the relationship between mental computation and computational estimation are explored in Section 2.6.

2.5 The Nature of Mental Computation

In developing an understanding of mental computation it is necessary not only to consider the ways in which it has been defined, but also to consider other terms which have been used to refer to the mental calculation of exact answers. Additionally, the links between mental computation as taught in schools and the methods used in non-classroom settings need to be analysed. Essential to this analysis are the characteristics of mental procedures which distinguish them from written methods.

2.5.1 Mental Computation Defined

Whereas a focus on approximate answers is a relative newcomer to the mathematics curriculum in primary schools, mental computation is "not a stranger" to the history of mathematics education (B. J. Reys, 1986a, p. 22). The use of the

term mental computation to refer to the process of producing an exact answer mentally without resort to calculators or any external recording device, usually with nontraditional mental procedures (Hazekamp, 1986, p. 116), based on conceptual knowledge (R. E. Reys et al., 1995, p. 324), is, however, relatively recent. Until the 1980s the most frequently used terms to describe such calculations were oral arithmetic and particularly mental arithmetic (or simply mental). These terms lack the preciseness of mental computation as their use often included the calculation of approximate as well as exact answers. Although not discounting the role paper-and-pencil and technological techniques may play, the process of mentally producing an answer that is not exact but sufficiently close so that the necessary decisions in a problem situation can be made is now commonly referred to as computational estimation. (R. E. Reys, 1984, p. 551). Mental computation is an "important component of estimation [see Figure 2.1] in that it provides the cornerstone necessary for the diverse numeric processes used in computational estimation" (R. E. Reys, 1984, p. 548). By focussing on the nature of the answers provided rather than solely on their accuracy, concern is shifted from the surface features of the computational process, such as the modes in which computational situations are presented, to the thinking strategies that are involved in calculating an answer. "Children's construction of increasingly powerful thinking strategies goes hand in hand with their development of increasingly sophisticated conceptual understandings" (Cobb & Merkel, 1989, p. 71), understandings that lead to children viewing mathematics as making sense. Although mental computation is now the term most commonly used in the mathematics education literature to refer to calculating exact answers mentally, other terms remain in use. Mental calculation is used by some writers (Cockcroft, 1982; Hope, 1986a; Hope & Sherrill, 1987). However, their usage is in contrast to that contained in the Curriculum and Evaluation Standards for School Mathematics (NCTM, 1989, p. 9), and by Hope (1986b). In the latter instances, mental calculation is employed as a general term for representing all mental procedures, whether producing exact or approximate answers (see Figure 2.1). Other writers continue to use mental arithmetic, albeit redefined to refer only to the process for calculating exact answers mentally (Allinger & Payne, 1986; McIntosh, 1990a, 1990b; Treffers, 1991). This redefinition is accompanied by a recognition that the way skill with mental computation should be developed needs to be a departure

from the "series of short, low-level unrelated questions to which answers are...calculated instantaneously and the answers written down with speed", characteristics of conventional mental arithmetic lessons (McIntosh, 1990a, p. 40).

2.5.2 Mental and Oral Arithmetic

Traditionally, mental arithmetic has been used as a generic term for all mental calculation, equivalent to the use of mental calculation in this study. It was typically defined as "arithmetic done without the aid of paper and pencil" (Flournoy, 1954, p. 148), and usually without a clear initial indication whether exact or approximate answers were being considered. Arithmetic problems presented verbally were also called mental problems or oral problems (Hall, 1947, p. 212). The term oral arithmetic was used in the 1952 and 1964 Queensland mathematics syllabuses to represent all mental calculations (Department of Public Instruction, 1952b, p. 2; Department of Education, 1964, p. 2). However, its usage referred primarily to calculating exact answers, with very few references to finding approximate solutions. Much discussion is contained in the professional literature from the first-half of this century, particularly in that originating in the United States, concerning the correctness of the terms oral arithmetic and mental arithmetic. As Hall (1954, p. 351) points out, many authorities attempted to avoid controversy by using the terms interchangeably, whereas others referred solely to oral arithmetic or to mental arithmetic. Some (Suzzallo, 1912; Thompson, 1917, cited in Hall, 1954; Thorndike, 1922), however, made a distinction between the two. Their concern was not simply with the method of calculation being a mental one. Of equal importance were the nature of presentation and the mode of response. Thompson (1917, p. 270, cited in Hall, 1954, p. 351) contended that oral arithmetic could be classified as mental arithmetic, but that all mental arithmetic is not necessarily oral. It could be classified as oral if the calculation was started by a spoken question, or its result was recorded in speech (Thompson, 1917, p. 270, cited in Hall, 1954, p. 351; Thorndike, 1922, p. 262). However, the stimulus for a mental calculation need not be oral. It could be something that has been read or merely thought about. Further, the solution could be recorded mentally or in writing, and not necessarily communicated orally.

From a somewhat different perspective, Suzzallo (1912, p. 75, cited in Hall, 1954) uses oral to refer to the situation where a "child works aloud, that is, expresses his procedure step by step in speech" (p. 352). Mental arithmetic was, to Suzzallo, the silent method. Thorndike (1922) believed that a devotion to oralness or mentalness per se was “simply fanatical...[as] oral, written, and inner presentations of initial situations, oral, written, and inner announcements of final responses, and oral, written, and inner management of intermediate processes have varying degrees of merit according to the particular arithmetical exercise, pupil, and context" (p. 263). Thorndike's (1922, p. 263) concern, however, was with pedagogical issues rather than with any real consideration of how a child may conduct the inner management of a calculation. In his view, the merit of a particular calculative method depended on such factors as: (a) the ease with which tasks could be understood, (b) the ease with which work could be corrected, (c) the ways in which cheating could be prevented, (d) the freedom from eyestrain, (e) the amount of practice a class would receive per hour spent on a task, and (f) the cheerfulness and sociability of the work to be undertaken. Given the restrictions that could be placed on the mental activities that could be classified as oral, Hall (1954) argued that mental arithmetic was the expression that should be used exclusively, and that it should have the following meanings:

(1) Arithmetic problems which arise (a) in an oral manner, (b) in a written form, or (c) "in the head" of the person who needs to solve the problem; (2) problems in which pencil and paper and other mechanical devices, such as calculators, are not used to record the intermediate steps between the statement of the problem and its answer; (3) problems in which pencil and paper are used, and problems in which they are not used to record the answer; and (4) problems in which quick estimations are made which either may or may not be verified by a written response. (pp. 352-353)

Except for point four, these criteria are ones which remain applicable to the use of mental computation to refer to the mental calculation of exact answers, as now used in the mathematics education literature and in this study. With concern shifting from modes of presentation and response to the cognitive processes involved, oral

now has a much broader meaning than the narrow usage outlined above. This is reflected in Cockcroft's (1982, pp. 92-93, paras 315-317) use of the expression mental mathematics to refer to the mental and oral work that should form a major part of the mathematics undertaken in classrooms. Its usage highlights that the promotion of mathematical discussion is as important as a consideration of the mental strategies employed, with both teachers and students benefiting from the discussion of strategies that ensues (Cockcroft, 1982, p. 93, para 317)

2.5.3 Mental Computation and Folk Mathematics

The issue of school mathematics versus ethno- or folk mathematics has as its focus the way mathematical tasks are accomplished. Central to this are the roles paper-and-pencil, calculators and mental calculation play in the calculative process. In particular, the concern is with the appropriate balance between, and the nature of, written and mental methods of calculating. School mathematics continues to be primarily concerned with standard paper-and-pencil methods of calculating (McIntosh, 1990a, p. 25; Willis, 1990, p. 12). Nevertheless, as French (1987, p. 41) points out, it is becoming increasingly unusual for these standard algorithms to be used anywhere except in the classroom. Reporting findings from the Bath and Nottingham studies which provided background information for his report, Cockcroft (1982, p. 20, para 71) indicates that in industry a range of idiosyncratic and "back of an envelope" methods is used, especially for long multiplication and division. For school mathematics to be useful, it should reflect the techniques used in everyday life (Willis, 1990, p. 9), and, in so doing, "assist children to cultivate and enlarge their inherent affinities and abilities for folk mathematics" (Maier, 1980, p. 21). Proficient folk mathematicians are skilled at calculating exact and approximate answers mentally ( Carraher et al., 1987, p. 94; Maier, 1980, p. 23) and in the use of calculators (J. P. Jones, 1988, p. 44). From an analysis of data from research undertaken in such every-day situations as a supermarket, a commercial dairy, and home kitchens, Lave (1985) concluded that not only are arithmetic and problem solving techniques situationally specific, but also "there are theoretically crucial ways in which people are similar in how they vary" (Lave, 1985, p. 172). One is that the arithmetic procedures used are

often more complex than those many of the users learnt in school. Secondly, making errors is a natural part of a process for arriving at correct arithmetic solutions. Terezinha Carraher et al. (1987) observe that in out-of-school situations, the results obtained by young Brazilian market sellers, "even when wrong, were sensible because there was a continuous monitoring of the quantities during the computation procedure; in [mental] procedures, children seem to ‘know where they are' at any given point" (p. 94). With respect to division strategies, Murray, Olivier, and Human (1991, pp. 50, 55) note that most children invent powerful non-standard algorithms, some of which are mental, in parallel with those learnt in school. Further, children are significantly more successful when they use their own procedures rather than the standard algorithms (Murray et al., 1991, p. 50). In school-like settings, "people [tend] to produce, without question, algorithmic, place-holding, school-learned techniques for solving problems, even when they could not remember them well enough to solve problems successfully" (Lave, 1985, p. 173). Ginsburg, Posner, and Russell (1981, p. 173) indicate that when schooled American and Dioula children added mentally, they frequently used written algorithmic procedures and were incorrect about twenty-five percent of the time, a finding which may reflect the lack of emphasis given to the development of personal procedures for computing mentally in classrooms. In Lave's (1985, p. 175) view, the standard written algorithms taught in schools de-contextualise arithmetic, are cumbersome and inappropriately require pencil and paper to be used most of the time. School mathematics should therefore be organised to provide opportunities for children to deal with mathematics in their own environments in ways similar to those of folk mathematicians (Maier, 1980, p. 23; Masingila et al., 1994, p. 13). In such situations people use a variety of techniques, often mental, and invent units with which to compute (Lave, 1985, p. 173). Gladwin (1985) suggests that:

Learners need to have a strong sense of the problem and the contradictions that must be resolved to reach a solution. Take away this strong sense of problem and we are left listening to the calculation...without knowing the purpose of the calculation. (p. 209)

This has implications for how problems are presented to children within classrooms. Murtaugh's (1985, p. 192) analysis of how shoppers mentally solve arithmetic problems in a supermarket suggests that, in many real-world situations problem formation and problem solving are likely to be integral parts of a single process. In contrast to the prepackaging of traditional school problems, with or without a veneer of real-world characteristics, "when people are free to formulate their own problems...the relevant inputs may be as negotiable as the eventual solution" (Murtaugh, 1985, p. 189). In making best-buy decisions, shoppers use qualitative features of items on the supermarket shelves to narrow the choice to two items for which some mental arithmetic calculation is undertaken to select the item for purchase (Murtaugh, 1985, p. 192). As a step towards closing the gap between folk and school mathematics, mental computation (with computational estimation and calculator usage) should receive greater prominence in the mathematics which children undertake in classrooms and should replace written methods as the basic computational skill of the computer age (MSEB & NRC, 1990, p. 19). The National Research Council (1989, p. 46) intimates that an emphasis on the development of number sense, with which mental computation is closely entwined, should move children beyond a narrow concern for school-certified computational algorithms. Such algorithms have no intrinsic merit (Willis, 1992, p. 11). They were developed to make use of the technology of paper-and-pencil and designed to be used without the need to think about the numbers involved (Jones, 1988, p. 42; McIntosh, 1992, p. 136). For this reason, Hope (1986a, p. 50) suggests that standard paper-and-pencil algorithms may contribute to a fragmentary view of numbers and number relationships and therefore not support the development of number sense. Individuals, who are proficient with their use, work with digits and book-keeping rules. They focus on "the written symbols, thereby losing track of both the meaning in the transactions they are quantifying and the meaning within the quantification system" (Carraher et al., 1987, p. 95). In contrast, meaning is preserved during mental calculations. Hence, the major differences between mental and written procedures arise from the degree of knowledge required of an individual about the problem situation (Nunes, Schliemann, & Carraher, 1993, pp. 53-54). In Skemp's (1976) terms, opportunities for the development of relational understanding are restricted. However, with the demise of standard computational algorithms

opportunities arise to combine the development of number sense with the teaching of computational procedures in useful and meaningful ways (Sowder & Schappelle, 1994, p. 344). To the degree that children continue to be required to perform paper- and-pencil calculations, McIntosh (1990a, p. 37) suggests that at least one of the following criteria should be met: (a) informal methods should be used to support and extend the use of mental procedures, (b) their development should occur in a problem solving context, (c) they assist in the development of number sense, and (d) they are of intrinsic interest to children. The calculator, in Hope's (1986a, p. 47) view, poses a greater threat to paper- and-pencil procedures than do mental ones. When folk mathematicians use calculators it is to replace the use of paper-and-pencil (Maier, 1980, p. 23). It therefore "seems very unlikely that any children at primary school in the 1990s [sic] will, as adults, make more than very sporadic use of [standard paper-and-pencil methods] of calculation in a society in which calculators are commonplace" (Curriculum Programmes Branch, Western Australia, 1989, p. 24). However, given the propensity for individuals to take the easy way out, it is possible that calculators could be regularly used in instances for which mental computation is more appropriate. For this reason, Atweh (personal communication, May 13, 1992) believes that calculators do pose a threat to mental procedures, as well as to paper- and-pencil ones. It is therefore essential for students to be assisted in making sensible choices about the planned method of calculation in particular situations─whether to use a calculator, paper-and-pencil, or mental computation (AAMT & CDC, 1987, p. 2).

2.5.4 Characteristics of Mental Procedures

Plunkett (1979, p. 3) proposes eleven characteristics of mental algorithms. He suggests that such algorithms are often fleeting, variable, flexible, active, holistic, constructive and iconic. They usually are not designed for recording, require understanding, and often provide early approximations of correct answers. Additionally, the range of contexts for which they are appropriate is limited. These features are in contrast to those of the standard written algorithms which are considered to be standardised, contracted, efficient, automatic, symbolic, general

and analytic. Further, they are not easily internalised and encourage cognitive passivity (Plunkett, 1979, pp. 2-3). Unlike written procedures which are, by definition, permanent and correctable, mental strategies are often transitory. As Hope (1985, p. 358) reports in his analysis of expert mental calculators, some appear incapable of explaining the processes used. This may be due to their ability to engage in automatic processing (Shriffrin & Schneider, 1977, cited in Jensen, 1990, p. 270) of information. Such processing is "fast, relatively effortless, and can handle large amounts of information and perform different operations on it simultaneously" (Jensen, 1990, p. 270). For non-expert mental calculators, their capacity for short-term mental storage is a critical factor in their ability to compute mentally and, by implication, in their ability to recall procedures used. The rate of forgetting information stored in working memory is directly proportional to the number of interpolated stages before its recall, and to the total amount of information simultaneously being held (Hitch, 1977, p. 337). These factors are a function of the particular procedure being employed for a mental calculation. Strategies for computing mentally are also characterised by their variability. This contrasts with school-authorised paper-and-pencil procedures which are standardised. This standardisation creates the impression that such procedures may be easily taught and leads to the belief that their use is the correct and only way in which calculations should be performed (Curriculum Programmes Branch, Western Australia, 1989, p. 28), a conventional goal of school mathematics programs. Plunkett (1979, p. 3) suggests that, for members of the public and many non-specialist mathematics teachers, the concept of a particular operation and its standard written algorithm are synonymous. This belief leads to the view that to teach an operation, a method, rather than an idea, needs to be taught. It is this assumption that confuses the issues relevant to the debate concerning the appropriate mathematical concepts, processes and skills with which children should become familiar. In instances where children are permitted to use the mathematics that they know, efficient mental procedures develop almost spontaneously (Sowder, 1990, p. 20). Such are personally meaningful and reflect the differences which exist in the knowledge that children bring to the task (Yackel et al., 1990, p. 13). The variations that can occur between strategies used by different children for the same task are

illustrated by the following finding of Cooper et al. (1992, p. 112). When calculating the sum of 52 and 24, some Year 2 children used a counting-on procedure (count on 24 with fingers by fives), whereas others used their knowledge of tens (50 + 20 = 70; 2 + 4 = 6). The latter strategy is based on a deeper understanding of number than the counting strategy. Cooper et al. (1992, p. 113) report that children who used tens in their strategies were able to progress to items beyond those which consisted solely of basic facts or simple algorithms involving teens. These children tended to exhibit diverse number understandings and hence were able to manipulate numbers in sophisticated ways. However, the knowledge, both conceptual and procedural (Hiebert & Lefevre, 1986), which an individual brings to a situation is not the only factor which governs the nature of the strategy applied. The features of the problem context also bear on a strategy's make-up. Mental strategies exhibit a flexibility which enables an individual to use a particular form of a strategy, one that is matched to the nature of the numbers involved. A general method─distributing─identified by Hope (1987, p. 334), for calculating products involves the transformation of one or both of the numbers to be multiplied into a series of sums or differences before calculating the product using one of three strategies, namely, additive distribution, subtractive distribution, and quadratic distribution. The particular strategy used is dependent upon the numbers involved. For example, Hope (1987, p. 334) found that his subject used additive distribution to calculate 16 x 72, namely 16 x (70 + 2) = 16 x 70 + 16 X 2 = 1120 + 32 = 1152. Whereas for 17 x 99, subtractive distribution was used: 17 x (100 - 1) = 17 x 100 = 1700 - 17 = 1683. In summary, the variability and flexibility of mental strategies are features that arise naturally from the process of calculating mentally. Representative of this is Jensen's (1990) observation of the procedures used by Shakuntala Devi, a prodigious mental calculator. Solutions are obtained:

Through exercising different routines drawn from an immense repertoire of numerical information and strategies, and the peculiarities of the problem itself determine the elements that are drawn upon from this repertoire to achieve the solution most efficiently. (Jensen, 1990, p. 270)

Standard paper-and-pencil algorithms discourage thinking (R. E. Reys, 1984, p. 551), or in Williams' (cited in Plunkett, 1979, p. 3) terms, encourage cognitive passivity, or suspended understanding. Such algorithms are designed to be automatic, which permits their being taught to and used by students who have little understanding of the processes involved (Plunkett, 1979, p. 2). Little, if any, thought is given to why the procedure is being carried out in the way that it is. Irrespective of the nature of the calculative situation, the same procedure is applied for each operation. The recognition and use of the structural relationships embedded in the situation is constrained, the use of alternate solution paths is deterred (R. E. Reys, 1984, p. 551). In situations where mental computation is appropriate, the use of mental methods allows an individual to select from a range of possible solution paths and strategies. This leads Plunkett (1979, p. 3) to suggest that mental methods are active ones. An individual is able to exercise some choice over the procedures used, even though the user may not always be totally conscious of the reasoning processes involved. Expert mental calculators give priority to selecting an appropriate strategy─one that allows an economy of effort. This is the fundamental decision in calculating mentally and governs all that follows during the computation process (Hunter, 1977a, pp. 36-37). In contrast to the use of the standard written algorithms, mental computation encourages thinking. Understanding is required to effectively arrive at a solution. Plunkett (1979, p. 3) suggests that children who get their mental calculations correct almost certainly understand what they are doing. However, care needs to be taken in assuming that children's self-devised strategies will be necessarily understood and that they will always provide correct answers. McIntosh (1991a, p. 5) found that when children from Years 2 to 7 used a working from the left strategy, more competent children produced incorrect answers on five occasions out of twenty- nine, whereas less competent children produced incorrect answers in fifteen instances out of forty. This is indicative of "a potentially valuable strategy [being] intuitively devised by different children, but in its embryonic form it is not yet efficient: not all the features of the situation are understood and therefore correctly controlled by the children" (McIntosh, 1991a, p. 5). Standard written procedures are not easily internalised because these methods do not match the ways in which people think about numbers (Plunkett,

1979, p. 3). In Reed and Lave's (1981, p. 442) terms, and supported by Carraher et al. (1985, p. 28), the focus is on the manipulation of symbols rather than on the manipulation of quantities as occurs during informal (folk) mental calculations. Such an analytic approach is "divorced from reality" (Reed & Lave, 1981, p. 442), as efficient use of the standard paper-and-pencil algorithms requires that the digits be dealt with separately without reference to their meaning or their relationship to real- world or representational models. This view remains valid under current approaches to developing children's skill with the written algorithms. These approaches are based on developing an understanding of the processes involved through relating each step to the manipulation of base-ten materials. However, the goal continues to be the automatic processing of written calculations. A manipulation-of-quantities approach, which characterises mental computation procedures, allows children to make meaningful alterations to the problems encountered and to work with quantities that can be easily manipulated (Carraher et al., 1987, p. 94). The approach is a holistic one (Plunkett, 1979, p. 3) in which people use convenient groups or invent units with which to calculate (Lave, 1985, p. 173). To calculate the cost of four coconuts at Cr$35.00 each, a 12 year- old street market seller in Brazil determined that "three will be 105, plus 30, that's 135...one coconut is 35...that is...140" (Carraher et al., 1987, p. 26). The calculation involved using the cost of a coconut as the calculative unit, and the knowledge that 30 plus 5 is 35. Such a procedure is a constructive one, as the correct answer is progressively built-up from an early approximation. This applies particularly where a left-to-right approach is used (Plunkett, 1979, p. 3). Such mental calculations occur in the context of complete numbers. For example, to calculate 24 x 50, a proficient mental calculator may simply calculate 12 x 100 (Hope, 1985, p. 362). The digits are not considered separately as occurs with the standard written algorithm, or its mental equivalent that non-proficient mental calculators tend to use (Hope & Sherrill, 1987, p. 108). Related to their meaningful use of the knowledge they have of numbers and number relationships, some children are aware of their reference to, and manipulation of, mental pictures (McIntosh, 1990c, p. 7; Olander & Brown, 1959, p. 100; R. E. Reys, B. J. Reys, Nohda, & Emori, 1995, p. 319). Plunkett (1979, p. 3) refers to mental strategies as often being iconic. Children have an overall picture of

the numbers involved, often relating them to an icon such as a number-line. A subject of McIntosh's (1990c) explains:

I just picture the numbers in my head somehow. I don't know how I do it....When I think of sums I sort of close my eyes a little bit and picture them all in my head...in a straight line....It's sort of like an arrow and when I want to I can imagine it moving down the many spaces. (pp. 7-8)

Although mental methods can be recorded if the need arises, they are not designed for doing so. This contrasts with written methods which Plunkett (1979) describes "as contracted in the sense that they summarise several lines of equations involving distributivity and associativity" (p. 2). Further, standard paper- and-pencil algorithms are general, being applicable to situations involving all numbers, large or small, whole number or decimal. Hence, there is a trade-off in meaning and generalisability for both mental and written procedures (Nunes et al., 1993, p. 54). Although mental algorithms preserve meaning, they are limited in their usage, becoming grossly inefficient when dealing with large numbers─for example, when multiplying through the chaining of successive additions, the preferred strategy for multiplication identified by Carraher et al. (1985). Nevertheless, Plunkett (1979, p. 3) has concluded that mental strategies are applicable to a greater range of mathematical situations than a cursory analysis of classroom number work might suggest.

2.6 Mental Computation and Computational Estimation

As illustrated in Figure 2.1, the distinction between mental computation and computational estimation derives from the degree of precision required in arriving at appropriate solutions. Threadgill-Sowder (1988) defines computational estimation as "the process of converting from exact to approximate numbers and mentally computing with those numbers to obtain an answer reasonably close to the result of an exact computation" (p. 182). In problem solving contexts, the approximate answer obtained needs to be sufficiently close to the exact answer to allow appropriate decisions to be made.

2.6.1 Components of Computational Estimation

Computational estimation is a complex task (Sowder, 1989, p. 26). It involves the co-ordination of two fundamental subskills: (a) converting from exact to approximate numbers using nearness judgements and number comparison skills, and (b) mentally computing with these numbers (Case & Sowder, 1990, p. 88). Sowder (1989, p. 27) reports that, even though students may be skilled at both approximation and mental computation, they may still not be able to arrive at reasonable estimates. Case and Sowder's (1990) study provides support for this view. In neo-Piagetian terms, it was hypothesised that children would not be capable of coordinating the two components of computational estimation until they were capable of vectorial thought at around age 12. During this stage children exhibit "growth in their ability to coordinate two or more complex and qualitatively different components of a task" (Sowder, 1989, p. 25). By around 17 years of age, students are capable of estimating 188 + 249 + 296 + 6 using the same level of significance for each approximation within the calculation (Sowder, 1989, p. 26). Mental computation, computational estimation, and number comparison share a common background characterised by factors essential to a well-developed sense of number (see Figure 2.2). These include: (a) an understanding of place value concepts related to whole numbers and decimals, (b) an ability to operate with multiples and powers of ten, (c) an ability to use the properties of operations, and (d) an understanding of the symbol systems used to represent numbers (Threadgill- Sowder, 1988, p. 195). The four constructs, mental computation, computational estimation, number comparison, and number sense do not develop in a strictly hierarchical order. Rather, Threadgill-Sowder (1988, pp. 194-195) suggests that each is dependent upon, while strengthening, the others in a spiral development, with computational estimation developing slightly later than mental computation and number comparison abilities associated with an increase in number sense.

Based on their study of the 1200 students in Years 7 to 12, Robert Reys, Bestgen, Rybolt, and Wyatt (1982, pp. 196,198) reported that a common characteristic of those who are proficient with computational estimation, is the

Prerequisite Fundamental Estimation Specific Concepts and Subskills Processes Estimation Skills Concepts

Convert Exact to Approximate Numbers

Approximate Reformulation numbers are Understand Compare used symbol system numbers by size

Understand place value The estimate is an approximation Compensation

Operate with multiples and Appropriateness powers of ten of estimate depends on desired outcome

Knowledge of MENTAL basic facts COMPUTATION More than one estimate is valid

Use properties Translation of operations

More than one estimation process is permitted

Figure 2.2. Components of computational estimation (Adapted from Threadgill-Sowder, 1988, p. 192)

ability to quickly and efficiently calculate exact answers mentally, a finding supported by Heirdsfield (1996, p. 128), particularly for addition by Year 4 children. All demonstrated skill with operating with a limited number of digits or with multiples of tens. The most proficient estimators displayed self confidence, used a variety of strategies and were able to mentally compute with larger numbers, with more digits, and with types of numbers other than whole numbers (R. E. Reys et al., 1982, pp. 198, 199). From a study of Year 8 students' performances on computational estimation tasks, Rubenstein (1985, p. 117) concluded that the ability to multiply and divide by powers of ten has an especially strong relationship with estimation performance. Further, students who lack this ability have a reduced understanding of the size of numbers. An analysis of Figure 2.2 suggests that it may be possible to be simultaneously competent at mental computation and very poor at computational estimation, a view supported by Robert Reys (1984, p. 549). The ability to convert exact to approximate numbers is one which needs to be developed before proficiency with computational estimation is achieved. This ability depends upon being able to make comparisons and to judge the relative size of numbers. Number comparison is defined by Threadgill-Sowder (1988) as "the ability to order real numbers [according to] size, such as in selecting the larger of two or more numbers, or by the ability to compare different magnitudes, such as selecting which of two numbers is closer to a third" (p. 183).

2.6.2 Computational Estimation Processes

Three key cognitive processes─reformulation, compensation, and translation─are evident in the methods used by proficient computational estimators (R. E. Reys et al., 1982; R. E. Reys et al., 1991). Reformulation, which is "heavily dependent on the ability to compare numbers" (Threadgill-Sowder, 1988, p. 191), is the one most often used to convert exact to approximate numbers. Although some aspects of compensation and translation are dependent upon number comparison skills and therefore inform the selection of appropriate approximate numbers, these strategies are more directly involved with the mental computation process (see Figure 2.2).

Traditionally, where the development of computational estimation skills has been included in the curriculum, it has been confined primarily to approximating numbers using the rounding convention, a reformulation strategy (R. E. Reys, 1984, p. 551). Sauble (1955, p. 37) has suggested that for children to develop competence in estimating they need to learn to round numbers and to recognise situations in which the use of approximate numbers is more meaningful and useful than exact numbers. This view is reflected by the finding of Schoen, Blume, and Hart (1987, pp. 22-23) that, irrespective of grade and ability levels, middle-school students equated estimation to a round to the closest approach, for whole numbers and decimals. The students were observed to round numbers to the leading powers of ten before mentally calculating. However, little use was made of compensatory strategies or compatible numbers, with few attempts to refine initial estimates. The estimation strategies used were not associated with conceptual understanding, even in instances where this was encouraged by the test item. Reformulation is defined as a "process of altering numerical data to produce a more mentally manageable form...[while leaving] the structure of the problem intact" (R. E. Reys et al., 1982, p. 187). It is characterised by two basic strategies. These are the front-end use of numbers and the substitution of numbers by more acceptable forms. The most common method for approximating numbers is rounding, usually to the nearest five, ten, hundred, thousand, and so forth (Trafton, 1978, p. 200). Robert Reys et al. (1982, p. 197) report that this ability is evident in all those who are proficient with computational estimation. Rounding is a variant of a front-end use of numbers. The other common front-end approach is truncation, in which the left-most digits of a number are used in a mental calculation─for example, 87,398 - 54,246 could be converted to 87 (thousand) - 54 (thousand). The substitution of numbers by more acceptable forms usual occurs in one of two ways. The first, using compatible numbers, involves rounding to more convenient multiples of numbers within a problem─for example, substituting 5 for the 3 in 63 ÷ 5. The use of this skill is less common than rounding to multiples of ten, but is still exhibited by a majority of those who are proficient with computational estimation (R. E. Reys et al., 1982, p. 197). A second common substitution strategy is the use of equivalent forms of numbers to simplify the mental calculation. For example, to find an approximate answer for 30% of $103, 30% could be replaced by

1/3. In this situation it is likely that $103 would be rounded to $105 to simplify the mental calculation further. This highlights that reformulation (and translation and compensation) strategies are not simply used in isolation. Proficient computational estimators use a combination of strategies appropriate to the context in which a mental calculation, designed to provide an approximate answer, is to occur. Figure 2.2 indicates that both compensation and translation also depend on an ability to compare numbers. This view is somewhat at variance with that of Threadgill-Sowder (1988) who does not postulate a link between translation and number comparison. Translation entails modifying the mathematical structure of a problem to a form which is more easily managed mentally (R. E. Reys et al., 1982, p. 188). Threadgill-Sowder (1988, p. 191) and Sowder and Wheeler (1989, p. 135) categorise averaging, classified as a translation strategy by Robert Reys et al. (1982, p. 188), as reformulation. Averaging is particularly useful for addition problems in which the numbers cluster around a common value. An approximate answer is calculated by multiplying the common value, which can be viewed as a reasonable group average, by the number of values in the group (R. E. Reys, 1984, p. 554). For example, for 6,146 + 7,200 + 5,300, an approximate answer could be calculated by selecting 6,000 as the group average and multiplying it by 3. Although a reformulation strategy─compatible rounding─is used, it can be argued that the rounding occurs in context with a desire by the estimator to change the structure of the problem and that this influences the particular approximations used in the calculation, a strategy dependent upon an individual's ability to compare numbers. Translation is also dependent upon factors relevant to the process of calculating mentally. The structure of the problem is changed to simplify the mental computation and this is contingent upon such factors as a child's knowledge of number relationships and confidence with various computational methods. Robert Reys et al. (1982) suggest that the process for translation is more flexible than that for reformulation: "The student seems to have a panoramic view of the problem and is less constrained by the numbers involved. In fact, very often the numbers and operations are simultaneously altered to result in more manageable forms" (p. 189). Trafton (1986, p. 25) suggests that an essential element of insightful estimation is the ability to sense the relationships between approximate and exact answers. In judging an estimate's appropriateness, it may be necessary to make adjustments to take account of any numerical variation that may have occurred through the process

of translation or reformulation (R. E. Reys et al., 1982, p. 189). Compensation, as this process is called, also relies on an ability to compare numbers, and is affected by (a) the manageability of the numerical data, (b) the context of the problem, and (c) the individual's tolerance for error (R. E. Reys et al., 1982, p. 189). Robert Reys et al. (1991) report that Japanese fifth- and eighth-grade children were more likely to adjust their estimates if compatible rounding, rather than straight rounding, had been used. This suggests that "compatible rounding is more likely to leave the user with a sense of the direction of the exact answer and therefore more likely to be accompanied by compensation" (R. E. Reys et al., 1991, p. 49). As one Grade 8 student explained for 347 x 6 ÷ 43: "That's about 347 x 6 ÷ 42 or 347 x 1/7. This is close to one-seventh of 350, which is 50; 350 is more than 347, so my estimate is 48" (R. E. Reys et al., 1991, p. 48). Of the three key cognitive processes, Threadgill-Sowder (1988, p. 194) suggests that compensation seems to be the one most closely linked to a conceptual understanding of the computational estimation process. Nonetheless, as Poulter and Haylock (1988, p. 28) have observed, adjustments often reflect an intuitive feeling for number and are ones for which a rationale may not be easily explained. Compensation need not only occur after an estimate has been calculated. Adjustments may also be made during intermediate stages of mental computation. Robert Reys et al. (1982, p. 190) report that some children, when calculating an estimate for the total attendance at six Super Bowl games, selected a group average for five of the attendance figures and ignored the sixth to compensate for rounding upwards. Such a strategy appears to be more difficult than compensating after an estimate has been calculated. In Robert Reys' et al. (1982, p. 197) study, intermediate compensation is classified as a Level 3 characteristic, being evident in only 20-30% of the competent estimators interviewed. In contrast, final compensation is classified as a Level 2 strategy and is one that is in the repertoire of a majority of proficient computational estimators.

2.6.3 Comparison of Mental Computation and Computational Estimation

Barbara Reys (1986a, p. 22) suggests that whereas an approximate answer can be obtained for every arithmetical problem through computational estimation,

the calculation of exact answers is limited to a subset of problems. The degree of arithmetical complexity which can be mentally processed is dependent upon such factors as an individual's (a) knowledge of useful strategies for mental calculation, (b) ability to recall a large number of numerical equivalents, and (c) ability to remember partial answers at various stages of a calculation (Hope, 1985, p. 358). These factors are relevant to calculating exact as well as approximate answers mentally. However, by definition, the degree of arithmetical complexity that has to be processed during computational estimation is less than that when calculating an exact answer. The first step in arriving at an approximate answer is to convert from exact to approximate numbers so that an estimate can be more easily calculated mentally (see Figure 2.2). It follows that the range of numbers dealt with when mentally computing approximate answers is a subset of those that may be required to be manipulated when calculating exact answers. Conventional rounding and truncation usually produce an even number of tens, hundreds or thousands with which to compute mentally. Conversion to compatible numbers typically results in an operation for which the result is able to be retrieved from long term memory, or is easily calculated. This applies not only to reformulation per se, but also when front- end and compatible number strategies are used in conjunction with translation and compensation. When calculating exact answers, all digits in each of the numbers to be operated on must be taken into account. Some of the strategies used, which are analysed in detail in Section 2.7.4, bear some relationship to the processes used to derive approximate numbers in computational estimation. In contrast to written procedures, those skilled in mental computation often proceed from left-to-right (Hope & Sherrill, 1987, p. 108; Carraher et al., 1987, p. 94), a front-end strategy─for example, 38 x 5 is calculated as: (30 x 5) + (8 x 5) = 150 + 40 = 190 (Hazekamp, 1986, p. 118). Some form of compensation is also used by some children when calculating exact answers mentally. Olander and Brown (1959, p. 99) outline the following method for calculating 34 - 9: 34 - 10 = 24, 24 + 1 = 25, albeit a confusing approach for those who have difficulty in working out which way to compensate (Sowder, 1992, p. 41). The similarity between these approaches and their computational estimation counterparts adds weight to the belief that mental computation, computational estimation, number comparison, and number sense develop in a spiral fashion, as suggested by Threadgill-Sowder (1988, pp. 194-195).

Some methods used to arrive at more precise estimates may aid children in their development of strategies for calculating exact answers. For example, in calculating a closer estimate for 378 + 236 + 442, children could consider the tens digit in each number, or the tens and ones in each, to adjust an initial estimate of 900, increasing it by 140 or 160, respectively. Addition examples such as this are possibly ones for which primary school children may not normally be expected to calculate exact answers mentally. Nevertheless, experiences with getting closer strategies during computational estimation may assist children to develop flexible approaches to calculating exact answers. Where the numbers and operations involved are within the capability of an individual to calculate mentally, getting closer strategies may ultimately result in turning estimates into exact answers (Irons, 1990b, p. 1). Strategies used by those skilled in mental computation, however, go beyond those which bear any direct relationship to those used in computational estimation (see Figure 2.1, Tables 2.1 & 2.4). A characteristic of highly skilled mental calculators is their ability to perceive number properties and relationships which may be useful for calculating an exact answer (Hope, 1985, p. 358; Hunter, 1977a, p. 36; B. J. Reys, 1986b, p. 3279-A). Numbers are decomposed and recomposed in ways which are not necessarily based on conventional base-ten place value relationships. For example, Hope (1987, p. 335) reports that Charlene, a highly skilled mental calculator, calculated 87 x 23 in the following steps:

87 x 23 = (29 x 3) x 23 = 29 x (3 x 23) = 29 x 69 = 69 x (30 - 1) = 69 x 30 - 69 = 2070 - 69 = 2001

Such an approach requires sufficient working memory capacity to be able to coordinate and monitor many interrelated calculation activities without losing track of the calculation (Hope, 1987, p. 335). In summary, mental computation and computational estimation are processes that are closely related. Each is performed mentally, taking advantage of the structural properties and relationships among numbers. Both are used to check

whether or not an answer from a calculator or paper-and-pencil calculation is reasonable (R. E. Reys, 1984, p. 551)─a reflective process that Shigematsu, Iwasaki, and Koyma (1994, p. 20) call metacomputation. As represented in Figure 2.1, some prerequisite skills are shared. As well, people proficient with mental computation and computational estimation exhibit similar cognitive and affective characteristics and use strategies which bear some similarity. Nonetheless, there are essential differences between mental computation and computational estimation. Although computational estimation produces many different solutions, all of which are reasonable and acceptable, mental computation is concerned with producing answers that are either correct or incorrect. Mental computation is a vital prerequisite to computational estimation (see Figure 2.2). However, in Robert Reys' view (1984, p. 551), the reverse does not apply, a belief that is in conflict with that of Irons (1990a, p. 31) referred to above. In the sections that follow a detailed analysis of a number of issues central to a deeper understanding of mental computation is presented. These issues include: (a) cognitive and affective components of mental computation, (b) the characteristics of proficient mental calculators, (c) the process of computing mentally, and (d) pedagogical issues.

2.7 Components of Mental Computation

As previously intimated, although approximate answers may be determined for all arithmetical problems, the range of problems for which exact answers may be found mentally is limited (B. J. Reys, 1986a, p. 22). This situation is not merely due to the numerical complexity of many arithmetical tasks. Of equal importance is a consideration of an individual's understanding of, and competence with, the essential components of mental computation. Little research is available on which to base a comprehensive analysis of the components of mental computation. Flournoy (1957) suggests that in situations requiring exact answers, a person must be able to:

• Recognise the problem and organise the facts of the problem. • Keep the numbers in mind as s/he thinks about the problem.

• Perform the necessary arithmetical process or processes, without paper and pencil, and reach a decision about the problem. (p. 148)

Flournoy's (1957, p. 148) first point is one that is not peculiar to mental computation. Whatever the mathematical task, an individual needs to be able to recognise what the problem is and be able to organise the perceived facts so that a solution can be determined. The second requirement depends on working memory capacity and the way in which it is used during processing─in Hunter's (1978, p. 339) terms, their memory for interrupted working. The degree to which working memory is burdened during mental calculation is largely dependent upon the particular strategy being used, an issue that is explored in Section 2.8.3. By arithmetical process, Flournoy (1957, p. 148) is simply referring to the particular operation─addition, subtraction, multiplication or division─that has to be applied. Of greater importance is a consideration of the particular mental strategies that are used to carry out these operations. In any analysis of the components of mental computation, not only should the range of strategies used be considered but also the factors that underpin their proficient use. In the absence of a detailed analysis of the components of mental computation in the literature, it is considered that Sowder and Wheeler's (1989, p. 132) model for specifying the components of computational estimation is one that can provide a framework for discussion. Sowder and Wheeler (1989, p. 132) classify the components into four categories: (a) Conceptual Components, (b) Related Concepts and Skills, (c) Skill Components, and (d) Affective Components. Conceptual components are defined as those that relate to a basic understanding of what the process of finding an estimate entails (Sowder & Wheeler, 1989, p. 131). For this analysis, conceptual components of mental computation are deemed to be those that relate to an understanding of the basis of the process for calculating exact answers mentally. Sowder and Wheeler (1989, p. 131) define related concepts and skills as those abilities that are known, or suspected, to indirectly influence an individual's proficiency with finding approximate solutions. With respect to computational estimation, the ability to compute mentally is classified as a related skill (Sowder & Wheeler, 1989, p. 132). Given the delineation of prerequisite concepts and skills relevant to mental computation in its relationship with computational estimation (see Figure 2.2), it is considered

appropriate that this category is also relevant to an analysis of the components of mental computation per se. Research into the categorisation of strategies for calculating exact answers mentally has not yet resulted in such a well-defined classification as that for the processes used to convert exact to approximate numbers. Sowder and Wheeler (1989) define these components as those "that seemed more appropriately listed as skills, even though they do not appear in students' work without accompanying evidence of the conceptual understanding necessary to employ them as skills" (p. 131). The classification of the components of mental computation presented in Table 2.1 highlights strategies classified as relying on relational understanding. Such strategies reflect a constructivist view of mental computation which posits that mental strategies are based on an individual's intuitive understanding of numbers and of the manipulations required (R. E. Reys et al., 1995, p. 305). It is hypothesised that mental computation strategies relevant to particular contexts cannot be devised, understood or performed successfully without the

Table 2.1 Components of Mental Computation

1. Affective components

• Display confidence in ability to calculate mentally. • Display confidence in ability to do mathematics. • Recognise that mental computation is useful. • Recognise that procedures for computing mentally can make sense.

2. Conceptual components

• Recognise arithmetical contexts for which mental computation is appropriate. • Accept more than one strategy for obtaining an exact answer mentally. • Recognise that the appropriateness of a strategy depends on the context of the calculation.

3. Related concepts and skills

An ability to: • Translate a problem into a more mentally manageable form.

Key questions: (a) How can the numbers be expressed to obtain questions which can be answered by recall? (b) How will the operational sequence proceed as result of the way that the numbers have been expressed?

• Understand and apply place value concepts. • Recall basic facts related to the four operations. • Operate with multiples and powers of ten. • Compose and decompose numbers and to express them in a variety of ways. • Link numeration, operation and relation symbols in meaningful ways. • Recall and use a wide range of relationships between numbers, including whole numbers, fractions, decimals and percents. • Use the associative and commutative properties of addition and multiplication. • Use the distributive properties of multiplication and division. • Recognise the need for and undertake compensations necessitated by transformations to the numbers involved.

Table 2.1 cont.

Components of Mental Computation

4. Heuristic strategies based upon relational understanding

• Add or subtract parts of the first or second number

• Use fives, tens and/or hundreds ◊ add-up ◊ decomposition ◊ compensation

• Work from the left ◊ organisation ◊ incorporation

• Work from the right ◊ mental analogue of standard written algorithm ◊ place-grouping

• use known facts

• Use factors ◊ general factoring ◊ half-and-double ◊ aliquot parts ◊ exponential factoring ◊ iterative factoring

• Use distributive principle ◊ additive distribution ◊ subtractive distribution ◊ fractional distribution ◊ quadratic distribution

Note. The four hypothesised categories for analysing the components of mental computation are based on Sowder & Wheeler’s (1989, p. 132) model for the components of computational estimation.

support of appropriate conceptual understandings, related concepts and skills, and affective characteristics. It is therefore proposed to analyse the components of mental computation using the following categories: (a) Affective Components, (b) Conceptual Components, (c) Related Concepts and Skills, and (d) Strategies for Computing Mentally.

2.7.1 Affective Components

When learning mathematics, "it is not the memorization of mathematical skills that is particularly important...but the confidence that one knows how to find and use mathematical tools whenever they become necessary" (NRC, 1989, p. 60). Mental computation is one such tool, one with which students generally do not display proficiency and prefer not to use, even in instances where mental calculation is appropriate (Carpenter et al., 1984, p. 487; B. J. Reys et al., 1993, p. 314; R. E. Reys, 1985, p. 15; R. E. Reys et al., 1995, p. 323). If children are to become proficient with mental computation, they need to develop confidence in their ability to calculate mentally (see Table 2.1 & Figure 2.1). It has been argued previously that mental computation, computational estimation and number sense are closely interdependent. The development of a good sense of number requires confidence in dealing with numerical situations (AEC, 1991, p. 107). Further, Sowder and Wheeler (1989, p. 132) suggest that such confidence is an important affective component of computational estimation. It is therefore reasonable to assume that the degree of confidence with which children approach mental computation tasks is influenced by the degree of confidence felt in their ability to perform mathematical tasks in general, particularly those related to numerical situations (see Table 2.1). Children gain confidence and ownership of the mental procedures they use if they are allowed to create, construct and discover mathematics for themselves (Payne, 1990, p. 3), particularly where the mathematics is encountered in meaningful situations. They come to view mathematics, and procedures for computing mentally, as making sense and useful, as evidenced in the findings of Carraher et al. (1985). It is therefore proposed that the affective components of mental computation should also include a recognition that (a) mental computation is

useful, and (b) procedures for computing mentally should make personal sense (see Table 2.1).

2.7.2 Conceptual Components

As highlighted in Figure 2.1, in any situation requiring a calculation, children need to be able to determine whether paper-and-pencil, mental or technological calculation is the most appropriate method. This depends on the nature of the operation, the degree of precision required, the availability of particular calculative tools, and the level of confidence felt with regard to the mathematics embedded in the situation. It is therefore hypothesised that a fundamental conceptual component of mental computation is the ability to recognise arithmetical contexts for which mental computation is appropriate (see Table 2.1). Given that in any particular context children display varying levels of confidence, with respect to their ability to compute mentally, there can be no fixed criteria for defining contexts appropriate for mental computation. This component needs to be operationalised with respect to particular individuals in particular contexts. Howden (1989) suggests that "students who can make judgements about the reasonableness of computational results and realize that more than one way can be used to arrive at a solution gain confidence in their ability to do mathematics" (p. 7). In Sowder and Wheeler's (1989) delineation of the components of computational estimation, consideration is given to multiple processes─that is, the "acceptance of more than one process for obtaining an estimate" (p. 132). In their classification, this component is not specifically tied to the need to convert exact to approximate numbers nor to the mental computation component of computational estimation. However, one of the defining characteristics of mental algorithms, as discussed beforehand, is their variability (Plunkett, 1979, p. 3). Those skilled at mental computation use a variety of strategies for calculating exact answers (Carraher et al., 1987, p. 91; Hope, 1987, p. 331; Hunter, 1978, p. 340). Accordingly, it seems reasonable to propose that an important conceptual component of mental computation is an acceptance of more than one strategy for obtaining an exact answer mentally (see Table 2.1).

Hope (1987) contends that "success in mental [computation] depends more on the ability to select the ‘right tool for the job' than upon the possession of some innately superior mechanism for processing information" (p. 339). The right tool─mental computation strategy─depends not only on an individual's store of conceptual and procedural knowledge, but also on the context in which the calculation is to be undertaken. In parallel with Sowder and Wheeler's (1989, p. 132) suggestion that appropriateness needs to be a consideration when formulating the components of computational estimation, a third conceptual component of mental computation is proposed, namely, people engaged in mental computation need to recognise that the appropriateness of a strategy for computing mentally depends on the context of the calculation (see Table 2.1).

2.7.3 Related Concepts and Skills

Barbara Reys (1989, p. 72) notes that Paul Trafton suggested during a conference on number sense that there are possibly two levels of mental computation, each distinguished by the nature of the strategies used to determine the exact answers and the nature of the numbers involved in the operation. The first of the two levels does not generally necessitate an invented strategy to be applied, but does require a method for determining place value of the answer (B. J. Reys, 1989, p. 72). Such computational problems often involve operating with powers of ten or multiples of ten, and are of the nature of the mental computations required during computational estimation. At this level, the mental operations required are essentially extended basic facts (B. J. Reys, 1989, p. 72), but can also involve numbers other than whole numbers. With respect to operations involving simple common fractions, ½ + ¼ for example, Barbara Reys (1989, p. 72) suggests that, although conceptual understanding is essential to obtaining the correct answer, it is unlikely that an invented strategy would be required─at least for individuals who have a sound understanding of common fractions. Trafton's second level of mental computation (cited by B. J. Reys, 1989, p. 72) concerns problems for which an in-depth knowledge of the properties of the numbers involved is generally required, as well as a self-developed procedure understood by the user─for example for 99 x 7 (B. J. Reys, 1989, p. 72). Sowder

(1992, p. 5) suggests that the ability to perform mental computations using invented strategies is a behaviour that indicates the presence of number sense. Problems which require the invention of strategies are those more likely to be encountered when the purpose of calculating mentally is the determination of exact answers (see Figure 2.1). For example, one method for calculating 7 x 99 relies on the application of the distributive law of multiplication: 7 x 99 = 7 x (100 - 1) = (7 x 100) - (7 x 1) = 700 - 7 = 693. This method also relies on a knowledge of, what Hazekamp (1986) calls, special products─"products that are easily found by multiplying by a power of 10 or a multiple of a power of 10" (pp. 117-118). Threadgill-Sowder (1988) approached an analysis of the components of mental computation from a somewhat different, though related, perspective to that of Trafton. A principal component of mental computation is the ability to translate a problem into a more mentally manageable form (Hunter, 1977a, p. 25; B. J. Reys, 1985, p. 46). Two key questions need to be asked and answered, albeit unconsciously in many instances. These are: (a) How can the numbers be expressed to obtain basic fact questions? and (b) How will the operational sequence proceed as result of the way that the numbers have been expressed? (Threadgill- Sowder, 1988, p. 184) Threadgill-Sowder (1988) suggests that "because there is never just one way to answer Question 1, that is, to choose a format for the numbers to be calculated, mental [computation] is a very creative, inventive act" (pp. 185-186). For those who are skilled with mental computation, it is likely that their aim is to express the numbers in a form which can be related to elements in their store of number knowledge and relationships that extends beyond the basic facts. Hope (1987, p. 335) reported that Charlene, an expert thirteen year-old mental calculator, calculated 16 x 72 by reasoning 16 x (70 + 2) = (16 x 70) + (16 x 2) = 1120 + 32 = 1152. It is likely that Charlene used place value knowledge to express 72 as 70 + 2 so that she could draw on her store of products that included 16 x 70 = 1120. This suggests that Threadgill-Sowder's (1988, p. 184) first question should be rephrased to increase its generality, namely, How can the numbers be expressed to obtain questions which can be answered by recall? (see Table 2.1) "Skilled mental calculation demands that the user ‘search for meaning' by scanning the problem for salient number properties and relationships" (Hope, 1986a, p. 52), a view supported by Barbara Reys (1986b, p. 3279-A). Olander and

Brown (1959, p. 99) report a number of ways in which children calculated 51 - 34. As suggested by Threadgill-Sowder (1988, p. 184), the operational sequence in each case is dependent upon the way in which the numbers to be operated upon are expressed. Included among the methods observed by Olander and Brown (1959, p. 99), for 51 - 34, were:

• 34 + ?(6) = 40, 40 + ?(11) = 51; Therefore 51 - 34 = 17 (6 + 11). • 34 + 10 = 44, 44 + ?(7) = 51; Therefore 51 - 34 = 17 (10 + 7). • 34 + ?(1) = 35, 35 + ?(15) = 50, 50 + ?(1) = 51; Therefore 51 - 34 = 17 (1 + 15 + 1). • 50 - 30 = 20, 20 - 4 = 16; Therefore 51 - 34 = 17 (16 + 1). • 51 = three 17s, 34 = two 17s; Therefore 51 - 34 = (one) 17.

During a calculation "numbers [obtained] are re-expressed in ways that lead to basic fact [or recall] questions, and the computations carried out as a result...present new mental calculations calling for a new cycle of questions" (Threadgill-Sowder, 1988, p. 185). For example, in relation to the first method, 34 + ?(6) = 40 was either recalled, or calculated using basic fact knowledge (10 - 4 = 6), and to move from 40 to 51 required question one to be reconsidered. As with the first step, 40 + ?(11) = 51 may have merely been recalled, or it may have been calculated using basic fact knowledge (1 - 0 = 1, 5 - 4 = 1). In each case it is evident that the way in which the numbers and the perceived relationships between them are expressed determines the way in which the answer is calculated. From the above analysis, for Threadgill-Sowder's (1988) first question to be answered successfully, children need to be able to:

• Recall basic facts. • Understand place value concepts (72 is 70 + 2). • Compose and decompose numbers to express them in a variety of ways─that is, to draw on their store of numerical equivalents (51 is 3 x 17 or 40 + 11). • Operate with multiples and powers of ten (Use 5 - 3 to solve 50 - 30, or recognise that 40 x 800 is 4 x 10 x 8 x 100). (Threadgill-Sowder, 1988, p. 185)

However, in view of the rephrasing of question one, the list of related concepts and skills (see Table 2.1) should be extended to include the ability to recall and use a wide range of relationships between numbers. Cockcroft (1982, p. 92, para 316) provides support for the first two components. He suggests that efficient mental procedures are based upon an understanding of place value in association with an ability to recall addition and multiplication facts. The ability to compose and decompose numbers is dependent upon well-developed part-whole relationships (Ross, 1989, p. 47) and is closely related to place value knowledge. An "understanding of place value and ownership of its essential features is crucially important in opening up more efficient and more simple [mental] strategies" (McIntosh, 1991a, p. 5). Sowder (1992, p. 4) lists the ability to compose and decompose numbers as one which demonstrates some presence of number sense. As their number sense grows, children demonstrate increased flexibility in the way they think about numbers. As Ross (1989) points out, ultimately "their thinking allows them mentally to compose wholes from their component parts, decompose whole quantities into parts, and perhaps rearrange the parts and recompose the whole quantity, confident all the while that the quantity of the whole has not changed" (p. 47). Hiebert (1989, p. 82) believes that written arithmetic symbols can function in at least two ways: (a) as records of something already known, and (b) as tools for thought. A well developed sense of number requires that numerals, operation and relation signs operate in both ways. This leads Sowder (1992, p. 5) to suggest that a key behaviour indicative of the presence of number sense and critical to mental computation (and numeration and computational estimation) is the ability to link numeration, operation and relation symbols in meaningful ways (see Table 2.1). Employing symbols as tools necessitates that they be treated as objects of thought. There is evidence to suggest that proficient mental calculators use symbols in this manner, as reflected by their physically-oriented language. They often refer to chopping or breaking numbers apart when describing their mental techniques (Hiebert, 1989, p. 83). Although the use of numbers as objects of thought contributes to the development of number sense, through advantage being taken of the properties of the system (Hiebert, 1989, p. 83), in Trafton's (1989) view, "number sense is more related to intuitions and insights associated with numbers as quantities, rather than

numbers as abstract, formal entities" (p. 74). As discussed previously, proficient mental calculators use a manipulation-of-quantities approach. Terezinha Carraher et al. (1987, p. 94) reported that children, when calculating mentally, altered the problems presented so that they were able to work with more manageable quantities. This is reflective of the approaches observed by Olander and Brown (1959) outlined above. Children were not operating with symbols per se, but with symbols given meaning by their relationships to place value and part-whole knowledge. With respect to the Threadgill-Sowder's (1988) second question─"How will the operational sequence proceed as result of the way that the numbers have been expressed?" (p. 184)─children should be able to:

• Regroup terms using associative and commutative properties of addition and multiplication: 54 - 31 = (50 + 4) - (30 + 1) = (50 - 30) + (4 - 1). Use the distributive properties of multiplication and division: 16 x (70 + 2) = (16 x 70) + (16 x 2). • Multiply by powers of 10 (as for question one). (Threadgill-Sowder, 1988, p. 185)

However, as previously described, each question is considered cyclically during the mental computation process. Skills listed by Threadgill-Sowder (1988, p. 185) as applying to Question 2 may in fact be used to aid in answering Question 1, as in Charlene's use of the distributive property (Hope, 1987, p. 335) to produce a form which could be related to her store of number knowledge. Hence, in providing a summary of essential concepts and skills in Table 2.1, each is not specifically related to Threadgill-Sowder's (1988) two key questions. An additional skill related to proficient mental computation is the ability to recognise the need for and undertake compensations necessitated by modifications to the numbers involved (see Table 2.1). One of the strategies observed by Olander and Brown (1959, p. 99) is classified as compensatory rounding, the fourth approach to solving 51 - 34 listed above. This approach requires that 1 be added to the partial result (16) to compensate for initially rounding 51 to 50 before sequentially subtracting 30 and 4. As discussed previously, this strategy is

conceptually similar to final compensations which may be necessary to derive closer estimates during computational estimation (see Figure 2.2). The ability to compensate during computational estimation and mental computation is indicative of a well developed sense of number (Sowder, 1992, p. 6). Behr (1989, p. 85) suggests that there are two aspects of, what he calls, variability: (a) a transformation of one of the operands in combination with a compensatory transformation of the answer, the case outlined above, and (b) compensatory transformations which are applied to each of the operands prior to an answer being calculated. Although Behr's (1989, p. 85) analysis is specifically related to number sense and computational estimation, Sowder (1992, p. 6) suggests that the ability to compensate for numerical transformations plays a critical role in mental computation.

2.7.4 Strategies for Computing Mentally

Since the mid-nineteenth century recognition has been given periodically to the importance of mental computation as a component of education, employment, and everyday living (R. E. Reys, 1984, p. 549). Nevertheless, it is only relatively recently that there has been any empirical interest in the mental processes that underlie the ability to calculate exact answers mentally (Vakali, 1985, p. 106). The research that has been undertaken focuses primarily on mental strategies associated with determining the basic facts─addition, subtraction, multiplication, and division operations with numbers from zero to nine. The identification, analysis and classification of mental computation strategies beyond the basic facts has received little attention (McIntosh, 1990b, p. 1; Vakali, 1985, p. 107). Of the studies that have been carried out, analogous to those associated with identifying and classifying basic fact strategies, most have focussed on addition and/or subtraction with whole numbers (Beishuizen, 1985, 1993; Cooper et al., 1992; Cooper, Heirdsfield & Irons (1996); Ginsburg et al., 1981; Hamann & Ashcraft, 1985; Heirdsfield, 1996; Hitch, 1978; Murray & Olivier, 1989; Olander & Brown, 1959; Resnick, 1983; Resnick & Omanson, 1987; Vakali, 1985; van der Heijden, 1995). Multiplication (Carraher et al., 1985; Gracey, 1994; Hope, 1985, 1987; Hope & Sherrill, 1987) and division, in particular, have received scant

consideration. Four studies considered all four operations (Carraher et al., 1987; Carroll, 1996; McIntosh, 1991a; B. J. Reys, 1986b). In some studies (Cooper et al., 1992; Ginsburg et al., 1981; Hamann & Ashcraft, 1985) operations with basic facts were included in the examples used. However, clear distinctions between the strategies used for these and the strategies that may have been used to mentally calculate with the larger numbers, should they be different, have not always been clarified (Ginsburg et al., 1981; Hamann & Ashcraft, 1985). Consequently, the following synthesis of available data to formulate a single system of strategies for mentally computing with numbers greater than nine (see Tables 2.1 to 2.4), using the work by McIntosh (1991a, 1990c) as a framework, needs to be considered preliminary, and therefore interpreted and applied with caution. In undertaking research to identify and classify mental strategies, McIntosh (1990b) believes that "it is very difficult to escape the need for subjective judgments on the part of the interviewer or the analyser of the protocols as to the interpretation of the child's description of its mental activity" (p. 13). Such identification is all the more difficult in instances where attempts to identify strategies are based on written reports of past performances by expert adult mental calculators. Hope (1985, p. 358) noted that many of these reports are vague, with the writer not giving a clear indication of the methods used by the calculator. Further, many professional mental calculators kept the strategies they used as closely guarded secrets. Others found difficulty in explaining their computational techniques. The analysis of data arising from the paucity of research is confused, not only by the subjective nature of the identified strategies, but also by the variations in sample characteristics. The samples vary with respect to age, grade, ability, and cultural background. Although the majority of studies centre on strategies used by randomly selected children, some focused on those used by children categorised according to their mental computation ability (Hope, 1985, 1987; Hope & Sherrill, 1987; McIntosh, 1991a; Olander & Brown, 1959; B. J. Reys, 1986). The latter sampling technique is the more useful approach, particularly where the aim is to identify efficient mental strategies, ones that could become the focus when providing learning experiences to develop children's mental computation abilities. The age of subjects ranged from approximately six years for children in Grade 1 (Hamann & Ashcraft, 1985) to adulthood (Hitch, 1977, 1978; Hope, 1985). Some studies investigated strategies used by children in various year-levels (Cooper et al., 1996,

Grades 2-4 longitudinally; Hope & Sherrill, 1987, Grades 11-12; Hamann & Ashcraft, 1985, Grades 1, 4, 7 & 10; McIntosh, 1991a, Grades 2-7; Olander & Brown, 1959, Grades 6-12). A further variable to be taken into consideration when analysing the research is the size of the numbers contained in the numerical situations. The majority of studies focused on operations with one- and two-digit numbers. Those that involved three-digit numbers and beyond are generally ones in which senior students or adults formed part or all of the samples (Ginsburg et al., 1981; Hitch, 1977, 1978; Hope, 1985; Hope & Sherrill, 1987; Olander & Brown, 1959; Petitto & Ginsburg, 1982). Exceptions to this are Hope's (1987) study of a 13 year-old skilled mental calculator and that of Carraher et al. (1987), which investigated the strategies used by third-grade Brazilian children, whose ages ranged from eight to thirteen years. Few studies have included examples employing common fractions, decimal fractions or percentages (R. E. Reys et al., 1995). Contexts used in the studies include informal out-of-school settings and formal school situations in which the operations are most commonly context-free. In some studies, the numbers to be operated upon were embedded in word problems in an endeavour to add a life-like quality to the operations (Carraher et al., 1987; Cooper et al., 1992; Vakali, 1985). Context-free examples may be presented in a horizontal or vertical format, thus providing an additional variable that may influence the use of a particular mental strategy. Operations presented vertically increase the likelihood of paper-and-pencil procedures being used to calculate mentally (Cooper et al., 1992, p. 105). Terezinha Carraher et al. (1987, p. 89) reported that children were more likely to use mental approaches when solving simulated store situations and word problems, with context-free computational exercises being more likely solved by written procedures.

Models for Classifying Mental Strategies

A number of researchers have proposed models for the classification of strategies used to calculate mentally beyond the basic facts. McIntosh's (1991a, 1990c) system is the most comprehensive of these. This model, based on data from children in Grades 2 to 7 who were presented with context-free examples

involving the four operations, classifies strategies into clusters dependent upon such factors as whether an understanding of place value is evident, or whether it was simply a matter of using known facts. Other classifications have been based, not only on the particular strategies used, but also on the observed general predispositions towards mental computation (Cooper et al., 1992). The results from a longitudinal study, initially with Grade 2 children's approaches to addition and subtraction, by Cooper et al. (1992), suggest that general predispositions may be considered on two continua, namely, (a) the propensity to visualise numbers, and (b) the way in which children perceive number situations. Although some children do not appear to have the ability to visualise numbers, others demonstrate "a strong ability..., particularly with arrays, and to use this visualisation in their counting" (Cooper et al., 1992, p. 114). The strategies used are hypothesised to be directed by three different approaches to thinking: (a) a feel for number, (b) a feel for strategy, and (c) a feel for process. Feel for number is characterised by verbal reports such as "eight is two less than ten" and reflects a tendency to compose and decompose the numbers to make the operation more manageable─a related concept and skill discussed above (see Table 2.1). Cooper et al., (1992) reported that "children with feel for strategy tend to be fixated on their strategy and attempt to apply it seemingly without regard for efficiency" (p. 115). Such children often rely on counting-on and counting-back by twos, fives and tens. These two approaches are not necessarily used in isolation. Each may be used with the other, or in association with the approach that reveals a more global conception of their task─feel for process (Cooper et al., 1992, p. 115). The researchers reported that the children who consistently use a feel for process approach are demonstrating the most marked growth in their ability to compute exact answers mentally. Such an approach indicates a more acute sense of number and operation, as reflected by such comments as: "I want the number to go up" and "Adding makes things bigger" (Cooper et al., 1992, p. 115). In subsequent research, Cooper, Heirdsfield, and Irons (1995, p. 9) considered the approach to operation and the method of calculation, together with a subject's attitude to numbers and approach to process, in determining the category into which a child's strategy would be classified. In categorising a child's approach to an operation, such factors as the method of subtraction─take-away, missing- addend─were considered. Consideration was also given to the method of

calculation. For example, whether a solution was achieved through recall or counting. Hamann and Ashcraft (1985, p. 54) distinguish between preparation strategies and solution strategies. Preparation strategies are those that relate to the way in which complex number situations are approached. In Hamann and Ashcraft's (1985, p. 54) terms, complex addition occurs in examples with sums greater than 18. Their complexity stems from there being at least one two-digit addend, sometimes requiring carrying. The most common preparation strategies are classified as: (a) begin with the 1's column, and (b) begin with the 10's column─right- to-left and left-to-right approaches, respectively. A solution strategy is one used to describe the way in which a sum, or component sum, is reached. Three solution strategies are reported: "counting, statement of a memory operation, and statement of an arithmetic fact" (Hamann & Ashcraft, 1985, pp. 66-67). However, the distinction between the last two strategies is unclear. Commenting on these two categories, McIntosh (1990b) asserts that "if I know a fact, then I hold it in memory: if I remember that 3 + 3 = 6, then I know it as a fact. The child's words may be different, but the strategy is the same" (p. 15). Whether an answer to a calculation can be retrieved from long-term memory depends upon an individual's store of numerical equivalents. Given that most people have access to number relationships that extend to varying degrees beyond the basic facts, it could be hypothesised that the strategies used to calculate beyond the basic facts evolve in a manner comparable to those related to the development of basic fact knowledge. Svenson and Sjoberg (1982), from a longitudinal study, over Grades 1 to 3, which analysed children's strategies for subtraction with numbers less than or equal to 13, reported that, in general, "the development of the children's cognitive processes involved a gradual shift from more primitive and less demanding memory strategies...to reconstructive memory processes...to retrieval processes" (p. 91), that is, a shift from using external memory aids to reconstructing basic facts in working memory, to recalling the answer from long-term memory. In identifying the components of mental computation (Table 2.1), it is the reconstructive memory processes that are of primary concern. These processes are the ones that should provide the focus in any attempt to develop children's mental computation skills. "In a reconstructive process the answer is reached through a series of more or less conscious derivations or manipulations in working memory to

reach the answer" (Svenson & Sjoberg, 1982, p. 91). The complexity of the manipulations increases proportionally to the complexity of the numerical situation (Vakali, 1985, p. 112). An increase in complexity may be due to a change in size of the numbers involved, a factor that influences the implementation of particular mental strategies (R. Cooper, 1984, cited in Murray & Olivier, 1989, p. 4). With increased number size, complexity may be added by the need to carry and borrow during addition and subtraction, respectively, procedures that are avoided by more proficient mental calculators (Hope & Sherrill, 1987, p. 108). As discussed previously, Trafton (cited in B. J. Reys, 1989, p. 72) proposes two levels of mental computation. The first level incorporates problems that primarily require routine operations with powers of ten and multiples of powers of ten. In contrast, the reconstructive strategies, which typify the second level, depend on a person being able to use self-developed techniques that are dependent upon a knowledge of the properties of numbers and operations (B. J. Reys, 1989, p. 72). Such strategies require adaptive expertise, rather than routine expertise (Hatano, 1988, cited in Sowder, 1992, p. 19). Adaptive expertise necessitates an understanding of how and why particular strategies work and how they can be modified to suit the characteristics of particular numerical situations. A key aspect of an individual's conceptual knowledge relevant to mental computation is an understanding of place value concepts (see Table 2.1). Murray and Olivier (1989, pp. 5-7) propose a model describing four increasingly abstract levels of computational strategies with two-digit numbers. The type of strategy at each level is linked to prerequisite number and numeration knowledge. It is suggested that the distinction between pre-numerical (count-all) and numerical (counting-on, bridging-the-ten) strategies evident in basic fact calculations remains for two-digit numbers (Murray & Olivier, 1989, p. 5). A further distinction, which may also apply to larger numbers, can be made between the numerical strategies based on counting and the heuristic strategies that do not entail counting─a distinction originally proposed by Carpenter (1980, p. 317) with respect to the acquisition of addition and subtraction concepts. Heuristic strategies are ones that "often involve the decomposition of one or more of the numbers in a problem in order to transform the given problem to an easier problem or series of problems" (Murray & Olivier, 1989, p. 5). Carraher et al. (1987) refer to the two types of mental strategies identified in their

study─decomposition and repeated grouping─"as heuristics, so as to emphasize the flexibility of the solutions" (p. 91). It is suggested by Murray and Olivier (1989, p. 4) that when children are required to compute with numbers outside their range of constructed numerosity, they regress to more primitive strategies─that is, they may rely on a counting strategy rather than on an heuristic one. With respect to two-digit numbers, and based on research involving context- free additions presented orally, written horizontally and written vertically to Grade 3 children, Level 1 of Murray and Olivier's (1989, pp. 5-6) model is characterised by the pre-numerical strategy of counting-all. Children who rely on this strategy have not yet acquired the numerosities of numbers within the range presented, although their names and numeral forms are known. No meaning is assigned to individual digits within a number, with, for example, "the symbol group 63...regarded as a way of ‘spelling' the number name" (Murray & Olivier, 1989, p. 5). At Level 2, the numerosities of numbers within the range under consideration have been acquired. This allows the use of counting-on strategies. However, such strategies, when used with larger two-digit addends, become laborious and prone to error (Murray & Olivier, 1989, p. 6). Based on chronometric data, and supported by verbal reports, Resnick and Omanson (1987, p. 66) distinguish between three forms of count-on strategies related to the addition of a two-digit number and a single-digit number. These are: min of addends, sum of addends and min of units. The thinking which characterises Murray and Olivier's (1989) Level 2, together with that for Level 3, is a necessary prerequisite for understanding the intricacies of two-digit numeration and computation (Murray & Olivier, 1989, p. 9). At the third level "the child sees a two-digit number as a composite unit, and can decompose or partition the number into other numbers that are more convenient to compute with, for example to replace 34 with 30 and 4" (Murray & Olivier, 1989, p. 6). These abilities provide the conceptual basis for the use of heuristic strategies and, at least for the Grade 3 children in the sample, are usually based on place value knowledge (see Table 2.1). Murray and Olivier's (1989, p. 6) research suggests that Level 3 understanding is sufficient for developing powerful strategies for computing mentally. This is before a full understanding of base ten numeration has been achieved which "necessitates the conceptualization of ten as a new abstracted repeatable (iterable) unit which can be used as a unit to construct other numbers" (Murray & Olivier, 1989, p. 3). Resnick and Omanson (1987, p. 70) have observed

that few primary grade children, in the United States of America, decompose and recompose counting numbers based on knowledge of their decimal structure. The conceptualisation of ten as an iterable unit typifies Level 4 understanding, as defined by Murray and Olivier (1989, pp. 6-7). Children are able to conceive of two-digit numbers as consisting of groups of tens and some ones. Murray and Olivier (1989, p. 6) point out that, at Level 3, children work with tens as numbers, but at Level 4 work with ten as an iterable unit. Hence, Level 3 children view 67, for example, as 60 and a 7 and not as 6 tens and 7 ones. Although McIntosh (1990b) considers that "the distinction, though defended by the authors, is not entirely clear for mental computation" (p. 17), Level 4 understanding, in Murray and Olivier's (1989) view, allows for "a progressive schematization (‘shortening') and abstraction of the Level 3 heuristic strategies" (p. 7). To illustrate this distinction, a Level 3 solution to 36 + 27 is: "Take the six and the seven away, thirty plus twenty is fifty; now add six, then add seven.” This compares to a Level 4 strategy: "Thirty plus two tens, that's fifty. Six plus seven is thirteen, that's sixty-three" (Murray & Olivier, 1989, pp. 6-7). Terezinha Carraher et al. (1987) have suggested that mental computation "can no longer be treated merely as idiosyncratic procedures nor inconsequential curiosities. It involves sophisticated heuristics that are general, revealing a substantial amount of knowledge about the decimal system and skill in arithmetic problem solving" (p. 96). Additionally, they should not be dismissed as inferior and irrelevant to formal learning. Teachers should become aware of the strategies children use and capitalise on these to enhance the development of children's mathematical abilities. As indicated previously, it is the reconstructive memory processes (Svenson & Sjoberg, 1982, p. 91) that are of primary concern in the formulation of the strategy components of mental computation. If teachers are to take advantage of the approaches used by children and adults to compute mentally, they need to be aware of the pre-numerical, numerical and, most importantly, the heuristic strategies (Carraher et al., 1987, p. 91) that have been identified as being in common use. This awareness needs to be accompanied by a realisation that mental strategies tend to be adapted to the specific characteristics of the numerical situations, particularly in out-of-school settings (Carraher et al., 1985, p. 95; Murtaugh, 1985, p. 192).

McIntosh (1990c, 1991a) provides a comprehensive analysis of mental strategies. His classification constitutes a system that encompasses all four operations for the basic facts and beyond. However, in identifying the breadth of commonly used strategies, this classification needs to be supplemented with data from other studies. In comparing the categorisations used by various researchers, recognition needs to be given to the absence of agreed descriptors for many of the commonly identified approaches. The categories into which mental strategies may be clustered are described by McIntosh (1990c, 1991a) as: (a) initial strategy, (b) elementary counting, (c) counting in larger units, (d) used place value instrumentally, (e) used place value relationally, (f) used other relational knowledge, (g) known fact, (h) related to known fact, and (i) used aids. Except for the last category, McIntosh (1990c, p. 7) views the classification as hierarchical. The strategies classified by McIntosh (1990c, 1991) as used aids─used fingers, and used a mental picture─are ones normally employed in conjunction with other strategies─for example, counting (Cooper et al., 1992, pp. 108-109). Svenson and Sjoberg's (1982, p. 99) first two stages in the evolution of children's strategies to solve basic subtraction facts highlight the use of fingers to serve as external memories. McIntosh (1990c, p. 7) reported the use of a wide range of finger techniques, some quite sophisticated. Additionally, many children appear to use mental pictures to aid their calculations. Although some appear to be unaware of their use, others are quite conscious of referring to, and manipulating, iconic representations such as number lines (McIntosh, 1990c, p. 7), and mental images of a soroban─Japanese abacus (R. E. Reys, B. J. Reys, Nohda & Emori, 1995, p. 319). Resnick (1983) suggests that:

The earliest stage of decimal number knowledge can be thought of as an elaboration of the number line representation so that, rather than a single mental number line linked by the simple "next" relationship, there are now two co-ordinated lines....Along the rows a "next-by-one" relationship links the numbers....Along the columns a "next-by-ten" relationship links the numbers. (p. 127)

This somewhat different view of decimal structure, as represented by a ninety- nine board, can be applied to mental addition and subtraction. For example, 72 - 47 can be calculated by moving down the 10 string from 72 four positions to 32 and then by moving down the ones string seven positions to 32 (Resnick, 1983, p. 133). This view is supported by Beishuizen (1993, p. 316) who found, particularly for Grade 2 children classified as demonstrating a lower-level ability, that the use of a hundredsquare had relatively positive effects on developing proficiency with addition and subtraction using an N10 strategy─a strategy in which the first number is kept whole while the second is decomposed. Although not included in Hamann and Ashcraft's (1985, p. 66) list of preparation strategies, the strategies categorised by McIntosh (1990c, p. 2) as initial strategies reflect ways in which complex number situations can be approached. These are classified as: (a) change subtraction to addition and division to multiplication, and (b) use the commutative laws of addition and multiplication. With respect to the latter, Barbara Reys (1986b, p. 3279-A) categorises the use of the commutative (and associative) properties of addition and multiplication as translation strategies. Translation is identified as one of the three most common approaches used by high and middle ability mental computers in Grades 7 and 8. To facilitate a comprehensive summary of the categories of strategies identified as being used to calculate exact answers mentally, a more detailed analysis is presented in the sections which follow. This analysis is organised under the following headings: (a) Counting Strategies, (b) Strategies Based Upon Instrumental Understanding, and (c) Heuristic Strategies Based Upon Relational Understanding.

Counting Strategies

McIntosh (1990c, pp. 2-3, 1991a, p. 2) distinguished between counting forwards or backwards in ones─elementary counting─and counting in both directions using larger units─counting in larger units (see Table 2.2). With respect to counting- on in ones, Resnick and Omanson (1987, p. 66), following the work of Resnick (1983), hypothesised four counting procedures for adding a one-digit number to a two-digit number: min of the addends, sum of the units, min of the units, and mental carry. These strategies have their origins with the work by Groen and Parkman (1972) who investigated addition methods involving two addends less than ten.

Chronometric models, such as these, "[assume] a ‘counter in the head' that [can] be set initially at any number, then incremented a given number of times and finally ‘read out'" (Resnick, 1983, p. 117). Different strategies necessitate a different number of increments. Hence, the time taken to provide an answer─reaction time─is the measure by which assumptions are made with respect to particular strategies used. From Resnick and Omanson's (1987) study, chronometric evidence supported by interview data, provides an indication for the use of each of the strategies, except sum of the units. The min of the addends strategy requires no place value knowledge as the single-digit number is added-on in increments of one after the mental counter is set to the two-digit number─for 23 + 9, the mental counter is set to 23, and the child counts 24, 25, 26...32. It follows from this example that an ability to count across the decade barrier is a prerequisite. To be able to use the min of the units strategy, a child needs to have acquired some ability with partitioning and recombining numbers (see Table 2.1), a Level 3 numeration skill, as proposed by Murray and Olivier (1989, p. 6). Resnick and Omanson (1987, p. 66) described this strategy as decomposing the two-digit number into a tens component and a ones component and then recombining the tens component with whichever of the two unit quantities is larger. The mental counter is then set to the reconstituted number and the smaller of the units digits is added in increments of one. For 23 + 9, the operation is recomposed to 29 + 3, with the mental counter set to 29 and then incremented three times to 32. Mental carry is a strategy that mimics the carrying procedure for the written algorithm. For addition, "mentally add the units digits, mentally carry a 1 if

Table 2.2 Counting Strategies

Elementary counting

• counting-on in ones • min of the addends 23 + 9: 24, 25, 26...32; 32

• min of the units 23 + 9: 29 + 3; 30, 31, 32; 32

• counting-back in ones 24 - 6: 23, 22, 21, 20, 19, 18

Counting in larger units

• counting-on in twos/fives/tens 80 + 60: 90, 100...140; 140 71 - 44: 54, 64, 74; minus 3; 27

• counting-back in twos/fives/tens 28 - 15: 23, 18, 13; 13

• counting-back to a second number in twos/fives/tens 140 - 60: 130, 120, 110...60; 80

• repeated addition 15 x 50: 50 + 50 + 50 + 50 + 50 = 250; 2 x 250 = 500; 500 + 50 + 50 + 50 + 50 + 50 = 750

• repeated subtraction 150 ÷ 30: 150 - 30 - 30 - 30 - 30 - 30 = 0; 5

Note. Adapted from: McIntosh (1990c, 1991a) and Resnick & Omanson (1987)

necessary, then mentally add the tens digit to the carry digit" (Resnick & Omanson, 1987, p. 67). The use of a mental form of a written algorithm is a strategy categorised by McIntosh (1990c, 1991a) as one relying on the instrumental use of place value (see next section). In a cross-cultural study by Ginsburg et al. (1981, p. 173) it is reported that, for both schooled and unschooled young American and Dioula children, counting was frequently observed for addition examples involving relatively small sums─12 + 7, for example. However, no indication of the particular counting strategies used was provided. Olander and Brown (1959, p. 99) observed that counting is a slow

procedure, with many low achievers forgetting what has been counted before the process is completed, an indication of their limited capacity for interrupted working (Hunter, 1978). Although decrementing─counting-back from the larger number─and incrementing─counting-on─models for subtraction have been proposed for basic addition and subtraction facts (Resnick, 1983, p. 119), few studies have investigated these models for operating beyond the basic facts. Counting-on for subtraction involves starting with the smaller number and counting on to the larger, usually by ones, twos, fives or tens (Cooper et al., 1992, p. 108; McIntosh, 1991a, p. 2; Olander & Brown, 1959, p. 99). Except for the first approach, McIntosh (1990c, p. 1; 1991a, p. 2) classifies these strategies as Counting in larger units (see Table 2.2). Olander and Brown (1959) related a count-on by tens approach for subtraction: "71 - 44: 54, 64, 74, minus 3, the answer is 27 because there are 3 tens minus 3" (p. 99), an approach that relies on a knowledge of basic subtraction facts, and an ability to compensate. Cooper et al. (1992, p. 108) reported a count-on strategy for the addition of two-digit numbers based on a knowledge of doubles. To add 23c and 12c, a child twice adds six to 23: 23 + 6 + 6. Other strategies classified by McIntosh (1990c, p. 1) as counting in larger units are: counted back in twos/tens, counted back to second number in two/tens, repeated addition, repeated subtraction, multiples, and recited tables. To the first of these can be added counting back in fives, for examples such as 28 - 15 (Cooper et al., 1992, p. 111). Whereas counting back by twos/fives/tens entails decrementing the larger number by a number of steps whose value is the smaller number, counting back to the second number requires that the value of the number of steps equals the difference between the two numbers─for 140 - 60, a child explained: "You have 140 and you counted backwards in 10s" (McIntosh, 1990c, p. 3). Some children have been observed to multiply using repeated addition. One of the strategies, classified by Carraher et al. (1987) as a repeated-grouping heuristic, involves repeatedly adding fifty to find 15 x 50: "[José] started by adding chunks of 50 five times to 250. He doubled this chunk, getting ten 50s, and then went back to adding individual 50s" (Carraher et al., 1987, p. 93). This procedure, as Carraher et al. (1985) point out, is one that "becomes grossly inefficient when large numbers are involved" (p. 28).

Repeated subtraction, the quotition aspect of the division process, is one strategy for mentally dividing one number by another that has been observed (Carraher et al., 1987, p. 93; McIntosh, 1990c, p. 3). Carraher et al. (1987) report an approach, classified as a repeated-grouping heuristic, that provides an example of repeated subtraction, albeit a complex one, to solve 75 ÷ 5:

The problem was solved by successively subtracting the convenient groups distributed while keeping track of the increasing share that each child received; 10 marbles were given to 5 children, which accounted for 50 marbles, and the remaining 25 were distributed among 5 children─5 to each, totalling 15 for each child. (p. 93)

The remaining two categories for counting in larger units, as defined by McIntosh (1990c, p. 1; 1991a, p. 2), Multiples and Recited tables, may be considered to apply more to basic fact calculations than to operations involving larger numbers. McIntosh's (1990c, p. 4) examples relate to division and multiplication basic facts, respectively.

