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8APMC 23(2) 2018

Narrative-first approach:

Teaching mathematics through

picture story books

Toby Russo

Bell Primary School, Vic.

<russo.toby.t@edumail.vic.gov.au

James Russo

Monash University, Vic.

<james.russo@monash.edu>

e benets of using children’s storybooks to support

mathematics instruction in primary schools is well

established. Muir et al. (2017) note that children’s

literature can be used to engage students in the topic or

lesson, contextualise important mathematical concepts

and promote mathematical reasoning. Typically, how-

ever, attempts to use children’s storybooks begin with

identifying a relevant aspect of the curriculum, and then

selecting books that connect to the mathematical area

in focus. Sometimes these children’s books have been

purposefully written to explore particular mathematical

ideas (e.g., e Greedy Triangle, by Marilyn Burns).

Other times the mathematical idea is explored explicitly

in the book, even though the book itself may have been

written for a broader purpose (e.g., e Doorbell Rang,

by Pat Hutchins). ere are many examples of teacher

educators and classroom teachers publishing articles

in journals such as Teaching Children Mathematics and

Australian Primary Mathematics Classroom sharing

lessons and units of work in which they have harnessed

children’s storybooks that possess these explicit links

to specic mathematical concepts (e.g., Malinsky &

McJunkin, 2008; Padula, 2004; Taber & Canonica,

2008).

An alternative means of exploiting the benets of

children’s storybooks when teaching primary mathemat-

ics is to be led by the story, rather than by the curricu-

lum. We refer to this as a narrative-rst approach, and

contrast it with a curriculum-rst approach (Russo &

Russo, 2017e). e narrative-rst approach involves

rst identifying rich narratives, for example, favourite

picture storybooks or novels, mapping out the key

components of the story (e.g., the characters, the plot),

and then developing rich problem-solving tasks that

connect to these key components. Curriculum links

are then made retrospectively.

We believe the benets of the narrative-rst approach

for supporting the integration of mathematics with

Figure 1. The four pillars of a narrative-first approach. Figure 2. Illustrating the narrative-first approach—examples of titles.

The four pillars of student engagement, teacher engagement, breadth of mathematics and

depth of mathematics are used to explain the benefits of a

narrative-first approach

for

supporting the integration of mathematics and children’s literature.

Narrative-first approach: Teaching mathematics through picture story books

9APMC 23(2) 2018

children’s literature are considerable. In addition to the

aforementioned benets which apply more generally to

using children’s literature to support teaching mathemat-

ics (e.g., contextualizing mathematical ideas, engaging

students), we believe this narrative-rst approach has

some additional advantages, that can be framed around

four ‘pillars’ (see Figure 1).

e narrative-rst approach can be eective for eng-

aging students and teachers, and for dierentiating math-

ematical experiences (i.e., teaching for depth), whilst

simultaneously covering a range of important mathemat-

ical concepts (i.e., teaching for breadth).e remainder

of the paper will provide examples of problem-solving

activities, that have all been delivered in a primary school

classroom, from four diverse children’s storybooks. For

emphasis, each example is used to illustrate one of the

four pillars of the narrative-rst approach (see Figure 2).

However, we would argue that any one of these examples

is built around all four of these pillars.

Student engagement: Fish Out of Water

Student engagement is paramount to the narrative-rst

approach. rough using the story as a ‘hook’, students

are highly motivated to engage with the associated

mathematical tasks. In addition, the rich ctional world

created by the narrative serves as an authentic imagina-

tive space through which students can engage in mean-

ingful and relevant mathematics. In our view, for mathe-

matics to be authentic, problems do not have to be built

around a real-world context, so as long as the ctional

world created by the author is ‘alive’ for the students.

e text Fish Out of Water forms the basis of an

investigation using the narrative-rst approach that

was undertaken in a Year 5/6 class, with a high level of

student engagement in the task (Russo & Russo, 2017e).

Fish Out of Water investigation

At the beginning of the story, the little sh, Otto, was only 5cm long but by the end of the story, Otto was

so big he was just over the length of a 50m Olympic size swimming pool! If Otto doubled in length every

ten minutes, how long did it take Otto to grow to this size?

