P e t e r We s t w o o d
What teachers need to know about
Numeracy
What teachers need to know about
Numeracy
What...

Author:
Peter Westwood

P e t e r We s t w o o d

What teachers need to know about

Numeracy

What teachers need to know about

Numeracy

What teachers need to know about

Numeracy

Peter Westwood

ACER Press

First published 2008 by ACER Press, an imprint of Australian Council for Educational Research Ltd 19 Prospect Hill Road, Camberwell Victoria, 3124, Australia www.acerpress.com.au [email protected] Text © Peter Westwood 2008 Design and typography © ACER Press 2008 This book is copyright. All rights reserved. Except under the conditions described in the Copyright Act 1968 of Australia and subsequent amendments, and any exceptions permitted under the current statutory licence scheme administered by Copyright Agency Limited (www.copyright.com.au), no part of this publication may be reproduced, stored in a retrieval system, transmitted, broadcast or communicated in any form or by any means, optical, digital, electronic, mechanical, photocopying, recording or otherwise, without the written permission of the publisher. Edited by Carolyn Glascodine Cover and text design by Mary Mason Typeset by Mary Mason Printed in Australia by Ligare National Library of Australia Cataloguing-in-Publication data: Author: Westwood, Peter S. (Peter Stuart), 1936– Title: What teachers need to know about numeracy / Peter Westwood. Publisher: Camberwell, Vic. : ACER Press, 2008. ISBN: 9780864319043 (pbk.) Notes: Includes index. Bibliography. Subjects: Numeracy—Study and teaching. Mathematics—Study and teaching. Dewey Number: 513.071

Contents

Preface

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1 Conceptualising numeracy Numeracy: important in its own right The evolving definition of numeracy The anatomy of numeracy Is numeracy the same as mathematics? Numeracy across the curriculum Affective aspects of numeracy 2 Numeracy in early childhood Pre-kindergarten Working with children in the preschool years Early childhood mathematics objectives Number sense 3 The development of number concepts Piaget’s theory Lev Vygotsky Jerome Bruner 4 The primary school years and beyond Transition from preschool to school Teaching in the primary years Positive intervention: the daily numeracy hour The key issue of teacher competence The secondary school years Adult numeracy 5 Calculating and problem solving The place of computational skills Number facts: the importance of automaticity v

1 2 4 6 8 9 10 13 14 17 18 19 24 25 29 30 33 34 35 38 39 41 42 46 47 49

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Use of a calculator Mental calculation Teaching problem solving Knowledge that learners need to acquire 6 Barriers to numeracy Teaching method as a cause of difficulty Other contributory causes The nature of students’ difficulties Dyscalculia

51 51 52 55 57 58 62 63 64

7 Assessment Purposes of assessment Approaches to assessment Assessing problem-solving skills Conclusion

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References Index

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69 71 76 78

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Preface

There can be no doubt that since the 1990s numeracy has been high on the agenda in many countries. There is an increasing need for numeracy skills in all aspects of life – at home, in employment, and in the community. Steen (2007, p. 16) states that, ‘Being numerate is one of the few essential skills that students absolutely must master, both for their own good and for the benefit of the nation’s democracy and economic well-being.’ Developments over the past two decades have seen a move toward less emphasis in schools on routine arithmetic teaching and more on application of number skills to problem solving. This change of emphasis is in response to recommendations for reform in the teaching of mathematics ema nating from influential bodies such as the National Council for Teachers of Mathematics in America and the Australian Association of Teachers of Mathematics. Departments of Education have also backed these reforms. We have reached a stage now when it is important to consider whether the changes in emphasis are proving effective. Have students gained a better understanding of mathematics than before; and do they have more positive feelings about the subject? Or have we moved too far away from teaching and practising computational skills so that now it is more difficult, rather than easier, for students to engage in problem solving and investigation? What degree of balance is needed between formal skills instruction and investigative approaches using those skills? This book explores some of the issues that are emerging in the domain of numeracy teaching. I have drawn on relevant literature from several different countries – notably the United Kingdom, the United States of America, Australia, New Zealand and parts of Asia – to provide a com prehensive overview. The issues range from those concerning children in the preschool and early school years through to those affecting adults with poor numeracy skills. I have provided many links to other sources of information, and I hope readers will find something of interest. vii

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My sincere thanks go to Carolyn Glascodine for her very efficient edit ing of this manuscript. Many thanks also to the staff at ACER Press for their continuing support. P e t er W e s t w o o d

Resources

www.acer.edu.au/need2know

Readers may access the online resources mentioned throughout this book through direct links at www.acer.edu.au/need2know

o n e

Conceptualising numeracy

Key issues ◗ Defining numeracy: The definition has expanded considerably over time. The process of change mirrors that of the evolution of the definition of literacy. We now acknowledge the existence of ‘multiple numeracies’. ◗ Relationship between literacy and numeracy: Numeracy should not be subsumed under literacy (as has happened in the past). Numeracy merits separate and serious attention. ◗ Relationship between mathematics and numeracy: The terms numeracy and mathematical competence are not synonymous. ◗ Core components of numeracy: These are difficult to specify because the demands on numeracy tend to be context-specific. ◗ Numeracy across the curriculum: Numeracy skills are needed in all subject areas. All school subjects can help to develop and generalise numeracy skills. ◗ Affective components of numeracy: The most recent descriptions of numeracy include reference to learners’ attitudes, confidence and disposition to use numeracy skills independently.

The term numeracy appears to have been coined officially many years ago in a report dealing with the education of students in upper secondary schools in the United Kingdom (the Crowther Report: Central Advisory Council 1

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for Education, 1959). In that document numeracy was presented as the companion skill to literacy. Numeracy was seen to be the ability to deal successfully with the quantitative aspects of everyday life, while literacy was the ability to cope with normal demands of reading and writing. Numeracy and literacy were thus seen at that time as separate but complementary domains of competence (O’Donoghue, 2002). In many ways it was unfortunate that this distinction between literacy and numeracy became blurred later, when a much broader interpretation of the concept of literacy emerged. Some educators and writers began to equate the term ‘literate’ with being ‘adequately educated’, not just able to read and write; and they considered that numeracy was therefore simply one aspect of being literate. Such a view was evident in 1990 when official documentation for the International Year of Literacy stated that literacy included numeracy (DEETYA, 1997). Similarly, in 1995 a major project under the Australian Language and Literacy Policy also included numeracy within its definition of literacy (Cumming, 1996). Even today, the two terms literacy and numeracy are often used together in an integrated way as if describing a single information-processing ability. The problem with taking literacy and numeracy closely together, for example in research studies and when funding is allocated for special projects, is that literacy (in the sense of reading) always seems to get the lion’s share of attention (e.g., Department of Education and Training, WA, 2006; Dymock, 2007; van Kraayenoord, Elkins, Palmer & Rickards, 2000). For many years, the amount of research and intervention in the literacy domain far outweighed research into numeracy. This tendency to subsume numeracy under literacy may also be one of the reasons why numeracy remained a relatively neglected area in education policy making until the mid-1990s.

Numeracy: important in its own right Common sense would suggest that literacy and numeracy are separate domains of competence; and by the end of the 1990s, the Australian Association of Mathematics Teachers (AAMT) was strongly expressing such a view. In its policy document on numeracy teaching in schools the AAMT (1998) made it clear that literacy and numeracy are fundamentally different areas of learning and each merits separate consideration.

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The recommendation to address students’ numeracy separately from literacy saw a flurry of interest and action, with many countries publishing guidelines for improving numeracy standards, and with a parallel increase in research interest and writing in this domain. For example, in the United States of America, the National Council of Teachers of Mathematics (NCTM) had first produced a document titled Curriculum and evaluation standards for school mathematics in 1989, but by 1997 a revision and updating of this material commenced. The outcome was the publication in 2000 of the influential document Principles and standards in school mathematics, in which numeracy was given a higher profile. In Australia, interest in numeracy, particularly among adults in the workplace, had started in the 1990s or even earlier ( Johnston, 2002). Activity in the field was given extra impetus in 1996 when the Common wealth, state and territory governments endorsed the National Plan for Literacy and Numeracy, followed soon after by the publication of a report titled Numeracy = Everyone’s business (DEETYA, 1997). National bench mark testing in numeracy was soon introduced for school students in Years 3, 5 and 7. From 2008, students’ numeracy skills in Years 3, 5, 7 and 9 are being tested under the National Assessment Program in Literacy and Numeracy (NAPLAN) (Curriculum Corporation, 2008). The Adult Literacy and Numeracy Australian Research Consortium (ALNARC) (2002) was established in 1999 and initiated several projects and organised forums on themes related to numeracy. The year 2000 saw the publication of a Commonwealth Government document Numeracy, a priority for all: Challenges for Australian schools. Since that time a number of studies have been conducted into various aspects of students’ learning and achievement in mathematics (e.g., Louden et al., 2000; van Kraayenoord et al., 2000). All education systems across the states and territories have drawn up numeracy development plans. A typical example of such a plan is the one operating in NSW (Department of Education and Training, NSW, 2006: see the Links box at the end of the chapter). In New Zealand, 2001 saw the introduction of the Numeracy Development Project, a well-documented and successful teacher and school development initiative that is still continuing (Annan, 2006; Irwin & Niederer, 2002). In the United Kingdom, the National Numeracy Strategy was launched in 1998 and implemented in schools from 1999. In that year, the UK commenced a daily period of intensive teaching of basic mathematical

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skills in primary schools (the ‘numeracy hour’) to complement the already existing ‘literacy hour’. The format of such sessions will be discussed in a later chapter. The year 2006 saw the Department for Children, Schools and Families in the UK issuing a revised Primary Framework for Literacy and Mathematics with renewed emphasis on numeracy and with some modifications to the original 1998 objectives (DfCSF, 2007a). The adult education sector has focused heavily on numeracy in recent years, presumably because schools have been far from successful in dev eloping the numeracy skills of an alarmingly high proportion of the student population (Boaler, 1997; Coben, 2003; Munn, 2005). It is reported that many adults lack the numeracy skills needed to function in a maximally effective manner in their vocational, civic and personal lives (Wiest et al., 2007). In the United Kingdom the National Research and Development Centre for Adult Literacy and Numeracy has been very active since 2001 with publications, resources and a major project called Maths4life, concerned with numeracy and what they termed ‘non-specialist’ mathematics. The main project, established for the Department of Education and Skills, ran from August 2004 to March 2007 and has now been transferred to the National Centre for Excellence in Teaching Mathematics (NCETM). There are positive signs that numeracy courses for adults yield good results, not only in terms of skill acquisition but also social and personal gains (Balatti et al., 2006; Dymock, 2007). More will be said about adult numeracy in Chapter 4.

The evolving definition of numeracy At first glance, the meaning of the term numeracy would appear to be simple and straightforward. Surely, it means the ability to apply number concepts and arithmetic skills for everyday purposes, together with the ability to interpret the quantitative information that bombards us daily from many different sources? Certainly this simple view accords well with that appearing in the Cockcroft Report Mathematics counts (1982) in which the writers proposed: We would wish ‘numerate’ to imply the possession of two attributes. The first of these is ‘at-homeness’ with numbers and an ability to make use of mathematical skills which enable an individual to cope with the practical mathematical demands of his [sic] everyday life. The second is an ability

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to have some appreciation and understanding of information which is presented in mathematical terms, for instance graphs, charts and tables or by reference to percentage increase or decrease. (p. 11)

In the United Kingdom, Askew et al. (1997) have defined numeracy as the ability to process, communicate and interpret numerical information in a variety of contexts. This definition is echoed in Australia in the AAMT (1998, p. 2) policy statement: ‘To be numerate is to use mathematics effectively to meet the general demands of life at home, in paid work, and for participation in community and civic life’. New Zealand’s definition of numeracy is very similar: ‘To be numerate is to have the ability and inclination to use mathematics effectively at home, at work and in the com munity’ (Ministry of Education, NZ, 2001). All this seems very straightforward, and surely not contentious? But Coben (2003, p. 9) warns us that, ‘Numeracy is a deeply contested and notoriously slippery concept’. Most of the difficulty relates to deciding exactly which specific areas of knowledge and skill together constitute numeracy. The concept of numeracy has undergone many changes in the years since the term was coined in the Crowther Report. Numeracy is now viewed as a ‘… multifaceted and sophisticated construct, incorporating mathematics, communication, cultural, social, emotional and personal aspects of each individual in context’ (Maguire & O’Donaghue 2002, cited in American Institutes for Research, 2006, p. 6). According to Turner (2007, p. 28), ‘Numeracy has become a personal attribute very much depen dent on the context in which the numerate individual is operating ... [and] numeracy will mean different things to different people according to their interests and lifestyles’. So how did such a simple concept become so complex? The changes that have occurred in conceptualising numeracy tend to parallel the changes that occurred in the past 25 years with the concept of literacy (Falk et al., 2002). Mainly as a result of ideas emanating from what has become known as ‘new literacy studies’, literacy is no longer regarded as simply being able to read and write. Literacy now embraces reading, writing, listening, speaking, viewing and critical thinking; and is said to exist in many forms described as ‘multiple literacies’ (Richards & McKenna, 2003). Examples include, ‘literacy of the workplace’, ‘consumer literacy’, ‘critical literacy’, ‘school literacy’, ‘mathematical literacy’, ‘financial literacy’, ‘literacy for the digital

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age’, and so on. Literacy acquisition is seen to be socially and culturally determined, and its nature, role and importance interpreted differently in different contexts – thus leading to the notion of ‘situated literacies’ (Barton et al., 2000). In very much the same way, the acquisition and application of numeracy is now regarded as culturally based, socially determined, situated and context-specific (Baker et al., 2001; Kerka, 1995). To illustrate the point, Steen (2000; 2007) provides examples to show how numeracy demands on individuals vary in a range of different real-life contexts. The term ‘multiple numeracies’ is now emerg ing in the professional literature (e.g., Baker et al., 2006; Gough, 2007; Grubb, 1996; Johnston, 1994). Examples include ‘community numeracy’, ‘critical numeracy’, ‘workplace numeracy’, ‘consumer maths’, and ‘street maths’. We can appreciate that numeracy skills are used for quite different purposes in different contexts. For example, Butcher et al. (2002) refer to ‘numeracy for practical purposes’, ‘numeracy for interpreting society’, ‘numeracy for personal organisation’ and ‘numeracy for knowledge’. Similarly, Steen (1997) suggests that numeracy, or ‘mathematical literacy’, needs to be: ◗ practical (for everyday use) ◗ civic (to understand issues in the community) ◗ professional (for employment) ◗ recreational (for example, understanding scoring in sports and games) ◗ cultural (as part of civilised persons’ deep knowledge and culture).

So, the precise meaning of numeracy will vary according to the context and purposes for which numeracy skills are used. This is perhaps why Coben (2003) suggests that numeracy is a slippery concept.

The anatomy of numeracy The fact that numeracy manifests itself in different ways according to the context in which it is applied makes it difficult to determine precisely what constitutes essential ‘core’ content in numeracy teaching. Steen (2000, p. 17) states that, ‘Numeracy has no special content of its own but inherits its content from its context’. But surely there must be some areas of know ledge and skill that are absolutely fundamental and would apply in all situations?

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Different writers have addressed in different ways the issue of which areas of knowledge and skill actually comprise numeracy. Gough (2007) tells us that numeracy embraces much more than ‘numberacy’ – meaning that the domain of numeracy includes much more than basic arithmetic. According to DEETYA (1997, p. 39) the mathematical underpinning of numeracy is ‘not restricted to working with numbers, but also includes work with space, data (statistical and measurement) and formulae’. In the adult numeracy domain, SAALT (2006) confirms that numeracy incorporates basic number skills, spatial and graphical skills, measurement and problem solving. Gough (2007) makes the interesting proposal that the content of numeracy is represented by most of what comprises the typical primary school mathematics course. Gough excludes all aspects of mathematics that the large majority of adults do not use – this would include much of what is contained in the typical secondary school academic maths curriculum. If his proposal is sound, the primary curriculum may serve as a guide to the core content needed to achieve numeracy. The Primary Framework for Mathematics in the United Kingdom (covering problem solving, reasoning and numeracy for Years 1 to 6) identifies seven strands that together make up the learning area. They are fairly similar to the six strands that form the mathematics curriculum in Australian schools (Curriculum Corporation, 1994). The strands in the UK are: ◗ counting and understanding number ◗ knowing and using number facts ◗ calculating ◗ understanding shape ◗ measuring ◗ handling data ◗ using and applying mathematics.

No doubt there could be endless debates concerning the precise level of com petence an individual would need to develop in each of these strands in order to be deemed ‘numerate’. But, the list above does represent a very reasonable starting point for giving some substance to the concept of numeracy. With a focus on teaching numeracy with adults, Ginsburg et al. (2006) suggest that courses should be organised around four key strands: ◗ number and operations sense ◗ patterns, functions and algebra

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◗ measurement and shape ◗ data, statistics and probability.

Within each of these strands, a learner needs to become proficient in four areas: ◗ conceptual understanding ◗ adaptive reasoning ◗ strategic competence for problem solving ◗ procedural fluency.

It is also seen as vital that the learner develops a positive attitude toward mathematics and acquires a ‘productive disposition’. More will be said later on the affective component of numeracy. DEETYA (1997) concludes that numeracy involves a combination of: ◗ mathematical concepts and skills from across the discipline (numerical, spatial, graphical, statistical and algebraic) ◗ mathematical thinking ◗ numerical strategies ◗ appreciation of context.

The above statement almost makes it seem that numeracy = mathematical ability. But is this so? Is numeracy simply another name for basic mathe matical competence?

Is numeracy the same as mathematics? At the beginning of this chapter it was argued that numeracy does not equal literacy; the two draw upon different bodies of knowledge and involve different processing skills. But numeracy and mathematics do draw upon the same body of knowledge and skills, so what is the relationship between the two? Perso (2006a) states that to be numerate you must know some mathe matics; but simply knowing some maths does not necessarily make a person functionally numerate. Martin (2007, p. 28) takes up the point and writes, ‘Just as knowing the definition of words does not make a person literate, knowing rules and algorithms to solve mathematics problems does not make a person mathematically literate.’ It is generally agreed that mathematical competence comprises more than numeracy (e.g., Lott, 2007).

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How teachers view numeracy in relation to mathematics is important because it will influence the way in which they structure the components of their classroom curriculum. Perso (2006a; 2007), who clearly accepts that numeracy is different from mathematics, suggests that this relationship is not well understood by some teachers, resulting sometimes in an imbalance in their approach – for example, placing a major emphasis on computation at the expense of task-based or investigative approaches; or vice versa. The official position (DEETYA, 1997, p. 11) states categorically that: Numeracy is not a synonym for school mathematics, but the two are clearly interrelated. All numeracy is underpinned by some mathematics; hence school mathematics has an important role in the development of young people’s numeracy.

Steen (2007), on the other hand, feels that the dichotomy between what is mathematics and what is numeracy should be eliminated, particularly the attitude that ‘abstract mathematics’ represents a more respectable level of academic study while numeracy is simply contextualised arithmetic for commercial and social purposes. Steen (2007, p. 18) says: ‘Unfortunately, numeracy is often characterised as watered-down mathematics – minorleague curriculum that schools offer to those who are unable to compete in the major league of algebra, trigonometry and calculus’. Steen feels that the everyday maths needed for understanding such events as elections, poll results, consumer finance, discounts, home management and clinical trials is just as important as the content in any academic-style secondary maths courses. It is suggested that recent UK reports and White Papers affecting numeracy in the post-school years are tending to reinforce the perception of a vocational/academic divide in their use of the term ‘functional mathematics’ (Hudson, 2006).

Numeracy across the curriculum The fact that students often do not spontaneously use their mathematical knowledge in other areas highlights the important role of all teachers in helping to facilitate this process of transfer and generalisation (Thornton & Hogan, 2005). Numeracy skills remain inert unless they can be readily applied in a variety of situations and for a variety of purposes. The official view is that ‘Numeracy is a proficiency that is developed mainly in mathematics but also in other subjects’ (Df EE, 2002, p. 9).

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Just as all teachers are said to be teachers of literacy, so too should all teachers endeavour to be teachers of numeracy by taking every opportunity to introduce students to the statistical and quantitative aspects of their subjects, and to relate these aspects to the real world (Posamentier & Jaye, 2007). Steen (2000) states that numeracy is not just one among many subjects but an integral part of all subjects. Perso (2006b, p. 27) suggests that: ‘Within each learning area two questions must be asked: ◗ How can numeracy contribute to enhanced learning outcomes in this learning area? ◗ How can this learning area enhance students’ numeracy?’

By integrating mathematical components into all school subjects, students are helped not only to strengthen and generalise their skills and under standings but also to appreciate the utility of numeracy in a wider sense (DETYA, 2000; DfCSF, 2001). Hogan et al. (2004) observe that knowledge of mathematics and its application in a range of contexts seems to provide students with the confidence to have a go, make mistakes and try again. In addition to infusing numeracy into school subjects and into learning projects, there are also very many events occurring regularly in any school day that provide authentic opportunities for students to exercise their numeracy skills for a genuine purpose – budgeting for school camps, concerts, field trips, bring-and-buy sales, fundraising events, sports days, inter-school matches, and many more (e.g., Rennie, 2006; Zawojewski & McCarthy, 2007). Effective teaching of numeracy will make full use of all such naturally occurring events.

Affective aspects of numeracy It is not only the contextual aspects of numeracy that have been stressed in recent definitions; increasing importance has also been placed on affective, as well as cognitive, aspects of ‘being numerate’. Ginsburg et al. (2006, p. 30) explain that, ‘The affective component of numeracy includes the beliefs, attitudes and emotions that contribute to a person’s ability and willingness to engage in, use, and persevere in mathematical thinking and learning, or in activities with numeracy aspects’. The terms ‘positive disposition’ and ‘positive inclination’ (to use mathematical skills) are appearing more

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frequently when the characteristics of a numerate person are described (e.g., Kilpatrick et al., 2001). Disposition and inclination in relation to numeracy would include an individual’s confidence, emotional comfort, interest and willingness to try to interpret and process quantitative data and solve problems. For example, Kemp and Hogan (2000, p. 3) state that, ‘Numeracy is having the disposition and critical ability to choose and use appropriate mathematical knowledge strategically in specific contexts.’ Similarly, Turner (2007) explains the role of inclination as having the desire and motivation to use numeracy skills. Turner further remarks that, ‘Having negative feelings about mathematics and one’s mathematical ability, implies a reluctance to use mathematics and hence a failure to be fully numerate’ (p. 33). Positive attitudes, interest in, and motivation for mathematics begin in the early years of childhood, and are either fostered and encouraged on entry to school or are snuffed out by lack of success. The following chapter looks at development of numeracy in these crucial early years.

L i nk s t o m o r e ab o u t c o nc e p t u al i s i n g n u m e r ac y ◗ Ginsburg, L., Manly, M., & Schmitt, M. J. (2006). The components of numeracy. Cambridge, MA: National Center for the Study of Adult Learning and Literacy. Available online at: http://www.ncsall.net/ fileadmin/resources/research/op_numeracy.pdf ◗ Coben, D. (2003). Adult numeracy: Review of research and related literature. London: National Research and Development Centre for Adult Literacy and Numeracy. Available online at: http://www.nrdc.org.uk/ uploads/documents/doc_2802.pdf ◗ American Institutes for Research. (2006). A review of the literature on adult numeracy: Research and conceptual issues. Washington, DC: US Department of Education. Available online at: http://www.eric.ed.gov/ ERICDocs/data/ericdocs2sql/content_storage_01/0000019b/80/29/ e3/66.pdf ◗ Steen, L. A. (2000). The case for quantitative literacy. Available online at: http://www.maa.org/ql/001-22.pdf >

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◗ The in-service training material Numeracy across the curriculum produced by DfCSF (2001) provides some very useful suggestions and activities to help teachers consider the role of numeracy in different subject areas. It also contains suggestions for specific content and skills that students should acquire by Year 9 – although some might argue that the list goes beyond what is normally considered basic numeracy. Available online at: http://www.standards.dfes.gov.uk/secondary/ keystage3/all/respub/numxc ◗ A good example of a Numeracy Plan is the one devised by Department of Education and Training, New South Wales. Available online at: http:// www.curriculumsupport.education.nsw.gov.au/primary/mathematics/ assets/pdf/numeracy_plan_15mar06.pdf ◗ Gough (2007) suggests that numeracy comprises the concepts, strategies and skills typically taught and developed in the primary years. For reference, a comprehensive list of objectives for mathematics covering Reception to Year 6 can be found online at: http://www.thegrid.org.uk/learning/maths/ks1-2/assessment/documents/ nnskeyobjectives.doc ◗ New Zealand’s curriculum statement on numeracy teaching can be located online at: http://www.tki.org.nz/r/literacy_numeracy/num_ practice_e.php

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Numeracy in early childhood

Key issues ◗ Early childhood as the foundation stage in numeracy development: Even before formal schooling begins, children are acquiring an informal understanding of quantitative relationships. They also develop their own strategies for dealing with number. And the early years shape a child’s feelings about engaging in number work and mathematics. ◗ The teacher’s role in the early years: The teacher is not only a facilitator but must also act as mediator, helping to interpret children’s quantitative experiences. ◗ Developing number sense: Number sense underpins genuine understanding of all numerical relationships and processes. Possessing number sense in mathematics is to some extent similar to possessing phonological sense and understanding the phonic concept in reading. ◗ Core objectives for early numeracy: In early childhood education it is essential to create a firm foundation of number concepts and skills. There is reasonable agreement on what needs to be taught.

The numeracy concepts and skills that most individuals possess in adol escence and adulthood had their beginnings in the very earliest stages of childhood, long before any formal instruction took place. It is even suggested that the human brain is in some way ‘pre-programmed’ before

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birth to process quantitative information – Butterworth (2005) describes it as ‘innate numerosity’. Beaty (1998, p. 227) remarks: Using the physical and mental tools they are born with, children interact with their environment to make sense of it, and in doing so they construct their own mental concepts of the world. The brain seems to be conditioned to take in information about objects and their relationships to one another.

Even before they enter kindergarten, children appear to acquire an intuition about number that enables them to tackle simple quantitative tasks and deal with everyday problems successfully (Koralek, 2007). There is evidence that infants not yet 4 months old can distinguish visually the difference between unequal quantities; and by the time they are 3 years old, most children have developed a pre-symbolic sense of number ( Jung et al., 2007). Griffin (2004) suggests that by the age of 4 years children develop two systems, one for making global comparisons and another for counting. By the age of 5 or 6 years these two systems have combined to provide a more powerful grasp of number relationships upon which subsequent numeracy concepts and skills develop. At around this age children begin to grasp that moving forward or back along a counting sequence is exactly the same as adding or subtracting. When applied to a numberline, this provides them with a mental model of addition and subtraction and marks a major step forward in their understanding of number relationships (Griffin & Case, 1997). The studies of Barth et al. (2006) and Gilmore et al. (2007) reveal that young children can carry out forms of quantitative comparison and simple addi tion and subtraction long before these skills are taught in school. Through exposure to more advanced individuals in their social setting children observe and acquire essential skills such as counting, sequencing and pattern making. Through play with others they meet concepts such as shape, relative size, capacity, sharing, sorting and classifying. They begin to compare and contrast groups of objects, and quantitative elements begin to appear in their early drawings, suggesting that they are able to invent simple ways of representing number relationships (Pound, 1999).

Pre-kindergarten In the United States of America, the document Principles and standards for school mathematics (NCTM, 2000) included coverage of children in the pre-

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kindergarten age range for the first time. The provision of Pre-K standards is a clear indication that numeracy development is now regarded as beginning soon after birth. The Pre-K objectives cover mainly counting, recognising ‘how many’, cardinal and ordinal number, connecting number words to numerals, sorting and classifying, recognising patterns and sequences, and identifying basic shapes. A similar stand is taken in the United Kingdom, where problem solving, reasoning and numeracy are identified as key learning areas for the very early years (birth to five) (Pimentel, 2007). The Department for Children, Schools and Families (DfCSF, 2005) has stressed the importance of stimulating children’s learning in the first three years of life and has prepared a guidance package titled Birth to three matters to help parents, caregivers and early-years educators support children’s learning, including numeracy, in the years before kindergarten. In Australia, where ‘early childhood’ spans the period from birth to age 8, the Australian Association of Mathematics Teachers joined forces with Early Childhood Australia to issue a joint position statement on Early childhood mathematics (2006). In relation to the preschool and beginning school years this statement highlights the importance of: ◗ engaging children’s natural curiosity ◗ using play and child-initiated activities as the focus ◗ dealing with quantitative issues relevant for a child’s age ◗ solving problems (in the sense of tackling real tasks that involve number concepts and skills) ◗ providing opportunities by supplying abundant materials, space and time ◗ using language to develop maths concepts and vocabulary ◗ encouraging mental manipulation of ideas ◗ assessing children’s level of development as a basis for planning activities.

Charlesworth (2005, p. 235) observes that: Pre-kindergarten mathematics focuses mainly on young children’s natur alistic explorations and the ability to provide informal scaffolding through questions and comments. Young children are developing mathematics concepts and skills at the initial level. These concepts include one-toone correspondence, number sense and counting, logic and classifying, comparing, parts and wholes, ordering and patterning, measurement and

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concrete addition and subtraction … [and] young children also begin to recognise number symbols and experiment with technology.

In the pre-kindergarten years, children who are exposed to normal experiences in the home and community develop an understanding of quantitative relationships. They acquire this awareness mainly through play, exploration and everyday experiences (Aubrey et al., 2003; Perry & Dockett, 2007; Tucker, 2005). For example, they are exposed to language that accompanies numerous informal quantitative experiences such as serving food at the table: (‘Give me two potatoes today, please, because they are rather small’; ‘Pour everyone half a glass of water’) or at the supermarket (‘Can you get two tins of peas please; not the big tins, the smaller ones’; ‘The apples are three for $1.00 today’) and so forth. Young children also hear and use number names as they join in with number rhymes and songs. They hear sister counting the steps as she walks upstairs, and they hear brother say, ‘Mum, I have four pages of homework to do tonight, but I have already finished two’. These incidental encounters with the language of everyday mathematics are laying an important foundation. Through these informal quantitative experiences, most children in early childhood begin to develop confidence with numbers – counting, sharing, comparing, and part–whole relationships – without direct teaching (Zaslavsky, 2001). Parents could do much to encourage children’s curiosity about numbers and number relationships simply by drawing attention in an interesting way to relevant quantitative situations, asking questions, making comments, and making more explicit their own daily use of numbers (Doig, McCrae & Rowe, 2003; Griffiths, 2007; Maher, 2007). Of course, not all children encounter a rich language and number environment at home and therefore do not enter school with the same depth of knowledge and experience. It is for children such as these that intervention programs in the kindergarten are needed. For example, Griffin (2004) describes one such intervention called ‘Number Worlds’. This is a research-based kindergarten maths program that provides rich experiences of investigating number in a variety of ways through a games approach. The program is reported to bring significant gains in children’s number knowledge. The activities seem to be particularly beneficial for disadvantaged children whose prior learning has not resulted in optimum development of number awareness before entry to school. Other early

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number programs include Mathematics Recovery (Wright, 2003), Numeracy Recovery (Dowker, 2001; 2005) and Early Numeracy: Assess ment and Teaching for Intervention (Wright, Martland & Stafford, 2006). Doig et al. (2003) provide a review of several programs designed for the early years.

Working with children in the preschool years It is recognised now that most young children enter school with a great deal of informal knowledge about number and mathematics (Thomson et al., 2005). The role of early childhood educators is to build upon this knowledge by creating an environment where children can continue to explore quantitative relationships with the aid of a teacher and with peers. Situations that invite quantitative investigation are most likely to arise in preschool classrooms where there is an abundance of materials and equipment available that will encourage children to play and investigate. In particular, every early childhood setting should have a ready supply of building blocks, boxes, counters, tiles, shapes, pattern boards, measuring tapes, calculators, squared paper, jars of beads, egg cartons and so on (Wallace et al., 2007). Adequate time and opportunity need to be made available for children to play and work with these materials. In the preschool years, direct and formal teaching of number skills is generally not recommended, although kindergarten programs in some countries (e.g. in parts of Asia) do introduce children to the beginning levels of arithmetic. Children of this age really need to discover number relationships for themselves and invent ways of representing or recording these, rather than having abstract processes taught to them in some formal manner. Most attempts at formal instruction end up destroying a child’s natural curiosity and confidence. Clements (2001) writes that preschools should capitalise fully on young children’s high level of natural motivation to learn in a self-directed manner. If early learning situations use children’s interests, they will help to promote a positive view of mathematics as an enjoyable, self-directed, problem-solving activity. None of the above is intended to suggest that a teacher must stand back and play no active part in fostering children’s early number development. It is believed now that early learning is greatly enhanced if adults help

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young children interpret their learning experiences (Baroody, 2000; Fleer & Raban, 2005; Kirova & Bhargava, 2002). Teachers should work with children’s interests and spontaneous activities, and also deliberately introduce many new mathematical situations into the curriculum (Epstein, 2003; Groves, Mousley & Forgasz, 2006). Skilled preschool teachers seize opportune moments to impose some degree of structure on naturally occurring events or situations in the day. This structuring is commonly achieved by supporting (scaffolding) the child’s own discoveries through giving encouragement, thinking aloud, making suggestions and asking questions. Warfield (2001) advises teachers to pose many questions and problems involving number throughout the school day, not just in maths lessons. The terms ‘mediated learning’ and ‘guided participation’ have become popular when describing the teacher–child shared interaction in such situations (Rogoff, 1995). The Russian psychologist Vygotsky (1978) presented the notion of a ‘zone of proximal development’. This relates to each child’s potential to move forward in his or her learning. All learners, when provided with relevant information or guidance at the right moment, can advance their conceptual development by building on what is already known and under stood. The zone of proximal development for a child represents the new learning that can take place if another person supports the child directly or indirectly in some way. The role of the educator is to supply information at the teachable moment and to build bridges between abstract ideas and the real world. Gradually, these supports are withdrawn so that the child is dealing with the task or problem independently. Learning activities that fall within a child’s zone of potential development have a high probability of success, whereas activities beyond the zone are usually too difficult for the child and may result in failure and frustration.

Early childhood mathematics objectives It is impossible (and probably undesirable) to be prescriptive about the precise knowledge, skills and attitudes relating to number work that should be addressed in the preschool years. However, some guidance may be found by combining suggestions from several different authorities. For example, from Sarama and Clements (2006) and Kehl et al. (2007) the following list could be offered as a possible core set of objectives:

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◗ building positive interest and confidence in working with numbers ◗ pattern making and sequences, for example with blocks or tiles ◗ verbal counting in sequence to 10, to 20 (or even to 100) ◗ using one-to-one correspondence in counting objects to 20 ◗ counting a collection of objects and knowing that the last count tells ‘how many’ ◗ recognising instantly how many objects are in a very small set (less than 5) without counting; this ability is known as subitising ◗ recognising numerals to 20 ◗ joining or separating small sets of objects, telling how many in each set and altogether ◗ sharing items equally among friends ◗ comparing small sets of objects and using terms such as same, equal, more, less ◗ knowing the number that is ‘one more than’ ◗ simple adding and subtracting below 10 ◗ recognising and naming shapes (circle, square, triangle, rectangle) ◗ building new shapes using other shapes ◗ recognising symmetry in shapes ◗ simple measuring ◗ grouping objects based on attributes ◗ making simple picture graphs, for example using shoes, pets, toys ◗ communicating ideas and information to others, based on the above contexts.

There will be significant variation among children in the extent to which they master the above skills. Preschool teachers need to assess each child’s achievement in order to plan appropriate learning experiences or to provide additional teaching for some children prior to school entry. Assessment is discussed more fully in Chapter 7.