Strategies Based Upon Instrumental Understanding

In the transition from counting strategies to the use of strategies based on relational knowledge, McIntosh (1990c, p. 1; 1991a, p. 2) has identified a cluster of approaches that reflect an instrumental understanding of place value. This involves the application of "rules without reason" (Skemp, 1976, p. 20). These strategies (see Table 2.3) entail "removing zeros without knowing why it worked, and calculating mentally by recreating mentally the standard written algorithm" (McIntosh, 1991a, p. 2). These techniques reflect the manipulation of symbols approach (Reed & Lave, 1981, p. 442) discussed earlier, an approach that embodies strategies that "are divorced from reality...[and give] no consideration [to]

Table 2.3 Strategies Based Upon Instrumental Understanding

Used place value instrumentally

• removed zero 90 - 70: 9 - 7 = 2; add a zero; 20

• used mental form of written algorithm 39 + 25: 9 + 5 = 14; carry the 1; 1 + 3 = 4 + 2 = 6; 64

For multiplication: • no partial product retrieved (No attempt to adapt written algorithm for mental use.)

• one partial product retrieved 25 x 48: 5 x 48 is 5 x 8 = 40, carry 4, 24, 240; 2 x 48 = 96, 960; 240 + 960 = 1200

• two partial products retrieved 12 x 250: 2 x 250 = 500; 1 x 250 = 250, 2500; 500 + 2500 = 3000

• stacking 8 x 999: 8 x 9 = 72, 72(0), 72(00)

Note. Adapted from: Hope & Sherrill (1987) and McIntosh (1990c, 1991a)

the relative value of the symbols" (Carraher et al., 1987, p. 90). With respect to the removed zero strategy, McIntosh (1990c) records the following dialogue between child and interviewer, and asserts that teaching mental computation through a focus on rules is as self-defeating as such an approach is for written computation (McIntosh, 1991a, p. 6; 1996, p. 273):

Child: Well 90 take away 70 ...well I just take it from 9 minus 7 equals 2 and add the zero. Interviewer: Right and how is it you are able to take these zeros on and off? Child: I always do it like that cause it seems easier for me. Interviewer: How did you find out how to do that. Do you remember when, how you know... Child: I think the teacher taught me to away the 0. You always take away the zero.

Interviewer: Why does it work? Child: Don't know. (McIntosh, 1990c, p. 4)

Ginsburg et al. (1981, pp. 173-174) have observed that as schooled American and Dioula children grow older, they tend to abandon counting strategies in favour of the standard paper-and-pencil algorithms applied mentally. These were found to be used approximately 75% of the time for addition by children categorised as middle, with respect to age. A number of researchers have observed the mental application of the standard written algorithms for addition, subtraction and multiplication to operations beyond the basic facts (Cooper et al., 1992; Ginsburg et al., 1981; Hope & Sherrill, 1987; Markovits & Sowder, 1994; McIntosh, 1990c, 1991a; B. J. Reys, 1985; Vakali, 1985). In other studies (Hamann & Ashcraft, 1985; Hitch, 1977, 1978; Resnick & Omanson, 1987), although the classifications have not been explicitly linked to the written algorithms, some of the strategies identified can be interpreted as being analogous to the standard procedures. One of the preparation strategies identified by Hamann and Ashcraft (1985) was classified as one's column, where an approach to solving 14 + 12 was recorded as "two plus four equals six and one plus one is two" (p. 67). It is where regrouping─borrowing or carrying─is required that it becomes most apparent that children are following the written algorithm, operating, in Ginsburg's et al. (1981, p. 171) terms, on the numbers as digits and not as tens or hundreds. One child explained their procedure for adding 39 and 25 as: "9 plus 5 is 14, carry the 1, 1 plus 3 is 4, plus 2 is 6,...64" (Vakali, 1985, p. 111). Vakali (1985, p. 111) observed that this ones-tens-organise strategy is the most frequently used approach for mental addition and subtraction by Year 3 children. In comparing the performances of children in Years 2 to 7, McIntosh (1991a, p. 5) indicated that many errors were made by children who used a mental form of the written addition and subtraction algorithms, particularly those classified as the least competent with mental computation. Similar results are reported by Markovits and Sowder (1994), and Hope and Sherrill (1987), with respect to multiplication. Students in Grades 11 and 12, unskilled with mental multiplication, preferred to use an analogue of the written algorithm, "[making] little attempt to examine the calculative task for even the most

transparent number properties that might aid in the calculation" (Hope & Sherrill, 1987, p. 104). Although heuristic strategies, reliant on a recognition of such properties, provide strong evidence for an understanding of the composition and partition principles, Resnick and Omanson (1987, p. 70) caution that the tendency to use school-taught algorithms may mask such an understanding. Four principal variations of the standard written algorithm for multiplication have been identified by Hope and Sherrill (1987, p. 101-105): no partial product retrieved, one partial product retrieved, two partial products retrieved and stacking (see Table 2.3). When using the first strategy, Grade 11 and 12 students "made no attempt to adapt paper-and-pencil methods to a mental medium. Each partial product was calculated digit by digit, and no numerical equivalent larger than a basic fact was retrieved during the calculation" (Hope & Sherrill, 1987, p. 101). This strategy, together with the second, was often guided by gestures that reflected a desire to write each stage of the calculation (Hope & Sherrill, 1987, p. 105). Markovits and Sowder (1994, p. 23) found that, following the implementation of instructional units on mental strategies, Grade 7 students tended to remain with standard paper-and-pencil algorithms in situations where they could easily be applied mentally, but looked for nonstandard methods where these could not so easily be used. By retrieving one or two of the partial products from long-term memory, the demands on working memory are reduced. However, Hope and Sherrill (1987, p. 105) observed difficulties with the organisation stage of the procedure not dissimilar to those observed by Vakali (1987, p. 112) for subtraction. As defined by Vakali (1985, p. 112), this stage involves combining partial answers into a final solution. Hope and Sherrill (1987) record that, following determining the partial products of 500 and 2500 for 12 x 250, one student said: "5, 0, 0, and 2, 5, 0, 0,...would be...0, 5, 7, 2...2, 7, 5, 0?" (p. 105), a right-to-left additive process. The organise stage, when using the stacking strategy relies on visualising the partial products as they would appear if they had been recorded in writing. This strategy was applied only in operations that involved multiplying a multiple-digit number by a single-digit number. For example, 8 x 999 was calculated by thinking: "8 times 72, 72, and 72, right across" (Hope & Sherrill, 1987, p. 102). Where skilled students used the paper-and-pencil analogue, they tended to retrieve larger

numerical equivalents than basic multiplication facts. In so doing, the number of calculation steps was minimised (Hope & Sherrill, 1987, p. 106).

Heuristic Strategies Based Upon Relational Understanding

The most powerful strategies are those that rely on relational knowledge; knowledge that allows an individual to "[know] not only what method worked but why" (Skemp, 1976, p. 23). This understanding depends on the nature of a person's propositional and procedural knowledge. One view of propositional knowledge is that it encompasses both conceptual and declarative knowledge (Sowder, 1991, p. 3). Conceptual knowledge is conjectured to be rich in relationships (Putman, Lampert, & Peterson, 1988, p. 83), part of which is the declarative knowledge of specific facts─numerical equivalents─stored in a long-term memory network (Hamann & Ashcraft, 1985, p. 52). An essential part of procedural knowledge, which consists of rules, algorithms, or procedures used to solve mathematical tasks (Putman et al., 1988, p. 83), is an individual's store of mental strategies or, in Hunter's (1977a) terms, calculative plans. Given the uniqueness of the patterns of each individual's knowledge, it is not unexpected that Vakali (1985) should comment that though "all students reported breaking down the problems into a series of elementary steps,...there were considerable individual differences with regard to the nature of the steps and the order of their execution" (p. 110). It follows that these differences are likely to be greater when heuristic strategies based on relational knowledge are employed. McIntosh (1991a, pp. 4-5) found a preponderance of place value strategies in his analysis of methods used to solve context-free addition and subtraction examples comprising two- and three-digit numbers. Strategies that use a relational understanding of place value are classified by McIntosh (1990c, p. 1) as: added/subtracted parts of second number, bridged tens/hundreds, used tens/hundreds, worked from left, or worked from right. In comparing this classification to the heuristic strategies identified by other researchers (Beishuizen, 1985, 1993; Carraher et al., 1987; Cooper et al., 1992, 1996; Flournoy, 1959; Ginsburg et al., 1981; Hamann & Ashcraft, 1985; Hazekamp, 1986; Heirdsfield, 1996; Hitch, 1978; Markovits & Sowder, 1994; Olander & Brown, 1959; Resnick,

1983; Sowder, 1992; Trafton, 1978; Vakali, 1985), those strategies that rely primarily upon place value knowledge can reasonably be placed into one of the McIntosh's (1990c, 1991a) categories without the need to use bridged tens/hundreds. Hence this strategy cluster is not included in the following analysis of heuristic strategies for calculating beyond the basic facts (see Tables 2.1 & 2.4). Strategies categorised as added/subtracted parts of second number are described by Vakali (1985), for the addition of two-digit numbers, as entailing:

The addition of the two-digit first addend to the tens of the second addend, then addition of this sum to the ones of the second addend....For 46 + 38, "46 plus 30 is 76, 76 plus 8 is 84.” (p. 111)

Beishuizen (1985, p. 252; 1993, p. 295) and Wolters, Beishuizen, Broers, and Knoppert (1990, p. 22) refer to this approach as the N-10 procedure, whereas Cooper et al. (1992, p. 108) classify it as a used tens strategy. Cooper et al. (1996, p. 149) and Heirdsfield (1996, p. 133) refer to this approach as left to right aggregation. A variation of this strategy is also reported by Cooper et al. (1992, p. 111). Rather than add the value of the tens digit first, the ones digit is used initially─for example, for 36 + 29; 36 + 9 = 45, 45 + 20 = 65. This is referred to by Beishuizen (1993, p. 295) as the u-N10 strategy, one which Cooper et al. (1996, p. 150) and Heirdsfield (1996, p. 133) term right to left aggregation. Though it appears customary to decompose the second addend, Flournoy (1959, p. 138) reported that, after a series of mental computation lessons, the majority of children were using two approaches to solving 34 + 48, one of which involved decomposing the first addend (34): 30 + 48 = 78, 78 + 4 = 82. To allow

Table 2.4 Heuristic Strategies Based Upon Relational Understanding

Add or subtract parts of the first or second number 34 + 48: 30 + 48 = 78, 78 + 4 = 82 46 + 38: 46 + 30 = 76, 76 + 8 = 84 33 - 16: 33 - 10 = 23, 23 - 6 = 17

Use fives, tens and/or hundreds

• add-up 317 - 198: 198 + 2 = 200, 200 + 100 = 300, 300 + 17 = 317; 119 51 - 34: 34 + 10 = 44, 44 + 7 = 51, 17

• decomposition 200 - 35: 200 = 100 + 100, 35 = 30 + 5, 100 - 30 = 70, 70 - 5 = 65, 165 252 - 57: 252 - 52 = 200, 200 - 5 = 195

• compensation 28 + 29: 30 + 30 = 60, 60 - 2 - 1 = 57 25 + 89: 89 + 11 = 100, 25 - 11 = 14, 100 + 14 = 114 86 - 38: 88 - 40 = 48

Work from the left

• organisation 58 + 34: 50 + 30 = 80, 8 + 4 = 12, 80 + 12 = 92 36 - 23: 30 - 20 = 10, 6 - 3 = 3, 10 + 3 = 13

• incorporation 39 + 25: 30 + 20 = 50, 50 + 9 = 59, 59 + 5 = 64 51 - 34: 50 - 30 = 20, 20 - 4 = 16, 16 + 1 = 17 43 - 26: 40 - 20 = 20, 20 + 3 = 23, 23 - 6 = 17

Work from the right

• mental analogue of standard written algorithm 58 + 34: 4 + 8 = 12, 5 + 3 = 8(0), 80 + 12 = 92 74 - 28: 14 - 8 = 6, 60 - 20 = 40, 6 + 40 = 46

• place-grouping 439 - 327: 39 - 27 = 12, 4(00) - 3(00) = 1(00), 112

Use known facts 29 - 14: 2 x 14 + 1 = 29, 1 x 14 + 1 = 15

Table 2.4 cont. Heuristic Strategies Based Upon Relational Understanding

Use factors

• general factoring 60 x 15: 60 x 3 x 5 = 300 x 3 = 900

• half-and-double 60 x 15: 30 x 30 = 900

• aliquot parts 25 x 48: 48 x (100 ÷ 4) = (48 ÷ 4) x 100 = 12 x 100 = 1200

• exponential factoring 32 x 32: (25)2 = 210 = 1024

• iterative factoring 27 x 32: 27 x 25; 27, 54, 108, 216, 432, 864

Use distributive principle

• additive distribution 64 ÷ 4: (60 ÷ 4) + (4 ÷ 4) = 15 + 1 = 16 21 x 13: (20 x 13) + (1 x 13) = 260 + 13 = 273

• subtractive distribution 8 x 999: 8 x (1000 - 1) = (8 x 1000) - (8 x 1) = 8000 - 8 = 7992

• fractional distribution 15 x 48: (10 + 5) x 48 = 10 x 48 = 480, ½ of 480 = 240, 480 + 240 = 720

• quadratic distribution 49 x 51: 50² - 1 = 2500 - 1 = 2499

Note. Adapted from: Carraher et al. (1987); Flournoy (1959); Hope (1987, 1985); McIntosh (1991a, 1990c); Olander & Brown (1959)

for this variation this cluster is referred to as: added or subtracted parts of the first or second number (see Tables 2.1 & 2.4). When this approach is used for subtracting two-digit numbers a strategy analogous to that described above for addition is used:

Subtraction of tens of the subtrahend from the minuend, subtraction of ones of the subtrahend from the above difference....For 33 - 16, "33 minus 10 is 23, 23 minus 6 is 17.” (Vakali, 1985, p. 111)

In some instances, as for addition, the ones digit of the subtrahend is subtracted initially: "For 36 - 23, 36 minus 3 is 33, 33 minus 20 is 13" (Vakali, 1985, p. 111). Whichever variation of the added or subtracted parts of the first or second number approach is used, it has the advantage of incorporating the result from each stage of the calculation into a single result, thus reducing the demands on short- term memory. Hitch (1978, p. 306) suggests that if partial results are not immediately used they will undergo rapid forgetting. Cooper et al. (1996, pp. 157- 158) found that these aggregation strategies were not as frequently used for addition and subtraction as were strategies that involved separating both numbers involved in the calculation. For subtraction, the used tens and/or hundreds approach typically relies on using the initial strategy of changing subtraction to addition so that an adding-up method can be used. This method is akin to the way in which change is often counted, except that the first step is to arrive at a multiple of ten or one hundred. Whereas Cooper et al. (1992, p. 109) refer to this approach as estimation, Olander and Brown (1959, p. 99) describe it as a rounding approach. They indicate that it often involves multiples of five or combinations of ten and five. Hence this strategy cluster is referred to as Use fives, tens and/or hundreds (see Tables 2.1 & 2.4). Cooper et al. (1996, p. 151) include this strategy, which incorporates what is described as N10 missing addend and u-N10 missing addend (Beishuizen, 1993, p. 295) approaches, within their aggregation categories referred to previously─for example, 362 - 128: 128 + 2 = 130, 130 + 70 = 200, 200 + 162 = 362, 2 + 70 + 162 = 234 (Heirdsfield & Cooper, 1995, p. 3). Sowder (1992, p. 41) indicated that the adding-up─counting up─method is one which works well for children: To find 317 - 198, count up 2 to 200, then 100 to get 300, then 17 more, or 119 in all. Olander and Brown (1959) referred to this strategy as "round [add] to a multiple of 10,” where two-digit numbers are being subtracted. A second variation of the use of ten is to "round by (add) a multiple of 10, e.g. in 51 - 34....’34 and 10, 44; 44 and 7, 51; 17'" (p. 99). These strategies, in Olander and Brown's (1959, p. 99) terms, entail rounding upward.

Another such approach, similar to that for counting change, involves arriving at a multiple of five at the first step, rather than ten as described above: For "51 - 34....’34 plus 1, 35; 35 plus 15, 50; 1, 51; 17'" (Olander & Brown, 1959, p. 99). It is suggested by Olander and Brown (1959, p. 99) that the advantage of these approaches is that the subtrahend can be forgotten following the initial step, although they require partial answers to be retained in working memory. A further advantage is that the need to borrow or regroup is eliminated, thus reducing the complexity of an operation. Strategies that involve decomposing the minuend and/or the subtrahend, so that multiples of fives, tens or hundreds may be used to simplify the calculation, are classified by Carraher et al. (1987, p. 91) as decomposition heuristics (see Tables 2.1 & 2.4). To solve 200 - 35, presented as a word problem, the following explanation was provided:

If it were thirty, then the result would be seventy. But it's sixty-five; one hundred sixty-five. (The 35 was decomposed into 30 and 5, a procedure that allows the child to operate initially with only hundreds and tens; the units were taken into account afterward. The 200 was likewise decomposed into 100 and 100; one 100 was stored while the other was used in the computation procedure.) (Carraher et al., 1987, p. 91)

Carraher et al. (1987, p. 92) point out that this approach for subtraction replaces the digits in one or more places with zero. The use of tens and hundreds is sought because round numbers, numbers that end in zero, are more likely to be related to known number facts─100 - 30 - 5, for example. Further, this procedure reduces the demands on short-term memory as the need to operate on hundreds, tens and units simultaneously is avoided. Where numbers without any zeros are involved, the search for round numbers becomes particularly apparent in some instances. To subtract 57 from 252 a child explained:

Take away fifty-two, that's two hundred, and five to take away, that's one hundred and ninety-five. (The children decomposed 252 into 200 and 52; 57 was decomposed into 52 + 5; removing both 52s, there remained another five to take away from 200.) (Carraher et al., 1987, p. 92)

A number of researchers (Cooper et al., 1996; Heirdsfield, 1996; B. J. Reys et al., 1993; Sowder, 1992; Trafton, 1978) have observed variations to these approaches which involve final compensations for initial rounding (see Tables 2.1 & 2.4), a holistic strategy (Cooper et al., 1996, p. 150). Such an approach is confusing for some children who find it difficult to determine in which direction the compensation should occur (Sowder, 1992, p. 41). However, for children proficient with such strategies, the complexity of the calculations is reduced by being able to calculate initially with multiples of fives, tens or hundreds. Two compensatory approaches to calculating 28 + 29 reported by B. J. Reys et al. (1993, p. 314) are:

• 30 + 30 is 60, so just take off 2 and 1 more. 60, 58, 57. • 25 + 25 is 50, plus 3 more is 53 and 4 more is 57.

In contrast to these undoing approaches, adjusting each number before calculating by levelling (Heirdsfield, 1996, p. 134) circumvents the need for final adjustments. This approach is exemplified by the following dialogue presented by McIntosh (1990c) to explain his used tens/hundreds classification:

Interviewer: What's 25 add 89 Child: 114...I took 89...I knew I needed 11 more to make 100 so I took 11 away...from 25...left me with 14...then I added 14 extra up to 100. Interviewer: How did you know you needed 11 more for 89? Child: Well 90 to 100 is 10 and 1 more. (p. 5)

Unlike unskilled mental calculators who tend to use a right-to-left approach, skilled mental calculators usually work from the left (Hope & Sherrill, 1987, p. 106). Although Hope and Sherrill's (1987) study involved multiplication, other researchers (Beishuizen, 1985, 1993; Cooper et al., 1992; Flournoy, 1954, 1959; Ginsburg et al., 1981; Heirdsfield, 1996; Hitch, 1977, 1978; McIntosh, 1990c; Olander & Brown, 1959; Vakali, 1985) report a common usage of strategies that involve working from the left (see Tables 2.1 & 2.4) for addition and subtraction. Observations from school practice and remedial teaching in The Netherlands indicate that a left-to-right

approach is one that children spontaneously use (Beishuizen, 1985, p. 253). A common form of this approach, categorised as left to right separated place value by Cooper et al. (1996, p. 155) and Heirdsfield (1996, p. 132), and organisation (Vakali, 1985, p. 111) in this study, is recounted by McIntosh (1990c):

Interviewer: 58 + 34 Child: 92...I added the 50 out of 58 and the 30 out of the 34 together and came up with 80, and then I added 8 and 4 and then got 12, so I added 12 onto [80]. (p. 57)

Beishuizen (1985, pp. 253, 256) and Wolters et al. (1990, p. 28) indicate that this strategy─the 10-10 procedure─produces more errors than the added/subtracted parts of second number (N-10) strategy discussed previously, a finding supported by Heirdsfield (1996, p. 133). In using the N-10 procedure fewer decisions have to be made as only one of the numbers is decomposed, thus resulting in fewer calculative steps. Arising out of an analysis of approaches to teaching addition in Grade 2 using concrete materials and children's spontaneous methods for addition and subtraction, Beishuizen (1985, p. 256) suggests that the added or subtracted parts of second number approach may be viewed as a more learned and algorithmic procedure than working from the left which is a more invented and heuristic procedure. However, the error rate for the latter strategy may be due, at least for Queensland children, to its being infrequently practised in classrooms (Heirdsfield, 1996, p. 133). Hope (1985, p. 359) indicates that there is some evidence to suggest that a left-to-right approach is less demanding on short-term memory than one that involves working from the right. The memory load, using a working from the left approach, can be reduced by progressively incorporating each interim calculation into a single result, a characteristic of an efficient mental strategy (Hope & Sherrill, 1987, p. 108). For example, "for 39 + 25, ‘30 plus 20 is 50, 50 + 9 is 59, 59 plus 5 is 64'" (Vakali, 1985, p. 99). This approach, designated the 10-10-N procedure by Beishuizen (1985), regrouping by Ginsburg (1982, p. 196), group by tens and ones: cumulating sums by R. E. Reys et al. (1995), and incorporation in this analysis, does not require an organisation stage (see Table 2.4) in which two sums are added to form the answer as the final step of the calculation (Vakali, 1985, p. 111).

With respect to subtraction, difficulties arise when the need to regroup occurs when using an approach analogous to that of McIntosh's (1990c, p. 57) example for addition outlined previously, which relies on subtracting the hundreds/tens then the ones. To overcome this need, an approach similar to the incorporation procedure for addition may be used (see Tables 2.1 & 2.4). Two variations of this method, referred to by Olander and Brown (1959, p. 99) as compensatory rounding, have been observed. The difference between the strategies pertains to the order in which the ones digits are brought into the calculation. The following examples characterise these approaches:

• For 51 - 34, "50 minus 30, 20; minus 4, 16; plus 1, 17" (Olander & Brown, 1959, p. 99) • For 43 - 26, "40 minus 20 is 20, 20 plus 3 is 23, 23 minus 6 is 17.” (Vakali, 1985, p. 111)

As discussed previously, unskilled mental calculators tend to use right-to-left methods which are essentially mental analogues of the standard written algorithms. Such methods are categorised as right to left separated place value by Cooper et al. (1996, p. 155) and Heirdsfield (1996, p. 132)─for example for 74 - 28: 14 - 8 = 6, 60 - 20 = 40, 46 (Heirdsfield & Cooper, 1995, p. 3). In some instances, however, work from the right strategies (see Tables 2.1 & 2.4) reflect a relational understanding of place value knowledge. However, in many cases they tend to lack a clear distinction from the mental application of the standard written algorithms. Nonetheless, an example of the use of relational place value knowledge is given by McIntosh (1990c):

Interviewer: 58 + 34. How did you get 92 as an answer? Child: Well I took the 4 and the 8 and added them together which went to 12, and then I took the 5 and the 3 which equalled 80 and added on the 12, which equalled 92. (p. 6)

This approach, classified as transition by Markovits and Sowder (1994, p. 14), is indicative of Murray and Olivier's (1989, pp. 6-7) Level 4 place value knowledge

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which allows a child to use ten as an iterable unit. It can be assumed that this child understands, implicitly at least, that 5 tens were added to 3 tens to give 8 tens (80). A strategy, which relies on an in-depth understanding of numeration to decompose the numbers so that the calculation is simplified, is referred to by Olander and Brown (1959, p. 99) as a place-grouping approach (see Tables 2.1 & 2.4). This approach is reflective of Jensen's (1990) observations of the methods used by Shakuntala Devi. Devi explained that when given a large number it "automatically ‘falls apart' in its own way, and the correct answer simply ‘falls out'. Each number uniquely dictates its own solution" (Jensen, 1990, p. 262). Five high achievers are reported by Olander and Brown (1959) as having used a place- grouping approach:

For example, in 439 - 327, when a pupil thought "39 minus 27, 12; 4 minus 3, 1; 112," he broke up the 439 into 400 and 39, and 327 into 300 and 27. (p. 99)

A complex variation of this strategy, which involves compensating for not regrouping during the initial subtraction, is also presented by Olander and Brown (1959): "6000 - 2249...."100 minus 49, 51; 60 minus 22, 38; minus 1, 37; 3751" (p. 99). In proposing a classification system for mental computation strategies, McIntosh (1990c, p. 1) distinguishes between strategies based upon the relational use of place value concepts and the use of other relational knowledge. Such a distinction, however, is not always clear-cut and may be considered somewhat artificial, given that place value knowledge is simply one aspect of an individual's conceptual knowledge. Mental computation strategies that rely on mathematical concepts and principles other than those related to place value, or on numerical equivalents beyond the basic facts, are likely to be ones used by those who demonstrate greater computational skill. With respect to the mental multiplication strategies used by skilled mental calculators identified by Hope and Sherrill (1987), McIntosh (1990b) remarked that "many of the students in this study were clearly at a more advanced level than those considered elsewhere in this paper" (p. 19), an observation relevant to the strategies hitherto discussed in this analysis of heuristic strategies.

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When using relational knowledge, other than that primarily based on place value, to calculate exact answers mentally, three categories are evident in the research literature. These are: (a) a number-relations approach (Olander & Brown, 1959, p. 99), and strategies based on (b) factoring and (c) distribution (Hope, 1985, pp. 358-366; Hope, 1987 p. 334; Hope & Sherrill, 1987, pp. 102-103). Although a number-relations approach is one appropriate for all four operations, factoring and distribution are ones more suited to multiplication and division. Based on an analysis of reports of strategies used by expert mental calculators, Hope (1985, pp. 358, 362) concludes that factoring is their favourite method of calculating, with multiplication their favourite operation. A number-relations approach relies on relating a calculation to another to which the answer is known (see Table 2.4)─that is, to use known facts (French, 1987, p. 39; McIntosh, 1991a, p. 2; Olander & Brown, 1959, p. 99). For example, to subtract 14 from 29, one method is to relate the operation to known multiplication facts: "two 14s plus 1, 29; one 14 plus 1, 15" (Olander & Brown, 1959, p. 99). The categories for classifying mental strategies are not mutually exclusive and the classification of particular strategies can only be achieved with any certainty if based on verbal reports. Of relevance to relating a calculation to a known fact, French (1987, p. 39) has suggested that many mental methods depend on relating a particular calculation to a simpler one. For example, to work out 80 x 7, it can be related to 10 x 7. In effect, the calculation becomes 10 x 7 x 8 and hence could also be classified as a factoring approach. Factoring involves transforming one or more factors into a series of products or quotients (Hope & Sherrill, 1987, p. 102). Hope (1985, pp. 362-266; 1987, p. 334) and Hope and Sherrill (1987, pp. 102-103) have identified five categories of factoring strategies into which strategies identified by other researchers (Carraher et al., 1987; Flournoy, 1954; B. J. Reys, 1985) are able to be placed. The five categories of mental strategies that involve factoring are: general factoring, half-and-double, aliquot parts, exponential factoring and iterative factoring (see Tables 2.1 & 2.4). General factoring involves factoring "one or more of the factors before applying the associative law for multiplication" (Hope & Sherrill, 1987, p. 103). Flournoy (1954) recounts one method used by an intermediate grade student to calculate 11 x 16: "Half of 16 is 8; 11 x 8 = 88. I doubled the 88 to get my answer of 176" (p. 149). Barbara Reys (1985), cited in a draft of B. J. Reys and Barger (1994),

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describes a similar process for finding the product of 60 x 15: "It's the same as 60 x 3 x 5 or 300 x 3 or 900.” In some instances, other strategies may be used to calculate intermediate steps. For example, Hope and Sherrill (1987, p. 103) noted that for 25 x 48 one senior high school student calculated "5 times 48 is 240, and 5 times 240 is 1200"; 25 had been factored into 5 x 5 and additive distribution applied to ascertain the intermediate calculations, 5 x 48 and 5 x 240: 5 x 48 = (5 x 40) + (5 x 8 ) and 5 x 240 = (5 x 200) + 5 x 40). This approach was classified by Hope and Sherrill (1987) "as general factoring rather than distribution because the computation was transformed initially into a series of products rather than sums" (p. 103). A strategy for solving 100 ÷ 4, considered by Carraher et al. (1987) to be a repeated-grouping heuristic, can also be classified as a general factoring strategy. A child explained that:

One hundred divided by four is twenty-five. Divide by two, that's fifty. Then divide again by two, that's twenty-five. (There was a factoring operation, two successive divisions by 2 to replace the given division by 4.) (p. 93)

The remaining types of factoring strategies can be viewed as special cases of general factoring as they have significant distinguishing characteristics that allow them to be regarded as separate categories (Hope & Sherrill, 1987, p. 103). Although a basic fact example is provided by McIntosh (1990c, p. 6) to illustrate doubling/halving as a use of other relational knowledge, this strategy is one of value when calculating beyond the basic facts. Half-and-double (see Tables 2.1 & 2.4) is particularly useful where at least one of the factors in a multiplication task is a multiple of two. To apply this strategy, the factor that is a multiple of two is halved and the other is doubled to compensate. This process continues until the answer can be determined by applying another strategy or by recall (Hope, 1987, p. 334). For example, to calculate 60 x 15 a seventh-grade child said: "It's the same as 30 x 30 (halve one factor, double the other) or 900" (B. J. Reys, 1985, cited in a draft of Reys & Barger, 1992, p. 13). Hope (1985, p. 334; 1987, pp. 364-365) and Hope and Sherrill (1987, p. 103) have defined a factoring strategy, which involves transforming one factor into a quotient, as factoring through the use of aliquot parts. "This strategy [can be]

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applied...to those computations where one factor (f) [is] a factor of a power of 10 (p) and the remaining factor [is] a multiple of the quotient p/f " (Hope & Sherrill, 1987, p. 103). Hope (1987) reports that Charlene, an expert 13 year-old mental calculator, multiplied 25 and 48 by reasoning "48 x 100/4 = 48/4 x 100 = 12 x 100 = 1200" (p. 334). This approach is particularly useful in instances where one of the factors is 25, 50 or 125, as the reference numbers are 100 (25 x 4), 100 (50 x 2) and 1000 (125 x 8), respectively (Hope, 1985, p. 364)─a useful method to simplify mental calculations. Strategies that rely on a knowledge of exponential arithmetic have been identified by Hope and Sherrill (1987, p. 103), and Hope (1985, pp. 365-368), namely, exponential factoring and iterative factoring, respectively (see Tables 2.1 & 2.4). Although these approaches are similar, Hope and Sherrill (1987) define exponential factoring as a strategy that relies on the application of an exponential rule. For example, to solve "32 x 32, one student reasoned, ‘I solved it by thinking powers of 2. 32 is 2 to the fifth, and squaring this is 2 to the tenth, which I just know is 1024'" (p. 103). Although iterative factoring also involves a knowledge of exponential arithmetic, this approach does not entail the use of the rules for operating with exponents. For example, 27 multiplied by 32 can be calculated by doubling the factor 27, 5 times: 54, 108, 216, 432, 864 (Hope, 1985, p. 366). However, this approach is rather cumbersome when finding the product of larger factors, given the number of iterations required. It is therefore useful primarily in situations where powers of two or three are involved. Whereas strategies that involve factoring entail the transformation of a calculation into series of products, other strategies depend on the calculation being transformed into a series of sums or differences. Hope and Sherrill (1987, p. 102) have identified four approaches that are based on the distributive properties of multiplication and division: additive distribution, subtractive distribution, fractional distribution, and quadratic distribution (see Tables 2.1 & 2.4). By convention, division is a worked from the left procedure. The most significant digits are considered first. A convenient mental approach to calculating 64 ÷ 4 is reported by Flournoy (1954): "I know that 60 ÷ 4 is 15 and one more 4 is 64 makes 16" (p. 149). This approach can be represented as: (60 + 4) ÷ 4 = (60 ÷ 4) +

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(4 ÷ 4) = 15 + 1 = 16. It relies on a knowledge, implicitly or explicitly, of the distributive property of division over addition. Hope (1985, pp. 358-359; 1987, p. 334) and Hope and Sherrill (1987, p. 102) refer to this procedure as additive distribution and is "considered the ‘calculative drafthorse' of [expert] mental calculators because it is suited to a wide range of calculation tasks" (Hope, 1985, p. 358). This strategy may involve the use of place value knowledge, as in Flournoy's (1954, p. 149) example, or depend on an individual's knowledge of numerical equivalents. For example, 174 ÷ 3 may be calculated by converting 174 to 150 + 24 so that the calculation becomes: (150 ÷ 3) + (24 ÷ 3) = 50 + 8 = 58 (Flournoy, 1959, p. 138). Although additive distribution strategies are usually implemented in a form that requires a final organisation stage, expert mental calculators often successively add each partial product to produce a running total, thus reducing the load on short-term memory. For example, 8 x 4211 is calculated by "8 times 4000 is 32 000; 8 times 200 is 1600; so it's 33 600. 8 times 11 is 88; so the answer is 33 688" (Hope & Sherrill, 1987, p. 102). Where one factor is near to a multiple of a power of 10, subtractive distribution is often applied (Hope, 1985, p. 360). To use the distributive principle of multiplication over subtraction, Hope and Sherrill (1987, p. 102) report that some skilled senior high school students use the following method for calculating 8 x 999: 8 x (1000 - 1) = 8000 - 8 = 7992. This strategy is one of two approaches to mental multiplication identified by Trafton (1978, p. 209) as ones that can be learned by students, the other being an additive distribution. Fractional distribution (see Tables 2.1 & 2.4) is an approach that is often "applied to those tasks in which at least one factor [contains] 5 as a unit digit" (Hope & Sherrill, 1987, p. 102). This allows partial products to be calculated from those that have already been determined. Hope and Sherrill (1987) indicate that one method for calculating 15 x 48 is: "10 times 48, 480, and half of 480 is 240; so it's 720" (p. 102). A common form of quadratic distribution (see Tables 2.1 & 2.4) is to calculate products through distributing by difference of squares. Hope (1985, p. 362) notes that even young children of average ability can stumble onto this method as evidenced by McGartland's (1980, cited by Hope, 1985, p. 362) finding that a fourth-

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grade child calculated 7 x 9 by squaring 8 and subtracting 1. This method, irrespective of number size but dependent upon an individual's store of number facts, is most useful when the two numbers to be multiplied are of a similar magnitude (Hope, 1985, p. 362). Hope (1987) reported that Charlene calculated 49 x 51 "by reasoning 50² - 1 = 2500 - 1 = 2499" (p. 334). Algebraically, this method is represented by: (x + y)(x - y) = x² - y², where x = 50 and y = 1. Other forms of quadratic distribution have been reported as being used by expert mental calculators to square large multi-digit numbers. These are based on the algebraic equivalences (a + b)² = a² + 2ab + b² and (a - b)² = a² - 2ab + b² (Hope, 1985, p. 361). Scripture (1891, cited in Hope, 1985, p. 361) noted that Henri Mondeux, a famous nineteenth-century lightning calculator, calculated the square of 2419 through an implicit use of the binomial theorem: 24² + 2 x (24 x 19) + 19². This procedure involves an initial use of the place-grouping approach outlined above, with mental adjustments being made to the partial products to account for the value of the 24 being 2400. Alexander Aitken, in Gardner's (1977) view "perhaps the best all-round mental calculator of recent times" (p. 70), used the algebraic identity a² = (a + b) x (a - b) + b² to square numbers. For example, 777² was calculated by transforming the calculation into [(777 + 23) x (777 - 23)] + 23², thus allowing him to use his vast store numerical equivalents, particularly of squares. Gardner (1977) points out that "the b is chosen to be fairly small and such that either (a + b) or (a - b) is a number ending in one or more zeros" (p. 75). Aitken (cited in Gardner, 1977) has pointed out that "the first step in any complicated calculation...is to decide in a flash on the best strategy" (pp. 73-74). This requires a knowledge of a range of computational strategies, such as those discussed in this section, combined with an ability to recall numerical equivalents and an ability to remember the numbers that are involved in the intermediate stages of a calculation. Strength in these characteristics distinguishes experts from non- experts (Hope, 1985, p. 358). These, together with other traits of proficient mental calculators, are to be given further consideration in Section

2.7.5 Short-term and Long-Term Memory Components of Mental Computation

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Whatever the strategies used to undertake mental calculations, Hunter (1978, p. 339) suggests that three types of demands on memory can be delineated. These are: memory for calculative method, memory for numerical equivalents, and memory for interrupted working. Although the first two place demands on an individual's long-term memory, the latter places demands on short-term memory capacity. People who are proficient at calculating exact answers mentally are able to draw upon greater than average resources in long-term memory, specifically, declarative and procedural knowledge related to stored facts─numerical equivalents─and heuristic strategies, respectively (see Figure 2.3). This enables them to devise efficient methods for approaching a problem and to carry out a

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PROBLEM TASK ENVIRONMENT

SHORT-TERM MEMORY

WORKING STORAGE EXECUTIVE PROCESSOR

Information about problem Information from Working Storage transformed using knowledge stored in Long-term Information about answer Memory

ANSWER

LONG-TERM MEMORY

CONCEPTUAL KNOWLEDGE

Declarative Knowledge (Network of stored facts)

PROCEDURAL KNOWLEDGE (Network of known strategies)

METACOGNITIVE KNOWLEDGE (Beliefs about ability to compute mentally)

REAL WORLD KNOWLEDGE (Prior knowledge about the Problem Task Environment)

Figure 2.3. A view of memory processes for computing mentally (Adapted from: Ashcraft, 1982, p. 229; Hitch, 1978, p. 320; Silver, 1987, p. 37)

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calculation in a rapid, but orderly, sequence of steps (Hunter, 1978, p. 343). In contrast, even expert mental calculators exhibit relatively slender resources with which to meet the demands of interrupted working (Hunter, 1978, p. 343; Jensen, 1990, p. 270). They tend to "shift the burden of calculation onto long-term memory while, at the same time, minimising their reliance on temporary memory" (Hunter, 1978, p. 340). Information processing models of mental computation are based on the assumption that human beings have only a limited capacity for processing concurrent activities. Consequently many tasks need to be performed as a series of steps rather than being accomplished all at once (Hunter, 1977a, p. 37). This implies that information concerning the problem task environment, "the structure of facts, concepts, and the relationships among them that constitute the problem" (Silver, 1987, p. 41), needs to be stored with partial answers derived during computation. Hitch's (1978, p. 320) hypothesised model for mental calculation of exact answers suggests that short-term memory consists of a working storage component and an executive processor (see Figure 2.3). The executive processor is responsible for the control of mental processes (Hitch, 1978, p. 303). It is also hypothesised to be the component that stores and retrieves information from long- term memory. Fayol, Abdi, and Gombert (1987, p. 199) observe that, as the storage load in short-term memory increases, the space devoted to reasoning is reduced. Consequently the number of errors made during mental computation increases. Hence, children need to be encouraged to develop proficiency with strategies that reduce the load on short-term memory, those in which partial results are incorporated into a single result (see Table 2.4). Long-term memory contains an individual's store of mathematical knowledge (Silver, 1987, p. 42). Hiebert and Lefevre (1986) distinguish between knowledge of mathematical concepts─conceptual knowledge─and knowledge of the formal language of mathematics and the strategies used to undertake mathematical tasks─procedural knowledge. Conceptual knowledge, of which declarative knowledge can be considered a part (see Figure 2.3), is viewed as an interrelated web of knowledge, with the links being as prominent as the distinct pieces of information (Hiebert & Lefevre, 1986, pp. 3-4). Each piece of information is considered to be part of conceptual knowledge only if its relationship to other pieces of information is recognised.

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In instances where strong relationships have been built between conceptual and procedural knowledge, recalling and effectively using particular computational procedures is enhanced. Recall is aided by strategies having been "stored as part of a network of information, glued together with semantic relationships...[with] the retrieval process...triggered by several external and internal cues" (Hiebert & Lefevre, 1986, p. 11). Conceptual knowledge and its links to procedural knowledge can enhance problem representations, monitor strategy operation, and promote transfer to structurally similar problems (Hiebert & Lefevre, 1986, p. 11). Representing a problem conceptually allows reasoning to focus on the quantities involved rather than on their symbol representations (Greeno, 1983, p. 228), a feature of out-of-school mental computation (Carraher et al., 1985, p. 28; Reed & Lave, 1981, p. 442). Although the demonstrated links between understanding and the improvement of algorithmic performance remain tenuous, Resnick (1984, cited in Nesher, 1986, p. 6) has suggested that such links depend upon how children represent a problem to themselves─whether they focus on manipulating the quantities involved or whether they focus the symbols used to represent them. Conceptual knowledge can therefore be thought of as a control structure for procedural knowledge. Such a view implies that conceptual knowledge, if linked to particular strategies, can be used to monitor strategy selection and execution, as well as to judge the reasonableness of the solution obtained (Hiebert & Lefevre, 1986, p. 12). Sowder (1994, p. 143) notes that the protocols of expert mental calculators illustrate the self-monitoring and self-regulation of the procedures employed during computation. It is these metaprocesses that characterise proficient mental calculators. Further, in Sowder's (1994, p. 143) view, flexibility in the selection of mental strategies is a result of this self-monitoring process. Aspects of both short-term memory and long-term memory exercise control over mental computation. Metacognitive processes are believed to involve two separate components: "(a) An awareness of the skills, strategies, and resources needed to perform a task effectively─knowing what to do; and (b) the ability to use self-regulatory mechanisms to ensure the successful completion of the task─knowing how and when to do what”(Woolfolk, 1987, p. 258). However, knowing what to do is not solely dependent upon relevant conceptual and procedural knowledge. Equally important is the metacognitive and real-world knowledge germane to the particular context that is also stored in long-term memory

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(Silver, 1987, p. 37), as depicted in Figure 2.3. In view of the findings that mental strategies tend to be adapted to the specific characteristics of the numerical situations, particularly in out-of-school settings (Carraher et al., 1985, p. 25; Murtaugh, 1985, p. 191; Scribner, 1984, p. 38), real-world knowledge, which can be applied to assist in understanding the problem task environment, has particular relevance for mental computation. With respect to the Woolfolk's (1987, p. 258) first component, metacognitive knowledge held in long-term memory concerns an individual's beliefs about mathematics and opinions of their ability to perform particular mathematical procedures in particular contexts (Silver, 1987, p. 42), affective components of mental computation discussed previously (see Table 2.1). Metacognition also refers to what Silver (1987, p. 37) designates meta-level processes, namely, the cognitive monitoring processes of self-monitoring, regulation and evaluation of the cognitive activity (Silver, 1987, p. 49), processes related to knowing how and knowing when─Woolfolk's (1987) second component. These processes are thought to be part of the executive control processes in short-term memory that operate on and control the flow of information through the memory systems (Woolfolk, 1987, p. 259). Based on chronometric evidence in relation to operations with basic facts, Ashcraft (1982) has postulated a fact retrieval model for mental computation. According to this model, older children and adults store an increasingly large number of numerical equivalents in long-term memory. The growth in declarative knowledge allows a gradual shift from "an initial reliance on procedural knowledge and methods such as counting...to retrieval from a network representation of basic facts" (Ashcraft, 1982, p. 213), a view in accordance with that of Svenson and Sjoberg (1982, p. 91) discussed previously. The fact retrieval model assumes that solution speed increases with an increased involvement of declarative knowledge (Ashcraft, 1982, p. 233). This assumption has its parallel in the cognitive characteristics of individuals proficient with mental computation beyond the basic facts, people who have been found to exhibit a large store of numerical equivalents rich in relationships. Ashcraft's (1982) model holds that all procedural processes are slow, an assumption that Baroody (1983) believes to be incorrect. In his view, efficient number fact knowledge is due not only to an increase in declarative knowledge, but

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more importantly to the development of procedural knowledge. "As the child learns rules, heuristics, and principles, these supplant less efficient procedural processes such as informal counting algorithms. Moreover, as these rules, heuristics, and principles become more secure and interconnected, their use becomes more automatic" (Baroody, 1983, p. 227). Hence, principled procedural processes are believed to be the key factor in the efficient production of number facts, even for adults, with the most important developmental shift being away from inefficient strategies that rely on counting to the use of spontaneous procedural knowledge (Baroody, 1983, p. 227). Whether this view has legitimacy when considering mental computation beyond the basic facts is yet to be determined. With respect to basic fact production, Baroody (1984, p. 152) cautions that whatever the relative importance that should be attributed to declarative and procedural knowledge it is too early to conclude that any one model of mental computation is clearly superior. Nonetheless, focusing on such knowledge, which these models entail, provides a useful framework for analysing the role of long-term memory components in mentally computing beyond the basic facts.

2.8 Characteristics of Proficient Mental Calculators

Little research has been undertaken by contemporary researchers into the cognitive processes involved in proficient mental computation (Hope, 1985, p. 331; McIntosh, 1990b, p. 1). That which is available indicates that significant differences exist in the nature of the strategies used by people who are competent with mental computation and those who are not so competent. Expert mental computers tend to use advanced arithmetic and algebraic techniques (B. J. Reys, 1985, p. 43), as evidenced by the heuristic strategies outlined previously. These strategies rely on a relational understanding of the properties of numbers. This usage is in contrast to that of low- and middle-ability mental calculators who tend to rely on an instrumental use of mental forms of the written algorithms (B. J. Reys, 1991, p. 4). Possibly through possessing knowledge that is richer in relationships, proficient mental calculators are able to comprehend the mathematical structure of a situation and rapidly classify the problem type so as to employ a strategy appropriate for that

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situation (Olander & Brown, 1959, p. 100; R. E. Reys, 1992, p. 63). High achievers, therefore, tend to use a variety of methods for computing mentally (Olander & Brown, 1959, p. 99). Based on a study of the strategies used by less and more competent children in Years 2 to 7, McIntosh (1991a, p. 3) observed that generally the latter start to use particular strategies earlier than the less able and in so doing make more active use of place value strategies (see Tables 2.3 & 2.4). Additionally, they have superior number fact knowledge. Although the non-proficient gradually increase their repertoire of strategies, they tend to rely more on elementary counting (see Table 2.2) and the use of external aids such as fingers. Strategies based on a relational use of place value, such as use fives, tens and/or hundreds and work from the left (see Table 2.4), were used as much by competent children in Years 2 to 4 as by the least competent in Years 5 to 7 (McIntosh, 1991a, p. 3). Except for children in Year 7, who often were able to answer a question by recall, those who displayed proficiency tended "to manipulate numbers and exploit the place value aspects of them in a dynamic way" (McIntosh, 1991a, p. 3), a finding supported by Olander and Brown (1959). Students classified as high achievers were more likely to use a place-grouping approach (see Table 2.4) than were those classified as low achievers (Olander & Brown, 1959, p. 98). The differences between the mental strategies used by proficient and non- proficient mental calculators reflect different uses of long-term and short-term memory systems (Hitch, 1977, p. 337; Hunter, 1978, p. 343). With respect to Shakuntala Devi, a skilled adult mental calculator, Jensen (1990, p. 270) suggests that an abnormally efficient encoding and retrieval system for long-term memory largely overcomes the basic information processing limitations of short-term memory. However, as discussed previously, there is considerable controversy in the chronometric literature (Ashcraft, 1982, 1983; Baroody, 1983, 1984) as to the way in which this information is encoded, particularly with respect to the relative importance of the role of declarative and procedural knowledge in the calculative process. Fayol et al. (1987, pp. 187-188) suggest that, for verbally presented arithmetic problems, arriving at a solution implies the use of three kinds of mental operations. These are: (a) storing of information as a whole until the task is understood, (b) searching long-term memory for appropriate schemata to organise and solve the

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problem, and (c) applying the problem solving process to the data as well as controlling its execution. This implies that arriving at a solution relies on the use of particular computational strategies or on prestored knowledge in long-term memory (Fayol et al., 1987, p. 188). Storing information during processing, together with the processing itself, places demands on working memory. However, there appears to be only a weak relationship between mental computation performance and short-term memory capacity (Hope & Sherrill, 1987, p. 110; Olander & Brown, 1959, p. 98). Hope and Sherrill (1987) suggest that "through the judicious selection of a calculative strategy, a skilled calculator can get by with fewer short-term memory resources than the selection of more inefficient strategies would necessitate" (p. 110). Individual differences in mental calculation can therefore be conjectured to depend upon differences in the choice of computational strategy, which are dependent upon a knowledge of useful numerical equivalents, and on the effects that such a choice has on the capacity to process arithmetical data (Hope, 1985, p. 358; Hope & Sherrill, 1987, p. 99).

2.8.1 Origins of the Ability to Compute Mentally

Most great adult mental calculators have indicated that their methods of calculation were largely self taught and arose from playful explorations of number patterns of interest to them. Hope (1986b, p. 73) suggests that their fascination with such patterns is the likely key to their remarkable calculative skill. The propensity to explore arithmetic patterns in their everyday environments characterises skilled student and adult mental calculators. Hope (1987, p. 337) reports that Charlene, a 13 year-old skilled in mental computation, attributes her largely self-taught ability to her practising and playing with numbers. Hunter (1962, p. 252) concludes that, while genetic predispositions may have been an important factor, Professor Aitken’s calculative prowess was largely due to prolonged and intensive practice, begun in his early teens. Proficient mental calculators appear to be more highly motivated and tend to enjoy their work more than children who are less proficient (Olander & Brown, 1959, p. 100). Those skilled at mental calculation are "driven by a passion for numbers" (Hope, 1985, p. 372). Shakuntala Devi attributes her career as a stage calculator to

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her love of numbers. As a child, "numbers and arithmetic were her favorite ‘toys'" (Jensen, 1990, pp. 269-270). In common with other skilled mental calculators, Devi perceives numbers in everyday situations in idiosyncratic ways─for example, interpreting the number 720 on a car's license plate as factorial six. "Each [number] evokes many associations and transformations, some more interesting than others" (Jensen, 1990, p. 271). Hunter (1977a, p. 38) has suggested that an increase in calculative ability, by whatever means, necessitates the acquisition of a system of calculative strategies. These are dependent upon the particular way in which numbers are viewed and therefore on the store of declarative knowledge in long-term memory. This knowledge "builds up, piece by piece, as a byproduct of its usefulness in pursuits which interest the individual. There is relatively little deliberate memorization bleakly undertaken for its own sake" (Hunter, 1978, p. 343). Different skilled mental calculators employ markedly different calculative systems. One characteristic of these systems is the direction in which the digits in the solution are usually produced─in a left-to-right or right-to-left order (Hunter, 1977a, p. 39), as presented in Table 2.4. Hunter (1977a, pp. 38-39) identifies two general properties of the systems of strategies used by proficient mental calculators. Firstly, solving problems in terms of the system saves effort. Problems can be solved in a shorter time. Additionally, the solving of problems that were previously too difficult can then be achieved. Secondly, effort is required to acquire this energy efficient system. The system of calculative plans or strategies is developed through experience with computing mentally (Hunter, 1977a, p. 39). The same degree of constant practice, undivided attention and knowledge, essential to the development of proficiency in other fields of human endeavour, is required (Hope, 1985, p. 372). With lack of practice, skill deteriorates (Hunter, 1977a, p. 40). Charlene's parents have suggested that she "was faster and more accurate in calculating mentally when she was younger because ‘she used to play and practice with numbers more [often]'" (Hope, 1987, p. 337). Some researchers have achieved some success in improving mental computation ability through extended periods of practice. College students, who were given three hundred hours of systematic practice with mental multiplication strategies used by expert mental calculators, reduced their average solution time for multiplying five- by two-digit numbers from around 130 to 30 seconds (Staszewski,

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1988, cited in Jensen, 1990, p. 272). However, as Jensen (1990, p. 272) notes, such students still do not demonstrate the skill of a lightning calculator such as Devi. For calculating prodigies, "it seems necessary to posit some initial, probably innate, advantage on which practice can merely capitalize" (Jensen, 1990, p. 273). Individuals with an exceptional ability for calculating exact answers mentally are characterised by diversity and widely different accomplishments. Hunter (1977a, p. 39) notes that such an ability has been evident in children, illiterates and gifted mathematicians, as well in those who have mental deficiencies. Some excel in a limited range of problems, while others are able to solve quite rapidly a wide range of numerical problems using a variety of ingenious techniques. Others attack reasonably simple problems by slow and conventional methods, but are remarkable for their ability to calculate without external aids. For some, average accomplishments may be considered exceptional, given their limited ability in other areas. With respect to Shakuntala Devi, "to all appearances...a perfectly normal and charming lady" (Jensen, 1990, p. 263), her exceptional skill may be due to her being able, as a child, to function in a "high fidelity attentional mode" (Jensen, 1990, p. 273) when calculating mentally. However, the source of such exceptional skill remains uncertain. It may prove to have more to do with motivation than primarily with attention or ability. As Jensen (1990, p. 273) points out, the biographies of exceptional performers in many fields record an extreme devotion to practice. From reports concerning skilled mental calculators, Hunter (1977a, p. 40) suggests that two general conclusions can be drawn: (a) Individuals gradually develop distinctive calculative systems, which arise from their experiences with numerical situations, and which facilitate solving particular types of problems with less effort; and (b) the limitations of short-term memory place restrictions on the accomplishments of mental calculators.

2.8.2 Memory for Numerical Equivalents

As suggested previously, in contrast to the average adult, expert mental calculators have an immense store of numerical equivalents which are drawn upon with speed and accuracy (Hunter, 1978, p. 370). In 1907, Mitchell (cited in Hope,

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1985, p. 367) concluded that an individual's vast store of declarative knowledge could only be built up through a deliberate attempt to commit extended tables to memory. This, however, is possibly an impossible task, given that many lightning calculators give the impression that they know the multiplication facts to 100 by 100 and beyond (Gardner, 1977, p. 73). As Hope (1985, p. 367) points out, such a view is in conflict with evidence suggesting that large amounts of information can be memorised unconsciously by those who are highly interested in a subject, a characteristic of skilled mental calculators. Further, to undertake deliberate memorisation for its own sake would most likely result in boredom and discouragement (Hunter, 1978, p. 343). However, given Baroody's (1983) proposition that principled procedural knowledge plays a key role in the production of number facts, it is possible that a similar process may be used by expert mental calculators to produce some of the vast store of numerical equivalents which they appear to possess. With respect to Aitken, Hunter (1962) notes that it was sometimes difficult to distinguish between recalling and thinking on the part of the mental calculator: "When he [attained] the answer to a problem with rapidity, good timing and a feeling of ‘all correct', then he [could not] easily say whether he calculated [the] answer or recalled it" (p. 256). This applied particularly to calculations for which Aitken was sure that he had previously derived answers. Nonetheless, expert mental calculators demonstrate an exceptional recall of some large numerical equivalents, especially 2- and 3-digit squares (Hope, 1984, cited in Hope, 1985, p. 368). Eighty-five percent of the 2-digit squares from 11² to 99², not including the squares of the multiples of 10, were successfully recalled during Hope's (1985, p. 368) analysis of Charlene's declarative knowledge. During computation, Charlene relied "heavily on an apparent ability to discern quickly number properties useful for a calculation and to complete the calculation by recalling large products from memory" (Hope, 1987, p. 335). To calculate 81 x 27, Charlene rearranged the calculation to 27² x 3 after realising that 81 was a multiple of 27. The answer was obtained by recalling and tripling the square of 27. Evidence provided by Hope and Sherrill (1987, pp. 104-105) indicates that marked differences between the abilities of skilled and unskilled mental calculators to retrieve numerical equivalents, useful for mental multiplication, are not always clearly apparent. In comparing the abilities of skilled and unskilled Grade 11 and 12

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students, Hope and Sherrill (1987, p. 109) reported that although the difference in their abilities to recall basic multiplication facts was statistically significant, it was not substantial. Hence, at least for Grade 11 and 12 students, basic fact mastery may not be an important factor contributing to differences in mental calculation performance. With respect to recalling larger numerical equivalents, although the ability of the skilled students to recall large 2-digit squares was markedly superior, their knowledge of the extended mental multiplication tables went only slightly beyond 10 x 10 (Hope & Sherrill, 1987, p. 109).

2.8.3 Memory for Interrupted Working

The use of procedural knowledge during mental computation necessitates a number of calculative steps and the temporary holding in working storage of details of the problem task environment and the outcomes of various subproblems (see Figure 2.3). Consequently, the capability for mentally solving arithmetic problems is confined by an individual's capacity for storing and processing information in short- term memory (Wolters et al., 1990, p. 21). According to this information processing view of mental computation, skilled mental calculators "should possess a superior mechanism for storing and processing numerical information" (Hope, 1987, p. 338). Keeping track of the results of a calculation is supposedly the greatest burden during mental computation (Hope, 1987, p. 338), with forgetting interim calculations a great source of error (Hitch, 1977, p. 336; Hope & Sherrill, 1987, p. 110; Wolters et al., 1990, p. 28). Many expert mental calculators exhibit superior memory resources for storing and retrieving numerical information from short-term memory, as measured by digit- span tests such as those in the Wechsler Intelligence Scale for Children used by Hope (1987) and the Wechsler Adult Intelligence Scale used by Jensen (1990) and Hope and Sherrill (1987). Hope (1987, p. 336) records that Charlene's digits- forward and digits-backward scores were 12 and 8, respectively, far in excess of the average child. Some of the 19th century adult lightning calculators studied by Alfred Binet also exhibited large digit-spans. Binet estimated Jacques Inaudi's forward memory span to be approximately 42 digits, whereas the accepted average adult's digit-span is seven (Hope, 1985, p. 370).

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Nevertheless, strong evidence to support the proposition that short-term memory capacity is a decisive factor in the superiority of the more competent mental calculators does not exist (McIntosh, 1991a, p. 4). Olander and Brown (1959, p. 98) report a correlation coefficient of .35 for the relationship between proficiency with mental subtraction and short-term memory, as measured by digit-span tests. Similar results were obtained by Hope and Sherrill (1987, p. 108) with correlations of .33 and .30 between mental multiplication and recalling digits forwards and digits backwards, respectively. Hope (1985, p. 371) noted that many people who demonstrated a superior memory for strings of digits and numbers in general did not necessarily demonstrate an above average ability for mental computation. As Mitchell (1907, p. 83, cited in Hope, 1985, p. 371) has cautioned, a distinction needs to be made with respect to a memory for figures per se and memory related to an ability to undertake calculations. Hope (1987) has concluded that "because Charlene could analyze a calculational task in terms of alternative subproblem formulations, the model of memory function implicit in digit-span assessments appears far too rudimentary and simplistic to have much explanatory power" (p. 338). The study of memory processes in skilled mental calculation, therefore, should not be limited to a model characterised by a sequential processing of numbers. Hope (1987, p. 339) suggests that the use of tasks devised to assess an ability to manipulate multiple pieces of information would be more appropriate for developing an understanding of highly proficient mental computation. One such test is Raven's Progressive Matrices, a nonverbal test of abstract reasoning involving the mental manipulation of complex nonrepresentational figures (Jensen, 1990, p. 265). However, with respect to Shakuntala Devi, Jensen (1990, p. 266) has concluded that her performance on this test was unexceptional, being similar to those of older, college-educated adults tested in earlier studies and within the range for university students. Devi, however, does exhibit a superior ability to encode digits in short-term memory, an ability made possible by a vast knowledge of numbers, and evident in her performance on memory-search tasks to determine the presence or absence of single digits held in short-term memory (Jensen, 1990, p. 269). Expert mental calculators search for meaning in the data presented. Aitken comprehended in terms of rich conceptual maps with his memory "intimately linked to discern multiple properties that were interwoven into distinctive patterns" (Hunter, 1977b, p. 157).

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2.8.4 Memory for Calculative Method

Although the relationship between short-term memory capacity and the ability to compute mentally remains somewhat equivocal, it is evident that enterprising mental calculators are able to find methods which are memory efficient. Such methods keep the need for information processing to a minimum. "Through the judicious selection of a calculative strategy, a skilled [mental] calculator can get by with fewer short-term memory resources than the selection of more inefficient strategies would necessitate" (Hope & Sherrill, 1987, p. 110). Expert mental computers draw upon a large repertoire of calculative methods (Hunter, 1978, p. 340) stored as part of their procedural knowledge. However, Owen and Sweller (1989, p. 326) caution that simply knowing and understanding a strategy is not sufficient to allow its efficient use in a new context, a view supported by van der Heijden (1995, p. 12). If at least elements of its use are not automated, sufficient cognitive resources may not be available for the controlled processing of the novel aspects of the solution process in short-term memory (Silver, 1987, p. 40). Jensen (1990, p. 270) has observed that the majority of the basic operations used by Devi probably became automatised during her childhood. Aitken often calculated using a "repeated cycle of operations on which he [had] a well practised grip...[and made] use of a strong, steady rhythm which [helped] him, as it were, to throw forward a loose end and then catch it at an appropriate later moment (Hunter, 1978, p. 343). From observing the way in which Aitken approached a mental calculation, Hunter (1977a, p. 36) concluded that the key decision was the selection of a strategy that allowed economy of effort, to lead to the answer in the shortest possible time and with the least difficulty (Hunter, 1977a, p. 37). Strategy selection is based on the nature of the properties and relationships between the numbers that are detected in the problem task environment (Hope, 1985, p. 358). This observation is supported by Barbara Reys (1986b). High ability seventh- and eighth-grade mental computers were found to make extensive use of number properties and equivalent number forms to translate an existing problem into one that is easier to compute mentally (B. J. Reys, 1986b, p. 3279-A). This enables

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appropriate declarative knowledge of numerical equivalents to be applied, thus reducing the load on short-term memory. Hence, in many instances, experts undertake very little calculation. For example, Aitken gave the following explanation for multiplying 123 by 456: "I see at once that 123 times 45 is 5535 and that 123 times 6 is 738; I hardly have to think. Then 55350 plus 738 gives 56088" (Hunter, 1978, p. 341). Numbers are thought of as distinctive entities (Hunter, 1978, p. 342), not as individual digits with particular place values. Hunter (1977a) notes that Bidder, a 19th century English lightning calculator, explained "that, for him, every number up to a thousand was but one idea, and every number between a thousand and a million was, to his regret, two ideas" (p. 43). In Hitch's (1977) view, the most efficient mental strategies are those which "minimize the effects of rapid forgetting, or at least localize such effects in less important components of the final answer" (p. 337). On this basis, work from the left strategies are preferable to work from the right strategies (see Table 2.4), as the latter may produce the highest errors with the most significant digits─for example, with the thousands in a four-digit whole number computation. Additionally, when calculating from right to left, the answer cannot be stated until the entire calculation is completed, thus increasing the likelihood that some feature will be forgotten (Hope, 1985, p. 359). Stage calculators have an additional reason for preferring to calculate from left to right. Calculating from left to right allows them to begin calling out an answer while still calculating, thus giving the impression that computing time is much less that it really is (Gardner, 1977, p. 70). Skilled mental calculators tend to use left-to-right approaches (Hope & Sherrill, 1987, p. 106; Olander & Brown, 1959, p. 99). In contrast, unskilled (Hope & Sherrill, 1987, p. 104; Olander & Brown, 1959, p. 99) and middle ability (B. J. Reys, 1986b, 3279-A) mental calculators make more use of mental analogues of the standard written algorithms, which except for division, are right-to-left approaches. Such strategies, which can involve many carry operations, are infrequently used by proficient mental calculators (Hope & Sherrill, 1987, p. 106). Carrying can be avoided, not only by using a left-to-right approach, but also by retrieving rather than calculating partial answers (Hope & Sherrill, 1987, p. 108). As well as working from left to right, highly proficient mental calculators tend to progressively incorporate interim results into a single solution (Hope, 1985, p. 360), thus obviating the need for an organisation stage (Vakali, 1985) in which partial

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answers are combined to form a final solution. As Hope and Sherrill (1987) have observed, when using distribution strategies "the retention of a single result [is] accomplished by continually retrieving a sum, updating by adding a newly calculated partial product, and storing the new sum. In the case of factoring, a running product rather than a sum [is] continually modified" (p. 109) ─ see, for example, additive distribution and general factoring in Table 2.4. This approach, together with the tendency of skilled mental calculators to work rapidly, reduces short-term memory load. When working rapidly, numbers set aside for subsequent use are brought into the calculation before much forgetting can occur (Hunter, 1977a, p. 43). Speed is gained by excluding all irrelevances. Writing of Professor Aitken, Hunter (1977a) observed that "when calculating, he is physically relaxed and inattentive to everything but the calculation....He solves a problem with concentrated and streamlined effort" (p. 37). In conclusion, efficient mental strategies are essentially those that (a) proceed from left-to-right, (b) eliminate the need for carrying, and (c) allow the progressive incorporation of partial answers into a single result (Hope & Sherrill, 1978, p. 108) (see Table 2.4). Hence, the fostering of an ability to compute mentally should centre on the development of strategies which enable an individual to make efficient use of a limited capacity for mentally handling data (Hunter, 1977a, p. 43). Hence:

It appears imperative that children be given a chance to build on their own natural skills by choosing and verbalising mental strategies, and by choosing when mental computation is appropriate. Less emphasis should be placed on traditional pen and paper algorithms and more emphasis on identifying and developing children's legitimate spontaneous strategies. (Cooper et al., 1996, p. 160)

Such an advocacy arises from the finding that teaching the standard written algorithms for addition and subtraction to Grade 2 and 3 children may account for the replacement of left-to-right strategies, based on out-of-school mathematical knowledge, by inefficient right-to-left approaches to calculating mentally (Cooper et al., 1996, p. 158).

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2.9 Developing the Ability to Compute Mentally

McIntosh (1990a, p. 25) has observed that primary school mathematics continues to be characterised by an emphasis on written computational procedures which find limited use outside the classroom, while the process used by all, mental computation, receives little consideration. For those who maintain that there has been a decline in numeracy, the de-emphasis of mental computation is considered by some to be a contributing factor (Ewbank, 1977, p. 28; Treffers, 1991, p. 343). Treffers (1991) suggests that mental computation (and computational estimation) "is a forceful means to prevent innumeracy, or better, to promote numeracy" (p. 343), numeracy being "the ability to comprehend and recognise the power and potential of mathematics..., and to direct that power towards solving personal and social problems of everyday life" (Handran, Toohey, & Luxton, 1993, p. 223). In as much as some proficiency with paper-and-pencil techniques needs to be maintained, such proficiency will be heightened through encouraging children to solve problems mentally. Children who become proficient with mental computation learn to think quantitatively with numbers (Carraher et al., 1985, p. 28; Reed & Lave, 1981, p. 442), "a much neglected but greatly needed arithmetic ability" (Koenker, 1961, p. 295). People who are mathematically effective in everyday life rarely use the standard written algorithms mentally, although such algorithms remain the focus of classroom instruction. Proficient mental computers tend to use personal adaptations of paper-and-pencil algorithms or idiosyncratic mental strategies (Cockcroft, 1982, p. 75, para 256). At present, these are largely self-acquired and are not well known by teachers (McIntosh, De Nardi, & Swan, 1994, p. 7). However, mental computation should not become a focus in mathematics classrooms simply for its social utility. Strategies for computing mentally are, in essence, strategies for thinking about mathematical tasks (Cobb & Merkel, 1989, p. 80). Fostering the development of thinking strategies that extend beyond the memorisation and recall of basic facts has a number of positive effects. Conceptual knowledge, to which procedural knowledge is linked, is enhanced. Building blocks for further learning are provided and children come to believe that mathematics makes sense (Cobb & Merkel, 1989, p. 80). This is a belief essential for gaining mathematical power and one that needs to become a goal of classroom mathematics (MSEB & NRC, 1990, p. 5). In short, through an emphasis on mental

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computation, ingenuity and resourcefulness in dealing with numbers are developed (Menchinskaya & Moro, 1975, p. 74; Sauble, 1955, p. 33), and number sense is enhanced (Markovits & Sowder, 1994, p. 23). However, significant changes are required in many mathematics classrooms to adjust the balance between mental and written arithmetic and to allow children to gain competency with mental computation (Willis & Stephens, 1991, p. 7). The climate in which mathematics is taught needs to be such that students' attempts at problem solution are valued and explored and "where mathematical conclusions are supported by reasoned argument rather than teachers or answer books" (Putman et al., 1990, p. 137). In harmony with this view is that of Heirdsfield and Cooper (1995, p. 5), which contends that the approaches taken to develop skill with mental computation should be such that the repression of children's spontaneous strategies should be circumvented.

2.9.1 Approaches to Developing Skill with Mental Computation

Robert Reys et al. (1995, p. 305) suggest that there are two broad ways in which to view the approaches for developing skill with mental computation: (a) a behaviourist approach, and (b) a constructivist approach. The former holds that mental computation is a basic skill and a prerequisite to paper-and-pencil computation, with proficiency gained through direct teaching and practice (Shibata, 1994, p. 17). This approach necessitates the prior determination of mental strategies to be taught, as well as a sequence for teaching them. In contrast, the constructivist view is that mental computation is a process of higher-order thinking in which "more than the mental application of an algorithm [is involved]. The acts of both generating and applying a strategy are significant" (R. E. Reys et al., 1995, p. 305). These acts constitute a personal application of idiosyncratic conceptual, procedural and metacognitive knowledge. Both these approaches, however, contrast with what may be considered the traditional approach to developing skill with mental computation. Whereas the behaviourist and constructivist approaches give recognition to the mental strategies employed, the traditional approach focuses primarily on the correctness or otherwise of the answer.

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Traditional Approach

Reminiscent of the role of mental arithmetic as a means of exercising the mind, as advocated by the proponents of faculty psychology in the 19th century, Davidson (1980) has called for a return to:

The good old days when our teachers hammered mental arithmetic into us as students....Starting a lesson with 10 "quickies" helps to sharpen up the mental processes and gets the students thinking about mathematics right from the start of the lesson. Answers should be corrected quickly so that the normal lesson can be started as soon as possible. (p. 24)

Such an approach, which tends to focus on one-step calculations or word problems (McIntosh, 1990a, p. 41), devalues the key role that mental computation plays in everyday life and in the development of mathematical ideas. Under this approach mental computation is not considered as real mathematics─a view in contrast to the demands of everyday living. It fails to consider the many important features of mental computation. These include (a) the variety of methods able to be used to arrive at an answer, (b) the ability of children to use an increasing network of connections between numerical equivalents and relationships between numbers, (c) the ability to check an answer by applying different mental strategies, and (d) the ability to carry out a string of logical operations mentally rather than a single-step process (McIntosh, 1988, p. 261). Mental computation taught in this traditional way constitutes testing rather than teaching (McIntosh, 1991b, p. 53). Ten quick questions at the commencement of a mathematics lesson may be a source of enjoyment and challenge for more able students, but for others with limited mathematical ability such questions are "much more likely to lead to a loss of confidence, increasing antipathy to mathematics and sometimes even to feelings of humiliation which [will] long be remembered" (Cockcroft, 1982, p. 75, para 254). For these children, the emphasis on being solely right or wrong results in mathematics lessons beginning with failure (Giles, 1986, p. 190). The reality of the classroom as a community of mixed abilities is neglected. Such traditional methods tend to exclude those who most need reinforcement and involvement (McIntosh, 1988, p. 261). In earlier advocacies for mental (oral)

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arithmetic, little consideration was given to individual differences. Spitzer (1948) contended that mental arithmetic was a useful means for keeping the class together and maintaining class spirit: "Suppose that, at the end of...thirty minutes [of written work], the teacher uses an oral exercise. All pupils become again one class giving attention to the same thing. When used in this way, oral arithmetic is a unifying experience" (p. 54). Biggs (1967, p. 220, cited in McIntosh, 1990a, p. 41) reported that there was no relationship between the allocation of time to this traditional approach to mental computation and attainment. Although the computational tricks that this approach encourages may have a place in the curriculum, they should not be emphasised at the expense of instruction that enhances a child's sense of number (Sowder, 1992, p. 15). It is essential that children be given opportunities to develop the ability to observe patterns, explain, generalise and to validate (McIntosh, 1978, p. 18). These abilities can be developed through a focus on the thinking strategies employed during mental computation rather than being solely concerned with correct or incorrect answers. As Colburn advocated in 1830, "if...teachers would have the patience to listen to their scholars and examine their operations, they would frequently discover very good ways that had never occurred to them before" (Colburn, 1830, reprinted in Bidwell & Clason, 1970, p. 34).

Alternative Approaches

Although the traditional approach to mental computation is dismissed as inappropriate for fostering efficient and flexible mental strategies, a preferred methodology for their development, based on a knowledge of how children learn mathematics, is as yet unclear (McIntosh, 1991, p. 6; Reys & Barger, 1994, p. 38; Zepp, 1976, p. 103). Referring to mental multiplication, but applicable to the other three operations, Hazekamp (1986, p. 117) has observed that little is known about how children develop strategies for mental computation. Nonetheless, there is evidence to suggest that gains are made where systematic instruction is provided (Flournoy, 1954, p. 153; Gracey, 1994, pp. 112-116; Josephina, 1960, p. 200; Markovits & Sowder, 1994, p. 22; Schall, 1973, pp. 365-366). Schall (1973, pp. 365- 366) concluded that short, frequent exercises in mental arithmetic seemed to be a

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worthwhile addition to the traditional paper-and-pencil oriented mathematics classroom. Further, Flournoy (1954, p. 153) has cautioned that although children in the intermediate grades can become proficient with mental computation in everyday situations, such will not occur unless children are provided with definite mental arithmetic experiences in school. Three approaches to developing the ability to compute mentally, which are worthy of further research and analysis, are delineated by Barbara Reys (1991, pp. 8-9). These approaches are characterised by the two opposing beliefs about how children learn mathematics referred to previously: (a) an authoritarian model based on teaching as the transmission of knowledge─a behaviourist approach; and (b) a child centred, constructivist approach in which the teacher assumes the role of intellectual coach (MSEB & NRC, 1990, p. 40). As suggested previously, the first of these approaches requires mental computation to be viewed as a topic within the mathematics curriculum. This implies that the focus should be on specifically teaching the strategies identified as those used by proficient mental calculators (see particularly Table 2.4). However, Hope (1987, p. 340) cautions that further research is required to ascertain whether it is legitimate to plan instruction on this basis. Robert Reys et al. (1995, p. 321) report that Japanese children, taught through this approach, exhibit a narrow range of mental strategies, with few of these strategies being idiosyncratic ones. The second approach, in which mental computation is not viewed as a topic, recognises that learners construct meaning from sensory and cognitive inputs by processing it through existing memory structures. Elements of these inputs are retained in long-term memory in forms─conceptual or procedural knowledge, for example─that allow for later retrieval to facilitate future processing (Good & Brophy, 1986, p. 229). This approach allows students to generate thinking strategies based on their own prior experience and knowledge (B. J. Reys, 1991, p. 8). Recognition of children's prior knowledge and understanding is one means for bridging the gap between in-school and out-of-school experiences (Masingila et al., 1994, p. 13), an essential factor in making the classroom more meaningful to students. Trafton (1989) suggests that mental computation should be viewed within a number sense sphere─to view mental computation "as an extension of children's abilities to compose and decompose numbers in ways that make sense to them in given situations" (p. 76). The focus is on the individual learner. Strategies are devised for

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solving problems that arise in natural settings or as part of planned problem solving experiences (Reys & Barger, 1994, p. 39). Children are given opportunities to build distinctive calculative systems, a feature that characterises skilled mental calculators (Hunter, 1977a, p. 40). The third approach suggested by B. J. Reys (1991, p. 9), and supported by Carroll (1996, p. 8), is an amalgam of these two approaches. The focus is on discussion and on sharing the devised mental strategies with peers and teachers. Such an approach leads children to think about alternative strategies as they listen to the ideas of others, to "negotiate shared meaning" (Masingila et al., 1994, p. 13). Although a teacher, under this approach, does not present a defined set of strategies to the children, at times it may be appropriate to introduce different ways of thinking about a problem and in so doing provide children with alternative paths to the solution of a particular problem, to introduce strategies which may not be spontaneously invented by children (Carroll, 1996, p. 8). For example, given that low proficiency students tend to use inefficient right-to-left procedures, Cooper et al. (1996, p. 159) suggest that such children may benefit from direct teaching of left-to- right strategies (see Table 2.4). However, in so doing it is essential that children do not gain the impression that one strategy is necessarily superior to another, merely because it is the focus of attention (Cockcroft, 1982, pp. 75-76, para 256; Trafton, 1992, p. 92). Hence, Carroll (1996, p. 8) has suggested that the dilemma over how to facilitate the development of number sense while providing students with powerful mental strategies is one for which a solution is not readily apparent. Greeno (1991, p. 173), in context with the development of number sense, has suggested that it is a natural, though inappropriate, response to treat mental computation as a topic in which a set of identified skills becomes the focus. Such an approach places value only on established mathematical techniques with the result that children "come to feel that their intuitive ideas and methods are not related to real mathematics" (Clements & Battista, 1990, p. 35). Further, strategies memorised without understanding do not aid the development of number sense, including an ability to flexibly think about numbers, a characteristic of skilled mental calculators, nor does rote memorisation aid the development of a comprehensive view of computation (Rathmell & Trafton, 1990, p. 171). It is likely that where standard strategies are taught, mental computation will lose many of its

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characteristics, as defined by Plunkett (1979), characteristics that provide mathematical power for the learner. "Mental computation is more appropriately viewed as an ongoing emphasis in a mathematics program than as a separate topic composed of subskills arranged in order and taught as separate lessons" (Rathmell & Trafton, 1990, p. 160). In situations where children are allowed to develop their own computational procedures, as evidenced in the second and third approaches outlined above, they gain experience in processes that are essential to doing mathematics: Decision making, creating problem solving strategies, refining home-made algorithms (Wheeler, 1977, cited in McIntosh, 1992, p. 133). Such approaches to mental computation take cognisance of the state of knowledge with respect to how children learn mathematics. Three interrelated aspects of learning are emphasised, namely, the belief that learning is a process of knowledge construction, that learning in knowledge dependent, and that learning is implicitly linked to the situation in which it occurs (Resnick, 1989a, p. 1). Constructivist instruction gives pre-eminence to the development of personal mathematical ideas. Children are encouraged to use their own methods for solving problems (Clements & Battista, 1990, p. 35). Whether or not particular individuals find particular problem solving settings problematic depends upon their store of conceptual and procedural knowledge, and their experience with the type of situation under investigation. Giving recognition to this in the classroom facilitates the individualisation of instruction (Yackel et al., 1990, p. 14). Children at different conceptual levels interpret tasks in different ways and use different solution strategies. Children are the best judges of what is a problem, what makes sense and what is helpful (Cobb & Merkel, 1989, p. 81). The third approach, in particular, with its emphasis on discussion, recognises that skills and the conceptual knowledge on which they are based are not independent of mental, physical, and social contexts (Resnick, 1989a, p. 3). Greeno (1991) has argued that an "environment that fosters curiosity and exploration" (p. 173) is a social construction in which students interact with the teacher and with each other about quantities and numbers. Whether an information processing perspective or a situated knowledge perspective of acquiring skill with mental computation is held, discussion plays an important role. As Putman et al. (1990, p. 138) have pointed out, it is through communication that individual

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knowledge is refined and revised in the social setting of the classroom, and through which children come to understand that which is implicit in particular mathematical situations. "When children are given opportunities to talk about their mathematical understandings, problems of genuine communication arise. These problems, as well as the mathematical tasks themselves, constitute occasions for learning mathematics" (Yackel et al., 1990, p. 12). In classrooms that encourage exploration, questioning, verification, and sense- making, the teacher's role is one of guide and moderator, rather than a dispenser of answers, the traditional role of the teacher (B. J. Reys, 1992, p. 95), thus making school mathematics more like folk mathematics (Maier, 1980, p. 23). The guidance provided by the teacher is a feature that distinguishes constructivism from unguided discovery (Clements & Battista, 1990, p. 35). Mathematical authority does not reside solely with the teacher, but in partnership with the children as an intellectual community (Yackel et al., 1990, p. 18). In summary, the preferred role of the teacher is that of an intellectual coach. The Mathematical Sciences Education Board and National Research Council (1990, p. 40) have outlined the various roles that an intellectual coach assumes and in so doing have described the ways in which mathematics teaching should be approached. Equally, the role descriptions describe the ways in which teachers should approach the task of developing skill with mental computation. At various times the teacher is required to act as: (a) a role model for the problem solving process, (b) a consultant, (c) a moderator who leaves much of the decision-making to the students, (d) an interlocutor who encourages self-reflection, and (e) a questioner who challenges students to define their strategies and conclusions.

2.9.2 General Pedagogical Issues

If children's performances with mental computation are to be improved, mental computation abilities must be developed in a regular and systematic manner (B. J. Reys et al., 1993, p. 314). Given that skilled mental calculators place an emphasis on opportunities for practice and playing with numbers as a key element in the development of their proficiency, the main object of each mental computation

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experience should be to provide children with an opportunity to explore numbers in a non-threatening environment.

[It] is not a drill session, not even a practice session. Still less is it a testing session. It is simply an opportunity for playing with the numbers and exploring the way they seem to relate to one another and the ways in which one can think about these relationships. (C. Thornton, 1985, pp. 9- 10)

McIntosh et al. (1994, p. 9) maintain that the aims are to help children to see how to calculate mentally and to realise that mental computation is a creative process, acceptive of many solution paths. Instruction needs to reflect the spirit of computing mentally─to explore different ways of reasoning and to share and justify solutions (Rathmell & Trafton, 1990, p. 158). Everyone needs to be involved and to find such involvement enjoyable. This can be facilitated by small-group work where the chance of dispiriting failure is reduced. Further, when working in small groups difficulties can more easily be perceived and handled sensitively, with children challenged according to their abilities (Cockcroft, 1982, p. 93, para 319). However, "mental work will be enjoyed if it is challenging only when the pupils are ready to be challenged" (Jones, 1988, p. 44). Children need to experience success with mental computation. Such success is self-fulfilling (Jones, 1988, p. 44) and contributes to the development of the affective components of mental computation (see Table 2.1), particularly the gaining of confidence. Although children cannot be made to think mathematically, their curiosity can be aroused through their participation in activities that require mental computation─activities that include games and puzzles (Ewbank, 1977, p. 28). Maier (1980) suggests that such experiences assist in the development of a "friendliness with numbers" (p. 23), a key aspect in the formation of a dense web of connections between numbers stored as declarative knowledge. Whether the problem context is one from real-life or purely mathematical, the aim is to encourage a flexible approach and to make explicit through discussion the advantages and insights that come from alternative strategies (French, 1987, p. 39), thus supporting the development of appropriate mental computation strategies and the Conceptual Components and Related Concepts and Skills that underpin them (see Table 2.1).

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Although there is a role for a weekly series of brief focus sessions, of around fifteen minutes each, Trafton (1978, p. 212) asserts that mental computation needs to become an integral part of classroom learning. This view is supported by Carraher et al. (1985) in their call for educators to "question the practice of treating mathematical systems as formal subjects from the outset and should instead seek ways of introducing these systems in contexts which allow them to be sustained by human daily sense" (p. 28). Given the idiosyncrasy of mental strategies, Barbara Reys (1985, p. 46) cautions that mental computation is not a topic that can fit into a certain grade level or sequence of the curriculum. Integrating mental computation into all relevant classroom activities should aid children in the realisation that mental techniques are legitimate computational alternatives for particular calculative needs. However, when focus sessions are conducted, it is better to discuss several approaches to a few problems rather than focussing on one approach for each of a large number of questions (McIntosh, 1988, p. 261). The emphasis can therefore be placed on how answers were calculated in a range of contexts with a range of number types and operations, rather than on whether they are simply correct or incorrect. "By merely reinforcing right answers and correcting wrong ones, traditional mathematics instruction unwittingly stifles children's ability to do their own thinking" (Kamii, 1990, p. 28). Both McIntosh et al. (1995, p. 36) and Robert Reys et al. (1995, p. 332) report, for Australian and Japanese students, respectively, that, for some items, the effect of mode of presentation on performance is significant─for example Year 5 children gained more correct responses for 182 + 97 when presented visually and for ½ + ¼ when presented orally (McIntosh et al., 1995, p. 64). It is likely that the varied modes of presentation lead to different strategies being employed by different students, resulting in differential performances levels (McIntosh, et al., 1995, p. 60). McIntosh (1988, p. 261) suggests that although many tasks may be presented visually, the answers should be oral. By so doing, children are able to focus on the mental strategy used rather than on writing the response, a view supported by Spitzer (1948, p. 215). In French’s (1987, p. 41) view, the oral presentation of problem contexts places these contexts closer to those faced in everyday situations, as well as eliminating the reading difficulties which some children may experience. The relationships between numbers more clearly become the focus (Hazekamp, 1986, p. 123).

