Content uploaded by James Russo

Author content

All content in this area was uploaded by James Russo on Apr 27, 2020

Content may be subject to copyright.

20 APMC 25(1) 2020

James Russo

Monash University, Vic

<james.russo@monash.edu>

Toby Russo

Spensley Street Primary School, Vic.

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

Movies through a

mathematical lens

Introduction and overview

Providing students with rich tasks that are both engag-

ing and mathematically important is a vital aspect of

teaching primary mathematics eectively. One means

of supporting the development of rich tasks that meet

these criteria is to embed them in a narrative. Narratives

can aid in the development of mathematical under-

standing through a number of mechanisms, including

by engaging students emotionally, promoting mental

imagery, and creating a meaningful context for a com-

munity of learners to explore ideas (Averill, 2018).

We have argued previously that the “more familiar,

enjoyable and deeply developed the narrative, the more

engaging the task is for students” (Russo & Russo,

2017, p. 78). Consequently, in line with many other

authors, we have suggested that children’s literature

oers an eective narrative hook for engaging students

(e.g., Muir et al., 2017; Schiro, 1997). However, rather

than use texts which explicitly introduce mathematical

ideas, our emphasis has been on beginning with rich

narratives and mapping on the mathematics. Our con-

tention is that a mathematical lens oers one perspective

on a narrative, and that educators attempting to develop

tasks connected to picture story books should consider

being led by the narrative, rather than the curriculum.

We have termed this a “narrative-rst approach” (Russo

& Russo, 2018).

We have run several workshops where we have

encouraged teachers to choose a favourite picture story

book and ask the question: Can the central aspects

of the story (e.g., the plot, the characters) be viewed

through a mathematical lens? If the answer to this ques-

tion is yes, we have encouraged and supported teachers

to develop rich tasks that arise from this text (see Minas,

2018, for an example).

e purpose of the current paper is to demonstrate

how this perspective can be extended to audio-visual

media, such as clips from movies, short-lms and tel-

evision shows. We present our experience of teaching

mathematics in this manner on two occasions to two

classes of Year 5 students with a focus on proportional

reasoning. We provide a detailed account of one particu-

lar lesson based on a clip from the movie Despicable Me,

before moving on to discuss student reactions to

the lessons.

Movies and mathematics

Visualisation is an important cognitive process that

can be used to support mathematical problem solving

(NCTM, 2000). Indeed, eective teachers of mathemat-

ics and science frequently use visual representations to

support student conceptual understanding. However,

Bell et al. (2012) contend that relying on static rep-

resentations, such as photographs, illustrations and

diagrams, can be limiting because they struggle to

adequately capture dynamic constructs. ey suggest

that in a number of topic areas a particular concept is

better captured by dynamic representations. is line of

reasoning was pursued by Pierce, Stacey and Ball (2005)

when they suggested that specically constructed moving

images (e.g., of a person juggling) can be used to explore

mathematical concepts (e.g., quadratic functions).

Although developing videos with the explicit purpose

of exploring a mathematical idea certainly has value,

we might suggest that it is analogous to writing a picture

story book with an explicit mathematical focus, such as

probability. See, for example, the excellent picture story

book by Tracey Muir, Heads or Tails (2018). In these

instances, the emphasis is on co-opting an engaging

A great deal of research has looked at the use of photographs, illustrations, and diagrams to

support student understanding of mathematical concepts. In this paper, the authors explore

some of the advantages that dynamic representations have over their static counterparts as

they put movies under a mathematical lens.

Movies through a mathematical lens

21APMC 25(1) 2020

medium with educative value to illustrate a mathemat-

ical concept. Even when a meaningful and compelling

narrative has been developed (as is the case with Heads

or Tails), it is primarily in service to the mathematics.

e vast majority of ctional books and movies have of

course not been developed for the purpose of represent-

ing abstract mathematical ideas; however, it is clearly

still possible for important mathematical ideas to be

embedded within these narratives. ese mathematical

ideas can be made explicit when the narrative is

re-examined through a mathematical lens.