Number sense Perhaps the most important development in the early years is the acquisi tion of ‘number sense’. Howell and Kemp (2006) observe that since the late 1980s the term number sense has gained much recognition and is now being used frequently within curriculum and policy documents to describe the

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informal – and often intuitive – understanding of number that all children need to develop if they are to succeed in mathematics. Steen (2000, p. 8) describes it as ‘… having accurate intuition about the meaning of numbers, confidence in estimation, and commonsense in employing numbers as a measure of things’. There is no widely accepted definition of the term number sense and Berch (2005) says that the way in which experts in mathematical cognition use the term differs from the way that it is understood and used by mathematics educators. Nor is there complete agreement among experts on the exact knowledge and skills that together make up number sense. Number sense develops from all the informal quantitative concrete experiences that a child encounters in the early years, and it eventually comes to underpin the child’s smooth entry into arithmetic in primary school. Number sense continues to expand through the school years as children engage with new tasks and solve new problems (Gersten & Chard, 1999; Wells, 2000). If children arrive in primary school lacking number sense, they are extremely likely to have difficulties when the more formal aspects of computation are introduced. Jordan et al. (2007) report that children’s number sense in kindergarten is highly correlated with maths achievement at the end of Grade 1. Griffin (2004) believes that acquisition of number sense follows a developmental path and can be enhanced in the early years by providing a learning environment in which quantitative concepts can be freely explored, interpreted and discussed. Similarly, Wells (2000) believes that number sense is developed most easily in situations where children are actively involved in their learning and where teachers encourage reflection and discussion on quantitative aspects of their activities. A teacher’s role is to encourage children’s number sense development by noticing at what level they are currently operating and helping them develop to the next level by questioning and challenging their thinking (Schwerdtfeger & Chan, 2007). Encouraging children to work mentally with numbers, rather than writing them down, is also considered important (Cutler, 2001). Number sense is least likely to develop fully in classrooms where written computation and memorisation of procedures or rules are introduced too early and become the prime focus of attention (Smith & Smith, 2006; Yang, 2005). Yang (2005) believes that too much emphasis on written computation narrows

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children’s thinking and reasoning at an early stage and causes them to begin to rely on rote memory, rather than meaningful learning. Howell and Kemp (2006) suggest that assessing young children’s number sense may be a valid way of detecting children who are likely to have difficulties later in primary mathematics. Early detection could then lead to appropriate forms of intervention to help these children develop the understanding and confidence with numbers they currently lack. A similar notion was put forward by Malofeeva et al. (2004) who devised a suitable test for this purpose for children in the 3- to 5-year age range. The test assessed six components of number awareness, namely counting, number identification, number-to-object correspondence, ordinality, comparison, and simple addition and subtraction. A study by Jordan et al. (2007) confirms that early testing of this type will indeed help to identify children at risk of learning difficulties in primary mathematics. The study by Howell and Kemp (2006) endeavoured to obtain consen sus on the precise components of number sense by soliciting opinions from mathematics experts in different countries. Eventually they gener ated a fairly daunting list of some 35 possible components; but most of these can be subsumed under broader categories such as counting, matching, comparing, ordering, combining groups, simple subtraction, numeral recognition and a sense of magnitude. Subitising (i.e., recognising the number of items in a small group without needing to count them) is also included by some experts (e.g., Clements, 1999). A few writers suggest that number sense also includes automatic recall of basic arithmetic facts (American Mathematical Association of Two-Year Colleges, 1995; cited in Ginsburg et al., 2006). It is worth noting that in addition to number sense, Steen (2000, p. 9) also refers to ‘symbol sense’ – that is, being comfortable using and inter preting signs and symbols. And other writers have introduced the notion of ‘operations sense’, meaning a deep understanding of how algorithms in arithmetic actually do model number operations. Obviously these areas of awareness develop a little later than the basic number sense described above. Cutler (2001) suggests that understanding the acquisition of number sense can be understood best by considering how concepts are developed. We are helped in this process by looking in the next chapter at the work

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of Piaget (1942; 1983), Vygotsky (1962) and Bruner (1960; 1966), who in different ways were all interested in children’s concept development.

L i nk s t o m o r e ab o u t e a r ly c h i l d h o o d n u m e r ac y ◗ Joint position statement by National Association for the Education of Young Children (NAEYC) and the National Council of Teachers of Mathematics (NCTM). (2002). Early childhood mathematics: Promoting good beginnings. Available online at: http://www.naeyc.org/about/ positions/psmath.asp ◗ Joint position statement by Australian Association of Mathematics Teachers and Early Childhood Australia. (2006). Early childhood mathematics. Available online at: http://www.aamt.edu.au/content/ download/722/19512/file/earlymaths.pdf ◗ Exemplars for early number concepts are presented in the New Zealand Curriculum Framework, Ministry of Education. (2006). Available online at: http://www.tki.org.nz/r/assessment/exemplars/maths/strategy/st_ overview_e.php ◗ Other useful information from New Zealand available at: http://www. nzmaths.co.nz/numeracy/Intro.aspx and at http://www.nzmaths.co.nz/ Numeracy/2006numPDFs/NumBk1.pdf ◗ A useful resource for early childhood educators and parents is Early childhood numeracy cards produced by Department of Education, Science and Training (Australia). (2006). These cards present ageappropriate photographs of real-life situations with quantitative elements. Discussion points and questions are presented on the reverse side. See samples online at: http://www.dest.gov.au/sectors/school_ education/programmes_funding/programme_categories/early_childhood/ learning_resources#The_resource_materials ◗ Fleer, M., & Raban, B. (2007). Early childhood literacy and numeracy: Building good practice. Canberra: DEST, Commonwealth of Australia. This is a booklet for early childhood educators, containing advice on creating an environment for mathematics, the role of language, and observation of children’s knowledge and strategies. It also contains guidance on concepts to be introduced in the early years.

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Available online at: http://www.dest.gov.au/sectors/school_education/ programmes_funding/programme_categories/early_childhood/learning_ resources#The_resource_materials ◗ The Commonwealth Department of Education, Science and Training website provides details of (and links to) a number of projects and reviews that have involved numeracy in Australian school settings. Available online at: http://www.dest.gov.au/sectors/school_ education?policy_initiatives_reviews/key_issues/literacy_numeracy/ numeracy_publications.htm

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The development of number concepts

Key issues ◗ How are number concepts developed? We form concepts as a result of engaging in, and interpreting, a variety of learning experiences. New information is assimilated and linked with prior knowledge as we categorise and make sense of our environment. ◗ The theories of Piaget, a developmental perspective: Piaget’s observations on children’s cognitive development at various stages have greatly influenced our views on developmentally appropriate practice. His notion of ‘schema’ helps us understand how new learning is linked with prior knowledge and how concepts are formed. ◗ The contributions of Lev Vygotsky: The zone of proximal development. The importance of scaffolding children’s learning. Learning as a social activity. ◗ Bruner’s views on learning: Learners must be actively involved in the learning process. Most learning progresses from concrete to abstract. The spiral curriculum.

A concept can be defined as a mental representation that embodies all the essential features of an object, a situation, or an idea. Concepts enable us to classify phenomena as belonging, or not belonging, together in certain categories. Concept formation is the means by which we mentally organise our environment into meaningful units of information that we can then use for future reference. 24

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The development of conceptual knowledge is achieved by the recog nition of relationships between different items of information. The process of forming concepts consists of linking items of information together because of common properties they possess. McInerney and McInerney (2005) suggest that by being exposed to a range of objects and experiences we begin to see common properties emerge. For example, our concept of ‘triangle’ embodies our knowledge of the number of sides, the properties of the angles, the different sizes and orientations of a triangle that are possible by varying the dimensions, and the different names that have been given to various triangular forms. We also discover that triangles can be seen occurring quite frequently within manufactured articles and in the built environment; finally, we can also classify triangles as falling within a larger concept group embracing ‘two-dimensional shapes’. Hiebert and Lefevre (1986, p. 3) believed that conceptual knowledge is characterised most clearly by ‘… knowledge that is rich in relationships’. Conceptual knowledge can be thought of as a connected web of information. The manner in which concepts develop from firsthand observation and from relevant information that we discover (or are given) is exactly what occurs when children experience quantitative situations in the early years and develop awareness of properties and relationships among numbers and shapes. Piaget, Vygotsky and Bruner are three pioneers in cognitive development research whose theories help to throw light on the way in which such concept development occurs in children.

Piaget’s theory We owe much of our understanding of how children develop number concepts from the work of the late Jean Piaget (1942; 1983). Piaget’s theory of cognitive development was derived from his close observation of children (mainly his own children) as they engaged in various tasks, including many involving quantitative and spatial relationships. He was interested in investigating how their perceptions, thinking and reasoning developed and changed over time. His ideas have influenced greatly the current ‘constructivist’ view of learning that places the learner rather than the teacher at the heart of the learning process. Piaget argued that children must continually construct and modify their own understanding of phenomena through their own actions and reflection. In Piaget’s theory,

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children’s active exploration of their environment, coupled with their increasing physical and neurological maturation, play the most important role in influencing conceptual development. An essential aspect of Piaget’s theory of cognitive development is his concept of ‘schemata’ (singular: ‘schema’). A schema is an integrated mental representation or ‘assembly of knowledge’ comprising everything one has learned about a particular concept over time. For example, in forming the ‘triangle’ concept above, we did not acquire all that knowledge on one occasion; the different items of information have been added at different times as a result of new experiences. Learners filter, interpret and adjust new information in terms of what they already know. Piaget used the term assimilation to describe the process of taking in new information and linking it with prior knowledge, and the term accommodation for the process involved in adjusting or revising the existing schema to reflect this advance in understanding. Thousands of schemata are developed over an indivi dual’s lifetime, and they are constantly changing, refining and expanding. Concept development is thus a process of creating mental structures and refining them over time. The acquisition of concepts continues throughout life and is the main characteristic of cognitive development. Piaget considered that children pass through four distinct stages on their way to mature cognitive functioning. At each stage, they become better able to process information accurately and less likely to develop misconceptions. In general, the sequence they follow begins at birth and continues into adulthood. An approximate age range for each stage has been suggested below, but children actually differ significantly in the age at which they pass through each Piagetian stage, due to factors such as maturity, mental ability, teaching and experience. The stages may be briefly described thus: ◗ Sensorimotor stage (birth to 18 months). During this stage, the young child develops motor and orienting responses or reactions to sensory input (e.g. focusing visual attention; reaching for and picking up an object; attending to sounds). At this stage, a child is rapidly coming to understand important features of his or her immediate environment, but is not aware, for example, that physical objects continue to exist even when they are out of sight; in other words, the child lacks an understanding of object permanence (Weiten, 2001). But, as stated in the previous chapter, even at this early age an infant can make some basic quantitative comparisons and judgements when real objects are present.

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◗ Pre-operational or intuitive stage (from age 18 months to 7+ years). Piaget used the term operation to mean the mental process of thinking something through. Children at the pre-operational stage tend not to be able to manipulate ideas mentally or deduce cause and effect relationships. As a result, they are often misled by what they see. An important example of this in the numeracy field is conservation of number. Children at the pre-operation stage at first do not understand that the number of items in a group does not change even though the spatial arrangement of the items may be altered. It is sometimes argued that until children can understand conservation of number there is little point in attempting any form of written recording. Studies have suggested that the concept of conservation of number is achieved by most children between the ages of 6 and 7 years (conservation of mass, length and area develop later). Children at the pre-operational stage of development tend to focus too much on one feature of a problem or task, and do not consider other aspects that may be important. They also have difficulty imagining an action reversed, for example, if 3 tokens are placed with 2 tokens to make 5, what would happen if the 3 tokens were then taken away from 5? ◗ Concrete operational stage (7 to 11+ years). During this stage, the child can begin to understand and process increasingly complex information if it can be experienced, acted upon and observed firsthand. During the concrete operational stage, the child becomes better able to handle symbolic repre sentation and carry out mental operations provided the symbols (e.g. numbers) can be easily related to reality. ◗ Formal operational stage (11 to adulthood). Finally, the normally developing individual becomes able to operate with abstract ideas, and to think and reason without the need for real objects or firsthand experience. Adolescents in the formal operational stage become more thoughtful and systematic in their problem solving; they reason things out rather than using a hit-or-miss approach.

Educators think it is important to consider Piaget’s four stages in relation to the types of mathematical experiences normally provided at different age levels, and how children can be supported best in learning number skills and concepts. According to Piaget, abstract reasoning and the use of purely symbolic representation cannot be forced on a child too early. His theory led educators to question, for example, the wisdom of attempting to teach young children formal arithmetic skills at an early age. When children are taught such rules, procedures and facts out of context, and too

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early, they can’t connect them to what they already know, so conceptual learning does not occur. According to Piaget (1942), the direct teaching of number knowledge and skills ahead of a child’s cognitive readiness to learn is largely a waste of time. Such teaching can also have negative impact on a child’s confidence and attitude toward number work. Over the years since Piaget’s works were first translated and published, numerous experimental studies have generally supported his description of the way in which cognitive development occurs. However several criticisms of his theory have emerged. The first is that he underestimated the learning capacity of preschool children (Case, 1991; Lutz & Sternberg, 1999; Mandler, 2004). Piaget placed heavy emphasis on the role of maturation and readiness, but more recent work appears to indicate that experience and instruction are as important as maturation. It is now believed that with appropriate experience and skilled teaching young children can actually learn very much more than Piaget thought possible. Developmental psychologists who have built on his earlier work (the neo-Piagetians) assert that the knowledge and processes needed to learn new skills and concepts and to solve problems are teachable, and we do not need to await biological maturation of the child. This suggests that instead of opting for the currently recommended ‘developmentally appropriate curriculum’ in early childhood settings, we should be seeking effective teaching methods for accelerating young children’s mathematical learning. It is argued that curriculum in the early years often underestimates children’s abilities and is therefore insufficiently challenging (Wright, 1994). In the United States of America, the National Child Care Information Center (2007) states that, in the domain of numeracy education, many researchers (and some early childhood educators) are now recognising the importance of complementing child-initiated learning with high-quality, teacher-directed instruction in the early years. But this notion does not sit comfortably with many contemporary guidelines on preschool teaching since they still advocate play and developmentally appropriate practice. The two differing views – constructivist vs instructivist – represent one of the ongoing debates in early childhood education (Katz, 1999). It is also believed now that Piaget overestimated what the average adolescent could do in terms of abstract reasoning. The age 11 years has often been suggested as the end of the concrete operational stage, but more recent studies have suggested that for the majority of adolescent students

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their thinking in subjects such as mathematics and science may remain at the concrete stage until at least age 15 years or 16 years (Collis & Romberg, 1992; Santrock, 2006). For this reason, much of the contemporary mathematics teaching in secondary schools misses the mark because it is conducted largely through textbook examples, ‘chalk and talk’, without hands-on experience. The relative absence of concrete experience and visual representation may well account for many of the learning difficulties evident in older students and their growing dislike of mathematics. Despite these limitations, Piaget’s major contribution to the field of numeracy teaching has been: ◗ to present the view of children as active and constructive learners ◗ to redefine the role of a teacher as facilitator of children’s self-initiated dis covery of new information ◗ to remind teachers to consider children’s level of cognitive maturity and readi ness for particular types of learning ◗ to highlight the futility of attempting to transmit predigested knowledge to young children by didactic methods.

Lev Vygotsky In the previous chapter, reference was made to Vygotsky (1962; 1978) and his notion of the zone of proximal development. To Vygotsky, optimum learning occurs when tasks or problems are correctly tailored to be just a shade above a child’s current level of ability but which the child can handle successfully with some support or guidance from an adult or a peer. This support has become known in education as ‘scaffolding’ and it takes the form of hints, suggestions, comments, questions, demonstrations and even direct explanations. Vygotsky and Piaget both see the role of the teacher as facilitator, but Vygotsky places much greater emphasis on the teacher actively guiding the child toward new knowledge construction. He was much more interested in helping to advance each child beyond his or her present level of understanding, rather than awaiting natural maturation and so-called readiness. He also believed in the importance of making topics or problems meaningful by situating them in real-life contexts. Vygotsky regarded language and social interaction as playing much greater roles in children’s concept development than Piaget had

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acknowledged. His recognition of the importance of discussion and ‘think ing aloud’ has greatly influenced the teaching of mathematics. Teachers are beginning to acknowledge now that talking is at the very heart of young children’s knowledge construction (e.g., Monaghan, 2006). The major messages for numeracy teaching that stem from Vygotsky’s work are as follows: ◗ The teacher must actively guide children towards better understanding by supporting (scaffolding) and mediating their thinking. ◗ It is essential to identify a child’s present level of understanding in order to provide guidance that will help him or her to progress further. ◗ Encouraging collaborative group work, peer assistance and discussion all foster concept development and learning. ◗ Schools should base much of the curriculum on real-life topics and problems.

Jerome Bruner Bruner (1960; 1966) was instrumental in raising educators’ awareness of the important role that learners themselves must play in constructing know ledge. In the domain of mathematics for example, he stressed the need for students to think mathematically for themselves instead of having a deconstructed and decontextualised version of mathematics presented for mally to them by the teacher and textbook. However, Bruner sees the role of the teacher to be more than simply a facilitator. Children do need opportunities to explore and discover on their own but they also need to interact positively with more knowledgeable adults and peers who can support their efforts, challenge them, and assist them in interpreting and assimilating new discoveries. According to Bruner, concept development progresses from the ‘enactive’ stage (in which learning should involve concrete experiences) through the ‘iconic’ stage (where pictorial and other graphic representations are used to move beyond the purely concrete) to the final ‘symbolic’ stage where abstract symbols and notation alone convey meaning to the learner. Applying Bruner’s three stages to early numeracy development, the first step that most children take in moving from the real world is to use pictorial recording of number relationships (for example, to draw three pet goldfish in a tank). At around the same time, children are also able to

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interpret pictorial representations of groups of objects to establish a number relationship, for example they can count the balloons in a picture. When situations are presented to children in pictorial form, or are recorded by them as pictures, they can easily relate to them even though they are not the real objects. This is the first stage in moving from concrete experience to symbolic representation. It might be called the beginning of the ‘semiconcrete’ stage. At the next (iconic) stage of development the child can use an object to ‘stand for’ some other real object. For example, a wooden block can stand for a car. Three blocks can stand for three cars moving along an imaginary road, and so forth. The blocks don’t look like cars, but the notion that one thing can be represented in a different way is established. At a later stage, tally marks (looking even less like the real object) can be used at the ‘semi-abstract’ stage, with an understanding of their one-toone correspondence with the original objects. It is not until a child has had these intermediate experiences of translating reality into different forms of semi-concrete and semi-abstract representation that they are ready to begin to use symbolic recording with understanding. It is believed that some children begin to experience difficulty in learning mathematics because they have been taken too quickly from the concrete stage to the abstract symbolic level of recording. A gap is created in children’s understanding if they are forced to operate too soon with symbols and mathematical notation. The use of structural apparatus such as Dienes MAB, Cuisenaire Rods, or Unifix can help bridge this gap by providing a visual link between real objects and the symbols that can eventually represent them. Bruner’s views are not incompatible with those of Piaget; but like Vygotsky, he is much less concerned with issues such as readiness and maturation. Instead, Bruner supports the view that young children can be taught many things if the quality of instruction is good and the teaching follows the sequence of concrete, through the semi-concrete, to the abstract levels. His claim is that any subject can be taught effectively in some intellectually honest form to a child of any stage of development – if the method is right. More recently, Watson (2004, p. 372), arguing against denying some students a right to a challenging and interesting maths curriculum because of poor ability, has echoed Bruner’s view, stating, ‘It is possible to engage nearly all students in some form of abstract and conceptual understanding.’

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In the domain of mathematics, Bruner’s suggestion that a curriculum should be spiral rather than linear in its progression is sound advice. A spiral curriculum implies that key ideas or operations that are first introduced at a simple level are revisited later at regular intervals to be expanded and enriched by application to new situations. In summary, Bruner’s influences on numeracy teaching include: ◗ the need for learners to be actively involved in investigating real problems and discovering information for themselves ◗ the need for children to work through concrete experiences before they are ready for abstraction ◗ the need to create learning environments that provide materials and situations necessary to stimulate inquiry ◗ the recognition that children who participate actively in their own learning are more able to use and generalise the knowledge and skills they acquire.

Many of the principles embodied in the theories of Piaget, Vygotsky and Bruner apply to the education of children beyond the early childhood period. The following chapter raises some of the issues involved in developing numeracy in the primary school years and beyond. L i nk s t o m o r e ab o u t c o nc e p t d e v e l o p m e n t ◗ Wikipedia provides a good overview of concept development. Available online at: http://en.wikipedia.org/wiki/Concept ◗ A summary of key mathematical concepts in the preschool years is presented in Early childhood today: Development of mathematical concepts. Available online at: http://teacher.scholastic.com/products/ ect/mathconcepts.htm ◗ For information related to concepts development and number sense, see Project Math Access: Teaching mathematical concepts at: http:// s22318.tsbvi.edu/mathproject/ch1.asp ◗ What is a mathematical concept? Available online at: http://www.emis. de/proceedings/PME28/SO/SO036_Jahr.pdf ◗ More on Vygotsky, Piaget and Bruner can be found at the North Central Regional Education Laboratory website. Available online at: http://www. ncrel.org/sdrs/areas/issues/methods/instrctn/in5lk2-4.htm

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The primary school years and beyond

Key issues ◗ Children commence formal schooling: A change of teaching approach. Higher expectations. Basic numeracy should be firmly established by the end of primary school. ◗ Effective teaching in the primary years: Several research studies have investigated teachers’ instructional skills. The results have provided a useful window on the type of teaching that produces the best results in numeracy and mathematics. ◗ Teacher competency: Some teachers have great difficulty implementing an interactive teaching approach. Many primary teachers lack expertise and confidence in teaching mathematics. Effective teaching principles are too rarely applied. ◗ Secondary school: Mathematics becomes increasingly abstract. The gap widens between high achievers and low achievers. Schools often resort to ability grouping to cope with this problem. ◗ Adult numeracy: An area of significant development. Recent years have seen an increase in new policies and provisions for this population.

Children enter formal schooling with a wide range of differences in their numeracy knowledge and skills. They also vary greatly in their feelings toward number work and their own ability to handle it. Some are eager, interested and confident in their own abilities; others are much less certain. 33

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Doig et al. (2003, p. 21) remarked that, ‘The difference between preschool and school is quite dramatic in terms of the aims, pedagogy, content of the numeracy program and in what is expected of the children’. What happens to them over the next few years will either strengthen their skills and confidence or will cause them to develop a distaste for mathematics, believing they have no aptitude for what appears to them now to be a difficult subject. Whitebread (1995, p. 11) observed that, ‘Far too many of our young children find learning mathematics in school difficult, lose their confidence in mathematics, and go on to join that large swathe of the adult population who panic at the first sight of numbers.’ This chapter looks at some of the issues involved in providing highquality teaching to strengthen children’s numeracy skills and enhance their interest in mathematics in the primary school years and beyond.

Transition from preschool to school Following on from the kindergarten years, a child’s entry into the reception class will not usually see a dramatic change in teaching methods for the first year. Although schools differ, teachers in most junior primary (infant) schools tend to subscribe to a child-centred philosophy with an emphasis on activity approach and avoidance of too much direct instruction. There is continuity therefore between preschool and beginning school experiences, providing an opportunity for some children whose early learning has not resulted in optimum development of number sense and skills to catch up during the first half-year. Teachers will be working to observe each child’s number ability and level of confidence, and will endeavour to plan and provide activities that build upon previous learning and reduce any significant gaps in children’s prior knowledge. In the years ahead, a teacher’s role is to help all children understand mathematics, compute fluently, develop concepts, solve problems, reason logically, and engage willingly with mathematics (Kilpatrick & Swafford, 2002). The primary years are vital for achieving this goal. Methods after the reception year soon become rather more structured and a little more teacher-directed to ensure that mastery of basic skills is achieved alongside learning with understanding. During the primary years children need to acquire an adequate pro ficiency in carrying out calculations and solving problems involving larger numbers. The informal numeracy strategies they developed in

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the earlier years are rarely adequate for this purpose, so new learning is required. Marmasse et al. (2000, n.p.) state that, ‘The strongest influence on arithmetical development is formal education, which can lead to the development of skills that would not have emerged in a more natural environment without formal education.’ The teacher’s role is to create a learning environment where there are opportunities for active investiga tion and problem solving by the children (Fleer & Raban, 2005). Teachers also have a responsibility to impart relevant mathematical information to children and to teach specific skills and strategies.

Teaching in the primary years There have been several important studies of the instructional approaches used by teachers of primary and secondary school mathematics (e.g., Askew et al., 1997; DEST, 2004; Lamb, 2004; Weiss & Pasley, 2004; Wilson et al., 2005). A few studies have looked at mathematics teaching across many different countries and cultures, for example the Third International Mathematics and Science Study (TIMSS) (Institute of Education Sciences, 1999). TIMSS even involved the videotaping of a number of mathematics lessons in action, allowing detailed analysis of the minute-by-minute inter actions between teachers, students and subject matter. TIMSS will be discussed in more detail in a moment. From the work of Askew et al. (1997) in the United Kingdom, a picture emerged suggesting that teachers of numeracy tend to reflect one of three possible orientations toward the teaching process. The orientation most in harmony with contemporary learning theory can be referred to as connectionist. Those with a connectionist orientation make every effort to link new learning to the children’s prior knowledge. Their aim is to encourage conceptual understanding. They make explicit connections within and across different mathematical topics and with real-life situa tions. Connectionists acknowledge and make use of (connect with) children’s own informal numeracy strategies and ideas as new topics and skills are introduced. The second orientation is towards direct teaching – a transmissionist approach. Those subscribing to this teacher-centred orientation believe it is important to teach explicitly the information, rules and procedures that students will need to acquire to become numerate. They also believe that it is important to practise essential skills until they are mastered. The third orientation is recognised in teachers who firmly

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believe that children must discover mathematical rules and concepts for themselves. This minimally guided discovery orientation leads the teacher to establish situations in which learners must investigate problems, find out information, and develop number skills and concepts for themselves. It is believed that children will learn basic computational skills such as adding, subtracting, multiplying and dividing through their regular engagement in exploratory quantitative activities. Of the three orientations, Askew et al. (1997) conclude that the connectionist seems to result in deeper learning than either of the other approaches. A study in the United States of America, involving students in the K to 12 age range, set out to explore in more detail what makes a difference in quality of mathematics instruction (Weiss & Pasley, 2004). Classroom observations were conducted during 364 mathematics lessons. The find ings revealed that the following variables were significant in ensuring that students make good progress: (a) using relevant and interesting subject matter (b) maintaining a high level of student engagement (c) using effec tive questioning to encourage children to reflect, and (d) assisting students to make complete sense of the subject matter (i.e., the teacher’s role as medi ator and guide). This study also reported that effective teaching employing these important principles was far from common. Also in the United States of America, a very much earlier study of mathematics teaching by Good and Grouws (1977) supported what they termed an ‘active teaching’ model. Good and Grouws found that effec tive learning in mathematics could be best achieved with a structured curriculum and a fair degree of direct teaching. Lessons were found to be maximally effective if the teacher introduces each new topic by explicitly linking it with previous work, provides clear process explanations and demonstrations, uses many illustrative examples, engages students in much guided and independent practice, and checks very frequently that students are understanding (Good et al., 1983). This work was one of the earliest examples of establishing ‘research-based practice’ in mathematics teaching, and it yielded powerful findings. However, the findings were never fully implemented across schools because that style of teaching was suddenly at odds with recommendations favouring a student-centred investigative approach with less, not more, direct instruction (NCTM, 1989; 2000). In Australia, a project initiated in the state of Victoria probed more deeply into the actual tactics employed by teachers of students in Prep

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to Year 4, and Years 5 and 6, as they interact with their students to bring about learning (DEST, 2004) (see the Links box at the end of the chapter). The researchers found that 12 tactics used by teachers were effective in promoting students’ learning in the numeracy domain. These tactics were classified as: ◗ Excavating: uncovering what students know ◗ Orienting: setting the scene; contextualising; reminding; linking ◗ Modelling: demonstrating; directing; explaining; instructing; showing; telling ◗ Collaborating: acting as co-learner; working closely with students ◗ Guiding: cueing; prompting; navigating toward understanding ◗ Noticing: being aware of how well students understand new work; identifying any gaps or misconceptions; providing coaching or re-teaching ◗ Probing: clarifying; monitoring; checking ◗ ‘Convince me’: seeking explanations and justification from students for their ideas ◗ Reflecting and reviewing: recounting; considering again; summarising; sharing ◗ Extending: challenging; taking students beyond simple ideas ◗ Apprenticing: encouraging peer assistance; mentoring.

Many of these effective teaching tactics are reminiscent of what was dis covered previously about highly effective teachers of mathematics in Japan. Students in Japan and other Asian countries usually do outstandingly well in international surveys of achievement in mathematics, so the teaching methods used with them are of great interest. In the TIMSS research, it was noted that effective teachers appeared to provide systematic instruction in a way that children not only master arithmetic skills and problem-solving strategies but also develop a genuine understanding of the subject matter. Japanese teachers teach in an interactive way and are seen to encourage their students’ participation, critical thinking and reflection at all points in a lesson in order to encourage a conceptual level of learning. The students spend more time devising and proving their strategies for solving problems and less time practising routine procedures (Stigler & Hiebert, 1997). The typical mathematics lesson in Japan involves four stages. First, the teacher presents a problem. Then, the students are given time to attempt a solution, often working collaboratively with a partner or in a small group. Next, their solutions are presented to the group and there is whole-class discussion to evaluate their ideas and the methods they used. Finally, there is a summing

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up by the teacher and an opportunity to apply the most effective strategies to similar problems for homework (Shimizu, 1995). One of the key features of effective lessons in Japan is the teacher’s final summary, providing an overview and consolidation of what students have discovered and how it can be applied (Benjamin, 1997). Benjamin (1997) reports that cohesion, thoroughness, and the emphasis on understanding as well as skill in calcu lation, are characteristic of Japanese teachers. The brisk pace of a lesson helps to motivate students and keeps them on task and productive. It is clear from the findings of these and other studies of teaching that effective teachers of numeracy and mathematics employ an interactive approach. They work closely with their students, guiding them as necessary and providing relevant input at appropriate moments. They are successful in engaging children fully in the business of learning; and as teachers, they actively teach rather than relying on a sequence of textbook exercises to enact the curriculum. They put students at the centre of each learning task, but they use their expertise to guide and support students’ construction of knowledge and their acquisition of skills and strategies.

Positive intervention: the daily numeracy hour As a key component of its National Numeracy Strategy in 1999, the UK government introduced the model of a daily mathematics lesson of 45 minutes to 1 hour for primary school students (DfCSF, 2007b; DfCSF, 2007c). This became known as the ‘numeracy hour’, to complement the already existing ‘literacy hour’. It has been mirrored in similar developments in some parts of Australia. The guidelines for operating the numeracy hour place emphasis on an interactive teaching approach, used within a whole-class context. Influenced, it seems, by the data from TIMSS and other studies of effective instruction, the session is to be conducted at a brisk pace, and there should be much use made of mental work and discussion. Children are to be encouraged to explain their thinking when they offer solutions and answer questions. The typical format for the numeracy hour comprises three parts: 1 Warm up: oral and mental work for about 5 to 10 minutes 2 Main teaching activity: Investigating a problem or introducing a new concept, for 30 to 40 minutes. This is still mainly conducted with the whole class, but some grouping may occur and level of work may be differentiated by ability.

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3 Plenary: summarising and consolidating the lesson, clearing up any miscon ceptions, setting homework.

Reviews by Brown (2002) and Kyriacou and Goulding (2004) indicated that the daily numeracy lesson has had some modest benefit in enhancing primary children’s confidence and competence in early maths. However, many teachers have not been particularly successful in running the sessions in the ‘interactive’ manner that is recommended; and there is some indi cation that the lesson is sometimes taught in a fairly formal manner using a whole-class teaching approach. Brown (2002) suggests that the lowerachieving students derive little benefit from the whole-class approach because much of the content is above their level of understanding. There is also evidence that teachers are not really effective in the final summarising and consolidating of the session, with many classes simply ending without summary as soon as the main teaching section has finished. The inability of a significant number of teachers to implement all aspects of the numeracy hour effectively undermines its potential value (Macrae, 2003). The National Union of Teachers (2002) expressed concern that teachers in reception classes were feeling under pressure to introduce a numeracy hour involving whole-class teaching with young children. It was felt that this degree of structured teaching, particularly for blocks of 45 minutes or more, was not appropriate at that age and is not in the children’s best interests. Watson (2004) suggests that the teaching methods recommended in the numeracy strategy and the numeracy hour are in danger of causing teachers to move too rapidly through the curriculum, spending too little time on each topic and changing tack too often, thus destroying continuity in students’ learning. It is notable that the numeracy hour has spawned a flood of resource materials (worksheets, books of ideas, charts and planners) from commer cial publishers. An online search under ‘numeracy hour’ will reveal many of these resources.

The key issue of teacher competence The inability of many teachers to implement fully an interactive and flexible numeracy lesson, together with findings from other studies that reveal poor quality teaching in many classrooms (e.g., Weiss & Pasley, 2004), raises a very important issue of the overall competence of primary teachers to teach

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mathematics. Stigler and Hiebert (2004) confirm that the teacher’s skills and the quality and type of interaction they have with students seem to be the most important variables in how well students learn mathematics (and therefore how well they acquire numeracy) and the feelings they develop about the subject. Teachers’ own attitude toward mathematics and their feeling of professional competence, as well as the depth of their subject knowledge, greatly influences the way they go about teaching the subject (Bonner, 2006; Thornton & Hogan, 2004). Those who lack confidence and have fairly limited subject knowledge may teach mathematics very poorly indeed. Unfortunately, it seems likely that this is the situation in many primary schools at this time. Part of the problem stems from the fact that in many countries, teachers in junior primary and primary schools are ‘generalists’ rather than specialist teachers of maths; maths is simply one of many subjects they must teach each day. Teachers who have no great interest in maths and no special exper tise to teach it tend to avoid a problem-based and open-ended approach because it is difficult and unpredictable to manage. They feel insecure with an investigative method so they avoid creating too many open-ended situ ations, fearing that they may not be able to answer students’ questions or deal with issues that may arise. Instead, they teach numeracy as if it only involves learning mechanical arithmetic through memorisation and repetitive practice. To achieve this narrow objective, the teachers are most likely to use a transmission mode of instruction and simply teach the operations without reference to children’s conceptual understanding. In other words, they will tend to teach as they were taught themselves in the primary school. This weakness was highlighted in an Australian study conducted by Lamb (2004), in which teachers’ understanding of the division algorithm was the focus of attention. Lamb concluded: ‘It would appear that the teachers do not have the depth of knowledge necessary to teach for conceptual understanding’ (p. 153). She also remarks that, ‘It is impossible to calculate the degree of student difficulty caused by teachers who remain ill-informed and fail to seek outside assistance yet continue to teach’ (p. 167). If teachers are to adopt the methods that research has shown to be effec tive, this situation will need to be addressed. It will be impossible to develop students’ numeracy skills to the full and to foster a positive attitude toward mathematics, if teachers are not confident enough to operate interactive,

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learner-centred methods rather than a textbook-centred approach. When teachers are not effective in the way in which they present mathematics they are reducing students’ opportunities to learn (Siemon et al., 2001). It is, of course, almost impossible for teachers to adopt a connectionist orientation if they themselves lack a conceptual understanding of the subject matter. It must be admitted that for teachers who lack a sound understanding of mathematics, it is far from easy to teach in the style displayed by expert teachers of the subject – for example, some of those depicted in Japanese classrooms in the TIMSS videos (Institute of Education Sciences, 1999). Chinese and Japanese teachers are reported to have a sound conceptual understanding of mathematics themselves, resulting in less reliance on procedural and algorithmic teaching (Ma, 1999). Fortunately, this problem of limited expertise in non-specialist teachers is being acknowledged now in countries such as Australia and the United Kingdom; and although there will be no quick solution, at least there has been an increase in the number of in-service professional development courses with a focus on teaching mathematics, and an increase in online and other resources available for teachers – for example, TeacherNet, provided by the Department for Children, Schools and Families in the United Kingdom, and SOFWeb in Victoria (see the Links box below). Projects have also been initiated to encourage teachers to support each other in building the pedagogical skills and understandings necessary to teach mathematics in a more realistic, flexible and effective way (e.g., Bonner, 2006; Carpenter et al., 1999; Clarke et al., 2000).

The secondary school years Many students entering secondary schools have already developed func tional numeracy as a result of their mathematical experiences in primary school, in the home and in the community. For these students, secondary school mathematics provides an opportunity to build on this foundation and explore more advanced concepts. Unfortunately, for some students, secondary school mathematics quickly becomes increasingly abstract, resul ting in a high proportion of them losing their confidence, motivation and initiative. They begin to doubt their own abilities and they come to rely much more on their teacher to transmit the knowledge which they then attempt to memorise (Watson, 2004). This in turn tends to make

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their teachers adopt a didactic approach rather than a student-centred investigative approach. Zawojewski and McCarthy (2007) have pointed out a very serious mismatch between the type of maths content taught in most secondary schools and the abilities really needed beyond school. It could be argued that most of the content of secondary school mathematics does little to help develop students’ numeracy. It is still very common to find that secondary schools organise the teaching of mathematics into ability groups (Turner, 2007). The mathe matically competent students find themselves in the ‘top set’, while the students with limited mathematical ability and little interest are placed in the ‘bottom set’. In theory, the focus of attention in the bottom set is on developing students’ numeracy to a level where they can function effec tively at work and in the community. Attempts are made to tailor the curriculum content and teaching method to suit the interests and learning rate of these students. Often these adapted mathematics courses are given catchy titles such as ‘Consumer Maths’, ‘Life Skills Maths’, ‘Workplace Maths’. Unfortunately, the students (who frequently refer to such courses as ‘veggie maths’) perceive them to be little more than watered down versions of the mainstream maths course, with heavy emphasis on practising routine arithmetic skills. In general, it can be stated that many secondary schools need to focus on making these courses much more attractive and much more relevant for lower-ability students by establishing a better balance between strengthening computational skills and applying such skills to authentic and interesting real-life issues and problems.

