Reading in Maths matters. But what does it mean to read like a Mathematician and how do we teach our students to do this?

In this post, I identify some of the challenges posed by word problems in Maths and offer action points to spark discussion. Bridging theory and practice, my aim is to inspire practitioners to think about how they can develop powerful tools that can support reading.

It’s a longer read to provide food for thought and several fascinating insights (I hope)!

I’ve based the post around some work I did for a local Maths hub and a short CPD session delivered to my own Maths department. 

Summary of action points

Read like a Mathematician

Before you read any further, take the active reading challenge! Sequence these sentences from a Maths word problem. Take time to read closely, like a Mathematician. What order would you put them in? Why?

If you are a Mathematician, what clues did you use to sequence this question? How did you use your knowledge of word problem structure to sort out the sequence? Could you explain to your students how you did it? Try this as a CPD task to see whether your colleagues arrive at the same answer using similar reasoning.

If you are not a Mathematician, how easy or difficult did you find this task? What clues did you use to piece this question together? Are you confident that your answer is correct? Why?

Whatever your response, if you stayed with the task, you probably found yourself reading and rereading the info closely, making links between the information given, asking questions, ordering and reordering. 

Mathematicians are skilful, active readers yet students will enter the Maths classroom with the misconception that reading is not something that they do in this subject. But in school Maths learners spend a lot of time interpreting word problems. They need to read and understand the information before they can relate it to the mathematical operation that’s required. Empowering students to be skilful, active readers – to read as Mathematicians – is therefore a crucial part of this process.

Action Point 

• Maths teachers can shape a reader identity with students through their classroom language. To create this mental model, explicit statements are important. 

Why focus on word problems – what does the research tell us?

There is a substantial amount of academic research into the language of school Maths and much of it foregrounds the challenges posed by word problems.

What’s happening in the classroom?

School Maths tends to focus on procedural knowledge – teaching students how to perform mathematical operations – largely because the literacy demands of word problems are not made explicit to teachers. 

The language demands of tasks also remains invisible in feedback from Awarding Bodies. I surveyed exam reports from a major UK exam board (for a single exam series) and found only one acknowledgement that reading skills were pivotal; there was a passing reference to the fact that some students ‘misread’ one particular question. When scrutinising the comments further, it was clear that word problems were often the discriminator between candidates as they struggled to mathematise the scenario given, or muddled references to different units in a questions – all of which points to the importance of teaching students to have a strategic approach to reading. 

How do word problems work?

This section will unpick the linguistic properties of word problems by analysing a question that is tough for students working around the Foundation / Higher Paper borderline. It was given to me by our local online Maths hub and is a useful illustration of text, sentence and word level features.

Text level

Understanding how word problems are structured is a powerful piece of disciplinary knowledge. Barwell (2011) notes that nearly all problems follow a three part structure. The first line sets up the scenario, then the key information is given, then the question. For those of you who aren’t maths specialists, here’s a quick survey of some other first lines, taken from a Year 7 Maths lesson. They set up different scenarios:

Sebastian buys a gift which costs £18.20

Jacqui’s basic pay is £8.65 per hour.

Chloe has a ribbon 8m long.

Peter is 1.64 metres tall.

Students often try to read word problems as a narrative – it has people and events – but it’s actually very different and trickier to comprehend. In a story, there is a sequence of linked and unfolding events. There’s a timescale, characters have motivations, thoughts and feelings. But word problems contain arbitrary information (40p per mile, 52 miles / gallon), timeless actions, no character development and no story!

They start by making general statements and then move on to specific information. The use of the present tense can be confusing, ‘Amal drives her car for work’; it refers to a habitual (repeated) action and situates the problem in the timeless present. Similarly, the effect of the present tense here is rather strange, ‘On one journey Amal drives 260 miles’. Arguably, the past tense would be less ambiguous, ‘Amal drove 260 miles.’ Another added challenge is that readers need to infer that she is using her car for work on this journey and that she claims money for this journey – we are not explicitly told.

All of the above may seem obvious to skilled readers with implicit awareness of how this genre operates, but to less confident students and some EAL students, it is far from clear. 

Action Points

• Teacher explanation. Talk to students about this formulaic structure. Break down a question. Explain how the genre works, how the present tense works, how it is different from a narrative. 

• Plan lesson activities that allow students to manipulate the three part structure. Very quickly, they will start to understand how it works. There are so many ways to go with this in the classroom. Here are some suggestions.

1. Sequencing the three parts.

Cut up a word problem into its three parts. Ask students to put it in the correct order and justify their choices. So, they need to identify the scenario, the information and the question. This is a straightforward reading task which draws attention to the structure. Ask them to explain their reasoning. 

To increase the reading challenge, students could be given several questions at one time on the current Maths topic, let’s say calculating percentage loss and gain. So, if there are three different problems, there will be nine cards to sort and sequence. This is excellent close reading practice for students as they will need to scan the text for particular clues, searching for patterns and links in the language. The activity also shows how the same Maths operation can be present in different scenarios.  My next post will show how this can work.

2. Reveal the question line by line.

Read out the question line by line but pause at the end of each and ask students to note down useful information. 

This is excellent listening practice and focuses attention on precise reading in order to extract the useful content. This would work well as a paired activity too. Modelling this process to show how to takes notes – arranging info clearly on the pages, visualising it – could also focus learners on writing skills in Maths. 

3. Sequence the whole question

Give students separate sentences from a question and ask them to work out the sequence, explaining their reasoning.

There are clues which students can use to assemble the text in a particular order. For example, the alternation of name and pronoun (Amal / she) establishes a logical order. It makes sense to name the person in the first line, and then refer back to her using a pronoun in line two. The pattern is repeated in the next two lines. The nouns in purple, green and teal also link the information in a logical sequence.

