Chapter 6: Mathematics in a Conceptual PlayWorld

Leigh Disney

Chapter goals

By reading and exploring the content of this chapter, you will learn:

  • how to teach mathematics in play from within the imaginary play of children
  • how to plan a Conceptual PlayWorld for mathematics
  • how to pedagogical position yourself within the imaginary situation

Introduction – Jumping into the imaginary situation

Within this chapter, we will continue to follow our preschool teachers, Charlotte, the student teacher, and Yuwen, an experienced teacher. This chapter will focus on embedding mathematics concept learning into children’s imaginative play using Fleer’s Conceptual PlayWorld model. The chapter will present a case study of Charlotte and Yuwen working collaboratively with their preschool children to solve a mathematical measurement-related problem.

However, before returning to our protagonists, it’s important to consider how we think about teaching mathematics to young children, a topic that Charlotte has given much thought to. When Charlotte reflects on her experience of learning mathematics, she thinks about the drill-and-practice style from her own primary school experience. Some of our earliest memories of mathematics reflect countless hours spent completing worksheets or working on pre-designed worded problems. Charlotte worries that such practices will not fit within a play-based program as delivered within Yuwen’s program.


Practice reflection 6.1: What are your earliest memories of learning mathematics? Like Charlotte, do you recall endless worksheets or a range of mathematics manipulatives (i.e. counting blocks, shapes, etc.)? Or, do you recall mathematics embedded in your play during everyday experiences (i.e. making biscuits with your parents, measuring quantities and making shapes)? Would you suggest that these experiences were playful? Do you think mathematics concepts can be intentionally taught during play? Reflect on your memories and thoughts in your Fleer’s Conceptual PlayWorld thinking book.

Supporting mathematics learning in the teaching program

During planning time, Yuwen and Charlotte discuss the following week’s teaching program. Charlotte notices that the mathematical focus is on the concept of measurement. Yuwen has already embedded several mathematical experiences into her program, mainly focusing on length measurement. This gives Charlotte an idea:


Charlotte: Yuwen, can we teach mathematical concepts using the Conceptual PlayWorld model?

Yuwen: Absolutely! What did you have in mind?

Charlotte: Well, I can see that in your teaching program, you focus on length measurement.

Yuwen: Do you remember the 5 characteristics of the Conceptual PlayWorld? Do you still have a copy of the Conceptual PlayWorld planning proforma?

Charlotte: I sure do! And I have a great story in mind!


Like other conceptual areas, when planning a Conceptual PlayWorld around mathematics, the chosen story must have a complex plot and a dramatic dimension. There are multiple mathematics-related children’s books specifically related to mathematical concepts, such as The Doorbell Rang by Pat Hutchins, which uses the notion of sharing to explain an equal distribution. Another is One is a Snail, Ten is a Crab by April Pulley Sayre and Jeff Sayre, a book focusing on number sense and skip counting. While great books for teaching children about specific mathematics concepts, stories like this do not always contain complex plots or a range of characters that lend to the multi-layered problem-solving scenarios inherent to Fleer’s Conceptual PlayWorld. Therefore, choose your story carefully.


Research reading 6.1: See Chapter 1 for detailed information about the 5 characteristics of Fleer’s Conceptual PlayWorld model, and Appendix B for an example of a planning proforma designed for infant and toddler children.



Practice reflection 6.2: What other books can you think of that may have inherent mathematical connections yet involve complex plots and a dramatic dimension? Record your ideas in your Fleer’s Conceptual PlayWorld thinking book.



Charlotte proposes her choice of story:

Charlotte: The book I think would be great is Room on the Broom by Julia Donaldson. As we get closer to Halloween, I have heard the children talking about witches and fairies and know they are very interested in this imaginative play. Plus, when I think about the book’s title, I think a link could be made to length measurement. You know, when we think about ‘room’, because, in the story, the broom needs to be fixed after getting broken. This got me thinking: how long would a broom need to be to fit everyone?

Yuwen: Great! I like many of Julia Donaldson’s books and think your choice will work well. What I also appreciate is that you have considered the children’s interests. I’ve also noticed the children getting excited about Halloween. This shows that you have thought about the first fundamental characteristic of the Conceptual PlayWorld.


Charlotte and Yuwen then spend time planning the Conceptual PlayWorld, considering each characteristic and how the concept of measurement can be learned within the meaningful conditions they intended to create. When designing the space, the teachers decided that the children would need room to move and manipulatives to work with. Adjacent to the main teaching space, the preschool had a large area where they decided to strategically place large items such as blocks (Figure 6.1), sticks, and milk crates so that when the children were deciding how to make a broom, they would have resources to choose from. The teachers also created badges of frogs, dogs, cats, and witches so the children would be helped to identify with their character roles within the Room on the Broom story.


Figure 6.1.

A range of large blocks used for gross motor manipulation.


In a classroom the are large wooden blocks stacked together to create a wall with a gateway.
Note. Mathematical manipulatives can come in a range of sizes, shapes, and quantities. Image provided by the author.