Adult numeracy In several countries, interest in numeracy within the adult population has been spurred by data from surveys such as those carried out by the Organisation for Economic Co-operation and Development (OECD, 1998), the National Assessment of Educational Progress (NAEP) that has collected data in the United States of America for more than 30 years (Institute of Education Sciences, 2007), the International Numeracy Survey of 1997, the National Child Development Study (Bynner & Parsons, 1997), the Moser Report A fresh start: Improving literacy and numeracy (Moser, 1999), and the 2003 Skills for Life Survey (Grinyer, 2006), all giving cause for

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alarm. For example, in the United States of America it was found that 35 per cent of students were scoring ‘below basic’ in the National Assessment of Educational Progress maths tests (American Institutes for Research, 2006). This led to the launch of the Adult Numeracy Initiative. Similar alarm was triggered in the United Kingdom by evidence indicating that some 7 million individuals in the 16 to 65 age-range exhibited very poor maths skills, and that poor numeracy had a major impact on an individual’s employability (Moser, 1999; Parsons & Bynner, 2005). Concern was also growing regarding the decline in the number of students over the age of 16 who opt to study mathematics and science. Studies were commissioned (e.g., Smith, 2004) and new policies made. The Qualifications and Curriculum Authority (QCA, 2004) developed National Standards for Adult Numeracy in 2000, and the Df EE prepared Skills for life: The National Strategy for improving adult literacy and numeracy skills in 2001. Since 2002, adult numeracy specifications have been introduced for the first time in England (DfCSF, 2006; Loo, 2007; QCA, 2004). The focus has been on providing opportunities beyond school for individuals to acquire knowledge and skills that they failed to acquire while at school, or that they need to update since leaving school. In the United Kingdom, the BBC, in collaboration with the Basic Skills Agency, ran a campaign in 1997 called Count Me In. It was designed to help adults with numeracy problems improve their skills and raise community awareness of the importance of mathematics in daily life. The titles Count Me In and Count Me In Too have been used in several other countries (notably Australia and New Zealand) in connection with numeracy intervention programs for children. In Australia, additional teaching for adults has usually been provided under ALBE (Adult Literacy and Basic Education) schemes, via further education centres or similar bodies. Provision for adult numeracy classes began in a low-key way in the 1970s but gained impetus from 1991 when the government endorsed the Australian Language and Literacy Policy (Cumming, 1996). More recently, instructors and tutors of adult numeracy classes are expected to have professional training in this area, and courses now exist for this purpose (e.g., Johnston, 2002; Johnston & Tout, 1995). Originally, adult numeracy classes tended to focus on arithmetic skills and the application of these skills to routine problems rather than attemp ting to link the mathematics taught to real-life contexts. There was a tendency to work methodically through the number and computation

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components of a typical primary or lower-secondary curriculum. More recently, every effort has been made to ensure that the content of adult literacy and numeracy courses has a more ‘social practices’ and ‘real life’ focus ( Johnston, 2002). To facilitate the linking of teaching to real life, efforts have been made to study the numeracy needs associated with particular areas of employment (e.g. Hagston & Marr, 2007; Hoyles et al., 2002). Studies in this area are continuing. In Australia, numeracy now features within the curriculum for the Certificate in General Education for Adults (CGEA) where it is organised under the framework of ‘numeracy for practical purposes’, ‘numeracy for interpreting society’, ‘numeracy for personal organisation’ and ‘numeracy for knowledge’. The teaching of numeracy to learners of any age requires a careful balance between ensuring that computational skills are mastered on the one hand, and that individuals become confident and competent in applying such skills to problem solving and to everyday situations on the other. The following chapter addresses this issue.

L i nk s t o m o r e o n n u m e r ac y i n t h e p r i m a r y s c h o o l an d b e y o n d ◗ A good summary of the numeracy hour, together with typical objectives for each year in the primary school, can be found online at: http://www. woodlands-junior.kent.sch.uk/Guide/ygroups/numer3.html ◗ Primary Framework for Literacy & Mathematics (UK). By entering this site and selecting ‘mathematics’ you will find information about numeracy topics for specific year levels: http://www.standards.dfes.gov. uk/primary/frameworks ◗ Researching numeracy teaching approaches in primary schools. DEST 2004. Available online at: http://www.dest.gov.au/sectors/school_ education/publications_resources/literacy_numeracy/researching_ numeracy_teaching.htm ◗ TeacherNet (2007). TeacherNet is sponsored by the Department for Children, Schools and Families in the United Kingdom. For information

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and resources for numeracy teaching, see: http://www.teachernet.gov. uk/teachingandlearning/subjects/maths/teachingnumeracy/ ◗ SOFWeb: Information on numeracy from State of Victoria, Australia. Available online at: www.sofweb.vic.edu.au/eys/num/index.htm ◗ Adult numeracy: Core curriculum. Department for Children, Schools and Families (UK). Available online at: http://www.dfes.gov.uk/curriculum_ numeracy/ ◗ Useful information on adult numeracy can be located at the SAALT website. Provides links also to ANAMOL project (Adult Numeracy and Maths Online). SAALT (Supporting Adults and Applied Learning and Teaching). (2006). Adult numeracy. Available online at: http://www. saalt.com.au/numeracy/background.html ◗ An excellent document surveying the development of adult numeracy research and practice, Johnston, B. (2002). Numeracy in the making: Twenty years of Australian adult numeracy. Sydney: Adult Literacy and Numeracy Australian Research Consortium, can be found online at: http://www.eric.ed.gov/ERICDocs/data/ericdocs2sql/content_storage_ 01/0000019b/80/1a/d0/c6.pdf

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Calculating and problem solving

Key issues ◗ The relative importance of computational skills: The proposed reforms in mathematics education have encouraged downplaying of the direct teaching of computational skills in favour of more time spent on investigation and problem solving. However, computational skills remain extremely important. ◗ Automaticity as an instructional goal: Number facts and operations need to be recalled quickly and easily by learners. Adequate practice is essential to automate such recall. ◗ Strategies for solving problems: Learners become better problem solvers if they are taught appropriate strategies to apply. ◗ Forms of knowledge that comprise numeracy: Declarative knowledge; procedural knowledge; strategic knowledge; conceptual understanding.

The ability to compute is only a part of being numerate; but it is an important part. Almost all numerical situations in everyday life require an individual to be able to add, subtract, multiply or divide, often with numbers that are too large to be manipulated mentally. It is for this reason that to be numerate, children still need to master computational processes as both paper-and-pencil and calculator skills. Equally important, they need to understand the principles and concepts underpinning the various algorithms (Booker, 2004; Thompson, 2007). 46

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Macrae (2003, p. 83) states that effective teaching of numeracy, as defined in the Effective Teachers of Numeracy Project, is teaching that helps children to: • acquire knowledge of and facility with numbers, number relations and number operations based on an integrated network of understanding techniques, strategies and application skills • learn how to apply this knowledge of and facility with numbers, number relations and number operations in a variety of contexts.

This chapter examines some of the issues related to teaching computational skills and problem-solving strategies.

The place of computational skills Interestingly, the place of instruction in computational skills remains a point of major contention in reforms of primary maths teaching. Perso (2006b) compares the debate over the relative importance of computational skills in numeracy to the ongoing debate over phonic skills in the literacy domain. Most reformers in mathematics education urge that much less attention and time be devoted to practising arithmetic operations; but practitioners maintain that explicit teaching and practice of computational skills is essential for more effective problem solving (Calhoon et al., 2007; Farkota, 2005; Westwood, 2003). Some authorities favour delaying the teaching of any arithmetic opera tions until students are ready to learn them with complete understanding. For example, in New Zealand, the Ministry of Education (2006, p. 8) suggests that: Students should not be exposed to standard written algorithms until they use part–whole mental strategies. Premature exposure to working forms restricts students’ ability and desire to use mental strategies. This inhibits their development of number sense.

It is very clear that at this time the United Kingdom places much more emphasis on developing children’s computational skills through direct teaching than does Australia, where the official message is to give much less attention to drill and practice and to keep everything contextualised. In the United States of America, a similar message seems to be creating

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conflict, because on the one hand there is a call for more investigative activities and less formal teaching of arithmetic, but on the other hand there are warning signs that, students’ skills in calculation are deficient (Kilpatrick & Swafford, 2002; Loveless & Coughlan, 2004). These poor computation skills are possibly due to a combination of factors such as inadequate training of primary maths teachers, greater use of calculators in the classroom, and education reforms that emphasise activity and problem solving over practice in basic skills (Loveless & Coughlan, 2004). It must be remarked, however, that a slight improvement in overall achievement in mathematics in United States of America was detected in the 2007 NAEP results (Grades 4 and 8) (Institute for Education Sciences, 2007). This is possibly due to an improvement in teacher preparation and a better balance between skill development and activity methods within the curriculum and teaching approach. In the United Kingdom, the official line is that teachers in primary schools should use approaches that involve both teaching for understanding and an element of memorisation (e.g., mastery of number facts, multi plication tables, signs and symbols) (DfCSF, 2007b). But the word ‘memorisation’ raises alarm bells in the minds of most maths education reformers who equate it with the rote learning typical of the very teaching approaches they are trying to replace. Some argue that the calculator can now perform in an instant every process that a student is likely to need, so why devote hours to paper-and-pencil arithmetic practice? (Watson, 2004). As typical of this viewpoint, Martin (2007) suggests that it is better to have students develop their own problem-solving strategies rather than memorise rules and procedures. And Boaler (1997) warns against leading students to develop inert procedural knowledge that is of limited use to them in anything other than textbook situations. There comes a time, however, when one has to question the wisdom of not teaching children to compute. How can a student really solve problems – other than at a very simple level, or with the help of a calculator – without having the necessary computational skills to use? The argument that children will learn these components of numeracy simply by creating their own mental strategies to handle number situations is attractive, but not entirely convincing. Surely effective teaching of mathematics (and therefore numeracy) must involve both the teaching of sound computational skills, and the opportunity to apply these skills in investigating and recording data

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and in solving authentic problems? Brown (1998) is undoubtedly correct when recommending that, in addition to engaging in investigative maths activities, students must, ‘Practise skills and consolidate their recall of basic facts’ (p. 84). The teaching of numeracy requires a sensible balance between instruc ted skills and discovered concepts, principles and applications. A purely problem-based approach often makes unreasonable assumptions concerning children’s ability to discover and remember mathematical relationships for themselves. In particular, major concern arises with such methods over the reduced attention given to developing children’s automaticity in essential arithmetic skills. The evidence seems to indicate that some children will not make good progress in skill development under such an approach (Ellis 2005; Farkota, 2005). These students make much better progress in mathematics when they are directly taught essential skills and strategies (Carnine et al., 1998; Farkota, 2005; Pearn, 1999; Pincott, 2004).

Number facts: the importance of automaticity ‘Number facts’ is the term applied to all the simple relationships among small numbers. Examples are, 7 + 3 = 10; 10 – 3 = 7; 10 – 7 = 3; or 3 × 7 = 21; 21 ÷ 3 = 7; etc. Knowing number facts is partly a matter of learning them through practice, and partly a matter of grasping a rule (e.g. that zero added to any number doesn’t change it: 3 + 0 = 3, 13 + 0 = 13, etc.; or if 7 + 3 = 10 then, 7 + 4 must be ‘one more than ten’, etc.). Some mathematics experts believe that knowledge of number facts should be regarded as one com ponent of number sense. These number relationships are so fundamental that all children should know them without having to work them out each time they need to apply them. Number facts should be recalled instantly, with a high degree of automaticity. The reason for this is obvious – recalling basic number facts automatically allows children to deal swiftly and effectively with calculations and the solution of problems. They are able to focus full attention and mental effort on the higher-order processes involved in addressing and working through the problem, rather than to the lowestorder steps in completing a calculation. Such automaticity is only acquired through frequent and successful practice. According to Sun and Zhang (2001, p. 28):

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A mastery of lower-order skills instills confidence in students and facilitates higher-order thinking. The ability to automatically recall facts strengthens mathematical ability, mental mathematics, and higher-order mathematical learning. Without this automation students have difficulty performing advanced operations.

In the United Kingdom, school inspectors now consider primary schools to be exemplifying best practice in early numeracy if, among other things, they are placing increased emphasis on ‘recall of number facts and on the ability to calculate quickly and accurately, both mentally and on paper’ (OFSTED, 1997, p. 4). Prior to 1997 – during the era when arithmetic was de-emphasised – teachers would have been criticised for attending too much to these very same aspects within their mathematics programs. But the pendulum has swung again, and in the United Kingdom at least, it is again respectable to teach computational skills and number facts. Of course, from the beginning, some exponents of problem-based maths have recognised the importance of mastering these basic skills. For example, Baker and Baker (1990, p. 103) indicated that all children should be able to give ‘snappy answers to number facts to ten and twenty’. Similarly, Mannigel (1992, p. 116) gave as one of the key objectives in early child hood mathematics that children be able to ‘recall number relationships instantly from memory’. However, these writers firmly believed that essen tial number facts would be discovered and learned incidentally through engaging in purposeful quantitative activities, rather than from drill and practice exercises. There is some evidence to support the notion that a few students do in deed acquire mastery of number facts and computational skills simply through activity, discovery and exploration (Baker & Baker, 1990; Thornton et al., 1997). But it is not at all certain, however, that they acquire necessar ily the same degree of facility in automatic recall as they might under more direct teaching and with more time devoted to intensive practice. Nor can we be sure that all students will acquire adequate mastery of basic number facts through incidental learning. Just as some students seem to need more direct teaching of basic literacy skills it seems equally evident that certain students require more than just casual exposure to number relationships through problem solving, discovery and discussion if they are to reach mastery (Fuchs & Fuchs, 2001). There is evidence that when students who

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are weak at recalling number facts receive additional guided practice through methods such as computer-aided instruction their fluency in recall and application improves significantly (Goldman & Hasselbring, 1997).

Use of a calculator As stated above, it can be argued that in this age of the pocket calculator it is pointless to spend time teaching children to recall number facts and perform paper-and-pencil arithmetic operations – after all, the answers are now at their fingertips. It is certainly true that the calculator has proved to be a boon for many students, allowing them to complete more work and spend more time, rather than less time, on problem solving (Clark, 1999; Dion & Harvey, 2001; Drosdeck, 1995). The calculator has been of particular value for students of high ability, enabling them to tackle complex problems or themes, and also for students of low ability, allowing them to bypass some of their computational weaknesses. There is no doubt at all that developing a student’s confidence and competence in using a calculator must be one of the main goals in numeracy teaching (Df ES, 2004; Huinker, 2002). However, there is a danger that children may use a calculator without necessarily understanding the operation they have performed. For this reason, it is advisable that, at first, calculator use for most children should follow or accompany other more concrete work that will build conceptual understanding of the four arithmetic processes.

Mental calculation In the past, mental calculation was regarded as important, but attention was devoted to it only briefly, often in the form of a test given the first five minutes of every lesson. This writer can recall that when he was teaching in primary schools in England in the 1950s he was provided with a book titled The daily ten, containing all the mental arithmetic items set for each day of the school year. Such a routine activity probably did little to help develop numeracy since it simply tested what children could do but did not teach them ways of improving their performance. Recently, teachers have been encouraged to place more emphasis on chil dren thinking about number relationships and working with them mentally rather than resorting immediately to written algorithms or the calculator

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(Filiz & Farran, 2007; McIntosh, 2005). It is argued that writing numbers down in that way reduces students’ opportunities to deepen their number sense and develop their own insightful strategies for adding, subtracting, multiplying and dividing (Kamii, 1994). Mardjetko and Macpherson (2007) even suggest delaying the formal teaching of paper-and-pencil algorithms until children have flexible mental computational strategies. Some students appear to have particular difficulty keeping numbers in mind (i.e., within their working memory space) long enough to complete a mental calculation. For example, if the teacher says, ‘Red team scored 9 points. Blue team scored 7 points. Green team scored 12 points. How many points scored altogether by the three teams?’ Before these children can add the three numbers together mentally, they have forgotten what the numbers were. One simple teaching tactic in such cases is to jot the numerals down anywhere on the whiteboard in random order (not in horizontal or vertical algorithm format) as the problem is presented. Having this key information available in visual form enables many more children to add the numbers mentally. Their problem was in retaining the auditory information, not in mentally calculating an answer. In the United Kingdom, the increased attention given to mental calculation appears to be having some benefits. Ineson (2007) found an improvement when comparing standards measured in the first year of introduction of the numeracy hour with standards obtained six years later. Ian Thompson’s website (see the Links box at the end of the chapter) provides more information related to helping children improve in mental calculation. His material is strongly recommended.

Teaching problem solving Reys et al. (2006) regard problem solving not so much as a subsection of the mathematics curriculum but rather as a method of teaching. Working with problems provides the most relevant way to help students engage in interesting learning and at the same time develop functional numeracy (OECD, 2002). When discussing computation, number facts and mental arithmetic above the impression may have been given that children need to acquire a foundation of arithmetic skills before they can even begin to engage in any ‘real’ problem solving. This is certainly not the case; the two developments

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must go hand in hand. Instead of being seen as something that you ‘move on to later’ after you have mastered arithmetic, problems should provide an interesting and motivating way for new skills to be acquired. But, in an appropriately balanced numeracy program, time also needs to be devoted to the teaching, practice and consolidation of computational skills. The topic of problem solving and how to teach it is vast; so it is beyond the scope of this book to go into problem-solving strategies in great detail. It can also be argued that higher-order problem solving really involves knowledge, skills and strategies that are beyond the generally accepted boundaries of what constitutes everyday numeracy. Here we will consider some of the more general issues involved in helping children approach maths problems strategically and with confidence. First, we need to recognise that solving an authentic problem is rarely as easy as simply applying a pre-taught algorithm. Real-life problems are often ‘messy’ in the sense that, to begin with, one is not sure which bits of available information are important and which are not. However, there are logical steps we can take in approaching most problems. A problem needs to be analysed, explored for possible actions to take, a decision made concerning procedures to use, calculations performed either mentally or by other means, and then the result checked. As well as the cognitive processes involved in this approach, such as identifying what is required and performing the necessary calculations, the individual solving the problem also needs to use metacognitive skills such as reflection, self-monitoring, and self-correction (Booker et al., 2004). For example, the list below identifies some of the self-directing questions that an individual could ask when approaching a problem. ◗ What needs to be worked out? (identification of goal) ◗ Can I picture this problem in my mind? (visualisation strategy) ◗ How will I try to do this? (selection or creation of a strategy; identification of the operations and steps required) ◗ Is this working out OK? (self-monitoring) ◗ How will I check if my solution is correct? (evaluation) ◗ Is my answer reasonable? (reflection and judgement) ◗ I need to correct this error and then try again. (self-correction)

Students will often generate their own strategies for tackling a particular problem, but those who find this process difficult need practice in sifting

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the relevant from the irrelevant information, identifying exactly what the problem requires, and deciding the best way of obtaining and checking the result. In other words, they need to be taught the very things that other students who are efficient and confident problem-solvers already know and do. To achieve this outcome, direct teaching in the early stages is a neces sary step toward later independence (Swanson, 1999). Much of what we already know about effective teaching has an impor tant place in teaching problem-solving strategies. In particular, the teacher needs to provide students with the following forms of guidance: ◗ clear modelling and demonstrating of effective strategies for solving a particular routine or non-routine problem ◗ ‘thinking aloud’ while identifying and analysing various aspects of the problem ◗ ‘thinking aloud’ while selecting and applying appropriate procedures for the solution ◗ reflecting upon the effectiveness of the procedure used and the plausibility of the solution obtained.

Once students have been shown an effective strategy, they need an opportunity to apply it themselves under teacher guidance with feedback. Finally, they are able to use the strategy independently and to generalise its use to other problem contexts. If all students are to develop problem-solving skills, time must be made available for discussing, comparing and reflecting on methods of solution with other individuals (peers, adults). Teaching in Japanese schools, as described earlier, reflects just such an approach. Xin and Jitendra (1999) and Swanson (1999) have reviewed results from a number of intervention studies designed to improve the problem-solving ability of students with learning difficulties. Their conclusion is that it is certainly possible to improve this area of performance using strategy train ing. Their analysis of the research indicates that longer-term interventions are very much more effective than short-term interventions, and students need to gain strong independent control of the strategies themselves if there is to be any likelihood of the learning being generalised. Gaining such control requires students to reason and reflect upon the procedures they use, not merely carry them out by rote. It is said that students who learn to monitor and regulate their own problem-solving behaviour show most improvement in problem solving (van de Walle, 2006).

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Knowledge that learners need to acquire From all that has been said in this and previous chapters it seems that in the domain of numeracy a learner needs to acquire four types of knowledge – declarative, procedural, conceptual and strategic. These different forms of knowledge are mutually supportive and are used together in different combinations when we calculate and solve problems. Declarative knowledge is represented by our fund of factual information; for example, we know that the numeral 14 is read as ‘fourteen’, that $ means ‘dollar’, that 2 x 8 = 16, and that a period of 60 minutes is called ‘one hour’. Declarative knowledge is (or should be) available for instant recall. Procedural knowledge involves knowing the most effective sequence of steps in performing an operation; for example, in the vertical algorithm for addition, we usually start on the right-hand side by adding units together, but in a division algorithm, we work from left to right, dividing thousands or hundreds first; and so forth. Conceptual knowledge represents all the connected knowledge and information we have acquired about different attributes of an object, a process or a situation; for example, being able to understand and visualise the division process as the equal partitioning of a group of items. Strategic knowledge represents our acquired repertoire of effective ways in which a task can be approached or a problem can be solved. To be numerate, an individual needs to possess these four types of knowledge to a level that enables him or her to function effectively at school and in the community. Not all students find it easy to achieve this level of numeracy, as the next chapter explains.

L i nk s t o m o r e o n calc u lat i n g an d p r o bl e m s o lv i n g ◗ Not all experts support a purely constructivist approach to mathematics teaching. Read a perspective from New York (2003) regarding curricula that lack adequate attention to the teaching of arithmetic skills. Available online at: http://www.nychold.com/pr-cf-030104.html ◗ A valuable website is provided by Ian Thompson. You will find many articles that can be downloaded covering topics such as teaching >

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mental calculation, place value, calculators, written calculation methods and much more. Available online at: http://www.ianthompson.pi.dsl. pipex.com ◗ A staff development seminar from New Zealand on teaching and assessing number facts is available online at: http://www.nzmaths. co.nz/Numeracy/Other%20material/Tutorials/BasicFactsNzmaths. ppt#291,1 ◗ DfES. (2004a). Guidance paper: The use of calculators in teaching and learning mathematics. Available online at: http://www.standards.dfes.gov. uk/primaryframeworks/downloads/PDF/calculators_guidance_paper.pdf ◗ DfES. (2004b). Approaches to calculation. Available online at: http:// www.standards.dfes.gov.uk/secondary/keystage3/downloads/ma_ study015604_mod1.pdf ◗ TeacherNet: http://www.teachernet.gov.uk/teachingandlearning/subjects/ maths/Numeracy ◗ Burns, M. (2007). 7 basics for teaching arithmetic today. Scholastic Website. Available online at: http://teacher.scholastic.com/professional/ teachstrat/arith.htm ◗ Resources and ideas for teaching number and place value. Available online at: http://teachingideas.co.uk/maths/contents04number.htm ◗ Resources for teaching the four operations. Available online at: http:// math.about.com/od/fouroperations/Add_Subtract_Multiply_Divide.htm ◗ Sherman, L., & Weisstein, E. W. (2004). Arithmetic. From MathWorld: A Wolfram Web Resource. Available online at: http://mathworld.wolfram. com/Arithmetic.html ◗ The New Zealand Numeracy Development Project (NDP) provides nine guidebooks for teachers covering most aspects of number work and computation. Details are available online at: http://www.nzmaths.co.nz/ Numeracy/2007numPDFs/pdf_updates.aspx ◗ McIntosh, R., & Jarrett, D. (2000). Teaching mathematical problem solving: Implementing the vision. Available online at: http://www.nwrel. org/msec/images/mpm/pdf/monograph.pdf ◗ Problem solving strategies: Math. Available online line at: http://math. about.com/library/weekly/aa041503a.htm

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Barriers to numeracy

Key issues ◗ The method of teaching is often a contributory cause of learning difficulty: For example, too much attention (or too little attention) given

to computational skills; lack of explicit instruction; loss of continuity; pace too rapid; poor communication. ◗ Other causes: Absence from school; changing schools; textbooks. ◗ Specific learning disability: Do some students have a disability that causes them to have major difficulties when processing quantitative data?

Studies already referred to (e.g., American Institutes for Research, 2006; Moser, 1999; OECD, 1998) revealed the unfortunate fact that many stu dents leave school with fairly limited numeracy skills and often a marked dislike for (or even fear of ) mathematics. Great concern has been expressed over this unacceptable situation, leading to some major new initiatives in providing additional opportunities for adults who wish to revisit mathe matics and gain more effective numeracy skills through evening classes and other provisions (Coben, 2003; McGlynn, 1999). What are the factors that have contributed to the failure of a significant number of students? Is the subject matter of mathematics simply too abstract for most students to understand and master? Can the problem be traced back to the method of teaching? Do these individuals who have great difficulty

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with almost all aspects of the subject have some form of learning disability? These issues will be addressed in this chapter.

Teaching method as a cause of difficulty It is not the case that the subject matter of mathematics is intrinsically impenetrable to all but an elite few students. Nor must it ever be assumed that all students who have difficulties with the subject have an innate disability that specifically prevents them from learning mathematical con cepts and operations. While it is true that such a disability does appear to exist, it affects only a very tiny percentage of the population, not the 35–40 per cent said to have significant difficulty with mathematics (American Institutes for Research, 2006). Instead, it can be inferred from points in previous chapters that a major contributory cause of learning difficulty in mathematics is poor quality teaching. Martin (2007) observes that innumeracy, or ‘mathematical illiteracy’, may not be the result of the subject matter taught, but the pedagogy used to teach it. Once children enter formal schooling, the effectiveness of the instruc tion they receive is the major influence on their progress toward numeracy. There are many ways in which the teaching of mathematics in primary and secondary schools falls short of the ideal (Boaler, 1999; Booker, 2004; Lamb, 2004; Weiss & Pasley, 2004). Often the problem relates to an imbalance between the amount of attention given to building fluency in computational skills compared to the time devoted to problem solving and to applying such skills. But other negative factors also contribute to poorquality teaching.

Too much discovery-type activity without adequate guidance and support Some learning problems occur when student-centred inquiry methods are used inefficiently. In attempting to implement an investigative approach, a teacher may fail to provide students with essential information to help them make complete sense of their discoveries and refine their existing strategies. The research evidence is strongly against the effectiveness of entirely openended and minimally guided discovery methods because most students need a great deal of support from their teacher in interpreting their findings and accommodating these within their existing mental schema (Kirschner et al., 2006; Mayer, 2004).

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Too little attention given to mastery of number facts and computational skills Inability to calculate quickly and accurately is one of the main characteristics of students with difficulties. Again, if teachers follow the recommendations from the education reforms, they may get the message that ‘doing arithmetic’ is unnecessary and represents the most boring and meaningless way to develop numeracy. Common sense, however, should tell us that proficiency in calculation is still an essential sub-skill of problem solving, as emphasised in the previous chapter. A problem-based approach is relatively ineffective if students can’t compute quickly and accurately. But computational skills need to be taught in a way that ensures students fully understand the principles on which the various algorithms operate, rather than simply learning them by rote as a set of steps to follow. Booker (2004, p. 139) has remarked: It is a focus on procedures learnt by rote which leads to most of the difficul ties that students experience in mathematics. If materials are not used, students may not be able to visualize the significance of the steps in the processes they are attempting to complete.

The ability to recall number facts and to perform basic addition, subtraction, division and multiplication needs to be firmly established. Automaticity in applying these skills can only be achieved with adequate practice (Westwood, 2003). Investigative approaches, if used alone, are generally deficient in this respect.

Too much attention given to computational skills This is the other side of the coin. Learning difficulties and a poor attitude to mathematics can occur if a teacher errs in the opposite direction by teaching mainly mechanical arithmetic and neglecting more motivating ways of making mathematics meaningful (Boaler, 1999; Martin, 2007). It has been noted already that this tendency to focus too much on arithmetic exists in primary schools when non-specialist teachers lack sufficient understanding of the scope, purposes and nature of mathematical learning. These teachers teach in the way that they were taught, with textbook and worksheets, because it is easy and it is secure. Most teachers of primary classes have not learned mathematics themselves in the way they are now expected to teach it. Of course, this problem is not confined to primary schools; secondary

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schools often resort to teaching mathematical concepts and operations in a highly formal manner, giving little attention to whether or not students understand fully what they are doing.

Curriculum content covered too rapidly It was noted previously in relation to teaching the numeracy hour that some teachers do not allocate sufficient time for studying each topic or concept before moving to something new (Kyriacou & Goulding, 2004; Watson, 2004). Under these fast-paced conditions, students do not really assimilate and master essential concepts and skills so learning difficulties can arise. The same thing can occur, of course, when covering the mathematics curriculum too quickly by any other teaching method.

Lack of continuity Within the domain of mathematics, essential concepts, strategies and skills tend to develop over time in a hierarchical and sequential manner, pro gressing from simple to more complex. If teachers ‘pick and mix’ topics without reference to their cognitive demands, some students will have diffi culties coping with them. Problems of continuity can occur, for example, with discovery-based methods because it is impossible to ensure that each new problem, topic or issue will involve number skills and concepts that are developmentally appropriate for students of a given age. Butterworth (1999, p. 298) has observed: There are many reasons for being bad at any school subject. But school maths is like a house of cards: the cards in the bottom layer must be firmly and accurately constructed if they are to support the next layer up. Each stage depends on the last.

Teachers’ less than perfect communication skills The learning of mathematics in school is greatly enhanced when teachers present information clearly, use language that is easily understood by stu dents, and make the subject more real and visual by using practical examples, diagrams, manipulatives, computer simulations and models. Silver and Hagin (2002) state that without concrete, visual, and experiential backup, verbal problem-solving approaches may leave many students lost. If

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teachers spend too much time ‘lecturing’, asking questions that are vague and poorly focused, and if they fail to define mathematical terms adequately, many students will have difficulties. Teachers need to be effective communicators when offering explan ations to students, clarifying students’ ideas, and answering students’ queries (Westwood, 1998). They also need to develop excellent skills in demon strating mathematical operations and problem-solving strategies on the board and with concrete materials. Much confusion arises in mathematics classes when teachers are poor communicators.

Moving to an abstract and symbolic level too soon In the preschool and early school years children acquire their understanding of quantitative relationships almost entirely from their real-life concrete experiences, and through seeing, handling and visualising objects. Gradually, they are ready to move from the concrete operational stage and begin to use symbols to represent quantities and the operations that can be performed with those quantities. This process of moving from concrete to abstract level of reasoning marks the point at which some children begin to have difficulty with mathematics (Booker, 2004; Heddens, 1986). If teachers abandon the use of materials and visual aids too soon and begin to teach new material using only chalk, talk, numerals and mathematical notation, some children will begin to lose the underlying meaning of number operations and will be forced to resort to rote memorisation of rules and procedures (Booker, 1999).

Inadequate review and revision Students will simply forget the mathematics they have been learning and will therefore fail to become fully numerate, if they are not given an opportunity to revisit concepts and skills frequently. Research outlined in Chapter 4 has shown clearly that one of the features of effective teaching of mathematics is regular review and revision of work that has been completed in previous weeks. Bruner’s (1966) notion of a ‘spiral curriculum’ applies here. Instead of presenting the mathematics curriculum in a linear manner, moving steadily on from topic to topic, teachers should regularly revisit previous concepts and skills. They can do this by creating new and more challenging tasks or problems where students can apply this prior learning

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again. Connectionist teachers make an effort to link new work with what students have experienced before.

Teachers failing to address students’ individual differences While teachers quickly become aware that some students are finding mathematics difficult, it is fairly rare that they make any significant adjust ments to attempt to tailor the teaching to students’ level or rate of learning. Fahsl (2007) suggests that the instructional needs of these students can be met more successfully if teachers make some simple modifications to the general approach used in the classroom. This obviously becomes increas ingly necessary if a class contains some students with special educational needs. She offers practical examples such as: ◗ encouraging the use of multiplication table-charts and calculators to bypass some of their weaknesses in computation ◗ representing concepts and operations in a visual and concrete manner by making greater use of structural apparatus, diagrams or sketches, in both primary and secondary schools ◗ breaking new work down into smaller units rather than presenting an infor mation overload ◗ distributing problems and tasks on printed sheets to avoid students wasting time and making errors when copying from the blackboard/whiteboard ◗ checking more closely for students’ understanding at each step in the lesson ◗ providing students who have untidy bookwork with squared paper instead of lined or blank sheets to help them keep figures and recordings correctly aligned ◗ varying homework tasks according to students’ needs (i.e., for extension, or for further practice).

Other contributory causes Absence from school Due to the hierarchical nature and interconnectedness of mathematical concepts, this subject is affected more than any other if a student is fre quently absent from school. Gaps in learning will occur. It is difficult for the student to understand and catch up with the work when he or she returns to the class.

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Moving to a new school Discontinuity resulting in confusion can also occur when a student moves from one school to another. Differences in curricula can result in learning difficulties.

Inappropriate textbooks Textbooks and other instructional materials can contribute to children’s learning difficulties in mathematics. In an ideal situation, a textbook will contain sufficient worked examples of particular problems and operations to enable a student to learn from these as well as from the teacher and peers. Many textbooks still fail in this respect and are therefore less than helpful to a student who needs to go over work again at home. It is also common to find that textbooks do not provide enough practice items to meet the needs of the students who require more than the usual amount of repetition in order to master new concepts and skills.

The detrimental effect of failure Finally, we must consider the impact of ongoing failure in mathematics on a student’s learning and motivation. Although this adverse impact is the result of learning difficulty rather than a primary cause of it, it must be considered here because it greatly exacerbates the learning problem. The impact of persistent failure produces a very damaging effect on a student’s self-esteem, self-efficacy, confidence, motivation and attitude towards the subject. In particular, ongoing failure impairs a student’s willingness to persevere in the face of difficulties (Chinn & Ashcroft, 1998). Learned helplessness and avoidance tactics are common among students who have difficulty mastering basic mathematics, because they attribute their failure to their own lack of ability (Houssart, 2002).

The nature of students’ difficulties All students who are weak in mathematics have a fairly characteristic pattern of difficulties. Some of these difficulties will be referred to again in Chapter 7. They can be summarised here as: ◗ poorly developed number sense ◗ general slowness and uncertainty in carrying out even routine calculations.

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This slowness reduces the amount of practice the student actually engages in during lessons and prevents the development of automaticity and fluency ◗ uncertainty in translating number words into correct numerals ◗ extremely untidy bookwork leading to errors. This weakness seems to be due either to fine-motor coordination difficulties and poor spatial ability, or to an attitudinal problem ◗ major difficulties in learning and recalling basic number facts, multiplication tables, and computational procedures ◗ difficulty appreciating the relative size of numbers ◗ very poor understanding of place-value (e.g. that in the number 111 the first numeral on the left represents 100 while the final numeral represents 1 unit) ◗ difficulty comprehending the exact meaning of specific mathematical terms ◗ major problems with understanding what is required in word problems and in selecting correct operations ◗ lack of effective strategies for approaching mathematical tasks and problems ◗ inability to recognise when an obtained answer is not reasonable ◗ reading difficulties associated with the textbook or worksheet.

A few students exhibit extreme difficulty in becoming numerate. Psych olgists believe that these students have a specific disability that impairs their capacity to deal with quantitative data and to master the abstract nature of mathematics. These students are said to have dyscalculia (Michaelson, 2007), a term that is now subsumed within the more recent classification ‘mathematics disability’ (Geary, 2005). Michaelson (2007, p. 21) says that dyscalculia ‘… is a debilitating disorder that affects a person’s ability to conceptualise operations and processes of fundamental mathematics’.

