# How picture books help maths teaching

When was the last time you read a maths-based picture book? I am guessing: not recently. I also assume that you are not altogether sure what I am talking about.

Let me enlighten you. Because what I am about to tell you could well change the way that you teach maths.

What is a mathematical picture book? Just like any story picture books, mathematical story picture books (MSPBs) have a plot, a cast of characters and page illustrations. What makes MSPBs unique is that mathematical concepts are explicitly or implicitly weaved into the plot to either demonstrate the concept or show how the concept can be used by characters to solve a problem found in the story.

Take, for example, Fractions in Disguise (Einhorn, 2014). This story is about George Cornelius Factor (who happens to share the same acronym, GCF, with - wait for it - the greatest common factor!). GCF invents a machine, called “Reducer”, to help him find a very sought-after fraction (5/9) that has been stolen from a fraction auction, and has been disguised as another fraction by the villainous Dr Brok.

While at Dr Brok’s mansion, GCF uses his knowledge of equivalent fractions (in the form of the Reducer machine) to reveal the true form of a range of fractions (eg, 3/21 is really 1/7; 34/63 is already in its true form; 8/10 is really 4/5; and so on).

Finally, GCF comes across 35/63, which is later revealed as the 5/9 fraction he has been looking for.

Happy ending.

When we examine the above story structurally, we will see that mathematical knowledge (ie, knowledge of equivalent fractions) is required to help the character solve the problem. The page illustrations also help readers visually to see how 35/63 is, in fact, the same as 5/9.

Mathematical story picture books are a specific genre of literature, and they are not (and should never be) maths textbooks or worksheets in disguise.

Moreover, as the above story shows, MSPBs are more than just counting books. There are several MSPBs for upper key stage 2 mathematics concepts (eg, any titles in the Sir Cumference series) and even for key stages 3 and 4 concepts (eg, What’s Your Angle, Pythagoras? (Ellis, 2014) for Pythagoras’ theorem, and Anno’s Magic Seeds (Anno, 1999) for exponential growth, among several others).

Why should we teach mathematics to older pupils using story picture books?

The idea of using MSPBs to enrich mathematics learning is not new. In fact, it has been around for almost three decades, particularly in the early years setting.

What is less common, particularly in the UK, is using MSPBs to enrich mathematics learning beyond the early years level.

I have been arguing - and will continue to argue - that the approach could also benefit mathematics learning of older pupils. Specifically, I would argue that the use of MSPBs could foster pupils’ conceptual understanding through multi-representation of mathematical concepts and variation of mathematical situations; develop language skills; and foster engagement with mathematics learning.

I’ll take each in turn.

## 1. Fostering conceptual understanding through multi-representation

We can all (hopefully) agree that we do not teach mathematics so that our pupils become a human calculator; that is, someone who is good at churning out correct mathematical answers but without understanding the concept behind them.

As part of one of my research projects, I asked Jack (pseudonym), a nine-year-old pupil, what 20 ÷ 5 was equal to, and he was able to give me the correct answer (4) almost instantly. Then, when he was asked to (contextually) represent 20 ÷ 5 using a word problem, this is what he came up with: “Spanish Yoda had a can of Coke and a bag of bananas and apples and paint. How much did it cost her? Coke: £1. Bag of bananas: £2. Apples: £8. Paint: £9. Total: £20.”

How Jack’s word problem is related to 20 ÷ 5 remains a mystery.

What Jack demonstrates is a classic example of a pupil whose procedural fluency (ie, the mechanical aspect of mathematical learning) in relation to division is good, but who has yet to fully grasp what the concept means.

As many mathematics education scholars have argued, in order to demonstrate conceptual understanding in mathematics, pupils must be able to represent mathematical concepts in different ways using different representations (eg, contextualisation, visualisation, etc).

Here, I would argue that key features of MSPBs, such as narrative and page illustrations, make learning mathematics conceptually effective as pupils get to learn mathematical concepts through these different representations.

## 2. Fostering conceptual understanding through variation

Another key strength of teaching maths using MSPBs is the development of pupils’ conceptual understanding through what I refer to as the variation of mathematical situations that are often found in well-written MSPBs.

Take Bean Thirteen (McElligott, 2007) as an example. The story follows two crickets, Ralph and Flora, who have collected 12 beans to bring home for dinner. When Flora decides to pick one more bean (ie, Bean Thirteen), Ralph is convinced it will bring bad luck. No matter how many friends they invite to try to share the 13 beans equally, it is always impossible.

- Situation 1: 13 beans to be shared between two crickets (Ralph and Flora), resulting in one remaining bean (six beans each).
- Situation 2: 13 beans to be shared between three crickets (Ralph, Flora and one friend), resulting in one remaining bean (four beans each).
- Situation 3: 13 beans to be shared between four crickets (Ralph, Flora and two friends), resulting in one remaining bean (three beans each).
- Situation 4: 13 beans to be shared between five crickets (Ralph, Flora and three friends), resulting in three remaining beans (two beans each).
- Situation 5: 13 beans to be shared between six crickets (Ralph, Flora and four friends), resulting in one remaining bean (two beans each).

