Understanding Mathematics for Young Children

A Guide for Teachers of Children 3-7

Derek Haylock, Anne D Cockburn

• Publisher: SAGE Publications Ltd
• Year: 2017
• ISBN:
  9781526410009
• Edition: 5
• Topic: Education
• Subtopic: Teaching Mathematics

Having a deep understanding of the mathematical ideas and conceptstaughtin the classroom is vital as a nursery or primary school teacher. In order for children to get to grips with these concepts, trainee teachersneed to be aware of how they come to interpret and understand them.

Now in its 5 th edition, this essential book helps trainee teachers develop their own knowledge of key mathematical ideas and conceptsfor thenursery and primary classroom. Now focusing specifically on ages 3-7, italsosupports trainees with several age-appropriate classroom activities.

As well as updates to further reading suggestions and research focuses, this revised edition includes new content on:

• Mastery in learning mathematics
• Simple fractions
• Roman numerals
• Money as a form of measurement

1 Understanding Mathematics

A Revealing Conversation

Gemma, aged 6 years, had no problems with questions like 2 + 3 = ◻ and even 8 + ◻ = 9. Her teacher said that she thought Gemma had a good understanding of the equals sign. But then the teacher asked Gemma how she did 2 + ◻ = 6. Gemma replied, ‘I said to myself, two (then counting on her fingers), three, four, five, six, and so the answer is four. Sometimes I do them the other way round, but it doesn’t make any difference.’ She pointed to 1 + ◻ = 10: ‘For this one I did ten and one, and that’s eleven.’

This conversation prompts us to ask the following questions:

• How does Gemma show here that she has some understanding of the concept of addition?
• What about her understanding of the concept represented by the equals sign?
• How would you analyse the misunderstanding shown at the end of this conversation?

In this Chapter

In this chapter we discuss the importance of teaching mathematics in a way that promotes the deep understanding that is necessary for mastery of the subject. So we aim to help the reader understand what constitutes understanding in mathematics. Our main theme is that understanding involves establishing connections. For young children learning about number, connections often have to be made between four key components of children’s experience of doing mathematics: symbols, pictures, concrete situations and language. We also introduce two other key aspects of understanding that will run through this book: equivalence and transformation.

Learning and Teaching Mathematics With Understanding

This book is about understanding mathematics. The example given above of Gemma doing some written mathematics was provided by a Key Stage 1 teacher in one of our groups. It illustrates some key ideas about understanding. First, we can recognize that Gemma does show some degree of understanding of addition, because she makes connections between the symbol for addition and the process of counting on, using her fingers. We discuss later the particular difficulties of understanding the equals sign that are illustrated by Gemma’s response towards the end of this conversation. But we note here that, as seems to be the case for many children, she appears at this point to perceive the numerical task as a matter of moving symbols around, apparently at random and using an arbitrary collection of rules.

Learning With Understanding

Of course, mathematics does involve the manipulation of symbols. But the learning of recipes for manipulating symbols in order to answer various types of questions is not the basis of understanding in mathematics. All our experience and what we learn from research indicate that learning based on understanding is more enduring, more psychologically satisfying and more useful in practice than learning based mainly on the rehearsal of recipes and routines low in meaningfulness.

For a teacher committed to promoting understanding in their children’s learning of mathematics, the challenge is to identify the most significant ways of thinking mathematically that are characteristic of understanding in this subject. These are the key cognitive processes by means of which learners organize and internalize the information they receive from the external world and construct meaning. We shall see that this involves exploring the relationship between mathematical symbols and the other components of children’s experience of mathematics, such as formal mathematical and everyday language, concrete or real-life situations, and various kinds of pictures. To help in this we will offer a framework for discussing children’s understanding of number and number operations. This framework is based on the principle that the development of understanding involves building up connections in the mind of the learner.

Two other key processes that contribute to children learning mathematics with understanding are equivalence and transformation. These processes also enable children to organize and make sense of their observations and their practical engagement with mathematical objects and symbols. These two fundamental processes are what children engage in when they recognize what is the same about a number of mathematical objects (equivalence) and what is different or what has changed (transformation).