Dyscalculia Dyscalculia (or more correctly, developmental dyscalculia) is a form of learn ing difficulty presumed to be of neurological origin, probably genetically determined and perhaps affecting up to 3 per cent of the population (Colwell, 2003; Landerl et al., 2004; Munro, 2003). The ‘ bible’ of psycho logical assessment – the Diagnostic and Statistical Manual of Mental Disorders (APA, 2000) – places the prevalence rate at about 1 per cent. Given that at least 35 per cent of all students do not achieve well in mathematics, dys calculia is obviously not the primary cause of learning difficulty in most of

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these cases. In the majority of cases learning difficulties are caused by the quality of teaching. Instead of using the term dyscalculia, the International Classification of Diseases (ICD-10) (WHO, 2007) prefers the term ‘specific disorder of arithmetic skills’ and applies the description to individuals who are not intellectually impaired and have received normal schooling but display major weaknesses in dealing with numbers and carrying out calculations. These ‘disorders’ are considered to stem from some type of subtle neurological dysfunction. Landerl et al. (2004) believe dyscalculia reflects a brain-based deficit that specifically affects numerical processing and is not due to weaknesses in other cognitive processes such as attention, memory or visual perception. The following characteristics are often reported for these students: ◗ poor mathematical concept development ◗ lack of understanding of mathematical terms ◗ confusion over printed symbols and signs ◗ extremely poor recall of number facts ◗ weak multiplication skills ◗ poor procedural skills ◗ inability to determine which processes to use in solving problems ◗ poor bookwork with misaligned columns of figures ◗ frequent reversal of single figures and reversal of tens and units (e.g. 34 written as 43) ◗ difficulties with reading text compound the student’s problem in maths.

Individuals with dyscalculia differ in the extent to which they exhibit these particular difficulties. Dowker (2005) says that dyscalculia may have a wide range of causes and therefore presents with different patterns of impairment. She suggests that the specific abilities of students with this difficulty are patchy, with some concepts or skills being relatively stronger or weaker than others. Of course, it can be argued that these difficulties are evident in very many students who do poorly in mathematics, and certainly they are not unique to dyscalculic students. However, it is the severity of the problems and their resistance to normal remedial intervention that set dyscalculic students apart from others with learning difficulties.

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It has been suggested that one of the main underlying weaknesses in students with dyscalculia is limited capacity in working memory (McLean & Hitch, 1999). Working memory is very important for efficient processing of information during problem solving and when completing mathematical calculations. Learners have to store verbal and numerical information in working memory while at the same time processing various steps within a procedure. They also need to be able to retrieve relevant information from memory quickly and efficiently. It is interesting that limitation in working memory in the kindergarten years has proved to be one of the predictors of possible future mathematics learning difficulties (Gersten et al., 2005). Lyon et al. (2003) suggest that within the dyscalculic population two sub-groups exist: (a) those individuals with significant difficulties in learning and retrieving number facts, and (b) those who have difficulty learning and applying the procedures involved in calculating. Geary (1993) concurs, and adds a third group (visuo-spatial subtype) with problems in misreading and misrepresenting place values, occasionally reversing and transposing numerals, and great untidiness in setting out bookwork. Dyscalculia is rarely identified early. Often students are not referred for assessment until the fifth year of school or later. For this reason, there has been a recent increase in interest in designing screening procedures to help detect children at risk in the kindergarten years (e.g., Butterworth, 2003; Mazzocco & Thompson, 2005). From these studies, predictors of potential mathematical disability have been identified. The main predictors include not knowing which of two digits is larger, lacking effective counting strategies, poor fluency in identification of numbers, inability to add simple single-digit numbers mentally and limitations in working memory capacity. It is important to state again that a specific disability in mathematics learning is relatively uncommon. Teachers should never assume that every student with poor results in mathematics has dyscalculia. It is far more likely that their learning difficulty is due to insufficient teaching, or reduced opportunity to learn in earlier years. Their problem is now being main tained by secondary emotional and attitudinal reactions such as learned helplessness, loss of confidence and motivation. The action that teachers need to take in relation to all students with learning difficulties, whether due to dyscalculia or other reasons, is to assess their current knowledge and skills as accurately as possible, and provide instruction that will build

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effectively on what they already know. The important issue of assessment is the focus of the following chapter.

L i nk s t o m o r e o n d i ff i c u l t i e s i n l e a r n i n g m at h e m at i c s ◗ For information on adapting mathematics for students with intellectual disability, see Planning, teaching and assessing the curriculum for pupils with learning difficulties: Mathematics (2001). London: Qualifications and Curriculum Authority. Available online at: http://www.nc.uk.net/ld/ dump/Ma_ld.pdf ◗ Yetkin, E. (2003). Student difficulties in learning elementary mathematics. ERIC Digest. ED482727. Available online at: http://www. ericdigests.org/2004-3/learning.html ◗ Wikipedia has a very good entry dealing with dyscalculia, including a comprehensive list of symptoms at: http://en.wikipedia.org/wiki/ ◗ For general information on dyscalculia, together with advice on improving computational skills, number facts and problem solving, see Dyscalculia defined. In NetNews 5, 4, n.p. (2005). Available online at: http://www.eric.ed.gov/ERICDocs/data/ericdocs2sql/content_storage_ 01/0000019b/80/1b/ed/84.pdf ◗ Practical advice on teaching basic number skills is provided in Geller, C. H. (2000). Strategies for teaching arithmetic: What are the facts? Learning Disabilities Journal, 10, 4, 15–19. Available online at: http:// www.ldam.org/pdf/journal/2000/11-00_arithmetic.pdf ◗ Dowker, A. (2004). What works for children with mathematical difficulties? Research Report RR554. London: DfES, at: http://www. dfes.gov.uk/research/data/uploadfiles/RR554.pdf

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Assessment

Key issues ◗ Assessment allows teachers to monitor the effectiveness of their teaching: Assessment should be linked closely with the objectives and

standards specified in the mathematics curriculum. ◗ Assessment enables teachers to modify their numeracy program if necessary: When teachers have precise information about students’

current knowledge and skills they can match their instruction and learning tasks more accurately to students’ ability. ◗ Assessment identifies which students are having difficulties and require additional assistance: Testing of a class will reveal which

students are making very good progress, which students are progressing at a satisfactory level, and which students need support. Additional diagnostic testing of the at-risk group will reveal the type of support needed. ◗ There are many ways in which teachers can assess students’ numeracy: Examples include observation, testing, examining work

samples, analysing students’ errors and conducting diagnostic interviews. ◗ Some forms of assessment are required at education system level: These tests are used to monitor and report standards across schools.

In order to monitor the development of numeracy in an individual student, in a whole class of students, or in all schools across an education system, it is necessary to conduct regular assessments. These assessments take many forms, ranging from the informal minute-by-minute observations 68

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that a teacher makes while conducting a lesson, through to large-scale testing projects such as the National Assessment Program in Literacy and Numeracy (NAPLAN) in Australia, or the National Assessment of Educational Progress (NAEP) in the United States of America. When the era of standards and accountability in education arrived it heralded an increase in the number of assessments of students’ learning that are made and reported each year. Lokan et al. (2000) describe many forms of such assessment in the numeracy domain. Similarly, the Numeracy assessment guide, produced by the Department of Education in Victoria, summarises several assessment procedures and discusses their applicability, strengths and limitations (Department of Education, Victoria, 2007). Different forms of assessment within mathematics serve a number of different purposes. For example, early assessment of young children is usually conducted to obtain a clear picture of each child’s knowledge and skills on entry to school and to identify any children who may require more than the usual amount of support in their learning (Commonwealth Department of Education, Science and Training, 2002; Doig et al., 2003). At the other end of the age range, senior students receive a very different form of assessment to determine their mathematical knowledge and skills in order to report such data accurately to potential employers or for entry into tertiary studies. Ysseldyke and Tardrew (2007) confirm that regular assessment of students’ progress makes it more feasible for a teacher to differentiate instruction and resources more effectively. In Australia, all states and territories have adopted some form of early assessment of children’s number knowledge. For example, New South Wales has the Schedule for early number assessment; Queensland includes numeracy within its Diagnostic Net covering the first three years of school; Tasmania assesses number skills against the Key Intended Numeracy Outcomes (KINOs). Details of these and other numeracy assessment programs can be found in Assessment of literacy and numeracy in the early years of schooling: An overview (Commonwealth Department of Education, Science and Training, 2002).

Purposes of assessment It has become popular to identify three main purposes of assessment in schools and elsewhere as assessment for learning, assessment as learning, and assessment of learning (Manitoba Education, Citizenship and Youth, 2006).

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1 Assessment for learning refers to using assessment data to improve the teaching program by finding out exactly what students know, where they are having difficulty, and how best to support their development. This form of assessment guides a teacher’s decision making and leads to action that will enhance learning.

2 Assessment as learning refers to assessment that causes students to monitor their own performance and think more deeply about their own learning needs. It causes them to examine more closely their thinking processes and strategies as they endeavour to construct knowledge and reflect upon their results. Students become more effective learners if they can self-assess and identify their own learning strengths, weaknesses and needs.

3 Assessment of learning refers mainly to assessment of students’ overall achievement relative to the goals, objectives or standards for the curriculum.

Classroom assessment of development in numeracy serves the same basic functions as assessment in other areas. In respect to assessments made by teachers on a regular basis, these functions include: ◗ checking the overall efficacy of the teaching program ◗ identifying any concepts, strategies or operations which may need to be retaught, reviewed or practised further with the whole class ◗ determining the stage of development any particular student has reached ◗ gaining information on an individual student’s specific weaknesses and special instructional needs.

In order to determine whether a teaching approach is effective it is necessary to assess students’ knowledge, skills and strategies on a regular basis (DfCSF, 2007a). Such assessment in numeracy is recommended to take one of three possible forms – short-term, medium-term and longerterm. Short-term assessment relates to the observations a teacher makes concerning children’s understanding and performance during any lesson. Short-term assessments are often referred to as formative. Such assessment allows for re-teaching of any concept or operation if necessary, or allocation of additional practice time. Short-term assessment can also lead to timely remedying of misconceptions. Medium-term assessment allows a teacher to appraise students’ progress over a longer time (e.g., every six weeks, or at half-term). Longer-term assessment relates to evaluation of learning over the school year and provides information for students’ next teacher, for

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parents, for the authorities and (in the case of secondary school students) sometimes for potential employers. This longer-term assessment is based mainly on the learning objectives or standards stated in the curriculum for students of a given age. Both medium- and longer-term assessments are often referred to as summative. Coben (2003, p. 66) summarises the three main forms of assessment in stating that: Assessment may be regarded as the sharp end of curriculum development; the point at which teachers endeavour to establish what an intending learner already knows (diagnostic assessment), devise or adjust programmes of study according to the progress the learner is making (formative assessment) or find out whether what has been taught has been learned (summative assessment).

In relation to using assessment to evaluate the effectiveness of teaching, McIntosh (2007) recommends that teachers should look for and record evidence of students’ improvements in the following areas: conceptual understanding, knowledge of facts, number sense, competence in applying skills, problem-solving ability, attitude and confidence. Such evidence may be collected by any (or all) of the methods described below.

Approaches to assessment In the numeracy domain, the following procedures are commonly used to obtain relevant information: ◗ observation of the students while engaged in mathematical activities ◗ questioning students individually, or within small-group contexts ◗ analysing samples of the students’ written work, including exercise books and portfolios ◗ applying teacher-made or published tests ◗ using an inventory or checklist of essential knowledge and skills ◗ diagnostic testing and individual interview (which may also include the use of any or all of the above procedures).

Observation A teacher will know in advance what to look for during a mathematics les son in terms of children’s understanding and application; direct observation

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of students at work can provide this evidence. Observing students as they answer and ask questions and as they work on tasks in the classroom pro vides a teacher with valuable insights related not only to their level of understanding of the lesson content but also affective aspects of their engagement in the work. For example, the teacher will notice if the stu dents are intrinsically motivated by the problems and tasks, and if they are confident, hesitant or anxious. It will also be evident whether the students are monitoring their own performance and self-correcting when necessary. Careful observation of students at work will reveal any difficulties students have in keeping on task and completing assigned work. The teacher obtains a fairly clear picture of which students are progressing well and which individuals require additional follow-up. Reys et al. (2006) advocate that teachers keep reasonably detailed records of what they have observed during lessons, particularly where the evidence suggests that something may need to be taught again, or where additional practice or different examples may be required. Booker et al. (2004) provide some very useful examples of concise record keeping at the individual student and whole-class levels.

Checklists One way to structure the process of observation and data collection from time to time is to use some form of checklist containing a concise summary of concepts, knowledge and skills that students are expected to have mastered by a certain age. Using representative curriculum content from the current and earlier years, teachers can construct their own informal numeracy checklist or inventory containing an appropriate bank of items. Such a list can be very useful in surveying quickly and effectively what particular students already know and what they still need to practise. Bahr (2007) recommends using a simple inventory of this type to obtain an overview of the range of ability in a new class.

Work samples Much diagnostic information can be gleaned from looking carefully through students’ exercise books, homework, test papers and mathematics projects. Samples of ‘rough’ workings, for example, may reveal not only faulty computation but may throw some light on the strategies the student has invented and tested while attempting a difficult or unusual problem. Such papers may also reveal clues that the student is still working at a

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semi-concrete level (e.g., the use of tally marks in the margin for adding or dividing; tiny drawings to help visualise the problem). Any areas of difficulty that are detected can then become the focus of a more in-depth diagnostic interview as described below. Work samples can include students’ portfolios. A mathematics portfolio might contain such items as homework samples, drawings, diagrams, test sheets, teacher’s checklists, solutions to problems that the student has attempted, interpretive writings and students’ self-appraisals (Koca & Lee, 1998). Van de Walle (2006) suggests that portfolios should also contain students’ own self-evaluation comments and reflections.

Testing Used alone, a test is not an adequate measure of a student’s ability; it is only one step in the process of collecting data to assist with decision making (McAsey, 1999). All data from testing needs to be supplemented with information obtained by different means. However, effective teachers do make good use of tests for both diagnostic purposes and to measure students’ overall achievement. Teacher-made tests should be directly linked to the objectives set for that particular unit of work and are often referred to as ‘curriculum based’ or ‘outcomes based’. Clear objectives make the design of assessment materials easier because they indicate not only what knowledge or skill the student must demonstrate but also the standard that is required. The ideal teacher-designed test should embody the following features: ◗ The test begins with a few easy items to allow even the least able students to experience some success. ◗ At least two, preferably three, items are provided at the same level of difficulty to enable the teacher later to differentiate random and careless errors from those that are persistent. ◗ A variety of question formats to make the test more interesting (e.g. some multiple-choice, some ‘missing numbers’, some calculations in which the work ings must be shown, some drawing or measuring, some word problems, etc). ◗ The concepts and skills tested should relate precisely to those covered in the teaching program. ◗ Unless the test is designed to assess only computational skills it should con tain problems that will allow the students’ conceptual understanding, strategic knowledge and reasoning to be appraised.

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Many published tests are available covering basic numeracy. Examples include Diagnostic mathematical tasks (DMT) (Schleiger & Gough, 2002), Numeracy progress tests (NPT) (Vincent & Crumpler, 2000), Booker profiles in mathematics: Number and computation (Booker, 1994) and, for younger chil dren, I can do maths (Doig & de Lemos, 2000). For details of these and other test materials, check the website for ACER Press (see the Links box at the end of the chapter).

Error analysis In the case of students with learning difficulties it is often helpful to examine in detail the nature of the errors they make in their written work. Within an adequate sample of a student’s paper-and-pencil calculations, a pattern sometimes emerges indicating a specific point of confusion in relation to a particular algorithm (Ashlock, 2005). Sometimes, however, errors appear to be fairly random and may reflect inaccurate recall of basic number facts or multiplication tables, a tendency to be distracted while working, or poor vertical alignment of figures on the page. Many different ways of categorising computational errors have been devised, some of them much too complicated and time consuming for use by teachers. One of the least complicated systems suggests that errors tend to fall into one or more of the following categories: ◗ wrong operation (e.g. adding instead of multiplying) ◗ defective algorithm (incorrect in one or more steps within the procedure) ◗ number fact error within the calculation (e.g. 3 x 7 recalled as 28) ◗ place value problem (forgetting that magnitude of a figure is indicated by its position) ◗ specific difficulty dealing with zeros during computation (for example, zeros within the top line in subtraction algorithm).

Error analysis is best implemented within the context of an individual diag nostic assessment interview, as described more fully in the next section. Unless error analysis leads to well focused intervention, there is no point in carrying it out. When the procedure does prove to be of value in identifying specific weaknesses that need to be remedied, it must still be remembered that the aim of intervention should be to help the student understand the process more effectively at a conceptual level, not merely replace one rotelearned computational trick with another.

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Closely related to error analysis are the many ‘diagnostic number tests’ that have been developed over the years. Most of these tests are based on an analysis of the specific steps involved in completing computational algorithms at different levels of difficulty. The test items begin at an easy level and progress gradually to more complex examples. The aim of using such tests is to identify precisely the point at which a student begins to make errors so that re-teaching can commence at that point.

Diagnostic interview The value of talking with individual students about their progress and their difficulties in mathematics has been more fully appreciated in recent years (e.g., Ministry of Education, NZ, 2003; Wright, 2003). The numeracy program Count Me In Too in New South Wales uses an interview procedure (Schedule for Early Number Assessment: SENA) as an essential component for data collection that guides program planning. A full description and evaluation of SENA can be found on the ACT Department of Education and Training website under ‘Assessable Moments in Numeracy’ (see the Links box at the end of the chapter). While an individual interview may be designed to monitor a particular student’s overall understanding of a topic, it is more likely to be carried out for diagnostic purposes when a student is observed to be having difficulties. An individual interview with a student is a powerful way to discover not only the knowledge he or she possesses but also the quality of his or her thinking and the manner in which the student copes with challenges (Reys et al., 2006). Assessment should also reveal whether the student approaches problems at a purely procedural level or whether he or she is developing sound conceptual understanding of key concepts and operations. Work samples and test papers can provide the basis for individual interviews with students. For example, using a test paper as the focus of an interview may involve asking the student to explain or to demonstrate how he or she obtained a particular answer or performed a particular operation. The teacher watches and listens as the student reworks a pre viously incorrect test item, and can identify the exact point of conf usion and provide corrective feedback. It is necessary to listen to the student’s own explanation of why he or she took some particular action at a specific point during the solving of a problem or the working of a calculation (Booker, 1999).

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Four key questions can provide teachers with a suitable framework for pro bing the knowledge, skills and strategies of an individual. The questions are: 1 What does the student know already; and what can the student do correctly without assistance? Answers to this question reflect the student’s current repertoire of concepts, skills and strategies.

2 What can the student do if given some degree of support or guidance? Answers to this question will reveal concepts and skills to be taught and scaffolded as priorities within the student’s ‘zone of proximal development’.

3 What gaps exist within the student’s previous learning? Often specific gaps can be detected in a student’s knowledge of certain operations, certain forms of notation, certain number facts or multiplication tables.

4 What does the student need to be taught next as a top priority in his or her program? Answers to this question need to take into account both the learning targets specified within the maths curriculum for the student’s age level, and the answers to the second and third questions above.

While some of the information to answer these diagnostic questions can be obtained from tests and work samples, teachers or tutors will also need to work closely with the student to obtain additional insights. This form of individual interview is sometimes referred to as ‘dynamic assessment’ because the teacher needs to be flexible in order to adapt or modify the tasks and questioning as the interview progresses. For example, during the assessment the teacher may need to move up or down between concrete to abstract levels of reasoning in order to determine the level at which the student is operating. The teacher may also decide to re-teach an operation or skill during the interview process, and then observe the extent to which this corrective feedback has been immediately understood and used cor rectly by the student.

Assessing problem-solving skills Assessment of a student’s numeracy must include his or her ability to inter pret and solve routine and non-routine problems. Routine problems are those where it is fairly obvious which operations to use to obtain a solution. Non-routine problems are more challenging because the procedures required for solution are not obvious.

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In assessment of problem-solving ability, a teacher is not only interested in the answer obtained but also the way in which it was obtained (i.e. the strategies that were applied). Bahr (2007) encourages teachers to use reallife problems that will reveal more about students’ quality of thinking and the flexibility of their approach. Again, the individual interview, rather than paper-and-pencil testing, is the most appropriate method for assessment of problem-solving skills. If a student has difficulty solving routine problems, the teacher needs to check: ◗ Can the student actually read the problem? ◗ Having read the words, does the student understand what is required? ◗ Is the student able to summarise and explain the problem to the tester? ◗ Can the student identify the appropriate operations to use? ◗ Is the student able to encode the correct algorithm? ◗ Can the student complete the algorithm correctly, swiftly and confidently? ◗ Does the student appear to have difficulty recalling basic number facts? ◗ Is the student able to check the reasonableness of the result obtained? ◗ Does the student self-correct when necessary?

For non-routine problems most of the questions above also apply. Extra attention must be given to the following issues: ◗ Does the student have a strategy for beginning the task? ◗ Can the student explain any steps to take that may help determine a solution? ◗ Can the student identify relevant information within the problem? ◗ Does the student benefit from hints the teacher might provide (e.g. ‘Maybe you could draw a sketch’; ‘Why not compare the two amounts’)? ◗ Does the student spontaneously make use of available aids (e.g. number line, table chart, or calculator)? ◗ What does the student do if the first attempt at a solution is unsuccessful? ◗ How long is the student willing to persevere with a challenging problem?

In the case of both routine and non-routine problems it is important to observe whether the students make careless errors through inaccurate encoding or untidy placement of the algorithm on the page.

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Conclusion Assessment is an essential component of effective teaching in all areas of the curriculum. In the case of numeracy development, the various forms of assessment are used to indicate precisely the knowledge, concepts, skills, strategies and beliefs an individual student has acquired. Assessment also reveals any misconceptions and gaps in students’ knowledge that need to be remedied. Assessment also provides teachers with a clear indication of the overall effectiveness of their teaching program, and should therefore lead to modifications where necessary. Assessment is only useful if it leads to appropriate action – at the level of the individual student, the class, the school, or the system.

L i nk s t o m o r e o n a s s e s s m e n t i n n u m e r ac y ◗ Advice on short-term, medium-term and longer-term assessment associated with the National Numeracy Strategy (UK) can be found at the DfCSF Standards website: http://www.standards.dfes.gov.uk/ primary/publications/mathematics/math_framework/assessment/ ◗ A very good summary of the purpose and nature of assessment in primary mathematics can be found online at: http://www.bristol-cyps. org.uk/teaching/primary/maths/pdf/assessment_summary.pdf ◗ The Department of Education (Victoria). (2007). Numeracy assessment guide. This document summarises several procedures for numeracy assessment, discussing their applicability, strengths and limitations. Available online at: http://www.eduweb.vic.gov.au/edulibrary/public/ teachlearn/student/numeracyasstguide.pdf ◗ The ACT Department of Education and Training operates a website (Assessable Moments in Numeracy) containing much useful information for teachers. In particular, it provides a description of the diagnostic interview known as Schedule for Early Number Assessment, associated with the NSW numeracy program Count Me In Too. Critiques are also provided of other forms of assessment such as portfolios, checklists and learning logs. Available online at: http://activated.act.edu.au/ assessablemoments/index.htm

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◗ Details of the Australian National Assessment Program in Literacy and Numeracy (NAPLAN) (2008) are available at the website for the Ministerial Council on Education, Employment, Training and Youth Affairs. Available online at: http://www.naplan.edu.au/about/about.html ◗ Information on the West Australian Literacy and Numeracy Assessment (WALNA) for students in Years 3, 5, and 7 is available online at: http:// www.det.wa.edu.au/education/walna/index.html ◗ A sample WALNA test paper for Year 7 numeracy is available online at: http://www.det.wa.edu.au/education/walna/pdfs/Yr7SampleNumeracy~ Test.pdf ◗ A general-purpose numeracy assessment instrument is presented by Hart, K., Ampiah, J. G., Nyirenda, D., & Nkhata, B. (2004) in Teacher’s guide to numeracy assessment instruments. Available online at: http:// www.cripsat.org.uk/downloads/numeracy_guide.pdf ◗ National Assessment Program for Literacy and Numeracy. NSW Department of Education and Training. Available online at: http://www. det.nsw.edu.au/media/downloads/dethome/yr2007/nafl_fact.pdf ◗ Secondary Numeracy Assessment Program (SNAP) for students in Year 7. NSW Department of Education and Training. Available online at: http://www.schools.nsw.edu.au/learning/7-12assessments/snaptest.php ◗ Literacy and Numeracy National Assessment (LANNA) for Years 3, 5 and 7. Conducted by Australian Council for Educational Research. See: http://www.acer.edu.au/lanna/ ◗ ACER Press can supply a variety of suitable tests and other assessment materials. Check the online education catalogue at: http://www.acer. edu.au/acerpress/edu-cat.html

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Index

automaticity 46, 49, 59, 64 avoidance 63

Main entries in bold ability grouping in mathematics 33, 38, 42 absence from school 57, 62 active teaching 36, 38 activity approach 34, 48, 50, 58 adaptive teaching 42, 62, 76 adult numeracy 3, 4, 7, 11, 33, 42–44, 45 affective outcomes from failure 1, 10–11, 72 algorithms 8, 21, 46, 47, 51, 52, 55, 59, 74, 75, 77 arithmetic v, 4, 7, 9, 17, 20, 21, 27, 37, 40, 42, 47–49, 50, 51, 55, 59 see also computation assessment 66, 68–78 curriculum-based 3, 73 diagnostic assessment 68, 71, 73, 74, 75–76, 78 dynamic assessment 76 formative 70, 71 observation 68, 71–72 outcomes-based 73 purposes of 69–71 summative 71 testing 3, 21, 68, 71, 73–74, 77 types of 71–76 attitude toward mathematics students’ 8, 28, 29, 33–34, 39, 40, 41, 59, 63, 64, 66, 71 teachers’ 40 Australian Association of Mathematics Teachers (AAMT) 2, 5

balance between skills development and problem solving v, 42, 44, 48, 49 bookwork untidy or careless 62, 64, 65, 66, 71, 72 Bruner, J. 25, 30–32 calculation 38, 48, 49, 52, 56, 59, 74, 75 see also computation calculators 17, 46, 48, 51, 56, 62, 77 changing schools 57, 63 Cockcroft Report: Mathematics counts 4 cognitive development 10, 24, 25–30, 65 cognitive maturity 29 communication teachers’ skills in 57, 60–61 computation 9, 20, 48, 56, 62, 72, 74 computational skills 9, 20, 36, 38, 48, 52, 56, 59, 62, 72, 74 importance of 47–49 teaching of 46–55 computer-aided instruction 51 concept development 22, 24–31, 65 conceptual understanding 8, 31, 35, 37, 40–41, 46, 51, 71, 73, 75 concrete materials 31, 61, 62 concrete operational stage 27, 28, 29, 30, 61 connectionist orientation 35–36, 41, 62 conservation of number 27 constructivist view of learning 25, 28, 55 97

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continuity in learning 34, 39, 57, 60 corrective feedback 54, 75, 76 counting 7, 14, 15, 16, 19, 21, 66 Crowther Report 1–2, 5 curriculum 1, 7, 9, 12, 18, 24, 28, 32, 36, 70 as a cause of difficulty 60 content 7, 9, 12, 30, 42, 44, 45, 60, 72 continuity 39, 60 declarative knowledge 46, 55 demonstrating as teaching technique 37, 54, 61 developmentally appropriate curriculum 24, 28, 60 diagnostic interview 68, 75–76, 78 Diagnostic Net (Queensland) 69 diagnostic testing 68, 71, 73, 74, 75–76, 78 differentiation 69 direct instruction 28, 34, 35, 36, 46, 47, 50, 54 discovery approach 29, 36, 50, 58, 60 dynamic assessment 76 dyscalculia 64–66 described 64 subtypes 66 early childhood 13–18, 22, 28, 32 enactive stage of learning 30 error analysis 74–75 fluency in number facts and computation 8, 51, 58, 64 formal operational stage 27 formative assessment 70, 71 generalisation in learning 1, 9, 10, 32, 54 guided participation 18 guided practice 36, 51, 54

homework 38, 39, 62, 72, 73 iconic stage in learning 30, 31 individual difference among students 33, 62 informal learning of number skills 13, 16, 17, 20, 34 innumeracy 58 inquiry method 58 interactive teaching 37–38, 39 investigative approach v, 9, 36, 40, 42, 48, 49, 58, 59 Key Intended Numeracy Outcomes (Tasmania) 69 learned helplessness 63, 66 learning difficulties 21, 29, 54, 59, 60, 63, 66, 74 learning disability 57, 58, 64–67 literacy 1–2, 3, 4, 5–6, 38, 43, 47 mathematical illiteracy 58 mathematics disability 64 see also dyscalculia Mathematics Recovery 17 Maths4life 4 maturation 26, 28, 29, 31 mechanical arithmetic 40, 59 mediated learning 18 memorisation 20, 40, 48, 61 mental calculation 27, 37, 50, 51–52, 56 methods see teaching methods minimally guided discovery 36, 58 modelling as teaching tactic 37, 54 multiple numeracies 1, 6 multiplication 48, 59, 62, 64, 65, 74, 76 National Assessment of Educational Progress (NAEP) 42, 43, 48, 69

index

National Assessment Program in Literacy and Numeracy (NAPLAN) 3, 69, 79 National Council of Teachers of Mathematics (NCTM) 3, 14–15, 22, 36 National Numeracy Strategy (UK) 3–4, 38, 78 National Plan for Literacy and Numeracy (Australia) 3 National Standards for Adult Numeracy (UK) 43 neo-Piagetians 28 number facts 7, 48, 50–51, 56, 64, 65, 66, 74, 76, 77 automaticity 46, 49 importance of 49–51, 59 numberline 14 number sense 13, 15, 19–22, 32, 34, 47, 49, 52, 63, 71 numeracy across the curriculum 1, 9–10, 12 adult numeracy 4, 7, 11, 33, 42–44, 45 content of 43–44 defined 5 issues of content 6–8 relationship to literacy 1, 2–3 Numeracy Recovery 17 objectives 4, 12, 13, 15, 44, 50, 68, 70, 71, 73 for the early years 18–19, 50 observation as an assessment process 68, 71–72 one-to-one correspondence 19 parents 15, 16, 22, 71 pattern making 14, 19 Piaget, J. 22, 24, 25–29, 32 place value 56, 64, 74 portfolio assessment 71, 73

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practice 36, 40, 46, 48, 49, 50, 59, 70, 72 importance of 46, 47, 53, 63 predictors of mathematics disability 66 pre-kindergarten 14–17 pre-operational stage 27 preschool development 15, 17–19, 28, 34 Primary Framework for Literacy and Mathematics (UK) 4, 44 primary school years 20, 33–38, 40, 44 Principles and standards in school mathematics 3 problem solving 7, 8, 15, 17, 27, 35, 48, 50, 56, 59, 66 assessment of 76–77 strategies for 27, 37, 46, 48, 53–54 teaching of 52–54, 61 problem-based approach 40, 49, 50, 59 procedural knowledge 46, 48, 55 questioning as a teaching tactic 20, 36, 71, 76 readiness 28, 29, 31 reading difficulties 64, 65 reflecting 25, 36, 37, 53, 54, 73 reforms in mathematics education v, 46, 47–48, 59 research into teaching 3, 33, 36–38, 40, 54 research-based practice 16, 36 reversal of numbers 65 review and revision 61–62 rote learning 48, 54, 59, 61 scaffolding 15, 18, 24, 29, 30 Schedule for early number assessment (New South Wales) 69, 75, 78 schema 24, 26, 58 secondary schools 7, 9, 29, 33, 41–42, 58, 59–60, 71 self-correction 53, 72 self-monitoring 53

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semi-abstract stage 31 semi-concrete stage 31, 73 sensorimotor stage of cognitive development 26 sequencing 14 spiral curriculum 24, 32, 61 strategic knowledge 46, 55, 73 strategies for problem solving 37, 48, 53–54, 56, 61 structural apparatus 31, 62 subitising 19, 21 summative assessment 71 symbol sense 21 symbolic stage 27, 30–31, 61 teacher-directed approach 28, 34 teachers attitude toward mathematics 40 competency 33, 36–38, 40, 59, 60–61 role of 9, 13, 17, 18, 20, 29, 30, 34, 35–36 teaching methods achieving a balance v, 42, 44, 48, 49 as a cause of difficulty 58–59

direct 17, 28, 34, 35, 36, 46, 47, 50, 54 discovery 29, 36, 50, 58–59, 60 inquiry 32, 58 testing 3, 21, 68, 69, 73–74 diagnostic 68, 71 published tests 71, 74 teacher-made tests 71, 73 textbooks 30, 38, 41, 57, 59, 63, 64 thinking aloud as a teaching technique 18, 30, 54 transmissionist orientation 35, 40 Trends in International Mathematics and Science Study (TIMSS) 35, 38, 41 visualising 53, 55, 61, 73 Vygotsky, L 18, 24, 25, 29–30, 32 whole-class approach 37, 38, 39 work samples assessment of 68, 72–73, 75, 76 working memory 52, 66 zone of proximal development 18, 24, 29, 76

What Teachers Need to Know About

Teaching Methods Peter Westwood ACER Press, 2008

The What Teachers Need to Know About series aims to refresh and expand basic teaching knowledge and classroom experience. Books in the series provide essential information about a range of subjects necessary for today’s teachers to do their jobs effectively. These books are short, easy-to-use guides to the fundamentals of a subject with clear reference to other, more comprehensive, sources of information.

About the author

Teaching Methods explains the different theories of teaching and learning, together with their underlying principles and methods. It defines the role of a teacher in the learning process and looks at the latest research on what contributes to effective practice. Teaching Methods deals with important key issues and provides a wealth of references for fur ther study and exploration in the subject.

To order What Teachers Need to Know About Teaching Methods

Peter Westwood has been an Associate Professor of Education and has taught all age groups. He holds awards for excellence in teaching from Flinders University in South Australia and from the University of Hong Kong. Peter has published many books and articles on educational subjects and is currently an educational consultant based in Macau, China.

Visit Other titles from Peter Westwood Visit

What Teachers Need to Know About

Spelling Peter Westwood ACER Press, 2008

The What Teachers Need to Know About series aims to refresh and expand basic teaching knowledge and classroom experience. Books in the series provide essential information about a range of subjects necessary for today’s teachers to do their jobs effectively. These books are short, easy-to-use guides to the fundamentals of a subject with clear reference to other, more comprehensive, sources of information. Spelling bridges the gap between knowledge accumulated from research on spelling acquisition and the practicalities of teaching spelling more effectively in schools. Current trends are examined, alongside community views on spelling standards because this is the context in which change is beginning to occur. Spelling contains practical suggestions on methods and activities applicable to all students, supplemented by specific advice on assessment, and links to additional resources.

About the author Peter Westwood has been an Associate Professor of Education and has taught all age groups. He holds awards for excellence in teaching from Flinders University in South Australia and from the University of Hong Kong. Peter has published many books and articles on educational subjects and is currently an educational consultant based in Macau, China. To order What Teachers Need to Know About Spelling Visit Other titles from Peter Westwood Visit

A Parent’s Guide to Learning Difficulties How to help your child Peter Westwood ACER Press, 2008

A Parent’s Guide to Learning Difficulties will help you help your child to learn. It provides parents with a clear explanation of the many causes of children’s problems in learning, and contains jargon-free and practical advice for helping children with reading, writing and mathematics. It also explains how previously proven and effective methods can be implemented in home-tutoring situations, as well as in school. While the focus is on ordinary children with general learning difficulties, the book also provides important information about teaching and managing children with intellectual, physical and sensory disabilities, as well as autism. A Parent’s Guide to Learning Difficulties is full of links to some great online information resources and references to books that you can use to help your child learn.

About the author Peter Westwood has been an Associate Professor of Education and has taught all age groups. He holds awards for excellence in teaching from Flinders University in South Australia and from the University of Hong Kong. Peter has published many books and articles on educational subjects and is currently an educational consultant based in Macau, China. To order A Parent’s Guide to Learning Difficulties Visit Other titles from Peter Westwood Visit

P e t e r We s t w o o d

The What Teachers Need to Know About series aims to refresh and expand basic teaching knowledge and classroom experience. Books in the series provide essential information about a range of subjects necessary for today’s teachers to do their jobs effectively. These books are short, easy-to-use guides to the fundamentals of a subject with clear reference to other, more comprehensive, sources of information. Other titles in the series include Teaching Methods,

Spelling, Learning Difficulties, Reading and Writing Difficulties, Personal Wellbeing, Marketing, and Music in Schools.

explores the issues that are emerging regarding the teaching of these skills, beginning with preschool and the early years of primary school through to adults with poor numeracy skills. It draws on research and relevant literature from several different countries to provide a comprehensive overview of the subject and contains many links to other sources of information and additional resources. Peter Westwood has been an Associate Professor of Education and has taught all

NUMERACY

There is an increasing need for numeracy skills in all aspects of life and Numeracy

What teachers need to know about

age groups. He holds awards for excellence in teaching from Flinders University in South Australia and from the University of Hong Kong. Peter has published many books and articles on educational subjects and is currently an educational consultant based in Macau, China.