Extension

If Mr Carp had not have dived into the pool and saved the day, Otto would have kept on growing and

growing. How long would it have taken Otto to grow so big he wouldn’t t into Albert Park Lake? How

long until he would have outgrown the length of the Yarra river? How long until he would have outgrown

the Pacic Ocean? How long until he would have outgrown the earth? How long until Otto was so long,

he would have stretched all the way to the moon? What about the sun? Pluto? Alpha Centauri?

e known universe?

e text tells the story of a boy who buys a pet

goldsh and, disobeying the instructions of Mr. Carp

the pet shop owner, overfeeds his new pet. e sh

grows and grows until it needs to be transported into

a swimming pool. When Mr. Carp nally returns he

saves the day and turns the sh back to normal.

To begin, the text was read to a class, who were then

asked to consider a mathematical problem: the Fish

Out of Water investigation below.

Following the lesson, students were invited to

complete a reection exploring both what they enjoyed

about the activity and their own learning. As we have

discussed in a previous paper (Russo & Russo, 2017e),

students responded very positively to the lesson. Whilst

some students emphasised the benets of the narrative

‘hook’ for engaging them in the corresponding prob-

lem-solving task, other students valued the narrative

because it supported them to make sense of exponen-

tial growth as a concept. Two exemplar quotes from

students are provided below:

I thought it was a really good idea starting a

maths lesson with a story because it opened up

my mind in a way and made me want to know

what this had to do with the lesson. So I was

feeling intrigued and sort of open. e lesson

was really fun and interesting as well.

It was fun to read the story at the start and then

go into maths because it helped explain some

things… it helped me make sense of it and

understand it better.

Student reections on their learning emphasized

two major themes: a developing appreciation for the

power of exponential growth and understanding the

Russo & Russo

10 APMC 23(2) 2018

relationship between dierent units of measurement.

Again, two illustrative student quotes are provided:

I learnt about how doubling again and again can

makes things big in such a short amount

of time.

I learnt about exponential growth which, I think,

is when something gets bigger and the size dou-

bles each time… and I got to practice converting

m-cm-km.

Teacher engagement:

Where the Wild Things Are

Many teachers feel restricted covering curriculum content

and this can be an impediment to developing engaging

tasks. We believe the narrative-rst approach allows for

greater exibility and creativity when planning maths

activities as teachers can primarily focus on developing

rich, authentic problem-solving tasks. is allows for

the focus to be on the four prociencies (understanding,

uency, problem solving and reasoning) when planning

tasks, rather than simply the content strands.

e narrative-rst approach also facilitates teacher

engagement as teachers can choose texts that they

personally enjoy and are excited to share with students.

is can be further enhanced by a teacher’s personal

connection to a text, which can motivate the develop-

ment of the associated task. For example, one of the

rst author’s favourite books as a child was Maurice

Sendak’s Where the Wild ings Are. is classic picture

storybook begins with the protagonist Max being sent

to bed without supper. His room transforms into a

mysterious world and he travels to the Land of the

Wild ings. Here he becomes king of many strange

creatures, only to travel back to his room where his

supper is waiting for him, still hot.

Part of the author’s fascination with this text was

the way in which so much time seemed to pass on Max’s

adventures, only for him to return home apparently

minutes later. is narrative opened up the concept of

alternative realities and the idea the time can be relative;

ideas that continue to fascinate into adulthood. e

author’s excitement has been harnessed and translated

into a mathematical investigation exploring these

concepts, which was undertaken with a Year 5/6 class

(Russo, 2018).

After reading the book, the class explored the sections

of text that specically state how much time Max spent

travelling to and from the Land of the Wild ings.

is could be done as a class, leading to the investiga-

tion outlined below, or as a stand-alone mathematical

investigation.

And an ocean tumbled by with a private

boat for Max

and he sailed o through night and day

and in and out of weeks

and almost over a year

to where the wild things are.

e class concluded that it took 365 days (‘almost

over a year’) to get to the Land of e Wild ings.

But Max stepped into his private boat

and waved goodbye

and sailed back over a year

and in and out of weeks

and through a day.

e class concluded that this period was equal to one

year plus two weeks plus one day, plus ve days staying

at the Land of e Wild ings or 385 days. is makes

a total of 750 days (365 + 385) that had passed in the

Land of the Wild ings.

