The role of generating questions in mathematical modeling

ABSTRACT

This study aims to clarify the role of generating questions in mathematical modeling and construct principles for teaching and learning mathematical modeling with an emphasis on generating questions. To achieve this purpose, the role of generating questions was determined by solving a sprinkler problem and considering the investigation process. Generating questions requires noticing the unknown and cognitive steps to elucidate the unknown and solve the problem. The process and outcome of generating questions were explored using an open-ended problem. This study demonstrates (1) the role in advancing mathematical modeling from various points of view, (2) the role of fostering mathematical modeling competencies, and (3) the role in facilitating the extension of the applicability of the mathematical model. Having established these roles, the principle of teaching and learning was constructed, which is expected to foster the ability to generate questions. This study demonstrates that, in mathematical modeling, generating questions is crucial for developing a clear sense of direction for modeling real-world problems. Furthermore, generating questions reflects on the structures of mathematical models, allowing the execution of the mathematical modeling process.

KEYWORDS:

• Mathematical modeling
• generating questions
• the roles of generating questions
• sprinkler problem

Acknowledgments

I would like to thank Editage (www.editage.jp) for English language editing.

The author is also grateful to the anonymous reviewers for their thorough and pertinent comments and suggestions, which have very much helped the revision of the manuscript and improved it considerably.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Notes

1 High-quality reclaimed water produced by further purifying treated wastewater with advanced membrane technology.

2 In the process of the procurement of raw materials from production, transportation, and disposal after consumption, the amount of greenhouse gases emitted by the product accumulates through the consumption of electricity and fuel that is converted to CO₂.

3 The X Junior High School in this study is a public junior high school in Saitama Prefecture in Japan. The sprinklers from Company H are used in that school, and the data were provided by Company H.

4 Assuming that the water droplet is a sphere, the coordinates of the droplet after t seconds are obtained as follows: $\left(\frac{v_0\cos \theta }{k}(1-e^{-kt})\text{, }-\frac{g}{k}t+\frac{1}{k}\left(v_0\sin \theta +\frac{g}{k}\right)(1-e^{-kt})\right)$where ($v_0$: Initial velocity of water droplet, $\theta$: Angle of nozzle, $k$: Resistance coefficient of sphere, $g$: Gravitational acceleration)