ISBN 978-0-86431-904-3

9 780864 319043

Westwood

Cover images: © Erengoksel | Dreamstime.com © Feng Yu | Dreamstime.com

Numeracy

What teachers need to know about

Numeracy

What teachers need to know about

Numeracy

What teachers need to know about

Numeracy

Peter Westwood

ACER Press

First published 2008 by ACER Press, an imprint of Australian Council for Educational Research Ltd 19 Prospect Hill Road, Camberwell Victoria, 3124, Australia www.acerpress.com.au [email protected] Text © Peter Westwood 2008 Design and typography © ACER Press 2008 This book is copyright. All rights reserved. Except under the conditions described in the Copyright Act 1968 of Australia and subsequent amendments, and any exceptions permitted under the current statutory licence scheme administered by Copyright Agency Limited (www.copyright.com.au), no part of this publication may be reproduced, stored in a retrieval system, transmitted, broadcast or communicated in any form or by any means, optical, digital, electronic, mechanical, photocopying, recording or otherwise, without the written permission of the publisher. Edited by Carolyn Glascodine Cover and text design by Mary Mason Typeset by Mary Mason Printed in Australia by Ligare National Library of Australia Cataloguing-in-Publication data: Author: Westwood, Peter S. (Peter Stuart), 1936– Title: What teachers need to know about numeracy / Peter Westwood. Publisher: Camberwell, Vic. : ACER Press, 2008. ISBN: 9780864319043 (pbk.) Notes: Includes index. Bibliography. Subjects: Numeracy—Study and teaching. Mathematics—Study and teaching. Dewey Number: 513.071

Contents

Preface

vii

1 Conceptualising numeracy Numeracy: important in its own right The evolving definition of numeracy The anatomy of numeracy Is numeracy the same as mathematics? Numeracy across the curriculum Affective aspects of numeracy 2 Numeracy in early childhood Pre-kindergarten Working with children in the preschool years Early childhood mathematics objectives Number sense 3 The development of number concepts Piaget’s theory Lev Vygotsky Jerome Bruner 4 The primary school years and beyond Transition from preschool to school Teaching in the primary years Positive intervention: the daily numeracy hour The key issue of teacher competence The secondary school years Adult numeracy 5 Calculating and problem solving The place of computational skills Number facts: the importance of automaticity v

1 2 4 6 8 9 10 13 14 17 18 19 24 25 29 30 33 34 35 38 39 41 42 46 47 49

vi c o n t e n t s

Use of a calculator Mental calculation Teaching problem solving Knowledge that learners need to acquire 6 Barriers to numeracy Teaching method as a cause of difficulty Other contributory causes The nature of students’ difficulties Dyscalculia

51 51 52 55 57 58 62 63 64

7 Assessment Purposes of assessment Approaches to assessment Assessing problem-solving skills Conclusion

68

References Index

81

69 71 76 78

97

Preface

There can be no doubt that since the 1990s numeracy has been high on the agenda in many countries. There is an increasing need for numeracy skills in all aspects of life – at home, in employment, and in the community. Steen (2007, p. 16) states that, ‘Being numerate is one of the few essential skills that students absolutely must master, both for their own good and for the benefit of the nation’s democracy and economic well-being.’ Developments over the past two decades have seen a move toward less emphasis in schools on routine arithmetic teaching and more on application of number skills to problem solving. This change of emphasis is in response to recommendations for reform in the teaching of mathematics ema nating from influential bodies such as the National Council for Teachers of Mathematics in America and the Australian Association of Teachers of Mathematics. Departments of Education have also backed these reforms. We have reached a stage now when it is important to consider whether the changes in emphasis are proving effective. Have students gained a better understanding of mathematics than before; and do they have more positive feelings about the subject? Or have we moved too far away from teaching and practising computational skills so that now it is more difficult, rather than easier, for students to engage in problem solving and investigation? What degree of balance is needed between formal skills instruction and investigative approaches using those skills? This book explores some of the issues that are emerging in the domain of numeracy teaching. I have drawn on relevant literature from several different countries – notably the United Kingdom, the United States of America, Australia, New Zealand and parts of Asia – to provide a com prehensive overview. The issues range from those concerning children in the preschool and early school years through to those affecting adults with poor numeracy skills. I have provided many links to other sources of information, and I hope readers will find something of interest. vii

viii pre f ace

My sincere thanks go to Carolyn Glascodine for her very efficient edit ing of this manuscript. Many thanks also to the staff at ACER Press for their continuing support. P e t er W e s t w o o d

Resources

www.acer.edu.au/need2know

Readers may access the online resources mentioned throughout this book through direct links at www.acer.edu.au/need2know

o n e

Conceptualising numeracy

Key issues ◗ Defining numeracy: The definition has expanded considerably over time. The process of change mirrors that of the evolution of the definition of literacy. We now acknowledge the existence of ‘multiple numeracies’. ◗ Relationship between literacy and numeracy: Numeracy should not be subsumed under literacy (as has happened in the past). Numeracy merits separate and serious attention. ◗ Relationship between mathematics and numeracy: The terms numeracy and mathematical competence are not synonymous. ◗ Core components of numeracy: These are difficult to specify because the demands on numeracy tend to be context-specific. ◗ Numeracy across the curriculum: Numeracy skills are needed in all subject areas. All school subjects can help to develop and generalise numeracy skills. ◗ Affective components of numeracy: The most recent descriptions of numeracy include reference to learners’ attitudes, confidence and disposition to use numeracy skills independently.

The term numeracy appears to have been coined officially many years ago in a report dealing with the education of students in upper secondary schools in the United Kingdom (the Crowther Report: Central Advisory Council 1

2 n u m erac y

for Education, 1959). In that document numeracy was presented as the companion skill to literacy. Numeracy was seen to be the ability to deal successfully with the quantitative aspects of everyday life, while literacy was the ability to cope with normal demands of reading and writing. Numeracy and literacy were thus seen at that time as separate but complementary domains of competence (O’Donoghue, 2002). In many ways it was unfortunate that this distinction between literacy and numeracy became blurred later, when a much broader interpretation of the concept of literacy emerged. Some educators and writers began to equate the term ‘literate’ with being ‘adequately educated’, not just able to read and write; and they considered that numeracy was therefore simply one aspect of being literate. Such a view was evident in 1990 when official documentation for the International Year of Literacy stated that literacy included numeracy (DEETYA, 1997). Similarly, in 1995 a major project under the Australian Language and Literacy Policy also included numeracy within its definition of literacy (Cumming, 1996). Even today, the two terms literacy and numeracy are often used together in an integrated way as if describing a single information-processing ability. The problem with taking literacy and numeracy closely together, for example in research studies and when funding is allocated for special projects, is that literacy (in the sense of reading) always seems to get the lion’s share of attention (e.g., Department of Education and Training, WA, 2006; Dymock, 2007; van Kraayenoord, Elkins, Palmer & Rickards, 2000). For many years, the amount of research and intervention in the literacy domain far outweighed research into numeracy. This tendency to subsume numeracy under literacy may also be one of the reasons why numeracy remained a relatively neglected area in education policy making until the mid-1990s.

Numeracy: important in its own right Common sense would suggest that literacy and numeracy are separate domains of competence; and by the end of the 1990s, the Australian Association of Mathematics Teachers (AAMT) was strongly expressing such a view. In its policy document on numeracy teaching in schools the AAMT (1998) made it clear that literacy and numeracy are fundamentally different areas of learning and each merits separate consideration.

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The recommendation to address students’ numeracy separately from literacy saw a flurry of interest and action, with many countries publishing guidelines for improving numeracy standards, and with a parallel increase in research interest and writing in this domain. For example, in the United States of America, the National Council of Teachers of Mathematics (NCTM) had first produced a document titled Curriculum and evaluation standards for school mathematics in 1989, but by 1997 a revision and updating of this material commenced. The outcome was the publication in 2000 of the influential document Principles and standards in school mathematics, in which numeracy was given a higher profile. In Australia, interest in numeracy, particularly among adults in the workplace, had started in the 1990s or even earlier ( Johnston, 2002). Activity in the field was given extra impetus in 1996 when the Common wealth, state and territory governments endorsed the National Plan for Literacy and Numeracy, followed soon after by the publication of a report titled Numeracy = Everyone’s business (DEETYA, 1997). National bench mark testing in numeracy was soon introduced for school students in Years 3, 5 and 7. From 2008, students’ numeracy skills in Years 3, 5, 7 and 9 are being tested under the National Assessment Program in Literacy and Numeracy (NAPLAN) (Curriculum Corporation, 2008). The Adult Literacy and Numeracy Australian Research Consortium (ALNARC) (2002) was established in 1999 and initiated several projects and organised forums on themes related to numeracy. The year 2000 saw the publication of a Commonwealth Government document Numeracy, a priority for all: Challenges for Australian schools. Since that time a number of studies have been conducted into various aspects of students’ learning and achievement in mathematics (e.g., Louden et al., 2000; van Kraayenoord et al., 2000). All education systems across the states and territories have drawn up numeracy development plans. A typical example of such a plan is the one operating in NSW (Department of Education and Training, NSW, 2006: see the Links box at the end of the chapter). In New Zealand, 2001 saw the introduction of the Numeracy Development Project, a well-documented and successful teacher and school development initiative that is still continuing (Annan, 2006; Irwin & Niederer, 2002). In the United Kingdom, the National Numeracy Strategy was launched in 1998 and implemented in schools from 1999. In that year, the UK commenced a daily period of intensive teaching of basic mathematical

4 n u m erac y

skills in primary schools (the ‘numeracy hour’) to complement the already existing ‘literacy hour’. The format of such sessions will be discussed in a later chapter. The year 2006 saw the Department for Children, Schools and Families in the UK issuing a revised Primary Framework for Literacy and Mathematics with renewed emphasis on numeracy and with some modifications to the original 1998 objectives (DfCSF, 2007a). The adult education sector has focused heavily on numeracy in recent years, presumably because schools have been far from successful in dev eloping the numeracy skills of an alarmingly high proportion of the student population (Boaler, 1997; Coben, 2003; Munn, 2005). It is reported that many adults lack the numeracy skills needed to function in a maximally effective manner in their vocational, civic and personal lives (Wiest et al., 2007). In the United Kingdom the National Research and Development Centre for Adult Literacy and Numeracy has been very active since 2001 with publications, resources and a major project called Maths4life, concerned with numeracy and what they termed ‘non-specialist’ mathematics. The main project, established for the Department of Education and Skills, ran from August 2004 to March 2007 and has now been transferred to the National Centre for Excellence in Teaching Mathematics (NCETM). There are positive signs that numeracy courses for adults yield good results, not only in terms of skill acquisition but also social and personal gains (Balatti et al., 2006; Dymock, 2007). More will be said about adult numeracy in Chapter 4.

The evolving definition of numeracy At first glance, the meaning of the term numeracy would appear to be simple and straightforward. Surely, it means the ability to apply number concepts and arithmetic skills for everyday purposes, together with the ability to interpret the quantitative information that bombards us daily from many different sources? Certainly this simple view accords well with that appearing in the Cockcroft Report Mathematics counts (1982) in which the writers proposed: We would wish ‘numerate’ to imply the possession of two attributes. The first of these is ‘at-homeness’ with numbers and an ability to make use of mathematical skills which enable an individual to cope with the practical mathematical demands of his [sic] everyday life. The second is an ability

c o n cep t u a l i s i n g n u m erac y

5

to have some appreciation and understanding of information which is presented in mathematical terms, for instance graphs, charts and tables or by reference to percentage increase or decrease. (p. 11)

In the United Kingdom, Askew et al. (1997) have defined numeracy as the ability to process, communicate and interpret numerical information in a variety of contexts. This definition is echoed in Australia in the AAMT (1998, p. 2) policy statement: ‘To be numerate is to use mathematics effectively to meet the general demands of life at home, in paid work, and for participation in community and civic life’. New Zealand’s definition of numeracy is very similar: ‘To be numerate is to have the ability and inclination to use mathematics effectively at home, at work and in the com munity’ (Ministry of Education, NZ, 2001). All this seems very straightforward, and surely not contentious? But Coben (2003, p. 9) warns us that, ‘Numeracy is a deeply contested and notoriously slippery concept’. Most of the difficulty relates to deciding exactly which specific areas of knowledge and skill together constitute numeracy. The concept of numeracy has undergone many changes in the years since the term was coined in the Crowther Report. Numeracy is now viewed as a ‘… multifaceted and sophisticated construct, incorporating mathematics, communication, cultural, social, emotional and personal aspects of each individual in context’ (Maguire & O’Donaghue 2002, cited in American Institutes for Research, 2006, p. 6). According to Turner (2007, p. 28), ‘Numeracy has become a personal attribute very much depen dent on the context in which the numerate individual is operating ... [and] numeracy will mean different things to different people according to their interests and lifestyles’. So how did such a simple concept become so complex? The changes that have occurred in conceptualising numeracy tend to parallel the changes that occurred in the past 25 years with the concept of literacy (Falk et al., 2002). Mainly as a result of ideas emanating from what has become known as ‘new literacy studies’, literacy is no longer regarded as simply being able to read and write. Literacy now embraces reading, writing, listening, speaking, viewing and critical thinking; and is said to exist in many forms described as ‘multiple literacies’ (Richards & McKenna, 2003). Examples include, ‘literacy of the workplace’, ‘consumer literacy’, ‘critical literacy’, ‘school literacy’, ‘mathematical literacy’, ‘financial literacy’, ‘literacy for the digital

6 n u m erac y

age’, and so on. Literacy acquisition is seen to be socially and culturally determined, and its nature, role and importance interpreted differently in different contexts – thus leading to the notion of ‘situated literacies’ (Barton et al., 2000). In very much the same way, the acquisition and application of numeracy is now regarded as culturally based, socially determined, situated and context-specific (Baker et al., 2001; Kerka, 1995). To illustrate the point, Steen (2000; 2007) provides examples to show how numeracy demands on individuals vary in a range of different real-life contexts. The term ‘multiple numeracies’ is now emerg ing in the professional literature (e.g., Baker et al., 2006; Gough, 2007; Grubb, 1996; Johnston, 1994). Examples include ‘community numeracy’, ‘critical numeracy’, ‘workplace numeracy’, ‘consumer maths’, and ‘street maths’. We can appreciate that numeracy skills are used for quite different purposes in different contexts. For example, Butcher et al. (2002) refer to ‘numeracy for practical purposes’, ‘numeracy for interpreting society’, ‘numeracy for personal organisation’ and ‘numeracy for knowledge’. Similarly, Steen (1997) suggests that numeracy, or ‘mathematical literacy’, needs to be: ◗ practical (for everyday use) ◗ civic (to understand issues in the community) ◗ professional (for employment) ◗ recreational (for example, understanding scoring in sports and games) ◗ cultural (as part of civilised persons’ deep knowledge and culture).

So, the precise meaning of numeracy will vary according to the context and purposes for which numeracy skills are used. This is perhaps why Coben (2003) suggests that numeracy is a slippery concept.

The anatomy of numeracy The fact that numeracy manifests itself in different ways according to the context in which it is applied makes it difficult to determine precisely what constitutes essential ‘core’ content in numeracy teaching. Steen (2000, p. 17) states that, ‘Numeracy has no special content of its own but inherits its content from its context’. But surely there must be some areas of know ledge and skill that are absolutely fundamental and would apply in all situations?

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Different writers have addressed in different ways the issue of which areas of knowledge and skill actually comprise numeracy. Gough (2007) tells us that numeracy embraces much more than ‘numberacy’ – meaning that the domain of numeracy includes much more than basic arithmetic. According to DEETYA (1997, p. 39) the mathematical underpinning of numeracy is ‘not restricted to working with numbers, but also includes work with space, data (statistical and measurement) and formulae’. In the adult numeracy domain, SAALT (2006) confirms that numeracy incorporates basic number skills, spatial and graphical skills, measurement and problem solving. Gough (2007) makes the interesting proposal that the content of numeracy is represented by most of what comprises the typical primary school mathematics course. Gough excludes all aspects of mathematics that the large majority of adults do not use – this would include much of what is contained in the typical secondary school academic maths curriculum. If his proposal is sound, the primary curriculum may serve as a guide to the core content needed to achieve numeracy. The Primary Framework for Mathematics in the United Kingdom (covering problem solving, reasoning and numeracy for Years 1 to 6) identifies seven strands that together make up the learning area. They are fairly similar to the six strands that form the mathematics curriculum in Australian schools (Curriculum Corporation, 1994). The strands in the UK are: ◗ counting and understanding number ◗ knowing and using number facts ◗ calculating ◗ understanding shape ◗ measuring ◗ handling data ◗ using and applying mathematics.

No doubt there could be endless debates concerning the precise level of com petence an individual would need to develop in each of these strands in order to be deemed ‘numerate’. But, the list above does represent a very reasonable starting point for giving some substance to the concept of numeracy. With a focus on teaching numeracy with adults, Ginsburg et al. (2006) suggest that courses should be organised around four key strands: ◗ number and operations sense ◗ patterns, functions and algebra

8 n u m erac y

◗ measurement and shape ◗ data, statistics and probability.

Within each of these strands, a learner needs to become proficient in four areas: ◗ conceptual understanding ◗ adaptive reasoning ◗ strategic competence for problem solving ◗ procedural fluency.

It is also seen as vital that the learner develops a positive attitude toward mathematics and acquires a ‘productive disposition’. More will be said later on the affective component of numeracy. DEETYA (1997) concludes that numeracy involves a combination of: ◗ mathematical concepts and skills from across the discipline (numerical, spatial, graphical, statistical and algebraic) ◗ mathematical thinking ◗ numerical strategies ◗ appreciation of context.

The above statement almost makes it seem that numeracy = mathematical ability. But is this so? Is numeracy simply another name for basic mathe matical competence?

Is numeracy the same as mathematics? At the beginning of this chapter it was argued that numeracy does not equal literacy; the two draw upon different bodies of knowledge and involve different processing skills. But numeracy and mathematics do draw upon the same body of knowledge and skills, so what is the relationship between the two? Perso (2006a) states that to be numerate you must know some mathe matics; but simply knowing some maths does not necessarily make a person functionally numerate. Martin (2007, p. 28) takes up the point and writes, ‘Just as knowing the definition of words does not make a person literate, knowing rules and algorithms to solve mathematics problems does not make a person mathematically literate.’ It is generally agreed that mathematical competence comprises more than numeracy (e.g., Lott, 2007).

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How teachers view numeracy in relation to mathematics is important because it will influence the way in which they structure the components of their classroom curriculum. Perso (2006a; 2007), who clearly accepts that numeracy is different from mathematics, suggests that this relationship is not well understood by some teachers, resulting sometimes in an imbalance in their approach – for example, placing a major emphasis on computation at the expense of task-based or investigative approaches; or vice versa. The official position (DEETYA, 1997, p. 11) states categorically that: Numeracy is not a synonym for school mathematics, but the two are clearly interrelated. All numeracy is underpinned by some mathematics; hence school mathematics has an important role in the development of young people’s numeracy.

Steen (2007), on the other hand, feels that the dichotomy between what is mathematics and what is numeracy should be eliminated, particularly the attitude that ‘abstract mathematics’ represents a more respectable level of academic study while numeracy is simply contextualised arithmetic for commercial and social purposes. Steen (2007, p. 18) says: ‘Unfortunately, numeracy is often characterised as watered-down mathematics – minorleague curriculum that schools offer to those who are unable to compete in the major league of algebra, trigonometry and calculus’. Steen feels that the everyday maths needed for understanding such events as elections, poll results, consumer finance, discounts, home management and clinical trials is just as important as the content in any academic-style secondary maths courses. It is suggested that recent UK reports and White Papers affecting numeracy in the post-school years are tending to reinforce the perception of a vocational/academic divide in their use of the term ‘functional mathematics’ (Hudson, 2006).

Numeracy across the curriculum The fact that students often do not spontaneously use their mathematical knowledge in other areas highlights the important role of all teachers in helping to facilitate this process of transfer and generalisation (Thornton & Hogan, 2005). Numeracy skills remain inert unless they can be readily applied in a variety of situations and for a variety of purposes. The official view is that ‘Numeracy is a proficiency that is developed mainly in mathematics but also in other subjects’ (Df EE, 2002, p. 9).

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Just as all teachers are said to be teachers of literacy, so too should all teachers endeavour to be teachers of numeracy by taking every opportunity to introduce students to the statistical and quantitative aspects of their subjects, and to relate these aspects to the real world (Posamentier & Jaye, 2007). Steen (2000) states that numeracy is not just one among many subjects but an integral part of all subjects. Perso (2006b, p. 27) suggests that: ‘Within each learning area two questions must be asked: ◗ How can numeracy contribute to enhanced learning outcomes in this learning area? ◗ How can this learning area enhance students’ numeracy?’

By integrating mathematical components into all school subjects, students are helped not only to strengthen and generalise their skills and under standings but also to appreciate the utility of numeracy in a wider sense (DETYA, 2000; DfCSF, 2001). Hogan et al. (2004) observe that knowledge of mathematics and its application in a range of contexts seems to provide students with the confidence to have a go, make mistakes and try again. In addition to infusing numeracy into school subjects and into learning projects, there are also very many events occurring regularly in any school day that provide authentic opportunities for students to exercise their numeracy skills for a genuine purpose – budgeting for school camps, concerts, field trips, bring-and-buy sales, fundraising events, sports days, inter-school matches, and many more (e.g., Rennie, 2006; Zawojewski & McCarthy, 2007). Effective teaching of numeracy will make full use of all such naturally occurring events.

Affective aspects of numeracy It is not only the contextual aspects of numeracy that have been stressed in recent definitions; increasing importance has also been placed on affective, as well as cognitive, aspects of ‘being numerate’. Ginsburg et al. (2006, p. 30) explain that, ‘The affective component of numeracy includes the beliefs, attitudes and emotions that contribute to a person’s ability and willingness to engage in, use, and persevere in mathematical thinking and learning, or in activities with numeracy aspects’. The terms ‘positive disposition’ and ‘positive inclination’ (to use mathematical skills) are appearing more

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frequently when the characteristics of a numerate person are described (e.g., Kilpatrick et al., 2001). Disposition and inclination in relation to numeracy would include an individual’s confidence, emotional comfort, interest and willingness to try to interpret and process quantitative data and solve problems. For example, Kemp and Hogan (2000, p. 3) state that, ‘Numeracy is having the disposition and critical ability to choose and use appropriate mathematical knowledge strategically in specific contexts.’ Similarly, Turner (2007) explains the role of inclination as having the desire and motivation to use numeracy skills. Turner further remarks that, ‘Having negative feelings about mathematics and one’s mathematical ability, implies a reluctance to use mathematics and hence a failure to be fully numerate’ (p. 33). Positive attitudes, interest in, and motivation for mathematics begin in the early years of childhood, and are either fostered and encouraged on entry to school or are snuffed out by lack of success. The following chapter looks at development of numeracy in these crucial early years.

L i nk s t o m o r e ab o u t c o nc e p t u al i s i n g n u m e r ac y ◗ Ginsburg, L., Manly, M., & Schmitt, M. J. (2006). The components of numeracy. Cambridge, MA: National Center for the Study of Adult Learning and Literacy. Available online at: http://www.ncsall.net/ fileadmin/resources/research/op_numeracy.pdf ◗ Coben, D. (2003). Adult numeracy: Review of research and related literature. London: National Research and Development Centre for Adult Literacy and Numeracy. Available online at: http://www.nrdc.org.uk/ uploads/documents/doc_2802.pdf ◗ American Institutes for Research. (2006). A review of the literature on adult numeracy: Research and conceptual issues. Washington, DC: US Department of Education. Available online at: http://www.eric.ed.gov/ ERICDocs/data/ericdocs2sql/content_storage_01/0000019b/80/29/ e3/66.pdf ◗ Steen, L. A. (2000). The case for quantitative literacy. Available online at: http://www.maa.org/ql/001-22.pdf >

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◗ The in-service training material Numeracy across the curriculum produced by DfCSF (2001) provides some very useful suggestions and activities to help teachers consider the role of numeracy in different subject areas. It also contains suggestions for specific content and skills that students should acquire by Year 9 – although some might argue that the list goes beyond what is normally considered basic numeracy. Available online at: http://www.standards.dfes.gov.uk/secondary/ keystage3/all/respub/numxc ◗ A good example of a Numeracy Plan is the one devised by Department of Education and Training, New South Wales. Available online at: http:// www.curriculumsupport.education.nsw.gov.au/primary/mathematics/ assets/pdf/numeracy_plan_15mar06.pdf ◗ Gough (2007) suggests that numeracy comprises the concepts, strategies and skills typically taught and developed in the primary years. For reference, a comprehensive list of objectives for mathematics covering Reception to Year 6 can be found online at: http://www.thegrid.org.uk/learning/maths/ks1-2/assessment/documents/ nnskeyobjectives.doc ◗ New Zealand’s curriculum statement on numeracy teaching can be located online at: http://www.tki.org.nz/r/literacy_numeracy/num_ practice_e.php

t wo

Numeracy in early childhood

Key issues ◗ Early childhood as the foundation stage in numeracy development: Even before formal schooling begins, children are acquiring an informal understanding of quantitative relationships. They also develop their own strategies for dealing with number. And the early years shape a child’s feelings about engaging in number work and mathematics. ◗ The teacher’s role in the early years: The teacher is not only a facilitator but must also act as mediator, helping to interpret children’s quantitative experiences. ◗ Developing number sense: Number sense underpins genuine understanding of all numerical relationships and processes. Possessing number sense in mathematics is to some extent similar to possessing phonological sense and understanding the phonic concept in reading. ◗ Core objectives for early numeracy: In early childhood education it is essential to create a firm foundation of number concepts and skills. There is reasonable agreement on what needs to be taught.

The numeracy concepts and skills that most individuals possess in adol escence and adulthood had their beginnings in the very earliest stages of childhood, long before any formal instruction took place. It is even suggested that the human brain is in some way ‘pre-programmed’ before

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birth to process quantitative information – Butterworth (2005) describes it as ‘innate numerosity’. Beaty (1998, p. 227) remarks: Using the physical and mental tools they are born with, children interact with their environment to make sense of it, and in doing so they construct their own mental concepts of the world. The brain seems to be conditioned to take in information about objects and their relationships to one another.

Even before they enter kindergarten, children appear to acquire an intuition about number that enables them to tackle simple quantitative tasks and deal with everyday problems successfully (Koralek, 2007). There is evidence that infants not yet 4 months old can distinguish visually the difference between unequal quantities; and by the time they are 3 years old, most children have developed a pre-symbolic sense of number ( Jung et al., 2007). Griffin (2004) suggests that by the age of 4 years children develop two systems, one for making global comparisons and another for counting. By the age of 5 or 6 years these two systems have combined to provide a more powerful grasp of number relationships upon which subsequent numeracy concepts and skills develop. At around this age children begin to grasp that moving forward or back along a counting sequence is exactly the same as adding or subtracting. When applied to a numberline, this provides them with a mental model of addition and subtraction and marks a major step forward in their understanding of number relationships (Griffin & Case, 1997). The studies of Barth et al. (2006) and Gilmore et al. (2007) reveal that young children can carry out forms of quantitative comparison and simple addi tion and subtraction long before these skills are taught in school. Through exposure to more advanced individuals in their social setting children observe and acquire essential skills such as counting, sequencing and pattern making. Through play with others they meet concepts such as shape, relative size, capacity, sharing, sorting and classifying. They begin to compare and contrast groups of objects, and quantitative elements begin to appear in their early drawings, suggesting that they are able to invent simple ways of representing number relationships (Pound, 1999).

Pre-kindergarten In the United States of America, the document Principles and standards for school mathematics (NCTM, 2000) included coverage of children in the pre-

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kindergarten age range for the first time. The provision of Pre-K standards is a clear indication that numeracy development is now regarded as beginning soon after birth. The Pre-K objectives cover mainly counting, recognising ‘how many’, cardinal and ordinal number, connecting number words to numerals, sorting and classifying, recognising patterns and sequences, and identifying basic shapes. A similar stand is taken in the United Kingdom, where problem solving, reasoning and numeracy are identified as key learning areas for the very early years (birth to five) (Pimentel, 2007). The Department for Children, Schools and Families (DfCSF, 2005) has stressed the importance of stimulating children’s learning in the first three years of life and has prepared a guidance package titled Birth to three matters to help parents, caregivers and early-years educators support children’s learning, including numeracy, in the years before kindergarten. In Australia, where ‘early childhood’ spans the period from birth to age 8, the Australian Association of Mathematics Teachers joined forces with Early Childhood Australia to issue a joint position statement on Early childhood mathematics (2006). In relation to the preschool and beginning school years this statement highlights the importance of: ◗ engaging children’s natural curiosity ◗ using play and child-initiated activities as the focus ◗ dealing with quantitative issues relevant for a child’s age ◗ solving problems (in the sense of tackling real tasks that involve number concepts and skills) ◗ providing opportunities by supplying abundant materials, space and time ◗ using language to develop maths concepts and vocabulary ◗ encouraging mental manipulation of ideas ◗ assessing children’s level of development as a basis for planning activities.

Charlesworth (2005, p. 235) observes that: Pre-kindergarten mathematics focuses mainly on young children’s natur alistic explorations and the ability to provide informal scaffolding through questions and comments. Young children are developing mathematics concepts and skills at the initial level. These concepts include one-toone correspondence, number sense and counting, logic and classifying, comparing, parts and wholes, ordering and patterning, measurement and

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concrete addition and subtraction … [and] young children also begin to recognise number symbols and experiment with technology.

In the pre-kindergarten years, children who are exposed to normal experiences in the home and community develop an understanding of quantitative relationships. They acquire this awareness mainly through play, exploration and everyday experiences (Aubrey et al., 2003; Perry & Dockett, 2007; Tucker, 2005). For example, they are exposed to language that accompanies numerous informal quantitative experiences such as serving food at the table: (‘Give me two potatoes today, please, because they are rather small’; ‘Pour everyone half a glass of water’) or at the supermarket (‘Can you get two tins of peas please; not the big tins, the smaller ones’; ‘The apples are three for $1.00 today’) and so forth. Young children also hear and use number names as they join in with number rhymes and songs. They hear sister counting the steps as she walks upstairs, and they hear brother say, ‘Mum, I have four pages of homework to do tonight, but I have already finished two’. These incidental encounters with the language of everyday mathematics are laying an important foundation. Through these informal quantitative experiences, most children in early childhood begin to develop confidence with numbers – counting, sharing, comparing, and part–whole relationships – without direct teaching (Zaslavsky, 2001). Parents could do much to encourage children’s curiosity about numbers and number relationships simply by drawing attention in an interesting way to relevant quantitative situations, asking questions, making comments, and making more explicit their own daily use of numbers (Doig, McCrae & Rowe, 2003; Griffiths, 2007; Maher, 2007). Of course, not all children encounter a rich language and number environment at home and therefore do not enter school with the same depth of knowledge and experience. It is for children such as these that intervention programs in the kindergarten are needed. For example, Griffin (2004) describes one such intervention called ‘Number Worlds’. This is a research-based kindergarten maths program that provides rich experiences of investigating number in a variety of ways through a games approach. The program is reported to bring significant gains in children’s number knowledge. The activities seem to be particularly beneficial for disadvantaged children whose prior learning has not resulted in optimum development of number awareness before entry to school. Other early

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number programs include Mathematics Recovery (Wright, 2003), Numeracy Recovery (Dowker, 2001; 2005) and Early Numeracy: Assess ment and Teaching for Intervention (Wright, Martland & Stafford, 2006). Doig et al. (2003) provide a review of several programs designed for the early years.

Working with children in the preschool years It is recognised now that most young children enter school with a great deal of informal knowledge about number and mathematics (Thomson et al., 2005). The role of early childhood educators is to build upon this knowledge by creating an environment where children can continue to explore quantitative relationships with the aid of a teacher and with peers. Situations that invite quantitative investigation are most likely to arise in preschool classrooms where there is an abundance of materials and equipment available that will encourage children to play and investigate. In particular, every early childhood setting should have a ready supply of building blocks, boxes, counters, tiles, shapes, pattern boards, measuring tapes, calculators, squared paper, jars of beads, egg cartons and so on (Wallace et al., 2007). Adequate time and opportunity need to be made available for children to play and work with these materials. In the preschool years, direct and formal teaching of number skills is generally not recommended, although kindergarten programs in some countries (e.g. in parts of Asia) do introduce children to the beginning levels of arithmetic. Children of this age really need to discover number relationships for themselves and invent ways of representing or recording these, rather than having abstract processes taught to them in some formal manner. Most attempts at formal instruction end up destroying a child’s natural curiosity and confidence. Clements (2001) writes that preschools should capitalise fully on young children’s high level of natural motivation to learn in a self-directed manner. If early learning situations use children’s interests, they will help to promote a positive view of mathematics as an enjoyable, self-directed, problem-solving activity. None of the above is intended to suggest that a teacher must stand back and play no active part in fostering children’s early number development. It is believed now that early learning is greatly enhanced if adults help

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young children interpret their learning experiences (Baroody, 2000; Fleer & Raban, 2005; Kirova & Bhargava, 2002). Teachers should work with children’s interests and spontaneous activities, and also deliberately introduce many new mathematical situations into the curriculum (Epstein, 2003; Groves, Mousley & Forgasz, 2006). Skilled preschool teachers seize opportune moments to impose some degree of structure on naturally occurring events or situations in the day. This structuring is commonly achieved by supporting (scaffolding) the child’s own discoveries through giving encouragement, thinking aloud, making suggestions and asking questions. Warfield (2001) advises teachers to pose many questions and problems involving number throughout the school day, not just in maths lessons. The terms ‘mediated learning’ and ‘guided participation’ have become popular when describing the teacher–child shared interaction in such situations (Rogoff, 1995). The Russian psychologist Vygotsky (1978) presented the notion of a ‘zone of proximal development’. This relates to each child’s potential to move forward in his or her learning. All learners, when provided with relevant information or guidance at the right moment, can advance their conceptual development by building on what is already known and under stood. The zone of proximal development for a child represents the new learning that can take place if another person supports the child directly or indirectly in some way. The role of the educator is to supply information at the teachable moment and to build bridges between abstract ideas and the real world. Gradually, these supports are withdrawn so that the child is dealing with the task or problem independently. Learning activities that fall within a child’s zone of potential development have a high probability of success, whereas activities beyond the zone are usually too difficult for the child and may result in failure and frustration.

Early childhood mathematics objectives It is impossible (and probably undesirable) to be prescriptive about the precise knowledge, skills and attitudes relating to number work that should be addressed in the preschool years. However, some guidance may be found by combining suggestions from several different authorities. For example, from Sarama and Clements (2006) and Kehl et al. (2007) the following list could be offered as a possible core set of objectives:

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◗ building positive interest and confidence in working with numbers ◗ pattern making and sequences, for example with blocks or tiles ◗ verbal counting in sequence to 10, to 20 (or even to 100) ◗ using one-to-one correspondence in counting objects to 20 ◗ counting a collection of objects and knowing that the last count tells ‘how many’ ◗ recognising instantly how many objects are in a very small set (less than 5) without counting; this ability is known as subitising ◗ recognising numerals to 20 ◗ joining or separating small sets of objects, telling how many in each set and altogether ◗ sharing items equally among friends ◗ comparing small sets of objects and using terms such as same, equal, more, less ◗ knowing the number that is ‘one more than’ ◗ simple adding and subtracting below 10 ◗ recognising and naming shapes (circle, square, triangle, rectangle) ◗ building new shapes using other shapes ◗ recognising symmetry in shapes ◗ simple measuring ◗ grouping objects based on attributes ◗ making simple picture graphs, for example using shoes, pets, toys ◗ communicating ideas and information to others, based on the above contexts.

There will be significant variation among children in the extent to which they master the above skills. Preschool teachers need to assess each child’s achievement in order to plan appropriate learning experiences or to provide additional teaching for some children prior to school entry. Assessment is discussed more fully in Chapter 7.