Students then considered “How much time do you

think passed at home?”, looking at this section of the

text:

Where he found his supper waiting for him

and it was still hot.

Student A: Well his dinner was still hot, so it couldn’t

have been all that long.

Student B: I reckon about ve minutes, no longer

than that.

It was agreed that ve minutes had passed in the ‘real

world’.

Where the Wild Things Are investigation

Using this information, how much time passes in

the Land of the Wild ings, compared with one

minute in the real world?

Extension

Max went to sleep at 9pm, after eating his supper,

and woke up at 6.30am. He then went straight

back to the Land of the Wild ings. How much

time has passed there?

e next time Max returned it was four years

later, at age 12. How much time had passed now?

Narrative-first approach: Teaching mathematics through picture story books

11APMC 23(2) 2018

Breadth of mathematics:

The Cat in the Hat Comes Back

In our experience, creating rich applied tasks and

investigations using the narrative-rst approach results

in numerous and various links to relevant curriculum

documents. After reading the book e Cat in the Hat

Comes Back to his Year 5/6 class, the rst author had

students engage with an investigation that involved a

range of mathematical ideas. is story is about the

return of the Cat in the Hat to the home of Sally and

her brother, who proceeds to make another horrendous

mess and then pulls out a series of small cats (Little

Cats A to Z) to help him clean up.

The Cat In the Hat Comes Back investigation

e Cat in the Hat is 1.6m tall and his hat is

another 65cm high.

Little Cat A is 1

5

the size of the Cat in the Hat.

Little Cat B is

1

2

the size of Little Cat A.

Little Cat C is

1

2

the size of Little Cat B,

and on it goes.

How tall is Cat C? What about Cat E?

Can you think of things in real life the same size

as these cats?

Extension

What is the size of Cat H?

Can you even see Cat Z?!

Are there any things in real life the same size as

these cats?

Do you think all these little cats would really t

inside the Cat in the Hat’s hat?

e curriculum links for this task are somewhat

dependent on the strategies used by individual students.

However all students recognised the initial need to

determine the size of Little Cat A and, in order to do

this eciently, their intuition was to convert 1.6 metres

to 160 centimetres to make the problem more workable

(VCMMG223; VCAA, 2017). Students were then re-

quired to explore fractions of an amount(VCMNA213;

VCAA, 2017), i.e., “If e Cat in e Hat is 160cm and

Little Cat A is one fth his size, then what is one fth

of 160?” “If Little Cat B is half the size of little Cat A,

what is one half of 32?”

As students worked through the extension, they

needed to create a sequence involving fractions or

decimals based on continuous halving (VCMNA219;

VCAA, 2017). Some students used multiplication of

decimals as a strategy for solving this problem, which is

within Level 7 of the curriculum (VCMNA244; VCAA,

2017), i.e., If Little Cat G is 0.5cm, then Little Cat F

is 0.5cm × 0.5cm. At this point, many students elected

to use digital technologies (i.e., calculators) for the

required division problems (VCMNA209; VCAA, 2017)

and some even had a go at using an algebraic expression

to nd a more ecient solution, which is within Level 7

or beyond (VCMNA252; VCAA, 2017). In order to solve

the ultimate question, students were required to add a

series of decimals (VCMNA214; VCAA, 2017) and then

compare this answer with the original size of the hat

(VCMMG224; VCAA, 2017).

Table 1 outlines the breadth of curriculum links

connected to this rich task when taught in a 5/6 class-

room. For the purpose of this table, all content descrip-

tors are from the Victorian Curriculum–Level 6, although

it should be noted that many connections could also

be made to the Level 5 and Level 7 curriculum.

Figure 3. Content descriptors from the Victorian Curriculum linked to The Cat in the Hat Comes Back investigation (VCAA, 2017).

Number and algebra (Level 6) Measurement and geometry

(Level 6)

Number and

place value Fractions and decimals Patterns and

algebra Using units of measurement

Select and apply ecient

mental and written

strategies and appropriate

digital technologies to

solve problems involving

all four operations with

whole numbers and

make estimates for these

computations.

