Mathematical Knowledge for Primary Teachers

Andrew Davis, Maria Goulding, Jennifer Suggate

• Publisher: Routledge
• Year: 2017
• ISBN:
  9781317219002
• Edition: 5
• Topic: Éducation
• Subtopic: Éducation générale

Now in its fifth edition, the best-selling text Mathematical Knowledge for Primary Teachers provides trainee teachers with clear information about the fundamental mathematical ideas taught in primary schools.

With rigorous and comprehensive coverage of all the mathematical knowledge primary teachers need, the text goes beyond rules and routines to help readers deepen their understanding of mathematical ideas and increase their confidence in teaching these ideas. The book has been updated to incorporate changes in the National Curriculum and the associated tests. In addition, Chapter 1 has been expanded to discuss mathematical understanding in the light of the challenges posed by the current changes. These include the re-introduction of traditional calculation methods for multiplication and division, the early coverage of abstract fractions calculations and much more.

Features include:

? 'Check' questions to test the reader's understanding

? 'Challenges' to increase teachers' confidence and stretch their mathematical abilities

? 'Links with the classroom' to emphasise the relevance of ideas to the classroom context

? Straightforward coverage from theory to practice for all aspects of the Mathematics Framework.

The book is accompanied by eResources which contains further visual activities and support, designed to scaffold and support the reader's own understanding. Essential reading for all practising and trainee primary teachers, this book is ideal for those who wish to increase their mathematical understanding and confidence in presenting mathematics in the classroom.

Part

1

Setting the scene

Much of this book looks systematically at specific areas of mathematical content. However, Part One differs in that it discusses a number of wider issues, some of which have already been raised in the introduction. The first three chapters examine aspects of mathematical knowledge in general, Chapter 1 taking up the theme of knowledge and understanding in mathematics, Chapter 2 dealing with language and Chapter 3 with problem solving and proof.

The nature and purpose of mathematics: brief remarks

If this were a very different work, a good deal of fascinating but abstruse discussion could be devoted to analysing the nature of mathematics. Despite the ‘absolute’-seeming character of the subject, there is no consensus of opinion as to its nature. Bertrand Russell remarks: ‘mathematics may be defined as the subject in which we never know what we are talking about, nor whether what we are saying is true’ (Russell, 1917, pp. 59–60). In the light of this, it is perhaps puzzling that he also observes in another essay in the same volume: ‘Mathematics, rightly viewed, possesses not only truth, but supreme beauty – a beauty cold and austere, like that of sculpture, without appeal to any part of our weaker nature, without the gorgeous trappings of painting or music, yet sublimely pure, and capable of a stern perfection such as only the greatest art can show’ (Russell, 1917, p. 49).

Knowledge, including knowledge of mathematics, is seen very differently by some writers in what is known as the ‘situated cognition’ tradition: ‘Learners … participate in communities of practitioners and … the mastery of knowledge and skill requires newcomers to move towards full participation in the sociocultural practices of a community’ (Lave and Wenger, 1991, p. 29).

In this tradition, learning mathematics might be compared to an apprenticeship, with all the human and fallible trappings that this process seems to imply. The writers of this book also see mathematics as a human construction. We view learning mathematics as a process in which pupils, partly through interaction with those who are in possession of fuller versions, are helped to construct an understanding of the subject. It is less clear than some claim that this wholly rules out Russell's perspective as expressed in the ‘supreme beauty’ quotation, but no more can be said about that here. Readers who are interested might try Ernest (1991) for one verdict on this argument.

For many children mathematics offers its own intrinsic satisfactions, but it also has essentially practical applications in everyday life. It enables us to communicate thinking and reasoning about number, quantity, shape and space. It embodies a precise language in which technological and scientific claims can be made. Those working in the field of the expressive arts also may make use of mathematics – design and architecture are two obvious examples.

We now turn to a consideration of the knowledge and understanding that we hope to develop when teaching primary mathematics, and of the knowledge and understanding needed by teachers to bring about such a development.