To illustrate, it is poignant to choose an example

from a related eld: physics. Video vignettes from

movies, television and commercial advertisements

have been used for decades to introduce and explore

concepts in the science classroom. To take one example,

Park and Lamb (1992) discuss how they used a scene

from Superman II involving the superhero rescuing a

boy falling from Niagara Falls to explore the notion of

acceleration due to gravity with their physics students.

One of the aspects they investigated was whether

Superman would actually have had sucient time to

transform from his alias Clark Kent into his costume

and rescue the boy before he hit the water, given what

we know about gravity and acceleration. Although this

scene from Superman II served as the narrative hook

for a scientic investigation, we would contend that

re-examining this aspect of the movie through the lens

of a physicist also served to enrich the narrative. Even

if students concluded after working on the task that

this scene from Superman II is “unrealistic” because

Superman had insucient time to react, the process of

considering super-heroism from a scientic perspective

might have paradoxically made the narrative “more real”

(or at least richer) for students. e students had been

invited to take Superman seriously.

e idea of using movies and television to contextu-

alise mathematical learning for students is a niche, yet

growing area of interest for researchers and practition-

ers. Niess and Walker (2010) argue that video can be

used in a variety of ways to support instruction in the

mathematics classroom, from introducing new con-

cepts, to exploring mathematics in real world contexts,

to providing students with images that can support

them in expressing their mathematical understandings.

Recently, there have even been several eorts to develop

and disseminate mathematical tasks and investigations

linked to well-known movies and television programs.

A notable example is Elana Reiser’s (2015) book

Teaching Mathematics Using Popular Culture. In this

text, the author provides a series of mathematics

activities for the secondary school classroom, built

around movies and television programs. Often the

activities included in her book focus on examples with

an explicit mathematical focus, for example using a

scene from e Da Vinci Code as an introduction to

cryptology. However, much like our approach, some

of the examples oered by Reiser use a movie’s narra-

tive as a context to explore mathematical ideas; such

as using a scene from Alice in Wonderland when Alice

drinks a shrinking potion to explore shrinking rates

using algebra and exponents.

In addition to Reiser’s book, a number of other

resources have been developed that use well-known

movies and television shows as a catalyst for math-

ematical investigations, such as from e Simpsons

(Greenwald & Nestler, 2004), and e Price is Right

(Butterworth & Coe, 2004). However, it is notable

that most of these resources are focussed on upper-

secondary or tertiary mathematics, rather than primary

mathematics. One of the purposes of sharing our

experience teaching these lessons to Year 5 students

is to demonstrate that this approach can be equally

powerful in a primary mathematics classroom.

The tasks and lessons

e two lessons were taught on consecutive days, and

both focussed on proportional reasoning. Each lesson

was built around a core challenging task, augmented

by enabling and extending prompts, and followed the

launch-explore-discuss structure (Sullivan et al., 2015).

Figure 1. Victor inspects the tiny toilet.

(Screen shot from

Dispicable Me

(Melendandri et al., 2010).)

We launched the rst lesson by asking students who

had seen the movie Despicable Me (Meledandri et al,.

2010). In both classes, almost all students indicated

that they had and responded enthusiastically. We asked

several students to briey describe the plot for the

benet of those who had not seen it. We then dis-

played a one minute clip from the movie; specically,

the scene where Victor gets hold of the Shrink Ray

22 APMC 25(1) 2020

Russo & Russo

and is playing with it in the bathroom. In this scene,

Victor carelessly shrinks several items in the bathroom,

including the toilet. e scene ends with Victor on his

knees, talking to the tiny toilet in a patronising tone

“Aw, look at you. A little, tiny toilet for a little, tiny

baby to…”. e toilet lid is then blown o and sprays

Victor in the face. He responds “Aghhh! Curse you,

tiny toilet”. We then presented the associated chall-

enging task to students (see textbox), alongside a

still frame from the end of the scene (see Figure 1).