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Nonetheless, there is some evidence to suggest that children perform better, as determined by the number of correct answers, in situations where written tasks are presented. Sister Josephina (1960), based on research involving workbook examples, has concluded that "teachers in presenting mental arithmetic problems strictly by the so-called ‘oral method' actually inhibit better productivity than if their pupils had the written problems before them" (p. 200). The difficulties that some children experience with oral presentation may stem from their lack of an ability to visualise numbers or from their inability to retain essential elements of the problem task environment in short-term memory while processing occurs. Olander and Brown (1959, p. 98) reported that the greatest difficulty that Grade 6 to 12 students experienced when calculating mentally was with the visualisation of numbers presented orally. Therefore there is a need for children to gain experience with a variety of representational models and to develop an ability to switch smoothly and quickly from one to the other (Giles, 1986, p. 191). Younger children should be allowed to use aids such as cubes and number lines when computing mentally to assist in the development of mental representations (Ewbank, 1977, p. 28).

2.9.3 Sequence for Introducing Computational Methods

Wherever practicable, children should be encouraged to perform mental calculations in preference to those reliant on paper-and-pencil (AEC, 1991, p. 109). Classroom experience suggests that children have difficulty with mental computation (and computational estimation) when paper-and-pencil skills are developed prior to an emphasis on mental procedures (Musser, 1982, p. 40). This is the sequence traditionally employed in mathematics classrooms (Irons, 1990a, p. 20 ; Rathmell & Trafton, 1990, p. 156), as represented in Figure 2.4. However, paper-and-pencil procedures for addition, subtraction, and multiplication conflict with those for mental computation. The former emphasise right-to-left processing compared to the left-to- right processing evident in many heuristic mental strategies. As Cooper et al. (1992, pp. 100-101) point out, there is a need to re-evaluate the sequence in which procedures for mental computation, computational estimation, and paper-and-pencil calculation are introduced.

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Although the research evidence is limited, focussing on mental computation strategies prior to introducing the standard paper-and-pencil algorithm for each operation "is both powerful and effective" (Rathmell & Trafton, 1990, p. 156), a view supported by Carroll (1996, p. 3) in context with the trial of The University of Chicago School Mathematics Project curriculum. Such a practice would assist children to make more appropriate use of the range of mental and written strategies which they have available; a flexible approach to calculating would result (Thornton, Jones, & Neal, 1995, p. 483). Biggs (1969, p. 25) suggests that mental agility with numbers should precede written practice, and that the "crucial test of readiness for practice in written computation with tens and units [is] the ability to add two 2-figure numbers mentally by an efficient method" (The Schools Council,

Concept of Operation

Basic Number Facts

Paper-and-Pencil Computation

Computational Mental Estimation Computation

Figure 2.4. Traditional sequence for introducing computational procedures for each operation (Adapted from Irons, 1990a, p. 20)

1966, cited in Ewbank, 1977, p. 29). Related to the development of this agility is a child's proficiency with computational estimation. In Lindquist's view (1984, p. 599), this should occur prior to the introduction of paper-and-pencil procedures, particularly before the standard written algorithm for each operation becomes the focus. Figure 2.5 presents an alternative view of the sequence in which computational skills should be developed.

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In implementing this sequence, recognition needs to be given to the role of mental computation as a fundamental component of computational estimation (see Figure 2.2). Hence, the development of selected strategies for mentally calculating exact answers needs to occur prior to a focus on computational estimation (Leutzinger, Rathmell, & Urbatsch, n.d., p. 10). Additionally, Case and Sowder (1990, p. 95) demonstrated that until around 12 years of age, children are unable to successfully co-ordinate the two qualitatively different procedures involved in finding approximate answers, namely (a) converting from exact to approximate numbers using nearness judgements, and (b) mentally computing with these numbers (Case & Sowder, 1990, p. 88). This finding has implications for the introduction of computational estimation. As Sowder (1992, p. 18) has observed,

Concept of Operation

Basic Number Facts

Computational Mental Estimation Computation

Paper-and-Pencil Computation

Figure 2.5. An alternative sequence for introducing computational procedures for each operation

this finding has led to the belief that a focus on computational estimation should not occur too soon after the introduction of a particular operation. Rather, the teachers should focus on developing (a) number size concepts, (b) mental computation strategies, and (c) on estimation-type problems that do not require the coordination of complex skills. With respect to written procedures, if standard paper-and-pencil algorithms are introduced after an initial focus on strategies for calculating both exact and

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approximate answers mentally, and following experiences with informal written techniques, it is likely that the standard algorithm for each operation will come to be viewed as one of many possible ways for calculating in particular contexts, rather than the way to calculate (Ross, 1989, p. 51). The ability of children to select appropriate calculative methods will be enhanced, a key computational outcome of classroom mathematics programs (AEC, 1991, p. 115; NCTM, 1989, p. 44). Nonetheless, "When should paper-and-pencil algorithms be introduced?" is, in Rathmell and Trafton's (1990, p. 171) view, a key question that remains. They point out that the placement of written algorithms for particular year-levels, the size of the numbers to be manipulated, and the manner in which the algorithms are taught will be fundamentally changed by the introduction of curricula in which mental computation and computational estimation skills are introduced early in combination with a ready-availability of electronic calculating devices.

2.9.4 Assessing Mental Computation

"What students learn and how they learn will be influenced by what they think teachers...value. Their view of what really counts in learning and doing mathematics will relate quite closely to what is assessed" (AEC, 1991, p. 21). For this reason Robert Reys (1992, p. 71; 1985, p. 16) has advocated the regular testing of mental computation skills. This would provide an overview of a student's performance with mental computation as well as a record of progress over time. Although recognising that such testing may take several different forms, Robert Reys (1985, pp. 15-16) concluded that, where testing occurs, emphasis appears to be given to traditional mental computation testing, with each test limited to between 10 and 20 context-free examples─a summative view of assessment. Such testing may "let both child and teacher know whether the child is ‘good' or ‘bad' at mental arithmetic. But does little or nothing to help the child acquire good mental methods" (McIntosh, 1991b, p. 53). Nevertheless, Robert Reys (1985, p. 16) does acknowledge that additional learning may occur in instances where particular examples are discussed. Assessment needs to be an integral part of the teaching cycle, the major purpose of which is to improve learning (AEC, 1991, p. 21). All components of mental computation (see Table 2.1), not merely the answers to particular items, need to be

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focussed upon at various times during assessment. The spotlight needs to be on the process of computing mentally. This requires that teachers go beyond traditional testing procedures to decisively measure mental computation skills (Sowder, 1992, p. 23). Opportunities for assessment, using observational techniques, occur during the discussion and sharing of strategies used. To gain deeper insights into the approaches that children use to compute mentally, structured interviews may be required. Self- reporting techniques can also provide useful information, particularly with respect to the level of confidence felt while computing mentally. In D. J. Clarke, D. M. Clarke, and Lovitt's (1990, p. 123) view, assisting students to develop the skills necessary for self-monitoring of their progress would be a useful and empowering educational goal. Although it is essential that teachers extend their repertoire of assessment techniques, a limited place remains for the traditional timed mental computation tests. To the extent that they continue to used, Robert Reys (1985, pp. 15-16) has remarked that such tests should be kept short, start by focussing on one operation with specific numbers, emphasise the mental nature of the tests, use oral and visual presentations of the problem-task environment, and build in groups of examples for which the strategy for one assists in solving another─for example, to be able to use the knowledge of 6 x 100 to calculate 6 x 99.

2.10 Summary and Implications for Mental Computation Curricula

As stated in Section 1.4.1, the purpose of this literature review was to "analyse the pedagogical, socio-anthropological and psychological literature relevant to mental computation" to provide a comprehensive understanding of key issues. In so doing, a framework for analysing past and contemporary beliefs and practices related to mental computation has been developed. This framework is reflected in the questions devised to guide the investigations discussed in Chapters 3 and 4. With respect to those which guided the analysis presented in this chapter, the following is a summary of principal findings.

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1. What are the recent developments in mathematics education of relevance to mental computation?

The developments in mathematics education of relevance to mental computation centre on issues related to redefined beliefs about numeracy, computation, number sense, and how children learn mathematics. The perceived interrelationships between these aspects are essential for children to gain mathematical power. A wide definition of numeracy encompassing mathematics beyond the development of competence and confidence in computational skills per se needs to be taken. Children need to be able to interpret, understand, use, apply and communicate numerical and spatial ideas associated with daily living skills, work tasks and recreational activities, many of which necessitate proficiency with mental computation. The impact of electronic calculating devices requires that the number strand of the mathematics syllabus be reassessed with a view to helping children develop appropriate methods for calculating (Willis, 1991, p. 3)─that is, the methods should be relevant to the tools available and to the contexts within which the mathematics is embodied. Many of these methods, whether mental, written or electronic, will be idiosyncratic to the individual. Hence, a realistic balance among the various forms of calculation needs to be encouraged (B. J. Reys & R. E. Reys, 1986, p. 4). Sowder (1992, p. 5) believes that an ability to calculate mentally using idiosyncratic strategies is an indicator of number sense, essential to which is an ability to compose and decompose numbers, an ability suggested as a component of mental computation (see Table 2.1). Number sense has been described as the demonstration, both implicitly and explicitly, of a "good intuition about numbers and their relationships" (Howden, 1989, p. 11). This is characterised by having an in- depth understanding of numbers and their relative magnitudes, the relative effects of operating on numbers, together with a well-developed recognition of multiple relationships between numbers (NCTM, 1989, p. 39). Characteristics such as these are essential to developing proficiency with mental computation (see Table 2.1). For children to gain mathematical power, learning needs to occur in meaningful, problem solving situations in which there is scope for active involvement and reflection (Literacy and Numeracy Diagnostic Assessment Project, 1991, p. 7). Such a child-centred focus allows the child to construct new knowledge from that

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already possessed. This is enhanced through social interaction, opportunities for which are believed to facilitate the development of multiple relationships between numbers and computational techniques. The belief that learning necessitates acquiring situated knowledge is interrelated with those that hold that learning is a process of knowledge construction and is knowledge-dependent (Resnick, 1989a, p. 1). Such beliefs reflect findings which suggest that, as individuals engage in mental computation, they modify the mathematics they use to suit the structure of a particular situation and their perception of its mathematical elements (Lave, 1985, p. 173). A holistic view is taken, one that is characterised by a focus on the quantities involved. Consequently, engaging children in mental computation is considered to be a means for linking school and folk mathematics, as well as a means for enhancing mathematical knowledge and confidence in its application.

2. What is the place of mental computation within the calculative process and particularly its relationships with computational estimation?

In any calculative situation, an individual needs to be able to decide what operations to use, to choose the method of calculation, and, once the calculation has been undertaken, to judge the reasonableness of the solution in terms of the characteristics of the problem. In choosing a calculation method, the Australian Education Council (1991, p. 109) suggests that mental computation should be the method of first resort, particularly for less complex calculations. In any given situation, the appropriateness of a method of calculation is dependent upon the nature of the operation, the degree of precision required, the availability of particular calculative tools, and particularly the confidence with which the situation is approached (see Figure 2.1). Nonetheless, as represented in Figure 2.1, irrespective of the method of calculation, some mental calculation occurs─number relationships need to be recalled or calculated, estimates need to be mentally determined to verify the reasonableness of solutions. Children need to be given opportunities to develop the ability to make sensible choices between mental computation, paper-and-pencil and technological methods (AAMT & CDC, 1987, p. 2). As noted by Robert Reys (1984, p. 58), mental computation is the cornerstone of the range of strategies used in computational estimation (see Figure 2.2).

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However, the mental strategies used, and the numbers manipulated, are subsets of those involved in situations where the mental calculation of exact answers is undertaken. The interrelationships between mental computation and computational estimation strategies, and their reliance on number comparison and elements of number sense has led Threadgill-Sowder (1988, pp. 194-195) to suggest that their development occurs in a spiral fashion, each feeding on and strengthening the others. Although mental computation is considered to be an essential prerequisite to computational estimation (see Figure 2.2), Robert Reys (1984, p. 551) has suggested that the reverse does not apply. This appears to be in conflict with Irons' (1990b, p. 1) view that the use of getting closer strategies during computational estimation may assist in the development of flexible approaches for calculating exact answers. Nonetheless, it is possible that although a limited range of mental computation strategies may be essential for initial experiences with computational estimation, strategies for calculating exact answers may be refined by additional experiences during the calculation of approximate answers.

3. What is the nature of mental computation as perceived by mathematics educators, both contemporaneously and historically?

Terms used to describe the mental calculation of exact answers during the past century have been characterised by their impreciseness. Although a concern for developing skill with the calculation of approximate answers is relatively new to primary school mathematics curricula (B. J. Reys, 1986a, p. 22), the use of such terms as mental arithmetic, oral arithmetic and mental to refer to the mental calculation of exact answers, often included references to approximate answers. Additionally, in using these terms, the focus was on the correctness of the answer rather than on the strategies used. Mental arithmetic was seen as a preamble to the real arithmetic, usually written, which was to follow. In an endeavour to distinguish between the mental calculation of exact and approximate answers, as well as to shift the focus from the answer to the processes used during calculation, it has become customary in the mathematics education literature to define mental computation as the process of producing an exact answer mentally without resort to calculators or any external recording device, usually with

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nontraditional mental procedures (Hazekamp, 1986, p. 116). Nonetheless, this term continues to be used imprecisely. For example, in Student Performance Standards in Mathematics for Queensland Schools (1994, p. 9) mental computation is used to refer to the calculation of both exact and approximate answers. In an endeavour to overcome the confusion in the use of the terms mental arithmetic and oral arithmetic, Cockcroft (1982, p. 92, para 316) employed the expression mental mathematics to refer to the mental─mental computation and computational estimation─and oral work essential to mathematics learning. Cockcroft (1982) was giving recognition to the role that discussion and explanation need to play in learning mathematics. Although oral arithmetic became equated with mental arithmetic as "arithmetic done without the aid of paper and pencil" (Flournoy, 1954, p. 148), oral has had a multiplicity of meanings during this century. These included instances where the problem was presented orally (Thorndike, 1922, p. 262) and where the calculation was worked aloud (Suzzallo, 1912, cited in Hall, 1954, p. 352). The concern, therefore, was not simply with the calculation being a mental one, but also with the nature of the presentation and mode of response. In common with the current use of the term mental computation, Hall (1954, p. 353) suggested that mental arithmetic should refer to arithmetic problems that arise orally, in writing or in the head, and where intermediate steps are typically recorded mentally, irrespective of how the answer is presented. By focusing on the approaches used to calculate mentally, the differences between mental and written computation can be clearly perceived. Mental strategies are considered to be fleeting, variable, flexible, active, holistic, constructive and iconic, but are limited in the range of contexts in which they may be applied. This contrasts with written procedures which are viewed as standardised, contracted, efficient, automatic, symbolic, general and analytic (Plunkett, 1979, pp. 2-3). Given that mental procedures require the mathematics involved to be understood, that thinking is encouraged, and that the focus is on the quantities involved rather than on mathematical symbols, the variability and flexibility of mental strategies arise naturally from the process of calculating mentally. That these are dependent upon an individual's declarative and procedural knowledge supports the belief that number sense and mental computation are inextricably bound.

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4. What are the roles currently perceived for mental computation within and beyond the classroom?

Although recognising that mental computation constitutes a valuable skill in everyday living, proponents for an increased classroom emphasis on mental computation suggest that it should have a broader role. Recognition is given to the central role that mental procedures occupy throughout mathematics (Cockcroft, 1982, p. 75, para 255). Additionally, as Reys and Barger (1994, p. 31) suggest, mental computation is believed to provide a vehicle for promoting thinking, conjecturing and generalising based on conceptual understanding. Such a provision contributes to the development of an in-depth understanding of the structure of numbers and their interrelationships, and hence to ingenious approaches to manipulating numbers (R. E. Reys, 1984, p. 549). Mental computation, therefore, may be viewed as a pedagogical tool which facilitates the meaningful development of concepts and skills associated with the number strand of the mathematics syllabus. Consequently, contrary to current practice, it is considered to be an essential prerequisite to written methods of calculation. Recognition is also being given to the gulf between learning and using school mathematics and that used outside the classroom (Masingila et al., 1994, p. 3). It has been demonstrated that the algorithms, particularly paper-and-pencil ones, traditionally taught in schools are generally not used once children leave the classroom (Carraher et al., 1987, p. 95; French, 1987, p. 41; Murray et al., 1991, p. 50). Willis (1990, p. 9) has asserted that the usefulness of school mathematics would be increased if it reflected the techniques used in everyday life. Folk mathematics is characterised by techniques that are often mental (Maier, 1980, p. 23) and frequently involve the manipulation of non-standard units using invented strategies (Lave, 1985, p. 173). This implies that there needs to be a shift in the way problems are presented in the classroom─from the presentation of prepackaged examples, with or without a veneer of real-world characteristics, to open situations in which children are free to formulate their own problems. In such situations the relevant inputs may be as negotiable as the solutions (Murtaugh, 1985, p. 189).

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5. What are the affective and cognitive components, including commonly used mental strategies, that constitute skill in computing exact answers mentally?

Little research is available to authoritatively present a concise, but comprehensive, model of the affective and cognitive components of mental computation. Nonetheless, Sowder and Wheeler's (1989) model for specifying the components of computational estimation was considered an appropriate starting- point from which to analyse and synthesise the available data. It was concluded that the concepts and skills, considered essential for mental computation to become a flexible, active and constructive process, could be classified under the following headings: Affective Components, Conceptual Components, Related Concepts and Skills, and Heuristic Strategies Based Upon Relational Understanding (see Table 2.1). Although other categories of mental strategies were identified, only those based on relational understanding were included in the proposed model. Such strategies are based on relational knowledge, as proposed by Skemp (1976), knowledge that allows an individual to understand why particular strategies are the most appropriate for particular situations. The thinking which occurs, a key element of mental computation, is based on an individual's declarative, conceptual and procedural knowledge (Sowder, 1991, p. 3), knowledge that is rich in relationships. The model recognises the role of affective components as facilitators of mental computation─having the confidence with which to select and apply particular strategies in particular contexts (see Table 2.1). Underpinning this process are what Sowder and Wheeler (1989, p. 131) have defined as conceptual components. These have been interpreted to refer to those elements that relate to an understanding of the factors on which the calculation of exact answers mentally is based (see Table 2.1). Critical to being able to engage efficiently and creatively in mental computation is the ability to recognise when it is appropriate to calculate mentally. Additionally, an individual needs to be able to recognise that the appropriateness of a chosen strategy is dependent upon the context of the calculation. The components classified as Related Concepts and Skills, in accordance with Sowder and Wheeler's (1989, p. 132) definition, are those that may be considered to indirectly influence an individual's proficiency with calculating exact answers

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mentally (see Table 2.1). Essential to this is an ability to translate a problem into a more mentally manageable form so that the demands on memory may be shifted from short- to long-term memory. Based on a proposal of Threadgill-Sowder (1988, p. 184), two key questions have been hypothesised, questions which are considered cyclically during the process of mental computation. These relate to the modification of the numbers embodied in the mathematical situation so that the resulting operations may be answered by recall, and to a consideration of how the operational sequence will proceed as a consequence of the way in which the numbers have been expressed (see Table 2.1). Underpinning these processes is an individual's conceptual knowledge related to, for example, place value, basic facts, numeration, arithmetical principles and their relationships. The classification of mental strategies for calculating exact answers beyond the basic facts was made difficult by a number of characteristics of the available research. Not only has such research been limited (McIntosh et al., 1995, p. 2), but also it has primarily focussed on addition and subtraction. In instances where studies have focussed on all four operations, clear distinctions have not always been adequately drawn between the strategies used for basic facts and those used to calculate with larger numbers (see Section 2.7.4). An added difficulty was that the data was obtained from a range of samples, varying with respect to age, grade, ability, cultural background, and contexts. Further, as McIntosh (1990b, p. 13) points out, the classification of strategies has often relied on subjective interpretations of interview protocols, and on written reports of past performances by expert mental calculators (Hope, 1985, p. 358), thus preventing certainty, on the part of the researcher, with respect to the use of particular strategies in particular contexts. Finally, these difficulties have been compounded by the absence of agreed descriptors for many of the commonly identified approaches. Nonetheless, the analysis presented in this chapter, centred on a model derived from one proposed by McIntosh (1990c, p. 1), provides a comprehensive summary of identified strategies. The following categories were used as organisers: Counting strategies (see Table 2.2), Strategies based upon instrumental understanding (see Table 2.3), and Heuristic strategies based upon relational understanding (see Table 2.4). Counting strategies, which primarily rely on counting-on or counting-back, do not entail the manipulation of complex numerical relationships, a skill indicative of a well-developed sense of number. The category

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of strategies that depends on instrumental understanding is viewed as an intermediary stage. These strategies, which include the use of the mental form of a written algorithm, are characterised by the application of, in Skemp's (1976) terms, "rules without reason" (p. 20). As noted previously, the use of strategies which rely on relational knowledge─for example: Working from the left, Using factors─are indicative of the view of mental computation as a process that involves the application of self- developed techniques which are dependent upon a knowledge of the properties of numbers and arithmetical operations (B. J. Reys, 1989, p. 72). Users of these strategies tend to take a holistic view of the numerical situation, modifying the numbers and operations involved to capitalise upon perceived relationships and known numerical equivalents, thus reducing the demands on short-term memory.

6. What are the affective and cognitive characteristics exhibited by skilled mental calculators, including the role that memory plays in the process of calculating exact answers mentally?

Individuals proficient at mental computation appear to enjoy mentally manipulating numbers and are more highly motivated than those who are less proficient (Olander & Brown, 1959, p. 100). They exhibit a passion for numbers, which is reflected in the degree to which they practice calculating (Hope, 1985, p. 372). Further, skilled mental calculators display a proclivity for exploring number patterns in their everyday environments, often in idiosyncratic ways. Such behaviours result in the largely unconscious build-up of systems of mental strategies based upon their understanding of numbers and number relationships which are stored as declarative knowledge (Hunter, 1978, p. 343). Nonetheless, the basis for the exceptional skill of expert mental calculators remains unclear. As Jensen (1990, p. 273) has observed, it may be motivational, or it may be attributable to superior powers of concentration, rather than to an ability variable, although it is likely to be a combination of all three. Hunter (1977a, p. 40) has concluded that the success of those engaged in mental computation is limited by their short-term memory capacities. However, strong evidence to support this capacity as a decisive factor in determining superiority in mental computation does not exist (McIntosh, 1991a, p. 4). In

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considering the demands placed on memory during mental computation, Hunter (1978, p. 339) has proposed three forms, namely, Memory for calculative method, Memory for numerical equivalents and Memory for interrupted working. Although the latter places demands on short-term memory, the first two draw on long-term memory components (see Figure 2.3). Given the limitations of short-term memory, proficient mental calculators tend to shift the burden of calculation onto long-term memory by accessing their superior store of declarative and procedural knowledge to devise efficient methods for approaching a problem (Hunter, 1978, p. 343). The key is to select a strategy from long-term memory that maximises the use of the properties and relationships between the numbers that are detected in the problem task environment (Hope, 1985, p. 358), thus reducing the number of partial answers that need to be determined. Further, the demands on short-term memory are reduced by their tendency to work quickly, thus allowing stored partial answers to be used before forgetting occurs.

7. What are the teaching approaches and sequence necessary for the development of mental computation skills?

The approaches recommended for developing skill with mental computation stem from the belief that the focus in classrooms should not be solely on its social utility, but should recognise that mental strategies are, in essence, ways of thinking about mathematics (Cobb & Merkel, 1989, p. 80). The focus needs to be on developing resourceful and ingenious approaches to working with numbers─to promote number sense. Hence, children's learning experiences should be such that their spontaneous calculative plans are recognised and valued (Heirdsfield & Cooper, 1995, p. 5), so that mental computation does not lose its essential characteristics, as described by Plunkett (1979, pp. 2-3). In contrast to what has been defined as the traditional approach to mental computation in this study, where the focus has been on the correctness of the answer, Robert Reys et al. (1995, p. 305) have identified two broad approaches in which mental strategies are of primary concern. The first, which may be classified as a behaviourist approach, involves the direct teaching and practice of a predetermined set of mental strategies in preparation for paper-and-pencil

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computation, whereas the second is a constructivist approach. This approach reflects the belief that mental computation is a process of higher-order thinking in which the act of generating a mental strategy is of equal importance to its application (R. E. Reys et al., 1995, p. 305). In the former, the teacher takes an authoritarian stance, whereas with the implementation of a constructivist approach the teacher assumes the role of an intellectual coach, assisting students to develop distinctive calculative systems, a characteristic of skilled mental calculators (Hunter, 1977a, p. 40). However, as Hazekamp (1986, p. 117) has observed, little is known about how children develop strategies for mental computation. Further, there is some evidence (Flournoy, 1954, p. 153; Gracey, 1994, pp. 112-116; Josephina, 1960, p. 200; Markovits & Sowder, 1994, p. 22; Schall, 1973, pp. 365-366) to suggest that gains are made, at least with respect to the correctness of the answer, where systematic instruction has been provided. Hence, a third approach suggested by Barbara Reys (1991, p. 9) and Carroll (1996, p. 8) is worthy of consideration. This approach, which contains elements of the first two, has as its focus discussion and the sharing of devised mental strategies with teachers and other students. Although the teacher, under this essentially constructivist approach, does not present a predetermined set of strategies to children, alternative solution paths may be presented to children in instances where, for example, the efficiency of particular procedures is of primary concern. More generally, as Rathmell and Trafton (1990) assert, "mental computation is more appropriately viewed as an ongoing emphasis in a mathematics program than as a separate topic composed of subskills arranged in order and taught as separate lessons" (p. 160). Children need to be given opportunities to play with numbers (C. Thornton, 1985, p. 9), using games and puzzles, for example. The aim is to assist children to see how to calculate mentally and to come to understand that mental computation is a creative process with many legitimate solution paths for particular problems. This understanding will be enhanced where opportunities are provided for children to share and justify their answers and methods of solution (Rathmell & Trafton, 1990, p. 58). Where focussed teaching occurs, it is recommended that several approaches to a few problems be considered, rather than one approach to each of a large number of questions (McIntosh, 1988, p. 261). It is essential that children find mental computation enjoyable and that they experience success.

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However, children should be challenged only when they are ready to be challenged (J. P. Jones, 1988, p. 44). Although this may be difficult for a teacher to determine, the critical indicator is the perceived confidence with which a child approaches mental computation in particular situations. The overriding emphasis needs to be on integrating mental computation into all relevant classroom activities in an endeavour to maximise the links to the real-world of the child, thus capitalising on their idiosyncratic mathematical knowledge. Nevertheless, consideration needs to be given to the emphasis on mental viz-á-viz written methods of computation at particular stages within a class mathematics program. There is some evidence to suggest that children have difficulty with mental computation when paper-and-pencil skills, particularly the standard written algorithms, are developed prior to a focus on mental strategies (Cooper et al., 1992, p. 100; Musser, 1982, p. 40). This has resulted in the suggestion that, for each operation, the focus should be on mental computation prior to the introduction of written procedures (see Figure 2.5). Carol Thornton et al. (1995, p. 40) have suggested that the consequences of such an approach would be the development of more flexible computational strategies, both mental and written. This contrasts with the inflexibility of the mental strategies which result from the traditional approach to teaching mental computation.

2.11 Concluding Points

For children to develop powerful mental computation strategies a second revolution, as proposed by McIntosh (1992, p. 134), is required. Although McIntosh's (1992, pp. 131-133) first revolution centres on changing the way in which mental computation is viewed by teachers, the second revolution is concerned not simply with the provision of opportunities for children to develop skill with mental computation, but, more importantly, that these opportunities should occur in association with teachers finding numbers a source of enjoyment for themselves. Should this occur, McIntosh (1992) asserts that "a window onto a wonderful, enjoyable and fascinating world─the world of numbers─[will be opened for children]" (p. 134). However, as Barbara Reys (1991) cautions, the changes in teacher attitudes and "in the content and methodology of teaching computation...will not

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occur without considerable teacher education, and [these] will take both time and a concerted effort by the entire mathematics education community" (p. 12). The nature of the teacher education required is not only dependent upon the current state of teacher knowledge and attitudes towards mental computation, but also on an understanding of the context within which these have developed. Hence, prior to the development of a proposed syllabus strand for mental computation (Chapter 5) and a consideration of the implications for professional development for Queensland teachers (Chapter 6), an understanding of their current beliefs and practices needs to be gained (Chapter 4). Additionally, an understanding of the nature of mental computation and how it has been taught under each of the Queensland syllabuses provides the background and context for current beliefs and practices. An analysis of mental computation in the mathematics syllabuses in Queensland from 1860 to 1965 is the focus of the next chapter (Chapter 3).

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CHAPTER 3

MENTAL COMPUTATION IN QUEENSLAND: 1860-1965

3.1 Introduction

As stated in Chapter 1, little has been documented about mental computation within Australian mathematics curricula. In an attempt to redress this situation, this study aimed to document the nature and role of mental computation, and associated pedagogical practices, under each of the Queensland mathematics syllabuses─the fourth principal purpose of this study outlined in Section 1.3. Further, by undertaking an analysis of mental computation prior to the introduction of the current Years 1 to 10 Mathematics Syllabus (Department of Education, 1987a), issues related to the refocussing of mental computation, as outlined in Chapter 2, can be placed in context with past beliefs and teaching practices. Such knowledge is essential to informed debate about the features of arithmetic teaching necessary for meeting the future needs of Queensland primary school children. As noted earlier, in order to introduce changes to an educational program, it is essential to know what has gone before (Skager & Weinberg, 1971, p. 50)─to be able to capitalise on past successes while avoiding the pitfalls. The analysis of mental computation and Queensland mathematics syllabuses has been divided into three time-periods: (a) 1860-1965, (b) 1966-1987, and (c) post-1987. These periods were selected on the basis of the perceived emphasis on mental computation, as embodied in syllabus documents and their implementation. The first of these, which is the focus of this chapter, is characterised by syllabuses which contained specific references to the mental calculation of exact answers beyond the basic facts. Hence, of the three periods, it is the one for which a depth of understanding of the nature and role of mental computation in Queensland primary school classrooms may be gained from primary documents.

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3.1.1 Method

In contrast to empirical research, “there is no single, definable method of historical inquiry” (Edson, 1986, cited by Wiersma, 1995, p. 234). Such inquiry is primarily qualitative, with the process being essentially a holistic one. Although the steps in undertaking historical research can be delineated, there is considerable overlap of the activities associated with each─for example, analysis and interpretation occurs during the identification of sources of evidence, during data collection, as well as during the synthesising and interpretation phases. Subsequent to the identification of “Mental Arithmetic in Queensland: 1860-1965” as the topic to be investigated in this study, the following steps were undertaken:

• The identification of relevant sources of evidence and collection of data. • The evaluation of the data for their authenticity and validity. • The synthesis of the data into a meaningful thematic pattern. • The reporting of the analysis and interpretation of the data, together with the conclusions drawn─this chapter. (Borg & Gall, 1989, p. 805; Wiersma, 1995, p. 235)

Sources of Evidence

During this study, the basic rule of historical research was followed, namely to place an emphasis on primary sources of evidence (Wiersma, 1995, p. 238). The primary data, upon which the issues of significance to mental computation in Queensland were analysed, were obtained from:

• Queensland mathematics schedules and syllabuses (1860-1964) published in the Queensland Government Gazette (1860-1904), The Education Office Gazette (1904), and as separate documents (1914-1964). • Annual reports of General Inspectors (1876-1904) and District Inspectors of Schools (1869-1965) published as part of the annual reports of the Board of General Education (1865-1875), Department of Public Instruction (1876- 1939), or as archival material (1940-1965).

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• The Education Office Gazette (1899-1965), the official publication to schools by the Department of Public Instruction, and from 1952 by the Department of Education in Queensland. • Queensland Education Journal (1895-1922) and Queensland Teachers’ Journal (1923-1965), published by the Queensland Teachers’ Union. • Text-books referred to in official publications─for example, Gladman (1904)─ and ones identified as being used by teachers to support their teaching of mental arithmetic─Larcombe (n.d.), for example. • First-hand accounts of experiences in Queensland primary schools─for example that of Hanger (1963).

Data were also obtained from such secondary sources as (a) the reports of studies into aspects of Queensland education─for example, Greenhalgh’s (1957) study of the 1952 syllabus; (b) books and articles on the history of Queensland education─Logan & Clarke (1984), for example; and (c) newspaper reports and articles of relevance to the teaching of mental arithmetic in Queensland primary classrooms─for example, “Scientific & Useful” (1882). Wherever possible, the data from these sources were cross-checked with the original references, or more generally against such primary sources of data as the reports of District Inspectors. The major sources of historical evidence were obtained from (a) the Queensland Department of Education’s History Unit, (b) the Queensland State Archives, and (c) the library of the Queensland Teachers’ Union. Data were also obtained from documents, books, and articles held in (a) the Fryer and Main Libraries, University of Queensland; (b) John Oxley Library and State Library of Queensland; (c) Mitchell Library, Sydney; and (d) private collections. During historical research sources of data need to be subjected to both external and internal criticism to determine their authenticity and validity with respect to the topic being investigated (Rodwell, 1992, p. 97). Given the nature of this study and its sources of information, determining the authenticity and validity of the primary documents was not as significant an issue as for those studies not primarily based on official documents and reports. The authenticity of the syllabus documents, District Inspectors’ reports and articles in the Queensland Teachers’ Journal─the documents on which this study was largely based─was without question.

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The validity of the beliefs espoused in the reports and articles was substantiated primarily through the completeness of the historical record and the consistency with which the various District Inspectors and writers expressed their views. All the reports of District Inspectors were accessed, either directly as archival material from 1940, or in the Annual Reports of the Board of General Education and Department of Public Instruction prior to 1940. Although only a summary of each District Inspector’s report was published in the Annual Reports from 1920 to 1939, it was evident that the majority of District Inspectors commented on mental arithmetic, as part of their reports on Arithmetic teaching and performance in the schools within their inspectorial districts. Nonetheless, the detail in the reports varied, with some District Inspectors─Macgroarty during the period 1879 to 1902 and Mutch from 1906 to 1925, for example─being particularly insightful. During the analysis of the data gathered and during the preparation of this study’s findings, cognisance was given to the inherent biases within the various reports and articles. It was recognised that key sources of data were often written from inherently conflicting perspectives about what occurred or should have occurred in classrooms. The impact of particular reports of District Inspectors and the articles in the journals of the Queensland Teachers’ Union was also unclear. However, a consideration of the opinions and recommendations expressed by different authors over time has enabled a clear understanding of the issues surrounding mental arithmetic to be ascertained and reported.

Research Questions

In historical research, a set of questions rather than an hypothesis is often devised to guide and focus the data collection and analysis (Cohen & Manion, 1994, p. 36). Such questions are effective for enhancing the continuity of the information presented (Wiersma, 1995, p. 236). In this study, to guide the analysis of past syllabuses, and to facilitate the identification of the major similarities and differences between current and past beliefs and practices, and those recommended as essential for gaining mastery of the calculative process, the following research questions were posed:

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These questions were derived from the fourth principal purpose of this research project, which was stated in Section 1.3, and emphasised in the introduction to this chapter.

Structure of Analysis

Reports in historical research do not have a standard format. They may be organised chronologically or thematically, with the format influenced by the nature of the questions posed (Wiersma, 1995, p. 832). In this study, the report of the research into past beliefs and practices also needed to be linked to the analysis of issues related to mental computation presented in the previous chapter. Such a link facilitated the formulation of the proposals for the future of mental computation in primary classrooms, as presented in Chapters 5 and 6. This chapter, therefore, has been structured thematically, on themes suggested by the analysis of the pedagogical, socio-anthropological, and psychological literature reported in Chapter 2, namely:

• Background issues related to syllabus development and implementation which may have influenced the role and nature of mental arithmetic within the classroom. • The terms used to describe the ability to calculate exact answers mentally.

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• The roles ascribed to mental arithmetic. • The nature of mental arithmetic from the perspectives of syllabus content, syllabus notes, textbooks, and psychological and pedagogical beliefs, and the constraints which impinged on syllabus implementation. • The teaching methods used to develop skill with calculating exact answers mentally.

3.2 Selected Background Issues Related to Syllabus Development and Implementation

It is evident that the importance of mental calculation has long been recognised by Queensland educators. District Inspector Mutch (1906), in his annual report for 1905, reiterating Fish's (1874, p. vii) position, rhetorically asked: "Should not all arithmetic be mental, and the pencil called into requisition only when the numbers are large? Are not slate and paper used merely to lessen the strain on the memory?" (Mutch, 1906, p. 63). From another perspective, it can be considered that all computation, including written and technological, involves a mental component. However, from a review of the primary sources of historical data, there appears to be little documentary evidence to link mental arithmetic specifically to issues which surrounded and underpinned the syllabuses formulated during the period being investigated. Nevertheless, it is reasonable to assume that such issues would have had some influence, at least indirectly, on the nature of mental arithmetic, as embodied in each of the syllabuses, and as presented to children. Among the key issues which impinged on the development and implementation of syllabuses were: (a) the function of primary education, including its role as a preparation for the after- life of the child; (b) the focus of teaching─the subject or the child; (c) the principles on which the syllabuses from 1905 were based; (d) syllabus interpretation and overloading; and (e) teacher freedom versus syllabus specificity. These issues and the links between them, which constitute the context within which mental arithmetic was taught, are to be explored using the following organisers: (a) Focus of Syllabus Development and Implementation, (b) Principles Underlying the Syllabuses from 1905, and (c) Syllabus Interpretation and Overloading.

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3.2.1 Focus of Syllabus Development and Implementation

Syllabus development and particularly its implementation were characterised by conservatism between 1860 and 1965 (Department of Education, 1978, p. 5; "Der Tag,” 1936, p. 3; Lawry, 1968, p. 568). This conservatism had its origins in the colonial period in which educational authority was excessively centralised (Creighton, 1993, p. 77), initially with the Board of General Education and subsequently with the Department of Public Instruction, following the passing of the Education Act of 1875. Furthermore, the Department was dominated in its early years by Under Secretary Anderson and General Inspector Ewart1 "whose administration was authoritarian and whose philosophy of education─based on mental discipline theory─[became] increasingly antiquated" (Creighton, 1993, p. 77). These officers were not greatly interested in educational innovations external to Queensland2, nor did they encourage teachers or District Inspectors to participate in the decision making processes (Barcan, 1980, p. 185). "So completely was the spirit of teacher-participation in the framing of school activities suppressed that the medieval idea of education, as the imparting of ‘correct knowledge,' almost completely dominated the curriculum" (“Professional Standards,” 1936, p. 1). The attitudes of Anderson and Ewart gave little scope for hope of rapid progress in state education, no matter how urgent the needs (Lawson, 1970, p. 228), needs that included syllabus revision to reflect the changing nature of Queensland's society and economy (Lawson, 1970, p. 215), and improved teacher training (“The Queensland State Department of Education,” 1966, p. 330; Wyeth, 1955, p. 156). Their resistance to change also occurred in an environment characterised by public apathy and opposition towards education (Lawson, 1970, p. 216), coupled with satisfaction with the state of education as felt by politically influential groups (Tyrrell, 1968, p. 154)─to such an extent that the systematic transformation of Queensland's education system was not urged by any influential group prior to the outbreak of World War 1 (Tyrrell, 1968, p. 155). The debate surrounding the passing of the State Education Act of 1875 resulted in the Department of Public Instruction, for reasons of political feasibility, being given the responsibility for controlling a narrow system of primary education (Lawry, 1975, p. 60). Sir Samuel Griffith, the architect of the 1875 Bill and the first Minister for Public Instruction, considered that "the duty of the state...was merely to give that

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rudimentary instruction which would enable the child to become a good member of society" (Griffiths, 1874, p. 395). This view reflected that of "Omega" (1871) who suggested that:

The...object [of education was] to develope (sic) and improve to the utmost all the powers and faculties, and to give instruction in all those branches of knowledge which are considered suitable and necessary to our probable circumstances and condition in after-life....[Education also included] the formation of good, and what is much more important, the eradication of bad habits [particularly for the poorer and working classes]. (p. 8)

The main focus of the 1876 Syllabus was therefore on reading, writing and arithmetic, although object lessons, drill and gymnastics, vocal music and needlework for girls were also included. In keeping with the rudimentary nature of the desired instruction, the higher mathematics subjects, Euclid and algebra, which were part of the 1860 Syllabus for the Fifth Class, were not considered necessary for a primary education. The frequency of syllabus change between 1876 and 1905 (see Table 3.1) reflected the indecision surrounding the nature of the curriculum for children in upper primary classes, particularly for those who were not able to win scholarships for a secondary education3 at one of the private fee-charging Grammar Schools (Dagg, 1971, p. 14). The modifications that were made were undertaken to accommodate pressures for the inclusion of practical subjects, drawing in 1894, for example, but without disturbing the predominance of the mental disciplinary approach to education (Lawry, 1968, pp. 635-636). Reports from District Inspectors indicating poor results in formal subjects, which included mental arithmetic, were ignored by Anderson (Lawry, 1968, p. 570). Despite the progressive outlook embodied in the 1904 Syllabus (Wyeth, 1955, p. 157), in essence it maintained a belief in a narrowly conceived primary school curriculum (Lawry, 1968, pp. 604, 606). As Ewart (1905) pointed out in his report for 1904: "What is needed...is to show how little ‘new' there is in it. Stand the ‘three R's' where they did? Undoubtedly" (p. 161). The Education Conferences4 of 1903 and 1904, attended by departmental officials, District Inspectors and teachers, thus avoided the task of a full curriculum revision, although reports were prepared by a

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range of sub-committees, including one which focussed on arithmetic5 (Lawry, 1968, p. 602). The 1930 Syllabus was considered by Chief-Inspector Edwards (1933) to be one that, in content and aims, embodied "a compromise between the extreme English theory of former years─knowledge for its own sake, and the extreme American preference for utility and efficiency" (p. 27). Edwards (1933) noted that whereas the English view tended to dismiss the value of practical work, a view previously espoused by Anderson and Ewart, the American "attitude tended to dismiss as mere ‘mental lumber' anything of which the practical value was not apparent" (p. 27). The philosophy that underpinned this syllabus, as well as subsequent ones, was encapsulated in the Report of the Secretary of Public

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Table 3.1 Queensland Mathematics Schedules and Syllabuses: 1860-1965

1860 Regulations for the establishment and management of primary schools in Queensland. (1860). Brisbane: Fairfax and Belbridge.

1876 Schedule V: Table of the minimum amount of attainments required from pupils for admission into each class in primary schools. (1876). Queensland Government Gazette, XVII(36), 825.

1892 Para 143: The course of instruction for each class. (1891). In State Education Act of 1875; together with the regulations of the Department (pp. 23-24). Brisbane: Government Printer. Para 144: Standards of proficiency. (1891). In State Education Act of 1875; together with the regulations of the Department (pp. 24-28). Brisbane: Government Printer.

1894 Schedule V: Course of instruction for each class. (1894). Queensland Government Gazette, LXII(34), 320-321. Schedule VI: Standards of proficiency. (1894). Queensland Government Gazette, LXII(34), 321-323.

1897 Schedule V: Course of instruction for each class. (1897). Queensland Government Gazette, LXVII(62), 798-799. Schedule VI: Standards of proficiency. (1897). Queensland Government Gazette, LXVII(62), 799-801.

1902 Schedule XIV: Course of instruction for each class. (1902). Queensland Government Gazette, LXXIX(18), 169-170. Schedule XV: Standards of proficiency. (1902). Queensland Government Gazette, LXXIX(18), 170-174.

1905 Schedule XIV: Course of instruction for each class. (1904). Education Office Gazette, VI(11), 204-211.

1915 Department of Public Instruction. (1914). The syllabus or course of instruction in primary schools with notes for the guidance of teachers. Brisbane: The Department.

1930 Department of Public Instruction. (1930). The syllabus or course of instruction in primary and intermediate schools. Brisbane: The Department.

1938 Department of Public Instruction. (1938). Amendments to course of instruction in primary and intermediate schools. Brisbane: The Department.

1948 Department of Public Instruction. (1948a). Course of instruction in primary and intermediate schools. Brisbane: The Department.

1952 Department of Public Instruction. (1952b). The Syllabus or course of instruction in primary and intermediate schools, Book 3: Mathematics. Brisbane: The Department.

1964 Department of Education. (1964). The syllabus or course of instruction in primary schools: Mathematics. Brisbane: The Department.

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Instruction for 1920: "Education authorities have come to recognise that education has a life-long relationship with the various aspects of citizenship─physical, social, industrial, and political─and this conception steadily widens" (Huxham, 1921, p. 21). Education had come to be "considered in terms of life rather than of purely intellectual equipment....[It was recognised] that the purpose of any educational scheme is to develop the full powers of [the] personality [of the child]" (J. A. Robinson, 1945, p. 12)─the "principle of social living" outlined in the 1930 Syllabus (Department of Public Instruction, 1930, p. vi) and the 1952 Syllabus (Department of Public Instruction, 1952a, p. 2). The centre of interest, therefore, had shifted from the subject to the child. Nonetheless, as an indicator of the conservative implementation of syllabuses, particularly from 1905, is the proposition in the 1952 Syllabus that "modern thought regards the child as the focal point in the educational process" (Department of Public Instruction, 1952a, p. 1). Influenced by the beliefs of John Dewey6, such a focus was first proposed in the 1904 Syllabus. The preface to the schedule indicated that the course of instruction was designed "to increase the influence of the school as an agent in the intellectual, moral, and social development of the child" (“Schedule XIV,” 1904, p. 200), a principle of utmost importance and potency (Roe, 1915, p. 29). Ewart (1907) stressed in his report for 1906 that:

[Teachers] must learn without loss of time that there is before them in the profession they have joined a new study they must tackle seriously, and that it will take them all their lives to learn. That subject is the CHILD. Hitherto, it has been arithmetic, languages, and the rest of it─how to know these, understand them, explain them; now it must be how to bring all these in turn or together to bear upon the life and wellbeing, the growth and development of the children placed in their charge. (p. 39)

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3.2.2 Principles Underlying the Syllabuses from 1905

The development of the 1904 Syllabus occurred during the early stages of a period, 1900 to 1914, noted for its considerable reform and progress7 in Queensland education (“Parliamentary Select Committee,” 1980, p. 4). This period signalled, in theory at least, an end to the emphasis on the theory of mental discipline, with its focus on teacher and subject, and heralded a concern for the child (Creighton, 1993, p. 77; Logan & E. Clarke, 1984, p. 3). Moreover, the 1904 Syllabus could be considered to be the ideological turning point for Queensland State school curricula (Creighton, 1993, p. 78), "an epoch in [Queensland's] educational history" (“The Revised Syllabus,” 1914, p. 81). However, the high ideals expressed in the preface to this syllabus─"an excellent ‘New Charter' for...primary schools" (“The New Syllabus,” 1904, p. 143)─were doomed to lie dormant, even following the passing of Anderson's influence in mid-1904 and that of Ewart in 1909. The preface to the 1904 Syllabus, the first to give teachers assistance in syllabus implementation, not only provided some guidance as to how mathematics should be taught, but, possibly more importantly, delineated the principles on which the schedule was based. In addition to indicating that the school needed to increase its influence on the pupils' social, moral and intellectual development, it was stated that:

[The syllabus was] designed to give practical application in the teaching work of schools to the principle of the correlation of the subjects of study, to make "the self-activity of the pupil the basis of school instruction," [and] to bring the work of the pupil into closer touch with his home and social surroundings. (“Schedule XIV,” 1904, p. 200)

These principles were espoused by advocates of the New Education Movement which, in the late nineteenth century, constituted an international reaction against the narrowness and formality of the prevailing subject-based pedagogy (Turney, 1972, p. 32). The nature knowledge introduced into the syllabus was to be the main vehicle for the implementation of these principles (Creighton, 1993, p. 79). However, it was also recognised that, in teaching mathematics:

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From the earliest stages the child [should be] led to deal with quantities of actual things and with the measurement of quantities, using various units....The youngest children should not only see but handle the quantities with which they deal, and actually make measurements on which they are to operate....It is valuable education for the pupil to be made to do things for himself, instead of merely seeing the teacher do them. (“Schedule XIV,” 1904, p. 201)

Proponents of aspects of the New Education believed that teaching through chalk-and-talk was anathema to the essence of the development of the young child who wanted to be active and to investigate material for himself (Ensor8, reported in “Wasted Time at School,” 1937, p. 15). The focus in Queensland, particularly prior to World War 1, however, was on extending basic education throughout the state, with the result that the impact of the implied reforms to teaching was tentative and incomplete (Tyrrell, 1968, p. 78), at least until the late 1970s when renewed efforts to reform the way mathematics was taught began to be reflected in classroom practice. Greenhalgh (1949a) observed that classrooms continued to be characterised by passivity rather than activity with "children [being] required to sit still and listen far too frequently" (p. 5). This, in Greenhalgh's (1949a, p. 5) view, was due to teachers trying to cover too much work with its concomitant cursory consideration of many aspects of the mathematics curriculum, including mental arithmetic. Nonetheless, Edwards (1929, p. 31) believed that, by 1928, teachers were making links between subjects to an extent not considered possible in 1905 when the revised schedule was introduced. In Roe's (1915) view, a focus on the correlation of subjects and the practical application of school knowledge to the features of daily life were "merely improvements of method by which a child's interest in his lessons [were] awakened and maintained" (p. 29). It was recognised that interest is the "mainspring in educational progress" (Roe, 1915, p. 29) and leads to a desire to become involved in the learning process. The General Memorandum on 1948 Amendments to the [1930] Syllabus (1948) gave recognition to this:

By making a child desire to solve a certain problem gets him to put his Heart into it; it gets him to use his Head in the collection and arrangement of the

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relevant facts; it gets him to use his Hand in the construction of apparatus to be used in the solution of the problem. (Department of Public Instruction, 1948b, p. 16)

This statement was presented in context with an advocacy for the Project Method, the principal manifestation of an emphasis on self-activity in the upper grades. As Greenhalgh (1957, p. 76) pointed out, the 1930 Syllabus virtually instructed teachers in small-schools and of upper grades in larger schools to use this method which allowed the teacher to "set problems going, so that the pupils, in order to solve them, must arrive at the knowledge of the [mathematical] principle or rule" (Department of Public Instruction, 1930, p. vii)─an emphasis on self-education was therefore advocated. The acceptance and implementation of the principles and teaching approaches outlined in the 1904 Syllabus relied on the training of a whole new generation of teachers (“The Queensland State Department of Education,” 1966, p. 330). Although the long awaited Teachers' College was opened in 1914, many teachers continued to be trained under "the old hard system" (Wyeth, 1955, p. 157), the pupil- teacher system not being phased out until 1935. Hence, teachers were generally not prepared for the changed expectations. Representatives of the Queensland Teachers' Union, during a deputation to the Minister of Public Instruction in 1946, asserted that few teachers were undertaking an activity approach to learning due to a lack of equipment and materials, a lack of time as well as not having an understanding of how to incorporate activity based learning into their teaching programs (“Deputation to Minister,” 1946, p. 4). Wyeth (1955, p. 158) asserted that it would be reasonable to state that the aspirations crystallised into specific statements of ideals in the 1904 Syllabus had not been realised by the mid-1950s. This applied particularly to the principles on which the syllabus was based, even though they continued to be emphasised in subsequent syllabuses (Department of Public Instruction, 1914, p. 6; 1930, p. vi; Edwards, 1951, p. 25). Further, their implementation was not supported by all senior officers of the Department. Director-General Watkin voiced his official antagonism to such education in his 1956 Report:

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In emphasising "Learning by Doing" [teachers] concentrated on activity, any activity provided it was not intellectual; self-activity developed into licence; self- discipline degenerated into lack of discipline, and objective standards were reduced if not entirely neglected. (Watkin, 1957, cited by Barcan, 1980, p. 308)

That the principles of correlation of subjects, self-activity and closer links between home and school were not being implemented was given recognition at various times during the 1920s and 1930s. Edwards (1922), on retiring as the president of the Queensland Teachers' Union, noted that although "the educational work was throbbing with new ideas, having as their objects the freedom of the child, the greater elasticity of the curriculum, and the closer connection of education with reality, ...the influence of these movements has scarcely been felt in Queensland" (p. 1). This observation was reiterated in 1939 in an editorial in the Queensland Teachers' Journal. While noting that the function of the school was to provide more than a grounding in the three Rs, it was concluded that: "Queensland [had] been in an educational backwater where the flood tide of this reform movement [had] scarcely rippled the surface" (“Educational Reform,” 1939, p. 1). District Inspector Pestorius (1940) noted that "the complacency and self-satisfaction of many teachers in their methods [should be undermined, and that]...for complete success...we must devise our methods or base them upon our own understanding of the individual children" (p. 15). In reality, it has only been since the 1970s that gains have been made in placing the child in his/her environment (Creighton, 1993, p. 78), the prime focus of the 1904 Syllabus principles. This has been facilitated by such factors as the cessation of the external Scholarship examination from 1963, the discontinuation of annual inspections for teachers in 1970, and improvements in the education of teachers, the influence of which on mental arithmetic is to be analysed later in this chapter. Cramer9 (1936) noted that "just as external examinations may make teachers think they are teaching arithmetic...rather than boys and girls, so inspectors may forget boys and girls and think of the machinery of the system and the teachers that they are visiting" (p. 8).

3.2.3 Syllabus Interpretation and Overloading

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The 1904 Syllabus was the first to be accompanied not only by the principles on which the syllabus was based, but also by suggestions concerning its implementation. These were intended to be suggestive rather than specific, namely, "to indicate the scope of treatment, and to lay down fundamental principles and lines of direction at various stages of the pupils' progress; but it [did] not prescribe detailed methods for teaching individual subjects" (“Schedule XIV,” 1904, p. 203). St. Ledger (1905) regarded the 1904 Syllabus as the "Teachers' Magna Carta" (p. 218). It gave freedom to the teacher. Importantly, it was accompanied by changes to the system of inspection. No longer were District Inspectors required to examine every child in every subject. This double change was considered to be "the most important advance...since the passing of the Education Act of 1875" (St. Ledger, 1905, p. 218). The Queensland Teachers' Union interpreted the 1904 Syllabus as being based on the principle of thought, whereas previous syllabuses were considered to have been based on pressure (“The New Syllabus,” 1904, p. 143), pressure from "the blighting influence of percentages" (Shirley, 1905a, p. 189). Recognition was given to the belief that "there is no absolutely best method of treating any subject; one method is best suited to one type of teacher, another is best for others10" (Department of Public Instruction, 1914, p. 10)─methods which were surrounded in controversy (“Teaching Hints,” 1908, p. 15). However, teacher freedom was not extended to the selection of course content. Director of Education McKenna (1936, p. 12) reported that he could not agree with the absence of definite schemes of work for teachers, as occurred in England. His disagreement, however, appears to have arisen primarily from administrative rather than from educational considerations. He stated that he would be disinclined to advocate the extension of freedom of this type for teachers as it "would demand a much more highly trained type of teacher than we have at present, and would necessitate the retention of head teachers in the same schools for considerable periods of time" (McKenna, 1936, p. 12). Nonetheless, Edwards (1937b, p. 7), on replacing McKenna as Director of Education, suggested that the greatest hope for educational progress lay with improving the quality of teachers. Cramer (1937, p. 8) observed that in the Australian states which he visited, of which Queensland was one, the preparation of teachers was referred to as teacher training rather than teacher educating. The focus of teacher training, Cramer (1937) observed, was the production of "a competent school-room technician whose training [was] largely restricted to the

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syllabus he [was] expected to teach" (p. 8), rather than the development of a well- educated individual able to adapt to various situations. Such a view was not in conflict with teacher beliefs about education, which held that education was the act of training, although recognition was given to the development of the various intellectual, physical, aesthetic and moral faculties (Bensted, 1924, p. 34). Paradoxically, although scope for teacher freedom in planning for syllabus implementation remained a feature of syllabuses subsequent to 1905, syllabus outlines, particularly from 1930, increased in detail. This potentially incompatible tendency arose from a belief that inexperienced teachers needed greater guidance. Edwards (1938a), in his memorandum to schools concerning the 1938 Amendments, emphasised that "such specification [was]...not [intended] as a restriction of the freedom of the more experienced. Scope [was] still afforded for originality and initiative in the inclusion of matter and the selection of method" (p. 2). However, in his reports for 1936 and 1947, Edwards (1937a, p. 25; 1948, p. 20) noted that the spirit of professional freedom was not always grasped by teachers. He suggested that this, together with the calls for a greater prescription of work, was perhaps due to:

A form of inertia─a disinclination to get off the beaten track and to depart from methods to which they have become accustomed. [Further,] it [was] more likely due to timidity. Many teachers [were] not prepared to accept the responsibility of embarking on some original line of thought or action. (Edwards, 1937a, p. 25)

The Queensland Teachers' Union believed that the conundrum of professional freedom in association with calls for greater prescription, particularly with respect to the 1930 Syllabus and its 1938 Amendments, was due to the system of teacher inspection (“The ‘Dead Hand’ of Inspection,” 1939, p. 1). Most teachers interpreted the detailed statements as prescriptive, believing that District Inspectors would use them as their guide for assessing teacher performance (Turney, 1972, p. 45). In what could be interpreted as an admonishment of District Inspectors, Edwards (1948) emphasised in his report for 1947 that high marks on a teacher's "‘yellow card' should not depend upon rigid conformity with standards or curricula set up by the central authority" (p. 20). Calls for greater prescription, and for standardised

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work books, were designed to place limits on the ways in which the syllabus had been interpreted. However, the suggestions for standardised workbooks were frowned upon by the Queensland Teachers' Union as it believed that this would make the individualisation of learning even less likely (Burge, 1946, p. 13). Although the 1930 Syllabus in mathematics was a very exacting one, with little to be added to meet the requirements of the secondary school Junior Public Examination (Greenhalgh, 1957, p. 285), it was not uncommon to find that the examinations of District Inspectors and Head Teachers exceeded the requirements specified for particular grades (“Syllabus Debate,” 1938, p. 15). As "The Commentator" (1947) observed in the Queensland Teachers' Journal, "teachers and inspectors went into operation when the 1930 Syllabus was published; [and this] is not the same thing as putting the 1930 Syllabus into operation" (p. 21), a consideration that was often overlooked by teachers who advocated syllabus change. A multiplicity of textbooks (see Table 3.2), all more or less unofficial, became available to teachers, many of which were criticised for not being trialed before

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

Bevington, W. F. (Ed.). (n.d.). Queensland syllabus mental arithmetic. Brisbane: H. Pole.

Class teacher's manual of oral arithmetic for grades VII & VIII in Queensland schools. (n.d.). Warwick: Premier Educational Publishers.

Department of Public Instruction/Education. Queensland. (1946-). Arithmetic for Queensland schools. Brisbane: Government Printer.

Henderson, T., Irish, C. A., Bowden, L. T., & Watkin, H. G. (1932-). New syllabus arithmetic (Brooks' Queensland School Series). Brisbane: William Brooks.

Henderson, T., Irish, C. A., Bowden, L. T., & Watkin, H. G. (1932-). New syllabus mental arithmetic (Brooks' Queensland School Series). Brisbane: William Brooks.

Larcombe, H. J. (n.d.). Speed tests in mental arithmetic: Senior book 1a (The Minute a sum series). London: Evans Brothers.

Mental arithmetic: Senior grades (The Moreton Series). (1926-). Brisbane: Moreton Printing Company.

Mutual Aid Society, Ipswich. (n.d.). Queensland syllabus mental arithmetic: Grades VI-VII. Sydney: Philip.

Olsen, F. J. (1953). Oral arithmetic (McLeod's School Series). Brisbane: McLeod.

Potter. (n.d.). Mental and intelligence tests in common-sense arithmetic.b London: Pitman.

Test papers based on model work books: Grades III to VI. (n.d.). Brisbane: Pole.

Thompson, F. C. (Ed.). (1930-). The Queensland mathematics. Brisbane: Ferguson.

Wisdom, A. (1932). Arithmetical dictation.a London: University of London Press.

Note. aIdentified as a text used in Queensland by the presence of a Queensland School's identification stamp in a copy held in the Oxley Library. bA text listed as a valuable guide to the intelligent treatment of the arithmetic in the 1930 Syllabus (Department of Public Instruction, 1930, p. 31).

publication ("Debunker,” 1940, p. 14). Some were compiled or, at least, supported by District Inspectors─for example, those prepared by the South Coast Syllabus Notes Committee under the direction of ex-District Inspector Bevington. It was asserted by "Debunker" (1940, p. 14) that some inspectors had insisted upon the use of particular texts. Some years prior to this, in an endeavour to promote the teaching of mental arithmetic, District Inspector Mutch had given the majority of schools in his district a copy of Burt's Mental Tests in Arithmetic11, which he

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regarded as exemplary (Mutch, 1924, p. 40). Model work books were also prepared by District Inspectors for commercial publication. For example, District Inspectors Router and Pascoe, assisted by District Inspectors Baker and Inglis, prepared a series of Programmes of Work (1932), which included suggestions for oral arithmetic for each term of the school year. These texts tended to provide exhaustive treatments, with their maximums becoming teachers' minimums (Darling Downs Branch, 1946, p. 21). The fear of teachers was that if the children in their classes did not achieve high percentages on the inspectors' test, they would be marked down (“Syllabus Debate,” 1938, p. 15). The Queensland Teachers' Union believed that "all concerned, with the exception of those most vitally concerned, the children, brushed aside the important General Introduction [to the 1930 Syllabus] in their eagerness to get at the actual requirements....The Syllabus was ruined by zealots with their fanatical shibboleth of ‘a bit ahead'" (“Editorial,” 1947, p. 2), an opinion shared by the Department in 1937 (Department of Public Instruction, 1937, p. 26). Pestorius (1942, p. 2) noted that some head teachers prescribed work from textbooks rather than planning independently from the syllabus. This was a criticism also levelled at the examiners for the annual Scholarship examination during the 1930s and 1940s─"What was found in a Scholarship Class text book could surely be used as part of the examination!" (Dagg, 1971, p. 57). Pestorius (1942) believed that "reliance upon the subject matter in textbooks rather than individual planning and initiative based upon Syllabus requirements [tended] to make the work stereotyped, and [was] likely to [have resulted in] cramming.” As Dagg (1971, p. 57) concludes, the textbooks which often went beyond the scope of the syllabus therefore set the standard. The General Introduction to the 1930 Syllabus recommended that "arithmetic, while emphasizing speed and accuracy, [should] not entail long and useless calculations" (p. vi). Further, the syllabus stressed that "the very practical purposes that mathematical work has to serve in the child's future life should regulate the character of its treatment in school" (Department of Public Instruction, 1930, p. 30). Nonetheless, "Der Tag" (1936) felt compelled to write that "nothing is left out [of the textbooks] even when ruled out by the Syllabus itself" (p. 3). In "Green Ant's" (1942, p. 17) view, the extensions of syllabus requirements were most flagrant for mental arithmetic. For example, in a textbook prepared by the Head Master of the Ascot Practising School as the main author, the mental work for Grade VI included such

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15 2 8 3 examples as: (a) Increase 3 by .37 of its value; and (b) Express 100 as a decimal (Henderson, Irish, Bowden, & Watkin, 1935, p. 10). This, in spite of the syllabus indicating that the oral work for Grade VI should be preparatory to the written work, with the notes accompanying the latter indicating that "the exercises [should] deal only with simpler fractions" (Department of Public Instruction, 1930, pp. 42, 43). Extensions to the syllabus were also encouraged by non-Queensland texts. Larcombe's (n.d., p. 2) Speed Tests in Mental Arithmetic, for example, a text referred to by Router and Pascoe (1932, p. ii), presented items such as the following for pupils aged 10½-12 years: 23,765 x 9 and 70,186 ÷ 7. Such instances prompted calls for the Department to publish official textbooks, the first of which was published for Grade 3 in 1946, as a means for gaining realistic interpretations of the syllabus (Darling Downs Branch, 1946, p. 21). With a layout similar to that of the commercial textbooks, these texts included sets of ten mechanical and problem arithmetic examples. However, it is unlikely that these textbooks proved to be a panacea. As the General Secretary of the Queensland Teachers' Union pointed out in 1951: "A teacher, a professional man, might be expected to do his work without a text book; a coach or a crammer is useless without one" (cited by Dagg, 1971, p. 33), a sentiment which echoed that of Edwards in his 1930 report (Edwards, 1931, pp. 28-29). Teacher discontent with the way in which the 1930 Syllabus had been interpreted, together with its perceived overloading, resulted in increased teacher input into syllabus development, to such an extent that, by 1952, the initiative for a new syllabus "came [right through], not from the administration, but from the teachers" (Greenhalgh, 1957, p. 84). Overloading was a traditional feature of Queensland mathematics syllabuses, and one that reached its climax in 1930 (Dagg, 1971, p. 32). Until 1915, overloading was, in Lawry's (1968) view, primarily due to the "absence of an articulated system of education from primary school to university" (p. 600). Nonetheless, mathematics tended to be "marked by the logical completeness of the schedule, based on an adult notion of what should be taught, as opposed to what the capacity of the average child could assimilate" (Greenhalgh, 1957, p. 77). Syllabus development to 1930 had been characterised by additions, such that, by 1928, the primary school pupil was regarded "as a polymath in comparison with his predecessor of twenty years [previously]" (Edwards, 1929, p. 31). However,

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Edwards (1929), the architect of the 1930 Syllabus, believed that no subjects could be omitted during the revision of the syllabus as "they [were] calculated to widen the outlook and to develop those qualities of body, heart, and mind that lead to the efficient and true fulfilment of function in life" (p. 31). It was emphasised in the General Introduction to the 1930 Syllabus that:

Complaints in regard to an overloaded curriculum usually come from those who make instruction and information the chief aims of school work, and who fail to regard added subjects, or branches of subjects, merely as further means of training and developing the mind of the child. (Department of Public Instruction, 1930, p. vi)

By 1933, the Queensland Teachers' Journal had cause to editorialise that the 1930 Syllabus was in urgent need of a thorough overhauling (“The Primary School Syllabus,” 1933, p. 1), a belief often repeated during the mid-1930s. Following a European study tour, Director of Education McKenna (1936, p. 13) gave recognition to the need for the syllabus to be revised. He noted that, in comparing Queensland's requirements with those expected overseas, the syllabus, particularly for the upper grades, demanded too high an academic standard. He doubted that one pupil in a hundred could have passed Queensland's Scholarship Examination at the age of 13½ years (McKenna, 1936, p. 13). Chief-Inspector Edwards also recognised the excessively high demands of the Queensland syllabus following his visit to the United States of America in 1935. He noted that:

The main fault of our educational system in Queensland is that we seem to be in a great hurry to cover the course. Children are required to gulp down rich material in a short space of time. They do not have sufficient time for mastication and digestion. They take this rather rich food too quickly and they suffer to some extent from mental indigestion. (Edwards, 1936, p. 17)

A study by Cunningham and Price (1934, p. 81) demonstrated that the majority of mathematics topics were introduced at lower ages in Queensland than in any other Australian state, as well as being below the age recommendations of the United States Committee of Seven12. This, and the length of time spent on

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mathematics13, particularly in the lower grades, were judged to be the chief contributing factors to the consistently high performances of Queensland primary school pupils in written arithmetic (Cunningham & Price, 1934, p. 93). However, as was noted in the press at the time of the release of the findings of a 1931 survey on arithmetic by the Australian Council for Educational Research, "thoughtful teachers see [this pre-eminence] as a menace rather than [as] an occasion for congratulations" (cited in “The Primary School Syllabus,” 1933, p. 1). Not only was this rush to impart knowledge "a negation of real teaching" (“The Primary School Syllabus,” 1933, p. 1), but as revealed by a curriculum survey of Australian schools published in 1950 by the Australian Council for Educational Research, Queensland children occupied the lowest position of all the states for each of the two reading tests (cited by Dagg, 1971, p. 26). In Queensland, the concentration had been on arithmetic, primarily written arithmetic, at the expense of reading. Edwards (1936, p. 17) also gave recognition to Thorndike's belief that individuals learn many things very laboriously when young that could be learned more easily and readily when older. This undoubtedly contributed to Edwards' giving the committee of departmental and union representatives "a definite mandate to secure relief" (Greenhalgh, 1957, p. 286) through the 1938 syllabus revision. It was characterised by omissions rather than additions. For example, for Grade V, mental exercises involving simple proportions were deleted from the syllabus, and "Ratio expressed as fractions and decimals" was moved to Grade VI (see Appendix A.10). However, the changes which were undertaken were not seen as final. It was stated during debate at the 1938 Annual Teachers' Conference that "the Syllabus is now in the transition stage and further amendments must come" (“Syllabus Debate,” 1938, p. 16). Nevertheless, there would appear to have been uncertainty in the late 1940s surrounding the development of a new syllabus. This uncertainty possibly arose from a lack of confidence on the part of Queensland syllabus designers to prepare a syllabus to match the needs of Queensland children without significantly relying on overseas trends, particularly since educational research across all curriculum areas specific to Queensland was limited (Creighton, 1993, p. 83). In a discussion paper, submitted to the Syllabus Committee in 1947, it was argued that a new syllabus should not be developed in the short term as there was:

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A need to await a clearer definition of the post-war international pattern and of the way of life which will be the lot of individuals therein....[Also] there was the need to await the further crystallization of general world education ideas and especially those within the Empire; [particularly there was a need to] await for a little more clarity concerning the innovations implied in the series of reports of the Consultative Committee on the English Board of Education and especially those suggested by the very next report of the Advisory Council in Scotland upon the curriculum of the Primary School. (Mathematics Subcommittee, 1947, p. 1)

The report of the Advisory Council on Education in Scotland supported the emphasis on the social utility of arithmetic. It recognised that arithmetic was required by all citizens in their daily lives, while stressing that an ability to undertake complicated problems was not one essential for all pupils. The Council also believed that children should be required to have an automatic recall of tables, and an efficiency and accuracy in simple calculations. Most importantly, it was deemed necessary for individuals to be able to correlate these skills with the needs of everyday life (Advisory Council on Education in Scotland, cited by Burge, 1947, p. 6). The latter constituted a reaffirmation of the third principle underlying each of the syllabuses since 1905, namely, that school work should be closely linked to activities outside the classroom. In accordance with this view, the Syllabus Committee received recommendations that calculation should be the core mathematics subject, and that speed and mechanical accuracy tests should receive greater prominence, particularly during Grade VI. Somewhat paradoxically, given the emphasis placed on social utility during this period, it was suggested that oral arithmetic had been given greater prominence than it deserved (Mathematics Subcommittee, 1947, p. 2). It was also proposed that:

Teachers should rest content mainly with teaching the rules listed in the Syllabus and applying them to simple mechanical operations of the same rule, stressing at all times accuracy and speed. The old type of "brain teasers" cannot justify its place in the modern Arithmetic Syllabus. The only "problems"

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to be taught will be those arising in Class Activities. (Mathematics Subcommittee, 1947, p. 3)

Given these beliefs, and the apparent hesitancy in drafting a new syllabus, it is not inconsistent that the goals for 1952 Syllabus were adopted from a dated publication of the Board of Education in England, namely, its Handbook of Suggestions (1937, pp. 499-500). Director-General Edwards (1951, p. 25) noted, in his report for 1950, that the Syllabus Committee did not find any fault with the 1930 educational objectives14. As a consequence, the general introductions to the 1930 and 1952 Syllabuses were essentially identical. With respect to mathematics, these objectives included an assertion that the role of the primary school was to "give [children] practice in the art of calculation" (Department of Public Instruction, 1930, p. v; 1952a, p. 1). In keeping with this belief, the 1952 Syllabus stated that the generally accepted goals for mathematics were: (a) "To help the child to form clear ideas about certain relations of number, time and space; (b) to make the more useful of these ideas firm and precise in his mind through practice in the appropriate calculations; and (c) to enable him to apply the resulting mechanical skill intelligently, speedily and accurately in the solution of every-day problems." (Department of Education, 1952b, p. 1) The response to the overloading and interpretations placed on the 1930 syllabus and its amendments, by District Inspectors, Head Teachers and teachers, constituted by the 1952 Syllabus, was therefore a re-emphasis on calculation, albeit in context with number, time and spatial concepts, as the principal focus in the primary grades. This occurred in conjunction with a suggested time allocation of five hours per week15 being placed on mathematics (Department of Public Instruction, 1952a, p. 6), the first occasion that such a specification had been made in a Queensland syllabus. Following the introduction of the 1952 Syllabus, District Inspector Hendy (1953, p. 2) hoped that mathematics had been deposed as the subject that dominated the curriculum and that no longer would classes be judged primarily on their arithmetical performances. Although the 1964 Syllabus did not include a precise set of goals, the emphasis on computational competency in relevant social situations remained. However, pre- empting the focus of subsequent syllabuses, the importance of the discovery and understanding of mathematical principles was highlighted for the first time in a

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Queensland mathematics syllabus. It was held that this "should prepare the child for more advanced studies beyond the primary school" (Department of Education, 1964, p. 1), this syllabus being introduced following the abolition of the Scholarship examination, but before discussions on the New Maths were sufficient to provide a basis for further syllabus development.

3.2.4 Summary of Background Issues

Although the historical record does not contain many specific links between the background issues discussed and mental arithmetic, the foregoing analysis does suggest that a number of the aspects highlighted are relevant to an understanding of mental arithmetic during the period 1860-1965. These include:

• The authoritarian atmosphere of the Department of Public Instruction in which students, teachers, Head Teachers and District Inspectors of Schools worked, albeit one that decreased as the twentieth century progressed. • The pervasiveness of elements of the theory of mental discipline into the 1950s, despite its tenets being discredited around the turn of the century. • The attempted shift from a focus on the subject to a focus on the child from 1905. • The tension between viewing arithmetic as essential knowledge for its own sake and as knowledge that is socially useful, with the ascendancy of the latter culminating with the 1952 Syllabus. • The advocacy for closer links between what is taught in schools and the life of children outside the classroom─the third principle espoused in the 1904 Syllabus. • The general passivity of children in classrooms, despite the various interpretations placed on the need for schools to make "the self-activity of the pupil the basis of school instruction" (“Schedule XIV,” 1904, p. 200). • The effect on classroom practice of the ways in which children and teachers were examined by Head Teachers and District Inspectors of Schools. • The influence of the Scholarship Examination on what and how children were taught.

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• The effect of the methods of teacher training on the ways in which children were taught. • The impact of textbooks and journal articles on syllabus implementation and teaching practices, particularly during the 1930s and 1940s.

These issues, although not of equal importance, are given further consideration in subsequent sections of this chapter, the first of which focuses on the terminology associated with the calculation of exact answers mentally during the period being investigated.

3.3 Terms Associated with the Calculation of Exact Answers Mentally

As discussed in Chapter 2, the practice of using the term mental computation to refer to the calculation of exact answers mentally, where the focus is on self- developed strategies based on conceptual knowledge (R. E. Reys et al., 1995, p. 324), is a relatively recent one. Its use has arisen from a perceived need to distinguish more clearly between the various aspects of mental calculation, particularly calculating exact answers vis-à-vis approximate answers, as well as to provide a distinction from the traditional approaches to teaching mental calculation. Based on an analysis of mathematics syllabus documents, textbooks, articles in the Queensland Education Journal, Queensland Teachers' Journal, and Education Office Gazette, and from reports of District Inspectors of Schools, it is evident that a number of terms were used, seemingly interchangeably, during the period 1860- 1965. These primarily were: mental arithmetic, oral arithmetic, mental exercises, mental, mental work, oral work, mental and oral work, and oral and mental work. Negligible recognition was given to computational estimation in Queensland syllabuses prior to 1966-1968. Although the period being investigated is characterised by a lack of preciseness in terminology, attempts were made to clarify the situation as early as 1881. Joyce (1881), in his Handbook of School Management and Methods of Teaching16, argued that:

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[Mental arithmetic] is commonly understood to mean merely a number of short rules (often called ‘short accounts'), which are usually found at the end of treatises....But this is only a portion of mental arithmetic; the term has wider significance, and means not only these technical rules but all kinds of numerical combinations performed mentally, from the common addition table up to the most complicated operations. (p. 210)

This usage is similar in meaning to the broad definition given to mental by the Board of Education (1937) in its Handbook of Suggestions. In the Board's view, "‘Mental' Arithmetic [included] all exercises in which pen or pencil [were] not used, except perhaps to record the answer" (Board of Education, 1937, p. 513). For all mathematics syllabuses except those of 1952 and 1964, in which the heading Oral Arithmetic was used for Grades II to VIII to indicate work related to mentally calculating exact answers, little consistency in terminology within each syllabus is evident. For example, in the Mathematics Schedule published in 1904 the following terminology was used: Mental Exercises (First Class), Mental Work (Second Class), Mental and Oral Work (Third and Fourth Classes), Oral and Mental Work (Fifth and Sixth Classes) (see Appendix A.7). This schedule marked the initial use of the term oral. In a general sense, oral referred to the explanation and discussion of arithmetic processes, as reflected in the notes on arithmetic accompanying the 1914 Syllabus. In these it was stated that, for the First Class (First Half-year), "the work should be exclusively oral, except that the children should learn to know and to make the digits including 0" (Department of Public Instruction, 1914, p. 62). The importance of oral questioning had been stressed previously by District Inspector McIntyre in his 1875 report. In decrying the "absurd habit, on the part of junior teachers, of writing down [on the blackboard] all arithmetic exercises...ready manufactured into the shape of ‘sums',” he advocated that "the method of dictating sums should be frequently practised" (McIntyre, 1876, p. 30). A narrower meaning of oral, one more closely related to mental arithmetic, is suggested by a statement in the 1904 Schedule with respect to the Fourth Class. This required the "oral statement of processes employed in written work of [the] class", a declaration that was omitted from the 1914 Syllabus as it was considered to be a valuable practice for all pupils, not simply for those in the Fourth Class17 (Department of Public Instruction, 1914, p. 71).

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Following the implementation of the 1904 Schedule from January 1905, District Inspector Shirley observed that teachers were placing an increased emphasis on the oral explanation of problems. He noted that "it is now quite common to find pupils able to state rules for working a problem, to make special application of the rule to the question in hand, and to work out the steps mentally" (Shirley, 1905b, p. 117). Hence mental arithmetic, slate arithmetic and the discussion of the processes involved were seen to be inextricably bound. However, the focus on the latter by teachers was not widespread. District Inspector Gripp noted that, in his inspectoral district in 1908, "oral work, particularly the explanation of the successive steps of a solution, deserves still more practice than it is at present receiving" (Gripp, 1909, p. 60). Although oral was initially used to highlight the need for teachers above the First Class to encourage explanations of arithmetic processes, usage of the term in syllabuses subsequent to that of 1904 made no apparent distinction between oral and mental arithmetic. In the notes on arithmetic accompanying the 1914 Syllabus, explanations of work required for each class concerning the calculation of exact answers mentally were presented under the headings Oral Arithmetic and Oral Work (Department of Public Instruction, 1914, pp. 64-70). This practice was continued in later syllabuses (see Appendix A.8). For example, in the 1948 amendments, which retained the headings used in the 1930 Syllabus and the 1938 Amendments, buying and selling came under the heading of Mental for Grade I and Oral Work for Grade II, whereas shopping transactions for Grade IV were outlined under Oral Arithmetic. Oral Work for Grade VI included a list of short methods of calculation with which children were to become familiar, reminiscent of Joyce's (1881, p. 210) reference to mental arithmetic. The blurring of the distinction between oral and mental was also encouraged by articles in the Queensland Teachers Journal. For example, in an article which provided sets of examples for mental arithmetic, although the heading was Mental Arithmetic, the subheading on the same page was Seventh grade oral mathematics ("J.R.D.,” 1931, p. 9), the implication being that the set of examples was to be presented orally, rather than by writing on the blackboard or on cards. Although oral arithmetic did not appear as a syllabus heading until 1930, it was used as early as 1906 by a district inspector to refer to the calculation of exact answers mentally. Canny (1907) observed in his report that "Oral arithmetic [italics added] is yet, as of old, mostly a failing subject" (p. 55). Nevertheless, mental

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arithmetic was the term most commonly used by District Inspectors in their reports prior to the introduction of the 1952 syllabus, by the authors of text books used in Queensland classrooms, and by the writers of curriculum articles in the journals of the Queensland Teachers' Union and in the Education Office Gazette. It therefore seems reasonable to assume that, despite the various descriptions used in the syllabuses from 1860 to 1965, mental arithmetic was the term habitually used by school personnel, at least until the 1950s when oral arithmetic and mental (personal knowledge) became the commonly used terms. This, despite the fact that the work associated with the calculation of exact answers mentally was rarely referred to as mental arithmetic in the various syllabuses. This occurred only for the Third and Fourth Classes in the 1860 Schedule and for the Fifth and Sixth Classes in the 1897 and 1902 Schedules (see Appendices A.1, A.5 & A.6). The inconsistency with which the terms were used undoubtedly contributed to the blurring of the distinction between oral and mental arithmetic, with an attendant loss of the intended meaning of oral work. In his report for the 1958 school year, District Inspector Searle (1959) noted that "the lack of practice in oral [italics added] arithmetic [that is, mental arithmetic] problems is reflected in the results in written work of this type, which is very seldom satisfactory" (p. 14). In contrast, in his report of the same year, District Inspector Costin (1959) suggested that there was "an undue reduction in oral [italics added] teaching and explanation" (p. 14), a view with which District Inspector Pyle (1959) concurred: "There [was] a tendency [for some teachers] to resort to the old ‘mental' types of examples rather than [use] the oral lessons as aids to mechanical accuracy on the one hand and processes on the other" (p. 9). The apparent confusion in the use of these terms reflects that referred to by Hall (1954), with respect to usage in the United States─an issue discussed in Chapter 2. Hall's (1954, p. 351) conclusion that many authorities attempted to avoid controversy by using a range of terms interchangeably may also be appropriate for the Queensland situation, at least prior to 1952. From an analysis of primary historical sources, there appears to have been little serious discussion in Queensland concerning the use of these terms. What did occur was largely based on overseas debate, particularly that from the United Kingdom, the most influential of which is likely to have been contained in Primary Education: A Report of the Advisory Council on Education in Scotland (1949). Greenhalgh (1949b), writing in

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the Queensland Teachers Journal, highlighted the following in his presentation of key aspects of this Scottish Report to Queensland teachers: "All arithmetic is mental; the kind so called would more correctly be termed ‘oral.' The artificial distinction thus made...encourages the giving of unnecessarily difficult and elaborate examples" (p. 11). Given that Greenhalgh was a member of the Department of Public Instruction's Syllabus Committee, which was responsible for overseeing the development of the 1952 Syllabus, it is not surprising that oral arithmetic was the term consistently used in reference to what was colloquially called mental arithmetic. In the forward to his text, oral arithmetic, to support the 1952 syllabus, Olsen (1953) stated that "mental arithmetic need not necessarily be oral arithmetic with its attendant air of hurry and consequent tension for some children.” It is apparent from this statement that Olsen attributed to oral arithmetic the characteristics traditionally associated with mental arithmetic, namely, a "series of short, low-level unrelated questions to which answers [were]...calculated instantaneously and the answers written down with speed" (McIntosh, 1990a, p. 40). Nevertheless, despite the concern expressed, his text consisted of daily sets of ten questions on which "a time limit [should] usually...[be] placed" (Olsen, 1953, Foreword). It is likely that the inconsistency in the use and interpretation of the various terms to describe the mental calculation of exact answers, and its attendant confusion with the oral aspects of a mathematics lesson, contributed to mental arithmetic being a facet of the mathematics curriculum that "never [seemed] to reach a standard above fair in many schools" (Lidgate, 1958, p. 7). This was a conclusion consistently reached by District Inspectors during the period under investigation.