Practice reflection 6.1: What are your earliest memories of learning mathematics? Like Charlotte, do you recall endless worksheets or a range of mathematics manipulatives (i.e. counting blocks, shapes, etc.)? Or, do you recall mathematics embedded in your play during everyday experiences (i.e. making biscuits with your parents, measuring quantities, and making shapes)? Would you suggest that these experiences were playful? Do you think mathematics concepts can be intentionally taught during play? Reflect on your memories and thoughts in your Fleer’s Conceptual PlayWorld thinking book.

Practice reflection 6.3: When you are next at an early learning centre, what manipulatives (large or small) do you have that could be used for mathematics learning? Make a list in your Fleer’s Conceptual PlayWorld thinking book of what you have, which may transform within the Conceptual PlayWorld setting to be used in new and interesting ways.



Between the main teaching space and the large area was a door. Yuwen found a witch’s hat in the storeroom and stuck it to the door to act as the entry and exit to the Conceptual PlayWorld space (see Figure 6.2).


Figure 6.2.

A transition device from the regular classroom, into the Conceptual PlayWorld space.


A witches hat is stuck on a classroom door.
Note. Image provided by the author.

Situating mathematics into the storyline

The children each chose a character role before entering the Conceptual PlayWorld space for the first time. Charlotte decided to play the role of ‘Itchy Witchy’, and as the children entered the Conceptual PlayWorld space, she would say a magic chant and wave a magic wand to support them as they entered through the door with the witch’s hat (Figure 6.3). Yuwen decided to take on the character role of a dog, calling herself ‘Detective Dog’ because she was always carrying around a notepad and pen (Figure 6.3). Yuwen did this during the Conceptual PlayWorld so she could first, stay in a character role and second, have easy access to documentation materials to note what the children said and did during the Conceptual PlayWorld. In this way, Yuwen was engaged in authentic documentation practices.


Figure 6.3.

The teachers in their character roles.

Yuwen is wearing a blue cowboy hat and writes something on a clipboard. Charlotte stands next to her wearing a witches hat and holding a wand.



Research reading 6.2: See Chapter 5 for insights into assessment practices capturing learning within the Conceptual PlayWorld.


After reading the story of Room on the Broom to the children, the teachers were keen to discover what aspect of the story the children were interested in and whether they could identify any problems that needed to be fixed. Much to the teacher’s delight, the children immediately identified that the broom needed to be fixed, and thus, the conceptual problem was formed: How long do we need to make the broom when we fix it?

To help dramatise the conceptual problem, the teachers crafted a letter, which they said was from ‘Ritchy Witchy’, the sister of Itchy Witchy (played by Charlotte). The letter (Figure 6.4) invited Itchy Witchy and all her animal friends (played by the children) to come to Ritchy Witchy’s house for a party, but they needed to come together on one broom.


Figure 6.4.

Helping to dramatise the conceptual problem: a letter from the witch’s sister, Ritchy Witchy.


The letter the teachers wrote. It has a picture of a cobweb, dog on a flying broom and bats across the top. On the bottom there are pictures of ghosts, a skull, and a cauldron.
Letter transcript: Dear Itchy Witchy. I would like to invite you and your friends to our favourite holiday of the whole year. Please come on your broom, I hope it is long enough to fit you and all of your friends. Also, please bring Detective Dog as there will be some Detective jobs for her to do. Love Ritchy Witchy. Image provided by the author.


The letter created the dramatised problem of creating a broom that could fit all the characters. We now pick up the Conceptual PlayWorld between the teachers and children as they collectively solve the problem, and we will explain how the teachers pedagogically positioned themselves to support the children during the Conceptual PlayWorld experience (see Chapter 5 for more information on pedagogical positioning).

Considering your pedagogical positioning to best support children’s mathematical discovery

As the children worked together, they used the large building blocks (see Figure 6.1) to create a new broom, where each block was a seat for a character. As this was a collective experience, the children were never in the primordial-we position, as all the children were building within the storyline of the Conceptual PlayWorld and understood their roles. As the children worked, Charlotte interacted with them, asking them simple questions and getting into the character role of a witch. During this time, both teachers were in the under position and the children in the above. In this way, the children and teachers made an emotional connection with their characters, which led to the children’s motivation to eventually solve the conceptual problem of making the broom long enough to fit all their friends.

While the children were certainly building a structure using the large blocks that was straight and resembling a new broom, many were more interested in their own character roles rather than solving the mathematical problem. For example, the frogs were having a conversation about needing water on their delicate skin for the long journey; hence, they needed a tap attached to their seat. At that moment, Charlotte feared that the mathematics concept of learning about length measurement was being lost, but then, a breakthrough occurred (Figure 6.5).


Figure 6.5.

Charlotte has a breakthrough in how to support children’s mathematics thinking – using multiple pedagogical positions.

Charlotte wears a witches hat and holds a wand. She is looking thoughtful.