Number sense Perhaps the most important development in the early years is the acquisi tion of ‘number sense’. Howell and Kemp (2006) observe that since the late 1980s the term number sense has gained much recognition and is now being used frequently within curriculum and policy documents to describe the

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informal – and often intuitive – understanding of number that all children need to develop if they are to succeed in mathematics. Steen (2000, p. 8) describes it as ‘… having accurate intuition about the meaning of numbers, confidence in estimation, and commonsense in employing numbers as a measure of things’. There is no widely accepted definition of the term number sense and Berch (2005) says that the way in which experts in mathematical cognition use the term differs from the way that it is understood and used by mathematics educators. Nor is there complete agreement among experts on the exact knowledge and skills that together make up number sense. Number sense develops from all the informal quantitative concrete experiences that a child encounters in the early years, and it eventually comes to underpin the child’s smooth entry into arithmetic in primary school. Number sense continues to expand through the school years as children engage with new tasks and solve new problems (Gersten & Chard, 1999; Wells, 2000). If children arrive in primary school lacking number sense, they are extremely likely to have difficulties when the more formal aspects of computation are introduced. Jordan et al. (2007) report that children’s number sense in kindergarten is highly correlated with maths achievement at the end of Grade 1. Griffin (2004) believes that acquisition of number sense follows a developmental path and can be enhanced in the early years by providing a learning environment in which quantitative concepts can be freely explored, interpreted and discussed. Similarly, Wells (2000) believes that number sense is developed most easily in situations where children are actively involved in their learning and where teachers encourage reflection and discussion on quantitative aspects of their activities. A teacher’s role is to encourage children’s number sense development by noticing at what level they are currently operating and helping them develop to the next level by questioning and challenging their thinking (Schwerdtfeger & Chan, 2007). Encouraging children to work mentally with numbers, rather than writing them down, is also considered important (Cutler, 2001). Number sense is least likely to develop fully in classrooms where written computation and memorisation of procedures or rules are introduced too early and become the prime focus of attention (Smith & Smith, 2006; Yang, 2005). Yang (2005) believes that too much emphasis on written computation narrows

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children’s thinking and reasoning at an early stage and causes them to begin to rely on rote memory, rather than meaningful learning. Howell and Kemp (2006) suggest that assessing young children’s number sense may be a valid way of detecting children who are likely to have difficulties later in primary mathematics. Early detection could then lead to appropriate forms of intervention to help these children develop the understanding and confidence with numbers they currently lack. A similar notion was put forward by Malofeeva et al. (2004) who devised a suitable test for this purpose for children in the 3- to 5-year age range. The test assessed six components of number awareness, namely counting, number identification, number-to-object correspondence, ordinality, comparison, and simple addition and subtraction. A study by Jordan et al. (2007) confirms that early testing of this type will indeed help to identify children at risk of learning difficulties in primary mathematics. The study by Howell and Kemp (2006) endeavoured to obtain consen sus on the precise components of number sense by soliciting opinions from mathematics experts in different countries. Eventually they gener ated a fairly daunting list of some 35 possible components; but most of these can be subsumed under broader categories such as counting, matching, comparing, ordering, combining groups, simple subtraction, numeral recognition and a sense of magnitude. Subitising (i.e., recognising the number of items in a small group without needing to count them) is also included by some experts (e.g., Clements, 1999). A few writers suggest that number sense also includes automatic recall of basic arithmetic facts (American Mathematical Association of Two-Year Colleges, 1995; cited in Ginsburg et al., 2006). It is worth noting that in addition to number sense, Steen (2000, p. 9) also refers to ‘symbol sense’ – that is, being comfortable using and inter preting signs and symbols. And other writers have introduced the notion of ‘operations sense’, meaning a deep understanding of how algorithms in arithmetic actually do model number operations. Obviously these areas of awareness develop a little later than the basic number sense described above. Cutler (2001) suggests that understanding the acquisition of number sense can be understood best by considering how concepts are developed. We are helped in this process by looking in the next chapter at the work

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of Piaget (1942; 1983), Vygotsky (1962) and Bruner (1960; 1966), who in different ways were all interested in children’s concept development.

L i nk s t o m o r e ab o u t e a r ly c h i l d h o o d n u m e r ac y ◗ Joint position statement by National Association for the Education of Young Children (NAEYC) and the National Council of Teachers of Mathematics (NCTM). (2002). Early childhood mathematics: Promoting good beginnings. Available online at: http://www.naeyc.org/about/ positions/psmath.asp ◗ Joint position statement by Australian Association of Mathematics Teachers and Early Childhood Australia. (2006). Early childhood mathematics. Available online at: http://www.aamt.edu.au/content/ download/722/19512/file/earlymaths.pdf ◗ Exemplars for early number concepts are presented in the New Zealand Curriculum Framework, Ministry of Education. (2006). Available online at: http://www.tki.org.nz/r/assessment/exemplars/maths/strategy/st_ overview_e.php ◗ Other useful information from New Zealand available at: http://www. nzmaths.co.nz/numeracy/Intro.aspx and at http://www.nzmaths.co.nz/ Numeracy/2006numPDFs/NumBk1.pdf ◗ A useful resource for early childhood educators and parents is Early childhood numeracy cards produced by Department of Education, Science and Training (Australia). (2006). These cards present ageappropriate photographs of real-life situations with quantitative elements. Discussion points and questions are presented on the reverse side. See samples online at: http://www.dest.gov.au/sectors/school_ education/programmes_funding/programme_categories/early_childhood/ learning_resources#The_resource_materials ◗ Fleer, M., & Raban, B. (2007). Early childhood literacy and numeracy: Building good practice. Canberra: DEST, Commonwealth of Australia. This is a booklet for early childhood educators, containing advice on creating an environment for mathematics, the role of language, and observation of children’s knowledge and strategies. It also contains guidance on concepts to be introduced in the early years.

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Available online at: http://www.dest.gov.au/sectors/school_education/ programmes_funding/programme_categories/early_childhood/learning_ resources#The_resource_materials ◗ The Commonwealth Department of Education, Science and Training website provides details of (and links to) a number of projects and reviews that have involved numeracy in Australian school settings. Available online at: http://www.dest.gov.au/sectors/school_ education?policy_initiatives_reviews/key_issues/literacy_numeracy/ numeracy_publications.htm

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The development of number concepts

Key issues ◗ How are number concepts developed? We form concepts as a result of engaging in, and interpreting, a variety of learning experiences. New information is assimilated and linked with prior knowledge as we categorise and make sense of our environment. ◗ The theories of Piaget, a developmental perspective: Piaget’s observations on children’s cognitive development at various stages have greatly influenced our views on developmentally appropriate practice. His notion of ‘schema’ helps us understand how new learning is linked with prior knowledge and how concepts are formed. ◗ The contributions of Lev Vygotsky: The zone of proximal development. The importance of scaffolding children’s learning. Learning as a social activity. ◗ Bruner’s views on learning: Learners must be actively involved in the learning process. Most learning progresses from concrete to abstract. The spiral curriculum.

A concept can be defined as a mental representation that embodies all the essential features of an object, a situation, or an idea. Concepts enable us to classify phenomena as belonging, or not belonging, together in certain categories. Concept formation is the means by which we mentally organise our environment into meaningful units of information that we can then use for future reference. 24

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The development of conceptual knowledge is achieved by the recog nition of relationships between different items of information. The process of forming concepts consists of linking items of information together because of common properties they possess. McInerney and McInerney (2005) suggest that by being exposed to a range of objects and experiences we begin to see common properties emerge. For example, our concept of ‘triangle’ embodies our knowledge of the number of sides, the properties of the angles, the different sizes and orientations of a triangle that are possible by varying the dimensions, and the different names that have been given to various triangular forms. We also discover that triangles can be seen occurring quite frequently within manufactured articles and in the built environment; finally, we can also classify triangles as falling within a larger concept group embracing ‘two-dimensional shapes’. Hiebert and Lefevre (1986, p. 3) believed that conceptual knowledge is characterised most clearly by ‘… knowledge that is rich in relationships’. Conceptual knowledge can be thought of as a connected web of information. The manner in which concepts develop from firsthand observation and from relevant information that we discover (or are given) is exactly what occurs when children experience quantitative situations in the early years and develop awareness of properties and relationships among numbers and shapes. Piaget, Vygotsky and Bruner are three pioneers in cognitive development research whose theories help to throw light on the way in which such concept development occurs in children.

Piaget’s theory We owe much of our understanding of how children develop number concepts from the work of the late Jean Piaget (1942; 1983). Piaget’s theory of cognitive development was derived from his close observation of children (mainly his own children) as they engaged in various tasks, including many involving quantitative and spatial relationships. He was interested in investigating how their perceptions, thinking and reasoning developed and changed over time. His ideas have influenced greatly the current ‘constructivist’ view of learning that places the learner rather than the teacher at the heart of the learning process. Piaget argued that children must continually construct and modify their own understanding of phenomena through their own actions and reflection. In Piaget’s theory,

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children’s active exploration of their environment, coupled with their increasing physical and neurological maturation, play the most important role in influencing conceptual development. An essential aspect of Piaget’s theory of cognitive development is his concept of ‘schemata’ (singular: ‘schema’). A schema is an integrated mental representation or ‘assembly of knowledge’ comprising everything one has learned about a particular concept over time. For example, in forming the ‘triangle’ concept above, we did not acquire all that knowledge on one occasion; the different items of information have been added at different times as a result of new experiences. Learners filter, interpret and adjust new information in terms of what they already know. Piaget used the term assimilation to describe the process of taking in new information and linking it with prior knowledge, and the term accommodation for the process involved in adjusting or revising the existing schema to reflect this advance in understanding. Thousands of schemata are developed over an indivi dual’s lifetime, and they are constantly changing, refining and expanding. Concept development is thus a process of creating mental structures and refining them over time. The acquisition of concepts continues throughout life and is the main characteristic of cognitive development. Piaget considered that children pass through four distinct stages on their way to mature cognitive functioning. At each stage, they become better able to process information accurately and less likely to develop misconceptions. In general, the sequence they follow begins at birth and continues into adulthood. An approximate age range for each stage has been suggested below, but children actually differ significantly in the age at which they pass through each Piagetian stage, due to factors such as maturity, mental ability, teaching and experience. The stages may be briefly described thus: ◗ Sensorimotor stage (birth to 18 months). During this stage, the young child develops motor and orienting responses or reactions to sensory input (e.g. focusing visual attention; reaching for and picking up an object; attending to sounds). At this stage, a child is rapidly coming to understand important features of his or her immediate environment, but is not aware, for example, that physical objects continue to exist even when they are out of sight; in other words, the child lacks an understanding of object permanence (Weiten, 2001). But, as stated in the previous chapter, even at this early age an infant can make some basic quantitative comparisons and judgements when real objects are present.

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◗ Pre-operational or intuitive stage (from age 18 months to 7+ years). Piaget used the term operation to mean the mental process of thinking something through. Children at the pre-operational stage tend not to be able to manipulate ideas mentally or deduce cause and effect relationships. As a result, they are often misled by what they see. An important example of this in the numeracy field is conservation of number. Children at the pre-operation stage at first do not understand that the number of items in a group does not change even though the spatial arrangement of the items may be altered. It is sometimes argued that until children can understand conservation of number there is little point in attempting any form of written recording. Studies have suggested that the concept of conservation of number is achieved by most children between the ages of 6 and 7 years (conservation of mass, length and area develop later). Children at the pre-operational stage of development tend to focus too much on one feature of a problem or task, and do not consider other aspects that may be important. They also have difficulty imagining an action reversed, for example, if 3 tokens are placed with 2 tokens to make 5, what would happen if the 3 tokens were then taken away from 5? ◗ Concrete operational stage (7 to 11+ years). During this stage, the child can begin to understand and process increasingly complex information if it can be experienced, acted upon and observed firsthand. During the concrete operational stage, the child becomes better able to handle symbolic repre sentation and carry out mental operations provided the symbols (e.g. numbers) can be easily related to reality. ◗ Formal operational stage (11 to adulthood). Finally, the normally developing individual becomes able to operate with abstract ideas, and to think and reason without the need for real objects or firsthand experience. Adolescents in the formal operational stage become more thoughtful and systematic in their problem solving; they reason things out rather than using a hit-or-miss approach.

Educators think it is important to consider Piaget’s four stages in relation to the types of mathematical experiences normally provided at different age levels, and how children can be supported best in learning number skills and concepts. According to Piaget, abstract reasoning and the use of purely symbolic representation cannot be forced on a child too early. His theory led educators to question, for example, the wisdom of attempting to teach young children formal arithmetic skills at an early age. When children are taught such rules, procedures and facts out of context, and too

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early, they can’t connect them to what they already know, so conceptual learning does not occur. According to Piaget (1942), the direct teaching of number knowledge and skills ahead of a child’s cognitive readiness to learn is largely a waste of time. Such teaching can also have negative impact on a child’s confidence and attitude toward number work. Over the years since Piaget’s works were first translated and published, numerous experimental studies have generally supported his description of the way in which cognitive development occurs. However several criticisms of his theory have emerged. The first is that he underestimated the learning capacity of preschool children (Case, 1991; Lutz & Sternberg, 1999; Mandler, 2004). Piaget placed heavy emphasis on the role of maturation and readiness, but more recent work appears to indicate that experience and instruction are as important as maturation. It is now believed that with appropriate experience and skilled teaching young children can actually learn very much more than Piaget thought possible. Developmental psychologists who have built on his earlier work (the neo-Piagetians) assert that the knowledge and processes needed to learn new skills and concepts and to solve problems are teachable, and we do not need to await biological maturation of the child. This suggests that instead of opting for the currently recommended ‘developmentally appropriate curriculum’ in early childhood settings, we should be seeking effective teaching methods for accelerating young children’s mathematical learning. It is argued that curriculum in the early years often underestimates children’s abilities and is therefore insufficiently challenging (Wright, 1994). In the United States of America, the National Child Care Information Center (2007) states that, in the domain of numeracy education, many researchers (and some early childhood educators) are now recognising the importance of complementing child-initiated learning with high-quality, teacher-directed instruction in the early years. But this notion does not sit comfortably with many contemporary guidelines on preschool teaching since they still advocate play and developmentally appropriate practice. The two differing views – constructivist vs instructivist – represent one of the ongoing debates in early childhood education (Katz, 1999). It is also believed now that Piaget overestimated what the average adolescent could do in terms of abstract reasoning. The age 11 years has often been suggested as the end of the concrete operational stage, but more recent studies have suggested that for the majority of adolescent students

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their thinking in subjects such as mathematics and science may remain at the concrete stage until at least age 15 years or 16 years (Collis & Romberg, 1992; Santrock, 2006). For this reason, much of the contemporary mathematics teaching in secondary schools misses the mark because it is conducted largely through textbook examples, ‘chalk and talk’, without hands-on experience. The relative absence of concrete experience and visual representation may well account for many of the learning difficulties evident in older students and their growing dislike of mathematics. Despite these limitations, Piaget’s major contribution to the field of numeracy teaching has been: ◗ to present the view of children as active and constructive learners ◗ to redefine the role of a teacher as facilitator of children’s self-initiated dis covery of new information ◗ to remind teachers to consider children’s level of cognitive maturity and readi ness for particular types of learning ◗ to highlight the futility of attempting to transmit predigested knowledge to young children by didactic methods.

Lev Vygotsky In the previous chapter, reference was made to Vygotsky (1962; 1978) and his notion of the zone of proximal development. To Vygotsky, optimum learning occurs when tasks or problems are correctly tailored to be just a shade above a child’s current level of ability but which the child can handle successfully with some support or guidance from an adult or a peer. This support has become known in education as ‘scaffolding’ and it takes the form of hints, suggestions, comments, questions, demonstrations and even direct explanations. Vygotsky and Piaget both see the role of the teacher as facilitator, but Vygotsky places much greater emphasis on the teacher actively guiding the child toward new knowledge construction. He was much more interested in helping to advance each child beyond his or her present level of understanding, rather than awaiting natural maturation and so-called readiness. He also believed in the importance of making topics or problems meaningful by situating them in real-life contexts. Vygotsky regarded language and social interaction as playing much greater roles in children’s concept development than Piaget had

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acknowledged. His recognition of the importance of discussion and ‘think ing aloud’ has greatly influenced the teaching of mathematics. Teachers are beginning to acknowledge now that talking is at the very heart of young children’s knowledge construction (e.g., Monaghan, 2006). The major messages for numeracy teaching that stem from Vygotsky’s work are as follows: ◗ The teacher must actively guide children towards better understanding by supporting (scaffolding) and mediating their thinking. ◗ It is essential to identify a child’s present level of understanding in order to provide guidance that will help him or her to progress further. ◗ Encouraging collaborative group work, peer assistance and discussion all foster concept development and learning. ◗ Schools should base much of the curriculum on real-life topics and problems.

Jerome Bruner Bruner (1960; 1966) was instrumental in raising educators’ awareness of the important role that learners themselves must play in constructing know ledge. In the domain of mathematics for example, he stressed the need for students to think mathematically for themselves instead of having a deconstructed and decontextualised version of mathematics presented for mally to them by the teacher and textbook. However, Bruner sees the role of the teacher to be more than simply a facilitator. Children do need opportunities to explore and discover on their own but they also need to interact positively with more knowledgeable adults and peers who can support their efforts, challenge them, and assist them in interpreting and assimilating new discoveries. According to Bruner, concept development progresses from the ‘enactive’ stage (in which learning should involve concrete experiences) through the ‘iconic’ stage (where pictorial and other graphic representations are used to move beyond the purely concrete) to the final ‘symbolic’ stage where abstract symbols and notation alone convey meaning to the learner. Applying Bruner’s three stages to early numeracy development, the first step that most children take in moving from the real world is to use pictorial recording of number relationships (for example, to draw three pet goldfish in a tank). At around the same time, children are also able to

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interpret pictorial representations of groups of objects to establish a number relationship, for example they can count the balloons in a picture. When situations are presented to children in pictorial form, or are recorded by them as pictures, they can easily relate to them even though they are not the real objects. This is the first stage in moving from concrete experience to symbolic representation. It might be called the beginning of the ‘semiconcrete’ stage. At the next (iconic) stage of development the child can use an object to ‘stand for’ some other real object. For example, a wooden block can stand for a car. Three blocks can stand for three cars moving along an imaginary road, and so forth. The blocks don’t look like cars, but the notion that one thing can be represented in a different way is established. At a later stage, tally marks (looking even less like the real object) can be used at the ‘semi-abstract’ stage, with an understanding of their one-toone correspondence with the original objects. It is not until a child has had these intermediate experiences of translating reality into different forms of semi-concrete and semi-abstract representation that they are ready to begin to use symbolic recording with understanding. It is believed that some children begin to experience difficulty in learning mathematics because they have been taken too quickly from the concrete stage to the abstract symbolic level of recording. A gap is created in children’s understanding if they are forced to operate too soon with symbols and mathematical notation. The use of structural apparatus such as Dienes MAB, Cuisenaire Rods, or Unifix can help bridge this gap by providing a visual link between real objects and the symbols that can eventually represent them. Bruner’s views are not incompatible with those of Piaget; but like Vygotsky, he is much less concerned with issues such as readiness and maturation. Instead, Bruner supports the view that young children can be taught many things if the quality of instruction is good and the teaching follows the sequence of concrete, through the semi-concrete, to the abstract levels. His claim is that any subject can be taught effectively in some intellectually honest form to a child of any stage of development – if the method is right. More recently, Watson (2004, p. 372), arguing against denying some students a right to a challenging and interesting maths curriculum because of poor ability, has echoed Bruner’s view, stating, ‘It is possible to engage nearly all students in some form of abstract and conceptual understanding.’

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In the domain of mathematics, Bruner’s suggestion that a curriculum should be spiral rather than linear in its progression is sound advice. A spiral curriculum implies that key ideas or operations that are first introduced at a simple level are revisited later at regular intervals to be expanded and enriched by application to new situations. In summary, Bruner’s influences on numeracy teaching include: ◗ the need for learners to be actively involved in investigating real problems and discovering information for themselves ◗ the need for children to work through concrete experiences before they are ready for abstraction ◗ the need to create learning environments that provide materials and situations necessary to stimulate inquiry ◗ the recognition that children who participate actively in their own learning are more able to use and generalise the knowledge and skills they acquire.

Many of the principles embodied in the theories of Piaget, Vygotsky and Bruner apply to the education of children beyond the early childhood period. The following chapter raises some of the issues involved in developing numeracy in the primary school years and beyond. L i nk s t o m o r e ab o u t c o nc e p t d e v e l o p m e n t ◗ Wikipedia provides a good overview of concept development. Available online at: http://en.wikipedia.org/wiki/Concept ◗ A summary of key mathematical concepts in the preschool years is presented in Early childhood today: Development of mathematical concepts. Available online at: http://teacher.scholastic.com/products/ ect/mathconcepts.htm ◗ For information related to concepts development and number sense, see Project Math Access: Teaching mathematical concepts at: http:// s22318.tsbvi.edu/mathproject/ch1.asp ◗ What is a mathematical concept? Available online at: http://www.emis. de/proceedings/PME28/SO/SO036_Jahr.pdf ◗ More on Vygotsky, Piaget and Bruner can be found at the North Central Regional Education Laboratory website. Available online at: http://www. ncrel.org/sdrs/areas/issues/methods/instrctn/in5lk2-4.htm

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The primary school years and beyond

Key issues ◗ Children commence formal schooling: A change of teaching approach. Higher expectations. Basic numeracy should be firmly established by the end of primary school. ◗ Effective teaching in the primary years: Several research studies have investigated teachers’ instructional skills. The results have provided a useful window on the type of teaching that produces the best results in numeracy and mathematics. ◗ Teacher competency: Some teachers have great difficulty implementing an interactive teaching approach. Many primary teachers lack expertise and confidence in teaching mathematics. Effective teaching principles are too rarely applied. ◗ Secondary school: Mathematics becomes increasingly abstract. The gap widens between high achievers and low achievers. Schools often resort to ability grouping to cope with this problem. ◗ Adult numeracy: An area of significant development. Recent years have seen an increase in new policies and provisions for this population.

Children enter formal schooling with a wide range of differences in their numeracy knowledge and skills. They also vary greatly in their feelings toward number work and their own ability to handle it. Some are eager, interested and confident in their own abilities; others are much less certain. 33

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Doig et al. (2003, p. 21) remarked that, ‘The difference between preschool and school is quite dramatic in terms of the aims, pedagogy, content of the numeracy program and in what is expected of the children’. What happens to them over the next few years will either strengthen their skills and confidence or will cause them to develop a distaste for mathematics, believing they have no aptitude for what appears to them now to be a difficult subject. Whitebread (1995, p. 11) observed that, ‘Far too many of our young children find learning mathematics in school difficult, lose their confidence in mathematics, and go on to join that large swathe of the adult population who panic at the first sight of numbers.’ This chapter looks at some of the issues involved in providing highquality teaching to strengthen children’s numeracy skills and enhance their interest in mathematics in the primary school years and beyond.

Transition from preschool to school Following on from the kindergarten years, a child’s entry into the reception class will not usually see a dramatic change in teaching methods for the first year. Although schools differ, teachers in most junior primary (infant) schools tend to subscribe to a child-centred philosophy with an emphasis on activity approach and avoidance of too much direct instruction. There is continuity therefore between preschool and beginning school experiences, providing an opportunity for some children whose early learning has not resulted in optimum development of number sense and skills to catch up during the first half-year. Teachers will be working to observe each child’s number ability and level of confidence, and will endeavour to plan and provide activities that build upon previous learning and reduce any significant gaps in children’s prior knowledge. In the years ahead, a teacher’s role is to help all children understand mathematics, compute fluently, develop concepts, solve problems, reason logically, and engage willingly with mathematics (Kilpatrick & Swafford, 2002). The primary years are vital for achieving this goal. Methods after the reception year soon become rather more structured and a little more teacher-directed to ensure that mastery of basic skills is achieved alongside learning with understanding. During the primary years children need to acquire an adequate pro ficiency in carrying out calculations and solving problems involving larger numbers. The informal numeracy strategies they developed in

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the earlier years are rarely adequate for this purpose, so new learning is required. Marmasse et al. (2000, n.p.) state that, ‘The strongest influence on arithmetical development is formal education, which can lead to the development of skills that would not have emerged in a more natural environment without formal education.’ The teacher’s role is to create a learning environment where there are opportunities for active investiga tion and problem solving by the children (Fleer & Raban, 2005). Teachers also have a responsibility to impart relevant mathematical information to children and to teach specific skills and strategies.

Teaching in the primary years There have been several important studies of the instructional approaches used by teachers of primary and secondary school mathematics (e.g., Askew et al., 1997; DEST, 2004; Lamb, 2004; Weiss & Pasley, 2004; Wilson et al., 2005). A few studies have looked at mathematics teaching across many different countries and cultures, for example the Third International Mathematics and Science Study (TIMSS) (Institute of Education Sciences, 1999). TIMSS even involved the videotaping of a number of mathematics lessons in action, allowing detailed analysis of the minute-by-minute inter actions between teachers, students and subject matter. TIMSS will be discussed in more detail in a moment. From the work of Askew et al. (1997) in the United Kingdom, a picture emerged suggesting that teachers of numeracy tend to reflect one of three possible orientations toward the teaching process. The orientation most in harmony with contemporary learning theory can be referred to as connectionist. Those with a connectionist orientation make every effort to link new learning to the children’s prior knowledge. Their aim is to encourage conceptual understanding. They make explicit connections within and across different mathematical topics and with real-life situa tions. Connectionists acknowledge and make use of (connect with) children’s own informal numeracy strategies and ideas as new topics and skills are introduced. The second orientation is towards direct teaching – a transmissionist approach. Those subscribing to this teacher-centred orientation believe it is important to teach explicitly the information, rules and procedures that students will need to acquire to become numerate. They also believe that it is important to practise essential skills until they are mastered. The third orientation is recognised in teachers who firmly

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believe that children must discover mathematical rules and concepts for themselves. This minimally guided discovery orientation leads the teacher to establish situations in which learners must investigate problems, find out information, and develop number skills and concepts for themselves. It is believed that children will learn basic computational skills such as adding, subtracting, multiplying and dividing through their regular engagement in exploratory quantitative activities. Of the three orientations, Askew et al. (1997) conclude that the connectionist seems to result in deeper learning than either of the other approaches. A study in the United States of America, involving students in the K to 12 age range, set out to explore in more detail what makes a difference in quality of mathematics instruction (Weiss & Pasley, 2004). Classroom observations were conducted during 364 mathematics lessons. The find ings revealed that the following variables were significant in ensuring that students make good progress: (a) using relevant and interesting subject matter (b) maintaining a high level of student engagement (c) using effec tive questioning to encourage children to reflect, and (d) assisting students to make complete sense of the subject matter (i.e., the teacher’s role as medi ator and guide). This study also reported that effective teaching employing these important principles was far from common. Also in the United States of America, a very much earlier study of mathematics teaching by Good and Grouws (1977) supported what they termed an ‘active teaching’ model. Good and Grouws found that effec tive learning in mathematics could be best achieved with a structured curriculum and a fair degree of direct teaching. Lessons were found to be maximally effective if the teacher introduces each new topic by explicitly linking it with previous work, provides clear process explanations and demonstrations, uses many illustrative examples, engages students in much guided and independent practice, and checks very frequently that students are understanding (Good et al., 1983). This work was one of the earliest examples of establishing ‘research-based practice’ in mathematics teaching, and it yielded powerful findings. However, the findings were never fully implemented across schools because that style of teaching was suddenly at odds with recommendations favouring a student-centred investigative approach with less, not more, direct instruction (NCTM, 1989; 2000). In Australia, a project initiated in the state of Victoria probed more deeply into the actual tactics employed by teachers of students in Prep

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to Year 4, and Years 5 and 6, as they interact with their students to bring about learning (DEST, 2004) (see the Links box at the end of the chapter). The researchers found that 12 tactics used by teachers were effective in promoting students’ learning in the numeracy domain. These tactics were classified as: ◗ Excavating: uncovering what students know ◗ Orienting: setting the scene; contextualising; reminding; linking ◗ Modelling: demonstrating; directing; explaining; instructing; showing; telling ◗ Collaborating: acting as co-learner; working closely with students ◗ Guiding: cueing; prompting; navigating toward understanding ◗ Noticing: being aware of how well students understand new work; identifying any gaps or misconceptions; providing coaching or re-teaching ◗ Probing: clarifying; monitoring; checking ◗ ‘Convince me’: seeking explanations and justification from students for their ideas ◗ Reflecting and reviewing: recounting; considering again; summarising; sharing ◗ Extending: challenging; taking students beyond simple ideas ◗ Apprenticing: encouraging peer assistance; mentoring.

Many of these effective teaching tactics are reminiscent of what was dis covered previously about highly effective teachers of mathematics in Japan. Students in Japan and other Asian countries usually do outstandingly well in international surveys of achievement in mathematics, so the teaching methods used with them are of great interest. In the TIMSS research, it was noted that effective teachers appeared to provide systematic instruction in a way that children not only master arithmetic skills and problem-solving strategies but also develop a genuine understanding of the subject matter. Japanese teachers teach in an interactive way and are seen to encourage their students’ participation, critical thinking and reflection at all points in a lesson in order to encourage a conceptual level of learning. The students spend more time devising and proving their strategies for solving problems and less time practising routine procedures (Stigler & Hiebert, 1997). The typical mathematics lesson in Japan involves four stages. First, the teacher presents a problem. Then, the students are given time to attempt a solution, often working collaboratively with a partner or in a small group. Next, their solutions are presented to the group and there is whole-class discussion to evaluate their ideas and the methods they used. Finally, there is a summing

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up by the teacher and an opportunity to apply the most effective strategies to similar problems for homework (Shimizu, 1995). One of the key features of effective lessons in Japan is the teacher’s final summary, providing an overview and consolidation of what students have discovered and how it can be applied (Benjamin, 1997). Benjamin (1997) reports that cohesion, thoroughness, and the emphasis on understanding as well as skill in calcu lation, are characteristic of Japanese teachers. The brisk pace of a lesson helps to motivate students and keeps them on task and productive. It is clear from the findings of these and other studies of teaching that effective teachers of numeracy and mathematics employ an interactive approach. They work closely with their students, guiding them as necessary and providing relevant input at appropriate moments. They are successful in engaging children fully in the business of learning; and as teachers, they actively teach rather than relying on a sequence of textbook exercises to enact the curriculum. They put students at the centre of each learning task, but they use their expertise to guide and support students’ construction of knowledge and their acquisition of skills and strategies.

Positive intervention: the daily numeracy hour As a key component of its National Numeracy Strategy in 1999, the UK government introduced the model of a daily mathematics lesson of 45 minutes to 1 hour for primary school students (DfCSF, 2007b; DfCSF, 2007c). This became known as the ‘numeracy hour’, to complement the already existing ‘literacy hour’. It has been mirrored in similar developments in some parts of Australia. The guidelines for operating the numeracy hour place emphasis on an interactive teaching approach, used within a whole-class context. Influenced, it seems, by the data from TIMSS and other studies of effective instruction, the session is to be conducted at a brisk pace, and there should be much use made of mental work and discussion. Children are to be encouraged to explain their thinking when they offer solutions and answer questions. The typical format for the numeracy hour comprises three parts: 1 Warm up: oral and mental work for about 5 to 10 minutes 2 Main teaching activity: Investigating a problem or introducing a new concept, for 30 to 40 minutes. This is still mainly conducted with the whole class, but some grouping may occur and level of work may be differentiated by ability.

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3 Plenary: summarising and consolidating the lesson, clearing up any miscon ceptions, setting homework.

Reviews by Brown (2002) and Kyriacou and Goulding (2004) indicated that the daily numeracy lesson has had some modest benefit in enhancing primary children’s confidence and competence in early maths. However, many teachers have not been particularly successful in running the sessions in the ‘interactive’ manner that is recommended; and there is some indi cation that the lesson is sometimes taught in a fairly formal manner using a whole-class teaching approach. Brown (2002) suggests that the lowerachieving students derive little benefit from the whole-class approach because much of the content is above their level of understanding. There is also evidence that teachers are not really effective in the final summarising and consolidating of the session, with many classes simply ending without summary as soon as the main teaching section has finished. The inability of a significant number of teachers to implement all aspects of the numeracy hour effectively undermines its potential value (Macrae, 2003). The National Union of Teachers (2002) expressed concern that teachers in reception classes were feeling under pressure to introduce a numeracy hour involving whole-class teaching with young children. It was felt that this degree of structured teaching, particularly for blocks of 45 minutes or more, was not appropriate at that age and is not in the children’s best interests. Watson (2004) suggests that the teaching methods recommended in the numeracy strategy and the numeracy hour are in danger of causing teachers to move too rapidly through the curriculum, spending too little time on each topic and changing tack too often, thus destroying continuity in students’ learning. It is notable that the numeracy hour has spawned a flood of resource materials (worksheets, books of ideas, charts and planners) from commer cial publishers. An online search under ‘numeracy hour’ will reveal many of these resources.

The key issue of teacher competence The inability of many teachers to implement fully an interactive and flexible numeracy lesson, together with findings from other studies that reveal poor quality teaching in many classrooms (e.g., Weiss & Pasley, 2004), raises a very important issue of the overall competence of primary teachers to teach

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mathematics. Stigler and Hiebert (2004) confirm that the teacher’s skills and the quality and type of interaction they have with students seem to be the most important variables in how well students learn mathematics (and therefore how well they acquire numeracy) and the feelings they develop about the subject. Teachers’ own attitude toward mathematics and their feeling of professional competence, as well as the depth of their subject knowledge, greatly influences the way they go about teaching the subject (Bonner, 2006; Thornton & Hogan, 2004). Those who lack confidence and have fairly limited subject knowledge may teach mathematics very poorly indeed. Unfortunately, it seems likely that this is the situation in many primary schools at this time. Part of the problem stems from the fact that in many countries, teachers in junior primary and primary schools are ‘generalists’ rather than specialist teachers of maths; maths is simply one of many subjects they must teach each day. Teachers who have no great interest in maths and no special exper tise to teach it tend to avoid a problem-based and open-ended approach because it is difficult and unpredictable to manage. They feel insecure with an investigative method so they avoid creating too many open-ended situ ations, fearing that they may not be able to answer students’ questions or deal with issues that may arise. Instead, they teach numeracy as if it only involves learning mechanical arithmetic through memorisation and repetitive practice. To achieve this narrow objective, the teachers are most likely to use a transmission mode of instruction and simply teach the operations without reference to children’s conceptual understanding. In other words, they will tend to teach as they were taught themselves in the primary school. This weakness was highlighted in an Australian study conducted by Lamb (2004), in which teachers’ understanding of the division algorithm was the focus of attention. Lamb concluded: ‘It would appear that the teachers do not have the depth of knowledge necessary to teach for conceptual understanding’ (p. 153). She also remarks that, ‘It is impossible to calculate the degree of student difficulty caused by teachers who remain ill-informed and fail to seek outside assistance yet continue to teach’ (p. 167). If teachers are to adopt the methods that research has shown to be effec tive, this situation will need to be addressed. It will be impossible to develop students’ numeracy skills to the full and to foster a positive attitude toward mathematics, if teachers are not confident enough to operate interactive,

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learner-centred methods rather than a textbook-centred approach. When teachers are not effective in the way in which they present mathematics they are reducing students’ opportunities to learn (Siemon et al., 2001). It is, of course, almost impossible for teachers to adopt a connectionist orientation if they themselves lack a conceptual understanding of the subject matter. It must be admitted that for teachers who lack a sound understanding of mathematics, it is far from easy to teach in the style displayed by expert teachers of the subject – for example, some of those depicted in Japanese classrooms in the TIMSS videos (Institute of Education Sciences, 1999). Chinese and Japanese teachers are reported to have a sound conceptual understanding of mathematics themselves, resulting in less reliance on procedural and algorithmic teaching (Ma, 1999). Fortunately, this problem of limited expertise in non-specialist teachers is being acknowledged now in countries such as Australia and the United Kingdom; and although there will be no quick solution, at least there has been an increase in the number of in-service professional development courses with a focus on teaching mathematics, and an increase in online and other resources available for teachers – for example, TeacherNet, provided by the Department for Children, Schools and Families in the United Kingdom, and SOFWeb in Victoria (see the Links box below). Projects have also been initiated to encourage teachers to support each other in building the pedagogical skills and understandings necessary to teach mathematics in a more realistic, flexible and effective way (e.g., Bonner, 2006; Carpenter et al., 1999; Clarke et al., 2000).