Lesson 1: Tiny toilet challenging task

How much smaller is the tiny toilet than the

regular toilet?

Enabling prompt:

If a regular toilet is 100cm high, get a ruler

and estimate how tall the tiny toilet might be?

How powerful is the shrink ray? Explain.

Extending prompt:

How big would you be if you were zapped?

Explain how you worked this out.

Choose something you would like to shrink.

Estimate how big it is before and after being shrunk.

main task), students were invited to collaborate

on the task with peers.

Most students chose to draw diagrams to support

their thinking (see Figure 2). Some students also

accessed a variety of equipment in the classroom (e.g.,

chairs, tables, rulers) to physically model the problem.

For example, Caitlyn and Ella had decided that the

tiny toilet was a little smaller than a human head, and

approximately the length of a pen. ey also determined

that a chair was a good proxy for the height of a toilet

before it had been shrunk. Using this information, they

were able to conclude that the tiny toilet was around

six times smaller than a regular toilet.

A (surprisingly) small percentage of students approa-

ched the task additively. For example, after accessing

the enabling prompt, Susie decided that the shrink ray

had made the tiny toilet 80cm smaller. When we asked

her: “What would happen if we zapped something else?”,

she initially replied “It would also become 80cm small-

er”. We challenged her reasoning by asking the question:

“When the toothbrush (another item Hector zapped)

got shrunk, did it shrink by 80cm?”. Susie realised

that the toothbrush could not have shrunk by this

much “because it’s only about the size of a whiteboard

marker to begin with”. We then asked Susie “How

many times bigger is her regular toilet compared with

her tiny toilet?”. She was then able to apply repeated

addition to work out that it was ve times bigger

(“20+20+20+20+20=100”).

Generally students conceptualised the shrinking

as occurring along one-dimension—length. However,

some students conceptualised the notion of “how much

smaller” in two (or even three) dimensions. A notable

example was William, who immediately decided that

the tiny toilet was approximately the size of his hand,

and then asked to visit the bathroom to “measure a

toilet”. With the help of a peer, he determined that the

area of a toilet was six hands high and four hands wide

(see Figure 3). He used this information to calculate that

the tiny toilet was 24 times smaller than a regular toilet.

Drawing on William’s example in the discussion enabled

us to challenge students as to why William’s Shrink

Ray ostensibly seemed so much more powerful than,

for example, Caitlyn and Ella’s. An even more dramatic

example was provided by Leo, who framed the problem

in three dimensions. After deciding (for expediency)

that the tiny toilet was 10% the length of the original

toilet, he was able to (with some teacher support) calcu-

late that the volume of the tiny toilet was only 0.1%

of the original toilet. He concluded, much to the

amazement of his peers, that his Shrink Ray made

things 1000 times smaller.

Figure 2. Students frequently used diagrams to support their thinking.

e lesson was structured such that students were

required to explore the task independently for at least

ve minutes. After this time, anyone struggling to

make any progress was invited to come to the front (if

they wished) and get themselves the enabling prompt.

Around one-quarter of the students across the two class-

es (11 out of 47) accessed the enabling prompt at some

point in the lesson. After another ve or so minutes

(deemed sucient time for those students to get them-

selves the enabling prompt, digest it, and reanalyse the

Movies through a mathematical lens

23APMC 25(1) 2020

Figure 3. William’s diagram conceptualising the shrinking

in two dimensions (Length x Width).

After a total of around 20 minutes exploring the task,

we facilitated a brief whole-class discussion (approx.

7–10 minutes) where several dierent approaches to the

main task were shared. By this stage, as most students

had made signicant progress with the initial task, all

students were invited to attempt the extending prompt.

After this plenary, students continued working on the

task for a further 10–15 minutes. We closed the lesson

with another brief whole-class discussion (approx. ve

minutes), where another couple of groups of students

shared their work on the task, particularly their progress

with the extending prompt.