3.4 Roles Ascribed to Mental Arithmetic

District Inspectors consistently reported that children did not demonstrate proficiency with the ability to calculate mentally, despite mental arithmetic, in its various guises, being considered an important aspect of all mathematics schedules and syllabuses from 1860 to 1965. This may be attributed to the attitudes towards mental arithmetic engendered by teachers. District Inspector Lidgate (1957) observed in his 1956 report that "too many teachers...caused their pupils to dislike Oral Arithmetic because they have treated the subject as one apart, or because they [have made] the examples too difficult" (p. 8). Such practices can, at least in part,

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be ascribed to the roles designated for mental arithmetic during this period. Mental arithmetic was seen to provide:

1. A tool for developing mechanical skill with the various arithmetic processes18. 2. A skill that was socially useful. 3. A means for "quickening the intelligence" (Bevington, 1926, p. 80; Robinson, 1882, p. 179).

Although the first two roles have an enduring validity, little concern for mental arithmetic as a means for developing a deeper understanding of the structure of numbers and their properties was evident. In contrast, such an outcome is essential to the current vision for mental computation (R. E. Reys, 1984, p. 549). Reflecting a utilitarian view of mental arithmetic, Cowham (1895) suggested that:

[Mental arithmetic] is of high value during the acquisition of a new rule, and is not less valuable in the practical application of any rule to calculations of everyday life. The scholar who has had large experience in mental calculations acquires thereby a facility in dealing with numbers which no amount of slate- work can yield. (p. 169)

Nonetheless, in instances where children mentally applied the operations learnt to the calculations of every-day life, this may, as an unforeseen consequence, have resulted in a deeper understanding of numbers. Such an outcome was dependent upon the nature of mental arithmetic as implemented─an issue investigated in the Section 3.5.3. However, it was the third role, albeit a fallacious one, that was largely responsible for mental arithmetic being an aspect of the curriculum that was not well taught nor well liked by children. This role originated with a belief in the educational theory of formal discipline, one form of the psychological theory of mental discipline, which held that "education consists in strengthening or developing the powers of mind by exercising them, preferably on difficult, abstract material, such as Latin, Greek and mathematics" (Kolesnik, 1958, p. 4). Lawry (1968, pp. 635-636) contends that there was general support for teaching methods based on this theory,

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particularly prior to 1905 while the Department of Public Instruction was dominated by Under Secretary Anderson and General Inspector Ewart, both of whom were strong proponents of formal discipline. In directing that the teaching methods used by teachers at the Central School in Brisbane were to provide the prototypes for all other vested schools, it was stated in the Regulations for the Establishment and Management of Primary Schools (1861) that "these methods...have for their object, not the mere cramming of a child's memory, but the training and development of his intellectual faculties" (p. 84). Hence, the primary school teacher was expected to teach his pupils to think, with a minimal reliance on learning by rote, albeit in context with providing exercises to strengthen the child's mind. Despite such beliefs, none of the syllabus documents, nor accompanying notes, made reference to this aspect of the role of mental arithmetic (see Appendix A). Additionally, the theory of formal discipline was discredited early this century. This was recognised by District Inspector Baker (1929) who stressed "that facility acquired in any particular form of intellectual exercise produces a general competence in all exercises that involve the same faculty is no longer accepted" (p. 273). Nonetheless, the belief in the value of mental gymnastics persisted to such an extent that, although not referring to mental arithmetic specifically, the Syllabus Committee overseeing the 1948 Amendments to the mathematics syllabus saw the need to assert that teachers "should be instructed that all school-subjects should now be approached from the stand-point of realism and practical utility and that the old idea of including subjects to ‘train the muscles of the mind' [was] now discredited" (Mathematics Subcommittee, 1947, p. 1).

3.4.1 Mental Arithmetic as a Pedagogical Tool

As discussed in Chapter 2, the usual sequence for introducing computational procedures entails mental calculation beyond the basic facts becoming a focus subsequent to the introduction of the standard written algorithm for each operation (see Figure 2.4). Musser (1982, p. 40) has suggested that such an approach results in children having difficulty with mental calculations. An aim of the current vision for mental computation is the fostering of a flexible approach to viewing numbers and their interrelationships. Hence it has been argued (Cooper et al., 1992, pp. 100- 101) that the sequence for introducing computational procedures needs to be

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revised so that mental computation (and computational estimation) becomes a focus prior to, rather than subsequent to, the development of written procedures (see Figure 2.5). This view of mental computation was foreshadowed early this century in an article published in The Education Office Gazette in 1908, an article adapted from the England Board of Education's Suggestions for the Consideration of Teachers published in 1905. In presenting hints for teaching arithmetic it was declared that:

It is important that arithmetic should be treated not merely as the art of performing certain numerical operations; it should be taught with a view of making the scholars think clearly and systematically about number. It is thus clear that written work should be an appendage to mental work rather than the reverse. (“Teaching Hints: Arithmetic,” 1908, p. 15)

The written-mental sequence for teaching mental and written computational procedures has arisen in the Queensland context despite statements such as the above. In various schedules and syllabuses prior to those of 1966-1968, and in their accompanying notes, it was suggested that a prime role of mental arithmetic was to provide an introduction to the written mechanical and problem work to be undertaken. Although the 1904 Schedule was the first to specify that "written work [should] be supplementary to the mental and oral work" (“Schedule XIV,” 1904, p. 206), this practice was one previously advocated by some of the recommended texts for teachers on school method19, and by District Inspectors. Gladman (n.d.), in School Method, a text authorised by the Department for use by pupil-teachers, indicated that "mental exercises, generally involving concrete examples, should be employed in introducing a new rule in arithmetic, and should increase in difficulty until the learner finds it necessary to resort to the use of slate and pencil" (p. 75). This view was supported by Robinson (1882): "Mental arithmetic should be taught with each of the rules, and from the very first" (p. 178). In his report for 1884, District Inspector Ross (1885) concluded that one reason for arithmetic not being successfully taught in schools was that teachers did not fully realise "the importance of mental arithmetic as initiatory and subsidiary to slate work" (p. 70), a comment reiterated in his 1889 report (Ross, 1890, p. 109), and supported by District Inspector Radcliffe in 1898 (p. 73). The preface to Schedule

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XIV of the 1904 Regulations of the Department of Public Instruction stated that "mental calculations should be the basis of all the instruction, and [that] the pupils should be made familiar by mental exercises with the principles underlying every process before the written work is undertaken" (“Schedule XIV,” 1904, p. 200). It was stressed to District Inspectors that "the value of mental arithmetic in familiarizing the pupils with the principles of new rules cannot be too strongly insisted upon" (“Circular to District Inspectors,” 1904, p. 1). This belief was a theme flowing through each syllabus from 1904 to 1964. The 1930 Syllabus, for example, stated that "the teacher should introduce...new [rules] through appropriate and simple mental exercises devised by himself,” not simply extracted from available textbooks (Department of Public Instruction, 1930, p. 31). However, only occasionally did District Inspectors comment favourably on the effective use of this approach by teachers─for example, Pyle (1954, p. 2). Annual reports were used regularly to reemphasise their belief in the importance of using mental arithmetic as preliminary to written methods. Representative of these were the comments of Bevington (1925, p. 83), Fewtrell (1914, p. 70), and Lidgate (1953) who asserted, "If children are sound at oral arithmetic there will be little to fear with written arithmetic" (p. 2). Such comments suggest that the use of mental arithmetic as an introduction to written work was not widely practised by Queensland teachers, even though it seems reasonable to assume that issues highlighted in their annual reports were ones that District Inspectors would have emphasised during their inspections of State schools. Factors which may have contributed to the development of this mismatch between the recommended teaching practices and their implementation are analysed later in this chapter, particularly those factors which influenced the teaching process─for example, large classes, poor teacher education, the inspectorial and examination systems. Echoing Ross's 1885 comment referred to previously, District Inspector Farrell, in 1929, noted that:

Failure to teach mental arithmetic so as to make it a definite preparation for the written work to follow and to aid the child generally in his Arithmetical calculations is another cause of the poor results obtained in [arithmetic]. Frequently it is found that...mental exercises [and tables and notation] are taught on the one hand with very little reference to the written work, and classes

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appear to waste time in working out a series of mental gymnastics in figures which lead nowhere. (p. 289)

These comments were reiterated by those of District Inspector Moorhouse (1939, p. 76), and by District Inspector Crampton (1954) who noted that:

Many teachers [were] slow to realize that the main use of oral arithmetic is to teach the various mathematics processes, and thus it becomes a lead-in to all written arithmetic. Oral arithmetic is a most valuable teaching medium, whereas it [was] too often confined to testing. Graded teaching exercises should be first used and, when the processes taught have been mastered, miscellaneous exercises based on the processes taught should be given to consolidate the work taught. (p. 3)

The importance of typing the mental arithmetic to be taught on the written arithmetic that was to follow, was stressed in various curriculum articles in the Queensland Teachers' Journal, particularly during the late 1920s and 1930s. "J.R.D." (1930, p. 13) suggested that if the written work was to include the example 48 plus 69 plus 77 plus 89 then the oral work should focus on examples such as 9 plus 7, 16 plus 9 and 25 plus 8. However, it is likely that such an analytic approach may have contributed to an emphasis being placed on gaining the correct answers, rather than on the methods used by children to arrive at those answers, in effect, testing rather than teaching. In comparison, this teaching principle was interpreted in Mental Arithmetic: A Few Suggestions (1910) at a global level─that is, the focus was on the structure of the written examples. It was suggested that a written Standard I problem such as "A man has 94 apples. He gives away 30, and shares the rest equally between 4 boys. How many does each boy get?" becomes for mental arithmetic: "A boy has 18 apples. Gives away 6. Shares rest equally between 4 boys. How many each?" (p. 176). However, neither of these approaches would have assisted children to think creatively and systematically about number. The mental-written sequence for teaching arithmetic operations was embodied in the 1930 Syllabus as a spiralled introduction to new processes and rules. For example, the oral (mental) work for Grade II included the "application of

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multiplication to easy one-step reduction of money" (Department of Public Instruction, 1930, p. 36), and in Grade III, "Money─Four rules and reduction of sums to £20" was specified as written work (Department of Public Instruction, 1930, p. 37). This approach was foreshadowed in the text recommended for teachers from 1912 as a replacement for Gladman's School Work, a "faithful servant to Queensland schools" (Roe, 1913, p. 33), namely, Cox and MacDonald's (n.d.) Practical School Method. In this it was emphasised that "in the last quarter of the school year [mental arithmetic (oral practice)] should deal chiefly with the work of the next highest standard" (p. 217), a practice included in the Instructions to His Majesty's Inspectorate in England (cited by Cox & MacDonald, n.d., p. 217). Besides the role of mental arithmetic as a preparation for particular written work, District Inspector Cochran (1960, p. 12) suggested that oral work could also serve a number of other purposes: (a) to give brisk drill in basic number facts, especially when they are presented in a wide variety of forms; (b) to cultivate speed and accuracy in new work; and (c) to revise work essential for sound progress. These were legitimate goals for an arithmetic lesson, previously highlighted in the Board of Education's Handbook of Suggestions (1937, p. 513), and by District Inspectors in their annual reports─for example, Mutch (1916, p. 62). With respect to revision work, it was recommended in the Education Office Gazette in 1927 that a distinction needed to be made between the revision of rules and the revision of mechanical operations for the purposes of speed and accuracy, with the former being "undertaken through mental problems involving different processes" (“Teaching of Arithmetic,” 1927, p. 292). Additionally, Farrell (1929) suggested that, through oral exercises in revision work, "at least four times the ground could be covered, the knowledge gained becomes conversational which is the knowledge mostly required in every-day life, and it also becomes habitual, thus adding greater facility to the written work" (p. 295). However, these goals, in combination with the nature of the textbooks used by teachers, may have contributed to the belief that mental arithmetic primarily involves the giving of a series of examples, often involving one-step, with little or no reference to the strategies used by the children. Of relevance to strategies for calculating mentally, was the recommended use of mental work to introduce short methods of calculation (Board of Education, 1937, p. 513; Jeffrey, 1923, p. 68), both mental and written. In the Introductory Notes to the 1952 Syllabus it was emphasised that the application of oral arithmetic should not

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be limited to such methods of calculation (Department of Public Instruction, 1952b, p. 2), a point also made by Cox and MacDonald (n.d., p. 217). Nevertheless, in contrast to the range of strategies used to calculate exact answers mentally, which were discussed in Chapter 2 (see Tables 2.1 to 2.4), it was also stated in the 1952 Syllabus that "the processes which are applied orally are the same as those used in written operations" (Department of Public Instruction, 1952b, p. 2), further emphasising the perceived role of mental exercises as an introduction to the written work, albeit one that would have restricted the development of a range of flexible idiosyncratic strategies for children to use. The narrowness of this statement on the role of mental strategies in the 1952 Syllabus, which does not appear in the 1964 Syllabus, may have been influenced by the Board of Education's (1937) Handbook of Suggestions, the text from which the syllabus aims were drawn. It stated, in reference to the mechanical rules of written arithmetic, that "[they]...are best regarded as forms of mental technique or as complex habits to be formed" (p. 506). Such does not necessarily imply that mental and written techniques are synonymous. However, it was held that where the goal was to develop skill with written computation, "the purpose [of mechanical work, as opposed to problem work, was]...to help the child to form the mental habits in which skill in computation...[was believed to have been] rooted" (Board of Education, 1937, p. 506). Such a view reflects that of Thorndike (1922), who stated that "[children] learn the method of manipulating numbers by seeing them employed, and by more or less blindly acquiring them as associative habits" (p. 71).

3.4.2 The Social Usefulness of Mental Arithmetic

Although the mathematics schedules and syllabuses prior to that of 1952 did not specifically refer to the social utility of mental arithmetic, it can be argued that this was a constant theme during the period under investigation, given statements within recommended textbooks and by District Inspectors. In defining "‘Mental' Arithmetic [as including] all exercises in which pen or pencil is not used, except perhaps to record the answer,” the Board of Education (1937) concluded that, in this sense, "much of the Arithmetic of everyday life is ‘mental'" (p. 513). The 1952 Syllabus asserted that "the fact that oral arithmetic is more commonly used in after-school life than written arithmetic [indicated] to teachers the importance attached to this section

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of the work" (Department of Public Instruction, 1952b, p. 2), a sentiment reaffirmed in the syllabus of 1964 (Department of Public Instruction, 1964, p. 2). This statement reflected those in textbooks authorised for use by school personnel during the early years of the Department of Public Instruction: Gladman (n.d., p. 74); Park (1879, p. 42); Joyce (1881, p. 209); Robinson (1882, p. 179); and Gladman (1904, p. 206). In stressing that mental arithmetic was a valuable preparation for the business of life, Robinson (1882) highlighted two aspects, namely those required during industrial endeavours and "those little domestic and mercantile transactions that are continually occurring to all of us" (p. 179). Whereas Robinson (1882) focussed the former on the usefulness of mental arithmetic in agricultural communities, Gladman (n.d.) emphasised its commercial usefulness:

The practical value of [mental arithmetic] in ordinary business is universally allowed, and those who have watched the employés (sic) in our London warehouses as they "extend" their invoices, hardly know which to admire most, the wonderful accuracy and rapidity of the calculations, or the extent to which business is facilitated by their skill. (p. 74)

Consequently mental arithmetic was considered a "subject of great practical importance" (Park, 1879, p. 42) within the primary school curriculum, a belief often reiterated by District Inspectors in their annual reports. District Inspector Macgroarty (1891) wrote in his report for 1890 that "the importance of mental arithmetic...in rendering instruction in [arithmetic] intelligent and interesting...can scarcely be overrated, entering as it does so largely into the everyday transactions of most people in all walks of life" (p. 75). A similar view was expressed by Caine (1878, p. 97), Bevington (1926, p. 80) and Router (1941, p. 2) in their annual reports, and by Farrell (1929, p. 283). That the social utility of mental arithmetic was a consistent concern of those who devised and supervised the mathematics schedules and syllabuses is a distinguishing feature of the mathematics taught in Queensland schools. In contrast, in the United States of America, as discussed in Chapter 1, the renewed emphasis on the social utility of mathematics in the 1930s and 1940s resulted in a revival of mental arithmetic, following its de-emphasis in American classrooms,

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which had occurred following the discrediting of the theory of formal discipline some twenty years previously. A concern for the social usefulness of mental arithmetic was declared within the wider context of the usefulness of arithmetic generally in the after-life of children, a view expressed by Cox and MacDonald (n.d., p. 180), and Gladman (n.d., p. 72). Both the 1914 and 1930 Syllabuses suggested that "the very practical purposes that mathematical work has to serve in the child's future life should regulate the character of its treatment in school" (Department of Education, 1914, p. 60; 1930, p. 30). It was suggested, in part, that "the teaching of...[arithmetic] should aim...at imparting facility in the working of concrete examples dealing with matters pertaining to the everyday life of the pupil" (Department of Education, 1914, p. 60; 1930, p. 30). However, these syllabuses also assisted in perpetuating the belief in the disciplinary powers of arithmetic by stating that another aim of teaching arithmetic was to "[develop] the intelligence" (Department of Education, 1914, p. 60; 1930, p. 30). This aim, although increasingly open to question, was not inconsistent with such beliefs as those expressed by Macgroarty (1886) in his 1885 report, who, although recognising arithmetic's social utility, was also concerned with its disciplinary powers:

Arithmetic, which comes into play in the ordinary transactions of business life, is of very great importance in its practical bearings, but the part it takes in developing the pupil's intellectual and reasoning faculties, when properly and intelligently taught, is hardly of less importance. (p. 63)

3.4.3 Mental Discipline and Mental Arithmetic

Writing in the Queensland Teachers' Journal in 1945, "X+Y=Z" asserted that there was a widespread belief among both educators and laymen "that a study of mathematics improves [a child's] all round reasoning powers irrespective of what he reasons about" (p. 14). This belief had its origins in the theory of formal discipline, the most pervasive educational theory of the nineteenth century (Burns, 1973, p. 1). This belief persisted throughout the first half of the twentieth century, despite its tenets having been seriously questioned, both philosophically and scientifically. The theory of formal discipline provided the basis for maintaining that "[mental arithmetic]

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improves the tenacity of the mind, strengthens the memory, cultivates the power of abstraction, and tends to promote clearness of conception....It developes [sic] such moral qualities as patience, readiness, activity, and presence of mind" (Wilkins, 1886, p. 40). Further,

If a pupil is induced, first to give close attention to what he is expected to do, secondly to keep to the terms of his question, and thirdly to give a result which is in accordance with the truth─it can not but react favourably on his moral character. (Martin, 1920a, p. 82)

The consideration of moral, in addition to the intellectual aspects of behaviour, distinguished formal discipline from the theory from which it originated─mental discipline (Kolesnik, 1958, p. 7). Mental discipline, Kolesnik (1958) contends, "signifies nothing more than the psychological view that man's mental capacities can somehow be trained to operate more efficiently ‘in general,' and the philosophical conviction that such training constitutes one of the chief purposes of schooling" (p. 3). Formal discipline, however, involved developing the powers of the mind by exercising them on difficult, abstract material such as Latin, Greek and Mathematics (Kemp, 1944, p. 29). In so doing transfer of learning was thought to be facilitated─to produce a mental gymnast of a general kind from undertaking mental gymnastics of a particular kind (Ballard, 1928a, p. 3). However, the terms mental discipline and formal discipline were not used precisely, particularly as now defined, during the 19th and early 20th centuries. Mental discipline was commonly used to refer to the characteristics of the three related, but essentially separate, theories. Underpinning formal discipline was a belief that the mind was composed of a number of distinct powers or faculties. These included memory, attention, observation, reasoning and will (Kolesnik, 1958, p. 6). Robinson (1882) suggested that the foremost value of such calculations as 78, 654, 931 multiplied by 6 was not arithmetical but educational:

Practically, a child will never be called upon to solve mentally so long a question....Its utility consists in the formation of a power of concentrating all the faculties on the performances of an allotted task; and the mind that can do so

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will soon prove capable of any amount of labour upon other tasks as well. (p. 178)

Although a similar view was taken by Park (1879, p. 43) and Gladman (1904, p. 200), they did not propose that children should operate with numbers as large as those suggested by Robinson (1882, p. 178), a recommendation in keeping with the Queensland mathematics schedules of that time. Park (1879, p. 43) advocated that, once a class had a fairly intelligent grasp of an operation, a few questions should be framed for the children to work mentally, with such questions being related to the business of commercial and every-day life. Under the 1904 Schedule, where a focus was on the application of mathematical principles, mental arithmetic ideally should have discarded large numbers, and consisted of exercises which did not require any great mechanical working, so much as an intelligent grasp of principles (“Mental Arithmetic: A Few Suggestions,” 1910, p. 176). Many colonial teachers, however, went beyond the requirements of the schedules. In their hands "mental arithmetic consisted principally in working certain hard numbers in the shortest time by the shortest method" (“Mental Arithmetic: A Few Suggestions,” 1910, p. 176). Nonetheless, Burns (1973, p. 4) questions whether such teachers fully implemented their espoused beliefs in the use of arithmetic to, in the words of one Queensland State school teacher, "help sharpen the wits and strengthen the mind's grasp" (“Scientific and Useful,” 1882, p. 301). The latter was a view in accordance with that held by Ewart (1890) who stated, in his report for 1889, that from the "intelligent teaching [of arithmetic accrues] the intellectual gymnastics necessary for bracing the mind to logical and continuous thought" (p. 67). Burns (1973) suggested that:

Despite their apparent lip service to the aim of mental discipline and to its basis in faculty psychology the practices of many colonial teachers implied a different kind of aim from the one ostensibly guiding their teaching. The real aim of most teachers centred on the inculcation and retention of information and if they concerned themselves at all with faculty development then it was almost exclusively with the faculty of memory. (p. 4)

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Although research had cast doubt on the validity of formal discipline and the concomitant theory of transfer of training, beliefs in these theories persisted well into the twentieth century in Queensland (Kemp, 1944, p. 29; Schonell, cited in “Teachers and the Syllabus,” 1949, p. 1), albeit in a modified form. Greenhalgh (1949a, p. 3) made use of Bassett's (1949) arguments in a further attempt to convince Queensland teachers to forgo their conservatism in the late 1940s. Bassett (1949, p. 110) had argued that the fall-back position of the proponents of formal discipline, which held that mental arithmetic, in the form in which it had been taught, had a special disciplinary value, was also questioned by research evidence. Prior to this, in 1947, Greenhalgh had asked:

How...may we justify many thousands of the mathematics exercises that have been administered to children in enormous doses...; and above all that mental arithmetic, the sole effect of which [was] to develop in both teachers and pupils that nervous strain more often associated with hospitals to which the same adjective is applied? (p. 11)

Only in the United States of America did educators take cognisance of these experimental findings ("X+Y=Z,” 1945, p. 14), the most influential of which were those of Thorndike obtained during the first quarter of the twentieth century (Kemp, 1944, p. 31). These findings were accepted to such a degree that mental arithmetic ceased to be an essential part of the primary curriculum (Reys & Barger, 1994, p. 33). However, with the discrediting of formal discipline, many psychologists and educators assumed that transfer of training and mental discipline had also been discredited. "The mind could not be trained ‘in general'. [Hence] whatever was to be learned had to be taught specifically, and the only things worth teaching were those for which there was some obvious and immediate use" (Kolesnik, 1958, p. 5). To use Greenhalgh's (1947, p. 11) term, the predominant educational philosophy in the 1930s and 1940s was realism. The only "justification [for] the inclusion of any subject rests in its usefulness to the child, the child as he now is and the man he is to be" (Greenhalgh, 1947, p. 11), a belief that permeated the 1948 Amendments and the 1952 Syllabus. From his 1901 study, Thorndike concluded that "it [was] misleading to speak in terms of sense discrimination, attention, memory, observations and quickness, since

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what these words refer to are ‘multitudinous, separate, individual functions' which may have very little in common" (Kolesnik, 1958, p. 33). Chief-Inspector Edwards (1936), reporting to Queensland teachers following his visit to the United States of America, indicated that "the Progressivist does not believe that there is any such thing as general mental training. Mental benefit is specific. The results obtained from one subject cannot be transferred to any other subject" (p. 16). Prior to the work of Thorndike, belief in the tenets of formal discipline and faculty psychology were shaken by the advocates of Herbartian psychology (Ballard, 1928a, p. 3). Johann Herbart, a German philosopher, "contemptuously [cast] aside the doctrine of inborn faculties or capacities for acquiring knowledge" (Fennell, 1902, p. v) when he declared, as cited by Kolesnik (1958), that those who "cherish the obsolete opinion that there resides in the human soul such certain powers or faculties which have to be trained have no psychological insight" (p. 24). The Herbartian, therefore, did not set out to train or exercise the mind, but rather aimed to present new information in a form that could be selected and assimilated with the old (Fennell, 1902, p. v). Herbart asserted that masses of ideas needed to be trained, not abstract powers or faculties. The continued belief by Queensland teachers in aspects of formal discipline, faculty psychology and transfer of training was attributable, in Burns' (1973, p. 4) view, to the teacher training received under the pupil-teacher system which tended to perpetuate the traditional beliefs and practices espoused by senior teachers, and to the system of examinations and inspections which placed a premium on knowledge and learning by rote. Hence the concern was for developing the memory, as Burns (1973, p. 4) has suggested. That teachers and District Inspectors retained a belief in the role of arithmetic, in general, and mental arithmetic, in particular, as a means for quickening the intelligence, developing judgement, improving reasoning (Baker, 1929, p. 281; Bevington, 1923, p. 64; Mutch, 1924, p. 40) and the "power of concentrating the mind upon the solution of a problem" (Martin, 1916, p. 135), may also be ascribed to their beliefs about the nature of mathematics. Cox and MacDonald (n.d.), the recommended text on school management from 1912, asserted that:

As a means of mental training, arithmetic is the most important subject of the elementary school curriculum. Nothing need be taken for granted; every truth is

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capable of demonstration, and each new truth is seen to grow out of what has preceded. Thus the child is trained to investigate, to think in logical sequence, and to advance step by step along a chain of reasoning, until the desired truth is demonstrated. In this process the child learns the value of methodical arrangement and clearness of statement; he is taught to discern the essential from the non-essential, and to seize on that which is useful for his purpose; he is trained to habits of close attention and fixity of purpose, knowing no rest till the end is attained. Each successful effort tends to make him more and more conscious of his powers, and implants a spirit of self-reliance and perseverance. (pp. 180-181)

Ballard (1928a, p. xi) pointed out that this English view of arithmetic represented arithmetic as logic, whereas the American view was that of arithmetic as habit, a view based primarily on Thorndike's (1922) associationist beliefs about how arithmetic should be taught. However, it is the English view that was most influential on Queensland mathematics education beliefs and practices (Clements, Grimison, & Ellerton, 1989, p. 51; Schildt, Reithmuller, & Searle, n.d., p. 8). Arithmetic as logic was also espoused by Gladman (n.d.): "Mathematics are studied, not so much for the practical worth of their facts as for the logical processes through which the mind must pass in learning them" (p. 73). Cox and MacDonald (n.d., p. 217) argued that problems worked mentally share the same disciplinary value as those worked with paper and pencil, but have the advantage of being worked quickly. Hence mental arithmetic was judged to be the means for training children to think and to reason. "Intelligence in Arithmetic should be secured through the medium of mental exercises...taken for a few minutes at the commencement of [each] arithmetic lesson" (Bevington, 1925, p. 83). Accuracy in thinking and reasoning was paramount, as was accuracy in each step taken in working towards a solution (Martin, 1920a, p. 81). Following the implementation of the 1904 Schedule, with its emphasis on the correlation of subjects and the self-activity of the pupils, Mutch (1907) commented that "now teachers regard arithmetic not only as a practical art, but also as an excellent means of intellectual discipline....With...more stress laid on oral arithmetic, there has been a distinct gain in thinking power" (p. 70). It is possible that Mutch was reflecting a modified view of mental discipline, one that discounted the beliefs in

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direct transfer of training and in faculties of the mind, as defined by the proponents of faculty psychology. This revised view of mental discipline held that a study of arithmetic, including mental arithmetic, enhances concentration and the ability to think critically:

By mathematics a power has been gained that is universally applicable in all mental operations. The activity of attention is fundamental to all intellectual life, and the power of concentration and application developed to a habit by mathematical work will lead a pupil to attack a new subject and to progress with it more effectively than if no such habit had been acquired. (Welton, 1924, pp. 409-410)

3.5 The Nature of Mental Arithmetic

In coming to an understanding of the characteristics of mental arithmetic during the period under investigation, cognisance needs to be taken of issues beyond the background issues related to syllabus development and implementation, and the roles of mental arithmetic discussed previously. Aspects which need to be analysed are: (a) the various interpretations placed on the term mental arithmetic, (b) the nature of mental arithmetic as embodied in the syllabus documents, and (c) the characteristics of mental arithmetic as implemented, given that the nature of mental arithmetic as reflected in the learning experiences of children, does not necessarily mirror syllabus content nor the recommended ways in which skill with mental calculation should be taught.

3.5.1 Interpretations of Mental Arithmetic

Reflecting the social usefulness of arithmetic skills, Joyce (1881) defined a good practical arithmetician as:

[One who] can perform mentally, with readiness, and with little danger of error, all those innumerable short computations that are met with in everyday life;

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and...who can execute on paper all sorts of elementary calculations, even when considerably extended with rapidity and certainty. (p. 203)

In keeping with this view, Ross (1905) suggested that "the desiderata of mental arithmetic are ‘speed' and ‘accuracy'" (p. 33). Additionally, the introduction to a mental arithmetic textbook20 commonly used by Queensland teachers early in the twentieth century highlighted that mental arithmetic questions should exercise the mind of the child so as to encourage dexterity with numbers ("An Inspector of Schools,” 1914, p. 3). This consideration alludes to current beliefs about mental computation, which stress a need for children to be able to perform mental calculations with nonstandard strategies by taking advantage of an ability to compose and decompose numbers (Resnick, 1989b, p. 36; Sowder, 1992, p. 4). Although little recognition was given in the various schedules and syllabuses to the need for children to be encouraged to invent short methods for themselves, as advocated by Joyce (1881, p. 215), such was occasionally given, in an ad hoc manner, by articles in the Queensland Teachers' Journal and The Education Office Gazette, and by District Inspectors. In Teaching Hints: Arithmetic (1908), it was stated that "the teacher [should] have no difficulty in devising questions in mental arithmetic which are easy by special methods but too difficult for mental work by ordinary rules" (p. 16). In providing advice to teachers, District Inspector Bevington (1925) did not necessarily interpret all short cuts as being based on the rote application of rules─that is, on instrumental understanding, as defined in Chapter 2. Bevington (1925, p. 83) advocated the use of a compensatory approach (see Table 2.4) for examples such as 99 + 87 and 100 books at 19s 11½d. A similar approach was recommended by District Inspector Kennedy (1903) for mentally calculating the cost of 24 articles at 19s.11d. each─by taking 24 pence from 24 pounds. Nevertheless, he did consider that his example was "a suppositious and extreme one" (p. 69), when placed in context with the usual method of calculating such examples using aliquot parts. A more detailed outline of flexible approaches to mental addition and subtraction was presented by Cox and MacDonald (n.d., p. 224). A range of work from the left strategies (see Table 2.4) was advocated for addition: For example, 25 + 37 = 2 tens + 3 tens = 5 tens 5; 5 tens 5 + 7 = 6 tens 2 = 62. For subtraction, a decomposition strategy (see Table

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2.4) was the approach suggested. It was pointed out that 45 - 18, for example, could be calculated as 45 - 10 = 35; 35 - 8 = 27 (Cox & MacDonald, n.d., p. 231). However, such advocacies for flexibility were counteracted not only by an emphasis on the application of the written methods of computation, but also by such rigid views, with respect to process, as those expressed in Mental Arithmetic: A Few Suggestions (1910):

Great stress should be paid to the correct method of obtaining answers. When asked how the first answer (12) is obtained [for "A boy has 18 apples. Gives away 6. Shares equally between 4 boys. How many each?"] such an answer as 12 and 6 make 18 is incorrect. The child must see that from 18 six has been subtracted. The rest of the sum should be solved by the use of correct operations─i.e., 12 is divided by 4, not that 4 x 3 = 12. This is important. (p. 176)

Effective mental methods cannot be acquired by rote learning (French, 1987, p. 39). This was recognised by District Inspectors Kennedy (1887, p. 82) and Caine (1878, p. 97) in the late nineteenth century. The former concluded that the unsatisfactory results in mental arithmetic were due to "this branch [appearing] to be looked upon as merely a matter of fixed rules, applicable to particular kinds of work─such as finding the price of a dozen, score or gross" (Kennedy, 1887, p. 82). Such a focus on rule of thumb methods, applied without an understanding of the principles involved, was one which Priestley (1912, p. 222) believed would be eradicated by the New Education movement, which at that time was rapidly gaining strength. However, although an emphasis on short cuts diminished as the century progressed (see Appendix A), the widespread implementation of methods which encouraged children's understanding of mathematical processes did not occur until late-1960s, a period during which mental arithmetic did not receive explicit recognition in syllabus documents (see Section 4.2). The poor performances were also attributed to "[mistaken efforts] to apply to mental calculation processes proper enough in slate arithmetic but quite out of place in [mental arithmetic]" (Kennedy, 1903, p. 69). Nonetheless, if this practice did receive reduced emphasis during the first half of the twentieth century, it gained renewed prominence in the 1952 Syllabus which emphasised that the procedures

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used in written arithmetic should be the ones applied orally (Department of Public Instruction, 1952b, p. 2). Robinson (1882), in advocating that mental arithmetic was simply a collection of rules, suggested that mental arithmetic was "not so much opposed to slate arithmetic as to mechanical arithmetic, though it is opposed to both" (p. 177). Computations were considered to be mechanical when a class of questions was solved using one fixed rule. Such an approach reduced the practical application of mental arithmetic as the rules were easily forgotten (Robinson, 1882, p. 177). In contrast,

When mental arithmetic is opposed to mechanical, it consists of a judicious modification of the common rules, or in the framing of rules entirely new....This sort of mental arithmetic, therefore, requires a quick intelligence, and active and retentive memory, a thorough acquaintance with each step in the ordinary rules, and a perfect knowledge of the principles upon which they are founded. (Robinson, 1882, pp. 178-179)

Others, during the era under investigation, took a more simplistic view of mental arithmetic. Lidgate (1954) suggested that "oral arithmetic [would] improve if children [were] induced to realise that it is merely the application of tables" (p. 2). Whereas this may be true for children in lower grades, as was asserted by Joyce (1881, p. 210) and in the notes accompanying the 1914 Syllabus (Department of Public Instruction, 1914, p. 64), for example, the beliefs of Robinson (1882, pp. 178-179) suggest that mental arithmetic involves much more than the mere application of tables. Such beliefs are consistent with the characteristics of mental computation discussed in Chapter 2. Nonetheless, Queensland teachers appear to have consistently taken a narrow view of mental arithmetic, a view that cannot be fully explained merely by reference to the apparent on-going belief in aspects of the theory of formal discipline. Essential to this analysis is a consideration of more general factors that impinged on teacher beliefs and practices. These include: (a) the beliefs about mental arithmetic embodied in the syllabuses, (b) the influence of the systems of examination and inspection, (c) the level of teacher training, and (d) the size and structure of classes taught.

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3.5.2 The Syllabuses and Mental Arithmetic

Following the separation of Queensland from New South Wales on 10 December 1859, the establishment of the Board of General Education, through the Education Act of 1860, provided the foundation for the development of primary education in Queensland. The system framed by the board was largely that of the national system of education in Ireland (“Our First Half-Century,” 1909, p. 78), this being the system on which the New South Wales National Schools had previously been based. The Education Act of 1860, however, did not specify the curriculum. This was published in the regulations adopted by the Board of General Education on 14 December 1860 (“Regulations,” 1860, pp. 15-17) as a Table of the Minimum Amount of Attainments Required From Each Class in Primary Schools (see Appendix A.1). This table outlined the attainments expected of children enrolled for one quarter of the school year in each of the classes. The expected attainments were identical to those established by the National Board in New South Wales (“Statement. Explanatory,” n.d., p. 1). In both of these syllabuses, children in First Class were expected "to perform mentally all the Elementary Arithmetical operations with numbers, not involving a higher result than 30" (“Regulations,” 1860, p. 15)─that is, to be able to add, subtract, multiply and divide whole numbers. Children in third and fourth classes were expected to become familiar with the "Rules of Mental Arithmetic" (“Regulations,” 1860, pp. 16- 17)─for example, with the rules for finding the cost of a dozen, score or gross of articles21. These rules, predominantly related to money calculations which were essential for "the business and duties of every-day life" (Park, 1879, p. 43), were usually listed in arithmetic and mental arithmetic texts, with the number of rules presented varying from text to text. With the passing of the Education Act of 1875, which established the Department of Public Instruction, education in Queensland passed from being controlled by a subordinate board under the control of Executive Council to being overseen by a responsible minister, the first of whom was Sir Samuel Griffith, whose beliefs about education prevailed as "they were administratively feasible and politically acceptable at the time" (Lawry, 1975, p. 58). The curriculum outlined in Schedule V of the Regulations of the Department of Public Instruction (“Schedule

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V,” 1876, pp. 825-826) was narrower in scope than that in the 1860 Schedule. This restriction was designed to prevent a clash between the curriculum offered by the free primary schools and that of the Grammar Schools (Lawry, 1975, p. 59). Schedule V presented a Table of the Minimum Amount of Attainments Required From Pupils for Admission Into Each Class in Primary Schools (“Schedule V,” 1876, p. 825) (see Appendix A.2). General Inspector Anderson noted in his report for 1876 that the publication of the standards of attainments ensured "that no teacher who [read the schedule] ...carefully [could] unintentionally commit the grave fault of promoting his scholars prematurely" (Anderson, 1877, p. 21). The implementation of this table was clarified in a revision to Schedule V published in 1885. It explained that "the work to be gone through in any class (the Fifth Class excepted) [was to] be found detailed in the column with the name of the class next above it" (“Schedule V,” 1885, p. 490). With respect to mental arithmetic, the changes contained in the 1876 Schedule set more realistic goals for each class than those embodied in that of 1860. Children were required "to perform mentally easy operations in the simple rules"─for addition, subtraction, multiplication and division, by the end of the Upper Second Class. In the First Class, instead of being required to master all four operations with results to 30, children were required under the 1876 Schedule "to perform mental addition up to a result not higher than thirty.” Subtraction was included for the Lower Second Class (“Schedule V,” 1876, p. 825). Although no specific reference was given to the rules of mental arithmetic in the 1876 Schedule, among the arithmetic texts furnished to schools (“List of Books,” 1880, p. 23) were ones that contained expositions of these rules─for example, that presented in Colenso's (1874) A Shilling Arithmetic. The importance for children to be able to calculate mentally in a range of contexts beyond whole numbers and money was also recognised in the 1876 Schedule. Work for the Third and Fourth Classes, and by implication for the Fifth Class, included easy mental operations in the "compound rules and reduction, including bills of parcels, rectangular areas, ...proportion, practice, vulgar fractions and simple interest, including miscellaneous problems"22 (“Schedule V,” 1876, p. 825) (see Appendix A.2). By the 1890s almost all children in Queensland were receiving some primary education, albeit to a standard "hardly appropriate for a developing urban society" (Lawson, 1970, p. 215). However, the compulsory attendance provision of the

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Education Act of 1875 could not be enforced until the early 1900s as attendance was irregular, and education was not held in high regard. It was Hanger's (1963) view that:

More than sixty per cent of parents of 1890 were suspicious of education for their children or hostile to it: they regarded it as useless and teachers as a nuisance, and felt that the sooner the children were at work and helping to support the home, the better for themselves and none the worse for the children. (p. 89)

With the publication of a revised schedule (“Para 143,” 1891, pp. 23-24), effective from January 1892, school head teachers gained additional guidance for the placement of children into classes. Not only were the six classes redesignated as First Class to Sixth Class, with a specified duration for each, but also the schedule was restructured to include the expected standard of proficiency for each class (“Para 144,” 1891, pp. 24-28) (see Appendix A.3). The main changes in mathematics concerned the extension into elementary geometry for boys, whereas girls were to devote more time to needlework (Lawry, 1968, p. 583). The standards of proficiency provided greater guidance for the treatment of the various aspects of arithmetic in each class. The work was delineated for each half-year of enrolment and provided a sequence for introducing particular learnings. For example, children in First Class were required "to add mentally numbers applied to objects to a result not greater than 10" during the first half-year, to 20 during the second six months, to 30 during the third half-year and "to a result not greater than 40" by the end of the fourth half-year (“Para 144,” 1891, pp. 24-25). Nonetheless, few changes were made to mental arithmetic. As previously, children were expected "to perform mentally operations in [the four] rules" by the end the second class with the standards of proficiency for the third half-year indicating mental addition and subtraction of money to one pound (“Para 144,” 1891, p. 25). Mental subtraction was to be introduced during the third half-year of First Class, so that by the end of that grade children were required "to add mentally numbers to a result not greater than 40; [and] to lessen mentally any number of two figures by any number of one digit" (“Para 144,” 1891, p. 25). Mental work for the Third Class included shopping transactions based on measurement tables and easy

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bills of parcel, as well as the application of the dozen and score rules. Mental arithmetic in the Fourth and Fifth classes was extended to include the unitary method23, decimals, percentages, and square and cube roots (see Appendix A.3). Although some changes to the syllabus were introduced in 1894, 1897 and 1902, their minor nature induced Senior District Inspector Platt (1905) to observe in his report for 1904 that "the course of instruction in our schools has been practically the same for the last thirteen years─that is, since 1st January, 1892" (p. 30). Although Platt was referring to the syllabus as a whole, his comments were particularly relevant to mental arithmetic. In 1894, minor changes were made to the standards for the First Class due to its length being reduced from two to 1½ years (see Appendix A.4): The end-of-class expectations remained the same, but the size of the numbers to be manipulated in the each half-year were increased─to 20, 30 and 40 for each of the three half-years, respectively (Schedule VI, 1894, p. 321). The State Education Act Amendment Act of 1897, effective from 1 July 1898, not only introduced algebra and Euclid to the Fifth and Sixth Classes, but also extended the First Class course to two years and reduced its requirements (see Appendix A.5) to lessen "the field of work in schools taught by one teacher" (Dalrymple, 1899, p. 5), an outcome of which Macgroarty (1900, p. 63) was not convinced. Mental arithmetic for the first class focussed on developing the ability "to add mentally numbers of one figure to a result not greater than fifty" (“Schedule V,” 1897, p. 798). During the Second Class the ability to mentally subtract, multiply and divide with abstract numbers was to be developed. Additionally, mental operations with the weights and measures considered to be of more use were designated for the Fourth, rather than the Third Class. The 1902 revised schedule (“Schedule XIV,” 1902, pp. 169-170), the first to provide an indication of the expected age ranges of children in each of the classes, did not introduce any explicit changes to mental arithmetic (see Appendix A.6). As in the 1897 Schedule V, Schedule XIV (1902) and Schedule XV (1902), the latter presenting the standards of proficiency, did not specifically specify the mental arithmetic for the Fifth and Sixth Classes. For example, the arithmetic for the third half-year of enrolment in the Sixth Class was listed as: "Commercial arithmetic; mensuration─Longmans' Chapters I to XI.; mental arithmetic" (“Schedule XV,” 1902, p. 173).

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However, some insights may be gained from the textbooks used in conjunction with the 1897 and 1902 schedules. The mental arithmetic text supplied to schools for the instruction of pupils was The “Practical” Mental Arithmetic, an English publication written by "An Inspector of Schools" (1914). One of the four arithmetic texts was A Shilling Arithmetic by Pendlebury and Beard, first published in 1899 (“Appendix B,” 1902, p. 95). This text contained a section devoted to the mental rules for "the calculation of prices" (Pendlebury & Beard, 1899, pp. 174-175), which constituted the only reference to mental arithmetic within the text. Unlike A Shilling Arithmetic, which was arranged by topics, The “Practical” Mental Arithmetic was organised into sets of questions for each of the classes in the schools in England (Standards I to VII). It included the advice that teachers should consider the work for Standards V to VII together, given that the numbers of children in upper standards were usually small24 ("An Inspector of Schools,” 1914, p. 103). The 195 questions for Standard V and the 160 for Standards VI and VII were relevant to the work set for the Fifth and Sixth Classes in Queensland schools in the 1897 and 1902 schedules, given their focus on vulgar and decimal fractions, percentages and proportion, in context with money, weights and measures. In implementing the prescribed course of instruction for each class there is evidence to suggest that teachers interpreted the order in which the learnings were presented in the schedule as the order in which they were to be taught. This may help to explain why mental arithmetic, "except in a few rare cases, [gave] poor results" (Radcliffe, 1898, p. 73), an observation typical of those made by District Inspectors during the period under investigation. In an analysis of the reports for 1897 in the Queensland Education Journal, it was noted that the schedule "[prescribed] mental arithmetic last in the order of the arithmetic work, [whereas] it ought to [have been]...first" (“District Inspectors' Reports for 1897,” 1898, p. 6). It was also observed that Radcliffe "[had] put his finger on the key to the [poor results]. He [suggested that] the teaching of arithmetic [should be based] on a thorough mental grasp of simple operations in the rules" (“District Inspectors' Reports for 1897,” 1898, p. 6), an affirmation of the importance of mental arithmetic as initiatory to written work. In accordance with this belief, stated officially for the first time in the preface to the 1904 Schedule (“Schedule XIV,” 1904, p. 201), the mental work for each class was listed separately and prior to the listing of the written work in mathematics for

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Classes 2 to 6. Although the mental work remained essentially the same for each class as that in the previous schedule, the presentation of the mental and oral work reflected beliefs about how mathematics should be taught (see Appendix A.7). Teachers were encouraged to take every opportunity to allow for the self-activity of the pupils, thus allowing them to interact with actual objects and to measure quantities using various units─“sticks, bundles, tens, dozens, feet, pence, ounces” (“Schedule XIV,” 1904, p. 201). The mental work for the Second Class, for example, included "concrete exercises involving the four simple operations, and falling within the range of the pupils' experience" (“Schedule XIV,” 1904, p. 206), a child-centred focus for the operations dealt with in Second Class under previous schedules. However, the syllabus for this class became more exacting with the inclusion of money tables, previously in Third Class, and halves, fourths, and eighths of an inch which were previously taught in Fourth Class. Proportion, however, was moved from Fourth Class to the Fifth Class during which cubic measure, previously a Sixth Class requirement, was also to be taught. Ratio was specifically included for the first time in a schedule, as a topic for the Fourth Class (see Appendix A.7). The syllabus which came into force from January 1915 was the first to be published as a separate document, and the first to include detailed notes on the teaching of mathematics delineated for each grade. In was noted in the preface to this syllabus that the changes in the spirit of teaching, which were associated with the 1904 Syllabus, "were so extensive that some years of practical experience as to its working were required by teachers and inspectors before its full advantages could be reaped or its deficiencies detected" (Department of Public Instruction, 1914, p. 5). As the departmental view of the 1904 Syllabus was that it was a "good Syllabus" (Department of Public Instruction, 1914, p. 5), and one that was an improvement on previous schedules, drastic changes were not made to the syllabus in its 1914 form. With respect to mental arithmetic, no changes were made to the requirements for the mental and oral work for any of the six primary classes (see Appendix A.8). In association with the introduction of the 1930 Syllabus, classes were reclassified into seven yearly grades preceded by a preparatory grade of 1½ years for children enrolled in July, the time for enrolment considered most appropriate for those born in the first half or early in the second half of the year (Department of

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Public Instruction, 1930, p. x). It was expected that most children would enter Grade II at approximately 7½ years of age, an age comparable to that for the Second Class under the 1914 Syllabus. Grades VI and VII, Forms I and II, respectively, of the Intermediate Schools, were designed to cater for the interests, capacities and attitudes of the over-twelves, with a study of algebra and geometry intended to provide an opportunity for determining those children who would benefit from a high school education (Greenhalgh, 1957, p. 75). The syllabus recognised, however, that many children would not receive a secondary education. It was stated that "it is generally agreed that the Primary School [is] the only school that some children will know" (Department of Public Instruction, 1930, p. v). The changes embodied in the 1930 Syllabus were formulated to:

Fulfil the demands for foundation work, [to apply] school Arithmetic to the solution of problems connected with ordinary business transactions and operations with numbers and quantities within the children's experience, and to give more definitely the standards required in the different grades of the course. (Farrell, 1929, p. 291)

The importance of mental and oral work was stressed in the belief that such work had a definite place on the daily time-table and was the means by which new work would be introduced (Farrell, 1929, pp. 294-295). Although major changes were made to the written work25, the spiral nature of the 1930 Syllabus resulted in the mental work under this syllabus being essentially the same as that for the 1914 Syllabus (see Appendix A.8), when allowances were made for the restructuring of classes (“New Syllabus,” 1929, pp. 460-462). "A new rule [was to be] introduced in one grade by means of mental exercises followed by easy mechanical exercises and easy problems in the next grade, and by harder mechanical work and problems in the succeeding grade" (Farrell, 1929, p. 292). For example, the mental and oral work with vulgar fractions in Grades I to III were followed in Grade IV by written work involving "simple exercises in finding fractional parts of quantities or numbers,” and in Grade V by the addition, subtraction, multiplication and division of vulgar fractions (Department of Public Instruction, 1930, pp. 34, 35, 37, 39, 41). In combination with the syllabus notes, which were set out in columns opposite the requirements for each grade, the mental work in the 1930 Syllabus was

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presented with greater specificity than that presented in previous syllabuses. Mental calculations were to be applied to the range of mathematics topics to be encountered during the seven years of primary school. These calculations included mechanical and easy problem work with whole numbers, vulgar fractions, decimals, money, weights and measures, ratio, proportion, percentages, simple interest, and mensuration (see Appendix A.9). An emphasis was placed on social utility: For Grade III, it was recommended that "the material for [simple exercises based on the four rules in money was to] be supplied by ordinary household accounts, as for example, the butcher's, the baker's, the grocer's or the draper's bills" (Department of Public Instruction, 1930, p. 37). In accordance with this approach, and in context with the deletion of obsolete matter from the syllabus, simple practice using aliquot parts of a pound was retained only for mental exercises─exercises such as finding the cost of 18 books at 2s. 6d each (Department of Public Instruction, 1930, p. 42), a Grade VI example. A study of short methods of calculation was recommended for Grades V to VII, "after the pupils [had] been thoroughly exercised in any rule" (Department of Public Instruction, 1930, p. 41). However, the short methods to be applied were not stipulated, except for the suggestion for Grade V that "the ‘dozens' and ‘scores' rules might be applied" (Department of Public Instruction, 1930, p. 41). Such a focus was designed to encourage initiative, particularly where different methods of solution to the same problem were required of children, methods that did not entail the rote application of rules. In some respects the 1930 Syllabus was a more exacting one than that introduced in 1915. Although the mental arithmetic for First Class in the 1914 Syllabus was limited to addition and subtraction to 50, this was extended to 99 in 1930, together with multiplication involving multipliers to 6. Multiplication was applied "to easy one-step reduction of money and of the weights and measures dealt with in the Tables" (Department of Public Instruction, 1930, p. 36) in the oral work for Grade II, in preparation for the written reduction of money to £20 in Grade III. Children in Grade I were also expected to find one-half, one-quarter and three- quarters of numbers and quantities, albeit following "practical exercises in dividing and measuring things" (Department of Public Instruction, 1930, p. 34). Nonetheless, the 1930 Syllabus was characterised by more clearly defined limits for mental operations with money, particularly for the lower grades. Exercises

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in buying, selling and giving change were limited to 1s. in Grade I, and £1 in Grades II and III. This contrasts with "exercises in domestic accounts and simple business transactions" for the Third Class under the 1914 Syllabus (Department of Public Instruction, 1914, p. 17), the first two terms of which equated with Grade III from 1930. From Grade IV, however, the requirements were not expressed so precisely, with work for this class being delineated as "mental exercises based on the compound rules, including household and shopping transactions familiar to the pupils" (Department of Public Instruction, 1930, p. 39). Such wording opened the 1930 Syllabus to criticisms similar to those of the 1914 Syllabus, which centred on its being "too vague and open to many constructions, even by common-sense persons" (“Arithmetic,” 1927, p. 17), with the result that there were always wide differences of opinion particularly regarding standards in mental work ("J.R.D.,” 1931, p. 9). The inappropriate interpretations placed on the 1930 Syllabus, together with complaints of its being overloaded, resulted in amendments being introduced, initially in 1938, and subsequently in 1948, as previously noted. Edwards (1938a, p. 2) pointed out that the 1938 amendments were more extensive for mathematics than for other subjects within the primary school curriculum. Many of the changes affected mental arithmetic, particularly that for the lower school where "greater stress than formerly [was]...placed...on the mental and oral Arithmetic prescribed" (Edwards, 1938a, p. 2). Whereas no specific reference was made to mental arithmetic for the Preparatory Grade in the 1930 Syllabus, limits were placed on the mental exercises for the preparatory grades26 under the 1938 amendments (see Appendix A.10). These exercises, including easy problems, were limited to numbers to 10 for Preparatory 1 and 2, to 19 for Preparatory 3 and to 99 for the fourth preparatory grade (Department of Public Instruction, 1938, pp. 10-12). Although some changes reduced the difficulty of the work to be undertaken, others centred on re-allocating requirements to higher grades as a consequence of changes to the written work, in context with the spiral nature of the syllabus. The difficulty of fraction work for young children was given some recognition by restricting a study of vulgar fractions in Grade I to one-half and one-quarter of numbers and quantities. Additionally "resolving easy numbers such as 24 and 36 to prime factors as preparation for the work in fractions and cancelling" was moved from Grade III to Grade IV (Department of Public Instruction, 1938, p. 15),

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concomitant with the work in vulgar fractions for Grade V altered to include "the four simple rules with easy practical application to concrete quantities" (Department of Public Instruction, 1938, p. 16). Similarly, expressing ratios as fractions and decimals, and simple proportion were moved from Grade V to Grade VI. As a consequence, Grade VI children were also required to undertake preparatory exercises to simple proportion which entailed expressing ratios of measures as vulgar and decimal fractions─for example, "1 rood to 1 acre = ¼ or .25" (Department of Public Instruction, 1938, p. 17). Even though the 1938 amendments were considered to be "better adapted to the mental ages of the children" (Edwards, 1939, p. 31), teachers continued to advocate for further reductions in the requirements, as hitherto discussed. The amendments introduced in January 1948 were designed to "give some relief where such [was] necessary [and were to]...remain in force until a projected New Syllabus [had] been drawn up" (Department of Public Instruction, 1948, p. 1), a project that was not completed until 1952. With respect to mental arithmetic, the most substantial reduction in requirements related to the preparatory grades (see Appendix A.11). Mental exercises in addition and subtraction, with easy problems, were limited to numbers to eight in Preparatory 2, to 13 in Preparatory 3 and to 19 in Preparatory 4. For the latter, mental exercises related to counting in ones and twos were restricted to 19, in contrast to counting in ones, twos, fives, and tens to 100 under the 1938 amendments. Halving and doubling were also reduced from numbers to 100 to numbers to 18, with the former limit being moved to Grade I (see Appendix A.11). Multiplication and division of whole numbers within the pupils' range, and finding halves and quarters of numbers and quantities became part of the oral work for Grade II, rather than for Grade I (Department of Public Instruction, 1948, pp. 5-7). During the discussions, initiated by District Inspectors of Schools during the mid- to late-1940s, concerning proposed syllabus amendments, the requirements for short methods of calculation were frequently discussed, although it is apparent that unanimity of opinion was not achieved. Based on recommendations to the Syllabus Committee (Mathematics Subcommittee of Syllabus Committee, 1947, p. 1), a number of short methods of calculation were specified for Grades VI and VII. In addition to the dozens and scores rules stipulated in the 1930 Syllabus, from 1948 children were also expected to apply the rules for: (a) finding the value of 240 and

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480 articles27, (b) calculating using the aliquot parts of one pound, (c) finding the squares of numbers with ½ and of those ending in 5, (d) dividing and multiplying by 25, and (e) finding the differences between pairs of square numbers (Department of Public Instruction, 1948, p. 18). It had been recommended to the Syllabus Committee that calculating the value of 960 articles, finding the square root by factors and determining the difference between two squares should be deleted from the prepared list (Mathematics Subcommittee of Syllabus Committee, 1947, p. 1). That the deletion of the scores, 240, 480 and 960 rules was also recommended by some teachers (McCormack, 1947, p. 1) suggests that short methods of calculation was an area of mathematics that exceeded the intent of the 1930 Syllabus and its 1938 amendments in their implementation. In association with the introduction of the 1952 Syllabus, changes were made to the classification of children to more closely align the structure of primary school classes with those of the other Australian states. The classes were restructured into Grades I to VIII preceded by a Preparatory Grade28 of one year, the purpose of which was to "be a real preparatory or settling-in grade" (Devries, 1951, p. 10). Focussing on kindergarten methods, this grade was designed to provide children with informal experiences in reading and number. Consequently, mental arithmetic was not specifically mentioned in the requirements for the Preparatory Grade in the 1952 Syllabus. The changes embodied in the 1948 amendments and in the 1952 mathematics syllabus aimed "at developing skill in those calculations which men and women have to make in daily life...[so that] children should come to see in [mathematics] one of the indispensable tools used in all crafts and trades" (Department of Public Instruction, 1948b, p. 17). Although additional assistance, through the notes accompanying the content for each grade, was given to teachers concerning how the various requirements were to be taught, few substantial alterations were made to the syllabus from that presented in the 1948 amendments (Dagg, 1971, p. 33). Those that were made centred on a further reduction of the level of difficulty of the mathematics for particular grades. Although the children entering Grade I in 1952 were of a comparable age to those of previous years, the oral exercises involving addition and subtraction were limited to numbers to 10, compared to 99 previously, the same limits as those placed on written addition and subtraction (see Appendix A.12). Such a reduction also gave recognition to the one- rather than two-

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year period in preparatory grades, fortuitously, given this grade's abandonment in December 1952. Mental multiplication and division were moved from Grade II to Grade III, as were shopping exercises involving the use of component parts of 3d., 6d. and 1s. Work with fractions previously undertaken in Grades I and II was deferred to Grade III, where finding a half and a quarter of numbers and quantities were to be first considered. The addition and subtraction of two vulgar fractions was to be first encountered in Grade VI, with multiplication and division in Grade VII, rather than in Grade V. A limit of 16 was also placed on the denominator, in an attempt to further reduce the complexity and artificiality of the mental calculations with vulgar fractions. One-step reductions of money to one pound were also moved from Grade III to Grade IV, as were the addition and subtraction of money on which definite limits were placed─the addition of two amounts in pence and halfpence, for example, was not to exceed 1s, and giving change was limited to 3s. The aim was to provide examples within the pupils' range of experience and capabilities (Department of Public Instruction, 1952b, pp. 12-25). Ratio, simple proportion and percentages were moved from Grade VI to Grade VII, with the former becoming the class in which exercises involving the use of finite decimal fractions were to be first introduced. The support given to teachers in the accompanying notes which outlined the limits for each of the four operations with decimals was characteristic of the additional specificity contained in the 1952 Syllabus (see Appendix A.12). A focus on mental mensuration calculations was included for Grades V to VIII. These were designed to reflect life-like activities, with Grade VII students, for example, required to find the areas and perimeters of paths, borders, and walls so that construction and painting costs could be calculated (Department of Education, 1952b, pp. 24, 26). Except for the suggestions that the multiplication and division of decimal fractions by 10, 100, 1000 should be undertaken by inspection (Department of Education, 1952b, p. 24; 1964, p. 25), neither the 1952 Syllabus nor the 1964 Syllabus made specific reference to short methods for calculating mentally. This was a position with which Crampton (1956, p. 5) did not concur. He recommended in his report for 1955 that some practical short methods should be prescribed in the syllabus to provide oral practice prior to the related written work.

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Although a revised syllabus was introduced in 1964, in essence, the 1952 Syllabus remained in use until the publication, during the late-1960s, of the Program in Mathematics for Primary Schools (Department of Education, 1966-1968). The 1964 Syllabus was an interim document which was necessitated by the transfer of Grade 8 to the secondary school and the planned introduction of decimal currency in February 1966, an event which was to have a major influence on the nature of arithmetic in the primary school. Consequently, these changes, together with revised beliefs about how children learn and what constitutes mathematics appropriate for primary school children, were to be of greater ultimate consequence than the limited changes introduced in 1964 (Department of Education, 1978, p. 3). However, in the syllabuses published during the New Maths era, mental arithmetic was not emphasised to the extent that it had been in the syllabuses from 1860 to 1964. The mental arithmetic for Grades 1 and 2 in the 1964 Syllabus was essentially the same as that in the previous syllabus, except that the limit for oral addition and subtraction exercises in Grade 1 was lowered from 10 to 9. Additional specificity was also provided for Grades 3 to 5 for extending addition and subtraction beyond the first extension, and for multiplication and division (see Appendix A.13). An emphasis was placed on combined multiplication and addition to "teach [the] processes that arise in actual multiplication and division sums" (Department of Education, 1964, p. 12). Work with easy factors was also to be introduced in Grade 3, while factors and multiples, squares and square roots were to receive attention in Grade 6, the former being a requirement for Grade VII under the 1952 Syllabus. In recognition of the impending obsolescence of calculating with pounds, shillings and pence, the mental multiplication and division of money, which had been limited to one step reductions for Grade 5 students, was included in the syllabus only for 1964 (Department of Education, 1964, p.17). In summary, it is evident that mental arithmetic, as embodied in Queensland mathematics syllabuses from 1860 to 1964, was characterised by:

• A gradual reduction in the complexity of the mental calculations required for most children, in concert with the various changes introduced to the written requirements.

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• Increased specificity as new syllabuses were developed, primarily to provide assistance to inexperienced teachers, but also to provide limits on the mental work to be undertaken. • A gradual lessening of an emphasis on mental arithmetic as the application short methods of calculation, despite the importance given to them during the 1930s and 1940s. • Little emphasis on children devising their own strategies, although the 1930 Syllabus, for example, did suggest that Grade VII children "should be given practice in devising short cuts and easy methods" (Department of Public Instruction, 1930, p. 45). • An emphasis on mental calculations involving the breadth of mathematical concepts dealt with in particular grades. • The application of these concepts to situations relevant to the children─for example, the "daily experience of children in the home and school" (Department of Public Instruction, 1938, p. 14).

When compared to the size of the numbers that children were required to operate with in their written work29, the limits that were placed on mental calculations could be considered to have been reasonable. This, despite the gradual increase in the limit for First Grade mental addition, for example, from 30 in the 1860 Syllabus to 99 in 1930, prior to its reduction to 10 in the 1952 Syllabus. Teachers advocated a limit of 20 for Class I mental addition during the 1920s (The Teacher’s Revised Syllabus, 1927, p. 17). However, this was opposed by some departmental officers. Representative of their opinions was that of Farrell (1929) who asserted that "tables and mental exercises [appeared] to have become old-fashioned. [Further,] the too frequent use of the concrete [had] made [teachers] sacrifice the substance for the shadow and forget also the end in the means employed" (p. 284). This was a view indicative of the resistance exhibited towards shifting the teaching focus from the subject to the child, a shift that was essential if teachers were to acknowledge the need for placing mental calculations in contexts of relevance to children.

3.5.3 Mental Arithmetic as Implemented

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The General Introduction to the 1930 Syllabus recognised that any curriculum may be viewed as (a) a course of study, and (b) as a program of experiences and activities. Further, it was recognised that "while it is possible to prescribe a definite course of study, it is not possible to specify what activities the teachers should devise for the educational benefit of the children" (Department of Public Instruction, 1930, p. vii). The use which teachers make of a prescribed syllabus varies from teacher to teacher (Edwards, 1937a, p. 25) and is dependent upon such factors as their understanding of syllabus aims and content, their use of appropriate teaching strategies, and on the constraints placed on teacher interpretation of these factors. This was perceived by "Scipio" (1943, pp. 13-14) in his attack on those teachers who were critical of the 1930 Syllabus in its failure to foster sustained changes in teaching. Their criticisms, which were anchored in the assumption that positive changes could be effected by merely adjusting Syllabus content, failed to appreciate "that the printed programme of work followed by a school is but one of the factors which determine the quality of work done therein" ("Scipio,” 1943, p. 13). Turney (1972, p. 45) suggested that, by the early 1930s, the syllabus and classroom practice as well as practice and syllabus ideals were characterised by considerable schisms, the former dichotomy being particularly applicable to mental arithmetic. Except for rare commendations─for example, those of Hendy (1953, p. 2), Moorhouse (1941, p. 1), Mutch (1907, p. 70), and Pestorius (1939, p. 64)─District Inspectors of Schools were consistently highly critical of the standard of mental calculation and of the teaching methods used during the period 1860-1965. In the first District Inspector's report to be published, it was noted that "mental arithmetic...[was] not generally advanced, [and that] this subject [cannot] be considered satisfactory till children [could] solve, mentally, a question─involving small numerical operations─in every rule through which they...passed" (Anderson, 1870, p. 14). District Inspector Benbow (1911, p. 67) considered that the absence of success in mental operations was rather remarkable, particularly considering that the examples given during his inspections were predominantly easy shopping transactions, examples in accordance with the third principle delineated in the 1904 Schedule: "The work of the pupil [should be brought] into closer touch with his home and social surroundings" (Department of Public Instruction, 1904, p. 52). This principle, reemphasised in the 1914 and 1930 Syllabuses, was one that District Inspector Hendren (1939, p. 55) would have liked to have seen more

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evidence of its implementation, a view foreshadowed by District Inspector Canny (1893), when noting that "the efforts to deal with the ‘easy shopping transactions' were feeble in the extreme" (p. 102). He stated that he had "little admiration for what were known as ‘calculation tricks,' and would [have preferred] to see pupils trained in the more solid mental work of dealing with practical problems, such as they [would] meet with in their everyday lives" (Canny, 1893, p. 102). Although it was not uncommon for inspectors to report that children lacked facility with the calculation of every-day transactions, it was more often noted that a greater proficiency was demonstrated with the difficult mechanical operations, involving fractions, for example (Fewtrell, 1914, p. 70). This observation was supported by Crampton, who, in 1955, noted that there was a "tendency on the part of some [teachers] to limit Oral Arithmetic to purely mechanical processes, [with] too few exercises...given of the problem type" (Crampton, 1955, p. 3), thus leading to poor performances on such examples (Searle, 1958, p. 15). In his report for 1927, District Inspector Palfrey (1928) observed that "mental arithmetic [was] the outstandingly weak subject of schools. Even where it [was] regularly practised, and where thought, care, and attention [were] bestowed upon the teaching, results [were] frequently disappointing to all concerned" (p. 106), a view supported by District Inspector Taylor (1928, p. 43). Although the effectiveness of the 1930 Syllabus and its amendments depended on the spiral nature of the arithmetic content, with new topics being introduced mentally in one class before preceding to the written in the next, Hendren (1939) was to conclude from his inspections that mental arithmetic had "not yet assumed the new (sic) significance envisaged in the amended syllabus" (p. 55), namely, as a preparation for written work, a significance previously emphasised in the 1904 and 1914 Syllabuses. That Hendren (1939, p. 55) was to word his conclusion in this manner suggests that this use of mental arithmetic was one with which teachers were not readily cognisant. Ross (1894, p. 75) and Benbow (1905, p. 7) had previously stressed that mental arithmetic should not be taken as a separate lesson, except for the purposes of testing, but should be taught in conjunction with the written arithmetic lesson. When reporting on mental arithmetic, it was not uncommon for District Inspectors to note that it was a difficult topic. Acting District Inspector Kemp (1913) explained that "when valuing results in this branch, one does not expect to find the same percentage of correct answers as in written work. A 50 per cent. result would

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be valued at more than ‘moderate'" (p. 59). A similar view was held by Benbow (1926, p. 52), who, for one-teacher schools, regarded a proficiency of forty percent as being quite satisfactory. District Inspector Kemp (1929, p. 47) believed that a fair or moderate result in mental arithmetic may actually be considered as being good or very good. In highlighting characteristics of mental computation, similar to those discussed in Chapter 2, he suggested that:

Teachers should not expect results in mental arithmetic to equal those obtained in written work; the two branches vary considerably. The former demanding not only method of working, but a good memory and a visualisation of the figures representing the quantities involved. (C. Kemp, 1929, p. 47)

In response to the renewed emphasis on developing the ability to calculate speedily and accurately that was embodied with the 1952 Syllabus, District Inspector Thistlethwaite (1954) reported:

Oral work was often weak [with] many children [taking] far too long to work simple types in mechanical addition, subtraction, multiplication, and division of numbers, money, and weights and measures....There was a tendency to neglect the achievement of speed and accuracy in the use of the three extensions and, often, little use was made of them in oral work after the children had passed through Grade III. (p. 2)

Prior to the introduction of the 1930 Syllabus, District Inspectors Trudgian (1929, p. 109) and Farrell (1929, p. 290) had decried that speed and accuracy in mental work were being neglected. In their view, this neglect arose from too much time being spent on trying to understand problems rather than focussing on achieving correct answers, and from the practice of doing unnecessary work on paper. Although children of Grades 3 to 7 in 1946-47 were spending approximately 22% of their mathematics time on mental arithmetic30 (ACER, 1949, pp. 20-21), it was neglect to which District Inspectors regularly attributed the poor results in mental arithmetic during the period under investigation (Bevington, 1926, p. 80; Farrell, 1927, p. 104; Fox, 1905, p. 9; Gutekunst, 1958, p. 8; Platt, 1901, p. 59; Scott, 1892, p. 92). District Inspector Johnston (1920, p. 98) considered the efforts

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of many teachers to be perfunctory, a view shared by Farrell (1927, p. 104) who deemed the work in mental arithmetic in some schools to be valueless. That mental arithmetic received little, if any attention, in some schools during the mid-1920s was evidenced, in Bevington's (1927) opinion, by the children's response to his saying, "Now we'll have a few tests in mental." He noted that "often the children [seemed] bewildered, [picked] up [their] slates, [attempted] to work on their slates, [and didn't] wait to be told to write answers down" (p. 77). Bevington (1927, p. 77) noted further that it was quite uncommon to hear a mental arithmetic lesson in progress on entering a school. A contributing factor, however, to the response reported by Bevington (1927) may have been the view of inspectors as "strict, dour, humourless characters of whom the teacher lived in dread because their salaries depended on [their reports]" (Connors, 1984, p. 95). Representative of their approach to inspection, at least during the early years of the Department, is the observation that usually there was: No cheery word [to encourage] the children when they [displayed] proficiency, accuracy, and readiness in...useful and practical calculations. Mark, on the contrary, the inspector's disparaging grimace, when they fail to solve some intricate problem dictated─once only─in a vague and puzzling form. ("Z,” 1879, p. 768)

For children to become proficient in mental arithmetic, "regular and systematic treatment" (Crampton, 1955, p. 6) was required, a view supported by District Inspectors George (1930, p. 56), Ross (1894, p. 75), and Anderson (1872, p. 11), the latter reporting that mental arithmetic was not sufficiently practised, particularly in the girls' school31. District Inspector Fox (1918, p. 61) noted that many teachers seemed to dislike the mental arithmetic lesson, their bête noir (“Mental Arithmetic,” 1927, p. 18), and that this was a reason for its not receiving the full time allotted to it in the timetables. Teachers were often criticised for not appreciating the usefulness of mental arithmetic (Canny, 1893, p. 102; Smith, 1914, p. 78)─its social utility as well as its "teaching value, when and if scientifically applied" (Woodgate, 1955, p. 2). Further, Fox (1908) had concluded in his report for 1907 that "it [would] evidently take a good deal of time to convince some teachers that a child's proficiency in arithmetic [was] not measured solely by his ability to add ‘five lines of five figures'" (p. 74).

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It was also recognised by some District Inspectors that mental arithmetic made larger demands on a teacher's time, energy, and ability than written work (Canny, 1910, p. 58), particularly in small schools where multiple grades had to be taught. In many instances, teachers were admonished in their annual reports, and by implication during their inspections of schools, for not sufficiently preparing for mental arithmetic lessons (Benbow, 1911, p. 67; Benbow, 1925, p. 51; Harrap, 1908, pp. 46-47; Inglis, 1926, p. 97; Inglis, 1929, p. 86; Kemp, 1917, p. 96; Radcliffe, 1898, p. 73). Representative of these comments is that of Searle (1956) who emphasised that:

In very few cases would teachers produce lists of examples that had been definitely taught to the class when they were asked to do so. [Such] teachers [imagined] that they [could give] mental exercises extempore, and [did] not stop to consider that the questions asked [were] of a very stereotyped nature calling for little reasoning. (p. 7)

Kemp (1917, p. 96) considered the preparation of examples of such importance that he gave consideration, in his report for 1916, to recommending that failure to do so should lead to a teacher being officially reprimanded. That examples were not "properly graded out of school hours" (Benbow, 1911, p. 67) and matched to the average mental ages of the children in the class (Mutch, 1925, p. 48) often resulted in the examples used being considered as "too promiscuous...[with] the aims of the lessons too indefinite or too imperfectly realised" (George, 1930, p. 56). This was a judgement in accordance with the position of the 1930 Syllabus, which considered that "promiscuous work [was] of little value" (Department of Public Instruction, 1930, p. 33). Given the lack of preparation considered appropriate, many teachers relied on questions taken directly from textbooks, "without plan or purpose" (Kemp, 1915, p. 98), a practice regularly criticised by district inspectors (Cochran, 1955, p. 2; Denniss, 1933, p. 31; Farrell, 1929, p. 286; Jeffrey, 1930, p. 68). Such a reliance precluded a teacher from making the mental work "a live part" of the curriculum (Denniss, 1927, p. 62). Searle (1958, p. 15) expressed concern for the brighter children in the classes of teachers who indicated that were working through the textbook, albeit the one supplied by the Department. In Searle’s (1958, p. 15) view, such children, society's

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future scientists and engineers, required a different approach to mathematics in the primary school, an approach which the New Maths curricula of the 1960s and 1970s had hoped to provide. The slavish use of the departmental written arithmetic cards32 was also seen as a reason for mental arithmetic being of a low standard in schools (Somers, 1928, p. 84). Moorhouse (1941) commented that "the best work was [observed] in schools which regularly [took mental arithmetic and were] working along planned lines with specified types of work daily taken" (p. 3). Such was the perceived need for carefully planned examples that District Inspector Fowler (1923) called for a series of "[practical] exercises in logical sequence" (p. 54) to be arranged for each of the classes. However, this suggestion was not contained in the model syllabus compiled in 1924 by the Queensland Teachers' Union, nor in the departmental syllabus introduced in 1930. Nonetheless, in Lidgate's (1959) view, it was the teaching approaches used which "may to a degree [have been] the cause of...very few schools [reaching] a standard above fair in oral arithmetic" (p. 5). It was held that teachers did not sufficiently recognise that each day's mental arithmetic should have been based on the written work to follow─to teach and test the types and processes involved. Further, many teachers wrote the examples on the blackboard allowing the children an unlimited amount of time to obtain the answers (Crampton, 1959, p. 9), thus ignoring the intended focus on the development of speed and accuracy (Department of Public Instruction, 1952b, p. 1). In contrast to this view, Kemp (1929) had advocated that "questions in mental arithmetic should be put on the blackboard much more freely than [was] usually done [as] in after life the child [would] nearly always be called upon to work mentally what is written" (p. 47). In instances where examples were presented orally to the class, Farrell (1929), reiterating the comments he made in his 1926 report (Farrell, 1927, p. 104), described the typical approach thus:

The usual practice [was] for the teacher to give a sum. In many cases the question [was] repeated. The bright pupils put up hands and shake them up and down, thereby distracting the attention of many earnest pupils who [were] trying faithfully to work the sum. The lazy children simply [did] nothing. After a time the answer [was] written. Then the teacher [started] to do all the work. He

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generally [worked] it on a board, which [was] little help for the visual process required. It [was] also the bright pupils who [did] the work and who already [had] the correct answer. The backward pupils again [did] nothing. After all this, the teacher [went] cheerfully on with another type of problem, and so the lesson [proceeded]. There [was] no attempt at individual work or at individual diagnosis to discover the causes of the failures, and the backward pupils [were] hardly ever called upon to do any of the work. (pp. 289-290)

By so doing, it was considered that teachers were not encouraging children to think clearly and systematically about number (Smith, 1917, p. 82), which was probably one of the causes for pupils simply taking it for granted that they could not do mental arithmetic (Lidgate, 1954, p. 2). In consequence of the way in which mental arithmetic lessons were taken, "many teachers, especially young ones, [were] apt to [have made] mental arithmetic an examination lesson instead of teaching the subject. It [was] thus made obnoxious, instead of being a pleasant effort" ("Howard,” 1899, p. 79). Despite the changes made to the syllabus, this view of mental arithmetic continued to be a common one during the period under investigation (Crampton, 1954, p. 3; George, 1934, p. 29; Kehoe, 1955, p. 2; Pyle, 1959, p. 9), to which previous references have been made. Instrumental in the maintenance of this view were factors primarily beyond the control of the teacher, namely, their level of training, and the size and structure of classes taught, in context with the systems of pupil examination and teacher inspection. It was not until the 1970s that the Department of Education recognised that changes in curriculum content and method could not be successfully imposed unilaterally and that their adoption by teachers requires the provision of adequate support (Department of Education, 1978, p. 4). Whenever a new syllabus is to be introduced, a large percentage of teachers and senior officers are in need of re- education (Greenhalgh, 1957, p. 327). With respect to the 1930 Syllabus, Greenhalgh (1957) noted that "little was done to assist teachers, other than to hold group meetings33, resided over by District Inspectors" (p. 39), at which the more highly qualified teachers assisted the less experienced (Edwards, 1931, p. 28), a questionable process for improving mental arithmetic, given the reports of District Inspectors as to its neglect in schools. Connor (1954, p. 19) suggested that the spirit of the 1952 Syllabus was not universally adopted as the majority of senior

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teachers had received their training up to 40 years earlier and had neither the opportunity nor the time to keep abreast of modern developments in mathematics education. Further, it was from this group of teachers that District Inspectors and Head Teachers were drawn, many of whom would have been trained under the pupil- teacher system. Although they may have become competent teachers, the early years of their training at least, would have been attained at the expense of the scholars (Department of Public Instruction, 1928, p. 41). Some were as young as 14 years of age, receiving their teacher training before and following the school day (Logan & Clarke, 1984, p. 2), thus providing an added difficulty to meeting the advocacy by District Inspectors for out-of-school preparation of mental arithmetic examples34. Many of the teachers employed were untrained, particularly in Provisional Schools where their disproportionate number prior to 190935 was instrumental in minimising the overall standard of teaching (Logan & Clarke, 1984, p. 2). As the Queensland Teachers' Journal editorialised in 1936, "it [was] still possible for Queensland children to be placed in charge of a teacher who...had no previous teaching experience, no technical training, nor been given any opportunity to study the theory of modern educational practice (“Professional Standards,” 1936, p. 1). Additionally, little professional development was gained from inspectors during their visits, in contrast to the hopes expressed following the changes in inspection procedures introduced in 1904: "[The inspector] can [now] be what he is intended to be, the professional adviser and assistant to the teacher. He can go where he is best needed, and can also leave quickly where he is not needed" (“The New Syllabus,” 1904, p. 142). Such did not occur until the compulsory inspection of teachers was discontinued from 1970. As noted by the staff of Albert State School in 1947: "The present hurried methods make it difficult for the inspector to do anything but inspect; he has little time to instruct teachers in new methods nor has he sufficient opportunity to listen to the teacher expressing his ideas" (p. 1). Not only were Queensland teachers faced with the inadequacy of their training, but also with the physical lay-out of the school, and the size36 and structure of the classes that they had to teach. The size of the classes was often exacerbated by the organisation of classes designed to reduce the numbers in the Scholarship Class (Fletcher, 1931, p. 51; Pizzy, 1950, p. 17). Mutch (1916) suggested that

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where a large school consisted of comparatively few classrooms, as was frequently the case, the quietness required for mental arithmetic was difficult to find: "The subject appeals through hearing, and [with] the majority of pupils being eye-minded and ear-minded, extra quietness [was] needed" (p. 62). One consequence of large classes was that teachers tended to maintain order by keeping the children "quiet and occupied in sedentary work, and to treat them in the mass rather than individually" (Connor, 1954, p. 19), factors which may have contributed to mental arithmetic either being neglected or being treated as a topic to be tested rather than taught. Such was particularly true for teachers of multiple grades, most notably those in one-teacher schools37 who tended to be the less experienced (Cramer, 1936, p. 8). In relation to such teachers, Edwards (1931) commented:

The inexperienced teacher [found] it difficult to organize his school properly, to limit the number of drafts, to distribute his time in such a way that all of the classes [received] a fair share of his attention, and to ensure that while he [was] engaged with one class, the others [were] profitably employed. (p. 27)

Benbow (1925) declared in his report for 1924 that it was impossible for teachers in one-teacher schools to devote time to mental arithmetic because "during the slate arithmetic lesson of one class the teacher usually is actively employed in the actual teaching of another subject to one or more of his remaining classes" (p. 51). One method suggested to teachers to enable children to practice mental arithmetic by themselves, was for an exercise of the form presented in Table 3.3 to be written on the blackboard, preferably prior to the commencement of class. It was maintained that such an approach allowed quick workers to work more examples and to proceed at their own pace; the advantage being that they did not have to wait for the slow children as in an oral lesson. Further, as the exercises "[were] not wordy they [could] be read more quickly than ‘mental' sums printed on cards or in books" (“Mental Arithmetic,” 1927, p. 18). However, although such exercises could focus on a range of number types, their use would not necessarily have contributed to the use of mental arithmetic as an introduction to the written work to follow, particularly where that work involved problems, nor would it have significantly contributed to developing a facility with everyday calculations.

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

The uncompromising nature of mental arithmetic as implemented─given the uncertainty that it provoked in the minds of children, with its emphasis on testing─was possibly enhanced by the generally authoritarian atmosphere of the classroom during the period being investigated.

Inspectors and teachers were subject to a long tradition which stamped the school as a heavily authoritarian institution. The accumulated experience of teachers in the system led them to believe that strict order, the threat of sanctions, repetition, drill and cramming were likely to achieve results in examinations and during the inspector's visit. (England, 1971, p. 193)

Contributing to this atmosphere was the rigid view of arithmetic and how it should be taught, characteristic of educators during the period being investigated. As previously discussed, children needed to be trained to think in logical sequence and to learn the value of setting out mathematical processes in precise, ordered steps (Cox & MacDonald, n.d., pp. 180-181). Such a conviction was reflected in the recommendations that the teacher of arithmetic needed to be a strong disciplinarian (Cox & MacDonald, n.d., p. 214; Gladman, n.d., p. 77).