4. Write your own questions – from picture prompts, or from a given scenario.

To further increase the challenge, students could write their own questions and then solve them. Tasking students to apply the ‘numbers’ to their own real-world scenarios can be a good way of assessing their understanding of the Maths operation. 

The activity may need scaffolding with picture prompts and sentence stems. I explore this further in my next post.

5. Teach students how to identify grammatical features that link together information from across the text.

Identify these important words and develop a reading routine which encourages students to find them (example given further down). In word problems, the determiner is an important grammatical feature that pinpoints precise details and links information across sentences.

Definition: a word placed before a noun to make the reference more precise.

In Maths, determiners such as each, her, his, this, these and numbers (one, two…), appear frequently.

Examples here:    each gallon        one journey         this journey

Students need to pay careful attention to these words in order to build up their mental model of the scenario. For example, it’s important to know how far the car travels ‘for each gallon of petrol’ and then to connect this to the specific situation ‘one journey’. The question then refers back to it, ‘For this journey.’

Background knowledge

The first line of the question assumes that students know what it means to drive a car ‘for work.’ It is likely that this will be an unfamiliar concept for some students and that they may even confuse this, on a first reading, with driving a car ‘to work’. 

Action Point

• A quick explanation of what it means to drive your car ‘for work’ would be helpful. To reinforce comprehension of the phrase ‘for work’ I would also provide examples from different contexts: Tom wears a gun shield for rugby; my teacher uses a visualiser for modelling writing.

Tier Two vocabulary

Understanding of vocabulary is also bound up with prior knowledge as the verb ‘claims’ illustrates here. This is a classic example of a tier two word that students will encounter in different subject areas yet it will have different meanings.

Action Point

• Start with the specific meaning of ‘claims’ in this question and present a brief, student-friendly definition. Follow it up with other examples. Can students generate their own examples?

• Then, discuss other meanings of the word. This could include an example from History, ‘The historian claims that the army lost the battle because ….’, media discourse, ‘The Royal family claims that….’  and one from everyday experience, ‘The student claims she’s lost her homework.’

• Keep explanations brief and efficient but involve students in discussions of meanings – that way it’s possible to ascertain what they know and understand already. It’s also important to consider other meanings because it gives learners word breadth which is an important part of vocabulary acquisition. Be mindful that many students will benefit from seeing these examples written on the board, not just delivered orally.

• If the department or whole-school already has a strategy for recording and embedding Tier 2 vocabulary, this can enacted here. For example, do students write a list of new vocabulary in their books? Is there a wall display of key words relating to tier 2 words that occur in particular units? Are these words organised onto a literacy mat or knowledge organiser? Are there routines which encourage students to revisit and use common subject vocabulary in their speech and writing?

Sentence level

There are distinct sentence structures that reoccur in word problems. 

Here, there is a standard Maths question frame, ‘How much more …………  than …………? 

Once students have paid attention to this, they can go back and work out what they need to do with the information that has been given. In other words, its function is to lead the student towards the operation that is required.  

Action Point

• Highlight the structure and talk about what it means here. Discuss the use of bold font to direct students’ thinking about what they need to do. Draw attention to the fact that it is a common Maths question frame. Ask students where else they have seen this structure or others that are similar. Note that this is a command phrase, not just a single command word.

As with vocabulary teaching, common sentence frames can be embedded on KOs, wall displays and retrieval work. 

From reading to writing – the reading goal

Mathematicians read word problems with a clear purpose: to perform calculations and find solutions. Any active reading activity will therefore be linked to this goal. Students need to interact with the text in a way that helps them to apply the Maths. 

Action Point

• Model and practise a reading routine which encourages active reading. Provide a frame that allows learners to interact with the text by highlighting, circling and making notes which will support the calculations they need to do. Here’s an example that I’ve designed as a starting point for teachers to discuss, adapt and trial. The template idea might work – teachers can drop in any question – or the routine could be printed as a bookmark and stuck into books. As always, there’s potential for indirect instruction via wall displays too.

Adding it all up…

Language has a key role to play in learning and doing school Maths. In fact, attention to reading and the linguistic properties of word problems is integral to success. Making this visible to learners is essential. In all probability, if we start this process from transition, there will be less work to do on it at KS4 – and more tangible benefits. Go figure!

A copy of the slides embedded in this post can be found here. Feel free to adapt and use for CPD if you wish but please include the references.

Training

The National Literacy Trust offer two fantastic in depth sessions on Literacy in Maths which cover: writing and explaining, reading, Mathematical talk and vocabulary. They’re delivered by consultants who have tried and tested classroom resources; the material is inspirational and will galvanise any Maths department to develop their practice.

References

Barwell, R (2011) ‘Word Problems. Connecting language, mathematics and life.’ WHAT WORKS? Research into Practice. #34

Beck, I., McKeown, M., & Kucan, L. (2013) Bringing Words to Life. New York: Guildford Press.

Driver, C. & Dowling, D. (2020) ‘Literacy in Mathematics’ National Literacy Trust workshop.

Hall, G (2018) ‘The experiences of secondary school students with English as an additional language: perceptions, priorities and pedagogy’ ELT Research Papers 18.03.

Majewska, D (2019) Expresso Research, Filtered by Cambridge Mathematics, Cambridge Mathematics. Issue 26.

Oakhill, J., Elbro C., & Cain, K. (2015) Understanding and Teaching Reading Comprehension. Oxford: Routledge.

Quigley, A. (2020) Closing the Reading Gap. Oxford: Routledge.