Alan and George adopted a simple yet particularly

powerful approach to the task that enabled them to

have signicant success with the extending prompt.

ese students used the enabling prompt and ruler to

decide that the tiny toilet was around one-quarter the

height of the original toilet. ey used this information

to calculate the height of the tiny toilet by taking the

height of the original toilet and “halving it (50cm),

then halving it again (25cm)”. ey used this “halve it,

and halve it” again approach to work out how tall they

would be if they were zapped (37.5cm), as well as the

tree outside (1m) and the tallest building in the

school (2m).

e work on the extending prompt also provided

an opportunity to contrast additive and proportional

reasoning. Two students working together, Sammy and

Finley, had determined that the Shrink Ray made things

10 times smaller. ey measured themselves and found

that Sammy was 150cm tall and Finley was “about 2cm

shorter” (148cm tall). When they turned the Shrink

Ray on themselves, they concluded that Sammy was

15cm tall (“because 150 divided by 10 is 15”), and that

Finley was 13cm tall (“because Finley is 2cm shorter

than Sammy”). To challenge this inappropriate applica-

tion of additive reasoning, a provocative question was

asked by the rst author: If we put the shrink ray in

reverse and made everything 10 times bigger, how big

would you be now? is question helped to highlight

to the students that something had gone wrong with

their mathematical reasoning. Eventually, they were

able to deduce that the dierence in their shrunken

heights should be 0.2cm and not 2cm (“because the

dierence has got 10 times smaller too”).

Our lesson concluded with another clip from

Despicable Me, this time of Gru using the Shrink Ray

to shrink the moon so it is small enough for him to

steal. We conducted a brief think-pair-share where stu-

dents were asked to consider: Is the ray more powerful

at the start of the movie (with the toilet) or towards

the end (with the moon)? Explain how you know this.

A possible extension problem for a follow-up lesson

would be to ask students: “Can you estimate how

much smaller the shrunken moon is than the origi-

nal?” (Hint: the moon is approx. 3,500km wide).

Our second lesson involved a short-lm, Coin

Operated (Dalman & Arioli, 2017)of approximately

ve minutes, rather than a movie-clip. e correspond-

ing challenging task also involved students exploring

and applying proportional reasoning, although it is

worth noting that this second task was more comp-

utational in focus and less open-ended (see Lesson

2 textbox).

Lesson 2: Coin operated challenging task

Imagine one 5c coin propels his rocket 50cm,

but this is clearly not enough. He needs more.

Space is 100km away. How many 5c coins does

he to need in order to become a real astronaut

who has been to outer space?

Enabling prompt:

Draw a table to help you with the problem.

Distance Coins

50cm 1

100cm or 1m

10m

100 m

1000m or ?

Extending prompt:

70 years passes in the lm and he charges

10 cents for lemonade. How many cups of

lemonade did he need to sell per day to travel

to outer space? What about if he wanted to

travel to the moon instead?

24 APMC 25(1) 2020

Russo & Russo

Student learning experience

Following the second lesson, all students were invited

to complete an anonymous on-line questionnaire, to

share their experiences of the two lessons. Forty-one

students across the two classes completed the question-

naire. e questionnaire contained three likert-type

statements that students needed to respond to (with

the scale ranging from 1-strongly disagree to 5-strongly

agree), and three follow-up open ended questions

(see Table 1). A summary of student responses to the

three likert-type statements is provided in Table 2. e

majority of students agreed or strongly agreed that the

lessons were enjoyable (83%), led to a lot of learning

(64%), and were mathematically challenging (78%).

We have also included our analysis of responses to

the open-ended student enjoyment question, as we

feel it is particularly powerful in conveying the value

of incorporating movie clips and short-lms into

mathematics classes.

Student enjoyment

Several themes emerged from our analysis of the

open-ended item which asked students what speci-

cally about the lessons they enjoyed (see Table 3

for a summary). e three most frequently coded

themes are elaborated on below, with illustrative

responses from students included.

e majority of students specically identied

the learning context as a reason for the lessons being

enjoyable. e prevalence of this theme is not sur-

prising, and consistent with similar research we have

undertaken investigating the reasons students enjoy

learning through inquiry-based pedagogies (Russo

& Russo, 2019). Exemplary quotes included:

I enjoyed working with others to work things

out, and also watching movies. Because you

know, who doesn’t love watching movies, it

was fun how we got to base our maths investi-

gations o movies.

