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Challenging Default Assumptions in Learning Mathematical
Modeling Through Word Problems
Uegatani, Yusuke; Ishibashi, Ippo

秋期研究大会発表集録 , 56 : 201 - 204

2023-11-18

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https://ir.lib.hiroshima-u.ac.jp/00054908

Copyright (c) 2023 日本数学教育学会

Challenging Default Assumptions in Learning
Mathematical Modeling Through Word Problems

Yusuke Uegatani
Hiroshima University High School, Fukuyama

Ippo Ishibashi
Okayama University

1. Introduction
There are two controversial opinions on the relationship between mathematical word problem-solving andmathematical modeling: The ability of mathematical modeling can be cultivated through word problem-solving (e.g.,Czocher et al., 2020) vs. mathematical modeling is essentially different and more complex tasks than word problem
solving (e.g., Maaß, 2006). To understand both opinions consistently, Verschaffel et al. (2020) hypothesize thatmathematical word problems are oversimplified versions of mathematical modeling tasks and that word problems andmodeling tasks are two poles of a continuum. In this paper, we call this the continuum hypothesis. However, there islittle evidence that they make a continuum. In other words, we have not yet built an empirically supported theory fordesigning intermediate tasks between traditional word problems and mathematical modeling tasks.
The purpose of this paper is, thus, to propose a key feature of intermediate tasks between traditional mathematicalword problem-solving and mathematical modeling. Following a design research methodological framework, the study
presented in the present paper will empirically support our proposal.
2. Theoretical issue
First of all, the authors believe the continuum hypothesis is valid because the difference between traditional word
problem-solving and mathematical modeling is the degree of their authenticity. Vos (2018) contrasts the natures ofmathematical modeling in real and school contexts: Professional responsibility requires mathematical modelers in realsocieties to provide appropriate results through their work, while educational purposes allow students to makemistakes in classroom mathematical modeling. In this sense, mathematical modeling in classrooms cannot becompletely real mathematical modeling in societies. Educational purposes rationalize to remove some authentic
aspects of mathematical modeling. Vos (2018) call this de-authentication.
If we take this perspective, the continuum hypothesis implies that we can create intermediate tasks between
traditional word problems and mathematical modeling tasks by controlling the degree of authenticity. Previous studies
support this argument. We show some exemplary studies in this paper. First, Degrande, van Hoof, Verschaffel, andvan Dooren (2018) propose the Snake Problem. Students can interpret this problem in multiple ways (e.g., somestudents regarded a snake’s growth as a difference, and others as a rate). It is an authentic condition of mathematicalmodeling that there are multiple possible interpretations of the given situation. Second, Ishibashi and Uegatani (2022)
propose the Sushi Problem, which requires students to validate their solutions based on their cultural standard of eatingsushi. Cultural relevance is not given by the problem itself and is assumed by its interpreters. In this sense, students
interpret such an ill-defined problem in multiple ways and need additional information to determine its solution.
ー201ーIshibashi and Uegatani (2022) report that the Sushi Problem can prompt the modeling activity to become a cyclicprocess. Third, Maaß (2006) proposes an authentic modeling task, which requires students to estimate the height of astatue from its photo, and Stillman and Brown (2021) offer a data-rich task. These modeling activities use non-text
materials or rich data. Such information drastically increases the complexity of the modeling tasks like those in realsocieties. Table 1, thus, summarizes four authentic aspects presented in the existing literature. Traditional word
problem-solving does not contain any authentic element (Level 1). Ill-defined word problem-solving allows students
to interpret the problems in multiple ways but does not necessarily require additional information to determine theirsolutions (Level 2). Cyclic modeling prompts students to use additional information to validate their solutions
repeatedly by its nature (Level 3). Authentic modeling is as complex as real modeling in society, except thatprofessional responsibility is removed (Level 4).
Based on Table 1, we can describe the vital mission of this study as follows: Building a design theory for building
cyclic modeling tasks. Cyclic modeling tasks (Level 3) are simple but contain minimum and essential aspects ofmathematical modeling activities. To validate the continuum hypothesis, we must identify the borderline between
word problem-solving and mathematical modeling. However, existing literature must provide more information on
how such tasks can be designed.
3. An empirical study
We adopt a design research methodological framework (Bakker, 2018). We ask: What task design contributes toclassroom cyclic modeling?
(1) Task design, participants, and method
The Sushi Problem created by Ishibashi and Uegatani (2022) is an example of a potential task design for cyclicmodeling. They utilize cultural prejudice for task design. In their experimental lesson, the students first built a model
based on their biased belief that plates are countable only by natural numbers. However, the students noticed thatpeople share two pieces of sushi on one plate and that half of a plate can be a unit of analysis. It is suggested that theneed to rebuild a model must take place so that a modeling activity becomes a cyclic process.
From this perspective, we propose the Popcorn Problem (Figure 1). In Task 1, students may offer four possibilities(Table 2). If we assume that Ken’s brothers and sister did not share a cup of popcorn, only these four options areoptions. However, Task 2 suggests that this naïve assumption can be false. If we consider the additional information(Ken’s sister had eaten one Size L cup of popcorn with caramel flavor), Ken’s brothers must share one cup. The criticaldifference between the Sushi Problem and the Popcorn Problem is the potential way of rebuilding mathematicalmodels. The former only requires students to reconstruct a model with a new unit of analysis, while the latter requires
them to rebuild a model by using an essentially different idea. Because the Popcorn Problem suggests infinitely many
possibilities about what proportion of popcorn Ken’s brothers shared, the students must give up using a cup as a unit
Table 1: Authentic aspects of mathematical tasks in each level of authenticity
Possible aspects
LevelsMultiple