Charlotte noticed how Yuwen was positioned, interacting with the children, seemingly noticing everything, and then asking a pointed question:

Yuwen: [Spotting 3 blocks joined together, asked the entire group] I wonder how many of our friends can fit on this section of the broom?

Charlotte: That’s such a good question Detective Dog, I’m not sure. Can anyone help figure this problem out?

Child A: [Walking over and pointing at each block] There are 1, 2, 3 blocks.

Charlotte: So, how many of our friends will fit on this section of the new broom?

Child B & C: [In unison] Three! Three friends!

Yuwen: So, how many blocks long does are new broom need to be? How many friends are here with us today?


In this scenario, the teachers had moved from being below the children to being equal with the children. They were now collectively problem-solving together, focusing on solving the mathematical problem originally presented in the letter from Ritchy Witchy, to make one broom that could fit everyone to fly to the birthday party. The children had to determine how many characters needed to fit onto the broom, which would let us know how long the broom needed to be. Yuwen (still in her character role as Detective Dog) asked the children to sit in a circle so that one of the children could easily count all of the friends in the group. At this stage, the teachers began moving into the above position. They were showing children strategies that they could use to help solve the conceptual problem.


Child A: [After moving around the circle and tapping each friend on the head, declared] 16, we have 16 friends.

Yuwen: 16; wow! That’s so many friends that need to fit onto the broom. So, if one block can only fit one friend, how many blocks do we need?

Multiple children: 16! 16 blocks!

Yuwen: Oh, I think I need to write this down in my Detective Dog book!

Charlotte: [Trying to capitalise on the teaching moment, in the above position, asked Child A] How do you know we need 16 blocks?

Child A: [Looking a little unsure of themselves] Everyone needs a seat…

Child B: [Stood up with a loud and enthusiastic voice] Because one block can only fit one friend!

Yuwen: According to my Detective Dog book, we have 16 friends. So, how many seats do we need on our broom, everybody?

Whole group: [Very loudly] 16!

Charlotte and Yuwen: That’s right!


The children then busily worked together to build a broom that was 16 blocks long. Once the children thought they were finished, the teachers asked them to count how long it was. The children then collectively counted the length of the broom. The first attempt was 14 blocks long. The children then added 3 more blocks. Oh no, too long! The children finally made a broom of 16 blocks, and the group took off on their new broom to Ritchy Witchy’s birthday party! This also acted as a transition into lunchtime, so the group exited the Conceptual PlayWorld setting.

In this way, the teachers had moved from the under (initially building an emotional connection to characters) to the equal (getting the children thinking mathematically) and finally to the above position (demonstrating mathematical problem-solving techniques) within the Conceptual PlayWorld experience. It is also important to note that the children were never in an independent position. This was because the teachers were inside the children’s play, fully aware of what the children were doing and part of the play scenario. Each pedagogical positioning allows the Conceptual PlayWorld to continue and grow, motivating children to solve mathematics problems through imaginative scenarios and their character roles.


Practice reflection 6.4: While the focus was on measurement, what other mathematics concept could you see happening in the Room on the Broom Conceptual PlayWorld?


Charlotte was so pleased with how the Conceptual PlayWorld had run, and while there were moments of worry, both Yuwen and Charlotte agreed that the Conceptual PlayWorld had met their teaching agenda and allowed for conceptual learning beyond their initial expectation.


In this chapter, Charlotte and Yuwen planned and implemented a Conceptual PlayWorld as part of their teaching program to specifically teach mathematics to children. By taking different pedagogical positions, both teachers could take different roles within the collective play to allow children to be mathematical problem solvers and teach specific mathematical concepts.


Practice reflection 6.5: Bring together the elements of your Fleer’s Conceptual PlayWorlds thinking book. Hopefully, not only can you help Charlotte on her journey, but your thinking will provide you with a springboard for embedding mathematics learning in Conceptual PlayWorlds. Carefully consider pedagogical positioning to engage children throughout the Conceptual PlayWorld to create an emotional connection to their character roles and, ultimately, solve the mathematical conceptual problem.


To find out more about Fleer’s Conceptual PlayWorld read the research on the Conceptual PlayLab website.



“Figure 6.1.” by Leigh Disney is licensed under CC BY-NC 4.0

“Figure 6.2.” by Leigh Disney is licensed under CC BY-NC 4.0

“Figure 6.3.” by Adriana Alvarez, based on an image by Anne Suryani, is licensed under CC BY-NC 4.0

“Figure 6.4.” by Leigh Disney is licensed under CC BY-NC 4.0

“Figure 6.5.” by Adriana Alvarez, based on an image by Anne Suryani, is licensed under CC BY-NC 4.0

About the author

Dr Leigh Disney’s research supports the Maths pillar of the Conceptual PlayLab. Leigh is a mixed methodological researcher who studies early years mathematical learning as well as the impact of digital technology within early years educational settings.


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Why play works Copyright © 2024 by Leigh Disney is licensed under a Creative Commons Attribution-NonCommercial 4.0 International License, except where otherwise noted.

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