Table 1. Questionnaire items.

Likert-item Open-ended question

I enjoyed the lessons What about the lessons did you enjoy?

I learned a lot from the lessons What about the lessons helped your learning?

I found the lessons challenging What about the lessons did you nd challenging?

Table 2. Student responses to likert statements.

Likert-item Strongly Disagree Disagree Neutral Agree Strongly Agree

I enjoyed the lessons 2% 0% 15% 46% 37%

I learned a lot from the

lessons

0% 5% 32% 37% 27%

I found the lessons

challenging

0% 7% 15% 39% 39%

Table 3. What about the lessons did you enjoy?

Theme Responses coded to this theme: Number (Percentage)

Learning context 21 (51%)

Level of challenge 11 (27%)

Learning new things in new ways 7 (17%)

Collaborating with peers 4 (10%)

Working individually 2 (5%)

Dierentiated 1 (2%)

Note: Several responses were coded to multiple themes. ree responses were too general to be coded to any

specic theme, and two students did not respond to the open-ended item.

Movies through a mathematical lens

25APMC 25(1) 2020

I liked how we didn’t get set tasks. We watched

videos and did maths in a more fun way.

It was made fun because it was a problem that

had context.

I liked how we could work on something

in a movie that we all have seen and it makes

maths funner.

Around one-quarter of students discussed how they

enjoyed the level of challenge in the lessons. Note

that all but one of these students emphasised the fact

that the lessons were more challenging than what they

were typically used to when learning mathematics.

is nding resonates with other research involving

primary-aged students learning mathematics through

challenging tasks, where similar proportions of

students have reported that the principal reason why

these lessons were enjoyable was because they were

challenging (Russo & Hopkins, 2017). e following

quotations are illustrative of student responses coded

to this theme:

I enjoyed the lessons because they provided a

challenge and they were dierent from normal

maths lessons.

ey were interesting. It’s fun to have a

challenge.

I enjoyed doing the maths and watching the

clips. My favourite was the rst one (lesson)

because I found it a lot more challenging.

Overlapping somewhat with the level of challenge

theme was the notion that students enjoyed learning

new things in new ways. is is consistent with the

idea that the medium (in this case, the use of mov-

ie-clips and short-lms), the task and mathematical

content can combine to generate a novel learning

experience that engages students (Ingram, Williamson-

Leadley & Pratt, 2016). Indicative student responses

coded to this theme included:

I like that it challenged me with stu I had

never done before.

I enjoyed the part of the sessions because it

made me work in dierent ways.

I enjoyed learning about something dierent than

from our regular maths, and it was fun watching

the short videos to go along with.

Concluding remarks

We believe that using age-appropriate movie-clips and

short-lms as a context for mathematical inquiry has

great potential to both enhance student engagement

in mathematics, and to bring to life abstract mathemat-

ical ideas (such as proportional reasoning).

Our own experience resonates with that of most other

educators in that students perceive mathematical tasks

as more interesting, meaningful and comprehendible

when they are contextualised. We have found contexts

that genuinely pique student interest to be particularly

powerful for supporting mathematical inquiry. is

appears to be the case whether these contexts are based

in reality (e.g., basketball, dance; Russo & Russo, 2019)

or ction (e.g., movies, picture story books; Russo &

Russo, 2018). We hope the approach outlined in this

article encourages teachers to experiment with using

movies to support mathematical problem solving in

their primary classrooms. Readers interested in access-

ing the tasks used in this article (and their associated

videos) can visit the short link: bit.ly/moviesandmaths

References

Averill, R. (2018). Examining historical pedagogies towards opening

spaces for teaching all mathematics learners in culturally responsive

ways. In J. Hunter, Perger, P., & Darragh, L. (Ed.), Making waves,

opening spaces (Proceedings of the 41st annual conference of the

Mathematics Education Research Group of Australasia), pp. 11-27.