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This authoritarian atmosphere was enhanced in Queensland primary schools by the emphasis placed on the Scholarship Examination (Schonell, 1955, p. 5), at least from the late 1920s, as the "be-all and end-all of education at the primary level" (Dagg, 1971, p. 47). Despite protestations to the contrary (Edwards, 1929, p. 32), it was a belief held by class teachers, Head Teachers, parents and by some senior departmental officers, including District Inspectors (Dagg, 1971, p. 47). To ensure that children passed Scholarship, they, in Pizzy's (1950) view, were "driven hard, bellowed at, scolded, caned, detained, overloaded with homework and crammed full with a host of useless facts, forgotten almost as soon as they were learnt, and denied a full and broad education" (p. 17). Although the effects of the need for children to pass the Scholarship Examination─an outcome on which both teachers and schools were judged (Connor, 1954, p. 19; Dagg, 1971, p. 47; "The Syllabus,” 1934, p. 1)─impacted primarily on the Scholarship Class, it indirectly affected the learning of children in lower classes, in ways other than its impact on class size. This, despite Director of Education Edwards' (1937a) proposition that the "examination [would] become an evil" (p. 6) if the examination were to dominate a school's outlook. One of its consequences, in association with the regular testing of classes by head teachers, was the concentration on skills which were readily testable (ACER, 1964, Annexure 2, p. 2), skills that were able to be tested by written rather than oral methods. Further, it is likely that this may have been a contributing factor to the interpretation of mental arithmetic as entailing testing rather than teaching, an interpretation supported by the nature of available text-books and Teachers' Notes contained in the Queensland Teachers' Journal in which the exercises presented were predominantly in the form of sets of questions38, often context-free. Representative of the recommendations to teachers was a method for stimulating mental arithmetic outlined by "J.R.D." (1928, p. 7), a regular contributor to the Queensland Teacher's Journal during the late 1920s and early 1930s. This method involved giving mental a few minutes before dismissal time and allowing the pupils to leave as they gained correct answers to questions. The exercises presented (see Table 3.4) were "those excellent examples which [had] been given of late years in the scholarship examinations [for the Sixth Class], differing from mental arithmetic only in the use of paper or slate in working" ("J.R.D.,” 1928, p. 7). Although some of the examples could be considered within the scope of the 1914

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Syllabus for the Fifth Class (see Appendix A.8), many─for example: .142857 times a certain number is 18.83, what is the number?─went beyond the spirit of its requirements, and were ones which would have proved very difficult. As "J.R.D." (1928) records, "The slower ones seldom completed them" (p. 8). The focus, at least with the decimal examples (see Table 3.4), was on mental calculation by the rote application of rules, as evidenced by his exhortation for teachers to "get from the pupils that these are division sums and [that] points must be moved accordingly" ("J.R.D.,” 1928, p. 8). The use of such procedures suggests that teachers went beyond the requirements of the syllabus, as previously discussed, and that the scholarship mathematics papers influenced the work undertaken in lower grades. The papers were considered by some to be the official interpretation of the Syllabus (“The Real Issue,” 1949, p. 2), at least for the Scholarship Class, and more generally exerted a firm control over the curriculum in the primary school (Barcan, 1980, p. 9; G. A. Jones, 1979, p. 20). "The...Commentator" (1947) concluded that "the curriculum for [grade seven] in our schools [was] the Scholarship examination. The curriculum for the other grades [was] that part of the 1930 Curriculum which the inspectors deal with in their annual examination of the school" (p. 21). Although notation was listed as a separate topic in syllabuses from 1930, its teaching could be categorised under the intended definition of oral arithmetic─as that in which explanation and discussion were of paramount importance. In some mental arithmetic texts available to teachers, some of the sets of questions contained notation items. For example, in a text designed to meet the requirements of the 1952 Syllabus, Grade VII children were asked to "Write 11½ million" and to "Make 101.03 ten times greater" (“Class Teacher's Manual,” n.d., p. 12). With the introduction of the 1948 Amendments it had been stressed that place value questions should be kept reasonably simple (Greenhalgh, 1947, p. 12).

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

Such a statement had become necessary as notation had "become a mental gymnastics exercise" (Radford, 1947, p. 1) with "some inspectors, head teachers and compilers of text-books [appearing] to delight in devising an almost infinite number of questions on ‘place value and relationships'" (“Comments,” 1947, p. 1) that went beyond the scope of the syllabus. For example, the Fassifern Branch of the Queensland Teachers' Union (1947, p. 1) noted that some inspectors were using such examples as:

101,101,101.00001 What decimal of b is a? b a (Presented orally. Time allowed: two seconds)

"Twenty-Keep List" (1942, p. 14) had protested against the practice of problemising place values. He suggested that it had been intensified and perpetuated by the monthly examinations set by head teachers. These examinations often "resulted in the disappointment and mystification of the pupils, [and] chagrin at failure to the teacher who...[had] not taken the ‘types' so cleverly and secretly worked out" (“Twenty-Keep List,” 1942, p. 14). Nonetheless, although the complexity of the calculations may have been beyond the 1930 Syllabus and may not have been

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appropriate as mental arithmetic, the nature of such examples was not in conflict with the syllabus. For Grade V, it had been suggested that children could be asked "How often is the least ‘one' contained in the greatest ‘one' [in 101.011]?" (Department of Public Instruction, 1930, p. 40), a question designed to assist teachers in "developing the intelligence [of children]" (Department of Public Instruction, 1930, p. 30).

3.6 Recommended Approaches to Teaching Mental Arithmetic

From the foregoing analyses, it is evident that there were three essential factors for effectively teaching mental arithmetic during the period 1860-1965. These were: (a) regular and systematic treatment, (b) prior preparation of graded examples, and (c) basing the mental examples on the written work which was to follow. In providing for mental arithmetic that was regular and systematic, teachers were advised that such work should form part of every arithmetic lesson, in contrast to being taught in isolation (Benbow, 1925, p. 52; Department of Public Instruction, 1914, p. 61; Joyce, 1881, p. 208; Kennedy, 1887, p. 82; "Successful Mental Arithmetic,” 1899, p. 79). The latter, however, was encouraged by the nature of many of the commercial texts. In one typical of those published from the 1930s, teachers were advised that the textbook provided "daily sets [of examples] for four days a week for nine months" (Olsen, 1953, p. i). Such a format did not encourage nor facilitate the linking of mental and written work, as recommended in the syllabuses. Nevertheless, some textbooks─The Queensland Arithmetic (Thompson, 1930), for example─recognised that "no text-book [could] take the place of a skilful teacher" (p. 1). Through the notes accompanying the 1914 Syllabus, teachers had been advised not to "trust to any text-book, but should prepare their own series of questions, adapted to local conditions and needs" (Department of Public Instruction, 1914, p. 70). Somewhat paradoxically, although the 1930 Syllabus did not prescribe any particular reference, it did recommend Pitman's Mental and Intelligence Tests in Common-Sense Arithmetic (Potter, n.d.)39 as an appropriate text for teachers to use. Under the 1860 Regulations for Queensland primary schools, teachers were permitted only to use the books40 sanctioned by the Board, unless its approval had been specially obtained (“Regulations,” 1860, p. 8). In practice, however, other books were used by teachers for lesson preparation. Margaret Berry, Head

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Mistress of the Central School for Girls, informed the 1874 Royal Commission on Education that although her teachers did not have children use any textbook not authorised by the Board, other texts were used for lesson preparation (“Minutes of Evidence,” 1875, p. 96). The list of authorised books published in the 1875 Regulations, which included Moffat’s Mental Arithmetic, was "intended to show what books teachers [were] empowered to place, when necessary, in the hands of their pupils and pupil-teachers" (“List of Books,” 1880, p. 23). It was further stated that although they were not confined to the listed books in preparing for their teaching, they would "be held responsible for the character of [their] lessons" (“List of Books,” 1880, p. 23). Effective mental arithmetic lessons were believed to be ones that were relatively short and conducted in the mornings when children's minds were fresh (Baker, 1929, p. 274; Drain, 1941, p. 2; “Teaching Hints,” 1908, p. 15). Mental calculation was considered to be taxing and therefore inadvisable for the strain to be maintained for long (Gladman, 1904, p. 207; Robinson, 1882, p. 180). Although some (Joyce, 1881, p. 212; “Mental Arithmetic,” 1910, p. 176) recommended five to ten minutes at a maximum, "Howard" (1899, p. 79) suggested that at least fifteen minutes should be spent early each morning on mental work, with children working ten or twelve sums. "The profit of the lesson [was judged to be] in proportion to the number of questions that [had] been answered correctly" (Joyce, 1881, p. 207; Gladman, 1904, p. 215), thus suggesting an emphasis on the answer rather than on the strategy used. Supporting this view, Taylor (1928, p. 43) reported that a method used by a successful teacher involved regular practice for short periods in which the examples were presented in constantly changing forms and where explanations were characterised by their brevity. Such was required if children were to develop the power of concentrating the mind, the development of which required a degree of effort which few children found easy to make (Martin, 1916, p. 135). Nonetheless, Gladman (1904), somewhat contradictory to his earlier statement, also promoted the cultivating of ingenuity through a focus on the strategy used: Children should be encouraged "to work by different methods. Get them to explain how they work; their explanation, with [the teacher's] comments, [would] do great good, especially if [the teacher could] show a readier method" (p. 208). Such an approach was alluded to in the 1930 Syllabus for Grade I, whereby pupils were required to give oral statements of the various steps involved in solving mental

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problems─steps that entailed stating the problem, providing reasons for successive operations, specifying the rules used for calculating the answer and describing their working (Department of Public Instruction, 1930, p. 33). Prior to this, it had been recommended in The “Practical” Mental Arithmetic, initially issued to schools in 1902, that the "composition of numbers and easy methods of dealing with numbers should be taught"─for example, in mentally calculating 99 + 67, children "should at once observe the facility of taking 1 from 67 to add to 99" ("An Inspector of Schools,” 1914, p. 5). As recognised by "J.R.D." (1928, p. 7), with the syllabuses from 1905 instructing teachers to use mental exercises as a preparation for written work, both mechanical and problem, it was expected that mental arithmetic lessons would be thoroughly planned. The 1930 Syllabus suggested, albeit for Grade I but applicable to other grades, as evidenced by inspectors' comments, that "exercises should be well graded and suited to the average intelligence of the class." (Department of Public Instruction, 1930, p. 33). Teachers were advised to record the planned examples in special notebooks for future reference and revision (Fewtrell, 1917, p. 74; Harrap, 1910, p. 56; Shirley, 1913, p. 37). Gladman (n.d., p. 75) had recommended that the difficulty of the examples should increase until the child needed to use slate and pencil, the implementation of which would have required an individual or group approach, one that was not seriously experimented with above Grade II in Queensland primary schools until the 1960s (Pyle, 1965, p.3). Acting District Inspector Martin (1916, p. 135) asserted that the use of carefully graded exercises that lead in steps from the easy to the more difficult would assist children to develop confidence and determination, key factors in being able to calculate mentally. Such gradations applied to problem as well as to mechanical work. It was considered important for children to master problems through their own working and in so doing gain the self-confidence necessary for attempting further problems even of unlike types (Farrell, 1929, p. 286). In consequence of the opinion that the imagination played an important part in mental arithmetic, it was suggested that questions should be framed so as to "fire the imagination of the children" (“Grade III Mental Arithmetic,” 1936, p. 14). This involved placing the arithmetic into detailed contexts. The long questions41 which resulted and the reduced number of questions treated during a session were not seen as difficulties. It was considered more important to exercise the imaginations

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of the children than "to give them a greater number and have [the pupils] guessing the answers" (“Grade III Mental Arithmetic,” 1936, p. 14). A countervailing view ("J.R.D.,” 1928, p. 7) was that teachers needed to be watchful of the wordiness of a problem so that the children were not distracted from focusing on the arithmetic within the setting provided. Nonetheless, it was considered important to place the numbers used within a context so as to familiarise pupils with the working of problems (Kennedy, 1889, p. 92). Significantly, the 1930 Syllabus, in accordance with Ballard's (1927) belief that "the most profitable form of oral arithmetic...is not that which consisted [of] casual questions" (p. 18), stated that promiscuous work was considered to be inappropriate (Department of Public Instruction, 1930, p. 33). Such work:

Nearly always [meant] asking the first questions that [came] into [a teacher's head]─questions that falsify the importance of tea or sugar, dozens or scores, in the general scheme of things, and that bear but little relationship either to the body of mathematical knowledge which the children have already acquired, or to the new material which they are about to study. As a mental exercise it [was] casual and fitful, and less closely related to the pupils' needs than to the grooves in the questioner's mind. (Ballard, 1928b, p. i)

It was necessary for every lesson to have some point to teach: a special rule to be learned, a short method of calculation or the revision of difficulties (C. Kemp, 1917, p. 96). Only one type, in contrast to the range of types generally presented in each of the sets of examples in textbooks, was to be the focus in any lesson (Bevington, 1922, p. 58; "J.R.D.,” 1928, p. 7; Moorhouse, 1927, p. 96; "Successful Mental Arithmetic,” 1899, p. 79), a recommendation that was made prior to the introduction of the 1930 Syllabus and the influence of a multiplicity of quasi-official textbooks. When teaching short methods of mental calculation, Park (1879, p. 13) stressed that rules developed through rote learning should be avoided. It was essential that:

Instruction in mental arithmetic should be imparted in such a way that it might be helpful to the practical business of life, and to prevent the possibility of the subject being taught in that worthless manner of which "exhibitions" are often

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given in public examinations. Every true teacher who knows "these tricks of the trade" gives the practice his unhesitating condemnation" (Park, 1879, p. 43)

Gladman (1904) believed that, when teaching short methods, oral exercises should not be limited to the "strict rule, [but] to go a little way on each side" (p. 208)─for example, when focusing on the rule for finding the cost of 100 articles it was considered useful to have children find the price of such quantities as 98 or 102 articles. Nonetheless, Macgroarty (1879, p. 71) cautioned that any focus on short methods of calculation should not be at the expense of suitable examples on the procedures necessary for slate work. As already implied, mental arithmetic was seen to have had a role in each of the three clearly marked stages in teaching new arithmetic, namely, (a) the theoretical or introductory phase in which children were to come to understand the processes involved in a new rule, (b) the practical or mechanical stage during which neatness, speed and accuracy were developed, and (c) the application stage during which the skills acquired were applied to problems (Board of Education, 1937, p. 504; “Teaching Hints,” 1908, p. 16). The role of mental examples during the first stage was to teach the "principle of the [operation],...worked out on the blackboard in such a way that the method of working [was] clearly seen" (Cox & MacDonald, n.d., 219). Following the introduction of the 1904 Syllabus, such teaching was to occur in conjunction with tangible objects, particularly for easier types of addition and subtraction, using examples involving subject-matter within the children's experiences (Baker, 1929, p. 274; “Teaching of Arithmetic,” 1927, p. 292). From this oral procedure, the method of working was to be determined before proceeding to the written (Gladman, n.d., p. 74). For each step in the process, teaching and test exercises were to be prepared. It was considered essential that such mental work should be based on that related to the application of tables, especially "the extended addition table, and the practical use of the Multiplication, Money, and Weights and Measures Tables" (Denniss, 1927, p. 62). In practice, however, the introduction of new work "largely centred on demonstration followed by the pupils' working of [written] examples" (Dagg, 1971, p. 75). Martin (1920b, p. 98) pointed out that this method was reasonable for the mere application of rules and formulae. However, he considered that such an approach provided for little mental development, which should have been the main aim of

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every lesson, besides providing for any additions to a pupil's knowledge. Farrell (1939) noted that there had been a tendency to test new work immediately after its presentation, at a time when "pupils have hazy ideas, and imperfect knowledge, and while they lack confidence" (p. 44). Although Farrell (1939) was referring to teaching in general, his recommendations to teachers during staff talks that discussion should be allowed for between presentation and testing is reminiscent of the intended meaning of oral arithmetic. Recommendations for the use of aids in mental calculation, which came with the espoused concern for the child from the 1890s, contrasts with Macgroarty's (1879, p. 71; 1886, p. 63; 1902, p. 67) abhorrence of the use of finger counting, in particular, and with Joyce's (1881, p. 213) rule for calculating mentally, another source of the authoritarian nature of mental arithmetic lessons. Joyce (1881) believed that:

The children, while calculating, should not be allowed to mutter audibly, or even to move the lips or distort the face; and remember not to let them count on their fingers. There should be, in fact, no exterior manifestation of the interior intellectual exertion; the first thing heard should be the answer. (p. 213)

As a method for catering for differing abilities, Olsen (1953, p. i) suggested that children could occasionally record partial answers as a prop during mental calculation, a procedure supported by Park (1881, p. 43) and Robinson (1882, p. 180). Farrell (1929), however, advised that the "practice of doing unnecessary work on paper [resulted] in loss of speed and deterioration in mental work generally" (p. 289). He also suggested that, on occasions, children could be compelled to perform all the calculations mentally. The Board of Education (1937) stressed that the rule "‘Never do work on paper that can be done mentally' often needs to be emphasised" (p. 514), a procedure with which District Inspector Brown (1901) would have concurred: The "best results in mental arithmetic [were] secured in schools in which the pupils [were] trained to perform mentally many of the operations for which the slate or blackboard [was] commonly used" (p. 85). Alternatively, Kemp (1914, p. 101) advocated using written arithmetic lessons for practice in mental arithmetic by not requiring children to record every elementary step in the working of a written calculation, a suggestion with which Gladman (n.d., p. 78) would have concurred,

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given his advocating an explicit connection between mental and written work through allowing children to use more efficient mental methods for particular examples set as written work─for example, by expecting the "elder boys to multiply by 25 and 125 in one line, and by 2,884,816 in three lines, exclusive of the answer" (Gladman, n.d., p. 78). The 1952 and 1964 Syllabuses made particular reference to the belief that effective oral arithmetic depended on the use of the blackboard and other graphical aids, particularly for more difficult examples (see Appendices A.12 & A.13). However, it was considered by some that the blackboard needed to be used judiciously. "Domas" (1952, p. 11) cautioned that, in instances where mental examples had been written on the board, mental arithmetic had a tendency to develop into written tests, a conclusion which may also be attached to the use of written examples on cards. Nonetheless, in recognising that some children had difficulty remembering significant details from examples presented orally, it was suggested in Grade III Mental Arithmetic (1936, p. 15) that cards, with five questions on each, could be used once a week. It was suggested that it was appropriate for slower children to do fewer than the five as was "it not better that they should successfully do these [few] than do none at all under the dictation system?" (“Grade III Mental Arithmetic,” 1936, p. 15). "An Inspector of Schools" (1914, p. 3) maintained that the tendency for mental arithmetic to become a written test was also manifested in instances where pupils recorded their answers to a series of oral questions in writing. Although such a procedure allowed the teacher to identify the lazy children, it did not facilitate the explanation of the operations necessary to calculate an answer. However, "the evil to be guarded against...[during the oral presentation of answers was] that it [afforded] much opportunity for lazy pupils to neglect making calculations, relying on the sharper and more industrious to satisfy the teacher with answers" ("An Inspector of Schools,” 1914, p. 3). Rapid question delivery was one of the means by which listlessness and inattention could be prevented (Robinson, 1882, p. 180). Ross (1893) believed that such exercises as "2s. 3d + 6d + 9d + 2s. 6d...with a long pause between each item, [were] neither instructive nor amusing. Speed and accuracy should both be aimed at" (p. 83). In instances where children were required to provide written answers, "J.R.D." (1928) cautioned that much valuable teaching time could be lost marking the slates of those who claimed to have the

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correct answer. Time was better spent focusing on those who were incorrect and "teaching them without delay" (p. 7). Given that the intention under each of the syllabuses was for children principally to become proficient in written arithmetic, little support was shown for Acting District Inspector Papi's (1912, p. 51) suggestion that teachers should devote the time spent on written arithmetic to mental. However, he believed that once proficiency with mental arithmetic was attained accuracy and speed in written work would necessarily follow. Nonetheless, with the general neglect by teachers of the recommendation that mental arithmetic should be a precursor to the written, it is not unexpected, in light of the analyses in Chapter 2, that Farrell (1929, p. 283) should have reported that children fail to calculate mentally on their leaving primary school. He noted further that "professional men and tradesmen [complained] that...the Arithmetical exercises [which could be interpreted as including those for mental arithmetic]...had no bearing on the problems met with in the affairs of every-day life" (Farrell, 1929, p. 283). Recognising that individuals relate to particular calculative situations in idiosyncratic ways, "Vigilate" (1950) asserted that:

The whole trouble seems to be that though in many ways arithmetic may be an exact science with regard to its truths, formulae and fundamental methods, it is not an exact science concerning its application, where the sky's the limit, without defined horizons and a comprehensible ceiling. (p. 11)

3.7 Conclusions and Summary

The foregoing analysis was designed to provide an understanding of nature and role of mental arithmetic during the period 1860 to 1965. It is evident that the historical record is substantially limited to the beliefs and opinions of two groups of stakeholders, namely, those of District Inspectors of Schools, as recorded in their annual reports, and of teachers, as expressed through publications of the Queensland Teachers' Union. Although these two groups of departmental officers often expressed differing opinions about issues which impinged on classroom practice, taken together, their recorded views, in conjunction with syllabus documents, have enabled a clear picture of mental arithmetic to emerge.

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Although generally there was unanimity in the views of teachers, albeit of those involved in the affairs of the Union, the inspectorate was not noted for its concordance on many issues. As the Director of Primary Education noted in his report for 1960, there tended to be "little unanimity of outlook among inspectors" (Guymer, 1961, p. 8). Nevertheless, as had been cautioned earlier, these differences of opinion were not to be taken as a house divided against itself (Edwards, 1931, p. 28; Ewart, 1901, p. 55), but as evidence of the inherent elasticity of the Department and of the Syllabus (Edwards, 1931, p. 28), albeit an elasticity that had resulted in teacher confusion with respect to syllabus implementation. However, in contrast to the differences of opinion which members of the inspectorate may have held on many educational issues, those associated with mental arithmetic were ones on which there was general agreement. Inspectors consistently reported that children lacked the ability to efficiently calculate mentally, that teachers did not plan their teaching effectively and that mental arithmetic did not receive the regular and systematic treatment which was required for meeting syllabus expectations. Although not always disagreeing with inspectors, teachers, particularly through their union and its publications, championed beliefs about mental arithmetic and associated Departmental procedures which they saw as essential to the improvement of teaching─not the least of which was the abolition of the Scholarship examination and the system of teacher inspection, which occurred in 1963 and 1970, respectively. To provide a focus for the presentation and analysis of the historical data, a number of research questions were posed. The following provides a summary of key points related to each.

1. What emphasis was given to mental computation in the various mathematics syllabuses for Queensland primary schools during the period 1860-1965?

In contrast to mathematics syllabuses in the United States of America during this period, each of the Queensland syllabuses contained specific references to the calculation of exact answers mentally beyond the basic facts (see Appendix A), the most common term for which was mental arithmetic, even though it was not often used in syllabus documents. As previously noted, the mental arithmetic embodied in the syllabus documents was characterised by a gradual reduction in the

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complexity of the mental calculations required, parallelling the greater specificity of syllabus requirements, requirements that covered the breadth of the mathematics curriculum. Mental arithmetic as the application of short methods of calculation decreased in emphasis during the early-20th century. However, short methods received increased attention during the 1930s and 1940s. This reversal was a reaction to the practice of many Head Teachers and District Inspectors of Schools presenting mental examples beyond syllabus requirements during their inspections of teachers and children. Nonetheless, little emphasis was given to children devising their own strategies for calculating mentally. Mental arithmetic retained its place in the syllabuses taught in Queensland schools for two key reasons. First, despite the doubt cast on the validity of the theories of formal discipline and the transfer of training, Queensland teachers and District Inspectors generally retained their belief in mental arithmetic as a means for providing "the intellectual gymnastics necessary for bracing the mind to logical and continuous thought" (Ewart, 1890, p. 67). However, although their belief in the mind as a complex of faculties may have diminished, Burns (1973, p. 4) questioned whether any faculty other than memory was ever developed as a consequence of the way in which mental arithmetic was taught─the focus in teaching was on the inculcation and retention of information, and the production of correct answers. Although recognising that the mind could not be trained in general, and believing that whatever was to be learned had to be specifically taught (Kolesnik, 1958, p. 5), a belief in arithmetic, in general, and mental arithmetic, in particular, as a means for concentrating the mind (Martin, 1916, p. 135) and developing the ability to think critically (Baker, 1929, p. 281; Bevington, 1923, p. 64; Mutch, 1924, p. 40) continued to be held. This, to such an extent that Greenhalgh (1947, p. 11; 1949b, p. 11) felt the need, during the late 1940s, to argue for teachers to forgo their conservatism and to reduce the complexity of the mental arithmetic examples given to children. Second, mental arithmetic retained its place in each of the syllabuses because of its recognised usefulness in the after-life of the child, even though prior to the 1952 Syllabus this was not specifically mentioned. However, textbooks issued to schools (Cox & MacDonald, n.d.; Gladman, n.d., 1904; Joyce, 1881; Park, 1879; Robinson, 1882) to support arithmetic teaching during the period investigated, stressed the social usefulness of mental arithmetic, a belief often reiterated by

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District Inspectors in their annual reports (Bevington, 1926, p. 80; Caine, 1878, p. 97; Macgroarty, 1891, p. 75; Router, 1941, p. 2). Although the degree to which the belief in the disciplinary powers of mental arithmetic remained fairly constant during this period, it is evident that the belief in its social usefulness gained ascendancy. The changes made to the mathematics syllabus in 1948, and continued in 1952, were aimed at ensuring that children developed skill in the calculations that needed to be made in daily life (Department of Public Instruction, 1948b, p. 17). The predominant educational philosophy from the 1930s was realism (Greenhalgh, 1947, p. 11). With the perceived discrediting of the transfer of training, it was believed that "the only things worth teaching were those for which there was some obvious and immediate use" (Kolesnik, 1958, p. 5).

2. What was the nature of mental computation as embodied in the various syllabuses and in the manner in which it was taught from 1860 to 1965?

The period 1860-1965 is characterised by a lack of preciseness in the way in which the mental calculation of exact answers was described, with each syllabus, prior to that of 1952, lacking consistency in the terminology used. Such calculation was variously called, in the 1904 Syllabus, for example, Mental exercises, Mental work, Mental and oral work and Oral and mental work (see Appendix A). Oral arithmetic was the term used in the 1952 and 1964 Syllabuses. However, mental arithmetic was the expression most commonly employed by District Inspectors, textbook authors, and contributors to the journals of the Queensland Teachers Union and the Education Office Gazette during the period being investigated. In practice, Queensland teachers took a narrow view of what constituted mental arithmetic. Encouraged by syllabus statements, comments by District Inspectors and the way in which mental examples were presented in textbooks, teachers tended to view mental arithmetic as the presentation of a series of oral questions, the aim of which was to obtain correct answers, speedily and accurately (Cochran, 1960, p. 12; Department of Public Instruction, 1930, p. vi, 1952, p. 1; Farrell, 1929, p. 289; "J.R.D.,” 1928, p. 7; Ross, 1893, p. 83, 1905, p. 33; Trudgian, 1929, p. 109). Given the impreciseness of the terminology used, and the poor quality of teacher training (“Professional Standards,” 1936, p. 1), the distinction between oral arithmetic, as encouraging explanation and discussion, and mental arithmetic was

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generally not exhibited in classrooms, nor in syllabus documents. The terms came to be used interchangeably. In the 1914 Syllabus, with respect to the First Class, oral arithmetic was viewed as "nothing more that a ready application of the tables" (Department of Public Instruction, 1914, p. 64), a view echoed by Lidgate (1954, p. 2) with respect to the oral arithmetic for all grades. Occasionally, textbook authors and District Inspectors of Schools attempted to broaden the view of mental arithmetic. Joyce (1881, p. 210), while defining an effective practical arithmetician as one who could mentally perform the short computations of everyday living, recognised that these required more than the rote application of short methods of calculation. In his view, individuals needed to be able to mentally perform "all kinds of numerical combinations..., from the common addition table up to the most complicated operations" (Joyce, 1881, p. 210). Recognition was also periodically given to a need for encouraging children to develop a "dexterity with numbers" ("An Inspector of Schools,” 1914, p. 3; Cox & MacDonald, n.d., p. 217; Joyce, 1881, p. 210; Mutch, 1924, p. 40). However, only Bevington (1925, p. 83) and Cox and MacDonald (n.d., pp. 224, 231) specifically referred to strategies similar to those identified in Chapter 2 as being based on instrumental understanding─ones that could be defined as compensatory and worked from the left. District Inspector Kennedy (1903) argued against the use of written methods for mental calculation, methods he considered as "quite out of place" (p. 69). Nonetheless, by 1952, as a consequence of the reaction to the misinterpretations of syllabus requirements by teachers, Head Teachers and District Inspectors during the 1930s and 1940s, coupled with the long-held view of mental arithmetic as initiatory to written arithmetic, this became the recommended approach to mental calculation─"The processes which are applied orally [that is, mentally] are the same as those used in written operations" (Department of Public Instruction, 1952b, p. 2). This view is one reason for the poor performances on mental arithmetic which were regularly criticised by District Inspectors in their reports, criticisms that continued into the 1950s and 1960s, even though there was increased specificity following criticisms of syllabus vagueness (“Arithmetic,” 1927, p. 17), and despite adjustments being made to grade placements of mental arithmetic tasks, in light of research findings concerning the written mathematical expectations of Queensland children (Cunningham & Price, 1934).

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District Inspectors of Schools identified a number of reasons for the poor performances of children on mental calculations, the most common of which was that mental arithmetic was often neglected in classrooms (Bevington, 1926, p. 80; Farrell, 1927, p. 104; Fox, 1905, p. 9; Gutekunst, 1957, p. 8; Platt, 1901, p. 59; Scott, 1892, p. 92). Where it was taught, teachers were often criticised for the perfunctory manner in which it was taken, often stemming from their failure to prepare sets of graded examples before the lessons (Benbow, 1911, p. 67, 1925, p. 51; Harrap, 1908, pp. 46-47; Inglis, 1926, p. 97, 1929, p. 86; Kemp, 1917, p. 96; Radcliffe, 1898, p. 73), or for using textbook examples "without plan or purpose" (Kemp, 1915, p. 98). Given the promiscuous manner in which examples were often presented, teachers tended not to base the mental work on the written work that was to follow (Hendren, 1939, p. 55; Lidgate, 1959, p. 5), despite syllabus recommendations for this to occur. Further, teachers were also castigated for limiting the work to mechanical processes to the neglect of problem types (Crampton, 1955, p. 2). Trudgian (1929, p. 109) had earlier complained that in aiming to assist children to understand problems speed and accuracy had been sacrificed. As a consequence of the way in which mental arithmetic lessons were usually conducted, with the teacher and bright children doing all the work (Farrell, 1929, pp. 289-290), many children came to believe that they could not calculate mentally. They felt that they were examined not taught during each lesson (Crampton, 1954, p. 3; George, 1934, p. 29; "Howard,” 1899, p. 79; Kehoe, 1955, p. 2; Pyle, 1959, p. 9). That teachers did not effectively implement mental arithmetic in the spirit of the syllabus nor as advocated by the inspectorate, was not only due to the nature of their pre-service and lack of inservice training, but also to such factors as: (a) the physical lay-out of the school and the size and structure of the classes taught, particularly in one-teacher schools (Benbow, 1925, p. 51), (b) the authoritarian nature of schools with their emphasis on "strict order, the threat of sanctions, repetition, drill and cramming...[as means for achieving] results in examinations and during the inspector's visit" (England, 1971, p. 193), and (c) the rigid view of arithmetic and how it should be taught. Cox and MacDonald (n.d.) encapsulated a linear view of arithmetic by declaring that "nothing need be taken for granted; every truth is capable of demonstration, and each new truth is seen to grow out of what has preceded" (p. 180). In this way, the child was trained to think logically─a form of

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mental training. Further, teachers of arithmetic needed to be strong disciplinarians (Cox & MacDonald, n.d., p. 214; Gladman, n.d., p. 77), not the least to ensure that copying did not occur (Cox & MacDonald, n.d., p. 214).

3. What was the role of mental computation within the mathematics curricula from 1860 to 1965?

The roles ascribed to mental arithmetic centred on its usefulness as: (a) a pedagogical tool, (b) a skill that was socially useful, and (c) a means for "quickening the intelligence" (Bevington, 1926, p. 80; Robinson, 1882, p. 179); issues that have been mentioned previously with respect to the first two research questions. As a pedagogical tool, mental arithmetic was seen to have had a role in each of the three stages for teaching new arithmetical ideas: (a) to assist teaching the processes involved, (b) developing speed and accuracy, and (c) applying the new skills to problems (Board of Education, 1937, p. 504; “Teaching Hints,” 1908, p. 16). Each syllabus from 1904 to 1964 highlighted that mental arithmetic should be initiatory to written work─"Mental calculations should be the basis of all the instruction, and the pupils should be made familiar by mental exercises with the principles underlying every [operation] before the written work is undertaken" (“Schedule XIV,” 1904, p. 201); a view previously expressed by Ross (1885, p. 70). The spiral nature of the syllabuses from 1930 embodied this belief in their allocation of requirements across the range of topics in mental and written arithmetic for each grade. Taking an associationist view, the Board of Education's (1937) Handbook of Suggestions, which influenced the preparation of the 1930 and 1952 Syllabuses, stated that the mechanical rules of written arithmetic are "forms of mental technique or...complex habits to be formed" (p. 506), thus implying the rote nature of written work; a belief transposed to mental calculation in the 1952 Syllabus by its statement that the "processes applied orally are the same as those used in written operations" (Department of Public Instruction, 1952b, p. 2). Mental work was also seen as a means for drilling basic facts, cultivating speed and accuracy in new work and revising work essential for sound progress (Board of Education, 1937, p. 513; Cochran, 1960, p.12; Mutch, 1916, p. 62). This approach enabled "at least four times the ground to be covered" (Farrell, 1929, p. 295) in revision work. However, it is likely that this role was one that contributed to the

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belief that mental arithmetic constituted the presentation of a series of examples, the focus of which was the gaining of correct answers. Although not specifically mentioned in a Queensland syllabus prior to that of 1952, the recognition given to the social usefulness of mental arithmetic was a distinguishing feature of the arithmetic recommended for Queensland schools, particularly through the reports of District Inspectors, and through textbooks and journal articles, as referred to under the first research question. This recognition was given in context with the social usefulness of arithmetic generally, a factor emphasised in the 1914 and 1930 Syllabuses. Nonetheless, teachers were often criticised for not giving sufficient recognition to its practical importance (Canny, 1893, p. 102; W. H. Smith, 1914, p. 78; Woodgate, 1955, p. 2), a criticism inextricably bound to District Inspectors' adverse reports on the quality of mental arithmetic teaching. As noted above, Queensland teachers maintained a belief in the role of mental arithmetic as a means for developing thinking power (Mutch, 1907, p. 70), particularly with respect to its role in promoting powers of concentration (Welton, 1924, p. 409). The development of an ability to concentrate on a task required educational effort (Martin, 1916, p. 135) and this gave "strength and activity to the mind" (Park, 1879, p. 43). However, not all children were willing to give the necessary mental effort, the consequence of which was to attribute the poor performances on mental arithmetic "simply to the lack of concentration and [the] ability to visualize" (Baker, 1953, p. 2). As recognised by the English Board of Education (1959) in the late 1950s, the use of mathematics, including mental arithmetic, as a means for developing concentration, accuracy and logical thinking, which in its crudest form encouraged the belief that such skills were automatically transferred to non-mathematical activities, had been discredited. Nonetheless, it was believed "that mathematics, well taught, may have an influence on children's general attitude to learning, and that ways of tackling problems in other situations are influenced for good by sound mathematical training" (Board of Education, 1959, p. 180).

4. What was the nature of the teaching practices used to develop a child's ability to calculate exact answers mentally during the period 1860-1965?

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During the period being investigated three tenets for teaching mental arithmetic were consistently advocated. These were: (a) regular and systematic treatment, (b) prior preparation of graded examples, and (c) the need to base the mental examples on the written work which was to follow. The profit of a lesson was believed to be in the number of questions answered correctly (Gladman, 1904, p. 215) in a period of five to ten minutes. Rapid question delivery was essential to prevent listlessness and inattention (Robinson, 1882, p. 180). Taylor (1928, p. 43), in contrast to Bevington (1922, p. 58) and Moorhouse (1927, p. 96), suggested that the examples given should constitute a range of forms and that explanations should be kept to a minimum. Such an approach was considered to have the effect of "concentrating the mind" (Martin, 1916, p. 135), with the examples selected having relevance for the goal of a lesson─a particular operation, a short method or revision (Kemp, 1917, p. 96). The Board of Education (1937, p. 514) recommended that children should never be required to work in writing those calculations that could be undertaken mentally. Prior to this, Gladman (n.d., p. 75) had suggested that children should work the prepared graded examples mentally to a point where pencil and slate were required. To encourage mental calculation, it was held that there was a need to "fire the imagination of the children" (“Grade III Mental Arithmetic,” 1936, p. 14) by placing the examples into contexts familiar to the child, a recommendation in keeping with the third principle espoused in the 1904 and subsequent syllabuses─"to bring the work of the pupil into closer touch with his home and social surroundings" (“Schedule XIV,” 1904, p. 200). However, given the tendency of some who followed this suggestion to present wordy examples, "J.R.D." (1928, p. 7) cautioned that it was essential for the arithmetic not to get misplaced in the complexity of the situation presented. The use of the blackboard was considered necessary for effective oral arithmetic, particularly for difficult examples (Department of Public Instruction, 1952b, p. 20; Department of Education, 1964, p. 11). However, as Farrell (1929, p. 289) noted, at least during the 1920s, there was a tendency for teachers to do all the working on the blackboard to the detriment of children's understanding of the processes involved. Additionally, it was argued that, where examples were written on the board prior to calculation, the propensity for mental arithmetic to become a written test was increased ("Domas,” 1952, p. 11). This added to the view that

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mental arithmetic constituted testing rather than teaching, an outcome that resulted in both teachers and students expressing apprehension with respect to mental arithmetic (Greenhalgh, 1947, p. 11).

5. What was the nature of the resources used to support the teaching of mental computation during the period 1860-1965?

With the introduction of the 1904 Syllabus in January 1905, teachers were encouraged to make "the self-activity of the pupil the basis of school instruction" (“Schedule XIV,” 1904, p. 200), a recommendation specifically contained in the 1914 and 1930 Syllabuses, and implied in those subsequently introduced. Teachers were encouraged to allow children to see and handle quantities of actual things─sticks in tens and hundreds, the foot-rule, money, weights. It was recommended that a child's introduction to number should be through the senses (Department of Public Instruction, 1930, p. 30). However, besides the use of the blackboard, little use appears to have been made of such aids to support a child's mental calculations. When observing that mental exercises were being neglected in schools, Farrell (1929, p. 284) suggested that the use of the concrete had diverted teachers and children from the all-important abstract manipulation of numbers. In instances where District Inspectors did refer to aids to support mental arithmetic, it was usually to the use of printed sheets and textbooks, the ad hoc use of which was identified as a reason for the poor performances on the mental arithmetic tests given during their inspections. The format of each of these texts (see Table 3.2) was predominantly in the form of sets of mechanical examples, which, although often referring to money and measures, did not sufficiently provide clear links to the "things by which the child is surrounded and [to] things in which he was interested" (Department of Public Instruction, 1930, p. 30). The format and content of the sets of questions supported the view that mental arithmetic lessons should involve the working of a number of questions in a relatively short period of time, the success of which was measured by the number of correct answers. Similar criticisms may be made of the teaching notes published in the Queensland Educational Journal and Queensland Teachers' Journal. Further, many of these articles were designated as mental arithmetic tests─for example, Mental Arithmetic Tests (1910), Mental Tests (1930)─thus reinforcing the view that the

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mental arithmetic lesson be equated with testing procedures. Further, even where testing was the legitimate goal of a lesson, the use of such examples did not allow teachers to implement the direction, as contained in the 1914 Syllabus, that "teachers should not trust to any text-book, but should prepare their own series of questions, adapted to local conditions and needs" (Department of Public Instruction, 1914, p. 70). However, given the conditions under which they had to operate─their inadequate training, their desire for greater specificity in grade requirements, the pressures exerted by Head Teachers, District Inspectors and Scholarship Examiners─it is not inconsistent that teachers should have embraced the examples provided as the published materials became available. Paradoxically, the consequence of this was for teachers, particularly during the 1930s and 1940s, to exacerbate the basis of their criticisms of the syllabus as being overloaded. As "Green Ant" (1942, p. 17) observed, the extensions to the syllabus requirements by the textbooks were most flagrant for mental arithmetic. Their requirements set the standard of work expected for each grade (Dagg, 1971, p. 57). Whether or not this remained a consequence of textbook use post-1965 was an avenue of investigation discussed in the next chapter, the purpose of which was to extend the understanding of mental arithmetic in Queensland schools beyond the era in which it was included as a specific branch of the mathematics syllabus.

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CHAPTER 4

MENTAL COMPUTATION IN QUEENSLAND: 1966-1997

4.1 Introduction

In contrast to the mathematics syllabuses introduced into Queensland primary schools during the period 1860-1965, a focus on mental computation was not a feature of those introduced during the New Maths era, which effectively occurred in Queensland between 1966 and 1987; nor is it considered explicitly in the current Years 1 to 10 Mathematics Syllabus (Department of Education, 1987a). Nonetheless, as discussed in Chapter 1, there has been a resurgence of interest in mental computation from the late 1980s, as a consequence of which is an emphasis on the calculation of exact answers mentally in national documents designed to guide syllabus development in Australia─A National Statement on Mathematics for Australian Schools (AEC, 1991), for example. Such a development is one that parallels that which has occurred in the United States, where educational authorities are attempting to implement the Curriculum and Evaluation Standards for School Mathematics (NCTM, 1989). Hence, to extend an understanding of mental computation within Queensland primary schools from 1966 to the present, it is not only necessary to analyse syllabus documents of the period, but also to consider the nature and effects of recent Queensland curriculum initiatives which have relevance for mental computation. Further, for recommendations to be formulated to enhance the teaching of mental computation, it is essential to gain an understanding of teachers’ current beliefs and teaching practices, the origins of which may be embedded in those of the past. This chapter, therefore, is structured around (a) an analysis of mental computation under the syllabuses from 1966-1987, (b) a survey of Queensland primary school personnel, and (c) an analysis of recent curriculum initiatives in Queensland state primary schools relevant to mental computation. 251

4.1.1 Background to the Research Strategy

In contrast to the period 1860-1965, documentary evidence, similar to that accessed during the research reported in Chapter 3, was unavailable for the post- 1964 period. The Queensland Teachers' Union had discontinued its practice of including articles on mental arithmetic in its journal, not only due to the changes to the mathematics incorporated in the Program in Mathematics (Department of Education, 1966-1968, 1975) but, more importantly, due to its emphasis on industrial rather than curriculum issues. The annual reports of the Queensland Department of Education also ceased to include those prepared by District Inspectors of Schools. Further, the thirty-year rule for the public release of government documents precluded direct access to the latter. Consequently, the use of a questionnaire to survey teacher beliefs and practices (see Section 4.3) was considered to be the most effective research method for this stage of the study. A postal, self-completion questionnaire allowed the opinions of a representative sample of school personnel to be obtained. Also, given that many teachers would have had experience teaching under each of the syllabuses from 1964, this was considered an appropriate method for extending the analysis of past beliefs and practices beyond the mid-1960s. The survey data support and extend issues raised in the discussion of the 1966 to 1987 Queensland Syllabuses (Section 4.2), and provides a background against which recent initiatives in mathematics education in Queensland (Section 4.4) may be considered for their relevance to mental computation.

4.1.2 Research Focus

Specific questions, which guided the construction of the questionnaire and the analysis of the data obtained, are delineated in Section 4.3.1. However, each of these is related to a broader question which provides a focus for all aspects of the analyses presented in this chapter, namely analyses related to the (a) 1966-1987 Syllabuses, (b) survey of school personnel, and (c) recent curriculum initiatives. These broader questions are: 252

1. What beliefs do teachers currently hold with respect to the nature and role of mental computation and how it should be taught? 2. What emphasis was given to mental computation in the period 1966-1987, with respect to both syllabus documents and teachers? 3. What emphasis is currently placed on developing the ability to compute mentally? 4. What are the characteristics of the teaching approaches currently used to develop the ability to calculate exact answers mentally? 5. What were the characteristics of the teaching approaches used to develop the ability to calculate exact answers mentally during the period 1966- 1987? 6. What need for inservice on mental computation is expressed by school personnel? 7. What was the nature of the resources used to support the teaching of mental computation during the period 1966-1987 and of those used currently? 8. What is the relevance to mental computation of recent initiatives in mathematics education in Queensland?

4.2 The Syllabuses and Mental Computation in Queensland: 1966-1987

As discussed in Chapter 3, the 1964 Syllabus was essentially an interim document necessitated by the introduction of decimal currency planned for February 1966 and the transfer of Year 8 to secondary schools from 1964. Although a new syllabus for Grades 1 to 3 was introduced in 1966, that of 1964 continued to be the syllabus for Grades 4 and 5 until 1967, and for Years 6 and 7 until 1969. The introduction of decimal currency in February 1966 was to have a major impact on primary school arithmetic, not least with the removal of the need for children to manipulate pounds, shillings, and pence in concert with an increased importance being attached to the ability to calculate with decimal numbers. Consequently, these changes, in addition to revised beliefs about how children learn mathematics, 253

were ultimately of greater consequence than the modifications made to the syllabus in 1964 (Department of Education, 1978, p. 3). Primarily through the influence of Piaget, "probably the most outstanding psychologist of the day" (Department of Education, 1966, p. i), it was recognised that an over-simplified behavioural theory of learning, using the stimulus-response model, was no longer adequate to describe mathematics learning (Hughes, 1965, p. 32). This recognition was also influenced by the work of European and English mathematics educators─that of Dienes, Gattegno, and Fletcher, for example─in which an emphasis was placed on the needs and interests of the child and how mathematical ideas grow in the minds of children (Keeves, 1965, p. 6). An emphasis was therefore placed on the discovery approach to teaching and learning (Department of Education, 1967, p. i), the rediscovery of which was, in Gordon Jones' (1967) view, "probably the most important development in pedagogy associated with modern mathematics" (p. 23). Following a series of conferences sponsored by the Australian Mathematical Society and the Australian Council of Educational Research in the early-1960s, Blakers (1978) was later to observe:

Leading educators and mathematicians accepted the rhetoric which stressed the need for precision of language and symbolism, the use of integrating concepts such as sets, function, algebraic structure and transformation geometry, and the need for children to justify each line of their mathematical reasoning by reference to an axiom or rule. (pp. 149-150).

In implementing these ideas, commencing in 1966 with the Program in Mathematics: Stages 1-4 (Department of Education, 1966), the Queensland mathematics syllabuses introduced during the New Mathematics era of the late- 1960s to mid-1970s (see Table 4.1) did not specifically refer to the development of an ability to calculate exact answers mentally beyond the basic facts, except with respect to: (a) the "application of higher decade addition and related subtraction to 254

Table 4.1 Queensland Mathematics Schedules and Syllabuses: 1965-1987

Department of Education. (1966). Program in mathematics for primary schools: Stages 1-4. Brisbane: The Department.

Department of Education. (1967) Program in mathematics for primary schools: Stages 5-6. Brisbane: The Department.

Department of Education. (1968). Program in mathematics for primary schools: Stages 7-8. Brisbane: The Department.

Department of Education. (1975). Program in mathematics. Brisbane: The Department.

Department of Education. (1987). Years 1 to 10 mathematics syllabus. Brisbane: The Department.

numbers up to 100,” (b) "combined multiplication and addition to 100,” and (c) "division with remainders within 100" (Department of Education, 1967, p. 13). This is reflected in the findings presented later in this chapter which indicate that the teachers surveyed believed that the Program in Mathematics (Department of Education, 1966-1968, 1975) placed considerably less emphasis on mental arithmetic than the 1964 Syllabus. Nonetheless, it is likely that, during implementation of the 1967, 1968, and 1975 Syllabuses, teachers continued to place an emphasis on mental arithmetic as a means for drilling basic facts, cultivating speed and accuracy, and for revision. Data from the survey reveals that, in contrast to their views about the importance of mental arithmetic embodied in the syllabuses, many teachers continued to place great importance on mental arithmetic, at least as traditionally defined (see Table 4.13). The various editions of the Program in Mathematics (Department of Education, 1966-1968, 1975) embodied a concern for developing an understanding of "underlying properties inherent in number systems and the discovery of underlying principles that enable...[the investigation of] many different areas of inquiry" (G. J. Jones, 1967, p, 20)─that is, the syllabuses embodied a concern for structure of the number system. However, stemming from an inadequate understanding of the intent of the syllabuses, teachers became focused on developing the standard 255

written algorithms through an over-emphasis on step-by-step proof, mathematical symbolism, and on the use of mathematical principles as ends rather than means (Boxall, 1981, p. 2). It is in this context that mental arithmetic was likely to have been devalued, although, in teaching the concepts outlined under the various syllabus topics, oral presentation and mental calculation would have been involved. However, it is likely that the focus would have remained on the correctness or otherwise of the answer, rather than with the mental strategies employed. As prophesied by G. J. Jones (1967), "it [was] all too possible for the content of [the new mathematics] course to receive such a formal, sterile and rigorous treatment, that it [became] far less appealing and meaningful than most traditional courses" (p. 63). Teachers were unable to cope with the "lofty ideas" contained in the syllabuses of this era (Clements, Grimison, & Ellerton, 1989, p. 70), due to a large extent to the inadequate inservice provided (Izard, 1969, p. 66). Powell (1968, p. 1) had noted that the Department of Education had not given any clear directives on how or when the programme was to be implemented. Further, Powell (1968) indicated that three days of inservice had been conducted for some teachers, but "no expert [had been] used to show us how to teach or intergrate (sic)" (p. 1), a situation in accordance with this researcher's experience. Ironically, mental computation, as now conceived, allows children to explore numbers and their interrelationships in ways that increase a child's awareness of the structure of the number system (Sowder, 1992, p. 15) in a learning environment similar to that recommended in the syllabuses of the New Maths era: "There should be frequent opportunities for the child to discuss his mathematical experiences and discoveries both with his teacher and with his classmates. Every opportunity should be allowed for expression, both oral and written" (Department of Education, 1975, p. iii). During a review of the Queensland mathematics syllabus during the early- 1980s, Murray (1981, p. 7) suggested that there was a need to investigate the implications of hand-held calculators and problem solving for the curriculum. In taking account of these recommendations and while redressing the causes of the rigid approach to teaching mathematics fostered by the 1966-1975 Syllabuses, although partially redressed in the 1975 Syllabus (G. A. Jones, 1979, pp. 24-25), the Years 1 to 10 Mathematics Syllabus (Department of Education, 1987a) placed an emphasis on how children learn mathematics, and, significantly for mental 256

computation, placed an emphasis on the thinking processes involved when using mathematics. Although it retains the development of the standard written algorithms as the major goal of the Number work, and makes no specific reference to mental computation, the 1987 Syllabus implicitly supports such a focus. Calculating is included as one of the general processes, and reference is made to "estimating and calculating...using mental and calculator procedures and written algorithms" (Department of Education, 1987a, pp. 15, 16). The development of a proficiency with mental strategies to extend the basic facts for each operation is recommended for "as many students as possible" in the Years 1 to 10 Mathematics Teaching, Curriculum and Assessment Guidelines (Department of Education, 1987b, p. 14). However, it is only in some of the accompanying sourcebooks, particularly in that for Year 5 (Department of Education, 1988, pp. 51-57), that it is clearly evident that some consideration should be given the development of flexible strategies for calculating exact answers mentally beyond the basic facts. Hence, the extent to which teachers were aware of these recommendations was one consideration for the survey of Queensland teachers and administrators reported next in this chapter.

4.3 Survey of Queensland Primary School Personnel

The procedures associated with undertaking the survey and the analysis of the data obtained have been presented using the following major headings: (a) Survey Method, (b) Survey Results, and (c) Discussion. The latter includes the conclusions drawn with respect to each of the specific research questions outlined in Section 4.3.1. The data from the survey reported in this chapter are designed to provide an understanding of:

• The beliefs about mental computation held by Queensland state primary school teachers and administrators. • The current status of mental computation in classroom mathematics programs. • The pedagogical practices related to mental computation presently employed, as well as those used during the period 1964-1987. 257

(Although this period provides an overlap with those previously discussed, it was considered that data from the questionnaire could inform the analyses relevant to the 1964-1987 Syllabuses, given that many current teachers began their careers during the early- to mid-1960s.) • The need for teacher inservice related to mental computation. • The resources used to support the development of the ability to calculate exact answers mentally beyond the basic facts.

An additional consideration for gaining insights into the status of mental computation at the time of the survey was the proposed implementation of Student Performance Standards in mathematics (“Student Performance Standards,” 1994) into Queensland primary schools (see Section 4.4). The introduction of these Standards required that mental computation be given a higher profile than that embodied in the Years 1 to 10 Mathematics Syllabus (Department of Education, 1987a). Mental computation was one of the seven Number sub-strands within the Standards, whereas it is not specifically mentioned in the 1987 Syllabus. If teachers are to effectively implement the recommended changes to the way in which mental computation should be taught, as discussed in Chapter 2, McIntosh's (1992, p. 134) two revolutions need to occur. Firstly, teachers need to enjoy manipulating numbers, but more importantly, teachers' understanding of the nature of mental computation, and of appropriate methods of teaching and assessing, needs to reflect those beliefs about computing mentally which are embodied in A National Statement on Mathematics for Australian Schools (AEC, 1991), the document on which the now discarded Student Performance Standards were based. The survey was therefore designed to determine whether the beliefs and practices of Queensland primary school teachers' and administrators' match with those currently espoused by mathematics educators─beliefs and practices previously analysed in the review of the mathematics education literature presented in Chapter 2.

4.3.1 Survey Method

In outlining the procedures used to undertake the survey, and their rationales, consideration has been given to: (a) the research questions used to guide the 258

development of the survey instrument and its analysis, (b) the instrument itself, (c) the sample of Queensland state school teachers and administrators, (d) the method of implementation, and (e) the procedures used to analyse the data.

Research Questions

To gain an understanding of the issues associated with the beliefs and teaching practices related to mental computation, as delineated above, questions were posed in relation to each. These are:

1. The beliefs about mental computation held by Queensland state primary school teachers and administrators. (a) What beliefs do teachers currently hold with respect to the various aspects of the nature of mental computation and how it should be taught? (I) Is skill in mental computation considered an important goal in mathematics education? (ii) Do the beliefs about the nature of mental computation reflect a nontraditional view? (iii) Do the beliefs about how mental computation should be taught reflect a nontraditional view? (b) What importance was placed on mental computation in the period 1964-1987, with respect to both syllabus documents and teachers? 2. The current status of mental computation in classroom mathematics programs. (a) What emphasis is currently placed on developing the ability to compute mentally? 3. The pedagogical practices related to mental computation presently employed, as well as those used during the period 1964-1987. (a) What are the characteristics of the teaching approaches currently used to develop the ability to calculate exact answers mentally? (i) Do teachers take a nontraditional approach to developing skill with mental computation? 259

(ii) Are the teaching approaches employed by middle- and upper- school teachers consistent with their stated beliefs? (b) What were the characteristics of the teaching approaches used to develop the ability to calculate exact answers mentally during the period 1964-1987? 4. The need for teacher inservice related to mental computation. (a) What need for inservice on mental computation is expressed by school personnel? 5. The resources used to support the teaching of mental computation. (a) What is the nature of the resources currently used to support the teaching of mental computation? (b) What was the nature of the resources used to support the teaching of mental computation during the period 1964-1987?

Instrument Used

The instrument used to undertake the survey of Queensland state primary school teachers and administrators was a postal, self-completion opinionnaire/questionnaire (Appendix C). In descriptive research, instrument items concentrate on the phenomenon to be described and on background characteristics, rather than on the identification of dependent and independent variables essential to explanatory research (de Vaus, 1991, p. 81). Hence, in keeping with the research purposes and questions, the survey instrument was divided into four sections: • Section 1 Beliefs About Mental Computation and How it Should Be Taught. • Section 2 Current Teaching Practices. • Section 3 Past Teaching Practices. • Section 4 Background Information.

The items in Sections 1 to 3, to which school personnel were asked to respond, are relevant to issues discussed in Chapters 1, 2 and 3, issues that are related to traditional and nontraditional beliefs about the role and nature of mental computation and how the ability to compute exact answers mentally should be developed. The traditional approach to teaching mental computation is characterised by an 260

emphasis on the teacher delivering a series of one-step questions. Speed of calculation and correctness of the answers are of paramount importance. In contrast, approaches which are defined as nontraditional are characterised by a concern for the mental strategies used to arrive at solutions. Recognition is given to the constructivist nature of mathematical development. Children are allowed to develop their own strategies for calculating mentally, and are provided with opportunities for their explanation and discussion. In Section 1, Items 1 to 9 are related to the nature of mental computation and its place within the curriculum, with Items 3, 5 and 8 reflecting a traditional view of mental computation. Items 10 to 23 concern issues related to the teaching of mental computation. Of these Items 15, 17, 20, and 21 concern teaching approaches advocated by educators who hold traditional beliefs about teaching mental computation (see Appendix C). Respondents were asked to record their beliefs on a four point Likert scale: strongly disagree, disagree, agree and strongly agree. To force respondents into expressing their beliefs, the scale did not include a neutral position. Section 2 of the survey instrument required respondents, who were currently teaching a class, to indicate how often particular pedagogical techniques were used in developing the ability to calculate exact answers mentally beyond the basic facts. It is recognised that the year level taught influences the nature of the responses to the items in this section. This confounding factor was taken into account in the data analysis. Of Items 24 to 38, traditional approaches to mental computation are represented by Items 29, 30, 33, 34, and 36. A four-point Likert scale was also used in this section: never, seldom, sometimes and often. Although a fifth point on the scale (always) would have provided a direct opposite to never and created a more balanced scale, always was omitted as it was considered that it is unrealistic to suggest that a particular approach would always be used when teaching a particular topic. In contrast, it is realistic to indicate that a particular approach is not within the repertoire of a particular teacher. As a means for gaining some insight into the teaching practices employed under the three syllabuses that do not specifically refer to mental computation, Items 39 to 46 in Section 3 required those teachers who were teaching at any time during the period 1964-1987 to respond to statements, similar to those in Section 2, in relation to each of the syllabuses. Items 39 and 40 relate to the degree of importance 261

placed on the ability to compute exact answers mentally by each of the three syllabuses, and by the respondents themselves during these periods. Items 43 to 46 reflect the traditional approach to developing mental computation skills. Given that responding to items in this section of the survey instrument required teachers to recall past teaching practices, an unsure category was added to the four-point Likert scale used in Section 2, and to that related to the importance of mental computation (see Appendix C). The items in the survey instrument were developed in association with Dr Calvin Irons, the researcher's supervisor, as well as with school personnel who have some expertise in educational research. Further revisions to the instrument resulted from its piloting with 10 teachers of Years 1 to 7 from an outer-Brisbane school which was not part of the sample of schools to be surveyed. Item analyses using correlational techniques were not undertaken in finalising the instrument. Such is in accordance with Tuckman’s (1988) view that these procedures are "not as critical for the refinement of questionnaires as they are for the refinement of tests. Questionnaire items are usually reviewed for clarity and distribution of responses without necessarily running an item analysis" (p. 226). During the revision of items based on trial data, an attempt was made to address the four assumptions on which survey research depends. Karweit (1982, p. 1837) describes these as:

• The respondent should understand the question in the same way as the researcher. • The respondent should be able to understand the question and to respond to it appropriately. • The responses should be able to be coded systematically. • Asking a particular question should not lead to a change in response.

Sample

It was intended that the data collected should be representative of the views of the 11,970 Queensland state primary school teachers and administrators. However, given the size of the population, direct sampling of school personnel would have been an impractical task. Further, data are not readily available to undertake the 262

identification of each teacher in advance by such factors as length of service, year level/s taught, location within the state and inservice undertaken. Hence the school was used as the sampling unit. To maximise the representativeness of the sample, a sampling frame (see Table 4.2), which categorised schools by Education Region and Band, was developed─Region classifies schools by geographic location, and Band classifies schools by student enrolment in combination with factors that impinge on the degree of complexity of school organisation and management. This procedure ensured that factors that influence the nature of educational programs at the school level, and their implementation, were considered, at least implicitly. These factors include school location, access to inservice, degree of teacher experience, socio-economic factors, and ethnicity of pupils. Although there is no fixed percentage of a population that determines optimum sample size (Best & Kahn, 1986, p. 16), a sample of approximately 10% of the 1075 state primary schools was considered adequate to produce a sample of teachers of sufficient size to permit meaningful conclusions to be drawn. It was also considered to be of sufficient size to allow for school personnel who chose not to participate in the survey. A sample of 10% of state schools was used by Warner (1981) to investigate the perceptions of Queensland state primary school teachers 263

Table 4.2 Sample of Schools by Band Within Educational Regions

Band of school

Region 4 5 6 7 8 9 10 Total

Sunshine 1 9a 6 10 27 18 6 77 Coast 1b 1c 1 3 2 1 9 11.7

Metropolitan 11 33 13 25 39 21 1 143 West 1 3 1 3 4 2c 14 9.8

Metropolitan 2 5 10 33 45 10 3 108 East 1c 1 3 5 1 1 12 11.1

Darling 19 37 26 12 20 3 2 119 Downs 2 4 3 1 2 1c 13 10.9

South 23 10 9 8 6 3 59 Western 2 1 1 1 1c 1 7 11.9

19 33 24 14 29 2 1 122 Wide Bay 2 3 2 1 3c 11 9.0

22 45 24 18 29 12 150 Capricornia 2 5c 2 2 3 1 15 10.0

16 16 15 7 15 11 80 Northern 2 2c 2 1 2 1 10 12.5

North 10 7 3 5 8 3 36 Western 1 1 1 1c 1 5 13.8

11 21 27 12 23 15 2 111 Peninsula 1 2 3 1 2 2c 11 9.9

South Coast 1 2 6 9 19 13 20 70 1c 1 1 2 1 2 8 11.4

135 218 163 153 260 111 35 1075 Total 13 24 17 16 28 13 4 115 9.6 11.0 10.4 10.4 10.8 11.7 11.4 10.7

Note. aTop line in each row: Number of schools in Region/Band. bBottom line in each row: Number of schools in sample. cOne school in sample was a trial school for SPS in 1993

264

with respect to aspects of the 1975 edition of the Program in Mathematics for Queensland Primary Schools (Department of Education, 1975). Alreck and Settle (1985) suggest that:

It is seldom necessary to sample more than 10 percent of the population to obtain adequate confidence, providing that the resulting sample is less than about 1000 and larger than [100]... For populations of 10,000 or more, most experienced researchers would probably consider a sample size between about 200 and 1000 respondents [appropriate]. (p. 89)

For each cell in Table 4.2 with 10 or more schools, 10% of the number of schools, rounded to the nearest 10, were randomly selected. To ensure that the percentage selected from each region and band was approximately 10%, further random selections were drawn from cells with fewer than ten schools. Prior to the random selection from each cell, given that the experiences of teachers trialing the Student Performance Standards in mathematics may have had an impact on their views of mental computation, one trial school from each region was randomly selected. These schools formed part of the 10% of schools selected from the cell in which they were categorised. On average, four primary schools were involved in the trial of Student Performance Standards in mathematics within each Educational Region. From the Directory of Queensland State Schools (Department of Education, 1991) it was determined that the 115 schools selected would provide a potential sample of approximately 1400 school personnel. With a potential sample of this magnitude, even allowing for a high nonresponse rate, 50% for example, it was projected that the actual sample of school personnel would fall comfortably within the range suggested by Alreck and Settle (1985, p. 89) referred to previously. It was recognised that in conducting a postal, self-completion survey control over who completes the questionnaire would be limited. The sample of teachers and administrators who participated within each of the schools was therefore largely self-selected. This introduced bias into the resultant sample (Karweit, 1982, p. 1839) of school personnel. However unavoidable this source of bias may be, it does necessitate cautious generalisation of the conclusions that are drawn from this study. 265

Research Procedure

The research was undertaken during Term 4 (October-December) of the 1993 school year. Three mailings to schools in the sample were undertaken. The initial mailing occurred on 4 October 1993, with follow-up correspondence being sent on 25 October and 28 November 1993 (see Appendix D). The initial mailing was timed so that the initial letter would not be received as part of the volume of mail usually received by schools at the commencement of a school term. Gay (1987) suggests that "it is sometimes worth the effort to do a preliminary check of potential respondents to determine their receptivity...and it is more productive to send the questionnaire to a person in authority" (p. 196), the latter also being the protocol where the purpose is to gain access to school personnel. As discussed below, this study was to be undertaken in a climate within schools not conducive to the willing involvement of teachers and administrators in the completion of survey instruments. This climate had arisen from the number of surveys, usually with short time-frames, that had been sent to schools from various sources during a period of rapid structural and curriculum change, change particularly associated with the management of schools and the concomitant increased expectations placed on teachers. Following Gay's (1987, p. 196) suggestion, in an endeavour to maximise participation by teachers and administrators, a letter was sent to each school principal explaining the purpose of the survey (see Appendices C.1 & C.2). The letter invited the principal, or another staff member, to act as Contact Person for the receipt and management of questionnaires within their school, and to indicate how many questionnaires they wished to receive. Cavanagh and Rodwell (1992, p. 286) suggest that obtaining contact persons is one method by which the response rate may be maximised. In the case of one-teacher schools, the principal was invited to participate in the study. This approach was also designed to reduce the cost of printing, packaging, and mailing questionnaires that would not have been returned. The initial mailing also contained a letter of authority from The Executive Director, Review and Evaluation Section of the Queensland Department of Education. This letter provided approval for the study to be conducted in the 115 schools nominated as the sample. Following the initial mailing on 4 October 1993, 52 schools agreed to participate in the survey and 27 schools declined the invitation to be involved. No response 266

was received from 36 of the 115 schools. The number of questionnaires requested by each school was dispatched in three batches as the participation forms were received. A deadline of three weeks from the date of dispatch was given for the return of the survey forms. Three hundred and ninety-five (395) questionnaires were sent to the 52 schools that had agreed to participate. Annotations on some participation forms returned by schools not wishing to receive the survey instrument provided reasons for their non-involvement. The annotations included:

• "This is an inordinately busy and taxing time for our staff." • "We do not have any time to complete any more surveys this year." • "The school is currently involved in too many surveys and projects to be able to accommodate yet another one." • "We are committed with LOTE [Language Other Than English] immersion and inclusive curriculum project."

These comments are reflective of the responses from the Queensland Department of Education's Senior Executive Forum to a request by the Minister for Education in August 1993 for comments related to the pace and scope of change occurring within the Department. The Forum indicated that schools were advocating that they should be left alone for some time to consolidate. With respect to the receipt of surveys, the Forum recommended that the number sent to schools should be reduced and that the distribution of surveys to schools had been too frequent (Senior Executive Forum, 1993, p. 9). It was in this environment that this study was undertaken and partly explains the relatively low initial positive response rate of 45%─52 of the 115 schools in the sample. As suggested by de Vaus (1991, p. 119), a follow-up mailing should be sent to schools that have not responded after three weeks. This mailing should contain a new letter, a replacement questionnaire and stamped return envelope. An estimation of the number of teachers in each of the 36 schools that had not responded was made from the Directory of Queensland State Schools (Department of Education, 1991). A maximum of 10 questionnaires was sent to each school with an accompanying follow-up letter (Appendix D.3) and reply-paid envelope. It was intended that there would be at least one questionnaire for teachers of each year level and one for each administrator. The identification of nonresponse schools was 267

enabled by the inclusion of a school identification number on each address label on the reply envelopes sent with the initial mailing. Two hundred and three (203) questionnaires were sent to schools in this mailing, resulting in a total of 598 questionnaires distributed. This represents survey instruments being sent to approximately 43% of the estimated 1387 teachers and administrators in the 115 schools in the sample. To further enhance the response-rate, a second follow-up letter was sent to the 15 Contact Persons at schools from which survey instruments had not been received by 25 November 1993 (Appendix D.4). Although agreeing to participate in the survey through a staff member nominating to be a Contact Person and requesting questionnaires, none were received from 12 of the 52 schools that had agreed to participate in response to the initial letter.

Methods of Analysis

The data from each respondent was coded in the form indicated below, and analysed using various subprograms of the Studentware version of the Statistical Package for the Social Sciences, SPSS/PC+ Studentware Plus (Norusis, 1991):

1. Identification number (001 through 201). 2. Beliefs about mental computation and how it should be taught: (a) Items 1 to 23─Strongly Disagree (1), Disagree (2), Agree (3), Strongly Agree (4), Missing (9). 3. Current teaching practices: (a) Items 24 to 38─Never (1), Seldom (2), Sometimes (3), Often (4) Missing (9). 4. Past teaching practices: (a) Items 39a to 40c─No Importance (1), Little Importance (2), Some Importance (3), Great Importance (4), Unsure (5), Missing (9). (b) Items 41a to 46c─Never (1), Seldom (2), Sometimes (3), Often (4), Unsure (5), Missing (9). 5. Background Information: (a) Item 47 (Educational Region)─Sunshine Coast (01), Metropolitan West (02), Metropolitan East (03), Darling Downs (04), South West 268

(05), Wide Bay (06), Capricornia (07), Northern (08), North West (09), Peninsula (10). South Coast (11), Missing (99). (b) Item 48 (Size of school)─Band 4 (1), Band 5 (2), Band 6 (3), Band 7 (4), Band 8 (5), Band 9 (6), Band 10 (7), Missing (9) (c) Item 49 (Years of teaching experience)─< 1 yr (1), 1-5 yrs (2), 6-10 yrs (3), 11-15 yrs (4), 16-20 yrs (5), 21-25 yrs (6), 26-30 yrs (7), 30+ yrs (8), Missing (9). (d) Item 50 (Teaching role)─Class Teacher (1), Teaching Principal (2), Principal (3), Deputy Principal (4), Missing (9). (e) Item 51 (Year Level/s taught)─For each of Years 1 to 7: Teaching a particular year level (1), not teaching a particular year level (0), Missing (9). (Data from the None box associated with this item were not coded. Using the highest class taught, the year-level data were used to categorise teachers as lower-, middle- or upper-school teachers.) (f) Items 52 to 53 (Student Performance Standards)─Yes (1), No (0), Missing (9). (g) Items 54 to 55 (Inservice)─Yes (1), No (0), Missing (9). (h) Item 56 (Inservice source)─Teaching colleague (1), Administrator (2), Mathematics Adviser (3), Tertiary Lecturer (4), Other (5), Missing (9).

Resources listed by respondents in Sections 2.2, 3.3, and 4 of the questionnaire were collated for inclusion in the results of the survey. Those used by middle- and upper-school teachers, as defined below, are the textbooks referred to in Table 4.18. Comments recorded by respondents in each of the above sections of the questionnaire were also collated. Where relevant, these are included in the presentation of data and their analysis. As indicated previously, the survey was conceptualised as descriptive research, for which univariate analysis provided appropriate statistical techniques. Nonetheless, given the non-randomness of the sample of school personnel (see below), the conclusions presented in Section 4.3.3 may only be considered suggestive of the beliefs and practices of Queensland teachers and administrators in general. The survey’s purpose was to gain insights into the beliefs and teaching practices of school personnel; to describe the status of mental computation, present and past, in Queensland state primary school classrooms. Hence, it was 269

considered sufficient to present the data for each item as a frequency distribution (see Tables 4.7 to 4.14). To facilitate an understanding of the degree to which school personnel espouse nontraditional beliefs and teaching practices concerning mental computation─those that are reflective of beliefs and practices currently advocated by mathematics educators─the mean ratings for selected items have been presented on continua (Figures 4.1 to 4.5). Although the theoretical measure of central tendency for ordinal data is the median, a scan of the data revealed that the means of item ratings would better reflect the spread of opinion in such graphical representations. For example, for the items represented in Figure 4.1 the median for each variable is 3, whereas the means range from 2.63 to 3.24, with standard deviations of .74 and .55, respectively. To present the percentage distribution for each item in Section 1 of the survey instrument (Beliefs), the items were grouped into four categories. These were:

1. Beliefs about the importance of mental computation (Table 4.8: Items 1, 2, and 7). 2. Beliefs about the nature of mental computation (Table 4.9: Items 3, 4, 5, 6, 8, and 9). 3. Beliefs about the general approach to teaching mental computation (Table 4.10: Items 10, 12, 13, 14, 18, and 23). 4. Beliefs about specific issues related to developing the ability to calculate exact answers mentally (Table 4.11: Items 11, 15, 16, 17, 19, 20, 21, and 22).

The mean rating was calculated for each item in Categories 2, 3, and 4 (see Figures 4.1 to 4.3), except for Items 12, 18, and 23 in Category 3. It was considered inappropriate to interpret these items in terms of their reflecting traditional or nontraditional beliefs, as, for example, conducting a series of mental computation sessions each week (Item 23) has relevance to both orientations. In calculating the means, the scoring for items that reflect a traditional approach was reversed to facilitate their placement on traditional-nontraditional continua. Hence, Items 3, 5, 8, 15, 16, 17, 20, and 21 (see Appendix C) were recoded as: Strongly Disagree (4), Disagree (3), Agree (2), and Strongly Agree (1). Appendix E presents the standard deviation of each mean. 270

The purpose of Section 2.1 of the survey instrument was to determine how frequently present classroom teachers use each of a range of teaching techniques to develop the ability to mentally calculate exact answers beyond the basic facts. The Years 1 to 10 Mathematics Teaching, Curriculum and Assessment Guidelines (Department of Education, 1987b, pp. 16-17) suggests that children should begin to experience mental strategies to extend the basic facts from Year 2 for addition, Year 3 for subtraction, Year 4 for multiplication, and from Year 5 for division. For each operation, it is expected that as many children as possible will develop proficiency by Year 7. The focus during Years 1, 2, and 3 is primarily on the development of basic fact knowledge. Hence, in analysing the responses to the items in Section 2.1, it was considered appropriate that the data to be analysed should be restricted to those from teachers of Years 4 to 7. It was anticipated that many teachers would be responsible for multi-age─multi- grade─classes. Additionally, it was assumed that, in responding to items related to developing the ability to calculate exact answers mentally beyond the basic facts, respondents would be likely to indicate their teaching practices with respect to the highest year level taught. Classroom teachers and teaching principals of multi-age classes were therefore classified by the highest year level taught and categorised as a lower-school (Years 1 to 3), middle-school (Years 4 and 5) or upper-school (Years 6 and 7) teacher. Teachers of single classes were classified similarly. For the purposes of the presentation of data from Section 2.1 (see Table 4.12) and their analysis, only data from middle- and upper-school teachers, with respect to current teaching practices, were considered. To gain insights into the degree to which current teaching practices of middle- and upper-school teachers reflect a nontraditional approach, means for items in Section 2 that referred to specific teaching practices were calculated and presented on a continuum (Figure 4.4). The coding for items reflecting traditional teaching practices was reversed, namely Never (4), Seldom (3), Sometimes (2), and Often (1). Items 25, 26, 29, 30, 31, 33, 34, and 36 (see Appendix C) were selected for analysis, with Items 29, 30, 33, 34, and 36 being recoded (see Appendix E). Section 3 of the survey instrument was designed to elicit opinions and past teaching practices related to mental computation within the context of Queensland mathematics syllabuses in use between 1964 and 1987 (see Table 4.1). As previously intimated, the Program in Mathematics for Primary Schools (Department of Education, 1966-1968) progressively replaced the 1964 Syllabus during the late 271

1960s, with the program for Grades 6 and 7 being introduced in 1969. This syllabus was revised in 1975 and remained in use until 1987 when the Years 1 to 10 Mathematics Syllabus (Department of Education, 1987a) became the basis on which mathematics teaching was to be planned for Queensland primary school pupils. Hence, the time periods used were: 1964-1968, 1969-1974, and 1975-1987. As the purpose of this section was to gain insights from school personnel who had teaching experience in Queensland classrooms during each of these three periods, responses were selected for analysis on the basis of their length of service (Item 49): (a) for the 1964 Syllabus, teaching for more than 25 years; (b) for the first edition of the Program in Mathematics (Department of Education, 1966-1968), teaching for more than 20 years; and (c) for the 1975 edition of the program, teaching for six years or more. It is recognised that this procedure may have resulted in some loss of data. It is possible that school personnel, with non- contiguous teaching careers, may have taught during one or more of these periods but not have a length of service matching the selection criteria. However, an advantage of applying these criteria is the exclusion of responses from teachers who did not teach during any of the periods under investigation. An analysis of the data revealed that at least two class teachers with less than one year's experience responded to this section of the survey. The mean for each item in Section 3.2 of the questionnaire was calculated to provide a procedure to gain insights into the degree to which teaching approaches classified as nontraditional were in use during each of the periods under investigation, namely 1964-1968, 1969-1974, and 1975-1987. Items 43 to 46, which reflect traditional teaching methods, were recoded as: Never (4), Seldom (3), Sometimes (2) and Often (1). The means for Items 41 to 46 were plotted on continua (Figure 4.5) to provide graphical representations of the data (see Appendix E).

4.3.2 Survey Results

Prior to presenting the data related to each of the sections of the questionnaire, analyses of the patterns of response and non-response have been undertaken. Such analyses are critical to the validity of the study's findings and to their generalisability. 272

Response Rate

Survey forms were received from 49 of the 115 schools in the sample, which constitutes a 42.6% response rate (see Table 4.3). Thirty-nine (39) of the 52 schools nominating Contact Persons, and 10 schools which did not reply to the initial letter, returned questionnaires. As described previously, copies of the survey instrument were sent to the 36 schools which did not reply to the initial invitation to participate in the survey. An analysis of the response rate of schools by band of school and educational region is provided in Table 4.4. Two (2) of the 11 trial schools for Student Performance Standards in mathematics, one from each of the Northern and North Western regions, returned questionnaires; three questionnaires (5.1% of those dispatched) were returned for analysis. In response to the initial letter, five trial schools indicated that they did not wish to receive the survey instrument. Fifty-nine (59) questionnaires were sent to the remaining six schools in the sample. This low response rate precludes a meaningful analysis of data with respect to school personnel who were involved with the trial of the Student Performance Standards. Table 4.3 Schools Returning Questionnaires

Number of schools in sample: 115 Schools refusing to participate in survey: 27

Schools requesting questionnaire after initial letter: 52 Schools returning questionnaires: 39 75.0%

Schools sent questionnaires in second mailing: 36 Schools returning questionnaires: 10 27.7%

TOTAL: 49 42.6%

As revealed in Table 4.5, 201 questionnaires were returned─a response rate of 33.6%. One hundred and seventy-one (171) questionnaires were returned from schools with Contact Persons and 29 from schools sent questionnaires in the second mailing. One (1) questionnaire remains uncategorised as the school from which it was returned was not able to be identified. 273

The response rate for questionnaires by educational region and band of school is presented in Table 4.6. Although the size of the sample of school personnel was potentially about 1400, a two-stage reduction in sample size was in operation. Firstly, not all schools in the sample chose to participate─either declining in response to the initial letter or by not returning any of the questionnaires sent. Secondly, not all teachers and administrators from each participating school chose to complete and return the questionnaire. As de Vaus (1991) comments: "Although we can ask the person who receives the mail questionnaire to pass it on to the appropriate person, we cannot be sure that this happens" (p. 108); nor can we be sure that the potential respondent is sufficiently interested to complete and return the survey form. Asterisks in Table 4.6 indicate cells from which no questionnaires were received from schools selected as part of the sample─26 of the 64 cells. However an analysis of Table 4.6 reveals that the spread of questionnaires received does encompass a range of geographic locations and sizes of schools. Despite this, an analysis of the data by educational region or band of school is precluded, given the magnitude of the row and column totals displayed in Table 4.6. Hoinville (1977, 274

Table 4.4 School Response Rate by Region and Band

Band of school

Region 4 5 6 7 8 9 10 Total

Sunshine 1 1 1a 3 2 1 9 Coast 1b 3 2 6 66.7

Metropolitan 1 3 1 3 4 2 14 West 1 1 1 3 6 42.9

Metropolitan 1 1 3 5 1 1 12 East 1 2 1 4 33.3

Darling 2 4 3 1 2 1 13 Downs 1 1 1 3 23.1

South 2 1 1 1 1 1 7 Western 2 1 3 42.9

2 3 2 1 3 11 Wide Bay 1 3 1 1 6 54.5

2 5 2 2 3 1 15 Capricornia 2 1 1 1 2 7 46.7

2 2 2 1 2 1 10 Northern 1 1c 1 3 30.0

North 1 1 1 1 1 5 Western 1 1c 2 40.0

Peninsula 1 2 3 1 2 2 11 1 1 1 1 1 5 45.5

South Coast 1 1 1 2 1 2 8 1 1 2 4 56.0

Total: Sent 13 24 17 16 28 13 4 115 Returned 10 8 5 6 13 4 3 49 Percent 76.9 33.3 29.4 37.5 46.4 30.8 75.0 42.6

Note. aTop line in each row: Number of schools in sample. bBottom line in each row: Number of schools returning questionnaires. cTrial school for Student Performance Standards in 1993. 275

Table 4.5 Analysis of Number of Questionnaires Returned

TOTAL questionnaires sent: 598

Number of questionnaires sent after initial mailing: 395 Number returned: 171 43.3%

Number of questionnaires sent in second mailing: 203 Number returned: 29 14.3%

Number returned of unknown origin: 1

TOTAL questionnaires returned: 201 33.6%

cited by de Vaus, 1991, p. 73), suggests that the smallest subgroup should have at least 50 to 100 cases for a meaningful subgroup analysis.

