The language of mathematics in problems

Teachers regularly highlight the difference between a pupil's mathematical ability and their ability to successfully read and comprehend a written problem. There is a distinction between being numerate, literate and mathematically literate.

Mathematical problems use a variety of language, including everyday language used to establish the context and domain-specific mathematical vocabulary. It is not enough that a pupil can complete an arithmetic procedure, they need to understand and parse the language to access and identify the mathematical structure underneath. With so much complexity in language and challenges in comprehension, how do we support all learners to ensure that they are able to access written mathematical problems? The goal is not to reduce the mathematical demand of problems, but to remove avoidable language barriers so that pupils can attend to the mathematical structure.

What makes mathematical language so challenging?

Mathematical problems use both technical vocabulary and everyday language, and pupils need to know when everyday words are carrying mathematical meaning. For example, worded problems in real-life contexts often contain lots of everyday language. Pupils need to be able to comprehend the context and identify the mathematical implications.

Where vocabulary with more than one meaning such as ‘prime’ is used, pupils need to remember the mathematical definition amidst the noise of other competing meanings. The cognitive load and effort may be overwhelming before they identify the underlying mathematical structure or procedure. This may mean solving a problem takes considerable time, they approach it incorrectly through lack of comprehension, or they give up even when the actual maths may be well within their grasp.

There are several actions we can take to resolve this issue, both as a task designer and when supporting pupils in the classroom.

Task design

When designing or adapting tasks, it is useful to focus on three key areas that may make a task less accessible:

Avoid ambiguity

Precision of language is incredibly important. It can be easy to misinterpret the intent of a task due to ambiguous wording. For example:

“Claire has twelve pairs of socks, but she loses three in her room. How many does she now have?”

This question can be interpreted in several different ways, and all leave a feeling of doubt that we’ve done it as the writer intended. Did Claire lose three pairs of socks, or just three socks? If she lost three pairs of socks, are we being asked how many pairs she has now, or how many socks she has now? Indeed, if she lost three socks, it would be impossible to know how many pairs she now has without further information regarding which socks were lost, and which socks made pairs!

Ambiguity matters because it makes the mathematical structure uncertain: pupils cannot know whether they are being asked to reason about individual socks, pairs of socks, or both.

Reduce the reading load

Maths tasks can include unnecessary language that pads out a narrative or overemphasises instructions. These require more effort to read than to solve, disadvantaging pupils with a lower reading age and some second-language speakers.

For example:

A large cube is made from small cubes. The outside of the large cube has been painted. Imagine the large cube is now broken apart into all the small cubes.

Some small cubes will have paint on 3 faces.

Some small cubes will have paint on 2 faces.

Some small cubes will have paint on 1 face.

Some small cubes will have no paint on any face.

For each of the following small cubes, decide where it would be found on the large cube:

A cube with paint on 3 faces.

A cube with paint on 2 faces.

A cube with paint on 1 face.

A cube with paint on 0 faces.

The task is presented with 112 words, 15 of which are ‘cube’. This increases the reading load without adding mathematical value, making it harder for pupils to hold the structure of the problem in mind.

If we reduce the number of words, we get:

This version has more than halved the text (53 words). The shorter version is not mathematically easier; it simply makes the relevant structure – the faces, edges and vertices of the large cube - easier to attend to.

Be careful with contexts

It is also important to consider the choice of words. This might involve the inclusion of a stylised or authentic context as part of a task, which can inadvertently introduce words that have little meaning to pupils, or worse, they have dual meanings depending on how they are used. For example:

"A theatre has Stalls and Circle seats. On a Saturday, adults and children attended. The ratio of adults to children was 5:2, 3/4 of the children were in the Stalls, and 117 children sat in the Circle. If the theatre holds 2,600 seats, were more than 60% of the seats occupied?"

The question assumes a pupil has cultural knowledge of theatres, and specifically refers to stalls and circles, but these words could confuse some pupils because they are used more frequently in other contexts. This is avoidable by considering what cultural knowledge the context assumes, and whether unfamiliar words are helping pupils access the proportional structure or obscuring it.

In the classroom

Teachers can support pupils’ understanding of problems in several ways. One useful routine is to ask: What is the situation? What quantities are involved? How are the quantities related? What representation would show that relationship?

Consider the language in the following KS2 assessment question from 2024:

The mathematical structure is multiplication and the equation required is 25 × 40 =. However, this requires some comprehension of the problem, specifically that there is one box and within this box there are 40 equal groups of 25. If pupils have not understood that each packet is an equal group, it becomes harder to recognise the multiplicative structure of the problem.

Using a model can support pupils’ comprehension:

Pupils can be asked to re-tell a problem in different contexts, helping to reveal the underlying structure:

A shelf holds 40 vases of roses. Each vase holds 25 roses. How many roses does the shelf hold?

It is also important to make connections between different problems to show that they have the same underlying structure. Teachers should select concrete or pictorial representations carefully to help pupils generalise when they encounter a problem with a structure they have seen before but perhaps using language they have not.

Connections also need to be made for technical vocabulary so that pupils can understand the word and associate it with an appropriate representation. For example, pupils who encounter prime numbers through different representations, such as numbers, arrays and factor bugs, can connect the word “prime” with its mathematical meaning, not just its everyday associations.

You can explore the Maths Horizons resources here.

Lisa Coe is an expert on the Maths Horizons specialist team.

Lisa is a Development Lead for Ark Curriculum Plus. Previously she worked as a trust-wide Primary Maths Lead supporting seven schools including a school implementing the Mathematics Mastery programme.