Auckland: MERGA.

Bell, L., Juersivich, N., Hammond, T. C., & Bell, R. L. (2012). e

TPACK of Dynamic Representations. In R. Ronau, C. Rakes, &

M. Niess (Eds.), Educational Technology, Teacher Knowledge, and

Classroom Impact (pp. 103-135). Hershey, PA: IGI Global.

Butterworth, W. T., & Coe, P. R. (2004). Come on down… e

prize is right in your classroom. Problems, Resources, and Issues in

Mathematics Undergraduate Studies, 14(1), 12–28.

Dalman, J. (Producer), Arioli, N (Director). (2017). Coin Operated

[Animated Short Film]. United States.

Greenwald, S. J., & Nestler, A. (2004). r dr r: Engaging students with

signicant mathematical content from e Simpsons. Problems,

Resources, and Issues in Mathematics Undergraduate Studies, 14(1),

29–39.

Ingram, N., Williamson-Leadley, S., & Pratt, K. (2016). Showing and

telling: using tablet technology to engage students in mathematics.

Mathematics Education Research Journal, 28(1), 123–147.

Lampert, M. (2001). Teaching problems and the problems of teaching.

New Haven: Yale University.

Meledandri, C., Cohen, J., & Healy, J. (Producers), Con, P., &

Renaud, C. (Directors). (2010). Despicable Me [Motion Picture].

United States: Universal Pictures.

26 APMC 25(1) 2020

Russo & Russo

Minas, M. (2019). Maths and children’s literature: Implementing the

narrative-rst approach. Prime Number, 34(2), 14–15.

Muir, T. (2018). Heads or tails. Little Steps.

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

Engaging with mathematics through picture books. Albert Park,

Australia: Teaching Solutions.

National Council of Teachers of Mathematics. (2000). Principles and

standards for school mathematics. Reston, VA: Author.

Niess, M. L. & Walker, J. M. (2010). Guest editorial: Digital videos as

tools for learning mathematics. Contemporary Issues in Technology

and Teacher Education, 10(1), 100–105.

Park, J. C., & Lamb, H. L. (1992). Video vignettes: A look at physics in

the movies. School Science and Mathematics, 92(5), 257–262.

Pierce, R., Stacey, K., & Ball, L. (2005). Mathematics from still and

moving images. Australian Mathematics Teacher, 61(3), 26–31.

Reiser, E. (2015). Teaching mathematics using popular culture: Strategies

for Common Core instruction from lm and television. Jeerson, NC:

McFarland.

Russo, J., & Hopkins, S. (2017). Student reections on learning with

challenging tasks: ‘I think the worksheets were just for practice,

and the challenges were for maths’. Mathematics Education Research

Journal, 29(3), 283–311.

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

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

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

conference of the mathematics association of victoria (pp. 78–84).

Melbourne, Australia: MAV.

Russo, J., & Russo, T. (2019). Teacher interest-led inquiry: unlocking

teacher passion to enhance student learning experiences in primary

mathematics. International Electronic Journal of Mathematics

Education, 14(3), 701–717. https://doi.org/10.29333/iejme/5843

Russo, T., & Russo, J. (2018). Narrative-rst approach: Teaching

mathematics through picture story books. Australian Primary

Mathematics Classroom, 23(2), 8–15.

Schiro, M. (1997). Integrating children’s literature and mathematics

in the classroom: Children as meaning makers, problem solvers,

and literary critics. Teachers College Press, Columbia University.

Spengler, P. (Producer), & Donner, R. (Director). (1980).

Superman II [Motion picture]. United States: Warner Bros.

Sullivan, P., Askew, M., Cheeseman, J., Clarke, D., Mornane,

A., Roche, A., & Walker, N. (2015). Supporting teachers in

structuring mathematics lessons involving challenging tasks.

Journal of Mathematics Teacher Education, 18(2), 123–140.