Provoking mathematical thinking through picture books

Most people have a favourite picture book. It may be the one you loved as a child, the one a parent or significant adult shared with you over and over again, or one your discovered as an adult. You may love it because it is funny, colourful, emotional or magical. It helped you find connections between sounds, words, gestures, intonation and phrases; it may have drawn you into a story which had a problem to be solved. If we consider mathematical thinking as a way of communicating about and with our world, and involving contextual problems requiring a range of strategies, then the link to exploring mathematics through picture books is one we can easily make. But it doesn't have to be a through"maths books", it can be any book at all!

In this section we will explicate links between picture books and mathematics, suggesting ways to use their contexts, language and connections to stimulate opportunities for mathematical thinking.

Reading picture books with a mathematical lens

From birth there is a dialogue between children and adults (or significant others) by way of sounds, gestures, movement and language. Reading picture books with and to children builds on these emerging communications skills.

Books should be enjoyed for many reasons, some of which are highlighted above.

We wonder how often you have thought of them as potentially rich in developing children's understanding of mathematical language, concepts and connections to their mathematical lives?

Can we look at picture books with a mathematical lens?

Amy and Louis by Libby Gleeson

Libby Gleeson is an Australian Children's author. Since 1984, she has written 20 books, including picture books, junior and young adult fiction.

Amy and Louis was written in 2006.

We thought this would be a great opportunity for you to listen to Libby read her book. She is delightful to listen to.

Click on the picture of Libby and enjoy.

We are sure you heard the central literary themes of friendship, distance, change and communication. The "coo-ee" is a very Australian call made up into the air so people can hear you over long distances.

It is interesting now, to consider the possibilities in terms of mathematics concepts, ones that can be derived from the central literary themes, and the beautiful illustrations.

What mathematics can you hear/see, think of?

What conversations do they promote?

What kinds of questions should we ask?

Field Building, Vocabulary Building and Targeted Questioning

Using a mathematical focus with this text, or other picture books like it, doesn't mean we have to change our approach completely. We are still reading for pleasure. We still want to connect with children's knowledge, interests and fascinations. We want them to wonder and ask questions.

To this end, we have found the connected processes of field building, vocabulary building and targeted questioning helpful with children of all ages:

FIELD BUILDING: To orientate children to a text, we prompt or ask a range of questions, depending on the ages of the children. They include:

Tell me about this book.
What do you notice? What can you see?
Who or what can you see on the front cover?
The title of this book is ___ . Why do you think the author call it that?
What do you think this book is about? Why?
Has anyone noticed...?
Is this the same as... or different to...?

Such noticing, exploring and talking opportunities build the field in relation to the story. They promote connection, logical thinking and exploration of children's everyday experiences, including mathematical ones. Sometimes what children notice will need to be guided, depending on the mathematics you want to explore as a result of this reading.

VOCABULARY BUILDING: Within this field building we want to promote the use of mathematical language, connecting words and ideas children know already to those that may be mentioned in the text or illustrations. This language scaffold will help children describe, explain, and reason about their mathematical ideas and strategies.

You may even ask children to focus on particular mathematical words in the text and let you know when they hear them. These could be recorded so that you can refer to them again. This would all be part of your planning.

TARGETED QUESTIONING: Asking a variety of questions during or after reading of the text will assist children to make sense of the story and provide insight into their understanding. It is good to ask both concrete and abstract questions. Concrete questions focus on literal comprehension. They require careful observation of the what children can see on the pages of the picture book . Abstract questions require children to consider beyond what is directly observable, using their background knowledge, experiences and thinking skills to infer information (https://prek-math-te.stanford.edu/overview/how-use-picture-books).

Concrete questions generally ask children to recall, label and describe.

Abstract questions generally ask children to predict, infer, explain and reason.

Putting it into practice: Amy and Louis

Let's return now to Amy and Louis by Libby Gleeson and the literary themes friendship, distance, change, and communication.

We can apply many of the FIELD BUILDING questions to the front cover of Amy and Louis. What is observable is the focus on the children and their closeness. They are leaning into each other and sharing like good friends do. Even the title suggests it is just about them. What we learn in the story is that this closeness is challenged as Amy moves "half a world away".

Reading the book with our mathematical lenses on, we can write down possible mathematical concepts for exploration and focus. In this case they could be:

Measurement: length, time
Spatial reasoning: position, 3D

To plan our mathematical VOCABULARY BUILDING focus, we can write down words and phrases worth highlighting.

