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Movies through a mathematical lens

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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.
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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 eectively. 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
oers an eective 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 oers 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, eective 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 specically 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 sucient 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 scientic 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 insucient time to react, the process of
considering super-heroism from a scientic 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 eorts 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 oered 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 briey describe the plot for the
benet of those who had not seen it. We then dis-
played a one minute clip from the movie; specically,
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 sucient 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 dierent approaches to the
main task were shared. By this stage, as most students
had made signicant 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 signicant 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 dierence in their shrunken
heights should be 0.2cm and not 2cm (“because the
dierence 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 specically identied
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%)
Dierentiated 1 (2%)
Note: Several responses were coded to multiple themes. ree responses were too general to be coded to any
specic 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 dierent 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 dierent ways.
I enjoyed learning about something dierent 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
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... In the context of the current study, it is postulated that learning mathematics through tasks that emerge authentically from rich narratives will lead to students experiencing autonomy. First, autonomy will be supported by working on problem solving tasks that have multiple possible solutions and/or can be feasibly solved in multiple possible ways [8,63]. Second, the rich contexts in which the tasks are embedded, that both connect to and validate children's imaginative worlds, will lead to the mathematics being perceived as relatively purposeful and valued. ...
... Note that the focus of the current paper is on the learning experience from the student perspective. As such, how the tasks elicited the targeted mathematical thinking from a teacher perspective, in particular, how they supported students to reason proportionally, is beyond the scope of the current paper; however, this is discussed elsewhere (e.g., see [63]). ...
... An adaptation to this lesson structure is noted elsewhere in the literature (e.g., see [79] and their instructional model incorporating (re)launch, re(explore), (re)summarize/review). Note that the core tasks used in each of the lessons are briefly summarized in Table 1, whilst the full tasks have been published elsewhere (Where the Wild Things Are, [80]; We're Going on a Bear Hunt, [81]; Despicable Me and Coin Operated, [63]). ...
Article
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Using children’s literature to support mathematics instruction has been connected to positive academic outcomes and learning dispositions; however, less is known about the use of audiovisual based narrative mediums to support student mathematical learning experiences. The current exploratory, qualitative study involved teaching three lessons based on challenging, problem solving tasks to two classes of Australian Year (Grade) 5 students (10 and 11 year olds). These tasks were developed from various narratives, each portrayed through a different medium (movie clip, short film, picture story book). Post lesson interviews were undertaken with 24 students inviting them to compare and contrast this lesson sequence with their usual mathematics instruction. Drawing on a self-determination theory lens, our analysis revealed that these lessons were experienced by students as both highly enjoyable and mathematically challenging. More specifically, it was found that presenting mathematics tasks based on rich and familiar contexts and providing meaningful choices about how to approach their mathematical work supported student autonomy. In addition, there was evidence that the narrative presentation supported student understanding of the mathematics through making the tasks clearer and more accessible, whilst the audiovisual mediums (movie clip, short film) in particular provided a dynamic representation of key mathematical ideas (e.g., transformation and scale). Students indicated an eclectic range of preferences in terms of their preferred narrative mediums for exploring mathematical ideas. Our findings support the conclusion that educators and researchers focused on the benefits of teaching mathematics through picture story books consider extending their definition of narrative to encompass other mediums, such as movie clips and short films.
... Little is written about dynamic and static representations beyond student use of games or teachers' technological pedagogical content knowledge. One example of the use of dynamic representations pertains to students engaged in using proportional reasoning to make sense of a movie clip where students reported enjoying the challenge and appreciating the context of the tasks (Russo and Russo 2020). In a summary of literature on the use of dynamic representations in mathematics with students, Bell et al. (2012) note that students benefited from having more active roles, engaging in discovery, and increasing knowledge when using dynamic representations. ...
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