Analysis of Nonresponse

From Table 4.5 it is apparent that the nonresponse rate for the return of questionnaires was high (66.4%). Nonresponse may lead to an unacceptable reduction in sample size, particularly when missing values are accounted for during analysis, and to bias (de Vaus, 1991, p. 73). With respect to the former, the number of valid cases for the belief Items ranged from a low of 186 for Item 15 to 201 for Items 4 and 11 (see Appendix C). The mean number of valid cases was 196.4. For Current Teaching Practices, the mean number of valid cases was 173.4, a lower mean as a consequence of this section not being relevant to all respondents. Nine non-teaching principals and six deputy principals returned questionnaires. Additionally, some respondents who identified themselves as class teachers may not have had responsibility for a class at the time of the survey's completion. The number of valid cases relating to Current Teaching Practices ranged from 170 for Item 38 to 176 for Items 25 and 26. Except for Items 4 and 11, the number of valid cases for each item was less than the optimum range─200 to 1000 cases for populations of 10,000 or more─suggested by Alreck and Settle (1985, p. 89). However, "the size of the population from which...the sample [was drawn] is largely irrelevant for the 276

Table 4.6 Questionnaire Response Rate by Region and Band

Band of school

Region 4 5 6 7 8 9 10 Total

4 14b 51 31 9 109 Sunshine Coast *a * 7c 16 26 * 49 45.0

Metropolitan 1 6 19 48 19 93 West 1 1 * 4 28 * 34 36.6

Metropolitan 8 16 51 8 83 East * * 2 12 * 8 22 26.5

Darling 1 2 4 24 30 61 Downs 1 2 4 * * * 7 11.5

South 2 2 2 5 8 8 27 Western 2 * * * * 1 3 11.1

2 7 8 16 16 49 Wide Bay 1 6 4 9 * 20 40.8

2 6 9 11 16 8 52 Capricornia 2 1 4 3 8 * 18 34.6

2 3 8 8 9 * 30 Northern 1 1 5 * * 7 23.3

North 1 2 5 8 Western 1 * * 2 * 3 37.5

1 3 6 6 16 9 41 Peninsula 1 3 1 5 3 * 13 31.7

17 9 19 45 South Coast * * * 8 4 12 24 53.3

Unknown 1

Total: Sent 12 31 49 95 261 114 36 598 Returned 10 14 18 30 77 31 20 201 Percent 83.3 45.2 36.7 31.6 29.5 27.2 55.6 33.6

Note. aAsterisk indicates cell in which no school in sample accepted/returned questionnaire. bTop line in each row: Number of questionnaires sent. cBottom line in each row: Number of questionnaires returned. 277

accuracy of the sample. It is the absolute size of the sample that is important" (de Vaus, 1991, p. 71). A perusal of Tables 4.7 to 4.11 suggests that, for most items, opinions and practices are skewed towards one end of the four-point Likert scale. Assuming representativeness of the sample (see below), this suggests a measure of homogeneity in the population of school personnel with respect to their beliefs and practices. In such instances, smaller sample sizes produce similar degrees of accuracy as compared to those required for samples from heterogeneous populations. It is therefore considered reasonable to conclude that the numbers of valid cases obtained are sufficient to allow meaningful conclusions to be drawn. Henry (1990) suggests that "nonresponse creates a potential for nonsampling bias that cannot be overlooked after the data has been collected" (p. 131). However, in Alreck's and Settle's view (1985, p. 76), it is not possible to completely eliminate such bias, and therefore some nonresponse bias needs to be tolerated. One method for determining the potential impact of nonresponse on the conclusions drawn involves detecting differences in the data obtained from questionnaires received at different times. As Henry (1990) intimates, "no differences in the ‘waves' of responses can indicate that response bias is less likely. This assumes that late responders may share characteristics with nonresponders" (p. 132). A search for differences in response was restricted to items related to Beliefs (Items 1 to 23) and Current Practices (Items 24 to 38). The receipt of questionnaires approximated four waves of responses, namely survey instruments identified as numbers 1 to 83 (Group 1), 84 to 129 (Group 2), 130 to 164 (Group 3) and 165 to 201 (Group 4). A one-way analysis of variance, together with Bonferroni multiple comparisons (Norusis, 1991, pp. 178-180), was undertaken to detect differences among the grouped responses for each of Items 1 to 38. This method is conservative in its approach to reducing the likelihood of a Type 1 error (Agresti & Finlay, 1997, p. 447; Myers & Well, 1995, p. 181). For each pairwise comparison, given that the survey responses were allocated to four groups, the observed significance level was set at .05 ÷ 6, or .008 approximately, for the difference to be considered significant at the .05 significance level. Table 4.7 reveals that statistically significant differences between pairs of group means at the .05 level were detected for Items 13, 18 and 28, the first two 278

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

Item Group Mean F Ratio F Prob

13. Strategies for calculating exact Grp 1: 3.07a 4.02 .01 answers mentally are best Grp 2: 3.16 developed through discussion Grp 3: 3.23 and explanation. Grp 4: 3.41a

18. Opportunities for children to Grp 1: 3.07b 2.94 .03 calculate exact answers Grp 2: 2.73b mentally need to be provided in Grp 3: 2.94 all relevant classroom activities. Grp 4: 3.06

28. Teach particular mental Grp 1: 3.31 2.47 .06 strategies and follow up with Grp 2: 3.56c practice examples. Grp 3: 3.13c Grp 4: 3.34

Note. a,b,cPair of groups significantly different at the .05 level.

being belief statements and the latter being one of the Current Teaching Practices to which teachers were asked to respond. For Item 13, statistically significant differences at the .05 level were detected between Groups 1 and 4. Given the progressive increase in group means (see Table 4.7), together with Henry's (1990, p. 132) assumption that late responders may share characteristics with nonresponders, it is possible that the latter may more strongly agree with this statement. For Items 18 and 28, the means for Groups 1 and 2, and Groups 2 and 3, respectively, were significantly different at the .05 level. Given this pattern, the opinion of nonresponders may not be markedly different to that of those who responded to the questionnaire. As statistically significant differences were detected for only 3 of the 38 items tested, it seems reasonable to assume that the opinion of nonresponders is likely not to be greatly different from that of respondents. Hence, although the nonresponse rate was 66.4%, the data collected may be cautiously considered indicative of the beliefs and practices of Queensland state primary school teachers.

279

Beliefs About Mental Computation and How it Should Be Taught

Queensland state primary school teachers and administrators agree that mental computation is a legitimate goal of mathematics education, with only 6% disagreeing and 0.5% strongly disagreeing with Item 1 (see Table 4.8). Complementing this finding, is the view that the ability to calculate exact answers with paper-and-pencil is not of more use outside the classroom than the ability to calculate mentally. Approximately 87% of respondents did not agree with Item 7 (see Table 4.8). However, with respect to the degree of importance on mental computation embodied in the Years 1 to 10 Mathematics Syllabus (Item 2), the opinion of school personnel is divided. Approximately 51% of respondents believe that the syllabus places little importance on mental computation, whereas 46.8% hold the opposing view. This pattern was reversed for middle- and upper-school teachers, with more─52.8%─ believing that the current syllabus places little importance on mental computation. The responses to Items 4, 6, and 9 (see Table 4.9) reveal an agreement with current beliefs about the role of mental computation with the respect to the development of number sense and ingenious methods for manipulating numbers. This is in contrast to the responses to Item 5 which indicate a relatively high percentage of respondents─approximately 37%─believing that mentally calculating exact answers involves applying rules by rote. Only 8.5% of respondents disagreed with the proposition that mental computation encourages children to devise ingenious computational short cuts (Item 4). Approximately 89% either agreed or strongly agreed that mental computation helps children to gain an understanding of the relationships between numbers (Item 6). Most respondents─93.1%─believe that children who are proficient with mental computation use non-standard mental strategies (Item 9). The responses for Item 8, however, reveal some inconsistency with the data for Item 9. Although the majority─72.6%─either disagree or strongly disagree with the 280

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

Item Strongly Disagree Agree Strongly Missing Disagree Agree

1. The development of the ability to calculate exact answers mentally is a 0.5 6.0 37.8 55.2 0.5 legitimate goal of mathematics education.

2. The Years 1 to 10 Mathematics Syllabus places little importance 6.5 44.8 43.3 3.5 2.0 on the development of 8.3a 38.0a 49.1a 3.7a 0.9a the ability to calculate exact answers mentally.

7. The ability to calculate exact answers with paper-and-pencil is 24.4 62.7 9.5 1.5 2.0 more useful outside the classroom than the ability to calculate mentally.

Note. N = 201. aPercentage related to middle- and upper-school teachers (n = 108).

proposition that children should use the written algorithms for mental calculations (Item 8), approximately 23% of respondents believe that children should use paper- and-pencil algorithms when computing mentally (see Table 4.9). In comparing the pattern of responses to Item 7 (see Table 4.8) and Item 3 (see Table 4.9), the overall pattern of responses is similar. However, a greater percentage of respondents─approximately 23%─consider written methods for calculating to be superior to mental methods (Item 3). This, despite the strong disagreement with the proposition that written methods are more useful outside the classroom than are mental procedures (Item 7). The mean responses for items relating to the beliefs of school personnel about the nature of mental computation were: Item 3─2.89, Item 4─3.17, Item 5─2.63, Item 6─3.17, Item 8─2.89, and Item 9─3.24 (see Appendix E for the standard deviations of these means). The coding for Items 3, 5, and 8 was 281

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

Strongly Strongly Item Disagree Disagree Agree Agree Missing

3. Written methods of calculating exact answers 16.4 56.7 18.9 4.0 4.0 are superior to mental procedures.

4. Mental computation encourages children to 8.5 65.7 25.9 devise ingenious computational short cuts.

5. Calculating exact answers mentally involves applying 9.0 50.2 30.8 6.5 3.5 rules by rote. 7.4a 52.8a 33.3a 5.6a 0.9a

6. Mental computation helps children gain an 10.0 61.2 27.4 1.5 understanding of the relationships between numbers.

8. Children should use the algorithms for written 14.4 58.2 21.9 1.5 4.0 computation when 14.8a 57.4a 24.1a 1.9a 1.9a calculating exact answers mentally.

9. Children who are proficient at mentally calculating exact 6.0 63.2 29.9 1.0 answers use personal adaptations of written algorithms and idiosyncratic mental strategies.

Note. N = 201. aPercentage related to middle- and upper-school teachers (n = 108).

reversed to facilitate their placement on a traditional-nontraditional continuum (Figure 4.1). The data suggest that Queensland state school personnel tend towards holding nontraditional beliefs about the nature of mental computation. As discussed in Chapter 2, mathematics educators believe that consideration should be given to focussing on mental computation prior to written computation (Biggs, 1969, p. 25; Carroll, 1996, p. 35; Cooper et al., 1992, pp. 100-101; Musser, 1982, p. 40; Rathmell & Trafton, 1990, p. 156). However, approximately 65% of

282

Traditional Nontraditional

Strongly Strongly Disagree Disagree Agree Agree

1 2 3 4 | | | | ─────────────────── ───────────X───X─── ─X─X───────────────

Items: 5 3 4 8 6 9

Recoded items: 3, 5 and 8

Figure 4.1. Position of means for items relating to the beliefs about the nature of mental computation on a Traditional-Nontraditional continuum

Queensland teachers and administrators either disagreed or strongly disagreed with the proposition that the written algorithms should be delayed so that mental strategies could be given increased attention (see Table 4.10, Item 10). The responses to Items 12 and 13 reveal that there is general agreement with the view that mental strategies need to be specifically taught─83.1%─and that such strategies are best developed through discussion and explanation─93.5%. However, approximately 14% of respondents did not believe that specific teaching of strategies was necessary (Item 12). This finding is complemented by data for Item 18 (see Table 4.10). Approximately 20% of respondents believe that opportunities for developing the ability to compute mentally should not be provided in all relevant classroom activities. However, 78.1% do support the integration of mental computation with other classroom experiences. A similar pattern of data is observed for Item 23 (see Table 4.10). The majority of respondents (77.6%) support the use of a series of focus lessons each week, whereas 19.4% do not agree with this approach. Paralleling the finding for Item 9 (see Table 4.9), which suggests a belief that proficient mental calculators use personal mental strategies, the data for Item 14 (see Table 4.10) indicate that school personnel believe that children should be allowed to develop and use their own mental strategies─63.2% agreed and 29.9% strongly agreed with this proposition. 283

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

Strongly Strongly Item Disagree Disagree Agree Agree Missing

10. Emphasis on written algorithms needs to be 5.5 59.7 25.9 5.5 3.5 delayed so that mental computation can be given increased attention.

12. Strategies for calculating exact answers mentally 14.4 57.2 25.9 2.5 need to be specifically taught.

13. Strategies for calculating exact answers mentally are 5.0 70.6 22.9 1.5 best developed through discussion and explanation.

14. Children need to be allowed to develop and use 6.0 63.2 29.9 1.0 their own strategies for 3.7a 63.9a 30.6a 1.9a calculating exact answers mentally.

18. Opportunities for children to calculate exact answers 1.5 17.9 60.2 17.9 2.5 mentally need to be provided in all relevant classroom activities.

23. A series of session that focuses on developing 19.4 60.2 17.4 3.0 strategies for computing exact answers mentally needs to be conducted each week.

Note. N = 201. aPercentage related to middle- and upper-school teachers (n = 108).

The mean responses for three of the items listed in Table 4.10 that may be considered on a Traditional-Nontraditional continuum were: Item 10─2.32, Item 13─3.18, and Item 14─3.24 (See Appendix E). These are presented in Figure 4.2 and indicate that, except in relation to delaying the focus on written algorithms, 284

Traditional Nontraditional

Strongly Strongly Disagree Disagree Agree Agree

1 2 3 4 |───────────────────|────X─────────────|─XX────────────────|

Items: 10 13 14

Figure 4.2: Position of means for items relating to beliefs about the general approach to teaching mental computation on a Traditional- Nontraditional continuum.

school personnel generally agree with propositions that reflect a nontraditional approach to developing the ability to calculate exact answers mentally. Table 4.11 reveals some inconsistencies in the beliefs of teachers and administrators about how mental computation should be taught. Although no respondent disagreed with the proposition that children should be given opportunities to discuss, compare, and refine their mental strategies (Item 19), there was sizeable support for key aspects of the traditional approach to calculating exact answers mentally in which the focus is on testing rather than teaching. Approximately 67% of respondents believe that children's mental processes are sharpened by starting a mathematics lesson with 10 quick questions (Item 15). This finding is supported by data for Item 16 which reveal that 57.2% agree and 7.5% strongly agree with the belief that children are encouraged to think about mathematics right from the start of a lesson when given 10 quick questions to solve mentally. Further, 47.7% of respondents believe that answers to the 10 quick questions need to be corrected quickly so that the mathematics lesson can begin (Item 17). Somewhat contrary to the latter finding, the responses to Item 20 indicate that the majority of respondents─85.1%─believe that the focus should not be on the correctness of the answer during mental computation sessions, but on the mental strategies used. There is general agreement that teachers need to be aware of the strategies used by those proficient at calculating exact answers mentally (Item 11)─59.7% of 285

Table 4.11 Percentage of Responses Related to Beliefs About Issues Associated with Developing the Ability to Calculate Exact Answers Mentally

Strongly Strongly Item Disagree Disagree Agree Agree Missing

11. Teachers need to be aware of the strategies used by those who are proficient at 4.5 59.7 35.8 calculating exact answers mentally.

15. Children's mental processes are sharpened by starting a mathematics 2.5 22.9 54.7 12.4 7.5 lesson with ten quick questions to be solved mentally.

16. Children are encouraged to think about mathematics right from the start of a 3.5 24.9 57.2 7.5 7.0 lesson when given ten quick questions to solve mentally.

17. Answers obtained for the "ten quick questions" need to be corrected quickly so 6.0 42.8 37.3 10.4 3.5 that the mathematics 6.5a 41.7a 40.7a 9.3a 1.9a lesson can begin.

19. Children should be given opportunities to discuss, compare and refine their 49.8 49.8 0.5 mental strategies for 53.7a 46.3a solving particular mental problems.

20. During mental computation sessions, the focus should be on the correctness of 21.9 63.2 10.0 3.0 2.0 the answer rather than on 14.8a 69.4a 9.3a 3.7a 2.8a the mental strategies used.

Note. N = 201 aPercentage related to middle- and upper-school teachers (n = 108)

286

Table 4.11 cont. Percentage of Responses Related to Beliefs About Issues Associated with Developing the Ability to Calculate Exact Answers Mentally

Strongly Strongly Item Disagree Disagree Agree Agree Missing

21. During mental computation sessions, one approach to each of a number of problems 18.9 52.7 23.4 1.5 3.5 should be the focus, rather than on several approaches to each of a few problems.

22. Children should be encouraged to build on 0.5 60.7 38.3 0.5 the thinking strategies 0.9a 71.3a 27.8a used to develop the basic facts.

Note. N = 201. aPercentage related to middle- and upper-school teachers. (n = 108)

respondents agreed and 35.8% strongly agreed (see Table 4.11). Ninety-nine percent (99%) of respondents believe that children should be encouraged to build on the thinking strategies used to develop the basic facts (Item 22). Complementing the findings for Item 23 (see Table 4.10), 52.7% disagreed and 18.9% strongly disagreed with the suggestion that it was better to focus on one approach to a number of problems rather than focus on several approaches to each of a few problems (see Table 4.11, Item 21). Figure 4.3 presents a graphical representation of the means for the items listed in Table 4.11, with the right-hand side of the continuum─Strongly Agree─reflecting a nontraditional approach to the issues raised. The means for each of the items were: Item 11─3.31, Item 15─2.16, Item 16─2.26, Item 17─2.45, Item 19─3.50, Item 20─3.06, Item 21─2.92, and Item 22─3.37 (see Appendix E). Items 15, 16, 17, 20, and 21 were recoded to facilitate their placement on the continuum. The data suggest that, except for issues related to introducing a mathematics lesson by giving 10 quick questions (Items 15 to 17), there is some agreement with constructivist approaches to developing skill with mental computation─that is, children should be given opportunities for discussing, comparing, and refining their strategies (Item 19), 287

and that their idiosyncratic strategies should be built upon those used to develop basic fact knowledge (Item 22).

Current Teaching Practices

Of the 201 respondents, 164 were class teachers and 19 were teaching principals. Fifty-seven (57) of these 183 teachers were teaching multi-age classes at the time of the survey. One hundred and eight (108) respondents were teachers of Years 4 to 7─ 47 middle-school and 61 upper-school teachers. The data presented in Table 4.12 indicate that mental computation is focussed upon to a varying degree in the classrooms of the majority of the middle- and upper- school teachers surveyed. Approximately 34% of teachers indicated that they often focus on developing the ability to calculate exact answers beyond the basic facts. A further 54.6% sometimes focus specifically on developing this ability (Item 24). Thirty-eight percent (38%) of teachers often teach particular mental strategies and follow up with practice examples (Item 28).

Traditional Nontraditional

Strongly Strongly Disagree Disagree Agree Agree

1 2 3 4

|───────────────────|─X─X──X───────────X|X───X───X──────────|

Items: 15 17 20 11 19 16 21 22

Recoded items: 15, 16, 17, 20, 21

Figure 4.3. Position of means for items relating to beliefs about specific issues associated with developing mental computation skills on a Traditional- Nontraditional continuum.

288

Table 4.12 Percentage of Responses Related to Current Teaching Practices for Developing the Ability to Compute Mentally for Middle- and Upper School Teachers

Item Never Seldom Sometimes Often Missing

24. Focus specifically on developing the ability to calculate exact answers mentally beyond the basic 6.5 54.6 34.4 4.6 facts.

25. Allow children to decide the method to be used to 2.8 37.0 56.5 3.7 arrive at an exact answer mentally.

26. Allow children to explain and discuss their mental 5.6 37.0 53.7 3.7 strategies for solving a problem.

27. Allow children to work mentally during practice of 0.9 3.7 50.0 39.8 5.6 written computation.

28. Teach particular mental strategies and follow up 11.1 46.3 38.0 4.6 with practice examples.

29. Give several one-step questions and simply mark 4.6 32.4 47.2 12.0 3.7 answers as correct or incorrect.

30. Require answers to problems solved mentally 10.2 56.5 29.6 3.7 to be recorded on paper.

31. Relate methods for calculating beyond the 7.4 50.0 38.0 4.6 basic facts to the thinking strategies used to develop the basic facts.

32. Use mental computation for revising and practising 3.7 48.1 43.5 4.6 arithmetic facts and procedures.

33. Emphasise speed when calculating exact answers 0.9 27.8 50.0 16.7 4.6 mentally.

Note. N = 108. 289

Table 4.12 cont. Percentage of Responses Related to Current Teaching Practices for Developing the Ability to Compute Mentally for Middle- and Upper School Teachers

Item Never Seldom Sometimes Often Missing

34. Insist on children using the procedures for the written algorithms when 38.0 38.9 18.5 0.9 3.7 calculating exact answers mentally.

35. Provide opportunities for children to appreciate how often they and adults use 0.9 14.8 49.1 30.6 4.6 mental computation.

36. Teach rules for calculating exact answers mentally 1.9 5.6 40.7 47.2 4.6 (e.g. divide by ten by removing a zero).

37. Have children commit to memory number facts 17.6 26.9 35.2 15.7 4.6 beyond the basic facts.

38. Use examples involving measures or spatial 12.0 47.2 34.3 6.5 concepts in problems to be calculated mentally.

Note. N = 108.

A further 46.3% sometimes employ this approach to develop the ability to calculate exact answers mentally. Approximately equal percentages of teachers allow children to decide the method of calculation (Item 25) and to explain and discuss their mental strategies (Item 26). Thirty-seven percent (37%) of teachers indicated that they sometimes use both of these teaching approaches, whereas 56.5% and 53.7%, respectively, often use these two approaches. The data for Item 25 is complemented by that for Item 34 (see Table 12). These reveal that, whereas 18.5% of teachers of middle- and upper year levels sometimes insist on children using the written algorithm for calculating exact answers mentally, the majority rarely place this requirement on their pupils─38.9% seldom have children use the written algorithm and 38% never insist on the use of this strategy. Nonetheless, the data for Item 36 (see Table 12) indicate that many teachers still teach rules as a means for developing the ability to calculate exact answers mentally. Only 1.9% of teachers indicated that this 290

approach was never used, with an additional 5.6% indicating that teaching rules for mental computation is undertaken infrequently. Approximately 90% of teachers allow children to work mentally during practice of written computation (see Table 12, Item 27). This unexpectedly high percentage suggests that many respondents may not have interpreted this statement as meaning that written answers may not always be required for children who are able to mentally calculate the operations planned as practice for the paper-and-pencil algorithms. The prime focus of the number strand of the Years 1 to 10 Mathematics Syllabus (Department of Education, 1987a) is on developing the ability of pupils to calculate using the standard written algorithm for each operation. Although 32.4% of teachers seldom give several one-step questions and simply mark the answers as correct or incorrect (Item 29), 47.2% sometimes, and 12% often, use this approach. Analogous to these findings, 27.8% of teachers of middle- and upper-school classes seldom emphasise speed during mental computation, whereas 16.7% often do so and 50% sometimes place this emphasis during mental computation sessions (Item 33). Most teachers require answers to problems solved mentally to be recorded on paper (Item 30)─56.5% sometimes require written answers, and 29.6% often emphasise this requirement. The responses to Item 31 reveal that an approach regularly used to develop mental proficiency is to ensure that the thinking strategies used to support basic fact development are used as a basis for calculating beyond the basic facts (Item 31). Fifty percent (50%) sometimes use this approach and 38% often relate strategies for calculating beyond the basic facts to the thinking strategies used to support their development. Many teachers─approximately 16%─ often encourage children to commit to memory number facts beyond the basic facts (Item 37). This occurs sometimes in the classrooms of 35.2% of middle- and upper-school teachers. However, 17.6% of teachers never require children to commit such facts to memory. When encouraging the development of the ability to calculate mentally, teachers often integrate the development of strategies for calculating exact answers mentally with other areas of mathematics (Items 32 & 38). Approximately 48% of teachers sometimes use mental computation as a means for revising and practising arithmetic facts and procedures (Item 32). No teacher indicated that this was a procedure that they never used. Further, the data for Item 38 reveal that 47.2% of teachers sometimes incorporate spatial and measurement concepts in problems to be calculated mentally. Opportunities are also provided for children to appreciate how often they and adults use mental computation (Item 35), with approximately 291

31% of teachers often endeavouring to lead children to an appreciation of the usefulness of mental computation. Of the items listed in Table 4.12, Items 25, 26, 29, 30, 31, 33, 34, and 36 relate to specific teaching practices which reflect either a traditional or nontraditional approach. The means for middle- and upper-school teachers for these items, after recoding Items 29, 30, 33, 34, and 36 to provide a nontraditional orientation, were: Item 25─3.55, Item 26─3.50, Item 29─2.30, Item 30─1.79, Item 31─3.32, Item 33─2.13, Item 34─3.18, and Item 36─1.60 (see Appendix E). Figure 4.4 presents these means on a traditional-nontraditional continuum. This graphical representation suggests that both traditional and nontraditional teaching strategies for developing the ability to calculate exact answers mentally are currently being used by middle- and upper-school teachers.

Traditional Nontraditional

Never Seldom Sometimes Often

1 2 3 4

|──────────X──X─────|X───X──────────────|─X──X───XX─────────| Items: 36 30 33 34 25 29 31 26

Recoded Items: 29, 30, 33, 34, 36

Figure 4.4. Position of means for selected current teaching practices related to developing mental computation skills on a traditional-nontraditional continuum.

Past Teaching Practices

The application of the selection criteria outlined in the Methods of Analysis section resulted in data being obtained for Section 3 of the survey instrument (see Appendix C) from 32 teachers who had experience teaching the 1964 Syllabus. Fifty-three (53) taught during the period 1969-1975 and 161 teachers taught using the 1975 edition of the Program in Mathematics (Department of Education, 1975). Table 4.13 presents the percentage of responses with respect to the importance placed on mental computation─mental arithmetic─by each syllabus and by teachers who taught during each of the syllabus periods. Section 3 of the survey 292

instrument required that respondents be able to respond to particular items based on their recollections. Hence, for this analysis, responses classified as unsure have been recoded as missing. The data for Item 39 (see Table 4.13) reveal that teachers perceive that the 1964 syllabus placed a greater emphasis on mental arithmetic that did either edition of the Program in Mathematics (Department of Education, 1966-1968, 1975). The majority, 88.5% of valid responses, view the 1964 Syllabus as having placed great importance on the ability to calculate exact answers mentally. This compares with 72.2% and 70.8% of valid responses believing that the first and second editions of the Program in Mathematics (Department of Education, 1966-1968, 1975), respectively, placed some importance on mental arithmetic. In contrast to Item 39a, the responses to Items 39b and 39c reveal that some teachers considered that both editions of the Program in Mathematics (Department of Education, 1966-1968, 1975) placed little importance on mental arithmetic─13.9% and 13.5% of valid responses, respectively. The pattern of results for Item 40a (see Table 4.13) is similar to that for Item 39a. The majority of teachers─81.5% of valid cases─indicated that they placed great importance on developing the ability to calculate exact answers mentally during the period 1964-1968. In contrast to the data for Items 39b and 39c, many teachers continued to place great importance on mental arithmetic during each of the periods 1969-1974 and 1975-1987. Sixty-one percent (61%) of valid responses indicated great importance during 1969-1974 and 47.7% during 1975-1987. However, 45.8% placed some importance during 1975-1987 compared to 293

Table 4.13 Percentage of Responses Related to Past Beliefs About Mental Computation

Degree of Importance

Item None Little Some Great Unsure Missing

39. For each period, how important did the syllabus consider the ability to calculate exact answers mentally?

(a) 1964a - 1968 (1964 Syllabus) 9.4d 71.9 6.3 18.8 11.5e 88.5

(b) 1969b - 1974 (PIM: 1st ed) 9.4 49.1 9.4 13.2 18.9 13.9 72.2 13.9

40. For each period, how important did you consider the ability to calculate exact answers mentally?

(a) 1964a - 1968 (1964 Syllabus) 15.6 68.8 3.1 12.5 18.5 81.5

(b) 1969b - 1974 (PIM: 1st ed) 1.9 28.3 47.2 1.9 20.8 2.4 36.6 61.0

Note. a1964 Syllabus: N = 32. b1968 Syllabus: N = 53. c1975 Syllabus: N = 161. dTop line in each row: Percentage of number of cases. eBottom line in each row: Percentage of valid cases, excluding unsure and missing categories.

36.6% during 1969-1974. Taking the percentage of teachers who placed some or great importance on mental arithmetic, the data suggest that teachers may have gradually placed less importance on this topic during these periods. The data for Item 41 (see Table 4.14) reveal greater percentages of respondents allowing children to decide the method for calculating exact answers mentally during the periods 1969-1974 and 1975-1987 than during 1964-1968, when the percentages for sometimes and often are considered together. This teaching practice was never used under the 1964 syllabus by 16% of valid cases compared to 5.3% and 1.9% under the first and second editions, respectively, of the Program in Mathematics (Department of Education, 1966-1968, 1975). 294

Table 4.14 Percentage of Responses Concerning Past Teaching Practices Related to Mental Computation

Item Never Seldom Sometimes Often Unsure Missing

41. Allowed children to decide the method to be used to arrive at an exact answer.

(a) 1964a - 1968 (1964 Syllabus) 12.5d 15.6 18.8 31.3 3.1 18.8 16.0e 20.0 24.0 40.0

(b) 1969b - 1974 (PIM: 1st ed) 3.8 3.8 39.6 24.5 7.5 20.8 5.3 5.3 55.3 34.2

42. Allowed children to explain and discuss their mental strategies for solving a problem.

(a) 1964a - 1968 (1964 Syllabus) 9.4 18.8 18.8 31.3 0 21.9 12.0 24.0 24.0 40.0

(b) 1969b - 1974 (PIM: 1st ed) 3.8 7.5 35.8 28.3 3.8 20.8 5.0 10.0 47.5 37.5

43. Gave several one-step questions and simply marked the answers as correct or incorrect.

(a) 1964a - 1968 (1964 Syllabus) 3.1 15.6 15.6 43.8 0 21.9 4.0 20.0 20.0 56.0

(b) 1969b - 1974 (PIM: 1st ed) 17.0 35.8 22.6 3.7 20.8 22.5 47.5 30.0

44. Emphasised speed when calculating exact answers mentally.

a 3.1 6.3 18.8 46.9 0 25.0 (a) 1964 - 1968 (1964 Syllabus) 4.2 8.3 25.0 62.5

b 7.5 35.8 30.2 3.8 22.6 (b) 1969 - 1974 (PIM: 1st ed) 10.3 48.7 41.0

Table 4.14 cont. Percentage of Responses Concerning Past Teaching Practices Related to Mental Computation

Item Never Seldom Sometimes Often Unsure Missing

45. Insisted that children use the procedures for the written algorithms when calculating exact answers mentally.

d (a) 1964a - 1968 (1964 Syllabus) 18.8 21.9 18.8 15.6 3.1 18.8 25.0e 29.2 25.0 20.8

b 15.1 30.2 22.6 5.7 5.7 20.8 (b) 1969 - 1974 (PIM: 1st ed) 20.5 41.0 30.8 7.7

46. Placed an emphasis on teaching rules for calculating exact answers mentally (e.g. divide by ten by removing a zero).

(a) 1964a- 1968 (1964 Syllabus) 3.1 28.1 43.8 3.1 21.9 4.2 37.5 58.3

(b) 1969b - 1974 (PIM: 1st ed) 1.9 37.7 35.8 3.8 20.8 2.5 50.0 47.5

However, the difference between the percentage of respondents who often used this approach during 1964-1968 and 1975-1987 is not marked, being 40% and 44.8% of valid responses, respectively. The data for Item 42 follows a similar pattern to those for Item 41. Most respondents, for each period, indicated that they sometimes or often allowed children to explain and discuss their mental strategies for solving a problem (see Table 4.14). However, the percentage of respondents indicating that they never or seldom used this approach decreased under both editions of the Program in Mathematics (Department of Education, 1966-1968, 1975). The percentage of teachers who never used this approach decreased from 12% for the period 1964- 1968 to 2.7% of valid responses under the 1975 edition. Items 43 to 46 (see Table 4.14) reflect teaching practices classified as traditional in this study. Although 56% of valid responses indicate that teachers often gave several one-step questions and simply marked the answers as correct or 296

incorrect (Item 43) during the period 1964-1968, this had decreased to 30% and 30.6% for the periods 1968-1974 and 1975-1987, respectively. This decrease was accompanied by a concomitant increase in the number of respondents who sometimes used this approach─from 20% of valid cases under the 1964 syllabus to 49.5% for the 1975 Syllabus. The emphasis on speed when calculating exact answers mentally (Item 44) decreased during the periods under investigation. An analysis of the percentage of valid cases indicates that 62.5% of respondents often emphasised speed under the 1964 syllabus (see Table 4.14). This had decreased to 33.6% during the period 1975-1987 and was accompanied by an increase, from 8.3% (Item 44a) to 19.6% (Item 44c) of valid cases, in the number of respondents who indicated that this emphasis was seldom applied. Although the data for Item 45 (see Table 4.14) reveal that many teachers insisted on the use of the procedures for written algorithms, the majority either never or seldom placed such an expectation on their pupils in each of the three periods of interest. Whereas the percentage of valid cases indicating that they sometimes insisted on the use of written procedures remained relatively constant─25%, 30.8%, and 27.6%, respectively─the percentage who often placed this expectation on children decreased from 20.8% under the 1964 syllabus to 12.4% during the period of the second edition of the Program in Mathematics (Department of Education, 1975). The emphasis placed on teaching rules for calculating exact answers mentally decreased slightly during the three periods under investigation. However, it remained an approach used by the majority of teachers. An analysis of data for Item 46 (see Table 4.14) indicates that whereas the percentage of teachers who often used this approach decreased from 58.3% to 36.8%, the percentage who sometimes taught rules to calculate mentally increased from 37.5% to 54.7% for the periods 1964-1968 and 1975-1987 respectively. Figure 4.5 presents a graphical representation, on a traditional-nontraditional continuum, of the means for the teaching practices focussed upon in Section 3.2 of 297

Traditional Nontraditional

Never Seldom Sometimes Often

1964 - 1968 1 2 3 4 |───────X─XX────────|──────────X──X─────|───────────────────|

Items: 46a 45a 41a 43a 42a 44a

1969 - 1974 1 2 3 4 |────────X─X───X────|───────────X───────|XX─────────────────|

Items 46b 43b 45b 41b 44b 42b

1975 - 1987

1 2 3 4 |───────────X───XX──|───────────X───────|────XX─────────────|

Items: 46c 43c 45c 41c 44c 42c

Recoded items: 43a-46a, 43b-46b, 43c-46c

Figure 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.

the survey instrument (see Appendix C). Following the recoding of Items 43 to 46 to reflect a nontraditional orientation, the means for the period 1964-1968 were: Item 41a─2.83, Item 42a─2.87, Item 43a─1.72, Item 44a─1.56, Item 45a─2.58, and Item 46a─1.46. Those for 1969-1974 were: Item 41b─3.18, Item 42b─3.17, Item 43b─1.92, Item 44b─1.69, Item 45b─2.74, and Item 46b─1.55. For the third period investigated, 1975-1987, the means were: Item 41c─3.31, Item 42c─3.27, Item 43c─1.89, Item 44c─1.86, Item 45c─2.69, and Item 46c─1.73 (see Appendix E). The data suggest that during 1969-1974 there was a slight increase in the use of teaching strategies that allowed children to decide the method of calculation (Item 41b), and to discuss and explain the mental strategies used (Item 42b). This trend 298

appears to have continued from 1975 to 1987 (Items 41c & 42c, respectively) (see Figure 4.5). Over the three periods investigated, there is also a suggestion that there was a slight decrease, represented by movement of item means towards the nontraditional end of the continuum, in the use of three of the teaching approaches classified as traditional, namely giving several one-step questions (Item 43), emphasising speed (Item 44), and teaching rules (Item 46). With respect to the remaining traditional approach─using the written algorithm to calculate mentally (Item 45)─little change in its use appears to have occurred across the three time periods.

Inservice on Mental Computation

Items 54 to 56 relate to teacher inservice on mental computation. An analysis of Table 4.15, which presents data with respect to the need for inservice (Item 54) and recent inservice attendance (Item 55), reveals a high percentage (57.7%) of missing data for the Item 54. This occurred due to respondents, who were at schools which were not trialing the Student Performance Standards in mathematics, being incorrectly directed to jump from Item 52 to Item 55 when completing the questionnaire (see Appendix C). However, of the 42.3% of school personnel who responded to Item 54, 88.2% indicated that inservice should be made available to teachers. Only 14.4% of the respondents had attended inservice in which mental computation was a specific topic for discussion during the three years prior to survey (see Table 4.15). The majority of these 29 teachers (65.5%) had received inservice from Departmental mathematics advisory teachers (see Table 4.16), the number of whom has now been greatly reduced in all educational districts. A further 24.1% had attended inservice sessions conducted by tertiary lecturers. These findings complement those for Item 2 (see Table 4.8). The divided opinion of school personnel concerning the degree of importance placed on mental computation by the Years 1 to 10 Mathematics Syllabus (Department of Education, 1987a) may be a consequence of the lack of inservice opportunities and/or participation.

Table 4.15 Percentage of Responses Related to the Importance of and Participation in Inservice Sessions on Mental Computation

299

Item Yes No Missing

54. Do you consider it important that inservice sessions on 37.3a 5.0a 57.7 mental computation be made 88.2b 11.8b available to teachers.

55. Have you attended, during the last three years, inservice 14.4 84.1 1.5 sessions in which Mental Computation was a specific topic for discussion.

Note. N = 201 aPercentage of total number of cases. bPercentage of valid cases.

Table 4.16 Sourcea of Inservice on Mental Computation During Period 1991-1993

Mathematics Tertiary Colleague Administrator Adviser Lecturer Other

3.4% 65.5% 24.1% 6.9%

Note. N = 29 aQuestion 56: “If you have attended inservice on mental computation, who conducted the inservice?”

Textbooks Used to Develop Skill with Mental Computation

Fifty-six (56) of the 108 respondents classified as middle- and upper-school teachers listed resources that they use to support current teaching practices related to mental computation (Section 2.2 of the survey instrument). Some of these were not able to be clearly identified owing to the inadequacy of bibliographic information. For example, five respondents identified the resource simply as "Mental arithmetic,” a title used by a number of different authors. Another respondent indicated that "various small ‘Mental Arithmetic' books" were used. Allowing for this ambiguity, 71 different resources were listed. An analysis of Table 4.17 reveals that 18.3% of these resources related specifically to mental computation (see Table 4.18 for a list of these resources). Resources to support basic fact development accounted for 11.3% of those listed. These included Basic Maths Facts kits (Department of Education, 1984), Nothing but the Facts (Baturo, 300

1988), and Maths in the Mind (Baker & Baker, 1991). The last text, however, is one which allows for the extension of its activities to mental calculation beyond the basic facts. Of the remaining resources, 25.3% were categorised as supporting the development of Problem Solving strategies and 24.0% were classified General Mathematics Texts. A range of resources relating to problem solving were listed, with no item being cited by more than one respondent. Those categorised as resources to support problem solving included Lateral Thinking Puzzlers (Sloane, 1991) and Creative Problem Solving in School Mathematics (Lenchner, 1983). Sixteen (16) respondents─28.6%─indicated that they used the Years 1 to 10 Mathematics Sourcebook (Department of Education, 1987-1990) relevant to the year level being taught. Other general mathematics resources commonly used were Rigby Moving Into Maths (Irons & Scales, 1982-1986) by 10.7% of respondents and Sunshine Mathematics (Baturo & English, 1983-1985) by 12.5%. Resources which did not fit into the four main categories presented in Table 4.17 were classified as Other (21.1%). This category included inservice that had been attended, calculator activities, concrete materials─multilink, for example─and more general resources such as Making the Most of 20 Minutes (Cain, 1989). Twenty-nine (29) of the 161 school personnel who responded to Sections 3.1 and/or 3.2 of the survey instrument (see Appendix C), concerning past teaching practices, listed resources that they had used during the period 1964-1987. Twenty- three (23) different resources were identified. Thirteen percent (13%) of these were classified as referring specifically to Basic Fact development (see Table 4.17), and 17.4% were classified as relating to mental computation (see Table 4.19). However, the majority─47.8%─were General Mathematics Texts. The resources categorised as Other─21.7%─were primarily items of mathematics equipment─ a bead frame, for example. The basic facts resources listed included 301

Table 4.17 Categorisation of Resources Listed by Respondents in Sections 2.2 and 3.3 of the Survey Instrument

Resource Categories

Teaching Mental Problem General Practices Computation Basic Facts Solving Texts Other

Currenta 18.3% 11.3% 25.3% 24.0% 21.1%

Pastb 17.4% 13.0% 47.8% 21.7%

Note. aN(Current) = 72. bN(Past) = 23

Table 4.18 Textbooks Specific to Mental Computation Currently Used by Middle- and Upper- School Teachers

Couchman, K. E., Jones, S. B., & Nay, W. (1992). Quick practice maths 2000 (Years 3-6). Melbourne: Longman Cheshire.

Lewis, B. (1991). Mental arithmetic and problem solving (Years 3-7) (2nd ed.). Melbourne: Longman Cheshire.

Nash, B., & Nightingale, P. (1992). Kookaburra mental activities (Books 1-6). Sydney: Nightingale Press.

Parkes, A. A., Couchman, K. E., Jones, S. B., & Green, K. N. (1982). Betty and Jim mental arithmetic (4th ed.). Sydney: Shakespeare Head Press.

Perrett, K., & Donlan, R. (1985). Breakthrough mental arithmetic (Years 3-6). Melbourne: Methuen.

Perrett, K., & Whiting, E. (1972). The "Dux" series mental arithmetic (Books 1-6) (2nd ed.). Melbourne: School Projects.

Petchell, D. L., & McDonald, M. K. (1980). Progress in mental arithmetic (Years 3-6). Sydney: Primary Education Publications.

Basic Maths Facts (Oostenbroek, 1976) and an audio tape simply referred to as a "tables tape.” With respect to general texts, of the 29 respondents, 65.5% indicated that they used texts published by William Brooks (Brisbane), particularly the Mathematics Guide (Willadsen, 1970-1976). As one respondent commented, "Were there any except Brooks [during this era]?" 302

Of the resources for which full bibliographic details were able to be determined from the information provided by respondents, Table 4.18 presents those that are currently being used to teach mental computation by middle- and upper-school teachers. Six (6) texts relating to mental computation, listed by 10 respondents, were not able to be definitively identified. The most commonly used text, by four respondents, is that by Bob Lewis, Mental Arithmetic and Problem Solving (1991). Except for two respondents, none indicated that they made use of more than one of the resources identified. Eight (8) of the 29 respondents (27.6%) who listed resources used under previous syllabuses listed resources specific to mental computation (see Table 4.19). None of these respondents indicated that they used more than one of the texts listed. Two listed Breakthrough Mental Arithmetic (Perrett & Donlan, 1985).

Table 4.19 Textbooks Specific to Mental Computation Used During the Period 1964-1987

Lewis, B. (1986). Mental arithmetic and problem solving (Years 3-7). Melbourne: Longman Cheshire.

Perrett, K. & Donlan, R. (1985). Breakthrough mental arithmetic (Years 3-6). Melbourne: Methuen.

Perrett, K., & Whiting, E. (1972). The "Dux" series mental arithmetic (Books 1-6) (2nd ed.). Melbourne: School Projects.

4.3.3 Discussion

As stated in Section 4.3, the survey of Queensland state primary school personnel was designed to provide insights into:

• Their beliefs about mental computation. • The current status of mental computation within the classroom curriculum. • The pedagogical practices used to develop skill with mental computation, both currently as well as under previous syllabuses. • Their inservice needs with respect to mental computation. • The resources used to support the teaching of mental computation. 303

Such insights enable the data from the analysis of the nature and role of mental computation within Queensland state school mathematics syllabuses, as presented in Chapter 3 and Section 4.2, to be extended to the present. The integration of the survey data with that from Chapters 2 and 3 is specifically undertaken in Chapter 6. The purpose of this section is to draw conclusions with respect to the survey questions, as delineated in Section 4.3.1. These conclusions form the basis for the extension of the historical analysis to the present.

Limitations of Findings

It was concluded previously in this chapter, that although the nonresponse rate was 66.4%, the data collected may be cautiously considered indicative of the beliefs and current practices of Queensland state primary school teachers (see Analysis of Nonresponse). This conclusion was drawn from an analysis of data obtained from the four waves of questionnaire return. It was also demonstrated that the numbers of valid cases obtained were sufficient to allow meaningful conclusions to be drawn, at least with respect to the beliefs and current practices of Queensland state primary school teachers. The data obtained from Section 3 of the survey instrument─Past Teaching Practices─can only be considered suggestive of the beliefs and practices which prevailed during the period 1964-1987, particularly given the relatively few respondents (32) for the 1964-1968 era. Although it was intended that the data be representative of the views of teachers who taught at any time during 1964-1987, it was not possible, prior to the sample being drawn, to identify those teachers who had taught under the three syllabuses of interest. It is recognised that self-selection introduces bias into the resultant sample of school personnel. Although this source of bias may have been unavoidable, it does necessitate cautious generalisation of the conclusions drawn from the survey. However, this caution should be viewed in context with the intent of the study─that is, to describe and to gain insights, rather than to explain.

Conclusions

The contemporary beliefs of, and the teaching practices employed by, Queensland state school personnel reflect aspects of traditional and nontraditional 304

approaches to mental computation. The view that mental computation helps to sharpen the mind appears to be one that may be commonly held. As one upper- school teacher expounded, "It is my experience that by ‘sharpening' the wit by Mental Arithmetic practice the children are more prepared and focussed on the tasks ahead. Often I use this strategy as an awakener during lessons." Nonetheless, recognition is given to the importance of allowing children to develop their own strategies for computing mentally: "To emphasise the process rather than getting the correct answer" (Teaching Principal). Insights into the relative balance between traditional and nontraditional beliefs currently held by school personnel may be determined by considering research Questions 1(a)(i), 1(a)(ii), and 1(a)(iii) (see Section 4.3.1). As Question 2(a) has relevance to Question 1(a)(i), these two questions are analysed together.

Question 1(a)(i): Is skill in mental computation considered an important goal of mathematics education? Question 2(a): What emphasis is currently placed on developing the ability to compute mentally?

The development of the ability to calculate exact answers mentally is seen as a legitimate goal of school mathematics by 93% of school personnel who responded to the survey (see Table 4.8, Item 1). That 89% of middle- and upper-school teachers at least occasionally focus on mental computation skills (Item 24, Table 4.12) provides additional support for the conclusion that school personnel do consider mental computation as a goal of some importance for school mathematics. However, this conclusion needs to be tempered by the recognition that only 34.4% of middle- and upper-school teachers often focus specifically on developing mental strategies for calculating exact answers beyond the basic facts (Item 24, Table 4.12). This finding may be a consequence of the belief, by approximately 50% of school personnel, that the Years 1 to 10 Mathematics Syllabus (Department of Education, 1987a) places little importance on the development of the ability to compute exact answers mentally (Item 2, Table 4.8). Some respondents provided anecdotal evidence of a belief in the need for an increased emphasis on mental computation: "I think this is an area that needs sharpening up on, and teaching in more depth and frequency" (Year 2 teacher). However, some teachers appear to conceptualise mental computation as a separate topic within the curriculum, rather than one which may be used as a means to 305

enhance and capitalise upon the links between various number, space and measurement concepts. Representative of this view is the following comment by a Year 6 teacher: "Although mental computation is important, it is only one of many mathematical strategies [sic] within the Queensland syllabus. Too many topics mean essential topics are not emphasised as much as they probably should."

Question 1(a)(ii): Do the beliefs about the nature of mental computation reflect a nontraditional view?

Figure 4.1 suggests that Queensland state primary school personnel tend towards holding a nontraditional view of mental computation, albeit one that does not appear to be strongly held. Traditional elements remain in their beliefs, particularly with respect to the use of rules for calculating mentally (Item 5). Although 91.6% of school personnel support the view that mental computation encourages children to devise ingenious computational short cuts (Item 4, Table 4.9), 37.5% believe that calculating exact answers mentally involves applying rules by rote (Item 5, Table 4.9). Approximately 92% of respondents supported the belief that proficient mental calculators use personal adaptations of written algorithms and idiosyncratic strategies (Item 9, Table 4.9). Nevertheless, approximately 23% of teachers agreed with the proposition that children should use the algorithm for written computation when calculating exact answers mentally (Item 8, Table 4.9).

Question 1(a)(iii): Do the beliefs about how mental computation should be taught reflect a nontraditional view?

Two categories of items are relevant to this question. The first consists of items concerned with the general approach to teaching mental computation (Figure 4.2, Table 4.10), whereas the second involves more specific issues (Figure 4.3, Table 4.11). With respect to the first category, it can be concluded that school personnel tend towards holding nontraditional views (see Figure 4.2). Table 4.10 indicates that most respondents agreed with all items in this category, except for that relating to the need for an emphasis on the written algorithms to be delayed so that mental computation can be given increased attention (Item 10). The focus of the number strand of primary school mathematics under the 1987 Syllabus is the development of the written algorithms. As discussed in Chapter 2, the traditional sequence for introducing computational procedures involves a focus on mental computation 306

following, rather than preceding, the introduction of the paper-and-pencil algorithm for each operation (Figure 2.3). Approximately 93% of school personnel indicated support for the proposition that strategies for calculating exact answers mentally are best developed through discussion and explanation (Item 13, Table 4.10). A similar percentage of support is evident for Item 14. This agreement with the belief that children should be allowed to develop and use their own strategies for calculating exact answers mentally is consistent with that expressed for Item 9 (see Table 4.9) concerning the use of idiosyncratic strategies by proficient mental calculators. The data displayed in Figure 4.3, concerning the second category of items, which relates to more specific teaching practices, provides additional evidence for the coexistence of traditional and nontraditional beliefs about mental computation. The traditional beliefs relate to the practice of introducing a mathematics lesson with 10 quick questions. Approximately 65% of respondents believe that such a practice sharpens children's mental processes and encourages them to think about mathematics right from the start of a lesson (Items 15 & 16, Table 4.11). The opinion of school personnel with respect to the need for the questions to be corrected quickly so that the mathematics lesson can begin was evenly divided (Item 17, Table 4.11). This division of opinion is conceivably related to the finding that 81.5% of respondents disagreed with the traditional view that the focus should be on the correctness of the answer rather than on the mental strategies used (Item 20, Table 4.11). One Year 5 teacher commented that, although sets of questions are given to children, they "are encouraged to discuss their methods to analyse either why they ‘messed up' or why they were right when everyone else ‘messed up'.” Item 19, the teaching practice that received greatest support (99.5%), pertained to the importance of providing opportunities for children to discuss, compare and refine their mental strategies for solving particular mental problems (see Table 4.11). However, some respondents expressed the opinion that the advantages of discussion and explanation are limited to the child whose strategy is being considered:

Yes, it's great fun listening to how some students calculate their answers to larger sums beyond the basic facts. The other students enjoy listening to them as well but I'm not sure [that] they then adopt other students' 307

strategies. When the pressure is on, you tend to go with what feels natural. (Year 4 teacher)

A somewhat contrasting view, but consistent with the reservations expressed concerning the use of the ideas of others to modify a strategy, is the comment from another respondent: "I find kids are interested in and willing to tell me how they calculated mentally, but the other 29 kids are really not interested in how that person solved the problem" (Year 7 teacher). Strong support was given to the practice of building on the thinking strategies used to develop the basic facts (Item 22) to generate mental strategies for calculating exact answers beyond the basic facts (see Figure 4.3). The degree of support for this nontraditional approach to mental computation may be confounded by the apparent focus on basic fact development, rather than on mental computation as defined for the survey, by some respondents. This assumption is supported by 11.3% of those who listed resources nominating texts that support basic fact development rather than mental computation per se (see Table 4.17). Further support for this assumption stems from the comments made by some respondents. For example, a Year 5 teacher stated: "I approach the teaching of basic facts [italics added] in as many ways as possible as I see it as an important component of the Year 5 curriculum in preparation for upper-school.” Whereas opinion with respect to the need to focus on several approaches to each of a few problems (Item 21) reflected a nontraditional view (see Figure 4.3), approximately 25% of teachers and administrators indicated a belief in the traditional view that one approach to each of a number of similar problems should be the focus─a focus on the teaching of types of problems. This belief is reminiscent of the advice given by "J.R.D." (1928) who suggested that teachers should plan to cover "one type only in one lesson" (p.7). In summary, from Figures 4.1, 4.2, and 4.3, it can be concluded that Queensland state school personnel tend towards holding nontraditional views about the nature of mental computation and its development. However, it is possible that such beliefs have arisen not so much from a depth of understanding of issues related to mental computation per se, but more from an acceptance of the recommended teaching practices that accompany the Years 1 to 10 Mathematics Syllabus (Department of Education, 1987a), particularly with respect to the development of the basic number facts and problem solving. As revealed in Table 4.15, the majority of school personnel have not recently undertaken inservice in 308

which mental computation was a specific topic for discussion (Item 55). "Current methods emphasise problem-solving strategies, estimation. Mental computation for exact answers is not a high priority" (Year 4 teacher). Many of the issues raised in the survey are not necessarily specific to mental computation, but relevant to mathematics teaching in general. As discussed in Chapter 2, current developments in mathematics education suggest, inter alia, that learning is enhanced where children are placed in problem solving situations, and where active involvement and reflection are encouraged─a focus on the construction of mathematical knowledge. The Years 1 to 10 Mathematics Teaching, Curriculum and Assessment Guidelines (Department of Education, 1987b, p. 4) advocates teaching mathematics through problem solving. This approach uses problem solving as a focal point for planning learning experiences in which children are required to apply their mathematical knowledge to resolve the problems encountered. Associated with this approach is the encouragement for children to discuss and reflect upon solutions and the strategies used. Teachers and children have been encouraged to accept a range of solutions and a range of strategies for arriving at particular solutions. Many respondents listed resources more specifically related to problem solving in general, rather than ones directly related to the development of the ability to mentally calculate exact answers beyond the basic facts (see Table 4.17). In commercial mathematics textbooks published during 1980s an emphasis has been given to the role of thinking strategies for developing number fact knowledge. This emphasis is also evident in the sourcebooks published by the Queensland Department of Education, texts that operationalise the beliefs about how children learn mathematics that are embodied in the Years 1 to 10 Mathematics Syllabus (Department of Education, 1987a). A focus on thinking strategies for obtaining the basic facts, before introducing drill and practice, is believed to make the facts more meaningful as well as easier for children to learn (Booker, n.d., p.11).

Question 2(b): What importance was placed on mental computation in the period 1964-1987, with respect to both syllabus documents and teachers?

From Table 4.13 it is apparent that the perceived emphasis placed on mental computation by the mathematics syllabuses diminished sharply with the introduction of the first edition of the Program in Mathematics for Primary Schools (Department of Education, 1966-1968). Whereas 88.5% of valid cases considered that the 1964 309

Mathematics Syllabus placed great emphasis on mental arithmetic (Item 39a), this had fallen to 13.9% for the first edition of the Program in Mathematics (Department of Education, 1966-1968) (Item 39b). Of the three syllabuses in operation during the period 1964-1987, only the 1964 syllabus made specific reference to mental arithmetic, as discussed in Section 4.2. A similar pattern of results was obtained for Item 40 (see Table 4.13) concerning the importance that teachers placed on mental computation during this period. However, the level of importance that teachers gave to mental computation remained relatively high under both the 1969 and 1975 editions of the Program in Mathematics. Sixty-one percent (61%) and 47.7%, respectively, indicated that they continued to place great importance on mental computation, even though the syllabuses made no specific reference to the need to develop the ability to calculate exact answers beyond the basic facts.

Question 3(a)(i): Do teachers take a nontraditional approach to developing skill with mental computation?

From Figure 4.4 it is evident that middle- and upper-school teachers employ both traditional and nontraditional teaching practices when developing the ability to calculate exact answers mentally. This finding parallels those that relate to the beliefs about mental computation expressed by all respondents as well as by middle- and upper-school teachers (see Tables 4.8, 4.9, & 4.10). Children taught by middle- and upper-school teachers are permitted to decide the method of calculation (Item 25, Table 4.12), and to explain and discuss the strategies used (Item 26), teaching practices classified as nontraditional. Links are made to the thinking strategies used to develop the basic facts. Thirty-eight percent (38%) of middle- and upper-school teachers indicated that they often use this approach (Item 31). Whereas 38% of middle- and upper-school teachers stated that they never insist on children using the procedures for the written algorithms (Item 34), 18.5% indicated that this requirement was sometimes placed on children (see Table 4.12). Teaching practices classified as traditional that remain in the repertoire of middle- and upper-school teachers include the practice of giving several one-step questions and simply marking the answers as correct or incorrect (Item 29). Twelve percent (12%) of respondents indicated that they use this strategy often, and 47.2% revealed that they use it sometimes. In excess of half of the middle- and upper- 310

school teachers continue to emphasise speed when calculating exact answers mentally (Item 33), with 16.7% of teachers often doing so. Most middle- and upper-school teachers require answers to problems solved mentally to be recorded on paper (Item 30). This practice and the teaching of rules for calculating exact answers mentally (Item 36) are the traditional practices most commonly identified by middle- and upper-school teachers as ones that are sometimes or often employed.

Question 3(a)(ii): Are the teaching approaches employed by middle- and upper- school teachers consistent with their stated beliefs?

Six of the current teaching practices delineated in Section 2 of the survey instrument─Items 25, 26, 29, 31, 34, and 36─may be directly related to particular belief statements in Section 1 of the questionnaire, namely Items 14, 19, 20, 22, 8, and 5, respectively (see Figure 4.6). Following the recoding of items representing beliefs and practices classified as traditional, namely Items 29, 34, and 36 (current teaching practices) and Items 5, 8, and 20 (beliefs), to reflect a nontraditional orientation, the means for each of the matched items, for middle- and upper-school teachers, were: Item 25─3.55 and Item 14─3.27, Item 26─3.50 and Item 19─3.46, Item 29─2.30 and Item 20─2.98, Item 31─3.32 and Item 22─3.25, Item 34─3.18 and Item 8─2.86, and Item 36─1.60 and Item 5─2.63 (see Appendix E for the standard deviations of these means). From Figure 4.6, it is apparent that, except for Items 29 and 36, there would appear to be some consistency between particular teaching practices and the beliefs that underpin them. Approximately 84% of middle- and upper-school teachers disagreed with the belief that the focus should be on the correctness of the answer rather than on the mental strategies used (Table 4.11, Item 20). However, most─60% approximately─continue to place an emphasis on giving several one-step questions and simply marking the answers as correct or incorrect (Table 4.12, Item 29). The continued use of this traditional teaching practice may be influenced by the ambivalence of middle- and upper-school teachers towards the belief that answers obtained for the 10 quick questions need to be corrected quickly so that the mathematics lesson can begin (Table 4.11, Item 17). A similar disparity between belief and practice is evident for Item 36 (see Table 4.12) and Item 5 (see Table 4.9). Although 60.2% of middle- and upper-school teachers do not support the 311

belief that calculating exact answers mentally involves applying rules by rote (Item 5), 87.9% continue to teach rules for calculating exact answers mentally (Item 36). The dissimilarity between some teaching practices and the beliefs that underpin them may stem, at least in part, from the nature of teacher inservice which accompanied the introduction of the Years 1 to 10 Mathematics Syllabus (Department of Education, 1987a). The inservice model was founded on Hall,

Teaching Practices

Traditional Nontraditional

Never Seldom Sometimes Often

1 2 3 4 |──────────X────────|────X──────────────|─X──X───XX─────────|

Items: 36 29 34 25 31 26

Beliefs

Traditional Nontraditional

Strongly Strongly Disagree Disagree Agree Agree

1 2 3 4 |───────────────────|──────────X────X──X|──XX───X───────────|

Items: 5 20 14 19 8 22

Recoded Items: 5, 8, 20, 29, 34, 36 Matched Items: 25-14, 26-19, 29-20, 31-22, 34-8, 36-5

Figure 4.6. Means for selected teaching practices and the beliefs that underpin them for middle- and upper-school teachers on a traditional- nontraditional continuum.

Wallace, and Dossett's (1973) Concerns-Based Adoption Model which posits that teachers implementing an innovation move through a series of levels before fully incorporating a particular teaching strategy, for example, into their repertoire of pedagogical techniques (Dunlop, 1990a, p. 4). For this to occur, teachers not only require knowledge about particular topics and associated teaching approaches, but collegiate support during its implementation. 312

Mental computation was not a specific focus during the inservice which occurred, nor was adequate in-classroom support available for teachers. Hence, it may be that, whereas teachers became convinced of the need for certain changes to occur─for example, that children should understand the processes involved when calculating, rather than simply applying rules by rote─the implementation of these beliefs has not occurred to any significant level, given the lack of knowledge and support during the change process. Traditional approaches to mental computation therefore remain in classroom practice. As one Year 4 teacher commented: "I enjoy mathematics, know a lot of short cuts and teach these to my class."

Question 3(b): What were the characteristics of the teaching approaches used to develop the ability to calculate exact answers mentally during the period 1964-1987?

As discussed previously, the findings that relate to this question should be interpreted with caution, given the nonrepresentativeness of the sample for each syllabus period between 1964 and 1987. Additionally, as one respondent pointed out, “It is difficult to recall exactly what strategies were used when; changes evolve gradually so cannot classify easily by year.” The data (see Table 4.14) that were obtained suggest that some teaching practices classified in this study as nontraditional may have been used during the period 1964-1987. Many teachers indicated that they allowed children to decide the method to be used to arrive at an exact answer (Item 41) and allowed children to explain and discuss their mental strategies (Item 42). The use of these strategies became more frequent during this period, particularly between 1975 and 1987 (Figure 4.5). This period was marked by a gradual disillusionment with the approaches recommended in the 1975 edition of the Program in Mathematics (Department of Education, 1975), particularly with those that related to the development of the written algorithms for the four operations. In Boxall's (1981, p. 2) view, there were errors in application of the intent of the syllabus that were partially exemplified by an over-emphasis on step-by-step proof in the development of the algorithm for each operation. The disillusionment that was being experienced was counterbalanced during the early- to mid-1980s by recommendations to place an emphasis on problem solving (Hickling, 1983; Salmon & Grace, 1984), and to reconsider how the four operations 313

might be taught meaningfully (Irons, Jones, Dunphy, & Booker, 1980). The recommended approaches placed an emphasis on understanding, discussion, and the need to explore alternative ways for deriving solutions. It is possible that the recent emphasis given to these approaches may have influenced respondents' recollections of the methods used to teach mental computation under earlier syllabuses. Although respondents indicated that nontraditional approaches were employed, strategies classified as traditional in this study were more commonly used. There was an emphasis on speed (Item 44), teaching rules (Item 46), and giving several one-step questions and simply marking the answers as correct or incorrect (Item 43) (see Table 4.14 & Figure 4.5). However, the use of these approaches decreased slightly under each edition of the Program in Mathematics (Department of Education, 1966-1968, 1975) (Figure 4.5), which paralleled the increased use of methods classified as nontraditional discussed previously.

Question 4(a): What need for inservice on mental computation is expressed by school personnel?

Queensland state primary school teachers─88.2% of valid cases─consider that it is important for inservice on mental computation to be made available to teachers (Item 54, Table 4.15). As one Year 5 teacher commented: "I feel mental computation is very important and I probably don't put enough emphasis on it in my teaching. I feel I need inservice on how to teach it. Different strategies etc."

Question 5(a): What is the nature of the resources currently used to support the teaching of mental computation?

The characteristics of the resources used to develop mental computation skills (see Table 4.18) are consistent with the finding that traditional teaching practices continue to be employed by middle- and upper-school teachers (Figure 4.4). Each of these texts is characterised by sets of predominantly one-step examples, many of which simply require the recall of facts─for example, "How many days in April?". Where teachers' notes are provided, no references are made to strategies for calculating mentally. Typical of the teachers' notes is that in the most commonly used resource, Mental Arithmetic and Problem Solving (Lewis, 1991). The "Note to the teacher" indicates, inter alia, that: 314

This series is designed to be of practical use to the teacher who wishes to reintroduce [italics added] "Mental" to the Maths programme, with exercises related to the main course. Each book [in the series] is set out in forty weekly units, a unit consisting of five sets of ten exercises. (p. i)

Each of the resources contains exercises from across the range of number, space and measurement topics within the syllabus, although none is specifically written to match the Years 1 to 10 Mathematics Syllabus (Department of Education, 1987a). Various structures for the sets of questions are evident in the texts. For example, Mental Arithmetic and Problem Solving: Year 6 (Lewis, 1991) employs a range of topics which includes number facts, extended facts, money, factors, measurement, and problems. Prominence is also given to the use of short methods─for example, to subtract by nine, subtract 10 and add one. In contrast, Progress in Mental Arithmetic: Year 6 (Petchell & McDonald, 1980) presents sets of examples based on a range of concepts grouped into four levels of difficulty within each unit. For example, Levels 3 and 4, for "more competent children,” include items such as: "48c x 4" and "How many ½m² tiles are required to tile an area 17m by 8m?" (p. 21). Indicative of the dominance of traditional texts to support mental computation are the comments of two teachers. A Year 7 teacher commented: "It is almost impossible to purchase books for this purpose. I rely on my own resources built up for many years.” Such a view gained support from a Year 6 teacher who noted that "the older the book the better" as a resource to assist in teaching mental computation. In contrast, another Year 6 teacher suggested that "any maths resource can be used. [The] teacher modifies the resource to meet current needs. Teachers should use common sense and make maths as ‘everyday' as possible.” Support for nontraditional teaching approaches to mental computation are to be found in some of the resources categorised as General Texts and in some supporting the development of the basic facts. Nonetheless, the majority of the resources classified as Other Than Specifically Relating to Mental Computation bear no direct relevance to developing the ability to calculate exact answers mentally. Sixteen (16) middle- and upper-school teachers indicated that they made use of the Years 1 to 10 Mathematics Sourcebook (Department of Education, 1987-1991) relevant to the year level taught. The sourcebook for Year 5 provides comprehensive support for a focus on developing thinking strategies for calculating mentally beyond the basic facts (Department of Education, 1988, pp. 51-57). Ideas 315

for assisting children to extend basic fact strategies to mental operations with larger numbers are included. However, the degree to which teachers actually apply these ideas is not able to be determined from this study. As discussed previously, opinion is divided (Item 2, Table 4.8) as to the importance placed on mental computation by the Years 1 to 10 Mathematics Syllabus (Department of Education, 1987a), and, by implication, by the Years 1 to 10 Mathematics Sourcebooks (Department of Education, 1987-1991). Maths in the Mind: A Process Approach to Mental Strategies (Baker & Baker, 1991), although primarily concerned with basic fact development, provides ideas for developing mental strategies for calculating beyond the basic facts. In approach, it is similar to the ideas presented by McIntosh (1988) in Volume 1 of the Mathematics Curriculum and Teaching Program (Lovitt & Clarke, 1988), nominated by one Year 7 teacher as a resource used to support mental computation. The nontraditional approach advocated by Baker and Baker (1991) emphasises that mental strategies need not be specifically taught. Rather teachers should:

Begin to understand the strategies that children invent themselves and encourage [their] use...in the classroom by demonstrating that it is not "wrong" to find a quick way of working out a number fact [mentally]; rather that it is valid and desirable. (p. 8)

Of relevance to developing the ability to calculate beyond the basic facts, A. Baker and J. Baker (1991) present games such as "Target numbers" in which children are required to find, for example, "How many times does five go into 1990, the target number?" (p. 95). Children are encouraged to discuss and compare the strategies, and to investigate ones that they think may be helpful in arriving at a solution.

Question 5(b): What was the nature of the resources used to support the teaching of mental computation during the period 1964-1987?

The degree to which conclusions can be drawn with respect to the resources teachers used between 1964 and 1987 is limited by information having been provided by only 29 of the 161 school personnel who responded to Section 3 of the survey instrument. As revealed in Table 4.17, 17.4% of the 23 resources listed for the period 1964-1987 related specifically to mental computation. Each of the texts 316

that was able to be identified from the bibliographic information provided (see Table 4.19) support traditional teaching approaches. These texts are ones that are currently being used by middle- and upper-school teachers (see Table 4.18). The text most commonly cited by respondents─55.2%─was a general mathematics text, the Mathematics Guide (Willadsen, 1970-1976), initially published to correspond to each stage of the Program in Mathematics for Primary Schools (Department of Education, 1966-1968). This finding is consistent with that by Warner (1981, pp. 76-77) who found that the Mathematics Guide (Willadsen, n.d.) was the text most widely used by teachers of Years 5-7. Although it made no specific reference to mental computation, its structure facilitated giving 10 quick questions in such areas as the four operations to 100 and numeration.

Concluding Points

In conclusion, although the findings from this survey may not authoritatively extend our understanding of mental computation in Queensland classrooms between 1964 and 1987, it is reasonable to conclude that those relating to the present, albeit late-1993, are representative of contemporary beliefs and practices. Given that many teachers are yet to fully embrace mental computation as currently envisaged, both in belief and practice, there is an urgent need for (a) significant documentary guidance for teaching mental computation, and (b) teacher inservice and classroom support, as recognised by the majority of respondents to the questionnaire. Of relevance to meeting these needs was the abortive introduction during the mid-1990s of the Student Performance Standards in Mathematics for Queensland Schools (1994), one of the initiatives discussed in the next section.

4.4 Mental Computation in Queensland: Recent Initiatives

Although the Years 1 to 10 Mathematics Syllabus (Department of Education, 1987a), supported by the Years 1 to 10 Mathematics Teaching, Curriculum and Assessment Guidelines (Department of Education, 1987b), continues as the syllabus for mathematics teaching in Queensland primary schools, two innovations, which were introduced in 1995, have impacted upon its continued implementation, innovations with some relevance for mental computation. These are: (a) Student Performance Standards in Mathematics for Queensland Schools (1994), which 317

introduced Queensland primary teachers to the concept of outcomes-based assessment, and (b) the Number Developmental Continuum (Queensland School Curriculum Office [QSCO], 1996b). Associated with the latter is the Year 2 diagnostic net, a performance-based mechanism proposed in the Report of the Review of the Queensland School Curriculum (Wiltshire, 1994, R7.1, p. xiv), for identifying children who have inadequate numeracy levels and for whom intensive intervention should be implemented. These innovations occurred in context with “national trends which [saw] students, parents and educators across Australia claiming for more uniformity, portability, accessibility, universality, and even higher standards to ensure Australia [remained] internationally competitive” (Wiltshire, 1994, p. 4). Entwined with the latter, was the belief, expressed particularly by employers, that many students were leaving school without reaching appropriate numeracy standards. Concern was expressed over the perceived inability of students “to add, subtract, multiply and divide, or to handle number facts, as well as an over-reliance on calculators to their detriment of mental calculations” (Wiltshire, 1994, p. 153). Within this atmosphere, reflective of that in other western democracies, and in context with a paucity of data to inform public opinion, has been a call for greater accountability on the part of schools, an exhortation which tends to gain prominence in times of economic crises (Boomer, 1989, cited by Clements, 1996, p. 6). In common with the implementation of the national curriculum in England and Wales in 1989, it was assumed that “education [could] play a major economic role, and, to do so, that the direction of education [could not] be left in the hands of educators” (Hughes, 1993, p. 143). Hence, as the New South Wales Director- General of Education observed in 1988, “the determination of educational policy has slipped largely from the hands of professionals...to reside firmly with governments, political parties with their educational policy committees, economists, management experts and their major advisers from business and large employee organizations” (Sharpe, 1988, p. 16, cited by Barcan, 1996, p. 20). The move towards a collaborative approach to curriculum development between the Commonwealth, State and Territory governments, with its attendant overt politicisation of curriculum issues (Ellerton & Clements, 1994, p. 49), was first given formal expression at the June 1986 meeting of the Australian Education Council. This collaboration, made difficult by political mistrust, by the “intransigence of states interests” (Baker, 1993, p. 1, cited in Wiltshire, 1994, p. 94), and by a perceived lack of consultation with tertiary mathematics educators, resulted in the publication, in 318

1990, of A National Statement on Mathematics for Australian Schools (AEC, 1991) and, in 1994, Mathematics: A Curriculum Profile for Australian Schools (AEC, 1994). Whereas the former is an input document designed as a framework for curriculum development, the latter focuses on outputs describing what it is expected of students at each competency level (Willis, n.d., p. 8). The decision by the Australian Education Council in July 1990, and supported by state Directors-General of Education, to endorse the concept of national profiles constituted, in Speedy’s (1992, p. 6) view, a significant move towards encouraging curriculum reform through a focus on outcome statements rather than on syllabus content. The Mathematics Profile was designed to provide the proposed national curriculum package with the leverage thought necessary to encourage curriculum reform (Clements, 1996, p. 9). Nonetheless, for the reasons outlined above, the Australian Education Council at its meetings in July and December 1993 chose not to adopt a national curriculum. Instead it was expected that the States and Territories would develop their own mathematics curricula based on the National Statement and Profile, a decision in accordance with preferences expressed to the Report of the Review of the Queensland School Curriculum (Wiltshire, 1994, p. 96). In Queensland, this decision confirmed the move towards a focus on outcomes- based education through the development of the Student Performance Standards in Mathematics for Queensland Schools (1994), a document that was based upon Mathematics: A Curriculum Profile for Australian Schools (AEC, 1994). The Student Performance Standards provided a means for responding to the increasing clamour from parents and the community for greater accountability for what is taught, and for how it is taught and reported (Department of Education, 1995, p. 1). In parallel with this development was the implementation of recommendations contained in the report on the school curriculum in Queensland (Wiltshire, 1994) particularly with respect to “the introduction of a Year 2 early-age ‘net’ whereby diagnosis (suggested by running records) of the...numeracy levels of all students after 18 months in the compulsory school system” and the introduction of Year 6 test in numeracy (Wiltshire, 1994, R7.1, p. xiv). Although the Year 6 test has had little direct positive relevance to mental computation, being a traditional paper-and-pencil test, it does represent an embracing of a statewide testing approach for gathering accountability data, as well as for providing information to teachers to support their judgements about student learning outcomes. Included among its aims is the provision of data to inform the Minister of Education on numeracy trends, and to “provide school and system level 319

information to guide decisions about learning and teaching and resource management” (Queensland School Curriculum Council, 1997, p. 1). The procedures used to develop the Year 6 test provides support for Clements’ (1996, p. 12) suggestion that the nationally-developed profiles provide an obvious basis for constructing statewide and national tests, with their inherent tendency to overvalue those components of the curriculum which can be easily tested (AAMT, 1997, p. 2). The Student Performance Standards continued to be used to inform the development of Year 6 test items during 1996 and 1997. This, despite their use being effectively discontinued in Queensland state schools from the end of the 1995 school year. Although earlier attempts at national collaboration to develop a national curriculum failed, a renewed effort was agreed to at the July 1996 and March 1997 meetings of the Ministerial Council of Education Employment Training and Youth Affairs. At the latter meeting it was agreed that a new national goal of schooling should indicate that the achievement of numeracy for every child leaving primary school is a national priority. In context with this goal, the state Directors-General of Education approved the development of benchmarks in numeracy, initially for Years 3 and 5. These are designed to: (a) “improve student learning in numeracy, and school performance,” and (b) “inform Australian governments and the community about student achievement in...numeracy” (“Draft Numeracy Standards,” 1997, p. 1). For this task, numeracy has been defined as “the effective use of mathematics to meet the general demands of life at school and at home, in paid work, and for participation in community and civic life (“Literacy & Numeracy Benchmarks,” 1997 p. 1). Given the importance of mental computation in these contexts, as discussed in Chapters 1 and 2, it is expected that it will receive some prominence in the benchmarks being developed. However, what impact they, and the associated Year 3 and 5 tests to be introduced in 1998 (Peach, 1997b, p. 2), will have on the development of mental computation in Queensland state primary school classrooms remains to be determined. Nonetheless, as outlined above, the recent innovations in Queensland, arising from the concern for accountability and the desire to provide effective early intervention, that have had some relevance for mental computation, are the Student Performance Standards in Mathematics for Queensland Schools (1994), and, to a lesser extent, the Number Developmental Continuum (QSCO, 1996a).

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4.4.1 Student Performance Standards and Mental Computation

As discussed in Section 4.3.1, teachers in approximately 80 schools in Queensland were involved in trialing the Student Performance Standards during 1993, an occurrence that was taken into consideration when designing the sample of schools for the survey of teachers and administrators. However, as a consequence of a range of concerns expressed by school personnel, and given voice particularly through the Queensland Teachers' Union, the Standards were not formally introduced until 1995 when teachers of Years 3 to 7 were required to report to parents on the Number, Space, and Measurement strands. Performances on the remaining strands, Chance and Data, and Working Mathematically, were to be reported from 1996. However, with the change of government in Queensland in late February 1996, the Director-General of Education, at the Minister of Education's direction, advised schools that the implementation of the Student Performance Standards in state schools would be suspended until advice was received from the Queensland School Curriculum Office with respect to their “future use and development” (Peach, 1996, p. 1), a direct outcome of the concerns which had been expressed by teachers. Consequently, the potential for the Standards, which made specific references to mental computation, to enhance the teaching of mental computation was short-lived. One of the misgivings held by teachers and administrators centred on perceived mismatches between the 1987 Syllabus and the Standards─a concern of relevance to mental computation. Included in the terms of reference for the Queensland School Curriculum Office project team, which was established to "accurately identify and analyse the concerns of teachers and school administrators about the implementation of SPS" (Quinn, 1996, p. 1), was the requirement to provide advice on: (a) "the extent to which the current SPS match the Years 1-10 Mathematics Syllabus,” and (b) "any changes that may need to be made to either SPS or the syllabus to improve the match" (Quinn, 1996, p. 1). However, the interim adjustments to the Years 1 to 10 Mathematics Syllabus (Department of Education, 1987a) recommended by the review did not refer to mental computation. The recommended adjustments were restricted to aspects of probability and statistics, algebra, and two- and three-dimensional shapes (QSCO, Interim Adjustments, 1996b. pp. 1-11). The intention of these recommendations was to 321

strengthen the link between A National Statement on Mathematics for Australian Schools (AEC, 1991) and the 1987 syllabus, and consequently, indirectly, its link with the Student Performance Standards. Significantly for mental computation, but to its detriment, it was considered that there was a reasonable match between key aspects of the Number strands in the national statement and the syllabus (QSCO, Interim Adjustments, 1996b, p. 1). Although there were differences in structure, there was an essential commonality among the documents which set out the broad directions for mathematics education in Queensland (“Student Performance Standards,” 1994, p. iii). These documents were: (a) the Years 1 to 10 Mathematics Syllabus (Department of Education, 1987a), (b) Years 1 to 10 Mathematics Teaching, Curriculum and Assessment Guidelines (Department of Education, 1987b), and (c) Student Performance Standards in Mathematics for Queensland Schools (1994), at least prior to the Minister for Education declaring, in November 1996, that “Student Performance Standards, as we know them are finished” (Office of the Minister for Education, 1996, p. 1), As discussed in Section 4.2, although no specific mention is made of mental computation in the 1987 Syllabus, mental computation was included as one of the seven sub-strands within the Number Strand of the Student Performance Standards. The Standards for Levels 2 to 5 were designed to cover Years 3 to 8, with the outcomes for Levels 3 and 4 of particular relevance for the majority of children in Years 4 to 7. These Standards were expressed as follows:

Level 2 Estimates and calculates mentally, including adding and subtracting numbers to 10 and making extensions based on place value. Level 3 Estimates and calculates mentally, including adding (sum to 100) and subtracting two-digit numbers and multiplying numbers to 10. Level 4 Estimates and calculates mentally, including adding and subtracting most two-digit numbers and multiplying and dividing multiples of 10 by one-digit numbers. Level 5 Estimates and calculates mentally with whole numbers, money and simple fractions, including multiplying and dividing some two-digit numbers by one-digit numbers. (“Student Performance Standards,” 1994, pp. 42, 58, 72, 88)

It is apparent from these outcomes that both computational estimation and mental computation, as defined in this study and the mathematics education 322

literature, were encompassed by the term mental computation in the Standards. This, despite distinctions being made between the calculation of approximate and exact answers in A National Statement on Mathematics for Australian Schools (AEC, 1991, pp. 108-109), the document to which the Standards were linked. Although the Standards primarily constituted a reporting framework, rather than a framework for planning, Willis (n.d.) suggests that one of their purposes was “to support improved teaching and learning through the identification of agreed desirable outcomes of learning” (p. 3)─that is, to supply the leverage for classroom change referred to previously. Therefore, it can be argued that if both mental computation and computational estimation are to receive the emphasis required for effective mathematical functioning, consideration needs to be given to distinguishing more clearly between the two forms of mental calculation. To some extent, this has been achieved in the holistic Strand Level Statements presented in the revised version of Student Performance Standards─Queensland Levels of Student Performance (QSCO, 1996b). Significantly, the descriptive nature of the statements for Number implicitly embeds the various computational procedures for each level with the development of number sense. Nonetheless, the term number sense, the construct on which the development and application of flexible mental strategies rely, is used only in the Level 7 statement which suggests that “students demonstrating Level 7 outcomes show a well developed number sense” (QSCO, Strand Level Statements, 1996b, p. 10). Of significance to mental computation is the elimination of all substrands and that no specific use is made of the term mental computation. However, the level statements for the Number Strand include clear references to both calculating exact and approximate answers. For Levels 2 to 5, which are comparable with the Student Performance Standards’ levels outlined above, both in grade comparability and content, the references to calculating exact answers mentally are:

Level 2: Students make and then test conjectures about operations on numbers, especially where mental calculations support their observations of relationships between sets of numbers. They work out that 48-30 is 18, suggesting that 49-31 would have the same answer (18)....Students calculate solutions using mental procedures. Level 3: [Students] use effective strategies when calculating mentally beyond the basic facts in addition, subtraction and multiplication to deal with: 323

(a) adding and subtracting two-digit numbers, and (b) multiplication of two-digit numbers by single digit numbers. Level 4: [Students] try mental arithmetic initially for most “one-off” calculations....They use a variety of strategies for calculating in their head in order to give exact values or approximate values. Level 5: [Students] use a range of mental and written methods to add, subtract, multiply and divide whole numbers, common fractions and decimal fractions, although for divisors with more than one digit they may use a calculator. (QSCO, Strand Level Statements, 1996b, pp. 1, 5)

Irrespective of the future of the Queensland Levels of Student Performance (QSCO, 1996b), as major revisions to the mathematics syllabus for Queensland schools are not scheduled to occur until after 2000, it is likely that teachers will not be encouraged to place an increased emphasis on mental computation in the short- term. This situation strengthens the conclusion that recent curriculum initiatives will have had little enduring relevance for mental computation (see Question 8, Section 4.1.2). The need for the syllabus to explicitly emphasise mental computation is essential for children to gain mastery of mental strategies for calculating exact answers, particularly given the finding that approximately 53% of middle- and upper- school teachers believe that the Years 1 to 10 Mathematics Syllabus (Department of Education, 1987a) places little importance on the development of the ability to calculate exact answers mentally (see Table 4.8). This situation is one that is unlikely to have altered since the survey was undertaken in light of the paucity of inservice related directly to mental computation, a conclusion based on anecdotal evidence obtained from some Education Advisers (Mathematics) during 1996. Further, opportunities for inservice were significantly reduced by the number of these advisers being cut by approximately two-thirds across the state from the commencement of the 1997 school year.

4.4.2 Number Developmental Continuum and Mental Computation

Although the Student Performance Standards and the Queensland Levels of Student Performance have been embroiled in disputes with both educational and 324

industrial connotations, the implementation of the Number Developmental Continuum (QSCO, 1996a) has continued. However, there are indications that Year 2 teachers, in particular, are finding that the performance-based validation tasks impact significantly on teaching time (Personal Observations, 1996-1997), an observation in accordance with that of Silvernail (1996, pp. 51, 59) concerning the impact of the original format of the Standard Assessment Tasks used in the national assessment program in primary schools in England in the early 1990s. Given that the number continuum applies to children in Years 1 to 3, its relevance to the mental calculation of exact answers beyond the basic facts is limited. Only two of the indicators provide an overlap with the mental computation statements for Levels 2 and 3 of the Student Performance Standards, namely:

Phase E Calculates mentally extended addition facts. Phase F Calculates mentally extended subtraction facts. (QSCO, 1996a, Indicators 8.3 & 12.3)

Nonetheless, it is while moving through the six phases of the continuum─Phases A to F─that children develop mental strategies, critical to basic fact development, strategies from which efficient mental procedures for calculating with larger numbers may be constructed.

