Children's Mathematics 4-15: Learning from Errors and Misconceptions

Children's Mathematics 4-15: Learning from Errors and Misconceptions

Children's Mathematics 4-15: Learning from Errors and Misconceptions

Children's Mathematics 4-15: Learning from Errors and Misconceptions


The phrase 'errors and misconceptions' has recently entered the vocabulary of mathematics teacher education and become prominent in the curriculum for initial teacher education. This follows several decades of academic study of children's errors and misconceptions. It is curious that the term 'misconception' is finding favour in the teaching profession just as the term is falling foul of the academic community.One approach to children's errors is to view them as potential windows into children's mathematics. Errors may 'diagnose' significant ways of thinking and stages in learning and so point to important opportunities for new learning. Often (mis)conceptions are acquired through reflection on experience in a limited context - for example, 'multiplication makes bigger' is a conception drawn from a context of whole numbers. When such a conception is erroneously extended and generalised to numbers less than 1, a misconception is diagnosed, and 'real teaching' can begin.The authors contest the popular view that errors and misconceptions should be corrected as soon as possible. On the contrary, such misconceptions may be supported by a child's generalised reasoning from experience and therefore require a focused 'treatment' that respects the child's intelligent behaviour.A positive view of misconceptions suggests respectful language like 'alternative frameworks' for children's thinking and concept formation. The most important consideration is to provide children with the conditions (including time) to articulate their reasoning, confront alternatives and make a rational decision to change their mind or not.This book makes use of recent and extensive original data from the authors' own researches on children's performance, errors and misconceptions across the mathematics curriculum, including standardised data from a large national sample of 4 to 15 year olds, and conversations involving children in argumentation, and work with teachers implementing change in their classrooms. The cycle of research includes: use of written diagnostic test items, children's reasoning as captured in group argumentation, details of the types of reasoning that help children to 'change their mind' and the development of tools for classroom teaching by practising teachers.The discussion of research is anchored in practical learning and teaching contexts in order to directly relate to mathematics teaching practice and teachers' expertise. The book progressively develops concepts for teachers to use in organising their understanding and knowledge of children's mathematics, concluding with an introduction to theoretical accounts of learning and teaching that can help make practical sense.The book bridges the gap between research in the psychology of learning mathematics and the reality of classroom practice. The book is ground-breaking in that it transforms research on diagnostic errors, argumentation and teaching strategies into knowledge for teaching. The voices in the research include those of children and teachers in classrooms, as well as the academic and research communities.


Learners, teachers, parents, academics, business leaders, politicians all say there is a serious problem with mathematics education, so it must be true. It is repeatedly asserted by government inquiries and reports in the UK (the most recent is the Smith Inquiry following the Roberts Review). Furthermore this has become a worldwide phenomenon, courtesy of international assessments such as PISA (the Organisation for Economic Co-operation and Development's Programme for International Student Assessment) and TIMSS (the International Association for the Evaluation of Educational Achievement's Trends in International Mathematics and Science Study); even in Japan there is a now a crisis over standards of mathematics.

In addition, there is growing evidence that state-mandated projects that claim significant improvements in standards are politically motivated and unsupported by rigorous research evidence. These projects typically tie accountability to performativity: performance on 'high stakes' state-wide tests, comparative pupil/school or teacher league tables, leading to the practice of 'teaching to the test'. The 'Texas miracle' and similar projects throughout the western world show that, typically, such efforts lead to short-term, superficial and in many cases illusory improvements in test scores without genuine, substantial gains in children's understanding.

Here are some data from our own analyses of a large, cross-sectional survey of some 15,000 children aged 4 to 15 years taken in the year 2005 in the UK (see Figure 1.1). Note the plateau in performance between ages 11 to 14 years, the very focus of government initiatives recently in this country. The slope of children's progress looks so slow that it appears to cease altogether for some years.

A little explanation about this data set is in order as we will be drawing on it periodically throughout the book (for more, see Appendix 1). The large representative sample of 15,000 children was drawn nationally for the purposes . . .

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