(VCMNA209)

Find a simple

fraction of a

quantity where

the result is a

whole number,

with and

without digital

technologies.

(VCMNA213)

Add and subtract

decimals, with

and without dig-

ital technologies,

and use estima-

tion and round-

ing to check the

reasonableness

of answers.

(VCMNA214)

Continue and

create sequences

involving whole

numbers, fractions

and decimals.

Describe the rules

used to create the

sequence.

(VCMNA219)

Convert between

common metric

units of length,

mass and

capacity.

(VCMMG223)

Solve problems

involving the

comparison of

lengths and areas

using appropri-

ate units.

(VCMMG224)

Russo & Russo

12 APMC 23(2) 2018

Figure 4. A student represented her thinking using Unifix

and associated number sentences.

Figure 5. Another student used a grid to represent the

house, and the initial of the animal to represent how much

space it is occupying.

Figure 6. A Year 5/6 student systematically identifies all

whole-number solutions to the Squash and a Squeeze

investigation.

All Year 5/6 students were able to solve this part of the

problem without manipulatives. Furthermore, some

students understood that by using decimals there would

be many more solutions, with some proposing innite

possibilities (see Figures 6 and 7).

Depth of mathematics: A Squash and

a Squeeze

In general, when developing a problem-solving task

using the narrative-rst approach, we aim to ensure

that the task is suitable for a variety of ability levels and

grade levels. e following investigation was developed

around the text A Squash and a Squeeze. is is a story

about a little old lady who feels her house is too small,

so a wise old man advises her to bring her animals into

the house, which eventually results in the lady realising

her house is relatively big once the animals are removed.

e investigation was taught with both a Year 2 class

and a Year 5/6 class. Whilst all Year 2 students attempt-

ed the initial investigation, around half the students

subsequently attempted the extension (Extension A).

By contrast, all Year 5/6 students attempted both the

initial investigation and extension (Extension A), with

around three-quarters of the class also engaging with

the additional extension (Extension B).

Initial investigation: A Squash and a Squeeze

investigation

e little old lady’s cottage was exactly 60 square

metres in area. Sharing her house with a hen, goat,

pig and cow did not leave a lot of space for the old

lady! To live comfortably:

• A hen needs exactly 5 square metres of space;

• A pig needs exactly 15 square metres of space;

• A goat needs at least 10 square metres of space,

but no more than 15 square metres;

• A cow needs at least 18 square metres of space,

but no more than 25 square metres.

When she was living with all four animals, how

much space might have been left for the old lady?

Show as many dierent possibilities as you can.

What is the least amount of space the old lady

would have to herself? What is the most amount

of space?

e initial investigation focuses on additive thinking,

in particular the part-whole idea, which is a central

concept in early primary school mathematics. Reect-

ing their developmental phase, many Year 2 students

attempted to represent their thinking using either

concrete materials, such as Unix (see Figure 4) or the

square grid on the back of a 100s chart to represent the

area of the house (see Figure 5).

Narrative-first approach: Teaching mathematics through picture story books

13APMC 23(2) 2018

Extension A: More part-whole and opportunities

for multiplicative thinking

e average Australian house is 240 square metres

in area. Given the above requirements for space,

what are some dierent combinations of hens,

goats, pigs and cows that could live comfortably

in the average Australian house?

e extended investigation provides further opportu-

nities to explore the part-whole idea (with larger num-

bers), whilst also providing opportunities for multiplica-

tive thinking (e.g., students can ‘scale up’ their answers

from the initial investigation). It was common for Year

2 students who attempted the extended investigation

to build on their grid representation employed for the

initial problem (see Figure 8).

Figure 7. This student has recognised that there are “infinite

possibilities between 12 and 0” due to possible decimal amounts.

Extension B: Contrasting additive thinking with

proportional reasoning

After doing the initial investigation and Extension

A, Josh and Jill were discussing the Squash and the

Squeeze problem.

Josh: Why didn’t the little old lady just buy herself

a bigger house to begin with? If the average

Australian house is 240 square metres, she

would have had a lot more space to herself.