Teaching With Understanding

This book has arisen from an attempt to help teachers to understand some of the mathematical ideas that children handle in the early years of schooling. It is based on our experience that many teachers and trainees in nursery and primary schools are helped significantly in their teaching of mathematics by a shift in their perception of the subject away from the learning of a collection of recipes and rules towards the development of understanding of mathematical concepts, principles and processes. So our emphasis on understanding applies not just to children learning, but also to teachers teaching, in two senses: first, it is important that teachers of young children teach mathematics in a way that promotes understanding, that helps children to make key connections, and that recognizes opportunities to develop key processes such as forming equivalences and identifying transformations; second, in order to be able to do this the teachers must themselves understand clearly the mathematical concepts, principles and processes they are teaching. Our experience with teachers suggests that engaging seriously with the structure of mathematical ideas in terms of how children come to understand them is often the way in which teachers’ own understanding of the mathematics they teach is enhanced and strengthened.

Mastery Through Understanding

In the context of the challenge to raise standards in mathematics in schools in England, the word ‘mastery’ has become prominent in the vocabulary of the English mathematics curriculum (see, for example, NCETM, 2014, www.ncetm.org.uk/public/files/19990433). This has developed from studying the comparative international successes of children with mathematics in Shanghai and Singapore in particular. These states redesigned their mathematics curricula around the turn of the millennium, when they realized that their previous dependence upon rote-learning, drill and frequent practice was not developing the fluent, creative mathematicians needed for competitive economic success in the 21st century. The curricula of these states have benefited particularly from the work of Bruner, who identified a progression from ‘concrete’ to ‘pictorial’ and then to ‘abstract’ in the development of understanding (Bruner, 1960).

The emphasis on mastery is consistent with the approach to children’s learning of mathematics that we adopt in this book. Mastery involves children developing fluency in mathematics through a deep understanding of mathematical ideas and processes. Teaching approaches for mastery should ‘foster deep conceptual and procedural knowledge’ and ‘exercises are structured with great care to build deep conceptual knowledge alongside developing procedural fluency’ (NCTEM, op.cit.). This is a key principle in teaching mathematics to young children: that mastery of the subject is not achieved simply by repeated drill in various procedures. Instead, the focus is on the development of understanding of mathematical structures and on making connections. Making connections in mathematics – a recurring theme in this book – ensures that ‘what is learnt is sustained over time, and cuts down the time required to assimilate and master later concepts and techniques’ (NCTEM, op.cit.). We shall see frequently in our exploration of young children’s understanding of mathematics that nearly all mathematical concepts and principles occur and can be applied in a wide range of contexts and situations. Because of this, the deeper understanding central to mastery in mathematics is facilitated by a wide variation in the experiences that embody particular mathematical ideas.

For example, mastery of the 5-times multiplication table is not just a matter of memorizing a chant that begins ‘one five is five, two fives are ten …’ – although that is part of it. It would also involve, for example:

• connecting each result in the table with a collection of 5p coins and the total value;
• articulating the pattern of 5s and 0s in the units position in the odd and even multiples of 5;
• explaining how to get from 4 fives to 8 fives by doubling;
• explaining how to get from 6 fives to 7 fives by adding 5;
• counting in steps of five along a counting stick;
• knowing that, say, ‘3 fives are fifteen’ is what you use for the cost of 3 books at £5 each;
• constructing patterns with linked cubes that show 1 set of five, 2 sets of five, and so on;
• filling in the missing number in number sentences like ‘6 × ◻ = 30’.

It goes without saying that to teach for this kind of mastery, even with young children, teachers themselves need a deep structural understanding of mathematics, an awareness of the range and variety of situations in which a mathematical concept or principle can be experienced, and confidence in exploring the connections that are always there to be made in understanding mathematics.

Concrete Materials, Symbols, Language and Pictures

When children are engaged in mathematical activity, as in the example above, they are involved in manipulating one or more of these four key components of mathematical experience: concrete materials, symbols, language and pictures. These four components will often provide a starting point for generating the variation in experience of mathematical concepts that is essential for developing mastery.

First, they manipulate concrete materials. We use this term to refer to any kind of real, physical materials, structured or unstructured, that children might use to help them perform mathematical operations or to enable them to construct mathematical concepts. Examples of concrete materials would be blocks, various sets of objects and toys, rods, counters, fingers and coins.

Second, they manipulate symbols: selecting and arranging cards with numerals written on them; making marks representing numbers on pieces of paper and arranging them in various ways; copying exercises from a work card or a textbook; numbering the questions; breaking up numbers into tens and units; writing numerals in boxes; underlining the answer; pressing buttons on their calculator; and so on.

Third, they manipulate language: reading instructions from work cards or textbooks; making sentences incorporating specific mathematical words; processing the teacher’s instructions; interpreting word problems; saying out loud the words that go wi...