If we focus on the concept of length, these would include:

as high as the sky
deep enough to bury bears
as wide as he could
tall buildings
far away
close
a long, long way away
too far away

If we consider words that address position, we hear:

across the yard
across the street
across the room
across the fence
through the gap
past the edge of the town
across the ocean
other side of the world.

A far more complex concept related to the distance and position (other side of the world) is time, i.e.

she is awake in the day, you are asleep at night.

TARGETED QUESTIONING could be developed with many of the mathematical words and phrases mentioned above in mind. They could also be designed by focusing on the illustrations and the many representations of boxes or buildings which are 3D objects. In the sandpit at the beginning of the text, Amy has built a castle with tall towers using a range of tools (bucket, truck, saucepan). When dressing up, Louis uses empty tissue boxes for his shoes and other boxes for his costume. There are many representations of "boxes" or rectangular prisms as houses in the streets and cities. Amy's sadness about her distance from Louie is represented by an empty packing box.

Considering all the above, we see many opportunities to connect with the mathematical language and concepts referred to in Amy and Louis. Building on this reading, we could plan rich investigative tasks that provoke childrens' mathematical thinking, providing opportunities for them to use the associated language and work with others to solve problems related to length, position, 3D, and/or time. Rarely when we are solving problems, can we limit thinking to just one!

These investigations may include the following challenges:

Can you build a tower that is "as high as the sky"?
Can you build a tower that is as tall as you?
- Who has the tallest tower? How could we work that out?
Tell the person next to you what you can you see across the room/garden/road? What can you see on the other side of ____ ?
If we lie on our back, what can we see?
- Let's stand up and look down, now what can you see? How is it the same or different?
Where would you be standing if you were close to your partner? Where could you be standing if you were far away?
- How many steps away from each other is "far away"?
- Can we stand a "long way" from each other? How far would that be?
Can you design a castle using playdo/sand with tall towers, and a fence with gaps to sneak through like Amy and Louis?

We are not forgetting here the call "coo-ee Lou-ee" and "coo-ee Am-ee" that the children use as a strategy for solving the problem of distance. This context could be an opportunity to discuss scientific and mathematical concepts about sound and how far it can travel.

Explicitly referenced, embedded and perceived mathematics

To frame our considerations of picture books and their potential for mathematical thinking, we can consider these three categories: explicitly referenced, embedded and perceived (Marston, 2010).

Books where the mathematics is most obvious is what we would categorise as explicitly referenced. These are often related to rote counting, counting collections, and 1:1 correspondence like this Roald Dahl book 1, 2, 3. These picture books have their place in early mathematical knowledge and skills. They can help invite children into the exciting world of counting. They may or may not have a 'story'.

Such books are also often associated with shape recognition skills like this My First Shapes book by DK Publications. They give names to the shapes children see and wonder about. It is worth saying here that care needs to be taken in terms of naming shapes that are only presented in one version and one orientation. For example, if a child saw the red triangle on the front of the My First Shapes book and only ever saw that version of a triangle (equilateral), they would not associate other three-sided 2D shapes with 'triangle'.

Embedded content refers to mathematical concepts that are embedded in the story of a picture book. The main purpose of the story, however, is to entertain. Amy and Loius by Libby Gleeson would fit into this category.

Perceived content in picture books is mainly written to entertain. Sometimes an incidental or unintentional mathematical concept is addressed in the story.

Each category of picture book provides and opportunity for exploring mathematics.

Room on the Broom by Julia Donaldson

We have chosen Julia Donaldson's Room on the Broom as our follow up example.

We have decided to read it to you because we love it so much! It is playful, has a lovely musical lilt, and beautiful illustrations by Axel Scheffler.

We would consider it a picture book with embedded mathematical content.

When you have listened to Room on the Broom, try the strategies we have mentioned in this article to provoke mathematical thinking, i.e:

Field building - what questions and prompts will you design to orientate the children and generate curiosity?
Vocabulary building - are there words and phrases that collectively lend themselves to explorations of mathematics concepts? Make some lists.
Targeted questioning: design some concrete and abstract questions to focus the children during and after the reading of the text
Using all of the above preparation, intentionally plan an open-ended investigation with potential to deepen the childrens' mathematical knowledge and problem solving strategies.

References:

Donaldson, J. (2001). Room on the Broom. Koala Books.

Gleeson, L. (2006). Amy and Louis. Scholastic.

Marston, J. (2010). Developing a framework for the selection of picture books to promote early mathematical development. In L. Sparrow, B. Kissane, & C. Hurst (Eds.), Proceedings of the 33rd Annual Conference of the Mathematics Education Research Group of Australasia (pp. 383-390). Mathematics Education Research Group of Australasia.