Jill: I actually think she is better o having lived

with the animals, and then getting rid of

them, than just buying a bigger house to

begin with. Her little house without the

animals feels bigger than moving to a bigger

house would have felt at the beginning of

the story.

Josh: But buying a bigger house would have given

her way more extra space! Instead of having

60 square metres, should would have had

240 square metres all to herself!.

Jill: But her little house still felt so much bigger!

With all the animals living there, she had no

more than 12 square metres of space. When

all the animals moved out, she had 60 square

metres all to herself! is change would feel

even bigger than if she had of just bought

a bigger house to begin with!

Who do you agree with, Josh or Jill? Can both

of them be right? Can you use mathematics to

prove that Josh’s reasoning is correct? Can you use

mathematics to prove that Jill’s reasoning is correct?

Some Year 5/6 students were limited to additive think-

ing when tackling this problem and therefore determined

that “Josh is correct” because a dierence of 180m2 is

larger than a dierence of 48m2. Others made the com-

parison between additive and multiplicative thinking,

although some of these students required

additional prompting questions from the

teacher to get to this point. Several Year

5/6 students used proportional reason-

ing to conclude that both Josh and Jill

could be correct, depending on how you

interpreted the problem (see Figure 9).

Figure 8. A student built on a grid representation to attempt the extended investigation.

e focus of Extension B is to invite students to

contrast additive thinking with multiplicative thinking

and, in particular, apply proportional reasoning.

Russo & Russo

14 APMC 23(2) 2018

Figure 9. A student demonstrated both multiplicative (top right)

and additive reasoning (bottom left) and in discussion compared

these to determine both Josh and Jill could be correct.

Concluding thoughts

It is common for children’s literature to be used in

connection with mathematical learning, but often

the maths is supercially linked to the text or a text is

chosen for its mathematical focus. is article outlined

a narrative-rst approach to lesson planning, whereby

key ideas, themes, and characters from well-known

children’s stories are reconstructed through a mathe-

matical lens.

We have attempted to demonstrate how the

narrative-rst approach can simultaneously engage-

teachers and students and energise the mathematics

classroom, whilst allowing a range of mathematical

skills and concepts to be covered across a variety of

ability levels. For other examples of our attempts to

employ this approach, see Russo & Russo (2017a,

2017b, 2017c, 2017d., 2018). If you’d like to nd

out more about this approach or any of the example

lessons, please feel free to email the authors.

References

Malinsky, M. A., & McJunkin, M. (2008). Wondrous tales of

measurement. Teaching Children Mathematics, 14(7), 410–413.

Muir, T., Livy, S., Bragg, L., Clark, J., Wells, J., & Attard, C. (2017).

Engaging with mathematics through picture books. Albert Park,

Australia: Teaching Solutions.

Padula, J. (2004). e role of mathematical ction in the learning of

mathematics in primary school. Australian Primary Mathematics

Classroom, 9(2), 8–14.

Russo, J., & Russo, T. (2017a). Harry Potter-inspired mathematics.

Teaching Children Mathematics, 24(1), 18–19.

Russo, J., & Russo, T. (2017b). Math and Mr. Men. Teaching Children

Mathematics, 24(2), 82–83.

Russo, J., & Russo, T. (2017c). One sh, two sh, red sh, blue sh.

Teaching Children Mathematics, 23(6), 338–339.

Russo, J., & Russo, T. (2017d). Problem solving with the Sneetches.

Teaching Children Mathematics, 23(5), 282–283.

Russo, J., & Russo, T. (2017e). Using rich narratives to engage students

in mathematics: A narrative-rst approach. In R. Seah, M. Horne,

J. Ocean, & C. Orellana (Ed.), Proceedings of the 54th Annual

Conference of the Mathematics Association of Victoria (pp. 78–84).

Melbourne, Australia: MAV.

Russo, T., & Russo, J. (2018). e narrative-rst approach: Room

on the broom investigation. Prime Number, 33(2), 10–11.

Russo, T. (2018). Challenging task: Where the wild things are.

Prime Number, 33(1), 16–19.

Taber, S. B., & Canonica, M. (2008). Sharing. Teaching Children

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Victorian Curriculum and Assessment Authority (VCAA) (2017).

e Victorian curriculum F-10: Mathematics.