Redefining Mathematical Modeling Examples: A Literature Review and Research Roadmap

Literature Review

The author initially thought a literature review merely introduced previous work and highlighted their own innovation, but realized that for a thesis on "analysis of mathematical modeling examples" the review must first clarify key concepts such as “mathematical model,” “mathematical modeling,” “example problem,” and “analysis of examples.” The review then surveys how earlier scholars defined these terms and notes where definitions are missing, providing the author’s own definitions as a contribution.

Key concepts discussed include:

Mathematical Model : Broadly, any mathematical knowledge can be called a model; narrowly, a fixed method or routine that solves a specific class of problems. Models are further divided into elementary (suitable for high‑school textbooks) and advanced models.

Mathematical Modeling : The process of converting real‑world situations into mathematical problems, solving them, and interpreting results. A common five‑step procedure is: problem formulation – analysis (variables/assumptions/goals) – model selection – model solving – model validation.

Modeling Example : Textbook problems marked as examples that demonstrate a problem and its solution. The author questions whether such examples must contain a real background and a clear five‑step solution strategy, and notes the lack of literature on this definition.

The author also distinguishes between application problems, open problems, and exploratory problems, emphasizing that true modeling examples should allow students to choose among different models and consider issues like sensitivity and robustness, which often require higher‑level mathematics.

Research Approach

The central research question is defining the concept of a "mathematical modeling example." The author proposes a three‑step understanding: (1) "modeling problem," (2) "elementary modeling problem," and (3) "elementary modeling example." The plan includes:

Identifying criteria that make a textbook problem a genuine modeling example and establishing a standard or benchmark for evaluating other problems.

Bridging the gap between university‑level modeling research and high‑school modeling by clarifying what constitutes "elementary mathematical modeling."

Investigating whether textbook examples can serve as modeling problems or if exercises must be used instead, focusing on the problem’s closeness to real practice and the flexibility it offers for model selection.

Research Outlook

The author envisions creating a "modeling problem affiliation model" to score the modeling attributes (difficulty, level) of selected textbook examples and exercises. By applying this model to a range of high‑school textbooks, the author aims to compute average scores, derive overall insights into the modeling nature of the materials, and ultimately propose guidelines for designing better modeling problems in curricula.

These findings could inform textbook authors and educators on how to embed mathematical modeling more effectively in high‑school education.