The secondary school years Many students entering secondary schools have already developed func tional numeracy as a result of their mathematical experiences in primary school, in the home and in the community. For these students, secondary school mathematics provides an opportunity to build on this foundation and explore more advanced concepts. Unfortunately, for some students, secondary school mathematics quickly becomes increasingly abstract, resul ting in a high proportion of them losing their confidence, motivation and initiative. They begin to doubt their own abilities and they come to rely much more on their teacher to transmit the knowledge which they then attempt to memorise (Watson, 2004). This in turn tends to make

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their teachers adopt a didactic approach rather than a student-centred investigative approach. Zawojewski and McCarthy (2007) have pointed out a very serious mismatch between the type of maths content taught in most secondary schools and the abilities really needed beyond school. It could be argued that most of the content of secondary school mathematics does little to help develop students’ numeracy. It is still very common to find that secondary schools organise the teaching of mathematics into ability groups (Turner, 2007). The mathe matically competent students find themselves in the ‘top set’, while the students with limited mathematical ability and little interest are placed in the ‘bottom set’. In theory, the focus of attention in the bottom set is on developing students’ numeracy to a level where they can function effec tively at work and in the community. Attempts are made to tailor the curriculum content and teaching method to suit the interests and learning rate of these students. Often these adapted mathematics courses are given catchy titles such as ‘Consumer Maths’, ‘Life Skills Maths’, ‘Workplace Maths’. Unfortunately, the students (who frequently refer to such courses as ‘veggie maths’) perceive them to be little more than watered down versions of the mainstream maths course, with heavy emphasis on practising routine arithmetic skills. In general, it can be stated that many secondary schools need to focus on making these courses much more attractive and much more relevant for lower-ability students by establishing a better balance between strengthening computational skills and applying such skills to authentic and interesting real-life issues and problems.

Adult numeracy In several countries, interest in numeracy within the adult population has been spurred by data from surveys such as those carried out by the Organisation for Economic Co-operation and Development (OECD, 1998), the National Assessment of Educational Progress (NAEP) that has collected data in the United States of America for more than 30 years (Institute of Education Sciences, 2007), the International Numeracy Survey of 1997, the National Child Development Study (Bynner & Parsons, 1997), the Moser Report A fresh start: Improving literacy and numeracy (Moser, 1999), and the 2003 Skills for Life Survey (Grinyer, 2006), all giving cause for

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alarm. For example, in the United States of America it was found that 35 per cent of students were scoring ‘below basic’ in the National Assessment of Educational Progress maths tests (American Institutes for Research, 2006). This led to the launch of the Adult Numeracy Initiative. Similar alarm was triggered in the United Kingdom by evidence indicating that some 7 million individuals in the 16 to 65 age-range exhibited very poor maths skills, and that poor numeracy had a major impact on an individual’s employability (Moser, 1999; Parsons & Bynner, 2005). Concern was also growing regarding the decline in the number of students over the age of 16 who opt to study mathematics and science. Studies were commissioned (e.g., Smith, 2004) and new policies made. The Qualifications and Curriculum Authority (QCA, 2004) developed National Standards for Adult Numeracy in 2000, and the Df EE prepared Skills for life: The National Strategy for improving adult literacy and numeracy skills in 2001. Since 2002, adult numeracy specifications have been introduced for the first time in England (DfCSF, 2006; Loo, 2007; QCA, 2004). The focus has been on providing opportunities beyond school for individuals to acquire knowledge and skills that they failed to acquire while at school, or that they need to update since leaving school. In the United Kingdom, the BBC, in collaboration with the Basic Skills Agency, ran a campaign in 1997 called Count Me In. It was designed to help adults with numeracy problems improve their skills and raise community awareness of the importance of mathematics in daily life. The titles Count Me In and Count Me In Too have been used in several other countries (notably Australia and New Zealand) in connection with numeracy intervention programs for children. In Australia, additional teaching for adults has usually been provided under ALBE (Adult Literacy and Basic Education) schemes, via further education centres or similar bodies. Provision for adult numeracy classes began in a low-key way in the 1970s but gained impetus from 1991 when the government endorsed the Australian Language and Literacy Policy (Cumming, 1996). More recently, instructors and tutors of adult numeracy classes are expected to have professional training in this area, and courses now exist for this purpose (e.g., Johnston, 2002; Johnston & Tout, 1995). Originally, adult numeracy classes tended to focus on arithmetic skills and the application of these skills to routine problems rather than attemp ting to link the mathematics taught to real-life contexts. There was a tendency to work methodically through the number and computation

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components of a typical primary or lower-secondary curriculum. More recently, every effort has been made to ensure that the content of adult literacy and numeracy courses has a more ‘social practices’ and ‘real life’ focus ( Johnston, 2002). To facilitate the linking of teaching to real life, efforts have been made to study the numeracy needs associated with particular areas of employment (e.g. Hagston & Marr, 2007; Hoyles et al., 2002). Studies in this area are continuing. In Australia, numeracy now features within the curriculum for the Certificate in General Education for Adults (CGEA) where it is organised under the framework of ‘numeracy for practical purposes’, ‘numeracy for interpreting society’, ‘numeracy for personal organisation’ and ‘numeracy for knowledge’. The teaching of numeracy to learners of any age requires a careful balance between ensuring that computational skills are mastered on the one hand, and that individuals become confident and competent in applying such skills to problem solving and to everyday situations on the other. The following chapter addresses this issue.

L i nk s t o m o r e o n n u m e r ac y i n t h e p r i m a r y s c h o o l an d b e y o n d ◗ A good summary of the numeracy hour, together with typical objectives for each year in the primary school, can be found online at: http://www. woodlands-junior.kent.sch.uk/Guide/ygroups/numer3.html ◗ Primary Framework for Literacy & Mathematics (UK). By entering this site and selecting ‘mathematics’ you will find information about numeracy topics for specific year levels: http://www.standards.dfes.gov. uk/primary/frameworks ◗ Researching numeracy teaching approaches in primary schools. DEST 2004. Available online at: http://www.dest.gov.au/sectors/school_ education/publications_resources/literacy_numeracy/researching_ numeracy_teaching.htm ◗ TeacherNet (2007). TeacherNet is sponsored by the Department for Children, Schools and Families in the United Kingdom. For information

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and resources for numeracy teaching, see: http://www.teachernet.gov. uk/teachingandlearning/subjects/maths/teachingnumeracy/ ◗ SOFWeb: Information on numeracy from State of Victoria, Australia. Available online at: www.sofweb.vic.edu.au/eys/num/index.htm ◗ Adult numeracy: Core curriculum. Department for Children, Schools and Families (UK). Available online at: http://www.dfes.gov.uk/curriculum_ numeracy/ ◗ Useful information on adult numeracy can be located at the SAALT website. Provides links also to ANAMOL project (Adult Numeracy and Maths Online). SAALT (Supporting Adults and Applied Learning and Teaching). (2006). Adult numeracy. Available online at: http://www. saalt.com.au/numeracy/background.html ◗ An excellent document surveying the development of adult numeracy research and practice, Johnston, B. (2002). Numeracy in the making: Twenty years of Australian adult numeracy. Sydney: Adult Literacy and Numeracy Australian Research Consortium, can be found online at: http://www.eric.ed.gov/ERICDocs/data/ericdocs2sql/content_storage_ 01/0000019b/80/1a/d0/c6.pdf

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Calculating and problem solving

Key issues ◗ The relative importance of computational skills: The proposed reforms in mathematics education have encouraged downplaying of the direct teaching of computational skills in favour of more time spent on investigation and problem solving. However, computational skills remain extremely important. ◗ Automaticity as an instructional goal: Number facts and operations need to be recalled quickly and easily by learners. Adequate practice is essential to automate such recall. ◗ Strategies for solving problems: Learners become better problem solvers if they are taught appropriate strategies to apply. ◗ Forms of knowledge that comprise numeracy: Declarative knowledge; procedural knowledge; strategic knowledge; conceptual understanding.

The ability to compute is only a part of being numerate; but it is an important part. Almost all numerical situations in everyday life require an individual to be able to add, subtract, multiply or divide, often with numbers that are too large to be manipulated mentally. It is for this reason that to be numerate, children still need to master computational processes as both paper-and-pencil and calculator skills. Equally important, they need to understand the principles and concepts underpinning the various algorithms (Booker, 2004; Thompson, 2007). 46

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Macrae (2003, p. 83) states that effective teaching of numeracy, as defined in the Effective Teachers of Numeracy Project, is teaching that helps children to: • acquire knowledge of and facility with numbers, number relations and number operations based on an integrated network of understanding techniques, strategies and application skills • learn how to apply this knowledge of and facility with numbers, number relations and number operations in a variety of contexts.

This chapter examines some of the issues related to teaching computational skills and problem-solving strategies.

The place of computational skills Interestingly, the place of instruction in computational skills remains a point of major contention in reforms of primary maths teaching. Perso (2006b) compares the debate over the relative importance of computational skills in numeracy to the ongoing debate over phonic skills in the literacy domain. Most reformers in mathematics education urge that much less attention and time be devoted to practising arithmetic operations; but practitioners maintain that explicit teaching and practice of computational skills is essential for more effective problem solving (Calhoon et al., 2007; Farkota, 2005; Westwood, 2003). Some authorities favour delaying the teaching of any arithmetic opera tions until students are ready to learn them with complete understanding. For example, in New Zealand, the Ministry of Education (2006, p. 8) suggests that: Students should not be exposed to standard written algorithms until they use part–whole mental strategies. Premature exposure to working forms restricts students’ ability and desire to use mental strategies. This inhibits their development of number sense.

It is very clear that at this time the United Kingdom places much more emphasis on developing children’s computational skills through direct teaching than does Australia, where the official message is to give much less attention to drill and practice and to keep everything contextualised. In the United States of America, a similar message seems to be creating

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conflict, because on the one hand there is a call for more investigative activities and less formal teaching of arithmetic, but on the other hand there are warning signs that, students’ skills in calculation are deficient (Kilpatrick & Swafford, 2002; Loveless & Coughlan, 2004). These poor computation skills are possibly due to a combination of factors such as inadequate training of primary maths teachers, greater use of calculators in the classroom, and education reforms that emphasise activity and problem solving over practice in basic skills (Loveless & Coughlan, 2004). It must be remarked, however, that a slight improvement in overall achievement in mathematics in United States of America was detected in the 2007 NAEP results (Grades 4 and 8) (Institute for Education Sciences, 2007). This is possibly due to an improvement in teacher preparation and a better balance between skill development and activity methods within the curriculum and teaching approach. In the United Kingdom, the official line is that teachers in primary schools should use approaches that involve both teaching for understanding and an element of memorisation (e.g., mastery of number facts, multi plication tables, signs and symbols) (DfCSF, 2007b). But the word ‘memorisation’ raises alarm bells in the minds of most maths education reformers who equate it with the rote learning typical of the very teaching approaches they are trying to replace. Some argue that the calculator can now perform in an instant every process that a student is likely to need, so why devote hours to paper-and-pencil arithmetic practice? (Watson, 2004). As typical of this viewpoint, Martin (2007) suggests that it is better to have students develop their own problem-solving strategies rather than memorise rules and procedures. And Boaler (1997) warns against leading students to develop inert procedural knowledge that is of limited use to them in anything other than textbook situations. There comes a time, however, when one has to question the wisdom of not teaching children to compute. How can a student really solve problems – other than at a very simple level, or with the help of a calculator – without having the necessary computational skills to use? The argument that children will learn these components of numeracy simply by creating their own mental strategies to handle number situations is attractive, but not entirely convincing. Surely effective teaching of mathematics (and therefore numeracy) must involve both the teaching of sound computational skills, and the opportunity to apply these skills in investigating and recording data

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and in solving authentic problems? Brown (1998) is undoubtedly correct when recommending that, in addition to engaging in investigative maths activities, students must, ‘Practise skills and consolidate their recall of basic facts’ (p. 84). The teaching of numeracy requires a sensible balance between instruc ted skills and discovered concepts, principles and applications. A purely problem-based approach often makes unreasonable assumptions concerning children’s ability to discover and remember mathematical relationships for themselves. In particular, major concern arises with such methods over the reduced attention given to developing children’s automaticity in essential arithmetic skills. The evidence seems to indicate that some children will not make good progress in skill development under such an approach (Ellis 2005; Farkota, 2005). These students make much better progress in mathematics when they are directly taught essential skills and strategies (Carnine et al., 1998; Farkota, 2005; Pearn, 1999; Pincott, 2004).

Number facts: the importance of automaticity ‘Number facts’ is the term applied to all the simple relationships among small numbers. Examples are, 7 + 3 = 10; 10 – 3 = 7; 10 – 7 = 3; or 3 × 7 = 21; 21 ÷ 3 = 7; etc. Knowing number facts is partly a matter of learning them through practice, and partly a matter of grasping a rule (e.g. that zero added to any number doesn’t change it: 3 + 0 = 3, 13 + 0 = 13, etc.; or if 7 + 3 = 10 then, 7 + 4 must be ‘one more than ten’, etc.). Some mathematics experts believe that knowledge of number facts should be regarded as one com ponent of number sense. These number relationships are so fundamental that all children should know them without having to work them out each time they need to apply them. Number facts should be recalled instantly, with a high degree of automaticity. The reason for this is obvious – recalling basic number facts automatically allows children to deal swiftly and effectively with calculations and the solution of problems. They are able to focus full attention and mental effort on the higher-order processes involved in addressing and working through the problem, rather than to the lowestorder steps in completing a calculation. Such automaticity is only acquired through frequent and successful practice. According to Sun and Zhang (2001, p. 28):

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A mastery of lower-order skills instills confidence in students and facilitates higher-order thinking. The ability to automatically recall facts strengthens mathematical ability, mental mathematics, and higher-order mathematical learning. Without this automation students have difficulty performing advanced operations.

In the United Kingdom, school inspectors now consider primary schools to be exemplifying best practice in early numeracy if, among other things, they are placing increased emphasis on ‘recall of number facts and on the ability to calculate quickly and accurately, both mentally and on paper’ (OFSTED, 1997, p. 4). Prior to 1997 – during the era when arithmetic was de-emphasised – teachers would have been criticised for attending too much to these very same aspects within their mathematics programs. But the pendulum has swung again, and in the United Kingdom at least, it is again respectable to teach computational skills and number facts. Of course, from the beginning, some exponents of problem-based maths have recognised the importance of mastering these basic skills. For example, Baker and Baker (1990, p. 103) indicated that all children should be able to give ‘snappy answers to number facts to ten and twenty’. Similarly, Mannigel (1992, p. 116) gave as one of the key objectives in early child hood mathematics that children be able to ‘recall number relationships instantly from memory’. However, these writers firmly believed that essen tial number facts would be discovered and learned incidentally through engaging in purposeful quantitative activities, rather than from drill and practice exercises. There is some evidence to support the notion that a few students do in deed acquire mastery of number facts and computational skills simply through activity, discovery and exploration (Baker & Baker, 1990; Thornton et al., 1997). But it is not at all certain, however, that they acquire necessar ily the same degree of facility in automatic recall as they might under more direct teaching and with more time devoted to intensive practice. Nor can we be sure that all students will acquire adequate mastery of basic number facts through incidental learning. Just as some students seem to need more direct teaching of basic literacy skills it seems equally evident that certain students require more than just casual exposure to number relationships through problem solving, discovery and discussion if they are to reach mastery (Fuchs & Fuchs, 2001). There is evidence that when students who

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are weak at recalling number facts receive additional guided practice through methods such as computer-aided instruction their fluency in recall and application improves significantly (Goldman & Hasselbring, 1997).

Use of a calculator As stated above, it can be argued that in this age of the pocket calculator it is pointless to spend time teaching children to recall number facts and perform paper-and-pencil arithmetic operations – after all, the answers are now at their fingertips. It is certainly true that the calculator has proved to be a boon for many students, allowing them to complete more work and spend more time, rather than less time, on problem solving (Clark, 1999; Dion & Harvey, 2001; Drosdeck, 1995). The calculator has been of particular value for students of high ability, enabling them to tackle complex problems or themes, and also for students of low ability, allowing them to bypass some of their computational weaknesses. There is no doubt at all that developing a student’s confidence and competence in using a calculator must be one of the main goals in numeracy teaching (Df ES, 2004; Huinker, 2002). However, there is a danger that children may use a calculator without necessarily understanding the operation they have performed. For this reason, it is advisable that, at first, calculator use for most children should follow or accompany other more concrete work that will build conceptual understanding of the four arithmetic processes.

Mental calculation In the past, mental calculation was regarded as important, but attention was devoted to it only briefly, often in the form of a test given the first five minutes of every lesson. This writer can recall that when he was teaching in primary schools in England in the 1950s he was provided with a book titled The daily ten, containing all the mental arithmetic items set for each day of the school year. Such a routine activity probably did little to help develop numeracy since it simply tested what children could do but did not teach them ways of improving their performance. Recently, teachers have been encouraged to place more emphasis on chil dren thinking about number relationships and working with them mentally rather than resorting immediately to written algorithms or the calculator

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(Filiz & Farran, 2007; McIntosh, 2005). It is argued that writing numbers down in that way reduces students’ opportunities to deepen their number sense and develop their own insightful strategies for adding, subtracting, multiplying and dividing (Kamii, 1994). Mardjetko and Macpherson (2007) even suggest delaying the formal teaching of paper-and-pencil algorithms until children have flexible mental computational strategies. Some students appear to have particular difficulty keeping numbers in mind (i.e., within their working memory space) long enough to complete a mental calculation. For example, if the teacher says, ‘Red team scored 9 points. Blue team scored 7 points. Green team scored 12 points. How many points scored altogether by the three teams?’ Before these children can add the three numbers together mentally, they have forgotten what the numbers were. One simple teaching tactic in such cases is to jot the numerals down anywhere on the whiteboard in random order (not in horizontal or vertical algorithm format) as the problem is presented. Having this key information available in visual form enables many more children to add the numbers mentally. Their problem was in retaining the auditory information, not in mentally calculating an answer. In the United Kingdom, the increased attention given to mental calculation appears to be having some benefits. Ineson (2007) found an improvement when comparing standards measured in the first year of introduction of the numeracy hour with standards obtained six years later. Ian Thompson’s website (see the Links box at the end of the chapter) provides more information related to helping children improve in mental calculation. His material is strongly recommended.

Teaching problem solving Reys et al. (2006) regard problem solving not so much as a subsection of the mathematics curriculum but rather as a method of teaching. Working with problems provides the most relevant way to help students engage in interesting learning and at the same time develop functional numeracy (OECD, 2002). When discussing computation, number facts and mental arithmetic above the impression may have been given that children need to acquire a foundation of arithmetic skills before they can even begin to engage in any ‘real’ problem solving. This is certainly not the case; the two developments

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must go hand in hand. Instead of being seen as something that you ‘move on to later’ after you have mastered arithmetic, problems should provide an interesting and motivating way for new skills to be acquired. But, in an appropriately balanced numeracy program, time also needs to be devoted to the teaching, practice and consolidation of computational skills. The topic of problem solving and how to teach it is vast; so it is beyond the scope of this book to go into problem-solving strategies in great detail. It can also be argued that higher-order problem solving really involves knowledge, skills and strategies that are beyond the generally accepted boundaries of what constitutes everyday numeracy. Here we will consider some of the more general issues involved in helping children approach maths problems strategically and with confidence. First, we need to recognise that solving an authentic problem is rarely as easy as simply applying a pre-taught algorithm. Real-life problems are often ‘messy’ in the sense that, to begin with, one is not sure which bits of available information are important and which are not. However, there are logical steps we can take in approaching most problems. A problem needs to be analysed, explored for possible actions to take, a decision made concerning procedures to use, calculations performed either mentally or by other means, and then the result checked. As well as the cognitive processes involved in this approach, such as identifying what is required and performing the necessary calculations, the individual solving the problem also needs to use metacognitive skills such as reflection, self-monitoring, and self-correction (Booker et al., 2004). For example, the list below identifies some of the self-directing questions that an individual could ask when approaching a problem. ◗ What needs to be worked out? (identification of goal) ◗ Can I picture this problem in my mind? (visualisation strategy) ◗ How will I try to do this? (selection or creation of a strategy; identification of the operations and steps required) ◗ Is this working out OK? (self-monitoring) ◗ How will I check if my solution is correct? (evaluation) ◗ Is my answer reasonable? (reflection and judgement) ◗ I need to correct this error and then try again. (self-correction)

Students will often generate their own strategies for tackling a particular problem, but those who find this process difficult need practice in sifting

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the relevant from the irrelevant information, identifying exactly what the problem requires, and deciding the best way of obtaining and checking the result. In other words, they need to be taught the very things that other students who are efficient and confident problem-solvers already know and do. To achieve this outcome, direct teaching in the early stages is a neces sary step toward later independence (Swanson, 1999). Much of what we already know about effective teaching has an impor tant place in teaching problem-solving strategies. In particular, the teacher needs to provide students with the following forms of guidance: ◗ clear modelling and demonstrating of effective strategies for solving a particular routine or non-routine problem ◗ ‘thinking aloud’ while identifying and analysing various aspects of the problem ◗ ‘thinking aloud’ while selecting and applying appropriate procedures for the solution ◗ reflecting upon the effectiveness of the procedure used and the plausibility of the solution obtained.

Once students have been shown an effective strategy, they need an opportunity to apply it themselves under teacher guidance with feedback. Finally, they are able to use the strategy independently and to generalise its use to other problem contexts. If all students are to develop problem-solving skills, time must be made available for discussing, comparing and reflecting on methods of solution with other individuals (peers, adults). Teaching in Japanese schools, as described earlier, reflects just such an approach. Xin and Jitendra (1999) and Swanson (1999) have reviewed results from a number of intervention studies designed to improve the problem-solving ability of students with learning difficulties. Their conclusion is that it is certainly possible to improve this area of performance using strategy train ing. Their analysis of the research indicates that longer-term interventions are very much more effective than short-term interventions, and students need to gain strong independent control of the strategies themselves if there is to be any likelihood of the learning being generalised. Gaining such control requires students to reason and reflect upon the procedures they use, not merely carry them out by rote. It is said that students who learn to monitor and regulate their own problem-solving behaviour show most improvement in problem solving (van de Walle, 2006).

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Knowledge that learners need to acquire From all that has been said in this and previous chapters it seems that in the domain of numeracy a learner needs to acquire four types of knowledge – declarative, procedural, conceptual and strategic. These different forms of knowledge are mutually supportive and are used together in different combinations when we calculate and solve problems. Declarative knowledge is represented by our fund of factual information; for example, we know that the numeral 14 is read as ‘fourteen’, that $ means ‘dollar’, that 2 x 8 = 16, and that a period of 60 minutes is called ‘one hour’. Declarative knowledge is (or should be) available for instant recall. Procedural knowledge involves knowing the most effective sequence of steps in performing an operation; for example, in the vertical algorithm for addition, we usually start on the right-hand side by adding units together, but in a division algorithm, we work from left to right, dividing thousands or hundreds first; and so forth. Conceptual knowledge represents all the connected knowledge and information we have acquired about different attributes of an object, a process or a situation; for example, being able to understand and visualise the division process as the equal partitioning of a group of items. Strategic knowledge represents our acquired repertoire of effective ways in which a task can be approached or a problem can be solved. To be numerate, an individual needs to possess these four types of knowledge to a level that enables him or her to function effectively at school and in the community. Not all students find it easy to achieve this level of numeracy, as the next chapter explains.

L i nk s t o m o r e o n calc u lat i n g an d p r o bl e m s o lv i n g ◗ Not all experts support a purely constructivist approach to mathematics teaching. Read a perspective from New York (2003) regarding curricula that lack adequate attention to the teaching of arithmetic skills. Available online at: http://www.nychold.com/pr-cf-030104.html ◗ A valuable website is provided by Ian Thompson. You will find many articles that can be downloaded covering topics such as teaching >

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mental calculation, place value, calculators, written calculation methods and much more. Available online at: http://www.ianthompson.pi.dsl. pipex.com ◗ A staff development seminar from New Zealand on teaching and assessing number facts is available online at: http://www.nzmaths. co.nz/Numeracy/Other%20material/Tutorials/BasicFactsNzmaths. ppt#291,1 ◗ DfES. (2004a). Guidance paper: The use of calculators in teaching and learning mathematics. Available online at: http://www.standards.dfes.gov. uk/primaryframeworks/downloads/PDF/calculators_guidance_paper.pdf ◗ DfES. (2004b). Approaches to calculation. Available online at: http:// www.standards.dfes.gov.uk/secondary/keystage3/downloads/ma_ study015604_mod1.pdf ◗ TeacherNet: http://www.teachernet.gov.uk/teachingandlearning/subjects/ maths/Numeracy ◗ Burns, M. (2007). 7 basics for teaching arithmetic today. Scholastic Website. Available online at: http://teacher.scholastic.com/professional/ teachstrat/arith.htm ◗ Resources and ideas for teaching number and place value. Available online at: http://teachingideas.co.uk/maths/contents04number.htm ◗ Resources for teaching the four operations. Available online at: http:// math.about.com/od/fouroperations/Add_Subtract_Multiply_Divide.htm ◗ Sherman, L., & Weisstein, E. W. (2004). Arithmetic. From MathWorld: A Wolfram Web Resource. Available online at: http://mathworld.wolfram. com/Arithmetic.html ◗ The New Zealand Numeracy Development Project (NDP) provides nine guidebooks for teachers covering most aspects of number work and computation. Details are available online at: http://www.nzmaths.co.nz/ Numeracy/2007numPDFs/pdf_updates.aspx ◗ McIntosh, R., & Jarrett, D. (2000). Teaching mathematical problem solving: Implementing the vision. Available online at: http://www.nwrel. org/msec/images/mpm/pdf/monograph.pdf ◗ Problem solving strategies: Math. Available online line at: http://math. about.com/library/weekly/aa041503a.htm

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Barriers to numeracy

Key issues ◗ The method of teaching is often a contributory cause of learning difficulty: For example, too much attention (or too little attention) given

to computational skills; lack of explicit instruction; loss of continuity; pace too rapid; poor communication. ◗ Other causes: Absence from school; changing schools; textbooks. ◗ Specific learning disability: Do some students have a disability that causes them to have major difficulties when processing quantitative data?

Studies already referred to (e.g., American Institutes for Research, 2006; Moser, 1999; OECD, 1998) revealed the unfortunate fact that many stu dents leave school with fairly limited numeracy skills and often a marked dislike for (or even fear of ) mathematics. Great concern has been expressed over this unacceptable situation, leading to some major new initiatives in providing additional opportunities for adults who wish to revisit mathe matics and gain more effective numeracy skills through evening classes and other provisions (Coben, 2003; McGlynn, 1999). What are the factors that have contributed to the failure of a significant number of students? Is the subject matter of mathematics simply too abstract for most students to understand and master? Can the problem be traced back to the method of teaching? Do these individuals who have great difficulty

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with almost all aspects of the subject have some form of learning disability? These issues will be addressed in this chapter.

Teaching method as a cause of difficulty It is not the case that the subject matter of mathematics is intrinsically impenetrable to all but an elite few students. Nor must it ever be assumed that all students who have difficulties with the subject have an innate disability that specifically prevents them from learning mathematical con cepts and operations. While it is true that such a disability does appear to exist, it affects only a very tiny percentage of the population, not the 35–40 per cent said to have significant difficulty with mathematics (American Institutes for Research, 2006). Instead, it can be inferred from points in previous chapters that a major contributory cause of learning difficulty in mathematics is poor quality teaching. Martin (2007) observes that innumeracy, or ‘mathematical illiteracy’, may not be the result of the subject matter taught, but the pedagogy used to teach it. Once children enter formal schooling, the effectiveness of the instruc tion they receive is the major influence on their progress toward numeracy. There are many ways in which the teaching of mathematics in primary and secondary schools falls short of the ideal (Boaler, 1999; Booker, 2004; Lamb, 2004; Weiss & Pasley, 2004). Often the problem relates to an imbalance between the amount of attention given to building fluency in computational skills compared to the time devoted to problem solving and to applying such skills. But other negative factors also contribute to poorquality teaching.

Too much discovery-type activity without adequate guidance and support Some learning problems occur when student-centred inquiry methods are used inefficiently. In attempting to implement an investigative approach, a teacher may fail to provide students with essential information to help them make complete sense of their discoveries and refine their existing strategies. The research evidence is strongly against the effectiveness of entirely openended and minimally guided discovery methods because most students need a great deal of support from their teacher in interpreting their findings and accommodating these within their existing mental schema (Kirschner et al., 2006; Mayer, 2004).

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Too little attention given to mastery of number facts and computational skills Inability to calculate quickly and accurately is one of the main characteristics of students with difficulties. Again, if teachers follow the recommendations from the education reforms, they may get the message that ‘doing arithmetic’ is unnecessary and represents the most boring and meaningless way to develop numeracy. Common sense, however, should tell us that proficiency in calculation is still an essential sub-skill of problem solving, as emphasised in the previous chapter. A problem-based approach is relatively ineffective if students can’t compute quickly and accurately. But computational skills need to be taught in a way that ensures students fully understand the principles on which the various algorithms operate, rather than simply learning them by rote as a set of steps to follow. Booker (2004, p. 139) has remarked: It is a focus on procedures learnt by rote which leads to most of the difficul ties that students experience in mathematics. If materials are not used, students may not be able to visualize the significance of the steps in the processes they are attempting to complete.

The ability to recall number facts and to perform basic addition, subtraction, division and multiplication needs to be firmly established. Automaticity in applying these skills can only be achieved with adequate practice (Westwood, 2003). Investigative approaches, if used alone, are generally deficient in this respect.

Too much attention given to computational skills This is the other side of the coin. Learning difficulties and a poor attitude to mathematics can occur if a teacher errs in the opposite direction by teaching mainly mechanical arithmetic and neglecting more motivating ways of making mathematics meaningful (Boaler, 1999; Martin, 2007). It has been noted already that this tendency to focus too much on arithmetic exists in primary schools when non-specialist teachers lack sufficient understanding of the scope, purposes and nature of mathematical learning. These teachers teach in the way that they were taught, with textbook and worksheets, because it is easy and it is secure. Most teachers of primary classes have not learned mathematics themselves in the way they are now expected to teach it. Of course, this problem is not confined to primary schools; secondary

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schools often resort to teaching mathematical concepts and operations in a highly formal manner, giving little attention to whether or not students understand fully what they are doing.

Curriculum content covered too rapidly It was noted previously in relation to teaching the numeracy hour that some teachers do not allocate sufficient time for studying each topic or concept before moving to something new (Kyriacou & Goulding, 2004; Watson, 2004). Under these fast-paced conditions, students do not really assimilate and master essential concepts and skills so learning difficulties can arise. The same thing can occur, of course, when covering the mathematics curriculum too quickly by any other teaching method.

Lack of continuity Within the domain of mathematics, essential concepts, strategies and skills tend to develop over time in a hierarchical and sequential manner, pro gressing from simple to more complex. If teachers ‘pick and mix’ topics without reference to their cognitive demands, some students will have diffi culties coping with them. Problems of continuity can occur, for example, with discovery-based methods because it is impossible to ensure that each new problem, topic or issue will involve number skills and concepts that are developmentally appropriate for students of a given age. Butterworth (1999, p. 298) has observed: There are many reasons for being bad at any school subject. But school maths is like a house of cards: the cards in the bottom layer must be firmly and accurately constructed if they are to support the next layer up. Each stage depends on the last.

Teachers’ less than perfect communication skills The learning of mathematics in school is greatly enhanced when teachers present information clearly, use language that is easily understood by stu dents, and make the subject more real and visual by using practical examples, diagrams, manipulatives, computer simulations and models. Silver and Hagin (2002) state that without concrete, visual, and experiential backup, verbal problem-solving approaches may leave many students lost. If

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teachers spend too much time ‘lecturing’, asking questions that are vague and poorly focused, and if they fail to define mathematical terms adequately, many students will have difficulties. Teachers need to be effective communicators when offering explan ations to students, clarifying students’ ideas, and answering students’ queries (Westwood, 1998). They also need to develop excellent skills in demon strating mathematical operations and problem-solving strategies on the board and with concrete materials. Much confusion arises in mathematics classes when teachers are poor communicators.

Moving to an abstract and symbolic level too soon In the preschool and early school years children acquire their understanding of quantitative relationships almost entirely from their real-life concrete experiences, and through seeing, handling and visualising objects. Gradually, they are ready to move from the concrete operational stage and begin to use symbols to represent quantities and the operations that can be performed with those quantities. This process of moving from concrete to abstract level of reasoning marks the point at which some children begin to have difficulty with mathematics (Booker, 2004; Heddens, 1986). If teachers abandon the use of materials and visual aids too soon and begin to teach new material using only chalk, talk, numerals and mathematical notation, some children will begin to lose the underlying meaning of number operations and will be forced to resort to rote memorisation of rules and procedures (Booker, 1999).

Inadequate review and revision Students will simply forget the mathematics they have been learning and will therefore fail to become fully numerate, if they are not given an opportunity to revisit concepts and skills frequently. Research outlined in Chapter 4 has shown clearly that one of the features of effective teaching of mathematics is regular review and revision of work that has been completed in previous weeks. Bruner’s (1966) notion of a ‘spiral curriculum’ applies here. Instead of presenting the mathematics curriculum in a linear manner, moving steadily on from topic to topic, teachers should regularly revisit previous concepts and skills. They can do this by creating new and more challenging tasks or problems where students can apply this prior learning

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again. Connectionist teachers make an effort to link new work with what students have experienced before.

Teachers failing to address students’ individual differences While teachers quickly become aware that some students are finding mathematics difficult, it is fairly rare that they make any significant adjust ments to attempt to tailor the teaching to students’ level or rate of learning. Fahsl (2007) suggests that the instructional needs of these students can be met more successfully if teachers make some simple modifications to the general approach used in the classroom. This obviously becomes increas ingly necessary if a class contains some students with special educational needs. She offers practical examples such as: ◗ encouraging the use of multiplication table-charts and calculators to bypass some of their weaknesses in computation ◗ representing concepts and operations in a visual and concrete manner by making greater use of structural apparatus, diagrams or sketches, in both primary and secondary schools ◗ breaking new work down into smaller units rather than presenting an infor mation overload ◗ distributing problems and tasks on printed sheets to avoid students wasting time and making errors when copying from the blackboard/whiteboard ◗ checking more closely for students’ understanding at each step in the lesson ◗ providing students who have untidy bookwork with squared paper instead of lined or blank sheets to help them keep figures and recordings correctly aligned ◗ varying homework tasks according to students’ needs (i.e., for extension, or for further practice).

Other contributory causes Absence from school Due to the hierarchical nature and interconnectedness of mathematical concepts, this subject is affected more than any other if a student is fre quently absent from school. Gaps in learning will occur. It is difficult for the student to understand and catch up with the work when he or she returns to the class.

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Moving to a new school Discontinuity resulting in confusion can also occur when a student moves from one school to another. Differences in curricula can result in learning difficulties.

Inappropriate textbooks Textbooks and other instructional materials can contribute to children’s learning difficulties in mathematics. In an ideal situation, a textbook will contain sufficient worked examples of particular problems and operations to enable a student to learn from these as well as from the teacher and peers. Many textbooks still fail in this respect and are therefore less than helpful to a student who needs to go over work again at home. It is also common to find that textbooks do not provide enough practice items to meet the needs of the students who require more than the usual amount of repetition in order to master new concepts and skills.

The detrimental effect of failure Finally, we must consider the impact of ongoing failure in mathematics on a student’s learning and motivation. Although this adverse impact is the result of learning difficulty rather than a primary cause of it, it must be considered here because it greatly exacerbates the learning problem. The impact of persistent failure produces a very damaging effect on a student’s self-esteem, self-efficacy, confidence, motivation and attitude towards the subject. In particular, ongoing failure impairs a student’s willingness to persevere in the face of difficulties (Chinn & Ashcroft, 1998). Learned helplessness and avoidance tactics are common among students who have difficulty mastering basic mathematics, because they attribute their failure to their own lack of ability (Houssart, 2002).

The nature of students’ difficulties All students who are weak in mathematics have a fairly characteristic pattern of difficulties. Some of these difficulties will be referred to again in Chapter 7. They can be summarised here as: ◗ poorly developed number sense ◗ general slowness and uncertainty in carrying out even routine calculations.

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This slowness reduces the amount of practice the student actually engages in during lessons and prevents the development of automaticity and fluency ◗ uncertainty in translating number words into correct numerals ◗ extremely untidy bookwork leading to errors. This weakness seems to be due either to fine-motor coordination difficulties and poor spatial ability, or to an attitudinal problem ◗ major difficulties in learning and recalling basic number facts, multiplication tables, and computational procedures ◗ difficulty appreciating the relative size of numbers ◗ very poor understanding of place-value (e.g. that in the number 111 the first numeral on the left represents 100 while the final numeral represents 1 unit) ◗ difficulty comprehending the exact meaning of specific mathematical terms ◗ major problems with understanding what is required in word problems and in selecting correct operations ◗ lack of effective strategies for approaching mathematical tasks and problems ◗ inability to recognise when an obtained answer is not reasonable ◗ reading difficulties associated with the textbook or worksheet.

