What is Math Modeling?

An abbreviated description of the process for newbies

Mathematical modeling refers to the process of creating a mathematical representation of a real-world scenario to make a prediction or provide insight. There is a distinction between applying a formula and the actual creation of a mathematical relationship. Some graphical illustrations of the modeling process can be seen on this one page flyer.

Real-world, messy problems can be approached with mathematics, resulting in a range of possible solutions to help guide decision making. Both students and teachers are sometimes uncomfortable with the notion of math modeling because it is so open-ended. So much unknown information seems prohibitive. And what factors are most relevant? But it is this open-ended nature of real-world problems that leads to building and applying problem solving skills, creativity, innovation, and mathematics.

Mathematical modeling can be thought of as an iterative process made up of the following components. (Note that the word “steps” is intentionally avoided to highlight the lack of a prescribed ordering of these components, as some may occur simultaneously and some may be repeated.)

• Identify the Problem Because modeling problems are open-ended, the modeler must be specific in defining what it is they would like to find out.
• Make Assumptions and Identify Variables Since it is impossible to account for all the important factors in a given situation, the modeler must make choices about what to incorporate in their representation of the real-world. Making assumptions helps reveal the variables that will be considered and also reduce the number of them by deciding not to include everything. Within this process, relationships between variables will emerge based on observations, physical laws, or simplifications.
• Do the Math Eventually, a relationship between input and output will allow for a solution to be found.
• Analyze and Assess the Solution When considering the results and insights gained from the model, one asks if the answer makes sense.
• Iterate Usually, the model can be refined and the process can be repeated to improve the model's performance.
• Implement the Model and Report Results A clear report on the model and its implementation makes the model understandable to others.

One of the biggest pitfalls in developing a reasonable model is time management. When modeling is new to students, it is easy for them to get overwhelmed. They may spend too much time “in the weeds.” To define a succinct problem statement, students need to brainstorm and it should be encouraged not to throw out any ideas. However, there are times that students may get caught up in trying to include variables or relationships in their model that are not tractable or where data is just unavailable. At this point, students should make an assumption and move on. They should reflect on those assumptions after a pass through the entire modeling process. Having said that, sometimes students include unneeded assumptions in their documentation that are never explicitly used in the modeling process. This can also take away from valuable time and detract from the presentation of the solution. Students can get off-track while creating models, in particular making choices or assumptions that undermine the solution quality.

When students are in a time crunch, apprehension may lead to mathematical relationships that are removed from reality. It is not uncommon to see nonsense math introduced. For example, students may form an additive relationship between the key variables they identified but the units are meaningless (for example, adding dollars to time to get a model for resources). Coefficients are often used in models that also do not reflect units properly or there is no justification as to why they were chosen.  Other times, students may have a sound idea for a mathematical relationship but then overcomplicate it to make the mathematics look more sophisticated (for example, introducing a triple integral when really addition is appropriate). This is another reason why leaving time for reflection is critical, so that a student can read over their entire solution and ask themselves “Does this make sense?”

Dealing with data can also be overwhelming. Students may have a brilliant idea for a model, but cannot find the data they need to move it forward (again, at this point they should make an assumption and stop wasting time looking). Other times, datasets may be prohibitively large and students are not equipped with the tools to interpret key trends. Linear regression or high degree polynomials are often used to fit data without any sound reason and then used as predictors. The connection to the underlying physical problem can get lost or the quality of the fit is ignored completely.

All of the above pitfalls (which is by no means an exhaustive list) can be amended if the team reflects on the quality of their work. If an assumption seemed way off base, students can honestly report out the identified weaknesses of their approach and point the way towards improvements even if they do not have the means or access the information to do so. Even better, a sensitivity analysis can help a student assess the robustness of their model and make comments on its applicability. Much of this circles back to time management.

With more experience modeling, students will naturally gain skills and confidence that can alleviate these issues.

*Written for a Workshop Summary, 2021 by Katie Kavanagh and Ben Galluzzo, Clarkson University and based on ideas presented in GAIMME: Guidelines for Assessment and Instruction in Mathematical Modeling Education, Second Edition, Sol Garfunkel and Michelle Montgomery, editors, COMAP and SIAM, Philadelphia, 2019.