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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
Mathematics, 15(1), 55–61.
Victorian Curriculum and Assessment Authority (VCAA) (2017).
e Victorian curriculum F-10: Mathematics.
