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1 Key understandings in mathematics learning Paper 1: Overview By Terezinha Nunes, Peter Bryant and Anne Watson, University of Oxford A review commissioned by the Nuffield Foundation 2 PaperSUMMARY 1: Overview – PAPER 2: Understanding whole numbers About this review In 2007, the Nuffield Foundation commissioned a Contents team from the University of Oxford to review the available research literature on how children learn Summary of findings 3 mathematics. The resulting review is presented in a series of eight papers: Overview 10 Paper 1: Overview References 36 Paper 2: Understanding extensive quantities and whole numbers Paper 3: Understanding rational numbers and intensive quantities About the authors Paper 4: Understanding relations and their graphical Terezinha Nunes is Professor of Educational representation Studies at the University of Oxford. Paper 5: Understanding space and its representation Peter Bryant is Senior Research Fellow in the in mathematics Department of Education, University of Oxford. Paper 6: Algebraic reasoning Anne Watson is Professor of Mathematics Paper 7: Modelling, problem-solving and integrating Education at the University of Oxford. concepts Paper 8: Methodological appendix About the Nuffield Foundation Papers 2 to 5 focus mainly on mathematics relevant The Nuffield Foundation is an endowed to primary schools (pupils to age 11 y ears), while charitable trust established in 1943 by William papers 6 and 7 consider aspects of mathematics Morris (Lord Nuffield), the founder of Morris in secondary schools. Motors, with the aim of advancing social w ell being. We fund research and pr actical Paper 1 includes a summar y of the review, which experiment and the development of capacity has been published separately as Introduction and to undertake them; working across education, summary of findings. science, social science and social policy. While most of the Foundation’s expenditure is on Summaries of papers 1-7 have been published responsive grant programmes we also together as Summary papers. undertake our own initiatives. All publications are available to download from our website, www.nuffieldfoundation.org 3 Key understandings in mathematics lear ning Summary of findings Aims Our aim in the review is to present a synthesis Pragmatic theories are usually not tested for their of research on mathematics lear ning by children consistency with empirical evidence, nor examined from the age of five to the age of sixteen years for the parsimony of their explanations vis-à-vis other and to identify the issues that are fundamental to existing theories; instead they are assessed in m ultiple understanding children’s mathematics learning. In contexts for their descriptive power, their credibility doing so, we concentrated on three main questions and their effectiveness in practice. regarding key understandings in mathematics. • What insights must students have in order to Our starting point in the review is that children need understand basic mathematical concepts? to learn about quantities and the relations between • What are the sources of these insights and how them and about mathematical symbols and their does informal mathematics knowledge meanings. These meanings are based on sets of relate to school learning of mathematics? relations. Mathematics teaching should aim to ensure • What understandings must students have in that students’ understanding of quantities, relations order to build new mathematical ideas using and symbols go together. basic concepts? Conclusions Theoretical framework This theoretical approach underlies the six main While writing the review, we concluded that there sections of the review. We now summarise the main are two distinct types of theor y about how children conclusions of each of these sections. learn mathematics. Explanatory theories set out to explain how Whole numbers children’s mathematical thinking and knowledge • Whole numbers represent both quantities and change. These theories are based on empir ical relations between quantities, such as differences research on children’s solutions to mathematical and ratio. Primary school children must establish problems as well as on experimental and longitudinal clear connections between numbers, quantities studies. Successful theories of this sor t should and relations. provide insight into the causes of children’s mathematical development and worthwhile • Children’s initial understanding of quantitative suggestions about teaching and lear ning mathematics. relations is largely based on correspondence. One-to-one correspondence underlies their Pragmatic theories set out to investigate what children understanding of cardinality, and one-to-many ought to learn and understand and also identify correspondence gives them their first insights obstacles to learning in formal educational settings. into multiplicative relations. Children should be 4 PaperSUMMARY 1: Overview – PAPER 2: Understanding whole numbers encouraged to think of number in terms of these • Two types of quantities that are taught in relations. primary school must be represented by fractions. The first involves measurement: if you want to • Children start school with varying levels of represent a quantity by means of a number ability in using different action schemes to solve and the quantity is smaller than the unit of arithmetic problems in the context of stor ies. measurement, you need a fraction; for example, They do not need to know arithmetic facts to a half cup or a quar ter inch. The second involves solve these problems: they count in different division: if the dividend is smaller than the divisor, ways depending on whether the problems they the result of the division is represented b y a are solving involve the ideas of addition, fraction; for example, three chocolates shared subtraction, multiplication or division. among four children. • Individual differences in the use of action • Children use different schemes of action in these schemes to solve problems predict children’s two different situations. In division situations, they progress in learning mathematics in school. use correspondences between the units in the numerator and the units in the denominator. In • Interventions that help children lear n to use their measurement situations, they use par titioning. action schemes to solve problems lead to better learning of mathematics in school. • Children are more successful in under standing equivalence of fractions and in ordering fractions • It is more difficult for children to use numbers to by magnitude in situations that involve division represent relations than to represent quantities. than in measurement situations. • It is crucial for children’s understanding of fractions Implications for the classroom that they learn about fractions in both types of Teaching should make it possible for children to: situation: most do not spontaneously transfer what • connect their knowledge of counting with their they learned in one situation to the other. knowledge of quantities • understand additive composition and one-to- • When a fraction is used to represent a quantity, many correspondence children need to learn to think about how the • understand the inverse relation between addition numerator and the denominator relate to the and subtraction value represented by the fraction. They must think • solve problems that involve these key about direct and inverse relations: the larger the understandings numerator, the larger the quantity, but the larger • develop their multiplicative understanding the denominator, the smaller the quantity. alongside additive reasoning. • Like whole numbers, fractions can be used to represent quantities and relations between Implications for further research quantities, but they are rarely used to represent Long-term longitudinal and intervention studies relations in primary school. Older students often with large samples are needed to suppor t curriculum find it difficult to use fractions to represent relations. development and policy changes aimed at implementing these objectives. There is also a need for studies designed to promote children’s Implications for the classroom competence in solving problems about relations. Teaching should make it possible for children to: • use their understanding of quantities in division situations to understand equivalence and order Fractions of fractions • Fractions are used in pr imary school to represent • make links between different types of reasoning quantities that cannot be represented by a single in division and measurement situations whole number. As with whole numbers, children • make links between understanding fractional need to make connections between quantities quantities and procedures and their representations in fr actions in order to • learn to use fractions to represent relations be able to use fractions meaningfully. between quantities, as well as quantities. 5 Key understandings in mathematics lear ning Implications for further research situations.This alternative approach has also not Evidence from experimental studies with larger been systematically assessed yet. samples and long-term interventions in the classroom are needed to establish how division situations relate • There is no research that compares the results to learning fractions. Investigations on how links of these diametrically opposed ideas. between situations can be built are needed to suppor t curriculum development and classroom teaching. Implications for the classroom There is also a need f or longitudinal
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