References

Ernest, P. (1991) The Philosophy of Mathematics Education. London: Falmer Press.

Lave, J. and Wenger, E. (1991) Situated Learning. Cambridge: Cambridge University Press.

Russell, B. (1917) Mysticism and Logic. London: Unwin Books.

Chapter

1

Mathematical understanding

Some who wish to teach mathematics to young children have the wrong kind of understanding of their subject. It is not that they know too little, though they may, but rather that their knowledge is not of an appropriate kind. Many years ago the psychologist Richard Skemp coined a distinction between two sorts of understanding. He described and contrasted what he called ‘relational understanding’ and ‘instrumental understanding’. His distinction can help us grasp the nature of the knowledge of mathematics that primary teachers should seek (Skemp, 1989).

Suppose Anna and Sarah are making their way to a wedding in a town which is unfamiliar to them. Each have been supplied by the host with a series of directions: ‘Left after the first roundabout on the way into town, under the railway bridge, third right, past the park and second left by the hypermarket. The church is on the right a few hundred yards down that road.’ Anna has no map of the town, whereas Sarah has one and she can read it. If Anna meticulously follows the directions supplied, she will make it to the wedding. However, it only needs one slip for her to be lost. Sheer chance may still bring success, but she is more likely to circle the roads in a state of increasing frustration and panic. How blessed is Sarah in comparison. She can follow the directions with confidence, knowing that if she strays, she has a remedy. This may mean that she is less likely to go wrong in the first place. If she does make a mistake, she can find her way back to the sequence of directions, or even pick out a new way to the church.

Anna's situation models the instrumental understanding that many of us possess in respect of our mathematics. We may know what to do to obtain the right answer, but we only know one way of doing so, and we have little idea why our sequence of moves is sensible. If we go wrong, we may not be able to recover. We have no other route to the solution. We lack the understanding that might enable us to solve the problem in our own way. We are likely to be in possession of a number of mathematical rules that we implement ‘without reason’. Certain examples have become classics and are usually quoted in this connection. Here are a few of them: to divide fractions, ‘turn upside down and multiply’; when dealing with equations, if you ‘change the side you change the sign’; and to multiply by 10, ‘add a zero’. (This one is not even universally correct, of course.)

In contrast, Sarah offers an analogy for those who have relational understanding of their mathematics. She has a map. Armed with a ‘cognitive map’ of the relevant mathematics, those with relational understanding are able to find a number of ways to solve a given problem. If they forget a particular procedure, or make a mistake in it, they can use and even construct an alternative route.

Some have understood Skemp to be extolling the virtues of relational understanding at the expense of instrumental understanding. In this book about subject knowledge in mathematics, we take the view that both kinds of understanding are important. Indeed, we do not assume that there is a sharp division between instrumental and relational understanding. Knowing rules and procedures can be extremely useful and efficient. However, they need to be embedded within a rich relational understanding of the concepts concerned. Without this understanding, as Skemp points out, much strain is placed on the memory, and feelings of frustration and panic are more likely. Moreover, people who are restricted to an instrumental understanding of mathematics cannot ‘use and apply’ it. They can only implement the procedure in contexts which closely resemble the situations in which they were taught. For instance, they can do ‘long division’ with paper and pencil, but are unable to work out how many dollars they will get for their pounds when they take a holiday in the States. This may be the case even though the calculation required would in fact be a division operation involving numbers for which a long division approach might be appropriate. Their mathematics is rigid and inert. Those whose knowledge is limited in this way may consider mathematics to be a pointless and tedious subject. Indeed, given that kind of knowledge, you might argue that they have every justification for such an attitude. This is hardly a desirable state of mind for a primary teacher who plays a significant part in shaping not only what young children know about mathematics but also how they feel about the subject.

It can be illuminating to link Skemp's distinction to constructivist approaches to learning, and to learning mathematics in particular. However, in order to grasp the nature of this link we have to face up to some challenges. The term ‘constructivism’ covers a multitude of sins and virtues. Accordingly, we devote some space in this chapter to ela...