In this example, while the number of crickets varies, the number of beans is invariant. Through this variation of situations, rich mathematical investigations are made possible. Pupils can be asked, for example, to continue the pattern to prove that 13 cannot be divided evenly by any other numbers except for 13 itself (and hence demonstrating the meaning of prime numbers in the process). I argue that such variation of mathematical situations is crucial to foster pupils’ conceptual understanding.

## 3. Develop language skills

From my earlier research (Trakulphadetkrai, Courtney, Clenton et al, 2017) and that of others, it has been found that children’s mathematical abilities are linked to their language abilities. What is exciting is how recent research (eg, Hassinger-Das, Jordan and Dyson, 2015; Purpura, Napoli, Wehrspann et al, 2017) has also uncovered the positive impact on the development of language abilities, particularly vocabulary knowledge, of using stories when teaching mathematics concepts to young children.

Why not kill two birds with one stone? Why not teach mathematics using MSPBs to develop both pupils’ mathematical and language development at the same time?

## 4. Engagement through emotional investment

Another key advantage of teaching mathematics using MSPBs is that pupils arguably do not see MSPBs in the same way that they see, for example, mathematics textbooks or worksheets with word problem after word problem to be solved. They are more likely to view MSPBs as something that they can emotionally invest in, and something that they can enjoy interacting with over and over again either together with the whole class or in their own time.

Research (eg, McAndrew, Morris and Fennell, 2017) has recently found that the use of stories in mathematics teaching can help to foster children’s positive attitude towards the subject.

## How to use story picture books in maths lessons?

Teachers could start their maths lesson by reading pupils a mathematical story to lure them in. This sets the scene and contextualises the mathematics to be taught.

I also know many teachers who prefer to wait until the end of the lesson to read the story, to consolidate the learning.

Alternatively, you might not want to finish reading your chosen story in one go. Quite often, there is a problem for the characters in the story to solve using their mathematical knowledge. Teachers could stop reading the story just before a solution is revealed and use the story’s plot to encourage pupils to solve the problem themselves through mathematical investigations.

The beauty of teaching mathematics using MSPBs lies in its flexibility: there isn’t one specific way of integrating these books into mathematics teaching. You can be as creative as you like.

## What about if pupils create their own mathematical story picture books?

Beyond reading MSPBs to pupils, a more innovative mathematics learning strategy that I have been trying to highlight to mathematics teachers (and curriculum developers) in the UK and abroad is the idea of getting pupils to develop their mathematical understanding through creating their own MSPB.

Here, I am not talking about asking pupils to create a full-feature, 30-page MSPB. As a mathematics learning activity, pupils can simply be asked to create their own mini MSPB with just 10 pages. For example, the first two pages set the scene and the problem to be solved by the characters; the next six pages can feature three variations (or attempts) in which the characters try to use their mathematical knowledge to solve the problem; and the story can come to a close on the last two pages.

With this activity, pupils need to carefully think about the storyline, which requires them to consider practical and meaningful applications of the mathematical concept in question. In brief, they need to contextualise abstract mathematical concepts.

Additionally, as the focus is on presenting the stories in the picture book format, pupils also need to actively think about page illustrations, and how best to communicate abstract mathematical concepts and situations visually to their readers.

As previously highlighted, not only could learning mathematics through storytelling benefit pupils mathematically, it could also develop their language and creative writing skills and provide a great cross-curricular teaching and learning opportunity.

Equally important, this approach would allow pupils to see mathematics in a different light - one that is less test-driven, and more fun and imaginative. This is crucial, especially if we want to improve pupils’ perceptions of the subject.

The preliminary findings of my pilot research with Year 4 pupils on the effectiveness of this mathematics learning activity is promising. Specifically, the results indicate that pupils in the intervention class (ie, those who were asked to create MSPBs on multiplication over a week period) had better conceptual understanding on multiplication (as measured through the study’s test) than their peers in the comparison class who learned multiplication the normal way (eg, worksheets and textbooks, etc).

From a distance, having pupils create their own MSPB might look like a cute, fun activity. However, when you carefully examine this approach, you see just how pedagogically powerful it can be. I’m surprised this approach hasn’t been used more often, because it costs nothing in terms of resources - just a few sheets of A4 paper, a pencil and a splash of imagination.

This mathematics learning activity can also save you time. For example, if the concept in focus is multiplication, you could start the day by getting your pupils in your maths lesson to consider the everyday situations in which having knowledge about multiplication can help to solve problems, and how the concept can be represented visually. Later in the literacy lesson, you could get your pupils to come up with the plot, characters and setting. You could also get them to work on their draft-writing, paying attention to things such as grammar.

After lunch, in the art lesson, you could get them to work on page illustrations, and putting their picture book together. Before home time, the pupils could read their MSPB, with the help of a visualiser, to their peers.

This one activity can be meaningfully integrated across different curricular subjects throughout the day. And what’s more, you would have just one set of work to mark.

*Dr Natthapoj Vincent Trakulphadetkrai is a lecturer in primary mathematics education at the University of Reading’s Institute of Education, and founder of MathsThroughStories.org. He tweets @NatthapojVinceT and @MathsStories*

*This article originally appeared in the 21 June 2019 issue under the headline “Maths story books help to put pupils in the picture”*

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