Making Connections in Primary Mathematics

Sylvia Turner, Judith McCulloch

• Publisher: Routledge
• Year: 2017
• ISBN:
  9781135397968
• Edition: 1
• Topic: Bildung
• Subtopic: Bildung Allgemein

First published in 2004. The main aim of this book is to identify the connecting ideas in primary mathematics in order to help teachers develop children's understanding of maths. The text emphasises the relationship between language, notation and representation; identifies misconception that can develop from false connections and provides activities to clarify understanding. Written for students training to be teachers and for teachers reconsidering their approach to maths as part of professional development.

CHAPTER 1

Language

Language and semantics

Introduction

The English language is said to be a ‘rich’ language in that it has an extensive vocabulary derived from a wide range of influences on its development. This ‘rich-ness’ means that thoughts and ideas can be expressed with clarity and subtlety, showing nuances of meaning. Take for example the word ‘mean’. The idea of mean when used as an adjective can be expressed by the synonyms selfish, nasty or unpleasant. We can express unpleasantness by saying that a person is mean but not that a smell is mean. Each of these words, however, has a slightly different connotation depending on the context and interpretation. Furthermore, the word ‘mean’ used as a verb can show intention, e.g. I mean to go home early today, or the interpretation of a word, e.g. What does ‘mean’ mean? ‘Mean’ used as a noun in a mathematical context as a technical term has yet another quite distinct definition: the arithmetic mean is a particular type of average used in statistics.

Mathematical language includes many examples of synonyms that children must be able to understand. In everyday language, they will need to develop an understanding of the subtlety in the difference in meaning depending on context and interpretation. In mathematics, children also need to develop an understanding of the differences between mathematical synonyms. The terms subtract, minus, take away, decrease by, the difference between and less than identify the operation of subtraction but do not all refer to the same subtraction structure. Children need to be aware of these structures in order to make sense of the words used to describe them. Such synonyms and phrases related to the operation of subtraction are discussed in detail in the next chapter (‘Calculation’, p. 36).

Issues of understanding related to language are identified particularly with respect to children for whom English is an additional language (EAL) but these issues can relate to all children to some degree. ‘Richness’ of language can lead to confusion in the initial stages when children are acquiring language and developing mathematical concepts. Two other factors compound further the problem of understanding the language of mathematics. The first is the infrequent use of many mathematical terms even within the subject itself (e.g. congruent) which results in a lack of regular reinforcement of understanding. Secondly, mathematical terminology is often used inaccurately, e.g. the bigger half.

Your mathematics

First, it is useful to consider just the words or phrases with equivalent meanings and when they are introduced to children. Taking vocabulary from the National Numeracy Strategy (NNS) (DfEE 1999), it can be seen in Figure 1.1 that much of the language related to equivalence is introduced in the reception class. Skemp (1991) identifies this as an early stage in children's development of language when connections between thoughts and the spoken word are stronger than between thoughts and the written or symbolic representation.

Figure 1.1

Language is the means by which we describe our mathematical experiences. It involves the use of both mathematical terminology and terminology that is associated with explanation and instruction. A teacher therefore needs to ensure that the language used has a shared meaning — without shared meaning children are likely to develop misconceptions by making false connections or not being able to access explanations. This means being aware of language that is ambiguous. Such ambiguity can be identified in the following ways:

• Meaning that differs according to whether the word is used as a noun or a verb. For example: the word note can mean an instruction [verb], a form of currency [noun] and a musical symbol [noun].

• Words that are derived from the same root but that have a different meaning in an everyday context, e.g. the net of a shape and a fishing net. Some words in this category have two or more mathematical meanings that are used in a mathematical context (e.g. left denotes direction and also identifies a numerical remainder).

• Homophones and homonyms, e.g. one/won; count/Count Ferdinand.

Think of some words that you use commonly when teaching mathematics and consider the alternative meanings that children could derive from them. It may be helpful to look at some planning material you have used.

The way in which words are contextualised gives meaning and understanding to the complex nature of the English language but this comes with experience and use of language. Again the majority of such words are introduced early in Key Stage 1 (see Fig. 1.3). Children with EAL will have even less experience to help them to discriminate between varying uses of words than those whose first language is English.

Your teaching

Children who are not hearing-impaired experience language initially by the spoken word. When language is used as the medium to instruct or explain and label mathematical concepts, rather than for acquisition, effective teaching will depend on variables such as children's preferred learning styles. In order to meet the needs of the majority of children in a class, they need to be offered a variety of ways to learn language.

Ambiguities

The following strategies aim to support the understanding of mathematical language to accommodate different learning styles and modes as well as to support children with EAL and those with a hearing impairment. Teaching needs to take into consideration misconceptions associated with language. Planning should include identification of possible ambiguity in the language that is to be used and teaching strategies should be devised that take such language into account. The topic of fractions can be used to illustrate strategies to deal with possible ambiguity:

• Avoid using an ambiguous word when introducing a concept, e.g. whole/hole. It is not helpful to completely avoid ambiguous terminology — children need to be aware of words that can have different meanings as this knowledge will eventually enable them to consider carefully a range of meanings and apply them appropriately:

I have a whole pizza [use of real or replica pizza for visual understanding]

How much pizza do I have?

What is another way of saying I have a whole pizza? [all of it; complete pizza etc.]

• Make explicit the false connection and invent rhymes to help children to be aware and remember the difference in meaning:

A whole egg doesn't have any holes.

Make up rhymes using homophones that can be associated with mathematics (see Fig. 1.3). It is useful to illustrate the r...

Table of contents

• Cover
• Half Title
• Related titles of interest
• Full Title
• Copyright Page
• Contents
• Acknowledgements
• Introduction
• 1 Language
• 2 Number
• 3 Shape, space and measures
• 4 Handling data
• 5 Cross-curricular connections
• Bibliography
• Index