4.4.3 Implications for Mental Computation Curricula

In summary, although the beliefs about mathematics education underlying the 1987 Syllabus are supportive of a focus on mental computation, the suspension of the implementation of Student Performance Standards in February 1996 is likely to have negated any momentum that may have been building towards a consideration of mental computation by Queensland primary school teachers. Nevertheless, irrespective of the future of the Standards in their modified form─Queensland Levels of Student Performance (QSCO, 1996b)─a renewed focus on mental computation is essential, not only for its social usefulness but also so that its role as a vehicle for promoting thinking, conjecturing and generalising (Reys & Barger, 1994, p. 31) may be fully realised. The mental strategies which are developed during the exploration of basic facts─a focus in the Number Developmental Continuum (QSCO, 1996a) 325

and in the Sourcebooks (Department of Education, 1987-1991) which accompany the 1987 Syllabus─need to be capitalised upon. However, as concluded with respect to the analysis of the survey data, the degree to which teachers are able to focus on mental computation is dependent upon their (a) gaining access to a syllabus that explicitly places an emphasis on mental computation, and (b) becoming aware of relevant issues, particularly with respect to the range of strategies used to calculate mentally, and the teaching practices considered essential for the development of flexible mental strategies. The former is addressed in the next chapter in which a syllabus component for mental computation is proposed, one that may inform the future revision of the 1987 Syllabus, and provide a basis for teacher inservice. Issues pertaining to the latter are explored in the final chapter. The professional development that needs to occur, should be planned in context with current teacher attitudes and practices. Although the survey data presented in this chapter provide relevant insights, the broader question as to the degree to which the philosophy and teaching practices embodied in the 1987 Syllabus have gained ownership by teachers is unknown. However, anecdotal evidence from Queensland Department of Education mathematics advisers (Personal communications, 1996-1997) suggests that many teachers, particularly those in the middle- and upper-schools, have yet to fully implement the spirit of the Years 1 to 10 Mathematics Syllabus (Department of Education, 1987a). One reason for this is that the 1987 Syllabus and Guidelines provide only very brief statements about content. The more detailed messages are contained in the year-level sourcebooks, resources that were never promoted as being prescriptive. Hence, “the more detailed messages about content coverage and teaching strategies were lost on many teachers, especially in those sectors of schooling where textbooks have been the prime engines of syllabus interpretation” (Grace, 1996, p. 11). 327

CHAPTER 5

MENTAL COMPUTATION: A PROPOSED SYLLABUS COMPONENT

5.1 Introduction

One conclusion that may be drawn from the analyses in preceding chapters is that a focus on mental computation is critical to a revitalisation of school mathematics. For school mathematics to be useful, it needs to reflect the computational techniques used in everyday life (Maier, 1980, p. 21; MSEB & NRC, 1990, p. 19; Willis, 1990, p. 9). Whereas school mathematics continues to be oriented towards paper-and-pencil techniques (McIntosh, 1990a, p. 25; Willis, 1990, p. 12), those used outside the classroom are predominantly mental (Carraher et al., 1987, p. 94; Wandt & Brown, 1957, pp. 152-157). Such methods, those used by folk mathematicians (Maier, 1980, pp. 21-23), differ with the context in which an arithmetical problem is to be solved (Carraher et al., 1987, p. 83; Lave, 1985, p. 172). Bridging the gap between computational techniques used within the classroom and those used beyond is critical to students developing confidence in their mathematical abilities (Case & Sowder, 1990, p. 100). Those who are proficient at mathematics in daily life, including the workplace, seldom make use of the standard computational techniques, particular the written algorithms, taught in schools (Carraher et al., 1987, p. 95; Murray et al., 1991, p. 50). Rather, idiosyncratic methods tend to be used or else unique adaptations of the written algorithms are developed (Cockcroft, 1982, p. 75 para 256; Lave, 1985, p. 172). Hence, it is concluded that the current emphasis on the standard paper-and-pencil algorithms needs to be reduced. Such a reduction is essential to dispelling the erroneous view of arithmetic as essentially involving linear, precise, and complete calculations. However, the impact of an emphasis on mental computation is not limited to its social utility. The research evidence suggests that such an emphasis significantly 328

contributes to the development of number sense through the fostering of ingenious ways in which to manipulate numbers. This depends upon, and contributes to, the development of a deeper understanding of the structure of numbers and their properties. Further, mental computation is now viewed as an essential prerequisite to the successful development of written algorithms (J. Jones, 1988, p. 44; McIntosh, 1990a, p. 37; Rathmell & Trafton, 1990, p. 157; R. E. Reys, 1984, p. 549). It is the concern for these aspects of mental computation that B. J. Reys and Barger (1994, p. 31) believe to be the novel facet of the resurgence of interest in mental computation, one which highlights mental computation as a means for promoting thinking, conjecturing, and generalising based on conceptual understanding. Nevertheless, despite the on-going advocacy for an increased emphasis on teaching rather than testing mental computation, this has yet to significantly translate into classroom practice (McIntosh, 1990a, p. 25; Reys & Barger, 1994, p. 46). In common with England and the United States, in particular, the place of mental computation in Australian mathematics curricula is only beginning to be seriously considered. To overcome the unwillingness on the part of many students to attempt to calculate mentally, and the concomitant low standard of mental computation (Carpenter et al., 1984, p. 487; McIntosh et al., 1995, pp. 36-37; B. J. Reys et al., 1993, p. 314; R. E. Reys et al., 1995, p. 323), requires school mathematics to become more meaningful to students, and more useful in non- classroom settings. Children need to be encouraged to value all methods of computation (Rathmell & Trafton, 1990, p. 156), and particularly to develop personal strategies for calculating mentally (AEC, 1991, p. 109). As emphasised by Barbara Reys et al. (1993, p. 312), teachers need to come to recognise the legitimacy of the development of mental skills as a major goal for school mathematics, and in so doing change the way in which mental computation is viewed─McIntosh’s (1992, pp. 131-134) first revolution. The second is of equal importance. This advocates that the opportunities provided for children to develop the understandings essential for effective mental computation should occur in association with teachers finding numbers a source of enjoyment for themselves (McIntosh, 1992, p. 134). However, teacher and curriculum change occur in association with such factors as recommendations of mathematics educators, historical context, and, significantly, teacher beliefs and classroom practices (Weissglass, 1994, p. 78). A consideration 329

of these factors in this chapter, and in Chapter 6, has been facilitated by the analysis of Queensland syllabuses from 1860 (Chapter 3) and the data from the survey of Queensland State school personnel (Chapter 4). Of relevance to the proposals presented in this chapter, these analyses reveal that Queensland teachers have tended to ignore syllabus recommendations with respect to calculating mentally, partly as a consequence of the impreciseness in the terminology used, the poor quality of teacher training, and the lack of appropriate professional development to accompany the introduction of new syllabuses. Further, arising from the nature of mental arithmetic embodied in the syllabuses prior to 1966, coupled with teacher understanding of the 1987 Syllabus, and a commitment to teaching the standard written algorithms, a tradition for encouraging students to devise their own mental and written strategies remains to be developed. These factors suggest that ongoing professional development and support will be required to empower teachers to emphasise mental computation, despite the inclusion of some nontraditional techniques such as encouraging discussion in their teaching repertoires (see Table 4.12),. This need, and that for significant documentary guidance, was revealed by the analyses of survey data (Section 4.3) and recent curriculum initiatives (Section 4.4) of relevance to mental computation in Queensland. Although aspects of the first of these needs are analysed in this chapter, the primary focus is on the formulation of proposals that may form a basis for incorporating specific references to mental computation into future mathematics syllabuses. The proposals centre upon (a) a sequential framework, embodying a mental-written sequence, for introducing mental, calculator and written procedures, and (b) mental strategies appropriate for focussed teaching at various year-levels. As a consequence of the critical nature of the environment in which curriculum change occurs, these proposals have been developed within a Queensland context. Nonetheless, given the similarities in school mathematics across national boundaries, as reflected in the research literature (see Chapters 1 & 2), it is considered that the proposals, and the contextual features, have relevance for all teachers of mathematics.

5.1.1 Context and Focus for Change

330

From a Queensland perspective, the need for professional development and documentary guidance is embedded in an environment characterised by a stalling of the potential for a renewed focus on mental computation through the discontinuance of the Student Performance Standards in Queensland State schools in 1996 (Office of the Minister for Education, 1996, p. 1). Not since the 1964 Syllabus has mental computation been specifically included. Further, major revisions to the mathematics syllabus are not scheduled to occur until after 2000, leaving the 1987 Syllabus to continue as the basis for mathematics learning in Years 1 to 10, a syllabus which, in the opinion of approximately 65% of the State primary school personnel surveyed, places little importance on the development of the ability to calculate exact answers mentally (Table 4.8, Item 2). Nonetheless, in common with syllabus development elsewhere, given that these revisions could be expected to address issues central to teachers’ beliefs about the nature of school mathematics and mathematics teaching, it is essential that debate, both professional and community, occurs well in advance of the preparation of a new syllabus. Such would not only provide an understanding of, and assist in the development of a commitment to, the recommended changes, it would, most importantly, serve to preserve the sense of efficacy teachers exhibit in their role as teachers of mathematics in the primary school. It is believed that the development of a syllabus component for mental computation─the focus for this chapter─could provide a stimulus for such debate. Key aspects of this debate centre on the sequence in which mental, written, and technological methods of calculation are introduced to students, the emphasis on self-generated mental and written computation strategies, and the legitimacy of a continued role for standard written algorithms. Of relevance to these issues are findings from the analysis of survey data. These indicate that, although Queensland State school personnel may tend towards holding nontraditional views about the nature of mental computation and its development (see Figures 4.1-4.3), aspects of the current view of mental computation (Chapter 2) are likely to threaten their confidence in their effectiveness as teachers of mathematics. Central to this threat is their disagreement with the proposition that an emphasis on the written algorithms should be delayed to facilitate increased attention to mental computation (Table 4.10, Item 10). To this can be added the proposal that students should be allowed to use self-generated written strategies, and, particularly, the questioning of the 331

place of standard written algorithms within the curriculum (Cockcroft, 1982, p. 75, para 256; Lave, 1985, p. 175; Lindquist, 1984, pp. 602). Additionally, in conflict with the proposition that mental computation involves the development of flexible mental strategies (R. Reys, 1992, pp. 65-66), is the belief of many teachers that calculating exact answers mentally involves applying rules by rote (Table 4.9, Item 5). Such a belief, however, is not inconsistent with that traditionally held by Queensland teachers, one that originated during the period 1860-1965 when mental arithmetic was viewed as the presentation of a series of one-step oral questions. The focus was on obtaining correct answers, speedily and accurately (see Section 3.7), rather than on the mental strategies employed. The success of a lesson was dependent upon the number of questions correctly answered (Gladman, 1904, p. 235). If students are to develop confidence with mental computation, the change process for school communities needs to be initiated without delay. Critical to this process is the need for teachers to gain a clear practical understanding of how any proposed changes might impact upon their beliefs and teaching practices (Lovitt, Stephens, Clarke, & Romberg, n.d., p. 2). The development of a syllabus component focussing on mental computation should assist teachers to ascertain the potential personal impact that may ensue from the recommended changes to the nature of computation within the primary mathematics syllabus. In formulating and implementing a mental computation syllabus component, mental computation needs to be considered from various perspectives. First, from a global perspective, mental computation needs to be viewed as an integral part of the whole school program, where the intent is the maximisation of links to the real-world of the children so as to capitalise upon their idiosyncratic knowledge (see Section 2.10, Conclusion 7). Second, from a mathematics learning perspective, a focus on mental computation needs to recognise that mental strategies are essentially ways of thinking about mathematics (Cobb & Merkel, 1989, p. 80). The development of flexible and resourceful approaches to manipulating numbers both reflects and develops number sense, a key goal for school mathematics. Third, from an operations curriculum perspective, the development of strategies for calculating mentally is implicitly enmeshed with the development of written and technological methods of calculation. Finally, from a calculation perspective, mental computation needs to be considered in terms of the 332

effectiveness, efficiency and accuracy of the various mental solution strategies. It is principally from the last two perspectives that the following analyses and proposals have been developed. However, in responding to the focus for this chapter (see Section 1.4.2), which particularly concerned proposals for restructuring the computation strand, and identifying mental strategies suitable for direct teaching, cognisance also has been taken of issues relevant to the second perspective.

5.1.2 Framework for Syllabus Development

From the analysis of recent issues in mathematics education in Queensland (Section 4.4), it is apparent that, for an outcomes-based approach to education to have a positive impact, outcome statements must be in accordance with a syllabus that emphasises the desired learnings. Merely including these in assessment and reporting documents such as Mathematics: A Curriculum Profile for Australian Schools (AEC, 1994) or Student Performance Standards in Mathematics for Queensland Schools (1994) with the intention of forcing curriculum change (Clements, 1996, p. 9; Speedy, 1992, p. 6) is ineffective, particularly where perceived mismatches between these documents and the syllabus are apparent. The alignment of outcome statements and curriculum content requires modifications to the curriculum development process. If the major focus is to be on what children are to do, rather than on what teachers are to do, the desired learning outcomes need to be clearly determined prior to, or at least in conjunction with, the design of appropriate mathematics syllabuses. Consequently, consideration needs to be given to (a) the desired learning outcomes, and (b) how their development can be effectively encapsulated in a syllabus, in context with relevant research data, particularly about the role of mental computation and how it should be taught. Hence, a principal purpose of this study has been “to formulate a mental computation component, encapsulating key elements of the research data, for inclusion as a core element in future mathematics syllabuses” (see Section 1.3). Recognition needs to be given to the nature of current mathematics syllabuses and that of syllabuses necessary to meet future needs. In primary schools, syllabuses, such as the Queensland’s 1987 mathematics syllabus, essentially provide outlines of scope and sequence. Whereas some recommendations about 333

teaching and assessment strategies are provided, little is included about expected student outcomes. Schools and teachers have been left to formulate their own standards for judging student progress (Grace, 1996, p. 11), with variability between teachers, both within and across schools, as a consequence. An approach to syllabus design which may assist in providing a focus on what students should be able to do as a result of their learning is presented in Figure 5.1. However, an analysis of the structure and organisation of future mathematics syllabuses (Stage 4) is considered beyond the scope of this study. Hence, this chapter is primarily concerned with the relationships between Stages 1 to 3 and mental computation.

Goal/s Sequential Outcomes Syllabus re student Framework for each Content learning for learning aspect of and sequential organisation framework

Stage 1 Stage 2 Stage 3 Stage 4

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

5.2 Mental, Calculator, and Written Computation

The discussion presented in Section 2.9.1 revealed that the research literature relating to the most appropriate ways to incorporate mental computation into the curriculum (Carroll, 1996, p. 8; McIntosh, 1991a, p. 6, 1996, p. 273; Rathmell & Trafton, 1990, p. 160; B. Reys, 1991, p. 9; R. Reys, 1995, p. 305) is characterised by a degree of equivocalness (see Section 2.9.1). Nonetheless, irrespective of the approach employed, teaching to promote the desired outcomes occurs within an overall sequence for introducing computational procedures. This sequence may be considered at two levels: (a) the order in which mental, technological, and paper- and-pencil techniques are taught to children; and (b) the order in which particular 334

procedures associated with each of these methods are learned. Of these, the first may have greater significance for mental computation.

5.2.1 Traditional Sequence for Introducing Mental, Calculator and Written Computation

The sequence in which computational procedures have traditionally been introduced (Figure 2.4) is inextricably bound with the purposes for learning school mathematics. Although current syllabuses exhibit broader emphases than those introduced prior to the 1960s, the prime focus of primary school mathematics continues to be written computation and the development of standard written algorithms. As concluded in Section 2.5.4, the goal continues to be the automatic processing of paper-and-pencil calculations, despite the concern for developing students’ understanding of the processes involved. Written algorithms are products of the needs of an industrial age which necessitated minimum competencies in arithmetic for all students, with higher mathematical training reserved for a few. The continued emphasis on written procedures has its origins in two outdated, but steadfastly held, assumptions, namely that “(a) mathematics is a fixed and unchanging body of facts and procedures, and (b) to do mathematics is to calculate answers to set problems using a specific catalogue of rehearsed techniques” (NSEB & NRC, 1990, p. 4). Although the sequence represented in Figure 2.4 has come to be considered as traditional, it was not one that was recommended in Queensland mathematics syllabuses prior to the New Maths era, syllabuses that explicitly referred to mental calculations. As discussed in Section 3.5.2, the focus in these syllabuses was on using mental arithmetic as an introduction to the written work that was to follow (“Schedule XIV,” 1904, p. 201). Nevertheless, in contrast to the current emphasis on mental strategies as a means for promoting number sense, the mental work under the syllabuses prior to 1966 was designed primarily as a means for rehearsing the steps to be undertaken in applying the standard written algorithms (see Section 3.5.2). However, it is evident, as reflected by the observations of District Inspectors of Schools (see Section 3.5.3), that teachers rarely satisfactorily achieved this syllabus requirement. Mental arithmetic tended to be thought of as a separate topic within 335

the mathematics curriculum, a view not inconsistent with that of many current Queensland primary school teachers (see Table 4.10). In the minds of teachers were syllabus goals that stressed written calculations─calculations that were to be achieved with speed and accuracy (Department of Public Instruction, 1952, p. 1). This belief has persisted to the present, despite the focus on teaching methods designed to promote an understanding of the concepts and processes being learned. To support this development, current mathematics syllabuses, of which the Queensland Years 1 to 10 Mathematics Syllabus (Department of Education, 1987) is representative, are characterised by: (a) a delay in the age at which particular aspects of the four operations are learned─a process first stimulated in 1930s in Queensland (see Section 3.2.3) by reports such as that of Cunningham and Price (1934), (b) clear distinctions being drawn between the concept of an operation and the written algorithm, (c) the memorisation of basic facts being supported by insights into the relationships between numbers gained from a focus on thinking strategies, and (d) the standard algorithms taught being ones which may be logically and mathematically linked to the use of appropriate materials, supported by the use of consistent language. Table 5.1 presents the operationalisation of these considerations, as embodied in Queensland’s 1987 Syllabus and its supporting documents. This 336

Table 5.1 Traditional Sequencea for Introducing the Four Operations with Whole Numbers as Presented in the Mathematics Sourcebooks for Queensland Schools

Yearb

1 2 3 4 5 6 7

ADDITION 1. Addition concept 2. Recording addition 3. Addition facts─Thinking strategies 4. The tens facts 5. Standard written algorithmc • 2-digit numbers, no regrouping • 2-digit numbers, regrouping • 3-digit numbers, regrouping • 4-digit numbers, regrouping

SUBTRACTION 1. Subtraction concept 2. Recording subtraction 3. Subtraction facts─Thinking strategies 4. The tens facts 5. Standard written algorithmc • 2- & 3-digit numbers, no regrouping • 3-digit numbers, regrouping • 4-digit numbers, regrouping

MULTIPLICATION 1. Multiplication concept · · 2. Recording multiplication 3. Multiplication facts─Thinking strategies 4. The tens facts 5. Standard written algorithmc • 2-digit by 1-digit number, regrouping • 2-digit by 2-digit numbers, regrouping

DIVISION 1. Division concept · · 2. Recording division 3. Division facts─Thinking strategies 4. Standard written algorithmc • 2- & 3-digit ÷ 1-digit number, no regrouping • 2- & 3-digit ÷ 1-digit number, regrouping

Note. aAdapted from Years 1 to 10 Mathematics Sourcebook: Year 5 (Department of Education, 1988, pp. 68-69, 72-73, 76-77, 82). bThe operations for Years 2 to 5 equate with Phases C to F of the Number Developmental Continuum (QSCO, 1996a, pp. 23-25, 27-28, 30-31, 33). cSelected levels of complexity of the standard written algorithm. 337

framework encapsulates the view expressed by the National Council of Teachers of Mathematics (1989) that the “long-standing preoccupation with [written] computation...has dominated what mathematics is taught and the way mathematics is taught” (p. 15). One consequence of this has been that a student’s view of mathematics has generally not reflected the subject’s vitality, an essential element in promoting the development of flexible mathematical thinking.

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

Critical to flexible mathematical thinking is an individual’s sense of number (see Section 2.3.3), the development of which is considered by the Australian Education Council (1991, p. 107) to be an essential goal for primary school mathematics. Number sense is, in part, characterised by an ability to perform mental computations with nonstandard strategies that take advantage of an ability to compose and decompose numbers (Resnick, 1989b, p. 36; Sowder, 1992, p. 5). In so doing, students with number sense tend to analyse the whole problem first to ascertain and capitalise upon the relationships among the numbers, and the operations and contexts involved, rather than merely apply a standard algorithm (Lave, 1985, p. 173). Hence, it was concluded in Section 2.10 that engaging children in mental computation can not only be considered as a means for linking school and folk mathematics, but also as a means for enhancing mathematical knowledge, and confidence in its application. The development of flexible mental strategies is influenced by the order in which mental and written techniques are introduced. Musser (1982, p. 40) has concluded that classroom experience indicates that children have difficulty with mental methods when written algorithms are taught prior to a focus on mental computation. Such a focus places an emphasis on symbols rather than on the quantities embodied in calculative situations (Reed & Lave, 1981, p. 442), thus reducing the opportunities for the development of number sense. Further, the right-to-left characteristics of the standard written algorithms for addition, subtraction, and multiplication contradict the holistic, left-to-right strategies frequently used by proficient mental calculators (see Table 2.4). Hence, as Cooper et al. (1992, p. 104) assert, the conventional written before mental sequence (Figure 2.4, Table 5.1) 338

needs to be reevaluated. One consequence of such a reevaluation would be the development of more flexible computational strategies, both mental and written (Thornton et al., 1995, p. 40). An additional consideration is the necessity to reassess the balance between the various forms of calculation─mental, calculator, and written─to calculate approximate and exact answers. Children need to be encouraged to select computational techniques appropriate to particular tasks in accordance with their store of declarative and procedural knowledge (see Figure 2.3). In so doing, recognition needs to be given to the profound effect that electronic calculating devices have had on society, and therefore on the features of school mathematics relevant to the present technological age (see Section 2.3.2). Consequently, less emphasis should be placed on developing paper-and-pencil algorithms (Rathmell & Trafton, 1990, p. 54), particularly the standard ones, which is the continuing principal concern of primary school mathematics, including that in Queensland classrooms (see Table 5.1). The continuing place for standard written algorithms, even those designed to enhance an understanding of the procedures involved, needs to be questioned, given that the focus for learning mathematics should be on the development of techniques and tools which reflect modern mathematics (AEC, 1991, p. 13). R. E. Reys (1984, p. 551) asserts that standard written algorithms discourage thinking as they are designed to be used automatically by students who require only a limited understanding of the processes involved (Plunkett, 1979, p. 2). Hence, they contribute little to the development of number sense, particularly where decontextualised examples are presented to students (Lave, 1985, p. 175). Further, as discussed in Section 2.5.3, there is evidence to suggest that the standard written algorithms are not used widely outside the classroom (Carraher et al., 1987, p. 95; Cockcroft, 1982, p. 20; French, 1987, p. 41; Murray et al., 1991, p. 50). As discussed in Section 2.5.4, when calculation occurs in settings in which the method of calculation is not imposed, the elements of problem situations are often changed to reflect the perceived numerical relationships. This facilitates the modification of calculation techniques to match the physical environment (Lave, 1985, p. 173). Consequently, school mathematics needs to be reorganised so that opportunities are provided for students to deal with mathematics in their own 339

environments in ways similar to those of folk mathematicians (Maier, 1980, p. 23; Masingila et al., 1994, p. 13). Nonetheless, a structured approach to developing operational proficiency, which a focus on standard written algorithms can provide, may be beneficial for some children. The language and representational models used may assist in the development of conceptual, procedural, metacognitive, and real-world knowledge (see Figure 2.3) of use when calculating in non-teacher-directed situations. This is an area for further research to inform the professional and community debate which needs to occur to support the recommended changes to the computation curriculum, research that could be modelled on that by Gracey (1994) and Shuard (1989), which focussed on teaching mental strategies and the impact of calculator use, respectively. As intimated previously, based on the model of the calculative process presented in Figure 2.1 and on statements in documents to guide syllabus development (AEC, 1991, pp. 115, 121; NCTM, 1989, p. 94), it is apparent that individuals need to be able to choose computational methods matched to context and personal strengths. Hence, a global outcome (Figure 5.1: Stage 1) for computational learning in the primary school could be expressed as: Select and apply a range of mental, written, and technological strategies for each operation─strategies appropriate to the context and the student. To operationalise a mental-written approach for introducing computational procedures, as represented in Figure 2.5, a sequential framework (Figure 5.1: Stage 2) encompassing each operation is presented in Table 5.2. An emphasis is placed on the use of mental and calculator procedures for each operation beyond the basic facts prior to the introduction of paper-and-pencil techniques. It assumes the availability of calculators for computations beyond those capable of being worked mentally, thus maximising opportunities for their becoming real computational tools, particularly for low-attaining children, an outcome supported by the PrIME Project in England (Shuard, 1989, p. 73). 340

Table 5.2 Revised Sequential Framework for Introducing Mental, Calculator and Written Procedures for Addition, Subtraction, Multiplication and Division

Year

1 -3 3 - 5 5 - 7

ADDITION 1. Addition concept 2. Non-written procedures (a) Addition facts─Thinking strategies (b) Addition beyond basic facts • Self-generated/shared mental strategiesa • Calculator strategies 3. Written procedures • Self-generated/shared strategies • Standard written algorithmb

SUBTRACTION 1. Subtraction concept 2. Non-written procedures (a) Subtraction facts─Thinking strategies (b) Subtraction beyond basic facts • Self-generated/shared mental strategiesa • Calculator strategies 3. Written procedures • Self-generated/shared strategies • Standard written algorithmb

MULTIPLICATI0N 1. Multiplication concept 2. Non-written procedures (a) Multiplication facts─Thinking strategies (b) Multiplication beyond basic facts • Self-generated/shared mental strategiesa • Calculator strategies 3. Written procedures • Self-generated/shared strategies • Standard written algorithmb

DIVISION 1. Division concept · 2. Non-written procedures (a) Division facts─Thinking strategies (b) Division beyond basic facts • Self-generated/shared mental strategiesa • Calculator strategies 3. Written procedures • Self-generated/shared strategies • Standard written algorithmb

Note. aMental strategies for calculating approximate as well as exact answers. bWhere appropriate, introduced using the traditional sequence (see Table 5.1). 341

Reflecting the place of mental calculation within the calculative process (see Figure 2.1), mental strategies refers to strategies for the calculation of both approximate and exact answers. Such strategies take advantage of the structural properties of numbers and the relationships between them. Their development and that of number comparison and number sense occurs spirally, each "feeding on and strengthening the others" (Threadgill-Sowder, 1988, p. 195). The ability to mentally compute with truncated and rounded numbers is a prerequisite for computational estimation (see Figure 2.2). Additionally, mental strategies used to refine estimates may assist in the development of flexible approaches for calculating exact answers─getting closer may ultimately result in turning approximate answers into exact answers (Irons, 1990b, p. 1). In concert with the analysis presented above and that in Section 2.5.3, the emphasis for written procedures is placed on self-generated strategies. However, reference to the standard written algorithm for each operation has been retained. Despite the arguments for such algorithms not to be included, from personal observations of previous syllabus changes, the reality of the classroom dictates that they will continue to be taught, even should a revised syllabus advise otherwise. The degree to which standard algorithms will continue to be taught is dependent upon the effectiveness of the professional debate, supported by further research into managing mathematics classrooms in which the focus is on self-generated mental, technological, and written strategies. By highlighting the position of the standard written algorithms within the sequential framework (see Table 5.2), the standard algorithm for each operation may come to be viewed as one of many possible ways for calculating in particular contexts, rather than the way (Ross, 1989, p. 51), thus enhancing a child's capacity to select appropriate methods for calculating in particular contexts. Most importantly, the focus on non-written and non-standard written procedures may assist in overcoming the belief of many adults and students, as identified by Plunkett (1979, p. 3), that the concept of a particular operation and its standard paper-and- pencil algorithm are synonymous. To support this focus, the placement of the standard written algorithms within the sequence accords with the recommendation of the Australian Education Council (1991, p. 109) that, in instances where these algorithms continue to be taught, such teaching should occur later in a child’s schooling (see Tables 5.1 & 5.2). 342

The framework presented in Table 5.2 recognises that schools need to (a) acknowledge that all students can learn, and (b) focus on differences with respect to the way they learn and the rate at which learning occurs (Peach, 1997a, p. 1). Such a focus requires a flexible approach to the allocation of particular learnings to particular time periods. Hence, the elements of the sequential framework are not year-level specific, an approach embodied in the Queensland 1987 Syllabus documents (Department of Education, 1987a, pp. 14-30; 1987b, pp. 12-40). This is in contrast to the traditional approach to syllabus design, which is characterised by estimating the range of topics that can be learned during a particular school year. However, it is an approach that is encapsulated in the Sourcebooks that accompany the 1987 Syllabus. The structure of the year-level bands (see Table 5.2) is based on recommendations contained in the Report of the Review of the Delivery of Curriculum in Primary Schools (Avenell, 1996, pp. 8, 10). It is intended that the overlapping bands provide for a smooth progression through the learnings related to computation, while recognising that each class is characterised by students who exhibit a range of achievement levels.

5.3 Mental Strategies: A Syllabus Component

It has been suggested that in formulating and implementing a mental computation syllabus strand, mental computation needs to be considered from a range of perspectives. The fourth of these has mental strategies that may be used to calculate mentally as its focus. The mental strategies included in the sequential framework for introducing mental, calculator, and written computational procedures (Table 5.2) refer to those for arriving at approximate as well as exact answers. The analysis presented in Section 2.6.1 indicated that mental computation is one of the two fundamental skills of computational estimation (see Figure 2.2). However, given that the second of these is the ability to convert exact to approximate numbers (Case & Sowder, 1990, p. 88), often a multiple of ten or hundred, mental computation as it applies to computational estimation involves the manipulation of a limited range of numbers. Nonetheless, students need to have developed some proficiency with the mental calculation of exact answers before they are able to find approximate ones. 343

It was concluded in Section 4.4.1 that if both mental computation and computational estimation are to receive the emphasis required for effective mathematical functioning, consideration needs to be given to distinguishing more clearly between the two forms of mental calculation. Whereas computational estimation is specifically featured in Queensland’s 1987 Syllabus, mental computation is not (see Section 4.2). The intent of this section, therefore, is to provide an outline of mental strategies for calculating exact answers which could form an essential component of the Number strand in future mathematics syllabuses─one aspect of Stage 3 of the proposed syllabus development process (Figure 5.1). However, the formulation of a complete mental computation substrand (Stage 4, Figure 5.1) is considered beyond the scope of this study. Such a strand would need to consider mental strategies, particularly the heuristic strategies based on relational understanding (Table 2.4), in context with the other components of mental computation─the Affective and Conceptual Components, and the Related Concepts and Skills, with the latter including those that mental computation shares with computational estimation (Figure 2.2, Table 2.1). The intent is for children to develop a range of holistic, flexible and constructive mental strategies embedded in a strong sense of number. The gaining of mathematical power is the desired outcome (see Section 2.1).

5.3.1 Background Issues

As revealed in Section 2.9.1, the delineation of a mental strategies component for inclusion in future mathematics syllabuses occurs in context with the recognition of the equivocal research evidence for such an undertaking. Hope (1987, p. 340) cautions that further research is required to ascertain whether it is legitimate to teach the strategies identified as those used by proficient mental calculators (Table 2.4). This caution is amplified by such findings as those of R. E. Reys et al. (1995, p. 321) that Japanese children taught using the behaviourist approach to mental computation exhibited a narrow range of non-idiosyncratic mental strategies. Nonetheless, the recommendations presented may provide a basis for the research which Hope (1987, p. 340) considers necessary, some of which has already been undertaken─for example that by Gracey (1994). 344

Additionally, as Carroll (1996, pp. 4, 8) suggests, the spontaneous invention of mental strategies may be difficult for some children. Hence, at times it may be appropriate for teachers to introduce particular strategies to students while recognising that strategies meaningful to one person may not be so for another (Heirdsfield, 1996, p. 146). Cooper et al. (1996, p. 159) suggest that low-proficiency children may benefit from direct teaching. The focus for these children tends to be on right-to-left mental strategies, which are often inefficient. However, the strategies taught need to be those matched to their level of mathematical development, particularly that related to place value (Murray & Olivier, 1989, p. 9). Further, there is some evidence (Flournoy, 1954, p. 153; Gracey, 1994, pp. 112- 116; Josephina, 1960, p. 200; Markovits & Sowder, 1994, p. 22; Schall, 1973, pp. 365-366) to suggest that gains can be made with respect to strategy use, and the correctness of the answer, where systematic instruction is provided. Markovits and Sowder (1994, p. 23) have reported that, following instruction, Grade 7 students exhibited a propensity for employing nonstandard strategies in instances where standard paper-and-pencil algorithms could not be applied easily to mental calculations. The strategies children construct have been found to relate to the nature of classroom experiences (Carpenter, 1980, p. 321), a finding not in conflict with the reasons provided for the tendency for many Queensland Years 2 and 3 students to use strategies based upon the use of the standard written algorithms for addition and subtraction (Cooper et al., 1996, p. 158; Heirdsfield, 1996, p. 132). Hence, it is considered legitimate to propose which mental strategies may become a focus at different stages of a child’s primary schooling. Nonetheless, when incorporating these suggestions into a mathematics syllabus (Figure 5.1: Stage 4), it is essential that this occurs in a way that encourages teachers to view the development of number sense as paramount. This may be assisted by highlighting B. J. Reys’ (1991, p. 9) approach to developing computational skill─an approach which is an amalgam of the behaviourist and constructivist approaches, one that focuses on discussion and on sharing the devised mental strategies with peers and teachers. Such an approach may be supported by the delineation of outcomes related to particular mental strategies. However, it is essential that teachers have an understanding of the range of mental strategies (Tables 2.2 - 2.4) that may be used by children (Carroll, 1996, p. 8). As McIntosh et al. (1994, p. 7) point out, these are not currently well known by teachers. Therefore, they constitute 345

one aspect of the professional development for teachers which needs to be conducted urgently.

5.3.2 Development Issues

In proposing a syllabus component which may form the basis of a mental computation strand, recognition needs to be given to the multiplicity of strategy variations arising from each individual’s unique long-term memory components (see Figure 2.3). Vakali (1985, p. 110) has noted that there are considerable individual differences in the way in which calculative plans are formulated and in the order in which their inherent steps are executed. In accordance with this view, Heirdsfield (1996, p. 135) has concluded that there is a diversity of mental strategies in use that are not prevalent in the literature. The variations in the mental strategies that have been observed are characterised by differences in the type of strategy used to solve particular problems, and by differences within particular strategy classifications. The more general the classification, the greater diversity of strategies within each category. Carroll (1996, p. 6) reports four different approaches used by Grade 5 students to correctly solve 426 + 75, namely

• 426 + 5 = 431 + 70 = 501 • 70 + 20 = 490 + (6 + 5) = 501 • 70 + 30 = 100 + 1 = 101 + 400 = 501 • 425 + 75 + 1 = 501

From the classification of strategies based on relational understanding presented in Table 2.4, the first of these may be classified as add (or subtract) parts of the (first or) second number, the second as incorporation, and the third and fourth strategies as compensation. However, given their idiosyncrasy, the second and third variants, in particular, are ones which may not be appropriate for inclusion in any recommendations for focussed teaching. The diversity of strategies also depends on the operation. Heirdsfield (1996, p. 131) observed a greater range of strategies for subtraction than for addition. For subtraction, the complexity of the mental strategies increases where carrying or 346

regrouping is required, particularly where an organisation─10-10─strategy is used (Beishuizen, 1993, p. 296). For example, one approach to solving 42 - 15 is: 40 - 10 = 30 e 20, 2 - 5 = ? e 12 - 5 = 7, 20 + 7 = 27. This contrasts with an (add or) subtract parts of the (first or) second number approach─N10 in Beishuizen’s (1993, p. 297) terms: 42 - 10 = 32, 32 - 2 - 3 = 27, an approach that reduces the load on working memory through the incorporation of partial differences. This diversity of mental strategies, both across and within categories, presents some difficulty for selecting strategies for inclusion in a proposed syllabus component. Nevertheless, the classification of heuristic strategies based on relational understanding, in particular, discussed in Section 2.7.4, provides an appropriate framework on which to base the recommendations (Table 2.4). Also included are selected counting strategies (Table 2.2) which are indicative of children who have at least gained an understanding of the numerosities of numbers (Murray & Olivier, 1989, p. 5). Strategies classified as relying on an instrumental understanding of place value (Table 2.3) have not been considered. These transitional strategies (McIntosh, 1991a, p. 2) rely on the rote application of rules and procedures, akin to the approach to mental arithmetic embodied in the pre-1966 syllabuses (see Section 3.5.2). Hence, the transitional strategies are not ones which significantly contribute to the development of flexible mental strategies. In formulating syllabus outcomes relating to mental computation (Figure 5.1: Stage 3) consideration needs to be given to the most effective way in which they may be expressed. Elements to be considered include the operation involved, the mental strategy, and the type and size of the numbers to be manipulated. This suggests a number of alternatives to structuring the outcomes─for example:

• Specify the operation and mental strategy. Use left-to-right addition to refine estimates obtained by using front-end digits (Ohio Department of Education, n.d., p. 91). • Specify the operation, and the type and size of the numbers. Add two 2-digit [whole] numbers mentally (Michigan State Board of Education, 1989, p. 26). • Specify the operation, the mental strategy, and the type and size of the numbers. 347

Use incorporation to add two 2-digit whole numbers.

Of these alternatives, the third is the preferred. However, it is essential that the selection of strategies for focussed teaching arises from the research literature, rather than from a theoretical analysis of possible number combinations that may appear in numerical situations, which appears to have occurred in the Framework for Numeracy: Years 1 to 6 (National Numeracy Project, 1997) for primary schools in England. The suggested mental strategies (Tables 5.3-5.5) are categorised by operation to provide a direct link to the revised sequential framework (Table 5.2), elements of which relate to self-generated and shared mental strategies for each operation. Although teachers should place an emphasis on self-generated mental strategies (Hunter, 1977a, p. 40), as implied by the second alternative, the purpose of the proposed syllabus component is to provide some structure and guidance for teachers to assist students to extend their repertoire of mental strategies (Carroll, 1996, p. 4; Cooper et al., 1996, p. 159; Heirdsfield, 1996, p. 146). By placing limits on the size of the numbers involved, teachers should be persuaded to allow students to extend their range of mental strategies through encouraging them to devise alternative methods, rather than principally maintaining the focus on a particular mental strategy to calculate with numbers of increasing size or complexity, an approach not inconsistent with that embodied in Queensland’s 1952 Syllabus (see Section 3.5.2). Although the intent of a syllabus Number strand should be to promote proficiency with a range of computational strategies with a range of number types, by limiting the proposed mental strategies to operations with whole numbers it is intended to encourage proficiency with particular mental strategies before they are applied to decimals, money, measures, or percentages. The essence of a mental strategy is not dependent upon the size or complexity of the numbers being manipulated. For example, the use of the strategy add parts of the second number to calculate 548 + 332 (548 + 300 + 30 + 2) or 5.48 + 3.32 (5.48 + 3 + .3 + .02) is not intrinsically different to its use when calculating 48 + 32 (48 + 30 + 2). Moreover, other aspects of an individual’s store of numerical equivalents and calculative plans (see Section 2.8.1) are developed and applied when alternative strategies are employed. When using a compensation strategy, for example, to find the sum of 48 and 32 this could be calculated as 50 + 30 = 80. 348

Such an approach has links to compensatory strategies used to find approximate answers.

5.3.3 Mental Strategies for Addition, Subtraction, Multiplication and Division

Although Tables 5.3-5.5 identify specific mental strategies which may be appropriate for focussed teaching, given that the variety used by students exceeds those which may be specifically taught (Heirdsfield, 1996, p. 131; Murray, Olivier, & Human, 1991, pp. 50, 55), and that a desired outcome of mathematics teaching is the development of an ability to respond flexibly and creatively to number situations (AEC, 1991, p. 107), it is appropriate for a learning outcome for mental computation to be expressed in general terms: Select and apply a variety of mental strategies in a range of numerical situations─strategies appropriate to the context and the student. The aim is for students to become proficient with a variety of heuristic strategies (Carpenter, 1980, p. 317), both self-generated and taught. As discussed in Section 2.7.4, such strategies are ones that rely on a well-developed sense of number. Hence the main focus for teaching mental strategies for each operation is the development of a proficiency with those that require one or more of the numbers to be decomposed in order to transform a problem into one that is more manageable, either through the use of a series of steps or through the application of known facts. Of the heuristic mental strategies discussed in Section 2.7.4, and summarised in Table 2.4, four which have been frequently reported in the literature in relation to addition and subtraction (see Tables 5.3 and 5.4) have been recommended for focussed teaching─add or subtract parts of the first or second number, incorporation, organisation, and compensation (Beishuizen, 1993, pp. 296-297; Cooper et al., 1996, p. 159; Heirdsfield, 1996, pp. 133-134; Olander & Brown, 1959, p. 99; Vakali, 1985, p. 111). Although Heirdsfield (1996, p. 132) found that using mental analogues of the standard written algorithms for addition and subtraction (see Table 2.4) were the most common strategies used by Year 4 students, these have not been included in the recommendations. The emphasis is on the more efficient left-to-right strategies. Add or subtract parts of the first or second number and incorporation have the advantage of producing a single result at each step, thus 349

reducing the load on working memory (see Sections 2.8.3 & 2.8.4). The former, together with organisation, is considered by Beishuizen (1993, p. 320) to be a basic mental strategy for calculating with 2-digit numbers. Cooper et al. (1996, p. 159) suggest that explicit exploration of holistic strategies should be undertaken to reduce the emphasis on paper-and-pencil procedures. The fourth heuristic strategy included for addition and subtraction, compensation (see Tables 5.3 & 5.4), is classified by Heirdsfield (1996, p. 134) as a holistic strategy─one that requires the manipulation of numbers as single entities. This strategy requires students to have a strong sense of the direction in which the answer needs to be compensated if an undoing approach is employed. Markovits & Sowder (1994, p. 15) report that this is particularly difficult for some Year 7 students when using a compensation approach for subtraction. The recommendations for addition and subtraction also include counting strategies selected from those discussed in Section 2.7.4─min of addends and min of units (Resnick & Omanson, 1987, p. 66) for addition, and counting-on in twos, tens and fives (Olander & Brown, 1959, p. 99) for subtraction. Except for min of units, these strategies (see Tables 2.2, 5.3, & 5.4) require students to be operating at least at Murray and Olivier’s (1989, p. 6) second level of understanding 2-digit 350

Table 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

Year

1 -3 3 - 5 5 - 7

Students should be able to use:

Counting strategies

• min of addends add 2-digit and 1-digit number 23 + 9: 24, 25, 26...32; 32

• min of units add 2-digit and 1-digit number 23 + 9: 29 + 3, 30, 31, 32; 32

Heuristic strategies

• add parts of the first or second number add two 2-digit numbers 46 + 38: 46 + 30 = 76, 76 + 8 = 84 46 + 38: 40 + 38 = 78, 78 + 6 = 84

• incorporation add two 2-digit numbers 39 + 25: 30 + 20 = 50, 50 + 9 + 5 = 64

• organisation add two 2-digit numbers 58 + 34: 50 + 30 = 80, 8 + 4 = 12, 80 + 12 = 92

• compensation add two 2-digit numbers 28 + 29: 30 + 30 = 60, 60 -2 -1 = 57 28 + 29: 30 + 27 = 57

numbers. Each builds upon strategies used to develop basic fact knowledge for addition and subtraction, particularly count-on, and on number knowledge developed through counting in twos, tens and fives. Min of addends, classified as an elementary counting strategy, constitutes one from which the various strategies classified as counting in larger units may be developed (see Table 2.2). It is a 351

Table 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

Year

1 -3 3 - 5 5 - 7

Students should be able to use:

Counting strategy

• counting-on in twos/tens/fives subtract 2-digit number from 2-digit number 38 - 34: 36, 38; 4 71 - 44: 54, 64, 74; minus 3; 27 57 - 35: 40, 45, 50, 55; plus 2; 22

Heuristic strategies

• subtract parts of the second number subtract 2-digit number from 2-digit number 33 - 16: 33 - 10 = 23, 23 - 6 = 17

• incorporation subtract 2-digit number from 2-digit number 51 - 34: 50 - 30 = 20, 20 + 1 = 21, 21 - 4 = 17

• organisation subtract 2-digit number from 2-digit number 46 - 23: 40 - 20 = 20, 6 - 3 = 3, 20 + 3 = 13 46 - 28: 40 - 20 = 20, 10; 16 - 8 = 8, 10 + 8 = 18

• compensation subtract 2-digit number from 2-digit number 86 - 38: 88 - 40 = 48

strategy to which many students are likely to regress when required to compute with unfamiliar numbers (Murray & Olivier, 1989, p. 4). Min of units is a strategy that relies on an ability to recognise the magnitude of the numbers of units involved, and to exchange these as necessary, a strategy that requires an understanding of numeration characteristic of Murray and Olivier’s (1989, p. 6) third level. It is this level which provides the conceptual basis for the use of heuristic strategies (Murray & Olivier, 1989, p. 6). Hence, this strategy may be difficult for some students in Years 1 to 3 (see Table 5.3). Further, using a count- on in twos, tens or fives strategy for subtraction may also be difficult for many 352

children, particularly where compensation is involved (see Table 5.4). This approach places greater demands on working memory, as not only does the number of iterations need to be tracked, but also the differential between the minuend and partial answer needs to be calculated. However, the variants of this strategy that require compensatory procedures to be applied may assist in developing a feel for the processes involved, processes essential to effectively using heuristic compensatory strategies for calculating exact (see Table 5.4) and approximate answers (see Section 2.6.2). Counting strategies for multiplication and division─repeated addition and repeated subtraction, respectively (see Table 2.2)─have not been recommended for focussed teaching. Although these strategies will be used by students as their understanding of multiplication and division is developed, they become inefficient with increased number size (Carraher et al., 1985, p. 28). More importantly, the models on which these strategies are based are not those commonly employed to develop basic fact knowledge for multiplication and division. Such knowledge is developed through thinking strategies based on array and partition models, respectively. As discussed in Section 2.7.4, the strategies recommended for multiplication and division (Table 5.5) do not have as wide a research base as those for addition and subtraction. Little research has been undertaken into mental strategies for division, and that which has been reported, in common with that for multiplication, has tended to focus on students in Year 7 at least (Hope, 1987; Hope & Sherrill, 1987; Markovits & Sowder, 1994; B. J. Reys, 1986b). Nonetheless, Gracey (1994, pp. 84, 90, 101) has successfully taught multiplication strategies, which may be classified as general factoring and half-and-double (Hope, 1987, p. 334), to Year 6 students. In common with the heuristic strategies recommended for addition and subtraction, factoring strategies, as discussed in Section 2.7.4, are based on a relational understanding of number. Consequently, their use depends upon, and contributes to, the development of numerical equivalents and calculative plans, which are part of an individual’s declarative and procedural knowledge, respectively (Hamann & Ashcraft, 1985, p. 52; Hunter, 1978, p. 339; Putman et al., 1988, p. 83). 353

Table 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

Year

1 -3 3 - 5 5 - 7

Students should be able to use:

MULTIPLICATION

• general factoring multiply 2-digit number by 1- or 2-digit number 15 x 8: 5 x 3 x 2 x 4, 10 x 12 = 120 60 x 15: 60 x 5 x 3, 300 x 3 = 900

• additive distribution multiply 2-digit number by 1- or 2-digit number 23 x 4: 20 x 4 + 3 x 4, 80 + 12 = 92 21 x 13: 20 x 13 + 1 x 13, 260 + 13 = 273

• subtractive distribution multiply 2-digit number by 1- or 2-digit number 98 x 8: 100 x 8 - 2 x 8, 800 - 16 = 784 19 x 13: 20 x 13 - 1 x 13, 260 - 13 = 247

• half-and-double multiply 2-digit number by 1- or 2-digit number 42 x 5: 21 x 10 = 210 60 x 15: 30 x 30 = 900

• aliquot parts multiply 2-digit number by 1- or 2-digit number 14 x 5: 14 x (10 ÷ 2), (14 x 10) ÷ 2, 140 ÷ 2 = 70 48 x 25: 48 x (100 ÷ 4) = (48 ÷ 4) x 100 = 12 x 100 = 1200

DIVISION

• general factoring divide 2-digit number by 1- or 2-digit divisor 90 ÷ 15: 90 ÷ 3 ÷ 5, 30 ÷ 5 = 6

• additive distribution divide 2-digit number by 1- or 2-digit divisor 64 ÷ 4: 60 ÷ 4 + 4 ÷ 4 = 15 + 1 = 16 78 ÷ 15: 60 ÷ 15 + 18 ÷ 15 = 4 + 1 rem 3 = 5 rem 3

• subtractive distribution divide 2-digit number by 1- or 2-digit divisor 76 ÷ 4: 80 ÷ 4 - 4 ÷ 4 = 20 - 1 = 19 354

A third factoring strategy, aliquot parts, has also been included in the recommendations (see Table 5.5). This strategy draws upon an individual’s understanding of special products, an essential component of mental multiplication, which “refers to those products that are easily found by multiplying by a power of 10 or a multiple of a power of 10” (Hazekamp, 1986, p. 117). This interpretation of aliquot parts contrasts with that expressed in the 1930 and 1952 Syllabuses (see Section 3.5.2) where, as traditionally included as one of the essential short methods of mental calculation, the emphasis was on aliquot parts of a pound, thus reflecting mental arithmetic’s use as a book-keeping tool. Aliquot parts, together with additive and subtractive distribution, was found to be the most difficult strategy for multiplication for Year 5 students by B. J. Reys et al. (1993, p. 310). Nonetheless, a form of the latter was one frequently used by Year 2 and 3 students who were encouraged to explore division through a constructivist approach which emphasised experimentation and discussion (Murray et al., 1991, p. 54). Both of these distribution strategies have been successfully taught to Year 6 students for mental multiplication (Gracey, 1994, pp. 68, 73). Despite the lack of research support, it is considered reasonable to propose that general factoring and additive distribution should also be a focus for the development of mental strategies for division (see Table 5.5). General factoring is closely linked to number fact knowledge for multiplication and division, and additive distribution is considered to be the “calculative drafthorse” of expert mental calculators (Hope, 1985, p. 358). Nonetheless, exponential and iterative factoring, and fractional and quadratic distribution (see Table 2.4) have not been included in the recommendations for focused teaching. Although these strategies are also used by exceptional mental calculators (Hope, 1985, pp. 361-362, 365-368; Hope & Sherrill, 1987, pp. 102-103), they are not widely reported in the literature. More importantly, their complexity removes them from those easily understood by primary school students.

5.4 Concluding Points

Prior to the implementation of the suggestions contained in the revised sequential framework (Table 5.2), and in the proposed mental strategies component (Tables 5.3-5.5), a number of points need to be taken into consideration. With respect to the former, as previously intimated, the implementation of a mental- 355

written sequence for introducing computational methods has the potential to severely threaten the sense of efficacy most teachers now hold towards written computation. It is imperative that teachers are informed about the proposed changes, and assisted in the actuation of the revised sequence─issues explored more deeply in the next chapter. Also, the mental strategies relating to each of the four operations proposed for direct teaching need to be validated through further research prior to their incorporation in a future mathematics syllabus─validation with respect to their appropriateness and year-level placement. Consideration needs to be given to the range of variants of a particular strategy, some of which may be inappropriate for focussed teaching for many children. However, the range of possible strategies should allow teachers to maintain a focus on student-generated strategies during focussed teaching, particularly where students are given opportunities to explore strategies using their intuitive knowledge, and are actively involved in the learning process through social discourse (Gracey, 1994, pp. 129-130). Data from such research would assist in the provision of specific guidance for teachers. Further information would be gained concerning factors which impinge on the use of the mental strategies recommended for focussed teaching. Such information could include: (a) the demands of particular strategies on working memory; (b) the effects arising from the need to regroup or carry, and the size and type of numbers being manipulated; (c) the level of numeration understanding required for particular strategies, one consequence of which would be to extend the work of Murray and Olivier (1989); (d) the nature of any difficulties experienced in balancing strategy development through using elements of the behaviourist and constructivist approaches; (e) the consequences arising from using visual and oral stimuli, and examples presented in and without context; (f) the appropriateness of various models which may be used to represent the calculative situation; and (g) the effectiveness of discussion as a means for encouraging strategy growth, and how it should be managed within the classroom, issues questioned by some respondents to the survey of Queensland school personnel (see Section 4.3.3).

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CHAPTER 6

MENTAL COMPUTATION: CONCLUSIONS and IMPLICATIONS

6.1 Restatement of Background and Purpose of Study

From the analyses presented, it is evident that, in contrast to the American experience, in particular, a continued emphasis has been placed on the calculation of exact answers mentally in Queensland State primary schools from 1860, albeit one that has fluctuated over time, due particularly to variations in teacher interpretations of the syllabus. Although the mathematics syllabuses from 1966 in Queensland have not explicitly given such calculation a high profile, mathematics educators, from the mid-1980s, have been taking a renewed interest in mental computation. While not neglecting the correctness of the answer, it is now recommended that the emphasis be placed on the mental processes employed. It is this which distinguishes mental computation from earlier considerations in which the correctness of the answer was of prime concern─that is, to distinguish mental computation from mental arithmetic. The resurgence of interest in mental calculation, as outlined in Chapter 1, has its origins in a number of sources. Calculating mentally remains a viable alternative, despite the availability of various electronic calculating devices. It continues to be the major form of calculation used in every-day life. Such mental calculations are typically undertaken using methods adapted to the particular characteristics of the situation in which they occur (Lave, 1985, p. 173). Coupled with this recognition is the realisation that the standard written algorithms─a major focus of primary school mathematics─are seldom used outside the classroom. School-taught algorithms are predominantly employed only in the solution of school-type problems (Carraher et al., 1987, p. 95). Further, as the Mathematical Sciences Education Board and the National Research Council (1990, p. 19) have observed, there is sufficient research

357

evidence now available to suggest that an overemphasis on written methods may reduce an individual's development of flexible mental strategies. Hence it is argued that paper-and-pencil skills should receive decreased attention in schools. Mental computation is no longer viewed as an end in itself, but rather as a means for promoting an individual's ability to think mathematically (R. E. Reys, 1992, p. 63). This, in Reys and Barger's (1994, p. 31) view, is the novel facet of the current resurgence of interest. However, if this facet is to be realised, teachers need to be persuaded to place an emphasis on encouraging children to devise personal mental strategies and to compare their strategies with those of others, so that they may be able to select the most appropriate method in accordance with their strengths and the particular mathematical context (AEC, 1990, p. 109). However, as McIntosh (1990a, p. 25) has pointed out, little attention is currently being given to actually teaching mental computation in Australian primary classrooms. The relevance of this observation to Queensland was one focus of the survey reported in Chapter 4. It is only in some of the sourcebooks, which operationalise the current mathematics syllabus in Queensland, particularly in that for Year 5, that specific references to mental computation have been included. However, the philosophy of the Years 1 to 10 Mathematics Syllabus (Department of Education, 1987a) is not in conflict with the recommendations of mathematics educators concerning how mental computation should be taught. Nevertheless, the significant debate necessary for curriculum change in Australian, and particularly Queensland, schools has not yet occurred. For mental procedures to be given pre- eminence over written procedures, a revolution in the way teachers and parents view mathematics will need to be achieved. In consequence, this study provides a comprehensive summary of the state of knowledge about mental computation. Key aspects of mental computation within primary school curricula have been analysed from past, contemporary, and futures perspectives. Based on these analyses, recommendations concerning how mental computation could be incorporated into the Number strand of future mathematics syllabuses have been delineated (Chapter 5). It is believed that by having undertaken this research from various perspectives, contributions have been made towards creating an enriched context for debate about future modifications to the mathematics taught in Queensland primary schools. Consequently, this chapter provides (a) conclusions about mental

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computation across the range of time periods considered, (b) an outline of the implications these conclusions hold for syllabus development, and (c) an analysis of the aspects of mental computation that require further theoretical and empirical investigation (see Section 1.4.2). The data on which these conclusions, recommendations, and implications have been based were obtained from the review of the pedagogical, socio-anthropological and psychological literature relevant to mental computation (Chapter 2), and analyses of mental computation in Queensland primary schools from past and contemporary perspectives─through an examination of archival material (Chapter 3), and the analysis of a postal, self-completion questionnaire (Chapter 4).

6.2 Mental Computation: Conclusions

This study has analysed the beliefs and practices pertaining to the nature, function, and teaching methods associated with the mental calculation of exact answers beyond the basic facts, from theoretical and Queensland perspectives. The conclusions which have been drawn need to be interpreted within the confines of the historical and survey data obtained. The former were derived from syllabus documents, textbooks, and the recorded beliefs and opinions of Departmental personnel, primarily the annual reports of senior Departmental officers and publications of the Queensland Teachers' Union. Taken together, these documents have provided a comprehensive source from which an understanding of both the intent and practice related to mental arithmetic has been obtained for the period 1860-1965. As noted in Chapter 4, the data applying to 1965-1987 may only be considered suggestive of the beliefs and practices which prevailed during that period, as few respondents had taught under the 1964 Syllabus or the first edition of the Program in Mathematics (Department of Education, 1966-1968). However, with respect to the beliefs and practices related to the current syllabus, which was formally implemented in 1988, it was concluded that the numbers of valid cases were sufficient to permit meaningful conclusions to be drawn, despite the non-response rate being relatively high (see Section 4.3.2). Further, based on anecdotal evidence obtained from some Departmental Education Advisers (Mathematics) during 1996, it

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is probable that the pattern of beliefs and practices in evidence during 1993 largely remains unchanged. It was their opinion that mental computation had received negligible attention during inservice related to the Student Performance Standards conducted in 1994 and 1995, despite such calculation being emphasised in the Standards for Number (“Student Performance Standards,” 1994, pp. 28, 42, 58, 72). In presenting the conclusions which may be drawn from this study, it is intended to provide a summary of key similarities and differences between the beliefs and teaching practices related to mental computation currently advocated by mathematics educators, and those of Queensland teachers across the eras investigated. Consideration has been given to: (a) the emphasis placed on the mental calculation of exact answers, (b) the roles ascribed to such calculation, (c) the nature of these calculations, and (d) the approaches to teaching mental computation.

6.2.1 The Emphasis Placed on Mental Computation.

Computational competence is now believed to involve more than the routine application of memorised rules. It encompasses an expertise in higher-order thinking, a sound understanding of mathematical principles, and an ability to know when and how to use a variety of procedures for calculating (NCSM, 1989, p. 44), a belief supported by the Years 1 to 10 Mathematics Syllabus (Department of Education, 1987a, pp. 7-11, 15) and one pre-empted by the 1964 and 1966-1968 Syllabuses. Nonetheless, paper-and-pencil calculation using standard algorithms continues to be the major focus of the number work undertaken by Queensland children. Although the 1987 Syllabus explicitly emphasises computational estimation, recognition of the ability to calculate exact answers mentally is implicit. Estimating is specified as one of the mathematical processes to be developed (Department of Education, 1987a, p. 8), whereas procedures for calculating exact answers mentally are incorporated under Calculating─"calculating with addition, subtraction and multiplication using mental and calculator procedures, and written algorithms...as appropriate for the context" (Department of Education, 1987a, p. 15). This implies that for mental computation to be considered as the method of first resort (AEC, 1990, p. 109), it will require greater prominence, both in syllabus

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documents and in the manner in which calculative methods are taught. The recommendations outlined in Chapter 5 are designed to aid in the achievement of this goal. Prominence was given to the calculation of exact answers mentally in all Queensland syllabuses from 1860 to 1964, albeit under various headings, which included mental exercises, mental, mental work, oral work, mental and oral work, oral arithmetic and mental arithmetic. From that of 1904, each syllabus emphasised that "mental calculations should be the basis of all instruction" ("Schedule XIV,” 1904, p. 201). The 1914 and 1930 Syllabuses suggested that "new types of problems should invariably be introduced in this way" (Department of Public Instruction, 1914, p. 61; 1930, p. 31), whereas the 1952 Syllabus stressed that "all written work should be preceded by introductory oral exercises" (Department of Public Instruction, 1952b, p. 2). Despite the prominence given to mental arithmetic in the syllabus documents and the exhortations of District Inspectors of Schools for teachers to implement their intent, the generally low standard of mental calculation was most commonly attributed to its not receiving sufficient regular and systematic treatment during the period 1860-1965. Mental arithmetic appeared to be the bête noir of many teachers ("Mental Arithmetic,” 1927, p. 18), an outcome of the characteristics ascribed to it and of the ways in which it was taught. Nonetheless, it can be concluded that mental arithmetic received some emphasis by teachers during the period 1860- 1965. Somewhat paradoxically, although the Program in Mathematics (Department of Education, 1966-68, 1975) did not give mental arithmetic the prominence of earlier syllabuses, the survey data suggest that middle- and upper-school teachers continued to place importance on such calculation. However, this emphasis progressively decreased from 1969 to 1987 (see Table 4.13). Additionally, whereas approximately 53% of middle- and upper-school teachers agreed that the 1987 Syllabus places little emphasis on the development of the ability to calculate exact answers mentally (see Table 4.8), 89% indicated that they at least sometimes focused specifically on developing the ability to calculate exact answers mentally beyond the basic facts (see Table 4.12). However, in so doing, teaching practices classified as traditional in this study continue to be integral to the range employed

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(see Figure 4.4), thus placing limits on the degree to which children are developing flexible, idiosyncratic mental strategies.

6.2.2 Roles of Mental Computation

The various roles attributed to mental computation during the period being investigated are associated with its perceived usefulness for strengthening the mind, and with its social and pedagogical usefulness. Undertaking mental calculations is not only the simplest method for performing many arithmetical procedures, it is also the main form of calculation used in everyday life. However, specific recognition was given to this only in the 1952 and 1964 Syllabuses, with the former acknowledging that "oral arithmetic is more commonly used in after-school life than written arithmetic" (Department of Education, 1952b, p. 2), a view with which present-day school personnel concur (see Table 4.8). Nonetheless, the social usefulness of mental computation was recognised by the authors of textbooks used in Queensland schools from 1860. Park (1879), for example, stressed that mental arithmetic was a "subject of great practical importance" (p. 42), a conviction often reiterated by District Inspectors in their annual reports. Representative of these was that of Bevington (1926) who commented that "in domestic and mercantile transactions, in calculating about farms, &c., mental exercises are so frequent that it seems to be absolutely essential that children [should] be trained to calculate quickly and accurately" (p. 80). However, it is in relation to the pedagogical usefulness of mental computation, that distinctions can more clearly be drawn between the beliefs presently held by mathematics educators and those of the past. In contrast to past beliefs and practices, which focused on gaining answers speedily and accurately, it is now recommended by mathematics educators that the focus should be on the mental processes involved. Such a focus would allow mental computation to be used as a tool to facilitate the meaningful development of mathematical concepts and skills─to promote thinking, conjecturing and generalising based upon conceptual understanding (Reys & Barger, 1994, p. 31). Hence, mental computation is closely linked to the development of number sense, to gaining mathematical power, a power that would be enhanced where the gap between learning and using school

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mathematics and that used outside the classroom is bridged (Willis, 1990, p. 9). An emphasis on the processes involved in mental computation could provide that bridge. Folk mathematics is essentially mental (Maier, 1980) and often involves the manipulation of non-standard units using invented strategies (Lave, 1985, p. 173). Hence, in McIntosh's (1990a, p. 37) view, mental computation is the most readily available means by which an understanding of how numbers generally behave may be gained, a belief supported by many current school personnel (see Table 4.9). While occasional recognition was given to mental arithmetic as a means for "making scholars think clearly and systematically about number" ("Teaching Hints: Arithmetic,” 1908, p. 15), the primary pedagogical function of mental work during the period 1860-1965 was to familiarise students with the arithmetical operations prior to an emphasis on paper-and-pencil calculation. This mental-written sequence was encapsulated in the 1904 Syllabus by its stressing that "the pupils should be made familiar by mental exercises with the principles underlying every process before the written work is undertaken" ("Schedule XIV,” 1904, p. 201), a sequence that was embodied in the spiral nature of the 1930 Syllabus. However, the failure on the part of teachers to sufficiently model their mental arithmetic examples on the written work that was to follow was a regular criticism of District Inspectors in their annual reports. Mental work was also considered to be an effective means for cultivating speed and accuracy in new work and for the revision of the arithmetic procedures (Board of Education, 1937, p. 513; Cochran, 1960, p. 12; Mutch, 1916, p. 62). However, such a focus possibly contributed to the belief that mental arithmetic entailed the presentation of a series of often one-step examples, the focus of which was the gaining of correct answers, the traditional view of mental computation, as defined in this study. It is with respect to the role of mental arithmetic as a means for "improving the tenacity of the mind" (Wilkins, 1886, p. 40) that sharp distinctions may be drawn between past beliefs and those currently advocated by mathematics educators. Nonetheless, aspects of this traditional view would appear to remain in the minds of many Queensland teachers and administrators. Approximately 67% of respondents supported the view that "children's mental processes are sharpened by starting a mathematics lesson with ten quick questions to be solved mentally" (see Table 4.10).

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Such a view has as its origins the tenets of formal discipline which were espoused particularly in the 19th century. This theory held that the mind was composed of a number of distinct powers or faculties, including memory, attention, observation, reasoning and will (Kolesnik, 1958, p. 6). Late last century, Robinson (1882) had maintained that the value of giving complex calculations, beyond the requirements of the various syllabuses of the period, was "the formation of a power of concentrating all the faculties on the performances of an allotted task...[so that] the mind... [would] prove capable of any amount of labour upon other tasks" (p. 178). Such a belief resulted in mental arithmetic being considered essentially as the "working [of] certain hard numbers in the shortest time by the shortest method" ("Mental Arithmetic: A Few Suggestions,” 1910, p. 176). Despite the discrediting of these beliefs early in the 20th century, and statements by senior Departmental officers affirming their belief that there is no such thing as general mental training, and that learning in one subject cannot be transferred to another (Edwards, 1936, p. 16), teachers and inspectors retained their beliefs in the role of mental arithmetic as a means for quickening the intelligence, developing judgement, improving reasoning (Baker, 1929, p. 281; Mutch, 1924, p. 40) and for enhancing an individual's ability to concentrate on mathematical tasks (Martin, 1916, p. 135). The maintenance of these beliefs was supported by the influence of the English view of arithmetic, which represented arithmetic as logic (Ballard, 1928a, p. xi). Hence, mental arithmetic was judged to be the means by which children were trained to think and reason: "Intelligence in Arithmetic should be secured through the medium of mental exercises" (Bevington, 1925, p. 83), with accuracy in thinking and reasoning of paramount importance (Martin, 1920a, p. 81). Nevertheless, Burns (1973, p. 4) concluded that most teachers were primarily concerned with imparting factual knowledge. Hence, their concern was almost exclusively with memory.

6.2.3 The Nature of Mental Computation

Essential to the nature of mental computation, as now conceived, is Plunkett's (1979, p. 2) notion of mental strategies as being fleeting, variable, flexible, active, holistic, constructive, and iconic. Such strategies are not designed for recording, and

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require an understanding of the mathematical relationships embodied in the problem task environment. These features contrast with those of the standard paper-and- pencil algorithms. Such procedures are believed to be standardised, contracted, efficient, automatic, symbolic, general, and analytic. Further, they are not easily internalised and encourage cognitive passivity (Plunkett, 1979, p. 3). The use of mental strategies facilitates a focus on quantities rather than on symbols, thus permitting an individual to make meaningful alterations to the problems encountered and to work with quantities that can be manipulated more easily (Carraher et al., 1987, p. 94), the latter being those that are dependent upon an individual's store of conceptual knowledge in long-term memory (see Figure 2.3). The survey data indicated that Queensland teachers supported the notion that mental computation encourages children to devise ingenious computational short cuts and helps children to gain an understanding of the relationships between numbers (see Table 4.9). Such a notion supports the belief that mental computation bears a reciprocal relationship with conceptual and procedural knowledge (see Figure 2.3). It was also recognised that children who are proficient at mentally calculating exact answers use personal adaptations of written algorithms and idiosyncratic mental strategies (see Table 4.9). While it can be concluded that Queensland State school personnel tend towards holding nontraditional views about the nature of mental computation (see Figure 4.1), this conclusion needs to be tempered by the finding that approximately 40% of teachers believe that calculating exact answers mentally involves applying rules by rote (see Table 4.9). As early as 1887, District Inspector Kennedy had argued against mental arithmetic being viewed as the mere application of fixed rules (Kennedy, 1887, p. 82), rules that were generally listed at the end of the treatises on mathematics used by teachers. Park (1879) had previously stressed that where rules were to be taught they should not be "got by rote" (p. 43). Nonetheless, the period 1860-1965 was characterised by an importance placed upon the rote application of short methods of mental calculation, although it was only in the 1930 Syllabus, and its amendments in 1938 and 1948, that specific references were made to such calculations in schedules and syllabuses. That the 1952 Syllabus did not refer to such methods was a situation with which District Inspector Crampton (1956, p. 5) did not approve, maintaining that practical short methods should have been prescribed.

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Little emphasis appears to have been placed on permitting children to devise their own mental strategies. Nevertheless, teachers were occasionally encouraged in articles in the Queensland Teachers' Journal and the Education Office Gazette to permit children to invent short methods for themselves. Some encouragement was also contained in the 1930 Syllabus and the 1948 Amendments. Complementing practice in the use of short methods, Grade V children were to be encouraged to devise different solutions for particular problems, but only "after the pupils [had] been thoroughly exercised in any rule" (Department of Public Instruction, 1930, p. 41; 1948, p. 13). The 1952 Syllabus appears somewhat equivocal about the nature of oral arithmetic. While stating that "its application should not be limited to short methods or similar devices,” it also maintained that "the processes applied orally [that is, mentally,] are the same as those used in written operations" (Department of Public Instruction, 1952b, p. 2). It is the latter view of mental arithmetic that was emphasised for all grades in all schedules and syllabuses from that of 1904, albeit a view that was not often realised, as recorded by District Inspectors in their reports. Rather, teachers tended to view mental arithmetic as the presentation of a series of often unrelated oral questions, the aim of which was to obtain correct answers, speedily and accurately. Such a conviction was supported by the format of textbooks and articles on mental arithmetic in the Queensland Teachers' Journal. As Lidgate (1954) noted, "There [was] a general tendency to test rather than to teach Mental” (p. 2). This, combined with the practice of providing examples beyond the requirements of particular grades by teachers, District Inspectors and Head Teachers, eventuated in Greenhalgh (1947, p. 11) condemning the way in which mental arithmetic was taught for the nervous strain that was being placed on children. However, this was a practice that continued at least into the late-1950s. District Inspector Kehoe (1957) recorded in his 1956 Annual Report that "some teachers were puzzling their pupils in tables and giving them mental gymnastics by ingenious and complicated methods they have evolved for working, for example, extensions tables and measurement tables" (p. 2).

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6.2.4 Approaches to Teaching Mental Computation

Robert Reys et al. (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 to be an essential prerequisite to written computation, with proficiency gained through direct teaching. However, 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. In both approaches, the emphasis is placed on mental strategies. This contrasts with the traditional approach to the mental computation. Although also behaviourist in nature, it focuses on the speed and accuracy with which answers are obtained─mental arithmetic, as defined in this study. From 1860 to 1965 teachers were impelled by District Inspectors and syllabus documents to make mental arithmetic a part of every lesson. The aim was for it not to be taught in isolation (Department of Public Instruction, 1914, p. 61), but as an introduction to all written work─for teaching the principle of an operation, for promoting speed and accuracy at the mechanical stage, and for applying the particular operation in problem situations. Hence, mental examples were to be based on the written work that was to follow, with these carefully graded prior to a lesson. Although mathematics educators would agree that mental computation needs to be a part of every mathematics lesson, Rathmell and Trafton (1990, p. 160) assert that it should not be considered as a separate topic with a set of ordered skills. Rather, a focus on mental calculation should receive an on-going emphasis throughout all situations requiring computation, thus leading children to view mental methods as legitimate computational alternatives (B. J. Reys, 1985, p. 46). Nonetheless, it was argued in Section 5.4.1 that, in context with developing proficiency with mental computation in a mental-written sequence for each operation (Table 5.2), it was legitimate to highlight particular mental strategies for direct teaching (Tables 5.3-5.5). This is not to suggest that mental computation should be considered as a discrete topic. Rather it recognises that, for some children, the spontaneous invention of mental strategies may be difficult (Carroll, 1996, p. 4). Further, it also recognises that those who demonstrate low proficiency with mental computation may benefit from a teacher-directed focus on selected mental

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strategies (Cooper et al., 1996, p. 159). The strategies highlighted are those counting and heuristic strategies (Tables 2.2 & 2.4) considered accessible for most children in each of the three year-level clusters (Tables 5.3-5.5). These strategies contribute to and draw upon a student’s sense of number. As B. J. Reys et al. (1993, p. 314) have observed, mental computation can provide numerous opportunities for the development of mathematical thinking. However, it requires regular and systematic practice. District Inspectors of Schools prior to 1965 often exhorted teachers to provide additional opportunities for practice in mental arithmetic. Preferably this was to be undertaken for short periods in the mornings when children's minds were fresh (Baker, 1929; Drain, 1941, p. 2; “Teaching Hints,” 1908, p. 15). Lessons were characterised by rapid question delivery, with the profit generally considered to be in the number of questions answered correctly (Gladman, 1904, p. 215). Martin (1916) advocated that explanations should be kept to a minimum as a means for ensuring a "concentrating of the mind" (p. 135). This contrasts with current beliefs about teaching mental computation, the essence of which is the focus on assisting children to see how to calculate mentally. The object is to play with numbers, to explore their relationships (C. Thornton, 1985, p. 10). Hope (1985, p. 372) and Olander and Brown (1959, p. 109) have observed that proficient mental calculators exhibit a passion for numbers, which is reflected in the degree to which calculating mentally is practised. Elements of the traditional approach remained in teachers' repertoires following the introduction of the Program in Mathematics (Department of Education, 1967a, 1968) in Grades 4 to 7 during 1968 and 1969, albeit with decreasing emphasis during the use of this syllabus and its 1975 revision. The survey data analysed in Chapter 4 revealed that middle- and upper-school teachers continued to emphasise speed and the use of rules to calculate mentally during the periods 1969-1974 and 1975-1987 (see Table 4.14). Similar findings were found for those who were classroom teachers in 1993, with the majority also indicating that they, at least sometimes, gave several one-step questions and simply marked the answers as correct or incorrect (see Table 4.12). Essential to the retention of these approaches was the nature of the textbooks identified by teachers as ones used to support mental computation (see Table 4.18).

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These findings with respect to traditional practices were tempered by middle- and upper-school teachers indicating that they also used constructivist approaches to developing skill with mental computation. The majority disclosed that they, at least sometimes, allowed children to decide on which mental strategy to employ, explain and discuss their strategies, and relate methods for calculating beyond the basic facts to the thinking strategies used in their development (see Table 4.12). Overall, the teaching practices of contemporary classroom teachers tend to reflect nontraditional approaches to developing mental computation skills (see Figure 4.4). Nonetheless, this conclusion needs to be considered cautiously. Although there was some consistency between their beliefs about how mental computation should be taught and the teaching practices that they employed, many respondents appeared to be focused on basic facts rather than on more complex mental calculations (see Table 4.17). Further, their apparent acceptance of constructivist approaches may derive, not from an understanding of mental computation per se, but from an acceptance of the principles of teaching mathematics embodied in the 1987 Syllabus and its supporting documents. "Discussion between the teacher and students and between the students themselves,” for example, is listed as one of six recommended approaches for teaching mathematics (Department of Education, 1987b, p. 3).

6.3 Implications for Decision Making Concerning Syllabus Revision

This study, as described in Section 1.3, has significance for (a) pedagogy, (b) the use of mathematics, (c) curriculum development, and (d) Queensland educational history. Key issues related to the nature and role of mental computation have been identified, together with an analysis of suggested teaching approaches. Essential to this analysis is the recommended sequence for introducing mental, written and technological methods for calculating (Table 5.2), and the proposed sequence for focusing on particular mental strategies during particular year-level periods (Tables 5.3-5.5). Their implementation, in context with teachers gaining an understanding of the breadth of mental strategies used by proficient mental

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calculators, should assist children to come to view mental computation as the method of first resort. More generally, this study provides a comprehensive source of data for mathematics educators, mathematics teachers and curriculum developers. By analysing mental computation from an historical perspective, cognisance may be given to the nature of past curricula and their implementation during future revisions to mathematics syllabuses. Syllabus development and implementation, however, need to be guided by research, in context with informed debate within society and within schools, in particular. Critical to the latter is adequate professional development for teachers of mathematics.

6.3.1 Fostering Debate about Computation

For children to become proficient calculators using a range of mental, technological, and paper-and-pencil strategies requires that syllabus developers, teachers, and parents become familiar with the issues surrounding computational methods. The decision by the Minister for Education (Office of the Minister for Education, 1996, p. 1) not to reintroduce Student Performance Standards into Queensland schools in 1997 has removed a potentially potent platform from which a focus on mental computation, and computational procedures in general, could have been fostered. Whereas an emphasis on mental strategies, and appropriate methods for teaching, can be incorporated within the approach to teaching mathematics embodied in the Years 1 to 10 Mathematics Syllabus (Department of Education, 1987a), the implementation of a revised sequence for introducing computational procedures, as proposed in Table 5.2, requires a long period of professional and public debate. Teachers, parents and employers will need to be convinced that a reduced emphasis on the standard written algorithms accompanied by an increased emphasis on non-standard mental, calculator and paper-and-pencil strategies should result in increased computational proficiency. Teachers will require extensive inservice to support the implementation of a mental-written computational sequence. Managing learning while allowing children to calculate using self-generated written strategies will require many teachers to broaden their repertoire of procedures for organising for mathematics learning.

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Conferencing techniques used to guide the development of children's writing may need to be adapted for the mathematics classroom. The use of self-generated strategies, whether mental, technological or written, individualises learning and therefore instruction, particularly where the goal of instruction is to promote an individual's ability to think creatively and independently. As revealed in Chapter 4, Queensland teachers believe that it is important for inservice on mental computation to be made available (see Table 4.15). It is through such professional development that teachers could become acquainted with the range of issues surrounding the nature, sequence, and role of mental, technological, and written computational procedures. Queensland teachers, while retaining many traditional beliefs about mental computation also hold some classified as nontraditional in this study (see Figure 4.1), an orientation conducive to their accepting many of the beliefs about the nature and role of mental computation currently espoused by mathematics educators. However, this acceptance does not at present extend to a change in the traditional sequence for introducing computational procedures─approximately 65% of teachers disagreed with the proposition that an emphasis on written algorithms needs to be delayed so that mental computation can be given increased attention (see Table 4.10). Skager and Weinberg (1971) suggest that an analysis of past syllabuses and their implementation enables curriculum developers to "know where [they] have been" (p. 50). The analysis presented in this study (Chapter 3) revealed that a frequent criticism of the various syllabuses from 1860 was that their implementation was usually accompanied by limited inservice opportunities for teachers, a key factor in teachers not teaching mathematics in the spirit intended. Teacher inservice, therefore, is crucial to effective syllabus implementation. However, as Blakers (1978) has pessimistically cautioned: "With virtually every primary school teacher teaching mathematics, the problems of adequate inservice preparation for significant changes at the primary level are well nigh insoluble" (p. 153). Nonetheless, if children are to gain mastery over of a range of flexible computational techniques and to learn mathematics that is significant and of value to their individual success, in both private and professional endeavours, it is essential that an emphasis is placed on facilitating teacher change with respect to their beliefs and practices related to the calculations of both exact and approximate answers. This

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needs to occur in the wider context of reanalysing their beliefs about mathematics and mathematics teaching. In common with the United States' Curriculum and Evaluation Standards for School Mathematics (NCTM, 1989, p. 9), a concern to encourage teachers to adopt conceptual rather than a calculational orientation (Thompson, Philipp, P. W. Thompson, & Boyd, 1994, p. 86) for teaching mathematics is embodied in the Years 1 to 10 Mathematics Syllabus (Department of Education, 1987a), the Years 1 to 10 Mathematics Teaching, Curriculum and Assessment Guidelines (Department of Education, 1987b) and in A National Statement on Mathematics for Australian Schools (AEC, 1990). Such an orientation is in harmony with the current view of mental computation and how it should be taught. Thompson et al. (1994) contend that "conceptually oriented teachers focus children toward a rich conception of situations, ideas and relationships among ideas" (p. 86)─key factors in developing and being able to apply flexible mental strategies. Making this shift requires a depth of reflection by teachers on the images and outcomes of mathematics and mathematics teaching. The Professional Standards for Teaching Mathematics (NCTM, 1991, p. 160) emphasises that opportunities need to be provided for teachers to "examine and revise their assumptions about the nature of mathematics, how it should be taught, and how students learn mathematics" (p. 160). Of significance for the implementation of new syllabuses, is the recognition that any recommended changes to teaching practices will not substantially impact upon what occurs in schools should the examination and revision of these assumptions not occur (D'Arcy, 1996, p. 1). This is particularly relevant during periods of system initiated change, which is usually perceived as being imposed upon teachers rather than teacher initiated. Opportunities for teachers to become familiar with the issues related to mental computation need to occur in context with viewing teacher change as a process of individual growth or learning (Clarke & Hollingsworth, 1994, pp. 158-159), a process encapsulated by Clarke and Peter’s (1993, p. 170) Dynamic Model of Professional Growth. This multidimensional model recognises, in accordance with that of Guskey (1985, p. 58), that significant changes to teacher beliefs and attitudes only occur after valued learning outcomes have arisen from experimentation with changed classroom practices. Of significance to gaining teacher acceptance of the recommended changes to teaching computation is the recognition that the desired

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learning outcomes, particularly the emphasis on self-generated mental and paper- and-pencil strategies, are in sharp contrast to the current emphasis on standard written algorithms. Although teachers adapt rather than adopt new teaching practices (Clarke, 1994, p. 46; Fennema & Franke, 1992, p. 162), Doyle and Ponder (1977, p. 8) caution that classroom practices radically different from common practice are unlikely to be successfully implemented. Additionally, it needs to be recognised that teachers have passions about their classroom practices and about the expectations for change placed upon them (Weissglass, 1994, p. 78). Hence, for the desired changes to occur, it is essential that teachers are assisted in the development of a clear vision (Lovitt et al., n.d., p. 2), in practical terms, of how a focus on mental and written idiosyncratic strategies within a mental-written sequence may impact upon their current teaching practices. This implies that an essential component of any professional development program is the facilitation of teacher reflection, a process for which time is typically limited during the normal school day (Clarke, 1994, p. 40). On-going support, including that in the classroom, is critical to the change process (Guskey & Sparks, 1991, p. 73; Joyce & Showers, 1980, p. 384). As Thompson et al. (1994, p. 90) note, once committed to a conceptual orientation, many teachers lose their reliance on resources such as worksheets and textbooks as well as on their repertories of familiar teaching practices. Such an experience is threatening due to their loss of a sense of efficacy, and consequently is a major obstacle to change. The design of professional development programs, therefore, needs to have as a key component procedures which recognise that success in making the desired changes of itself brings about a reconceptualisation of a teacher's sense of efficacy (Smith, 1996, p. 394). This is a view supportive of the belief that improvements in student learning arising from changed practices are critical to the adoption of the beliefs and attitudes which underpin those practices. For teachers to become committed to the recommended changes to the approach to developing computational skill, these beliefs and attitudes must become an essential part of a teacher’s professional belief system (Clarke & Peter, 1993, p. 174). This necessitates ongoing classroom assistance (Guskey, 1985, p. 59), and enhanced collegial relationships to provide emotional support during the change process. Change is not only a source of anxiety for teachers arising from the threat to their sense of efficacy, but also may occur in context with other sources of anxiety

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with which they may be faced (Weissglass, 1994, p. 71)─sources such as those related to an increased emphasis on school-based management, and using technology in classrooms, the latter being a source that has implications for what mathematics is to be learned in school, and how it is to be taught.

