Linking Cubes and the Learning of Mathematics

Ideas for linking cubes to be used meaningfully from KS2 through to university

Key Stage suitability • Explanation

Linking Cubes and the Learning of Mathematics

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A variety of activities with linking cubes.

There will be few primary schools or mathematics departments without a stock of linking cubes. This book offers the activities you have always needed to exploit this simplest of all resources to its full. Whatever age you teach, you will find a variety of activities that will both enhance your teaching, your students’ learning and your own understanding of mathematics.

This book aims to offer ideas that will allow linking cubes to be used regularly and meaningfully from upper primary through to undergraduate. In this way they become an everyday and accepted tool of the mathematics classroom which, therefore, helps prevent the claims made by many learners that they are the preserve of those younger or less talented than them.

What does this book contribute to mathematical understanding?

Much mathematics is about structure which is something that linking cubes can expose in ways that teaching based on, say, number patterns or algorithms is unlikely to achieve. Learners who understand, and have an aware of, structure will be snore successful learners of, and see more purpose in, mathematics than those who do not. An understanding and awareness of structure will make transparent many mathematical results which otherwise, despite being visibly algebraically sound, remain opaque in terms of insight and understanding.

104 pages, full colour throughout.

ISBN 1 898611 14 9

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Making algebraic structure and mathematical thining accessible to learners of all ages.

What’s in this book?

This book contains activities for the teaching of mathematics based on the use of linking cubes. The simplest and most cost-effective equipment a school can buy, linking cubes, as this book shows, can be used to offer learners concrete experiences of a wide range of mathematical ideas. These range from notions of pre-algebra through linear sequences and their general forms, non linear sequences, elementary number theory based on an analysis of triangular and square numbers, algebraic identities and techniques, to probability, sampling, permutations and combinations. In short, it is hoped that colleagues will find worthwhile uses for linking cubes irrespective of the age or alleged sophistication of their learners.

Why is it in this book?

It is my hope that this book will enable teachers at all levels to make productive and worthwhile use of linking cubes in their teaching. My experience is that most colleagues have a small repertoire of activities that they could use, but too often they don’t, due to perceived threats to the smooth running of their classrooms. It is not without irony that one observes that where children use such materials regularly few such problems are likely to occur. Of course, the first time learners are given cubes to use, they make models of Tracey Island or My Little Pony.

However, this reaction is not the preserve of the adolescent. Undergraduates, postgraduate teacher trainees, and teachers on in-service courses behave similarly; the only difference being that children’s models are more sophisticated.

What does this book contribute to mathematical understanding?

Almost all of the activities encourage high level thinking and many have been included solely for the contribution they might make to children’s understanding of, and ability to engage with, proof. A substantial number allude, although how transparently this is done will depend on the ways in which individual teachers chose to use them, to notions of mathematical induction and can be used as pre- inductive activities.

Many of the tasks could be viewed as simple investigations. However, the explicit focus on structure is intended to move learners' thinking beyond an inductive approach to mathematics - the systematic collection of numerical data from which generalities are inferred - to more deductive forms of argument.

Additionally, many of the activities are clearly focused on learners' gaining access to conventional results of algebra and elementary number theory, although it is argued that the manner in which they are presented makes them accessible to younger learners than might ordinarily be the case.

What isn’t in this book?

This isn’t a book to be given to learners. The activities are not offered as worksheets to be copied but presented in ways that invite colleagues to work with them, explore them, gain ownership over them and then decide how they might use them and with whom. I know, for example, that an idea that works for me may need changing for someone else. I know, also, that the same activity, if altered slightly, will be appropriate for different ages and abilities. That is, my reasons for using any activity will vary according to particular circumstances - nothing is pre-determined and nor should it be. I don’t know your classes and you don’t know mine.

What is the teacher’s role?

All the activities shown in this book expect colleagues to make some form of intervention. For many years the use of rnanipulatives (as things like linking cubes are more generically known) has received a poor press. This is almost entirely because teachers have assumed that just working with them will be sufficient for the learner to make sense of the task he or she has been given. Such assumptions are nonsensical. Teachers need to intervene, with individuals and with the class, in order to ensure that sense-making takes place. As indicated above, many of the activities have a simple script offered alongside them. This is intended as no more than a set of prompt questions to inform colleagues' sense- making of the task and preparation of their own scripts which will be focused on classes and individuals whom they know well.