A few students exhibit extreme difficulty in becoming numerate. Psych olgists believe that these students have a specific disability that impairs their capacity to deal with quantitative data and to master the abstract nature of mathematics. These students are said to have dyscalculia (Michaelson, 2007), a term that is now subsumed within the more recent classification ‘mathematics disability’ (Geary, 2005). Michaelson (2007, p. 21) says that dyscalculia ‘… is a debilitating disorder that affects a person’s ability to conceptualise operations and processes of fundamental mathematics’.

Dyscalculia Dyscalculia (or more correctly, developmental dyscalculia) is a form of learn ing difficulty presumed to be of neurological origin, probably genetically determined and perhaps affecting up to 3 per cent of the population (Colwell, 2003; Landerl et al., 2004; Munro, 2003). The ‘ bible’ of psycho logical assessment – the Diagnostic and Statistical Manual of Mental Disorders (APA, 2000) – places the prevalence rate at about 1 per cent. Given that at least 35 per cent of all students do not achieve well in mathematics, dys calculia is obviously not the primary cause of learning difficulty in most of

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these cases. In the majority of cases learning difficulties are caused by the quality of teaching. Instead of using the term dyscalculia, the International Classification of Diseases (ICD-10) (WHO, 2007) prefers the term ‘specific disorder of arithmetic skills’ and applies the description to individuals who are not intellectually impaired and have received normal schooling but display major weaknesses in dealing with numbers and carrying out calculations. These ‘disorders’ are considered to stem from some type of subtle neurological dysfunction. Landerl et al. (2004) believe dyscalculia reflects a brain-based deficit that specifically affects numerical processing and is not due to weaknesses in other cognitive processes such as attention, memory or visual perception. The following characteristics are often reported for these students: ◗ poor mathematical concept development ◗ lack of understanding of mathematical terms ◗ confusion over printed symbols and signs ◗ extremely poor recall of number facts ◗ weak multiplication skills ◗ poor procedural skills ◗ inability to determine which processes to use in solving problems ◗ poor bookwork with misaligned columns of figures ◗ frequent reversal of single figures and reversal of tens and units (e.g. 34 written as 43) ◗ difficulties with reading text compound the student’s problem in maths.

Individuals with dyscalculia differ in the extent to which they exhibit these particular difficulties. Dowker (2005) says that dyscalculia may have a wide range of causes and therefore presents with different patterns of impairment. She suggests that the specific abilities of students with this difficulty are patchy, with some concepts or skills being relatively stronger or weaker than others. Of course, it can be argued that these difficulties are evident in very many students who do poorly in mathematics, and certainly they are not unique to dyscalculic students. However, it is the severity of the problems and their resistance to normal remedial intervention that set dyscalculic students apart from others with learning difficulties.

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It has been suggested that one of the main underlying weaknesses in students with dyscalculia is limited capacity in working memory (McLean & Hitch, 1999). Working memory is very important for efficient processing of information during problem solving and when completing mathematical calculations. Learners have to store verbal and numerical information in working memory while at the same time processing various steps within a procedure. They also need to be able to retrieve relevant information from memory quickly and efficiently. It is interesting that limitation in working memory in the kindergarten years has proved to be one of the predictors of possible future mathematics learning difficulties (Gersten et al., 2005). Lyon et al. (2003) suggest that within the dyscalculic population two sub-groups exist: (a) those individuals with significant difficulties in learning and retrieving number facts, and (b) those who have difficulty learning and applying the procedures involved in calculating. Geary (1993) concurs, and adds a third group (visuo-spatial subtype) with problems in misreading and misrepresenting place values, occasionally reversing and transposing numerals, and great untidiness in setting out bookwork. Dyscalculia is rarely identified early. Often students are not referred for assessment until the fifth year of school or later. For this reason, there has been a recent increase in interest in designing screening procedures to help detect children at risk in the kindergarten years (e.g., Butterworth, 2003; Mazzocco & Thompson, 2005). From these studies, predictors of potential mathematical disability have been identified. The main predictors include not knowing which of two digits is larger, lacking effective counting strategies, poor fluency in identification of numbers, inability to add simple single-digit numbers mentally and limitations in working memory capacity. It is important to state again that a specific disability in mathematics learning is relatively uncommon. Teachers should never assume that every student with poor results in mathematics has dyscalculia. It is far more likely that their learning difficulty is due to insufficient teaching, or reduced opportunity to learn in earlier years. Their problem is now being main tained by secondary emotional and attitudinal reactions such as learned helplessness, loss of confidence and motivation. The action that teachers need to take in relation to all students with learning difficulties, whether due to dyscalculia or other reasons, is to assess their current knowledge and skills as accurately as possible, and provide instruction that will build

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effectively on what they already know. The important issue of assessment is the focus of the following chapter.

L i nk s t o m o r e o n d i ff i c u l t i e s i n l e a r n i n g m at h e m at i c s ◗ For information on adapting mathematics for students with intellectual disability, see Planning, teaching and assessing the curriculum for pupils with learning difficulties: Mathematics (2001). London: Qualifications and Curriculum Authority. Available online at: http://www.nc.uk.net/ld/ dump/Ma_ld.pdf ◗ Yetkin, E. (2003). Student difficulties in learning elementary mathematics. ERIC Digest. ED482727. Available online at: http://www. ericdigests.org/2004-3/learning.html ◗ Wikipedia has a very good entry dealing with dyscalculia, including a comprehensive list of symptoms at: http://en.wikipedia.org/wiki/ ◗ For general information on dyscalculia, together with advice on improving computational skills, number facts and problem solving, see Dyscalculia defined. In NetNews 5, 4, n.p. (2005). Available online at: http://www.eric.ed.gov/ERICDocs/data/ericdocs2sql/content_storage_ 01/0000019b/80/1b/ed/84.pdf ◗ Practical advice on teaching basic number skills is provided in Geller, C. H. (2000). Strategies for teaching arithmetic: What are the facts? Learning Disabilities Journal, 10, 4, 15–19. Available online at: http:// www.ldam.org/pdf/journal/2000/11-00_arithmetic.pdf ◗ Dowker, A. (2004). What works for children with mathematical difficulties? Research Report RR554. London: DfES, at: http://www. dfes.gov.uk/research/data/uploadfiles/RR554.pdf

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Assessment

Key issues ◗ Assessment allows teachers to monitor the effectiveness of their teaching: Assessment should be linked closely with the objectives and

standards specified in the mathematics curriculum. ◗ Assessment enables teachers to modify their numeracy program if necessary: When teachers have precise information about students’

current knowledge and skills they can match their instruction and learning tasks more accurately to students’ ability. ◗ Assessment identifies which students are having difficulties and require additional assistance: Testing of a class will reveal which

students are making very good progress, which students are progressing at a satisfactory level, and which students need support. Additional diagnostic testing of the at-risk group will reveal the type of support needed. ◗ There are many ways in which teachers can assess students’ numeracy: Examples include observation, testing, examining work

samples, analysing students’ errors and conducting diagnostic interviews. ◗ Some forms of assessment are required at education system level: These tests are used to monitor and report standards across schools.

In order to monitor the development of numeracy in an individual student, in a whole class of students, or in all schools across an education system, it is necessary to conduct regular assessments. These assessments take many forms, ranging from the informal minute-by-minute observations 68

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that a teacher makes while conducting a lesson, through to large-scale testing projects such as the National Assessment Program in Literacy and Numeracy (NAPLAN) in Australia, or the National Assessment of Educational Progress (NAEP) in the United States of America. When the era of standards and accountability in education arrived it heralded an increase in the number of assessments of students’ learning that are made and reported each year. Lokan et al. (2000) describe many forms of such assessment in the numeracy domain. Similarly, the Numeracy assessment guide, produced by the Department of Education in Victoria, summarises several assessment procedures and discusses their applicability, strengths and limitations (Department of Education, Victoria, 2007). Different forms of assessment within mathematics serve a number of different purposes. For example, early assessment of young children is usually conducted to obtain a clear picture of each child’s knowledge and skills on entry to school and to identify any children who may require more than the usual amount of support in their learning (Commonwealth Department of Education, Science and Training, 2002; Doig et al., 2003). At the other end of the age range, senior students receive a very different form of assessment to determine their mathematical knowledge and skills in order to report such data accurately to potential employers or for entry into tertiary studies. Ysseldyke and Tardrew (2007) confirm that regular assessment of students’ progress makes it more feasible for a teacher to differentiate instruction and resources more effectively. In Australia, all states and territories have adopted some form of early assessment of children’s number knowledge. For example, New South Wales has the Schedule for early number assessment; Queensland includes numeracy within its Diagnostic Net covering the first three years of school; Tasmania assesses number skills against the Key Intended Numeracy Outcomes (KINOs). Details of these and other numeracy assessment programs can be found in Assessment of literacy and numeracy in the early years of schooling: An overview (Commonwealth Department of Education, Science and Training, 2002).

Purposes of assessment It has become popular to identify three main purposes of assessment in schools and elsewhere as assessment for learning, assessment as learning, and assessment of learning (Manitoba Education, Citizenship and Youth, 2006).

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1 Assessment for learning refers to using assessment data to improve the teaching program by finding out exactly what students know, where they are having difficulty, and how best to support their development. This form of assessment guides a teacher’s decision making and leads to action that will enhance learning.

2 Assessment as learning refers to assessment that causes students to monitor their own performance and think more deeply about their own learning needs. It causes them to examine more closely their thinking processes and strategies as they endeavour to construct knowledge and reflect upon their results. Students become more effective learners if they can self-assess and identify their own learning strengths, weaknesses and needs.

3 Assessment of learning refers mainly to assessment of students’ overall achievement relative to the goals, objectives or standards for the curriculum.

Classroom assessment of development in numeracy serves the same basic functions as assessment in other areas. In respect to assessments made by teachers on a regular basis, these functions include: ◗ checking the overall efficacy of the teaching program ◗ identifying any concepts, strategies or operations which may need to be retaught, reviewed or practised further with the whole class ◗ determining the stage of development any particular student has reached ◗ gaining information on an individual student’s specific weaknesses and special instructional needs.

In order to determine whether a teaching approach is effective it is necessary to assess students’ knowledge, skills and strategies on a regular basis (DfCSF, 2007a). Such assessment in numeracy is recommended to take one of three possible forms – short-term, medium-term and longerterm. Short-term assessment relates to the observations a teacher makes concerning children’s understanding and performance during any lesson. Short-term assessments are often referred to as formative. Such assessment allows for re-teaching of any concept or operation if necessary, or allocation of additional practice time. Short-term assessment can also lead to timely remedying of misconceptions. Medium-term assessment allows a teacher to appraise students’ progress over a longer time (e.g., every six weeks, or at half-term). Longer-term assessment relates to evaluation of learning over the school year and provides information for students’ next teacher, for

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parents, for the authorities and (in the case of secondary school students) sometimes for potential employers. This longer-term assessment is based mainly on the learning objectives or standards stated in the curriculum for students of a given age. Both medium- and longer-term assessments are often referred to as summative. Coben (2003, p. 66) summarises the three main forms of assessment in stating that: Assessment may be regarded as the sharp end of curriculum development; the point at which teachers endeavour to establish what an intending learner already knows (diagnostic assessment), devise or adjust programmes of study according to the progress the learner is making (formative assessment) or find out whether what has been taught has been learned (summative assessment).

In relation to using assessment to evaluate the effectiveness of teaching, McIntosh (2007) recommends that teachers should look for and record evidence of students’ improvements in the following areas: conceptual understanding, knowledge of facts, number sense, competence in applying skills, problem-solving ability, attitude and confidence. Such evidence may be collected by any (or all) of the methods described below.

Approaches to assessment In the numeracy domain, the following procedures are commonly used to obtain relevant information: ◗ observation of the students while engaged in mathematical activities ◗ questioning students individually, or within small-group contexts ◗ analysing samples of the students’ written work, including exercise books and portfolios ◗ applying teacher-made or published tests ◗ using an inventory or checklist of essential knowledge and skills ◗ diagnostic testing and individual interview (which may also include the use of any or all of the above procedures).

Observation A teacher will know in advance what to look for during a mathematics les son in terms of children’s understanding and application; direct observation

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of students at work can provide this evidence. Observing students as they answer and ask questions and as they work on tasks in the classroom pro vides a teacher with valuable insights related not only to their level of understanding of the lesson content but also affective aspects of their engagement in the work. For example, the teacher will notice if the stu dents are intrinsically motivated by the problems and tasks, and if they are confident, hesitant or anxious. It will also be evident whether the students are monitoring their own performance and self-correcting when necessary. Careful observation of students at work will reveal any difficulties students have in keeping on task and completing assigned work. The teacher obtains a fairly clear picture of which students are progressing well and which individuals require additional follow-up. Reys et al. (2006) advocate that teachers keep reasonably detailed records of what they have observed during lessons, particularly where the evidence suggests that something may need to be taught again, or where additional practice or different examples may be required. Booker et al. (2004) provide some very useful examples of concise record keeping at the individual student and whole-class levels.

Checklists One way to structure the process of observation and data collection from time to time is to use some form of checklist containing a concise summary of concepts, knowledge and skills that students are expected to have mastered by a certain age. Using representative curriculum content from the current and earlier years, teachers can construct their own informal numeracy checklist or inventory containing an appropriate bank of items. Such a list can be very useful in surveying quickly and effectively what particular students already know and what they still need to practise. Bahr (2007) recommends using a simple inventory of this type to obtain an overview of the range of ability in a new class.

Work samples Much diagnostic information can be gleaned from looking carefully through students’ exercise books, homework, test papers and mathematics projects. Samples of ‘rough’ workings, for example, may reveal not only faulty computation but may throw some light on the strategies the student has invented and tested while attempting a difficult or unusual problem. Such papers may also reveal clues that the student is still working at a

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semi-concrete level (e.g., the use of tally marks in the margin for adding or dividing; tiny drawings to help visualise the problem). Any areas of difficulty that are detected can then become the focus of a more in-depth diagnostic interview as described below. Work samples can include students’ portfolios. A mathematics portfolio might contain such items as homework samples, drawings, diagrams, test sheets, teacher’s checklists, solutions to problems that the student has attempted, interpretive writings and students’ self-appraisals (Koca & Lee, 1998). Van de Walle (2006) suggests that portfolios should also contain students’ own self-evaluation comments and reflections.

Testing Used alone, a test is not an adequate measure of a student’s ability; it is only one step in the process of collecting data to assist with decision making (McAsey, 1999). All data from testing needs to be supplemented with information obtained by different means. However, effective teachers do make good use of tests for both diagnostic purposes and to measure students’ overall achievement. Teacher-made tests should be directly linked to the objectives set for that particular unit of work and are often referred to as ‘curriculum based’ or ‘outcomes based’. Clear objectives make the design of assessment materials easier because they indicate not only what knowledge or skill the student must demonstrate but also the standard that is required. The ideal teacher-designed test should embody the following features: ◗ The test begins with a few easy items to allow even the least able students to experience some success. ◗ At least two, preferably three, items are provided at the same level of difficulty to enable the teacher later to differentiate random and careless errors from those that are persistent. ◗ A variety of question formats to make the test more interesting (e.g. some multiple-choice, some ‘missing numbers’, some calculations in which the work ings must be shown, some drawing or measuring, some word problems, etc). ◗ The concepts and skills tested should relate precisely to those covered in the teaching program. ◗ Unless the test is designed to assess only computational skills it should con tain problems that will allow the students’ conceptual understanding, strategic knowledge and reasoning to be appraised.

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Many published tests are available covering basic numeracy. Examples include Diagnostic mathematical tasks (DMT) (Schleiger & Gough, 2002), Numeracy progress tests (NPT) (Vincent & Crumpler, 2000), Booker profiles in mathematics: Number and computation (Booker, 1994) and, for younger chil dren, I can do maths (Doig & de Lemos, 2000). For details of these and other test materials, check the website for ACER Press (see the Links box at the end of the chapter).

Error analysis In the case of students with learning difficulties it is often helpful to examine in detail the nature of the errors they make in their written work. Within an adequate sample of a student’s paper-and-pencil calculations, a pattern sometimes emerges indicating a specific point of confusion in relation to a particular algorithm (Ashlock, 2005). Sometimes, however, errors appear to be fairly random and may reflect inaccurate recall of basic number facts or multiplication tables, a tendency to be distracted while working, or poor vertical alignment of figures on the page. Many different ways of categorising computational errors have been devised, some of them much too complicated and time consuming for use by teachers. One of the least complicated systems suggests that errors tend to fall into one or more of the following categories: ◗ wrong operation (e.g. adding instead of multiplying) ◗ defective algorithm (incorrect in one or more steps within the procedure) ◗ number fact error within the calculation (e.g. 3 x 7 recalled as 28) ◗ place value problem (forgetting that magnitude of a figure is indicated by its position) ◗ specific difficulty dealing with zeros during computation (for example, zeros within the top line in subtraction algorithm).

Error analysis is best implemented within the context of an individual diag nostic assessment interview, as described more fully in the next section. Unless error analysis leads to well focused intervention, there is no point in carrying it out. When the procedure does prove to be of value in identifying specific weaknesses that need to be remedied, it must still be remembered that the aim of intervention should be to help the student understand the process more effectively at a conceptual level, not merely replace one rotelearned computational trick with another.

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Closely related to error analysis are the many ‘diagnostic number tests’ that have been developed over the years. Most of these tests are based on an analysis of the specific steps involved in completing computational algorithms at different levels of difficulty. The test items begin at an easy level and progress gradually to more complex examples. The aim of using such tests is to identify precisely the point at which a student begins to make errors so that re-teaching can commence at that point.

Diagnostic interview The value of talking with individual students about their progress and their difficulties in mathematics has been more fully appreciated in recent years (e.g., Ministry of Education, NZ, 2003; Wright, 2003). The numeracy program Count Me In Too in New South Wales uses an interview procedure (Schedule for Early Number Assessment: SENA) as an essential component for data collection that guides program planning. A full description and evaluation of SENA can be found on the ACT Department of Education and Training website under ‘Assessable Moments in Numeracy’ (see the Links box at the end of the chapter). While an individual interview may be designed to monitor a particular student’s overall understanding of a topic, it is more likely to be carried out for diagnostic purposes when a student is observed to be having difficulties. An individual interview with a student is a powerful way to discover not only the knowledge he or she possesses but also the quality of his or her thinking and the manner in which the student copes with challenges (Reys et al., 2006). Assessment should also reveal whether the student approaches problems at a purely procedural level or whether he or she is developing sound conceptual understanding of key concepts and operations. Work samples and test papers can provide the basis for individual interviews with students. For example, using a test paper as the focus of an interview may involve asking the student to explain or to demonstrate how he or she obtained a particular answer or performed a particular operation. The teacher watches and listens as the student reworks a pre viously incorrect test item, and can identify the exact point of conf usion and provide corrective feedback. It is necessary to listen to the student’s own explanation of why he or she took some particular action at a specific point during the solving of a problem or the working of a calculation (Booker, 1999).

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Four key questions can provide teachers with a suitable framework for pro bing the knowledge, skills and strategies of an individual. The questions are: 1 What does the student know already; and what can the student do correctly without assistance? Answers to this question reflect the student’s current repertoire of concepts, skills and strategies.

2 What can the student do if given some degree of support or guidance? Answers to this question will reveal concepts and skills to be taught and scaffolded as priorities within the student’s ‘zone of proximal development’.

3 What gaps exist within the student’s previous learning? Often specific gaps can be detected in a student’s knowledge of certain operations, certain forms of notation, certain number facts or multiplication tables.

4 What does the student need to be taught next as a top priority in his or her program? Answers to this question need to take into account both the learning targets specified within the maths curriculum for the student’s age level, and the answers to the second and third questions above.

While some of the information to answer these diagnostic questions can be obtained from tests and work samples, teachers or tutors will also need to work closely with the student to obtain additional insights. This form of individual interview is sometimes referred to as ‘dynamic assessment’ because the teacher needs to be flexible in order to adapt or modify the tasks and questioning as the interview progresses. For example, during the assessment the teacher may need to move up or down between concrete to abstract levels of reasoning in order to determine the level at which the student is operating. The teacher may also decide to re-teach an operation or skill during the interview process, and then observe the extent to which this corrective feedback has been immediately understood and used cor rectly by the student.

Assessing problem-solving skills Assessment of a student’s numeracy must include his or her ability to inter pret and solve routine and non-routine problems. Routine problems are those where it is fairly obvious which operations to use to obtain a solution. Non-routine problems are more challenging because the procedures required for solution are not obvious.

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In assessment of problem-solving ability, a teacher is not only interested in the answer obtained but also the way in which it was obtained (i.e. the strategies that were applied). Bahr (2007) encourages teachers to use reallife problems that will reveal more about students’ quality of thinking and the flexibility of their approach. Again, the individual interview, rather than paper-and-pencil testing, is the most appropriate method for assessment of problem-solving skills. If a student has difficulty solving routine problems, the teacher needs to check: ◗ Can the student actually read the problem? ◗ Having read the words, does the student understand what is required? ◗ Is the student able to summarise and explain the problem to the tester? ◗ Can the student identify the appropriate operations to use? ◗ Is the student able to encode the correct algorithm? ◗ Can the student complete the algorithm correctly, swiftly and confidently? ◗ Does the student appear to have difficulty recalling basic number facts? ◗ Is the student able to check the reasonableness of the result obtained? ◗ Does the student self-correct when necessary?

For non-routine problems most of the questions above also apply. Extra attention must be given to the following issues: ◗ Does the student have a strategy for beginning the task? ◗ Can the student explain any steps to take that may help determine a solution? ◗ Can the student identify relevant information within the problem? ◗ Does the student benefit from hints the teacher might provide (e.g. ‘Maybe you could draw a sketch’; ‘Why not compare the two amounts’)? ◗ Does the student spontaneously make use of available aids (e.g. number line, table chart, or calculator)? ◗ What does the student do if the first attempt at a solution is unsuccessful? ◗ How long is the student willing to persevere with a challenging problem?

In the case of both routine and non-routine problems it is important to observe whether the students make careless errors through inaccurate encoding or untidy placement of the algorithm on the page.

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Conclusion Assessment is an essential component of effective teaching in all areas of the curriculum. In the case of numeracy development, the various forms of assessment are used to indicate precisely the knowledge, concepts, skills, strategies and beliefs an individual student has acquired. Assessment also reveals any misconceptions and gaps in students’ knowledge that need to be remedied. Assessment also provides teachers with a clear indication of the overall effectiveness of their teaching program, and should therefore lead to modifications where necessary. Assessment is only useful if it leads to appropriate action – at the level of the individual student, the class, the school, or the system.

L i nk s t o m o r e o n a s s e s s m e n t i n n u m e r ac y ◗ Advice on short-term, medium-term and longer-term assessment associated with the National Numeracy Strategy (UK) can be found at the DfCSF Standards website: http://www.standards.dfes.gov.uk/ primary/publications/mathematics/math_framework/assessment/ ◗ A very good summary of the purpose and nature of assessment in primary mathematics can be found online at: http://www.bristol-cyps. org.uk/teaching/primary/maths/pdf/assessment_summary.pdf ◗ The Department of Education (Victoria). (2007). Numeracy assessment guide. This document summarises several procedures for numeracy assessment, discussing their applicability, strengths and limitations. Available online at: http://www.eduweb.vic.gov.au/edulibrary/public/ teachlearn/student/numeracyasstguide.pdf ◗ The ACT Department of Education and Training operates a website (Assessable Moments in Numeracy) containing much useful information for teachers. In particular, it provides a description of the diagnostic interview known as Schedule for Early Number Assessment, associated with the NSW numeracy program Count Me In Too. Critiques are also provided of other forms of assessment such as portfolios, checklists and learning logs. Available online at: http://activated.act.edu.au/ assessablemoments/index.htm

Assessment

◗ Details of the Australian National Assessment Program in Literacy and Numeracy (NAPLAN) (2008) are available at the website for the Ministerial Council on Education, Employment, Training and Youth Affairs. Available online at: http://www.naplan.edu.au/about/about.html ◗ Information on the West Australian Literacy and Numeracy Assessment (WALNA) for students in Years 3, 5, and 7 is available online at: http:// www.det.wa.edu.au/education/walna/index.html ◗ A sample WALNA test paper for Year 7 numeracy is available online at: http://www.det.wa.edu.au/education/walna/pdfs/Yr7SampleNumeracy~ Test.pdf ◗ A general-purpose numeracy assessment instrument is presented by Hart, K., Ampiah, J. G., Nyirenda, D., & Nkhata, B. (2004) in Teacher’s guide to numeracy assessment instruments. Available online at: http:// www.cripsat.org.uk/downloads/numeracy_guide.pdf ◗ National Assessment Program for Literacy and Numeracy. NSW Department of Education and Training. Available online at: http://www. det.nsw.edu.au/media/downloads/dethome/yr2007/nafl_fact.pdf ◗ Secondary Numeracy Assessment Program (SNAP) for students in Year 7. NSW Department of Education and Training. Available online at: http://www.schools.nsw.edu.au/learning/7-12assessments/snaptest.php ◗ Literacy and Numeracy National Assessment (LANNA) for Years 3, 5 and 7. Conducted by Australian Council for Educational Research. See: http://www.acer.edu.au/lanna/ ◗ ACER Press can supply a variety of suitable tests and other assessment materials. Check the online education catalogue at: http://www.acer. edu.au/acerpress/edu-cat.html

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Index

automaticity 46, 49, 59, 64 avoidance 63

Main entries in bold ability grouping in mathematics 33, 38, 42 absence from school 57, 62 active teaching 36, 38 activity approach 34, 48, 50, 58 adaptive teaching 42, 62, 76 adult numeracy 3, 4, 7, 11, 33, 42–44, 45 affective outcomes from failure 1, 10–11, 72 algorithms 8, 21, 46, 47, 51, 52, 55, 59, 74, 75, 77 arithmetic v, 4, 7, 9, 17, 20, 21, 27, 37, 40, 42, 47–49, 50, 51, 55, 59 see also computation assessment 66, 68–78 curriculum-based 3, 73 diagnostic assessment 68, 71, 73, 74, 75–76, 78 dynamic assessment 76 formative 70, 71 observation 68, 71–72 outcomes-based 73 purposes of 69–71 summative 71 testing 3, 21, 68, 71, 73–74, 77 types of 71–76 attitude toward mathematics students’ 8, 28, 29, 33–34, 39, 40, 41, 59, 63, 64, 66, 71 teachers’ 40 Australian Association of Mathematics Teachers (AAMT) 2, 5

balance between skills development and problem solving v, 42, 44, 48, 49 bookwork untidy or careless 62, 64, 65, 66, 71, 72 Bruner, J. 25, 30–32 calculation 38, 48, 49, 52, 56, 59, 74, 75 see also computation calculators 17, 46, 48, 51, 56, 62, 77 changing schools 57, 63 Cockcroft Report: Mathematics counts 4 cognitive development 10, 24, 25–30, 65 cognitive maturity 29 communication teachers’ skills in 57, 60–61 computation 9, 20, 48, 56, 62, 72, 74 computational skills 9, 20, 36, 38, 48, 52, 56, 59, 62, 72, 74 importance of 47–49 teaching of 46–55 computer-aided instruction 51 concept development 22, 24–31, 65 conceptual understanding 8, 31, 35, 37, 40–41, 46, 51, 71, 73, 75 concrete materials 31, 61, 62 concrete operational stage 27, 28, 29, 30, 61 connectionist orientation 35–36, 41, 62 conservation of number 27 constructivist view of learning 25, 28, 55 97

98 i n d e x

continuity in learning 34, 39, 57, 60 corrective feedback 54, 75, 76 counting 7, 14, 15, 16, 19, 21, 66 Crowther Report 1–2, 5 curriculum 1, 7, 9, 12, 18, 24, 28, 32, 36, 70 as a cause of difficulty 60 content 7, 9, 12, 30, 42, 44, 45, 60, 72 continuity 39, 60 declarative knowledge 46, 55 demonstrating as teaching technique 37, 54, 61 developmentally appropriate curriculum 24, 28, 60 diagnostic interview 68, 75–76, 78 Diagnostic Net (Queensland) 69 diagnostic testing 68, 71, 73, 74, 75–76, 78 differentiation 69 direct instruction 28, 34, 35, 36, 46, 47, 50, 54 discovery approach 29, 36, 50, 58, 60 dynamic assessment 76 dyscalculia 64–66 described 64 subtypes 66 early childhood 13–18, 22, 28, 32 enactive stage of learning 30 error analysis 74–75 fluency in number facts and computation 8, 51, 58, 64 formal operational stage 27 formative assessment 70, 71 generalisation in learning 1, 9, 10, 32, 54 guided participation 18 guided practice 36, 51, 54

homework 38, 39, 62, 72, 73 iconic stage in learning 30, 31 individual difference among students 33, 62 informal learning of number skills 13, 16, 17, 20, 34 innumeracy 58 inquiry method 58 interactive teaching 37–38, 39 investigative approach v, 9, 36, 40, 42, 48, 49, 58, 59 Key Intended Numeracy Outcomes (Tasmania) 69 learned helplessness 63, 66 learning difficulties 21, 29, 54, 59, 60, 63, 66, 74 learning disability 57, 58, 64–67 literacy 1–2, 3, 4, 5–6, 38, 43, 47 mathematical illiteracy 58 mathematics disability 64 see also dyscalculia Mathematics Recovery 17 Maths4life 4 maturation 26, 28, 29, 31 mechanical arithmetic 40, 59 mediated learning 18 memorisation 20, 40, 48, 61 mental calculation 27, 37, 50, 51–52, 56 methods see teaching methods minimally guided discovery 36, 58 modelling as teaching tactic 37, 54 multiple numeracies 1, 6 multiplication 48, 59, 62, 64, 65, 74, 76 National Assessment of Educational Progress (NAEP) 42, 43, 48, 69

index

National Assessment Program in Literacy and Numeracy (NAPLAN) 3, 69, 79 National Council of Teachers of Mathematics (NCTM) 3, 14–15, 22, 36 National Numeracy Strategy (UK) 3–4, 38, 78 National Plan for Literacy and Numeracy (Australia) 3 National Standards for Adult Numeracy (UK) 43 neo-Piagetians 28 number facts 7, 48, 50–51, 56, 64, 65, 66, 74, 76, 77 automaticity 46, 49 importance of 49–51, 59 numberline 14 number sense 13, 15, 19–22, 32, 34, 47, 49, 52, 63, 71 numeracy across the curriculum 1, 9–10, 12 adult numeracy 4, 7, 11, 33, 42–44, 45 content of 43–44 defined 5 issues of content 6–8 relationship to literacy 1, 2–3 Numeracy Recovery 17 objectives 4, 12, 13, 15, 44, 50, 68, 70, 71, 73 for the early years 18–19, 50 observation as an assessment process 68, 71–72 one-to-one correspondence 19 parents 15, 16, 22, 71 pattern making 14, 19 Piaget, J. 22, 24, 25–29, 32 place value 56, 64, 74 portfolio assessment 71, 73

99

practice 36, 40, 46, 48, 49, 50, 59, 70, 72 importance of 46, 47, 53, 63 predictors of mathematics disability 66 pre-kindergarten 14–17 pre-operational stage 27 preschool development 15, 17–19, 28, 34 Primary Framework for Literacy and Mathematics (UK) 4, 44 primary school years 20, 33–38, 40, 44 Principles and standards in school mathematics 3 problem solving 7, 8, 15, 17, 27, 35, 48, 50, 56, 59, 66 assessment of 76–77 strategies for 27, 37, 46, 48, 53–54 teaching of 52–54, 61 problem-based approach 40, 49, 50, 59 procedural knowledge 46, 48, 55 questioning as a teaching tactic 20, 36, 71, 76 readiness 28, 29, 31 reading difficulties 64, 65 reflecting 25, 36, 37, 53, 54, 73 reforms in mathematics education v, 46, 47–48, 59 research into teaching 3, 33, 36–38, 40, 54 research-based practice 16, 36 reversal of numbers 65 review and revision 61–62 rote learning 48, 54, 59, 61 scaffolding 15, 18, 24, 29, 30 Schedule for early number assessment (New South Wales) 69, 75, 78 schema 24, 26, 58 secondary schools 7, 9, 29, 33, 41–42, 58, 59–60, 71 self-correction 53, 72 self-monitoring 53

100 i n d e x

semi-abstract stage 31 semi-concrete stage 31, 73 sensorimotor stage of cognitive development 26 sequencing 14 spiral curriculum 24, 32, 61 strategic knowledge 46, 55, 73 strategies for problem solving 37, 48, 53–54, 56, 61 structural apparatus 31, 62 subitising 19, 21 summative assessment 71 symbol sense 21 symbolic stage 27, 30–31, 61 teacher-directed approach 28, 34 teachers attitude toward mathematics 40 competency 33, 36–38, 40, 59, 60–61 role of 9, 13, 17, 18, 20, 29, 30, 34, 35–36 teaching methods achieving a balance v, 42, 44, 48, 49 as a cause of difficulty 58–59

direct 17, 28, 34, 35, 36, 46, 47, 50, 54 discovery 29, 36, 50, 58–59, 60 inquiry 32, 58 testing 3, 21, 68, 69, 73–74 diagnostic 68, 71 published tests 71, 74 teacher-made tests 71, 73 textbooks 30, 38, 41, 57, 59, 63, 64 thinking aloud as a teaching technique 18, 30, 54 transmissionist orientation 35, 40 Trends in International Mathematics and Science Study (TIMSS) 35, 38, 41 visualising 53, 55, 61, 73 Vygotsky, L 18, 24, 25, 29–30, 32 whole-class approach 37, 38, 39 work samples assessment of 68, 72–73, 75, 76 working memory 52, 66 zone of proximal development 18, 24, 29, 76

What Teachers Need to Know About

Teaching Methods Peter Westwood ACER Press, 2008

The What Teachers Need to Know About series aims to refresh and expand basic teaching knowledge and classroom experience. Books in the series provide essential information about a range of subjects necessary for today’s teachers to do their jobs effectively. These books are short, easy-to-use guides to the fundamentals of a subject with clear reference to other, more comprehensive, sources of information.

About the author

Teaching Methods explains the different theories of teaching and learning, together with their underlying principles and methods. It defines the role of a teacher in the learning process and looks at the latest research on what contributes to effective practice. Teaching Methods deals with important key issues and provides a wealth of references for fur ther study and exploration in the subject.

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Peter Westwood has been an Associate Professor of Education and has taught all age groups. He holds awards for excellence in teaching from Flinders University in South Australia and from the University of Hong Kong. Peter has published many books and articles on educational subjects and is currently an educational consultant based in Macau, China.

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What Teachers Need to Know About

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About the author Peter Westwood has been an Associate Professor of Education and has taught all age groups. He holds awards for excellence in teaching from Flinders University in South Australia and from the University of Hong Kong. Peter has published many books and articles on educational subjects and is currently an educational consultant based in Macau, China. To order What Teachers Need to Know About Spelling Visit Other titles from Peter Westwood Visit

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A Parent’s Guide to Learning Difficulties will help you help your child to learn. It provides parents with a clear explanation of the many causes of children’s problems in learning, and contains jargon-free and practical advice for helping children with reading, writing and mathematics. It also explains how previously proven and effective methods can be implemented in home-tutoring situations, as well as in school. While the focus is on ordinary children with general learning difficulties, the book also provides important information about teaching and managing children with intellectual, physical and sensory disabilities, as well as autism. A Parent’s Guide to Learning Difficulties is full of links to some great online information resources and references to books that you can use to help your child learn.

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The What Teachers Need to Know About series aims to refresh and expand basic teaching knowledge and classroom experience. Books in the series provide essential information about a range of subjects necessary for today’s teachers to do their jobs effectively. These books are short, easy-to-use guides to the fundamentals of a subject with clear reference to other, more comprehensive, sources of information. Other titles in the series include Teaching Methods,

Spelling, Learning Difficulties, Reading and Writing Difficulties, Personal Wellbeing, Marketing, and Music in Schools.

explores the issues that are emerging regarding the teaching of these skills, beginning with preschool and the early years of primary school through to adults with poor numeracy skills. It draws on research and relevant literature from several different countries to provide a comprehensive overview of the subject and contains many links to other sources of information and additional resources. Peter Westwood has been an Associate Professor of Education and has taught all

NUMERACY

There is an increasing need for numeracy skills in all aspects of life and Numeracy

What teachers need to know about

age groups. He holds awards for excellence in teaching from Flinders University in South Australia and from the University of Hong Kong. Peter has published many books and articles on educational subjects and is currently an educational consultant based in Macau, China.

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Numeracy

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