6.4 Recommendations for Further Research

The analyses of the various aspects of mental computation presented in this study were structured around a number of hypothesised models. Each of these requires refinement and validation, processes that are critical to changing teacher beliefs and attitudes about mental computation within the wider context of how computation is viewed. Hence, research needs to be directed towards:

• The model of the components of mental computation (Table 2.1). This model, based on that for computational estimation delineated by Sowder and Wheeler (1989, p. 192), was developed to provide a mechanism for discussing the concepts and skills essential to the mental calculation of exact answers. • The categorisation of mental strategies presented in Tables 2.2 to 2.4. This categorisation needs further consideration to determine its appropriateness for providing a framework for synthesising the disparate data from the range of studies relevant to identifying and defining mental strategies. Of relevance to their categorisation is the need for researchers to standardise the labels ascribed to particular approaches to facilitate the analysis of the data and the drawing of meaningful conclusions. • The alternative sequence for introducing computational procedures, as represented by Figure 2.5 and Table 5.2. Although it is believed that an enhanced understanding of arithmetic can be developed through a curriculum that focuses on mental computation, computational estimation, and calculators, with a concomitant reduction in the emphasis on written computation (MSEB & NRC, 1990, p. 19), Barbara Reys (1991, p. 11) points out that it is essential that the effects of introducing such changes are

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identified and analysed─effects not only related to student learning, but also those related to teacher and parent attitudes. • The hypothesised mental strategies syllabus components for the four operations beyond the basic facts (Tables 5.3-5.5) that were based on the above models. Prior to the incorporation of these components in any future mathematical syllabus or professional development programs, research, similar to that by Gracey (1994), for example, needs to be undertaken to determine their appropriateness─to the child, to the development of proficiency with mental computation, and to the models, and data which underpin them.

By undertaking such research, insights would be gained into issues that impinge on the mental strategies that have been recommended for focussed teaching (see Chapter 5). These issues include:

• The strategies used by proficient mental calculators to divide mentally. • The nature of difficulties experienced in balancing strategy development through using elements of the behaviourist and constructivist approaches. • The way in which self-generated mental strategies mesh with formally taught algorithms. • The level of place value understanding required for particular strategies. • The demands of particular strategies on working memory. • The effects arising from the need to regroup or carry, and the size, and type of numbers being manipulated. • The consequences arising from using visual and oral stimuli, and examples presented in and without context. • The appropriateness of various models which may be used to represent the calculative situation. • The effectiveness of discussion as a means for encouraging strategy growth, and how it should be managed within the classroom.

A consequence of this research would be an enhancement of the aims of this study, namely to analyse key aspects of mental computation and to formulate

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recommendations to inform future revisions of the Number strand in mathematics syllabuses for primary schools. The implementation of the outcomes of this research should therefore strengthen the possibility that mental computation will receive priority in mathematics classrooms, and become the method of first resort in situations requiring calculations to be made.

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REFERENCES

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Wolters, G., Beishuizen, M., Broers, G., & Knoppert, W. (1990). Mental arithmetic: Effects of calculation procedure and problem difficulty on solution latency. Journal of Experimental Child Psychology, 49, 20-30.

Woodgate, W. J. L. (1955). Report as district inspector. 1954. Queensland State Archives. EDU/A738, File. No. 52837/55.

Woolfolk, A. E. (1987). Educational psychology (3rd ed.). Englewood Cliffs: Prentice-Hall.

Wyeth, E. R. (1955). Education in Queensland: A history of education in Queensland and in the Moreton Bay District of New South Wales (Australian Council for Educational Research Series No. 67). Melbourne: Australian Council for Educational Research.

"X+Y=Z". (1945). The myth of mathematics. Queensland Teachers' Journal, 50(5), 14-15.

Yackel, E., Cobb, P., Wood, T., Wheatley, G., & Merkel, G. (1990). The importance of social interaction in children's construction of mathematical knowledge. In T. J. Cooney (Ed.), Teaching mathematics in the 1990s (pp. 12-21). Reston: National Council of Teachers of Mathematics.

"Z". (1879, June 7). Letter to the editor. The Australian, p. 768.

Zepp, R. (1976). Algorithms and mental computation. In M. Suydam & A. R. Osborne (Eds.), Algorithmic learning (pp. 97-104). (Educational Resources Information Centre Document, Microfiche Edition, ED 373 960).

419

APPENDIX A

Summary of Mental Arithmetic in Queensland Mathematics Schedules and Syllabuses (1860-1964)

A.1 1860 Schedule

Minimum attainments for children enrolled one Quarter in each class:

First class: Perform mentally all the elementary arithmetical operations with numbers, not involving a higher result than 30.

Second class: Know the multiplication tables.

Sequel to second class: Know the most useful arithmetical tables.

Third class: Know the easier rules of mental arithmetic.

Fourth class: Know the rules of mental arithmetic.

Fifth class: (Be acquainted with the whole Theory of Arithmetic, and its application, to the Mensuration of Superficies1.)

A.2 1876 Schedule

Minimum attainments2 required from pupils for admission into each class. (Specific details were not provided for the First class in 1876. The following were issued by Under Secretary Anderson in April 1879):

1 Mensuration of surfaces.

2 The minimum attainments for admission into any class are the maximum attainments expected from the class below it. The schedule can therefore be regarded as a Programme of Instruction for pupils attending Primary Schools. The work to be gone through in any class (the Fifth Class excepted) will be found detailed in the column with the name of the class next above it.” (Regulations of the Department of Public instruction in Queensland. (1985). Queensland Government Gazette, XXXVI(30), p. 490). 420

First class (Division I): To add mentally two figures to a result not greater than 10.

First class (Division II): To add mentally three numbers to a result not greater than 20.

First class (Division III): To add mentally any numbers to a result not greater than 30.

(General Instructions for the guidance of teachers. April 1879, Queensland State Archives, EDU/A602.)

Lower second class: Perform mental addition up to a result not higher than thirty.

Upper second class: Perform easy mental operations in addition and subtraction.

Third class: Perform mentally easy operations in the simple rules.

Fourth class: Perform mentally easy operations on the compound rules and reduction, including bills of parcels, rectangular areas, and other practical problems.

Fifth class: Perform mentally simple operations in proportion, practice, vulgar fractions and simple interest, including miscellaneous problems.

A.3 1891 Schedule (Effective 1 January 1892)

First class (Course 2 years) (In infant schools 3 years): To add mentally numbers of one figure to a result not greater than 40, and to subtract any number of one figure from any number of two figures.

Standards of proficiency:

1st half-year of enrolment (9 months - Infant Schools): To add mentally numbers applied to objects to a result not greater than 10.

2nd half-year of enrolment: To add mentally numbers applied to objects to a result not greater than 20.

3rd half-year of enrolment: To add mentally numbers to a result not greater than 30, and to separate numbers not greater than 30 into two component parts. 421

4th half-year of enrolment: To add mentally numbers to a result not greater than 40; to lessen mentally any number of two figures by any number of one digit.

Second class (Course 1½ years): To perform mentally operations in the following rules: Addition, subtraction, multiplication and division of abstract numbers, and addition and subtraction of money.

Standards of proficiency:

First half-year: Mental - applications of addition, subtraction, and multiplication.

Second half-year: Mental - applications of addition, subtraction, multiplication and division.

Third half-year: Mental - to add any two sums of money (excluding parts of a penny) to a result not greater than £1; to subtract any sum of money as before from £1; to express £1 by combinations of current coins; to subtract any sum of money as before from £1; to express £1 by combinations of current coins.

Third class (Course 1½ years): To perform mentally operations in the following rules: The "more useful" weights and measures; Addition, subtraction, multiplication, division, and reduction of money; Bills of parcels.

Standards of proficiency:

First half-year: Mental - to find the price of a dozen the price of one being given, and conversely; easy shopping transactions.

Second half-year: Mental - to find the price of a score or a gross, the price of one being given, and conversely; easy shopping transactions.

Third half-year: Mental - shopping transactions.

Fourth class (Course 1½ years): To perform mentally operations in the following rules: The unitary method, vulgar fractions, practice, decimals, proportion.

Standards of proficiency:

First half-year: Mental - exercises in easy vulgar fractions and the unitary method. Second half-year: Mental - exercises in easy decimals and practice. 422

Third half-year: Mental - exercises in vulgar fractions, decimals and proportion.

Fifth class (Course 1½ years): To perform mentally operations in the following rules: Interest; discount; square root; percentages. Revision of the course for first, second, and third classes.

Standards of proficiency:

First half-year: Mental - easy exercises in interest, discount, and square root;

Second half-year: Mental - easy exercises in percentages.

Third half-year: Mental - easy exercises in the preceding rules.

Sixth class (Course 2 years): Revision of the course for fourth and fifth classes, with mental exercises.

A.4 1894 Schedule

Minor changes in standards of proficiency from 1892 resulting from the First and Sixth Class being reduced from 2 years to 1½ years.

First class (Course 1½ years)

Standards of proficiency:

First half-year: To add mentally numbers applied to objects to a result not greater than 20.

Second half-year: To add mentally numbers applied to objects to a result not greater than 30, and to separate numbers not greater than 30 into two component parts.

Third half-year: To add mentally numbers to a result not greater than 40, and to lessen mentally any number of two figures by any number on one figure.

423

A.5 1897 Schedule

First class (2 years): To add mentally numbers of one figure to a result not greater than fifty.

Standards of proficiency:

First half-year: To add mentally numbers applied to objects to a result not greater than 10.

Second half-year: To add mentally numbers applied to objects to a result not greater than 30.

Third half-year: To add mentally to a result not greater than 40.

Fourth half-year: To add mentally numbers to a result not greater than 50.'

Second class (1½ years): To perform mentally operations in addition, subtraction, multiplication, and division of abstract numbers.

Standards of proficiency:

First half-year: Mental - applications of addition and subtraction.

Second half-year: Mental - application of addition, subtraction, and multiplication.

Third half-year: Mental - application of the four simple rules.

Third class (1½ years): Mental operations in the rules for reduction, addition, subtraction, multiplication, and division of money and easy bills of parcels.

Standards of proficiency:

First half-year: Mental - easy practical applications of reduction, addition, and subtraction of money (mechanical operations only).

Second half-year: Mental - easy practical applications of reduction, addition, subtraction, multiplication, and division of money (mechanical operations only).

Third half-year: Easy problems in money, mentally. 424

Fourth class (1½ years): Mental operations in the rules for reduction, addition, subtraction, multiplication, and division of the more useful weights and measures; easy vulgar and decimal fractions; simple practice and simple proportion.

Standards of proficiency:

First half-year: Mental - exercises in reduction, addition, subtraction, multiplication, and division of the more useful weights and measures.

Second half-year: Mental - exercises in easy vulgar fractions.

Third half-year: Mental - exercises in simple practice, and simple proportion.

Fifth class (1½ years): Mental arithmetic3. (Compound practice and compound proportion; vulgar and decimal fractions; interest and discount; square root; mensuration of the parallelogram, triangle, and circle)

Standards of proficiency (Fifth Class):

First half-year: Mental arithmetic. (Compound practice, and compound proportion; square root; mensuration of the parallelogram)

Second half-year: Mental arithmetic. (Vulgar and decimal fractions, with applications to concrete quantities; mensuration of the triangle)

Third half-year: Mental arithmetic. (Simple and compound interest; discount; mensuration of he parallelogram, triangle and circle)

Sixth class (1½ years): Mental arithmetic. (Percentages; miscellaneous problems; cube root; mensuration of plane surfaces and solids)

3 For the fifth and sixth classes, the mental arithmetic to be taught is not directly specified. Following the specification of other work, which is presented here in brackets, the schedule simply states "mental arithmetic". 425

Standards of proficiency:

First half-year: Mental arithmetic. (Profit and loss; miscellaneous problems, with special reference to vulgar fractions;, cube root; mensuration of the polygon, prism, and cylinder)

Second half-year: Mental arithmetic. (Stocks; miscellaneous problems, with special reference to decimal fractions; mensuration of the cone and pyramid)

Third half-year: Mental arithmetic. (Misc. problems; mensuration of planes and solids)

A.6 1902 Schedule (Minor changes from 1897)

Fifth class (Course 1½ years. Age 11½ - 13 years): Mental arithmetic4 (Commercial arithmetic; compound practice and compound proportion; vulgar and decimal fractions; interest and discount; square root; Longman's Junior School Mensuration, Part I.)

First half-year: Mental arithmetic. (Commercial arithmetic; compound practice, and compound proportion; square root; mensuration - Longman's Junior School Mensuration, Chapters I to III)

Second half-year: Mental arithmetic (Commercial arithmetic; vulgar arithmetic and decimal fractions, with applications to concrete quantities; mensuration, Longman’s, Chapters I to V)

Third half-year: Mental arithmetic. (Commercial arithmetic; simple and compound interest; discount; mensuration, Longmans', Chapters I to VII)

Sixth class (Course 1½ years. Age 13 - 14½ years): Mental arithmetic (Commercial arithmetic; percentage; cube root; Longman's Junior School Mensuration, Parts I and II)

First half-year: Mental arithmetic (Commercial arithmetic; profit and loss; cube root; mensuration - Longman's, Chapters I to IX)

4 For the fifth and sixth classes, the mental arithmetic to be taught is not directly specified. Following the specification of other work, which is presented here in brackets, the schedules simply states "mental arithmetic". 426

Second half-year: Mental arithmetic (Commercial arithmetic; stocks; mensuration - Longman's, Chapters I to X)

Third half-year: Mental arithmetic (Commercial arithmetic; mensuration - Longman's, Chapters I to XI)

A.7 1904 Schedule (Effective 1 January 1905)

First class (Course, 2 years. Age 5-7 years): Mental exercises in addition and subtraction to 50.

Second class (Course 1½ years. Age 7-8½ years): Mathematics - Mental Work. Concrete exercises involving the four simple operations, and falling within the range of the pupils' experience. Simple factors. Exercises in finding ½, ¼, 1/8 of given quantities and numbers. Construction by pupils of multiplication table to 12 times 12. The money tables. Measurement with foot-rule in yards, feet, inches, and halves, fourths, and eighths of an inch. The symbols of operations required in addition, subtraction, multiplication, and division, and their use in representing the processes employed in concrete exercises.'

Third class (Course 1½ years. Age 8½-10 years): Mathematics. - Mental and Oral Work. measures of length (yard, feet, and inches), weight (avoirdupois), capacity (gallons, quarts, pints), and time. Concrete applications of the fractions ½, 2/3, 3/4, 11/12. Concrete exercises in these and in domestic accounts and simple business transactions. Measurements of objects and distances about the school premises. Symbols of operation - their use continued

Fourth class (Course 1½ years. Age 10 to 11½ years) Mathematics - Mental and Oral Work. Tables of length, area, weight, capacity, and time. Mental operations in the practical use of these and in the extended use of fractions, including decimal fractions. Measurements of furniture, rooms, school buildings, and play ground continued. Oral statement of processes employed in written work of class. Ratio. Symbols of operation - their use continued.

Fifth class (Course 1½ years. Age 11½ - 13 years): Mathematics - Oral and Mental Work. Mental operations in common business transactions of wider range than in the Fourth Class. Easy calculations in area. Exercises involving the use of simple fractions, decimals, and easy percentages. percentages applied to interest, ordinary retail discount, profit and loss, and proportion. Cubic measure.

Sixth class (Course 1½ years. Age 13 - 14½ years) 427

Mathematics - Oral and Mental Work. As before, with the addition of easy transactions in stocks. Easy operations in algebra and geometry, at the discretion of the teacher.

A.8 1914 Syllabus (Effective 1 January 1915)

First class (Course, 2 years. Age, 5-7 years): Mental exercises in addition and subtraction to 50.

First half-year: "Work should be exclusively oral, except that children should learn to know and to make the digits including 0."

Second half-year: Add mentally any numbers to a result not greater than 19, and to reduce any number not greater than 19 by any smaller number.

Third half-year: Mental addition and subtraction should involve, answers included, numbers no larger than 30.

Fourth half-year: Mental exercises in addition and subtraction to 50.

Second class (Course, 1½ years. Age, 7-8½ years): Mental Work. Concrete exercises involving the four simple operations, and falling within the range of the pupils' experience. Simple factors. Exercises in finding ½, ¼, 1/8 of given quantities and numbers. Construction by pupils of multiplication table to 12 times 12. The money tables and to know the value of current coins.

Measurement with foot-rule in yards, feet, inches, and halves, fourths, and eighths of an inch. The symbols of operations required in addition, subtraction, multiplication, and division, and their use in representing the processes employed in concrete exercises.'

Third class (Course, 1½ years. Age, 8½-10 years): Mental Work. Measures of length (yard, feet, and inches), weight (avoirdupois), capacity (bushels, pecks, gallons, quarts, pints), and time. Concrete applications of halves, thirds, fourths, eights, twelfths. Exercises in domestic accounts and simple business transactions. Measurements of objects and distances about the school premises; weighing common objects and measuring liquids, if practicable.

Fourth class (Course, 1½ years. Age, 10 to 11½ years): Mental Work. Tables of length, area, weight, capacity, and time. Mental operations in the practical use of these and in the extended use of fractions, including decimal fractions. Measurements of furniture, rooms, school buildings, and playground. Ratio as expressed by fractions and decimals.

Fifth class (Course, 1½ years. Age, 11½ - 13 years) 428

Oral Work. Mental operations in common business transactions of wider range than in the Fourth Class. Easy calculations in areas. Exercises involving the use of simple fractions, decimals, and easy percentages, interest, ordinary retail discount, profit and loss, and proportion. Cubic measure.

Sixth class (Course, 1½ years. Age, 13 - 14½ years) Oral Work. As before, with the addition of easy transactions in stocks. Easy operations in algebra and geometry, at the discretion of the teacher.

A.9 1930 Syllabus

Preparatory Grade: Mental arithmetic not mentioned specifically.

Grade I: Mental. (a) Exercises in addition, subtraction, and multiplication with numbers and quantities within the pupil's range. Very simple examples in division.

Notes: • Work in mental should precede and be preparation for written. The maximum amount of work should be thrown upon the pupils. An oral statement of the various steps to be followed in the solving of each mental problem should be given by the pupils • Suggested steps: (1) Statement of problem, (2) reasons for successive operations (3) rules employed, and (4) actual working. By this method teachers able to discover individual weaknesses. • The exercises should be well graded and suited to the average intelligence of class. Promiscuous work is of little value. One difficulty at a time should be mastered and constant revision is essential. • Exercises in addition and subtraction should involve numbers to 99. In multiplication, deal with multipliers to 6.

(b) To find ½, ¼ and 3/4 of numbers and quantities

Notes: • Applied to things, measurement of size or value; to numbers. • Recognition of parts should follow practical exercises in dividing and measuring things, and exercises in finding fractions of numbers should follow after practice in tables, thus: 4 times 6 = 24; ¼ of 24 = 6; 2/4 or ½ of 24 = 12; 3/4 of 24 = 18.

(c) Further practice in buying, selling, and giving change with sums to 1s. Component parts of 1s. taken two at a time, as 5d. and 7d., etc. Coin equivalents of 1s. in current coins.

Notes: • Tender a shilling for a pound of sugar at 5d. and loaf of bread at 6d. Purchase a notebook at 3d., a 1d. stamp, and a lead pencil at 2d. out of a shilling, &c.

429

Grade II: Oral work. (a) Finding half, quarter, and eighth of numbers and quantities

Notes: • Introductory exercises in paper-folding. • Fractional parts could be taught in connection with multiplication and division tables, e.g. 8 times 12 = 96, then 12 = 1/8 of 96 • Halves and fourths of an inch─taught and applied to measurements • "Half" and "quarter" should also be applied to "Time".

(b) Application of multiplication to easy one-step reduction of money and of the weights and measures dealt with in the Tables (quarts, pints, pounds, minute, hour day week, inches, feet, yard).

Notes: • For example, 1 quart = 2 pints; how many pints in 3 quarts? • Shillings to pence and pence to shillings (whole numbers of shillings only)

Notes: • Shopping transactions as suggested in previous grades. (d) Simple exercises in buying and selling with current coins up to £1.

Notes: • Spending a florin, a shilling, and a sixpence = total. Spending a two shilling piece and a shilling out of £1. (Actual or imitation coins could be used.)

(e) Easy problems in the four simple rules, with numbers within the pupils' range.

Notes: • These exercises should be a preparation for the written work.

Grade III: Oral Work. (a) Fractions; applications of fractions (one-half, one-third, one-quarter, etc) to numbers, money, and quantities. 430

Notes: • The exercises outlined in the previous grade could be continued and extended to harder examples in their application. These exercises are intended only as an introduction to the study of vulgar fractions at a later stage. • Resolving easy numbers such as 24 and 36 to prime factors as preparation for the work in fractions and cancelling.

(b) Simple mental problems based on the tables.

Notes: • These exercises are intended to show the practical application of the tables, and to serve as an introduction to the compound rules.

Notes: • The material for these exercises will be supplied by ordinary household accounts, as for example, the butcher's, the baker's, the grocer's, or the draper's bills.

Grade IV: Oral Arithmetic. (a) Exercises involving the use of vulgar fractions (denominators not to exceed 12) and of decimals to tenths.

Notes: • Fractional parts will now include fractions such as 3/8, 5/6, 11,12, 4/9, 6/7, 3/5 applied to numbers, money, and to tables - e.g. 3/8 of £1; 7/10 of £1; 3/7 of 1 quarter (in lb); • Decimals - .7 of £1; .9 of 1 ton, etc.

(b) Mental exercises based on the compound rules, including household and shopping transactions familiar to the pupils.

Notes: • Shopping transactions continued, and in addition, exercises as preparation for the written work in weights and measures.

Grade V: Oral Arithmetic. (a) Exercises involving the use of vulgar and decimal fractions

Notes: • Simple exercises involving the four rules as introduction to the written work, e.g. ½ + 1/3; 3/8 + 1/4; ½ of 8/9; .2 + .3 + .4; .8 ÷ 2; 2 ÷ .5. • The conversion of easy vulgar fractions to decimals and vice versa: ½ = .5; ¼ = .25; 7/8 = .875; 3/5 = .6, etc.

(b) Ratio expressed as fractions and decimals 431

Notes: • Preparatory exercises as introduction to Simple Proportion, e.g. 1 florin to £1 = 1/10 or .1; 1 furlong to 1 mile = 1/8 or .125; 1 rood to 1 acre = ¼ or .25; 1 ton 10 cwts. to 2 tons 10 cwts = 3/5 or .6. etc. (See Practical School Method, Cox & MacDonald, 269-171)

Notes: • Variety of mental exercises involving use of simple aliquot parts is essential.

(d) Mental exercises dealing with money, wights and measures, vulgar and decimal fractions (applied to concrete quantities), simple proportion. Finding areas of squares and rectangles.

Notes: • Preliminary work in mensuration should be confined entirely to oral and mental exercises.

(e) Practice in working by short methods (e.g. "dozens" and "scores" rules might be applied).

Notes: • After the pupils have been thoroughly exercises in any rule, short methods of calculation could be introduced. To encourage initiative, different solutions of the same problem might be required from the pupils. The 'dozens' and 'scores' rules might be applied.

Grade VI (1st Form, Intermediate Schools): Oral Work. (a) Exercises preparatory to the written work, including operations in simple business transactions of a wider range than those taken in Grade V.

Notes: • The work in this Grade will extend that of Grade V., and will include exercises in simple proportion, percentages, and simple interest; squaring and taking the square root; exercises preliminary to written work in Mensuration. While written work in Practice is not required, simple exercises, formerly regarded as questions in this rule, should be given and worked by appropriate methods. These will include problems in use of aliquot parts in such examples as - 18 books at 2s 6d each; 40 sheep at £1 12s. 6d. each.

Grade VII (2nd Form, Intermediate Schools): Oral Work. (a) Mental exercises preparatory to the written work in arithmetic and mensuration.

(b) Practice in short methods of calculation.

Notes: 432

• Pupils should be given practice in devising short cuts and easy methods.

A.10 1938 Amendments

Preparatory 1 and 2 (One Year): Mental exercises to 10. Easy problems.

Preparatory 3 (Six months): Mental exercises, including easy problems in numbers of 19.

Preparatory 4 (Six months): Mental exercises to cover: Count to 100 in ones, twos, fives, tens; Notation and numeration to 99; Addition tables to 99; Subtraction tables to 99; Halving and doubling to 100; Recognise coins - half-penny, penny, threepence, sixpence, shilling; The foot rule (inches only); Symbols +, -, and = ; Meaning of terms: pair, couple, dozen, score. Easy problems.

Grade I: Mental. (a) Exercises in addition, subtraction, and multiplication with numbers and quantities within the pupils' range. Very simple examples in division.

Notes: • Work in mental should precede and be a preparation for the written work. The maximum amount of work should be thrown upon the pupils. • The exercises should be well graded and suited to the average intelligence of the class. one difficulty at a time should be mastered, and constant revision is essential. Exercises in addition and subtractions should not involves numbers beyond 99. • At every stage the mechanical work should be applied to easy mental problems. teachers should not depend exclusively on text-books, but should prepare their own examples adapted to local conditions and needs.

(b) Fractions: To find one-half and one-quarter.

Notes: • Applied to things, measurement of size or value, and to numbers. • Recognition of parts should follow from practical exercises in dividing and measuring things, and exercises in finding fractions of numbers should follow after practice in tables - thus, 4 times 6 = 24; ¼ of 24 = 6.

(c) Further practice in buying, selling, and giving change with sums to 1s.; Component parts of 1s. taken two at a time as 5d. and 7d. etc.; Coin equivalents of 1s. in current coins. No farthings.

Grade II: Oral Work. As for 1930 syllabus.

Notes (Additional): 433

• Material for mental problems will be found in daily experience of children in home and school. In framing questions, teachers should be careful that the price they place on commodities should approximate those which obtain locally. • Examples should not exceed in difficulty those given in the syllabus.

Grade III: Oral Work. (a) Fractions; applications of fractions (one-half, one-third, one-quarter, etc. to one-twelfth) to numbers, money, and quantities.

Notes: • Reference to "Resolving easy numbers such as 24 and 36 to prime factors as preparation for the work in fractions and cancelling" was deleted.

(b) Simple mental problems based on the tables.

Grade IV: Oral Arithmetic. As for 1930 syllabus

Notes: • "Resolving easy numbers such as 24 and 36 in to prime factors as a preparation for the work in fractions and cancelling" moved from Grade III.

Grade V: Oral Arithmetic. As for 1930 Syllabus except that "Ratio expressed as fractions and decimals" and "Simple proportion" were deleted.

Grade VI: Oral Arithmetic.

(a) Exercises preparatory to the written work, including operations in simple business transactions of a wider range than those taken in Grade V.

(b) Ratio expressed as fractions and decimals (From Grade V in 1930 Syllabus).

Grade VII: Oral Work. As for 1930 Syllabus.

434

A.11 1948 Amendments

Preparatory 1 (Six months) No mental exercises specified.

Preparatory 2 (Six months) Mental exercises to 8. Terms - pair. Easy problems.

Preparatory 3 (Six months) Mental exercises including easy problems in number to 13. One operation only.

Preparatory 4 (Six months) Mental exercises to cover: Count to 19 in ones, twos; Notation and numeration to 19; Addition tables (including first extension) to 19; Subtraction tables to 19 (based on addition tables); Halving and doubling to 18; Recognise coins - half- penny, penny, three pence, six pence, shilling; Symbols +, -, and = ; Terms: pair, couple, dozen.

Grade I: Mental. (a) Exercises in addition and subtraction with numbers and quantities within the pupil's range

Notes: • Exercises in addition and subtraction should not involve numbers beyond 99.

(b) Fractions: Doubling and halving to 100

Notes: • Examples should not involve carrying or remainder, e.g. Double 42 or 21. Halve 84, 68

Notes: • Continue shopping practice as suggested in previous grade. Tender 1s.

Grade II: Oral Work. (a) Easy problems on the four simple rules, with numbers within the pupils' range.

Notes: • Children should have extensive practice in addition.

(b) Component parts of 1s. taken two at a time, such as 2½d. 9½d.; 1s.-10½d. etc.

(d) Terms - Pair, couple, dozen, score, gross.

435

(e) Finding halves and quarters of numbers and quantities

Grade III: Oral Work. (a) Simple exercises based on the four simple rules, and on the tables taught, (limited to two steps)

Notes: • These exercises are intended to show the practical application of the tables, and to serve as an introduction to compound rules

(b) Fractions; applications of fractions (one-half, one-third, one-quarter, etc. to one-twelfth) to numbers, money and quantities. (Multiples of denominators only)

Notes: • The exercises outlined in previous grade could be continued and extended to harder examples in their applications. The exercises are intended only as an introduction to the study of vulgar fractions at a later stage.

(c) Buying and selling, saving and spending with sums to £1. Very simple exercises based on reduction and the four rules in money.

Notes: • The material for these exercises supplied by the ordinary household accounts, as for example, the butcher's, the baker's, the grocer's, or the draper's bills.

Grade IV: Oral Arithmetic. (a) Mental exercises based on compound rules, including household and shopping transactions familiar to the pupils.

Notes: • Shopping transactions continued and, in addition, exercises as a preparation for the written work in weights and measures.

Grade V: Oral Arithmetic. (a) Exercises involving the use of vulgar and decimal fractions

Notes: • Simple exercises involving the four rules as introduction to the written work, e.g. ½ + 1/3; 3/8 + 1/4; 3/4 - 1/8; 11/12 - 1/6; 3/4 x 2/3; ½ of 8/9; 4/5 ÷ 1/10; 8/9 ÷ 2/3; .2 + .3+ .4; .9 - .1 - .3; 7 times .5; 6 times .9; .8 ÷ 2; 2 ÷ .5; 2 - .7; .3 x .2; .6 ÷ .2; The conversion of easy vulgar fractions to decimals and vice versa: e.g. ½ = .5; ¼ = .25; 3/4 = .75; 1/8 = .125; 7/8 = .875

(b) Prime factors of numbers to 100

436

Notes: • Variety of mental exercises involving the use of simple aliquot parts is essential.

(d) Exercises dealing with money, weights, and measures, and vulgar and decimal fractions (applied to concrete quantities)

(e) Practice in working by short methods.

Notes: • After the pupils have been thoroughly exercises in any rule, short methods of calculation could be introduced. To encourage initiative, different solutions of the same problem might bed required from the pupils. The 'dozens' and 'scores' rules might be applied.

Grade VI: Oral Work. (a) Exercises preparatory to written work

(b) Exercises of practical value in teaching principles involved in written arithmetic and those used in every day commercial transactions

Notes: • The dozen rule; The score rule; The value of 240 articles; The value of 480 articles; Aliquot parts of one pound; The square of numbers with ½; also square of numbers ending in 5; Division and multiplication by 25.

Grade VII: Oral Work. (a) Mental exercises preparatory to the written work in Arithmetic and Mensuration

(b) Continued practice in short methods of calculation

A.12 1952 Syllabus

Preparatory Grade: Mental arithmetic not mentioned specifically.

Grade I: Oral exercises to 10.

Notes: • These must involve one operation only. Correlate with number facts, games, songs, number experiences, projects, &c. 437

Grade II: Oral arithmetic. (a) Addition and subtraction to 18.

Notes: • Problems should be limited to two operations

(b) First extension to 99.

Notes: • One operation only is required.

Notes: • Practical elementary measuring with foot rule and tape measure, e.g. length of reader, slate, desk; heights of seat, table, &c. (Approximate measurements to an inch are sufficient.) Estimating lengths to 1 foot. Reading clock faces - hours only.

Grade III: Oral Arithmetic. (a) Exercises on the four simple rules with numbers within the pupils' range.

Notes: • These exercises should be a preparation for the written work.

(b) Shopping exercises involving use of component parts of 3d, 6d, and 1s (to meet shopping needs within the pupils' experience).

Notes: • Highest amount from which change will be required is one shilling. (Only one operation: 1½d. out of 3d.; 4½d. out of 6d.; 8½d. out of 1s.) (Change from recognised current coins only.)

Notes: • Introductory exercises in paper-folding. Fractional parts should be based on multiplication and division tables: 4 times 9 = 36, then ¼ of 36 = 9. • 'Half' and 'quarter' should be applied to 'Time': half-hour and quarter-hour.

(d) Terms: Pair, couple, dozen, score.

(e) Problems involving not more than two operations.

Grade IV: Oral Arithmetic. (a) Exercises based on the four simple rules and on the tables taught (limited to two steps) 438

Notes: • Give abundant practice in such exercises as: 26+17; 43-17; 34+19; 41-16

(b) Fractions: Application of fractions (½,1/3,¼, &c. to 1/12) to numbers, money, and quantities (multiplies of denominators only)

Notes: • The exercises are intended only as introduction to the study of vulgar fractions at a later stage. 1/6 of 72; 1/9 of 54; 1/5 of £1 = ?s.; 1/7 of 1s. 2d. = ?d.; ¼ lb = ?ozs.; ¼ of a gallon = ? pts.

Notes: • Reduction - one step - £ to s.; s. to £; s. to d.; d. to s.; d. to ½d.; ½d. to d.; s. to 6d. or 3d. and vice versa. • Addition: Any two amounts in pence and halfpence not to exceed 1s., e.g. 3½d. and 5½d. Any three amounts in pence only, not to exceed 2s, e.g., 2d. + 8d. + 9d. (Answer in s.d.). Any two amounts each composed of an exact number of shillings, the sum not to exceed £2, e.g. 18s. + 19s.; 27s. + 9s. (Answer in £ and s.) • Subtraction: Exercises based on giving change from sums of money not exceeding 3s.

(d) Problems involving not more than two operations.

Grade V: Oral Arithmetic. (a) Exercises based on the four simple rules.

(b) Money: Reduction and four rules.

Notes: • Reduction: One operation with amounts to £1, e.g. 16s.4d. to pence. Two operations: shillings, pence, and halfpence to halfpence and viced versa; limit of 4s., e.g. 3s.9½d. to halfpence • Addition: Add three amounts expressed in pence and halfpence e.g. 9½d. + 10½d. + 8½d. Add two amounts expressed in shillings and pence, e.g. 3s. 6d. + 5s. 6d.; 2s 11d + 8½d. • Subtraction: Practical problems in giving change, e.g. Docket, 2s.7d.; coins offered 2 florins; what change? • Multiplication: Limited to one reduction, e.g. 9½d. x 12; 2s. 9d. x 6; £1 7s. x 5. • Division: Sum to be divided not to exceed £2, and not exceed one step in reduction: 3s.4d. ÷ 8; 15s.7½d. ÷ 5 (No remainders)

Notes: Exercises introductory to written work. (d) Exercises in finding perimeters of rectangular figures.

439

Notes: • Exercises to be regarded as practical applications of Long measure table. Dimensions of one denomination only; answer may be expressed in two denominations.

(e) Fractions: Application of fractions (½,1/3,¼, &c. to 1/12) to numbers, money, and quantities

Notes: • Exercises are intended as introduction to the study of fractions at a later stage.

Grade VI: Oral Arithmetic

(a) Exercises preparatory to written work

Notes: • In the initial stages use blackboard freely in setting examples

(b) Exercises involving the use of vulgar fractions. Denominators not to exceed 12.

Notes: • Easy exercises on fractional equivalents up to 12ths. e.g. 3/4 = 1/8; 4/6 = 2/?; 4½ = 4 ?/10 • Exercises in reduction to lowest terms. • Reduction of improper fractions to mixed numbers and vice versa. • Reduce one number or amount to the fraction of another, e.g. What fraction is 6 of 8? Reduce 1s.3d. to the fraction of 2s. What fraction of 1 mile is 20 chains? • Addition and subtraction of two fractions - LCD not to exceed 12. • Multiply a proper fraction or a mixed number by whole number to 12. • Multiply two fractions. • Divide proper fractions by a whole number. • Divide whole number by a fraction. • Divide a proper fraction by a proper fraction.

Notes: • Conversion of vulgar fractions to decimals and vice versa: involving halves, quarters, fifths, eights, and tenths, e.g. ½ = .5; ¼ = .25 • Addition and subtraction: Limited to two decimal places, e.g. .2+.3; .8+.5; .08+.2; .9-.2; 3-.6; 2-.05 • Multiplication: Product not to exceed two decimal places, e.g. 7 x .5; .3x.2; 8x1.2 • Division: Quotient not to exceed two decimal places, e.g. 3÷4; 6÷8; .2÷5. • Easy decimalization of money and quantities, e.g. £6 8s. = £6.4; 3 tons 16 cwt = 3.8 tons

440

(d) Aliquot parts of £1 and 1s. - both vulgar and decimal fractions.

Notes: • Denominators of vulgar fractions not to exceed 12.

(e) Practical exercises dealing with money, weights and measures, vulgar and decimal fractions

(f) Practical application of tables to find: perimeters of squares and rectangles; Areas of squares and rectangles.

Notes: • Dimensions limited to whole numbers. Reduction limited to one step.

Grade VII: Oral Arithmetic. (Exercises preparatory to written work)

Notes: • Teachers should make use of the blackboard for the more difficult exercises which cannot be readily visualized without this aid.

(a) Factors and multiplies

(b) Vulgar fractions: Addition, subtraction, multiplication and division (Limited to two terms; Denominators to 12)

Notes: • Addition or subtraction; two terms to hundredths. Multiply or divide by 10, 100, 1000 (by inspection). In other examples of multiplication and division one of the terms should be limited to one figure and the other to two figures.

(d) Vulgar and decimal fractions applied to concrete quantities.

(e) Exercises involving everyday commercial and domestic transactions, including examples bearing on local industries.

Notes: • Exercises should be practical.

(f) Ratio. 441

Notes: • Give much oral practice in expressing ratios as vulgar fractions and vice versa. (g) Simple proportion.

(h) Percentages and their application to Profit and Loss and Simple Interest.

Notes: (See written work): • Express percentages as vulgar and decimal fractions. • Express fractions and percentages. • Express money as percentage of a £, and vice versa. • Find percentages of numbers and quantities. • With percentages given, find full values of numbers and quantities. • Find what percentage one number or quantity is of another. • Increase or decrease a number or quantity by a given percentage • Find what percentage numbers or quantities are increased/decreased.

(i) Mensuration

Notes: • Exercises preparatory to written work. • Area and perimeter of square and rectangle. • Areas and cost of paths and borders. • Areas of walls and costs of painting. • No reverse processes.

Grade VIII: Oral Arithmetic. Exercises preparatory to the written work in Arithmetic and Mensuration.

Notes: • Teachers should make use of the blackboard for the more difficult exercises which cannot be readily visualized without this aid.

A.13 1964 Syllabus

Grade 1: Oral exercises to 9.

Notes: • Correlate with number facts, games, songs, number experiences, projects, &c. Children should be given practice in putting mechanical exercises into problem form and vice versa. The expression 5 - 3 = ? might produce he response - I had 5 oranges and I ate 3. How many oranges have I left? 442

Grade 2: Oral Arithmetic.

Notes: • Teachers should make use of the blackboard for the more difficult exercises. Jotting should be permitted where children cannot readily visualize without this aid.

(a) Exercises based on number facts to 18

Notes: • Problems should be limited to two operations.

(b) First extension to 99.

Notes: • Frequent practice should be given in working examples such as: • 16+2, 21+4, 60+1, 90+4; 14+5, 24+5, 63+5, 96+3; • 18-16, 25-21, 61-60, 94-90; 19-14, 29-24, 68-63, 99-96. • One operation only

Notes: • Use extensively in various number situations, e.g. 13-7=, 6+8-7=. Continue giving practice in putting such mechanical exercises into problem form (and vice versa).

Grade 3: Oral Arithmetic.

Notes: • Teachers should make use of the blackboard for the more difficult exercises. Jotting should be permitted where children cannot readily visualize without this aid.

(a) Extended addition and subtraction to 99.

Notes: • Frequent practice should be given in working examples such as: Second extension - completing the 10; Third extension - bridging the ten

(b) Easy factors.

Notes: • A knowledge of simple factors follows from an intelligent acquaintance with the multiplication table, and, when pupils learn a table, exercises in finding the factors of a product should be given.

Notes: • These exercises teach processes that arise in actual multiplication and division sums: 7 x 3 + 2, 23 ÷ 3; 12 x 4 + 3, 51 ÷ 4.

(d) Practical activities: Reading the clock (In divisions of five minutes); Measuring length (In feet and inches). Notes: • The necessity for practice for each member of class is stressed.

(e) Exercises on the four simple rules with numbers within the pupils' range.

Notes: • Exercises should be a preparation for written work.

(f) Shopping exercises involving use of component parts of 3d., 6d., and 1s. (to meet shopping needs within the pupils' experience)

Notes: • Highest amount from which change will be required is one shilling.

(g) Finding half and quarter of numbers and quantities.

Notes: • Introductory exercises in paper-folding. • Fractional parts should be based on multiplication and division tables: 4 times 9 = 36, then ¼ of 36 = 9. • "Half" and "Quarter" should be applied to "time": half-hour and quarter- hour.

(h) Problems involving not more than two operations.

Notes: • Mechanical work involved in these problems should be simple.

(i) Terms: Pair, couple, dozen, score, half, double, plus, minus, equals. Pound, pint, minute, hour, day, week.

Grade 4: Oral Arithmetic.

Notes: • Teachers should make use of the blackboard for the more difficult exercises. Jotting should be permitted where children cannot readily visualize without this aid.

(a) Extended addition and subtraction. 444

Notes: • Types such as 28+7; 49+6; 35-28; 55-49 need special attention.

(b) Combined multiplication and addition. Division with remainder.

Notes: • These applied tables teach processes that arise in multiplication and division sums: 8 x 7 + 3, 59 ÷ 7; 5 x 9 + 6, 51 ÷ 9. (c) Easy factors and multiples.

(d) Exercises on the four simple rules on tables taught (limited to two steps)

Notes: • Give practice in such exercises as: 26+17; 34+19; 43-17; 61-16.

(e) Fractions: Application of fractions (½, 1/3, ¼, &c. to 1/12) to numbers, money and quantities (multiples of denominators only).

Notes: • The exercises are intended only as an introduction to the study of vulgar fractions at a later stage: 1/6 of 72; 1/9 of 54; 1/5 of £1 = ?s.; 1/7 of 1s.2s. = ?d.; ¼lb. = ?ozs; ¼ of a gallon = ?pts.

(f) Money: Exercises based on addition, subtraction, reduction.

Notes: • Addition: Any three amounts in pence only, not to exceed 2s., e.g. 2d.+8d.+9d. (answer in s.d.) • Any two amounts each composed of an exact number of shillings, the sum not to exceed £2, e.g. 18s.+19s.; 27s.+9s. (answer in £ and s.) • Subtraction: Exercises based on giving change from sums of money not exceeding 3s. • Reduction: One step - Pounds to shillings and reverse; shillings to pence and reverse; shilling to sixpences and threepences and reverse;

(g) Problems involving not more that two operations.

Notes: • Mechanical work involved in these problems should be simple.

Grade 5: Oral Arithmetic.

Notes: • Teachers should make use of the blackboard for the more difficult exercises. Jotting should be permitted where children cannot readily visualize without this aid.

(a) Extended addition and subtraction. 445

(b) Combined multiplication and addition. Division with remainder.

Notes: • To follow directly from multiplication tables.

(d) Money to 144 pence and 200 shillings.

Notes: • Applied to intermediate numbers (1964 only)

(e) Weights and measures.

Notes: • Exercises in estimating weights and measures and checking estimates by practical weighing and measuring are required.

(f) Exercises based on the four simple rules.

(g) Money: Reduction and four rules.

Notes: • Reduction: One operation with amounts to 10s., e.g. 7s.9d. to pence; £2 17s. to shillings. • Addition: Add three amounts expressed in pence and halfpence, e.g. 9d.+10d.+8d. • Add two amounts expressed in shillings and pence, e.g. 3s. 6d. + 5s. 9d.; 2s 11d + 8d. • Subtraction: Practical problems in giving change, e.g. Docket, 2s.7d.; coins offered 2 florins; what change? • Multiplication: Limited to one reduction, e.g. 9d. x 12; 2s. 9d. x 6; £1 7s. x 5. (1964 only.) • Division: Sum to be divided not to exceed £2, and not exceed one step in reduction: £1 15s ÷ 5; 3s.4d. ÷ 8; 15s.6d. ÷ 3. (1964 only) (No remainders)

(h) Practical application of tables of weights and measures.

Notes: • Simple exercises in reduction, addition, subtraction. Reduction limited to one step. Addition and subtraction limited to two denominations.

(i) Fractions: Application of fractions (½, 1/3, ¼, &c. to 1/12) to numbers, money and quantities.

Notes: • The exercises are intended only as an introduction to the study of fractions at a later stage. 446

(j) Mensuration (Square and rectangle):

Notes: • Exercises in finding perimeters. Teach reverse processes. (Exercises to be regarded as practical applications of Long Measure table. Dimensions of one denomination only. Limited to whole numbers.)

(k) Problems involving not more than two operations. Notes: • Mechanical work involved in these problems should be simple.

Grade 6: Oral Arithmetic.

Notes: • Teachers should make use of the blackboard for the more difficult exercises. Jotting should be permitted where children cannot readily visualize without this aid.

(a) Extended addition and subtraction.

(b) Combined multiplication and addition. Division with remainder.

Notes: • Applied to intermediate numbers.

(d) Weights and measures.

Notes: • Exercises in estimating weights and measures and checking estimates by practical weighing and measuring are required.

(e) Exercises preparatory to written work.

(f) Exercises involving the use of vulgar fractions. Denominators not to exceed 12.

Notes: • Easy exercises on fractional equivalents up to 12ths; e.g. 3/4 = ?/8; 4/6 = 2/?; 4½ = 4 ?/10. • Exercises in reduction in lowest terms. • Reduction of improper fractions to mixed numbers and vice versa. • Reduce one number or amount to the fraction of another, e.g. What fraction is 6 of 8? Reduce 1s.3d. to fraction of 2s. What fraction of 1 mile is 20 chains? • Addition and subtraction of two fractions - LCM not to exceed 12. • Multiply a proper fraction or a mixed number by any whole number to 12. • Multiply two fractions. • Divide a proper fraction by a whole number. 447

• Divide a whole number by a fraction. • Divide a proper fraction by a proper fraction.

(g) Factors and multiples.

Notes: • Introduce terms:- Prime number, prime factor, common multiple, lowest common multiple. • Frequent practice should be given in finding prime factors of numbers to 144, and in building multiples to 144 from two or more numbers.

(h) Squares and square roots.

Notes: • Introduce terms square and square root. Square whole numbers to 12 and reverse.

(i) Exercises involving the use of finite decimal fractions.

Notes: • Conversion of vulgar fractions to decimals and vice versa: involving halves, quarters, fifths, eights, and tenths, e.g. ½ = .5; ¼ = .25; 3/4=.75; 1/8=.875; 3/5=.6 • Addition and subtraction: Limited to two decimal places, e.g. .2+.3; .8+.5; .08+2; .9-.2; 3-.6; 2-.05 • Multiplication: Product not to exceed two decimal places, e.g. 7x.5; .3x.2; 8x1.2 Division: Quotient not to exceed two decimal places, e.g. 3 ÷ 4; 6 ÷ 8; 2 ÷ 5. • Easy decimalization of money and quantities, e.g. £6 8s. = £6.4; 3 tons 16 cwt = 3.8 tons

(j) Aliquot parts of £1 and 1s. - both vulgar and decimal fractions.

Notes: • Aliquot parts of 1s. Limited to: 6d. = ½s.; 3d. = ¼s. • Aliquot parts of £1. Limited to: 10s. = £½; 5/0d. = £¼; 4/0d. = £1/5; 2/0d. = £1/10

(k) Practical exercises dealing with money, weights and measures, vulgar and decimal fractions.

(i) Mensuration (Square and rectangle).

Notes: • Dimensions limited to whole numbers) • Reduction limited to one step • Find perimeter. Teach reverse process. • Find area. No reverse process.

448

Grade 7: Oral Arithmetic.

Notes: • Teachers should make use of the blackboard for the more difficult exercises. Jotting should be permitted where children cannot readily visualize without this aid.

(a) Extended addition and subtraction.

(b) Combined multiplication and division. Division with remainder.

Notes: • Applied to intermediate numbers.

(d) Weights and measures.

Notes: • Exercises in estimating weights and measures and checking estimates by practical weighing and measuring are required.

(e) Exercises preparatory to written work.

(f) Factors and multiples. (See note for Grade VI.)

(g) Squares and square roots. (See note for Grade VI.)

(h) Vulgar fractions: Addition, subtraction, multiplication, and division.

Notes: • Limited to two terms─denominators not to exceed 12.

(i) Decimal fractions; (Finite decimals only; limited to thousandths)

Notes: • Addition or subtraction; two terms to hundredths. • Multiply or divide by 10, 100, 1000 (by inspection). • In other examples of multiplication and division one of the terms should be limited to one figure and the other to two figures.

(j) Vulgar and decimal fractions applied to concrete quantities.

Notes: • To include simple exercises in multiplication and division of weights and measures.

(k) Exercises involving everyday commercial and domestic transactions, including examples bearing on local industries.

Notes: 449

• Exercises should be practical.

(l) Ratio.

Notes: • Give much oral practice in expressing ratios as vulgar fractions and vice versa.

(m) Simple proportion

(n) Percentages.

Notes: • Express percentages as vulgar and decimal fractions. • Express fractions and percentages. • Express money as percentage of a £, and vice versa. • Find percentages of numbers and quantities. • With percentages given, find full values of numbers and quantities. • Find what percentage one number or quantity is of another. • Increase or decrease a number or quantity by a given percentage. • Find by what percentage numbers or quantities are increased or decreased.

(o) Mensuration.

Notes: • Exercises preparatory to written work.

(p) Problems.

Notes: • Mechanical work...should be simple.

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APPENDIX B

Additional Notes: Chapter 3

1. Following the 1887 Civil Service Commission's interim report in 1888, The Brisbane Courier (“Editorial,” 18 August 1888) suggested that the Department of Public Instruction was little more that "a euphonious synonym for Messrs. Anderson and Ewart..., the almighty Siamese twins of the Education Department". Real authority lay with them rather than with the responsible minister (Barcan, 1980, p. 184).

2. After a tour of European schools in 1897, Ewart reported that "nothing in the professional working of the schools struck me as note-worthy compared with the ordinary conditions of our own schools; but I was somewhat impressed with the buildings and their equipment" (“Second Progress Report,” 1889, p. 959).

3. The Department of Public Instruction did not enter the field of secondary education until 1912. From 1913 the State provided free secondary education for all who qualified; "a dramatic reversal of the longstanding concept of the state's responsibility being limited to the provision of primary education" (Lawry, 1972, p. 25).

4. These conferences were effectively forced on the Department by Minister Barlow who was himself under political pressure, particularly from the Queensland Teachers' Union: "The conferences continued to press the position, and assisted by professional opinion outside, the work of revision was forced on the Department" (St. Ledger, 1905, p. 218).

5. The Sub-committee on Arithmetic recommended two minor changes to the mental arithmetic requirements: "In Class I.1. - To add and subtract mentally numbers applied to objects to a result not greater than 10. In Class II.1, 2, 3. - Easy practical applications of the Rules." These recommendations constituted an earlier introduction of mental subtraction and division (see Appendix A), and were made under a belief that "the...course of instruction in [Arithmetic was] calculated to give [the] pupils a fair knowledge of all that may be required of them in the ordinary business of life" (Gripp, Mutch, & McKenna, 1904, p. 20).

6. At a lecture given in 1899, and reprinted in 1915, Dewey stated that "the child [is] the sun about which the appliances of education revolve; he is the center about which they are organized" (p. 35).

7. The period 1909 to 1914 was, in Wyeth's (1955, p. 178) view, one of the most progressive periods for education in Queensland, which, probably not coincidently, coincided with Ewart's retirement in 1909. The University of Queensland opened in 1911, State High schools from 1912, and a Teachers' College in 1914.

451

8. Mrs Beatrice Ensor, founder of the New Education Fellowship, visited Australia in 1937 to attend conferences of the Fellowship, the first of which was held in Brisbane.

9. Cramer (Superintendent of Schools, The Dalles, Oregon) toured Australian schools in 1934 and published his observations in an Australian Council of Educational Research publication, Australian Schools Through American Eyes (1936).

10. This belief is reiterated in the General Introduction to the 1930 Syllabus:

Teachers will be expected to adapt the curriculum to levels of growth and to individual differences, and to organise activities that, while realising the purposes of the curriculum, will suit local circumstances...Where teachers probe below the letter to catch the spirit of the Syllabus, we can expect the most permanent results. (Department of Public Instruction, 1930, p. xi)

11. A copy of Burt's Mental Tests in Arithmetic was not found during the research for this study.

12. The Committee of Seven, under the Direction of Carleton Washburne (Superintendent of Schools, Winnetka, Illinois), set out to determine the stage of mental development at which specific mathematics topics could be most effectively mastered. Its report was published in the 29th Year Book of the National Society for the Study of Education (1932).

13. Prior to the 1952 syllabus, times for each subject were not specified. Arithmetic was considered by many to be "a subject that [would] poach time from every other subject in the syllabus" (Darling Downs Branch, 1946, p. 21), particularly in the scholarship class. Cunningham and Price (1934, p. 61) found that Queensland teachers of Grades II to VII devoted an average of approximately 5 hr per week on Arithmetic, compared to 4hr 23min in South Australian schools, for example. By 1949 the average time spent in Queensland was 5½hr per week (ACER, 1949, p. 12).

14. In the General Introduction to the 1930 Syllabus it was stated that:

It is generally agreed that the Primary School - the only kind of school some children will know - should teach its pupils to speak, to read, and to write; introduce them to what men think and do, and to what men have thought and done; enable them to gain some general ideas of the natural world; give them practice in the art of calculation; train them in habits of observation; develop their manual dexterity; introduce them to the beauty of form, colour, and sound; and improve their physique - in short, it should develop a sound mind in a sound body. (Department of Public Instruction, 1930, p. v; 1952a, p.1)

15. Five hours per week for mathematics was suggested as an appropriate time allocation for Grades III to VII in the General Memorandum issued to schools

452

concerning the 1948 Amendments (Department of Public Instruction, 1948b, p. 17). Advocating a further reduction on the time spent on mathematics, Primary Education: A Report of the Advisory Council on Education in Scotland suggested that 5 periods of 40 minutes (3 hours 20 minutes) per week were ample (cited by Greenhalgh, 1949a, p. 11).

16. An earlier version was a text recommended for pupil-teachers in The State Education Act of 1875 Together With the Regulations of the Department and General Instructions for the Guidance of Teachers and Others. (“List of Books,” 1880). Brisbane: Government Printer.

17. The committee, under District Inspector Skelton's chairmanship, constituted to make recommendations for the revision of the 1904 syllabus, considered the oral statement of processes of such importance that it recommended that the following be inserted in the syllabus for the First Class: "Attention is directed to the following extract from the preface of the Syllabus: `Pupils should be made familiar by mental exercises with the principles underlying every process before the written work is undertaken'". However, this recommendation was not accepted by the District Inspectors responsible for writing the 1914 Syllabus. It was deemed unnecessary as the matter would be dealt with in the explanatory memorandum. (Skelton, 1912, p. 1).

18. Process, in this historical context, refers to an arithmetical procedure or algorithm (e.g. for addition); a much narrower definition than that used in the Years 1 to 10 Mathematics Syllabus in which a process is defined as a "[way] of operating, both cognitively and physically, with or on mathematical concepts" (Department of Education, 1987a, p.10)

19. Gladman's School Method, Park's Manual of Method and Joyce's School Management were three of the texts listed in The State Education Act of 1875. Together With the Regulations of the Department and General Instructions for the Guidance of Teachers and Others (“List of Books,” 1880, p. 24). It was noted for head teachers that "this list of authorised books is chiefly intended to show what books teachers are empowered to place, when necessary, in the hands of...pupil-teachers" (p. 23). Gladman's School Work was included as one of the recommended texts for pupil-teachers in List II of the 1895 publication of the State Education Act, as was Robinson's Teacher's Manual of Method and Organisation.

20. The “Practical” Mental Arithmetic, written by "An Inspector of Schools" (1914). An earlier impression of this book was the mental arithmetic text "supplied to schools for the instruction of pupils" (Department of Public Instruction, 1902, p. 95).

21. To find the cost of: (a) a dozen articles. Calculate the given cost of one article in pence, and call the pence shillings: 1 dozen @ 1s. 4d. = 16s.

453

(b) a score of articles. Calculate the given cost of one article in shillings, and call the shillings pounds: 1 score @ £1. 2s. 7d. = £22 7 = £22. 11s. 8d. 12 (c) a gross of articles. As a gross is a dozen times a dozen, the cost of 1 gross @ 1s. 4d. = cost of 12 at 16s. = £9. 12s. (Pendlebury & Beard, 1899, p.174)

22. (a) Compound rules involved the addition, subtraction, multiplication or division of compound quantities; that is, of quantities that are expressed in terms of different denominations: e.g. 2s. 6d. x 4. (Pendlebury & Beard, 1922, p. 29) (b) Reduction was the process of converting a quantity from one denomination to another: e.g. inches to feet or vice versa. (Pendlebury & Beard, 1922, p. 27) (c) "Practice [was] a method of calculating by the ‘addition of aliquot parts' the value of a simple or compound quantity, when the value of a unit of one denomination [was] given. The method [was] ‘simple' or ‘compound', according as the quantity considered [was] simple or compound". (Pendlebury & Beard, 1922, p. 105)

For example: (i) to calculate the cost of 18 articles at 10s. each, using simple practice, knowing that 10s. is one-half of one pound, the cost of the 18 articles would be ½ of 18, which is £9; (ii) to calculate the cost of 60 yards of material at 5d. per yard, find the cost if it was 1d. per yard (5s.) and multiply this result by 5 (£0 5s. 0d. x 5 = £1 5s. 0d.).

23. The unitary method for manipulating proportions involved finding the cost of one unit from given information from which the value of the required number of units could be calculated. For example, to calculate the cost of 7 yards of material, knowing that 5 yards cost 15s: The cost of 1 yard is 3s. Therefore, the cost of 7 yards is 21s. or £1 1s.

24. While this advice was directed at the situation in England, it may have been relevant to Queensland, given that the Report of the Secretary of Public Instruction for 1898 indicated that the number of Sixth Class children in Queensland was 1,668 (Dalrymple, 1899, p. 5).

25. The changes to the written work contained in the 1930 Syllabus, for children of comparable ages given the changes in class structure, included:

Grade I: "Simple introductory exercises in division" rather than working with divisors to 6. Grade II: Divisors limited to one-digit numbers rather than to 100. Grade III: Four rules and reduction of money with sums limited to £20 rather than £100. Grade VI: Complex fractions, previously studied during the second term of Fifth Class were removed from syllabus. ("New Syllabus,” 1929, pp. 460-462)

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26. In association with the implementation of the 1938 Amendments, the Preparatory Grade was extended to two years, with Preparatory 1 and 2 occupying one year and Preparatory 3 and 4 each occupying six months. The work for Preparatory 1 and 2 was presented as a combined syllabus in the 1938 Amendments and separately in the 1948 Amendments.

27. To calculate the cost of:

(a) 240 articles. Change the price of one article to pence and call these pounds. The cost of 240 at 1s. 5d. = £17. (b) 480 articles: Change the price of one article to half-pence and call these pounds. 480 articles at 3s. 2d. = 480 at 76 half-pence each = £76. (c) 960 articles: Change the price of one to farthings and call these pounds. 960 articles at 6¼d. each = 960 at 25 farthings each = £25. (“Mental Arithmetic: Book II,” 1926, p. 7)

28. An acute shortage of teachers resulted in this grade being omitted from January 1953 (Education Office Gazette, 1952, p. 251, cited by Dagg, 1971, p. 35).

29. The first explicit direction concerning the limit of written addition in First Class was given in 1879: "To add three numbers of three figures" (Department of Public Instruction, 1879, p. 14). With the increase in the length of the First Class from 1½ to 2 years in 1892, the limit was extended to: "To add six numbers of five figures on slates" ("Para 143,” 1891, p. 23). In the 1904 Syllabus, addition was limited to sums to 200, but was extended to 999 in 1930.

30. The percentage of time spent on oral/mental work included work done during time tabled periods in arithmetic, homework, and in incidental and indirect instruction in other subjects (ACER, 1949, p. 10).

31. Under the 1860 Syllabus, mental arithmetic was to be conducted on Thursdays between 3:15pm and 4:00pm for girls, and for boys on Mondays, Wednesdays and Fridays between 2:30pm and 3:00pm ("Board of General Education,” 1866, p. 5).

32. The sums on the Departmental arithmetic cards were arranged, in sequence, into specific rules and types for each half-year, which tended "to divide the teaching of Arithmetic into watertight compartments" (Somers, 1928, p. 84).

33. Edwards (1931) reported, with apparent satisfaction, that the costs for the meetings organised by District Inspectors were "borne by the teachers themselves" (p. 28).

34. District Inspector Ross (1884) concluded that pupil-teachers "as pupils...[were] burdened with lessons and exercises of preposterous length; and as teachers...[were] left to blunder through their wearisome duties with little of either direction or encouragement" (p. 68).

455

35. In 1909 the minimum number of children in attendance required for classification as a State School was reduced from 30 to 12. This created 1059 State Schools with only 79 classified as Provisional (Logan & Clarke, 1984, p. 3).

36. In 1899, 11 pupil-teachers at the Central Brisbane State School taught classes from 28 to 76 pupils (Lawry, 1968, p. 607).

37. During the 1930s, in excess of 70% of schools were one-teacher schools (Edwards, 1930; p. 27; 1937a, p. 23; "Instructions to Inspectors,” 1935, p. 7).

38. The following items are from the Grade III text in the series─Brooks's Queensland School Series─most commonly used by teachers in the late 1940s (ACER, 1949, Appendix 1, p. 3):

(a) (13 x 8) - (14 x 7) (b) 12 x AB = 204. Find the missing figures A and B. (c) (32 - 17 + 15) ÷ 15 (d) The dividend is 288 and the divisor is 12. What is the quotient? (e) A man had four bags. The first held 19 pennies and each of the others 15 pennies. How many pennies were there in the four bags? (“New Syllabus Mental Arithmetic for Third Grade,” 1932, p. 5)

39. A copy of Pitman's Mental and Intelligence Tests in Common-Sense Arithmetic was not found during the research for this study.

40. The books sanctioned for arithmetic in the 1860 Regulations were listed simply as: (a) Arithmetic, (b) Arithmetic in Theory and Practice, and (c) Set Tablet Lessons, Arithmetic, 60 sheets ("Regulations,” 1860, p. 7).

41. "May, Nellie and Joan were three sisters, all of whom had been promised a new beret by their Aunt. Last Thursday she took them to town to buy the berets. The ones they liked were 1/6 each, so their Aunt allowed them to have the ones they chose. When paying for them she found that she had to give a ten-shilling note. Of course, they had to wait for change; how much was it?" ("Grade III Mental Arithmetic,” 1936, p. 14).

456

APPENDIX C

Self-Completion Questionnaire

COVER PAGE

Focus: The focus for this survey is Mental Computation (Mental/Oral Arithmetic). Mental computation is considered to be the calculation of exact answers mentally. Of particular interest is calculating beyond the basic facts. For the purposes of this study, Mental Computation needs to be considered as something distinct from Computational Estimation.

Purpose: This survey seeks to gain an understanding of the beliefs about Mental Computation held by Queensland State primary school teachers and administrators. It also aims to gather information about teaching practices. The data gathered will supplement that gained from archival material to provide a history of mental computation in Queensland State Primary Schools since 1859.

Instructions: Please do not put your name or that of your school on the questionnaire as there is no valid use for this information in the analysis and reporting of the data supplied.

Sections 1, 2 and 3 of the survey contain statements about Mental Computation to which you are asked to respond. Section 4 obtains data about you, your school, and inservice with respect to Mental Computation.

Section 1 is for all respondents. Please circle a number to show the extent to which you Agree or Disagree with each of the statements.

Section 2 is for those currently responsible for a particular class. Please circle a number to show the frequency over a week that you use each of the teaching techniques. Space is also provided for you to list resources that you have found useful.

Section 3 is for those who had the responsibility of a class at anytime in the period 1964 - 1987. Please circle a number to show how important you considered mental computation to be, and how frequently each of the listed teaching techniques was used. Space is also provided for you to list resources that you have found useful during this period.

Section 4 is for all respondents. Please tick the boxes that are appropriate for you and your school.

On completion, please return this questionnaire to the contact person in your school. This will enable all questionnaires from your school to be returned in the stamped, addressed packet which has been sent to your school's contact person.

Thanking you

Geoff Morgan Lawnton State School

457

SECTION 1

BELIEFS ABOUT MENTAL COMPUTATION AND HOW IT SHOULD BE TAUGHT

Circle a number on each line to show the extent to which you Agree or Disagree with each statement. Please remember that this part of the survey is interested in the mental calculation of exact answers beyond the basic facts.

Strongly Strongly Disagree Disagree Agree Agree

1. The development of the ability to calculate exact answers mentally is a legitimate goal of 1 2 3 4 mathematics education.

2. The Years 1-10 Mathematics Syllabus places little importance on the development of the 1 2 3 4 ability to calculate exact answers mentally.

3. Written methods of calculating exact answers are superior to mental procedures. 1 2 3 4

4. Mental computation encourages children to devise ingenious computational short cuts. 1 2 3 4

5. Calculating exact answers mentally involves applying rules by rote. 1 2 3 4

6. Mental computation helps children gain an understanding of the relationships between 1 2 3 4 numbers.

7. The ability to calculate exact answers with paper-and-pencil is more useful outside the 1 2 3 4 classroom than the ability to calculate mentally.

8. Children should use the algorithms for written computation when calculating exact answers 1 2 3 4 mentally.

9. Children who are proficient at mentally calculating exact answers use personal 1 2 3 4 adaptations of written algorithms and idiosyncratic mental strategies.

10. Emphasis on written algorithms needs to be delayed so that mental computation can be 1 2 3 4 given increased attention.

11. Teachers need to be aware of the strategies used by those who are proficient at 1 2 3 4 calculating exact answers mentally. 458

SECTION 1 (BELIEFS) cont.

Strongly Strongly Disagree Disagree Agree Agree

12. Strategies for calculating exact answers mentally need to be specifically taught. 1 2 3 4

13. Strategies for calculating exact answers mentally are best developed through discussion 1 2 3 4 and explanation.

14. Children need to be allowed to develop and use their own strategies for calculating 1 2 3 4 exact answers mentally.

15. Children's mental processes are sharpened by starting a mathematics lesson with ten quick 1 2 3 4 questions to be solved mentally.

16. Children are encouraged to think about mathematics right from the start of a lesson 1 2 3 4 when given ten quick questions to solve mentally.

17. Answers obtained for the "ten quick questions" need to be corrected quickly so that the 1 2 3 4 mathematics lesson can begin.

18. Opportunities for children to calculate exact answers mentally need to be provided in all 1 2 3 4 relevant classroom activities.

19. Children should be given opportunities to discuss, compare and refine their mental 1 2 3 4 strategies for solving particular mental problems.

20. During mental computation sessions, the focus should be on the correctness of the answer 1 2 3 4 rather than on the mental strategies used.

21. During mental computation sessions, one approach to each of a number of problems 1 2 3 4 should be the focus, rather than on several approaches to each of a few problems.

22. Children should be encouraged to build on the thinking strategies used to develop the basic 1 2 3 4 facts.

23. A series of sessions that focuses on developing strategies for computing exact answers mentally 1 2 3 4 needs to be conducted each week.

Please continue to Section 2 over page. 459

SECTION 2

CURRENT TEACHING PRACTICES

If you're not a class teacher at this time, please go to Section 3 on Page 5.

SECTION 2.1

Please circle a number on each line to show how frequently you use each teaching technique to develop the ability to mentally calculate exact answers beyond the basic facts.

Never Seldom Sometimes Often

24. Focus specifically on developing the ability to calculate exact answers mentally beyond 1 2 3 4 the basic facts.

25. Allow children to decide the method to be used to arrive at an exact answer mentally. 1 2 3 4

26. Allow children to explain and discuss their mental strategies for solving a problem. 1 2 3 4

27. Allow children to work mentally during practice of written computation. 1 2 3 4

28. Teach particular mental strategies and follow up with practice examples. 1 2 3 4

29. Give several one-step questions and simply mark answers as correct or incorrect. 1 2 3 4

30. Require answers to problems solved mentally to be recorded on paper. 1 2 3 4

31. Relate methods for calculating beyond the basic facts to the thinking strategies used 1 2 3 4 to develop the basic facts.

32. Use mental computation for revising and practising arithmetic facts and procedures. 1 2 3 4

33. Emphasise speed when calculating exact answers mentally. 1 2 3 4

34. Insist on children using the procedures for the written algorithms when calculating 1 2 3 4 exact answers mentally.

35. Provide opportunities for children to appreciate how often they and adults use 1 2 3 4 mental computation. 460

SECTION 2 (CURRENT PRACTICES) cont.

Never Seldom Sometimes Often

36. Teach rules for calculating exact answers mentally (e.g. divide by ten by removing a zero) 1 2 3 4

37. Have children commit to memory number facts beyond the basic facts. 1 2 3 4

38. Use examples involving measures or spatial concepts in problems to be calculated mentally. 1 2 3 4

SECTION 2.2

Resources:

Please list any resources (e.g. texts) currently used for developing the ability to calculate exact answers mentally.

Comments:

Please write any comments you may care to make:

Please continue to Section 3 over page. 461

SECTION 3

PAST TEACHING PRACTICES

If you were not a class teacher at any time during the period 1964 - 1987, please go to Section 4 on Page 8.

SECTION 3.1

If you taught during a particular period, please circle a number to show the level of importance placed on mental computation during each period. If you are unsure, please place a tick on the line provided.

No Little Some Great Importance Importance Importance Importance Unsure

39. For each period, how important did the syllabus consider the ability to calculate exact answers mentally to be?

(a) 1964 - 1968 (1964 Syllabus) 1 2 3 4 ___

(b) 1969 - 1974 (PIM: 1st ed) 1 2 3 4 ___

40. For each period, how important did you consider the ability to calculate exact answers mentally to be?

(a) 1964 - 1968 1 2 3 4 ___

(b) 1969 - 1974 1 2 3 4 ___

SECTION 3.2

If you taught during a particular period, please circle a number on each line to show how frequently you used each teaching technique to develop the ability to mentally calculate exact answers beyond the basic facts. If you are unsure, please place a tick on the line provided.

Never Seldom Sometimes Often Unsure

41. Allowed children to decide the method to be used to arrive at an exact answer mentally.

(a) 1964 - 1968 1 2 3 4 ___

(b) 1969 - 1974 1 2 3 4 ___

SECTION 3 (PAST PRACTICES) cont.

Never Seldom Sometimes Often Unsure

42. Allowed children to explain and discuss their mental strategies for solving a problem:

(a) 1964 - 1968 1 2 3 4 ___

(b) 1969 - 1974 1 2 3 4 ___

43. Gave several one-step questions and simply marked the answers as correct or incorrect:

(a) 1964 - 1968 1 2 3 4 ___

(b) 1969 - 1974 1 2 3 4 ___

44. Emphasised speed when calculating exact answers mentally.

(a) 1964 - 1968 1 2 3 4 ___

(b) 1969 - 1974 1 2 3 4 ___

45. Insisted that children use the procedures for the written algorithms when calculating exact answers mentally.

(a) 1964 - 1968 1 2 3 4 ___

(b) 1969 - 1974 1 2 3 4 ___

46. Placed an emphasis on teaching rules for calculating exact answers mentally (e.g. divide by ten by removing a zero).

(a) 1964 - 1968 1 2 3 4 ___

(b) 1969 - 1974 1 2 3 4 ___

463

SECTION 3 (PAST PRACTICES) cont.

SECTION 3.3

Resources:

Please list any resources (e.g. texts) that were used to develop the ability to calculate exact answers mentally. If any were used for a limited period only, please indicate.

Comments:

Please write below any comments you may care to make:

Please continue to Section 4 over page. 464

SECTION 4

BACKGROUND INFORMATION

For each question, please place a tick (√) in the appropriate box.

47. In which Educational Region is your school located?

Sunshine Coast Metropolitan West Metropolitan East Darling Downs ┌─┐ ┌─┐ ┌─┐ ┌─┐ └─┘ └─┘ └─┘ └─┘ South West Wide Bay Capricornia Northern ┌─┐ ┌─┐ ┌─┐ ┌─┐ └─┘ └─┘ └─┘ └─┘ North West Peninsula South Coast ┌─┐ ┌─┐ ┌─┐ └─┘ └─┘ └─┘

48. What is the size of your school?

Band: 4 5 6 7 8 9 10 ┌─┐ ┌─┐ ┌─┐ ┌─┐ ┌─┐ ┌─┐ ┌─┐ └─┘ └─┘ └─┘ └─┘ └─┘ └─┘ └─┘

49. How many years teaching experience have you had?

< 1 yr 1-5 yrs 6-10 yrs 11-15 yrs ┌─┐ ┌─┐ ┌─┐ ┌─┐ └─┘ └─┘ └─┘ └─┘

16-20 yrs 21-25 yrs 26-30 yrs 30+ yrs ┌─┐ ┌─┐ ┌─┐ ┌─┐ └─┘ └─┘ └─┘ └─┘

50. Are you a: Class Teacher Teaching Principal? (Go to next question) ┌─┐ ┌─┐ └─┘ └─┘

Principal Deputy Principal? (Go to Question 53) ┌─┐ ┌─┐ └─┘ └─┘

51. If a class teacher, which year level/s are you currently teaching?

1 2 3 4 5 6 7 None ┌─┐ ┌─┐ ┌─┐ ┌─┐ ┌─┐ ┌─┐ ┌─┐ ┌─┐ └─┘ └─┘ └─┘ └─┘ └─┘ └─┘ └─┘ └─┘

52. Is your school a trial school for Student Performance Standards in mathematics?

Yes No (If NO, go to Question 55) ┌─┐ ┌─┐ └─┘ └─┘ 465

SECTION 4 (BACKGROUND INFORMATION) cont.

53. If your school is a trial school and you are currently a class teacher, are you trialing the standards in your classroom?

Yes No ┌─┐ ┌─┐ └─┘ └─┘

54. Do you consider it important that inservice sessions on mental Computation be made available to teachers?

Yes No ┌─┐ ┌─┐ └─┘ └─┘

55. Have you attended, during the last three years, inservice sessions in which Mental Computation was a specific topic for discussion?

Yes No (If NO, skip the next question) ┌─┐ ┌─┐ └─┘ └─┘

56. If you have attended inservice on Mental Computation, who conducted the inservice?

Teaching colleague Administrator Mathematics Adviser ┌─┐ ┌─┐ ┌─┐ └─┘ └─┘ └─┘

Tertiary Lecturer Other ┌─┐ ┌─┐ └─┘ └─┘

If Other, please specify:

Comments:

Please write any further comments you may care to make:

THANK YOU for completing this survey. Please return it, in the envelope provided, to the contact person in your school. 466

APPENDIX D

SURVEY CORRESPONDENCE

D.1 Initial Letter: One-teacher Schools

Dear Colleague

As detailed in the attached "Memorandum to Principals" from the Executive Director, Review and Evaluation, approval has been granted for me (Geoff Morgan, Deputy Principal, Lawnton State School) to approach you to invite your participation in a survey of beliefs and teaching practices related to Mental Computation (Mental Arithmetic). Your school has been randomly selected as part of a sample of 115 State Primary Schools from all Education Regions and bands of schools.

The attached Research Outline indicates that the survey is designed as a culmination to historical research currently being undertaken. The aim of the project is to document the nature and place of Mental Computation in Queensland primary classrooms from 1860 to the present. The research is part of a doctoral program supervised by Dr Calvin Irons of the Queensland University of Technology's School of Mathematics, Science and Technology.

Rather than simply send a copy of the questionnaire to you in this mailing, the purpose of this letter is to determine whether you would be prepared to participate in the survey. The questionnaire should take about 15-20 minutes to complete, with not all respondents having to complete all sections. The information to be gathered concerns: Current beliefs about Mental Computation and how it should be taught; Present teaching practices; and Past teaching practices related to the 1964 Syllabus, and to the 1966-68 and 1975 "Programs in Mathematics". I am also interested in identifying the resources used by teachers to support the development of Mental Computation skills.

While some personal background information is asked for, all information obtained will be treated confidentially. In fact, you are asked not to identify yourself or your school on the questionnaire itself as this information does not have any valid use during data analysis. Some questions relate to recent inservice opportunities with regard to Mental Computation. Besides being able to document the status of Mental Computation, it is also hoped to be able to provide recommendations to the Department on any inservice needs which may be identified. /2 467

Your participation in this survey will be greatly appreciated.

Please tear off the form below and return in the envelope provided, preferably by Monday, 18 October. A questionnaire will then be dispatched should a preparedness to be involved be expressed.

Thanking you,

Yours faithfully

MENTAL COMPUTATION SURVEY

Please complete the appropriate section below and return in the envelope provided (by Monday, 18 October) to:

Geoff Morgan Lawnton State School Phone: (07) 285 2968 P.O. Box 2 Fax: (07) 285 6506 LAWNTON 4501

Please SEND the questionnaire on Mental Computation.

Name of school: ______State School

Your name (optional): ______(For addressing envelope to send questionnaire.)

Please DO NOT SEND the questionnaire on Mental Computation.

468

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

Dear Colleague

As detailed in the attached "Memorandum to Principals" from the Executive Director, Review and Evaluation, approval has been granted for me (Geoff Morgan, Deputy Principal, Lawnton State School) to approach you to invite your participation, and that of your staff, in a survey of beliefs and teaching practices related to Mental Computation (Mental Arithmetic). Your school has been randomly selected as part of a sample of 115 State Primary Schools from all Education Regions and bands of schools.

Rather than simply send copies of the questionnaire to you in this mailing, the purpose of this letter is to determine whether you and/or members of your staff would be prepared to participate in the survey. The questionnaire should take about 15-20 minutes to complete, with not all respondents having to complete all sections. The information to be gathered concerns: Current beliefs about Mental Computation and how it should be taught; Present teaching practices; and Past teaching practices related to the 1964 Syllabus, and to the 1966-68 and 1975 "Programs in Mathematics". I am also interested in identifying the resources used by teachers to support the development of Mental Computation skills.

While some personal background information is asked for, all information obtained will be treated confidentially. In fact, respondents are asked not to identify themselves or their school on the questionnaire as this information does not have any valid use during data analysis. Some questions relate to recent inservice opportunities with regard to Mental Computation. Besides being able to document the status of Mental Computation, it is also hoped to be able to provide recommendations to the Department on any inservice needs which may be identified.

/2 469

Your school's participation in this survey will be greatly appreciated. To facilitate the distribution of the questionnaire to staff members, I am wondering if you (or one of your staff) would be prepared to act as Contact Person for the receipt of the questionnaires, their distribution to other staff members, and their return after completion for which a postage-paid envelope will be provided.

It is hoped that at least one teacher from each year level, as well as school administrators, will be prepared to complete the questionnaire.

Could the form attached please be returned in the envelope provided, preferably by Monday, 18 October. Questionnaires will then be dispatched should a preparedness to be involved be expressed.

Thanking you,

Yours faithfully

470

MENTAL COMPUTATION SURVEY

Please complete the appropriate section below and return in the envelope provided (by Monday, 18 October) to:

Geoff Morgan Lawnton State School Phone: (07) 285 2968 P.O. Box 2 Fax: (07) 285 6506 LAWNTON 4501

Please SEND the questionnaire on Mental Computation.

(a) Name of School: ______State School

(b) Name of Contact Person: ______

(d) Number of Questionnaires required: _____

(e) Information to assist data analysis:

Number of class teachers on staff: _____

Number of Deputy Principals: _____

Please DO NOT SEND the questionnaire on Mental Computation.

471

D.3 Letter to Contact Persons Accompanying Questionnaires

Dear < Contact Person/Principal >

Thank you for your offer to act as Contact Person for the survey on Mental Computation. As requested, I have enclosed < number > copies of the questionnaire. Also enclosed is a Reply Paid envelope in which the completed questionnaires may be returned.

Hopefully the survey will not take too long to complete. For those who have been teaching since the mid-1960s, to which all sections may apply, it is anticipated that it should take around 15-20 minutes of their time.

As indicated in my original letter, all information received will be treated confidentially. It will constitute a valuable contribution to the knowledge about the teaching of Mental Computation in Queensland during the last quarter century.

Your assistance, and that of the other members of your school's staff who complete the survey, is greatly appreciated. Could the questionnaires please be returned by Friday, < Date > November.

Thanking you

Yours sincerely

472

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

Dear Colleague

I am writing to you with respect to the survey of Queensland state primary school teachers and administrators on Mental Computation that I am currently undertaking, to which documentation sent earlier this term referred.

A couple of schools have indicated that they wish to participate in the survey, but have mislaid my initial letter. On the off-chance that your school may be in a similar situation, I have enclosed 6 copies of the questionnaire. If you, or any of your staff members, are prepared to complete the questionnaire it would be greatly appreciated. I have enclosed a Reply Paid envelope in which any completed questionnaires may be returned.

Once again, your assistance, and that of the other members of your school's staff who complete the survey, is greatly appreciated. Could any completed questionnaires please be returned by Friday, 26 November.

Thanking you,

Yours faithfully Geoff Morgan

P.S. Should you have already posted the form, which was included with my previous letter, indicating that your school did not want to receive any questionnaires, please disregard this mailing. 473

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

Dear < Contact Person/Principal >

With this school term fast drawing to a close, I'm writing to once again express my thanks to you for agreeing to act as contact person for the mental computation survey which I'm currently undertaking. Could you please pass on my thanks to those at your school who completed the questionnaire. A preliminary look at the information so far received suggests that it will prove very useful in mapping the status of mental computation in Queensland primary classrooms.

I hope to begin analysing the data during the forthcoming vacation. To this end, it would be appreciated if the questionnaires that have been completed could be sent to me at your earliest convenience. Recognising how busy we all have been during this term, any that have been completed will be gratefully accepted.

Thanks once again for your assistance in conducting this survey. Wishing you an enjoyable vacation.

Yours sincerely Geoff Morgan 474

APPENDIX E

MEANS AND STANDARD DEVIATIONS FOR SURVEY ITEMS IN FIGURES 4.1 - 4.6

Figure 4.1: Beliefs about the nature Figure 4.4: Selected teaching of mental computation. practices of middle- and upper-school teachers.

Item Mean Std Dev. Item Mean Std Dev. 3a 2.89 .72 25 3.55 .55 4 3.17 .56 26 3.50 .60 5a 2.63 .74 29a 2.30 .75 6 3.17 .59 30a 1.79 .61 8a 2.89 .65 31 3.32 .61 9 3.24 .55 33a 2.13 .70 34a 3.18 .77 36a 1.60 .69

Figure 4.2: Beliefs about the general Figure 4.5: Teaching practices used approach to teaching mental during periods 1964-1968, 1969-1974, computation. and 1975-1987.

Item Mean Std Dev. Item Mean Std Dev. 10 2.32 .67 41a 2.78 1.08 13 3.18 .50 42a 2.78 1.02 14 3.24 .55 43aa 1.64 .87 44aa 1.58 .80 45aa 2.62 1.05 Figure 4.3: Beliefs about specific 46aa 1.47 .62 issues associated with developing mental computation skills. 41b 3.14 .76 42b 3.09 .81 Item Mean Std Dev. 43ba 1.81 .73 11 3.31 .55 44ba 1.64 .65 15a 2.16 .68 45ba 2.65 .88 16a 2.26 .65 46ba 1.53 .55 17a 2.45 .76 19 3.50 .50 41c 3.31 .72 20a 3.06 .66 42c 3.27 .77 21a 2.92 .70 43ca 1.89 .70 22 3.37 .51 44ca 1.86 .72 45ca 2.69 .95 46ca 1.73 .66

475

Figure 4.6: Selected teaching practices and beliefs of middle- and upper- school teachers.

Current Teaching Practices Current Beliefs

Item Mean Std Dev. Item Mean Std Dev. 25 3.55 .55 14 3.27 .52 26 3.50 .60 19 3.46 .50 29a 2.30 .75 20a 2.98 .63 31 3.32 .61 22 3.26 .50 34a 3.18 .77 8a 2.87 .67 36a 1.60 .69 5a 2.63 .70

Note. aItems representing traditional beliefs and teaching practices recoded to reflect a nontraditional orientation to facilitate placement of item means on traditional-nontraditional continua.