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Shanghai maths: elaboration theory vs traditional sequencing in the teaching of fractions

Over the past decade, international comparisons of education have assumed an increasingly important role in domestic policymaking. The big international surveys – principally PISA and TIMSS – have come to dominate thinking, not least through league tables that rank countries based on their pupils’ performances. Despite the fact that these league tables are the least analytical element of the surveys, they still have an almost magnetic appeal among policymakers. In recent years, focus has especially been directed at the Asian countries, and the ways in which they conduct mathematics education.

Xuhua Sun’s paper, ‘“Variation problems” and their roles in the topic of fraction division in Chinese mathematics textbook examples’, focuses on the way in which fractions are taught in Shanghai. It presents one of those seemingly small insights, which nonetheless have significant implications for curriculum thinking and practice. It focuses on ‘elaboration theory’: the way in which practice problems are varied to encourage children’s understanding of underlying mathematical relationships, which enable them to attain secure, and early, mastery of important mathematical operations and techniques.

The paper presents a fascinating illustration of how Chinese children work on all four functions with fractions – addition, subtraction, multiplication, and division – simultaneously in the last two years of primary education. The author does not locate this approach in a new or emerging theory, but in a thousand-year old philosophy of mathematical reasoning. To make children work on all four functions at the same time contrasts with traditional sequencing in English education, where children instead work on the functions in sequences that span over several years – with only a minority of children even getting to, let alone mastering, the division of fractions.

In Shanghai, rather than learning separate techniques, pupils are encouraged to see how adding, subtracting, multiplying, and dividing fractions are related. Previous research on mathematical misconceptions in England shows that many secondary children struggle with the division of fractions. For example, Jeremy Hodgen’s penetrating analysis of English children’s understanding of proportional reasoning shows a decay of facility over the last two decades.

And here, in this small study of Shanghai mathematics, we find a well-theorised, carefully-refined curriculum practice, which challenges our own approach to sequencing of content, age expectations, and practice activities. It contains important insights, which present profound challenges to the ways in which we approach mathematics education in our schools.


Tim Oates, CBE, is Group Director of Assessment Research and Development at Cambridge Assessment. He is a member of Ofqual’s Standards Advisory Group and following the publication of his paper ‘Could do better’, offering a framework for reform, was chair of the Expert Panel for Review of England’s National Curriculum.

This comment piece originally appeared in the CMRE Annual Research Digest 2015. The piece discusses a paper by Xuhua Sun, ‘“Variation problems” and their roles in the topic of fraction division in Chinese mathematics textbook examples’ published in the January 2011 edition of Educational Studies in Mathematics, the published version of which may be downloaded here.

CMRE also produce a Monthly Research Digest, edited by our Research Director, Gabriel Heller Sahlgren. You can download back copies of the digest here.

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