Additional

Non-text materials

Professional

interpretations

informationor rich data

responsibility

1

Traditional word problem solving

2

Ill-defined word problem solving

X

3

Cyclic modeling in classrooms

XX

4

Authentic modeling in classrooms

XX

X

5

Real modeling in societies

X

X

X

ー202ーXKen went to see the movie with his brother and his sister. Popcorn with salt and caramel flavors was sold at
the following price at the theater.
SaltCaramelSize S

Size M

Size L

Weight per one cup

60g

120g

150g

Price per one cup

400yen500yen550yenWeight per one cup

150g

300g

375g

Price per one cup

400yen500yen550yenKen’s brother bought much popcorn. When they measured the weight of the popcorn he purchased, it totaled
810g. Ken received 270g of popcorn from his brother and ate it. Then, his brother said, “Please pay me for thepopcorn you ate.” Ken obeyed with reluctance.
Task 1 What flavor and how much popcorn did Ken eat? How much should he pay his brother? Explain what wecan know about these questions.
Taks 2 Ken’s sister had eaten one Size L cup of popcorn with caramel flavor. What flavor and how much popcorn
did Ken eat? How much should he pay his brother? Explain what we can know about these questions.
(Note: Task 2 should be presented after the students solve Task 1.)
Figure 1: The Popcorn Problem
Table 2: Four possibilities students may propose when engaging with Task 1
Salt
Two Size S cups

CaramelOne Size S cup

Two Size S and one Size L cups
One Size M cup

One Size S cup

One Size M and one Size L cups

Total weight

Total price

270g

1200yen270g

1350yen

270g

900yen270g

1050yenof analysis and introduce a variable to describe infinitely many possibilities simultaneously.
We conducted an experimental lesson with the Popcorn Problem at a junior high school attached to a Japanesenational university. The teacher of this lesson was the first author, who sometimes teaches this class. The participants
were 19 eighth-grade students. We video-recorded the whole lesson from the rear of the classroom. Using document
cameras, we also video-recorded the two students’ real-time problem-solving processes. We gathered all the worksheets
the students used.
(2) Results
When engaging with Task 1 in the first half of the lesson, the students noticed the four possibilities listed in Table
2. All of them agreed that they knew the existence of the four possibilities in a mathematical sense. However,interestingly, when the teacher asked what the best solution to the problem was, some students did not regard the four
possibilities as equally likely. For example, they argued that Ken did not need two Size S cups of popcorn with salt flavor
because one Size M cup was more reasonable. As another example, they also questioned whether Ken liked too much
popcorn only with salt flavor; he would probably want to eat popcorn with caramel flavor, too. The students distinguishedbetween mathematical and likely solutions to the word problem.
Through engaging with Task 2 in the second half of the lesson, the students noticed that none of the four
possibilities was satisfied with the given additional condition that Ken’s sister had eaten one Size L cup of popcornwith caramel flavor. In that case, Ken’s brother had to eat 165g of popcorn, but they could not make 165g with thegiven cups of popcorn. When the students noticed this strange fact, they started looking around. The teacher considered
their behaviors a sign that they wanted to discuss their ideas with their neighbors. He allowed them to discuss, andー203ーthey started to assume that Ken’s brothers share cups of popcorn. They let Xg be the amount of popcorn Ken’s brotherate and tried to describe how Ken should pay his brother. This episode indicates that they rebuilt their model for thesituation.
4. Discussion and conclusion
As we expected, when the students started to engage with Task 2, they noticed the possibility that Ken’s brothershared popcorn. The students had a default assumption that a cup of popcorn was a unit of analysis but doubted its
validity when they engaged with Task 1. Let us introduce subtly different four axes for describing this experimental
lesson. First, essential information (popcorn is sharable) was experiential rather than given by the problem. Second,
the problem situation was inauthentic in Vos’s (2018) sense because the information was not certificated as sociallyoriginated. Third, the problem was fictional rather than real. Forth, the problem situation was unrealistic rather thanrealistic because the students considered the possibility that Ken bought too much popcorn. From this perspective, we
can argue that a modeling activity can be cyclic even though the problem is inauthentic, fictional, and unrealistic.
Students must rebuild their model by considering the validity of their default assumption because the word problem
requires them to refer to the given information and their experiential knowledge. Therefore, we propose a crucial
feature of intermediate tasks: Essential reference to experiential knowledge for solving the problems and contrast
between their default assumptions to determine which is better. Our empirical data succeeded in corroborating thecontinuum hypothesis.
Acknowledgments
This work was supported by the JSPS KAKENHI (Grant Number 21K13587 and 23K12847). The empirical study
reported in this paper is a part of studies approved by the Research Ethics Review Board of the Graduate School of
Humanities and Social Sciences, Hiroshima University (Registration Number HR-ES-000286).
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