Using Mathematical Language

Mathematics is a creative and highly inter-connected discipline that has been developed over centuries, providing the solution to some of history’s most intriguing problems. It is essential to everyday life, critical to science, technology and engineering, and necessary for financial literacy and most forms of employment. A high-quality mathematics education therefore provides a foundation for understanding the world, the ability to reason mathematically, an appreciation of the beauty and power of mathematics, and a sense of enjoyment and curiosity about the subject.

Aims

The National Curriculum for mathematics aims to ensure that all pupils:

•  become fluent in the fundamentals of mathematics, including through varied and frequent practice with increasingly complex problems over time, so that pupils have conceptual understanding and are able to recall and apply their knowledge rapidly and accurately to problems
• reason mathematically by following a line of enquiry, conjecturing relationships and generalisations, and developing an argument, justification or proof using mathematical language
• can solve problems by applying their mathematics to a variety of routine and non-routine problems with increasing sophistication, including breaking down problems into a series of simpler steps and persevering in seeking solutions.

In order for this to happen children need to understand key mathematical words and phrases if they are to make good progress in their mathematics.

There are three main ways in which children’s failure to understand mathematical vocabulary may show itself:

• children do not respond to questions in lessons
• they cannot do a task they are set
• they do poorly in tests.

Their lack of response may be because they:

• do not understand the spoken or written instructions, such as ‘draw a line between…’, ‘ring…’ or ‘find two different ways to…’
• are not familiar with the mathematical vocabulary, that is, words such as ‘difference’, ‘subtract’, ‘divide’ or ‘product’
• they may be confused about mathematical terms, such as ‘odd’ or ‘table’, which have different meanings in everyday English
• they may be confused about other words, like ‘area’ or ‘divide’, which are used in everyday English and have similar, though more precise, meanings in mathematics

There are, then, practical reasons why children need to acquire appropriate vocabulary so that they can participate in the activities, lessons and tests that are part of classroom life. There is, however, an even more important reason: mathematical language is crucial to children’s development of thinking. If children don’t have the vocabulary to talk about division, or perimeters, or numerical difference, they cannot make progress in understanding these areas of mathematical knowledge.

Children should be introduced to appropriate mathematical language in planning and teaching sequences. The following vocabulary checklists for each year group should be referred to as a core list of essential words and phrases but are not intended to be exhaustive. The lists have been extended from those provided by the National Numeracy Strategy to line up with the National Curriculum 2014. They are organised in four strands: Number, Measurement, Geometry and Statistics, using and applying is integrated throughout. In addition Year 6 has a section for Algebra and Ratio and proportion.

The words listed for each year include vocabulary from the previous year, with new words for the year printed in purple from Year 2 onwards. Some words may appear under different strands in different years, as their meaning is expanded or made more specific.

Class teachers can use these lists to identify the vocabulary relating to a series of lessons they are planning. They can make provision for the introduction of new vocabulary and the consolidation of familiar terms. They can ask support staff and parents to emphasise this vocabulary for an appropriate period.

Spoken Language

The national curriculum for mathematics reflects the importance of spoken language in pupils’ development across the whole curriculum – cognitively, socially and linguistically. The quality and variety of language that pupils hear and speak are key factors in developing their mathematical vocabulary and presenting a mathematical justification, argument or proof. They must be assisted in making their thinking clear to themselves as well as others and teachers should ensure that pupils build secure foundations by using discussion to probe and remedy their misconceptions.

The National Curriculum sets out specific expectations in relation to the use of vocabulary and its link to spelling to be met at the end of Year 2, 4 and 6:

Years 1 and 2

Pupils should read and spell mathematical vocabulary, at a level consistent with their increasing word reading and spelling knowledge at key stage 1.

Years 3 and 4

Pupils should read and spell mathematical vocabulary correctly and confidently, using their growing word reading knowledge and their knowledge of spelling.

Years 5 and 6

Pupils should read, spell and pronounce mathematical vocabulary correctly.

Teachers often use informal, everyday language in mathematics lessons before or alongside technical mathematical vocabulary. Although this can help children to grasp the meaning of different words and phrases, a structured approach to the teaching and learning of vocabulary is essential if children are to move on and begin using the correct mathematical terminology.

New words should be introduced in a suitable context, if possible with relevant real objects, mathematical apparatus, pictures and/or diagrams. Explain their meanings carefully and rehearse them several times as referring to new words only once will do little to promote learning. Encourage their use in context in oral sessions, particularly through a range of open and closed questioning.

Use every opportunity to draw attention to new words or symbols with the whole class, in a group or when talking to individual pupils. The final stages are learning to read and write new mathematical vocabulary in a range of circumstances, ultimately spelling the relevant words correctly.

Regular, planned opportunities for development

All children throughout Key Stages 1 and 2 needs to experience a cycle of oral work, reading and writing.

They should have opportunities to:

• listen to adults and other children using the words correctly
• acquire confidence and fluency in speaking, using complete sentences that include the new words and phrases, sometimes in chorus with others and sometimes individually
• describe, define and compare mathematical properties, positions, methods, patterns, relationships, rules
• discuss ways of tackling a problem, collecting data, organising their work
• hypothesise or make predictions about possible results
• present, explain and justify their methods, results, solutions or reasoning, to the whole class or to a group or partner
• generalise, or describe examples that match a general statement

Reading aloud and silently, sometimes as a whole class and sometimes individually for example, reading:

• numbers, signs and symbols, expressions and equations in blackboard presentations
• instructions and explanations in workbooks, textbooks, CD-ROMs
• texts with mathematical references in fiction and non-fiction books and books of rhymes during the literacy hour as well as mathematics lessons
• labels and captions on classroom displays, in diagrams, graphs, charts and tables
• definitions in illustrated dictionaries, including dictionaries that they themselves have made, in order to discover synonyms, origins of words, words that start with the same group of letters (such as triangle, tricycle, triplet, trisect)

December 4, 2013 MATHEMATICS VOCABULARY

Writing and recording in a variety of ways, progressing from words phrases and short sentences to paragraphs and longer pieces of writing, for example:

• writing prose in order to describe, compare, predict, interpret, explain, justify…
• writing formulae, first using words, then symbols
• sketching and labelling diagrams in order to clarify their meaning
• drawing and labelling graphs, charts or tables, and interpreting and making predictions from the data in them, in mathematics and other subjects

The use of questions is crucial in helping children to understand mathematical terms correctly. Asking questions in different ways and asking a range of open and closed questions are all important. Closed questions allow children to practise number facts and will be asked more often than open questions which require a higher level of thinking and present the opportunity for children to explain their thinking.