"Let’s create a post-calculus course in mathematical modeling!" I couldn’t have known that this one decision would drive my teaching philosophy moving forward. In late 2000, after attending the weeklong National Computational Science Education Consortium, I approached my department chair at Hanover High School in New Hampshire about creating a class that would build on this “new” idea of mathematical modeling. At the time, Hanover students who had completed calculus would typically take upper-level math classes at Dartmouth College, which is within walking distance of the school. But this often presented scheduling conflicts, so a high school course on mathematical modeling seemed like the next logical step. Such a class would give students the opportunity to encounter messy, real-world problems that require the formulation and application of assumptions, then iterate their processes to refine results. In the spring of 2002, Hanover High School offered math modeling for the first time.
Why Teach Modeling?
Guidelines for Assessment and Instruction in Mathematical Modeling Education (GAIMME), a report written by education advocates, university professors, and K-12 teachers, defines mathematical modeling as “a process that uses mathematics to represent, analyze, make predictions, or otherwise provide insight into real-world phenomena” . When I first started teaching, students regularly asked me when they would use mathematical concepts in the “real world.” I spent a lot of time trying to convince them of the many fields that employ algebra, geometry, or calculus. I racked my brain for applications that would speak to their interests and reflect the class material’s practicality. While some students accepted my explanations at face value, many indicated that they were not planning to pursue any related fields. Additionally, most high school concepts bear little resemblance to their real-world applications, especially at the lower levels. Students could thus tell when I was “stretching” the connection between their classroom math and any future utilization.
Over time, I realized that the question “When are we going to use this?” was actually about students’ frustration or anxiety with the difficulty of mathematical topics. It is easier for them to make peace with their math troubles if they convince themselves that math has no relevance to their daily lives. Modeling gives students that relevance by providing context to the mathematics in question and letting them make real, impactful decisions about a problem. They learn how to prioritize issues, make assumptions, and analyze their solutions. For example, my current modeling class has been investigating lunch lines because of frequent congestion in the cafeteria. We first studied queue theory—made tangible by the context of cafeteria lines—to understand line functionality. Students have also had to contend with real issues like budgets, scheduling, and building codes.
I recently attended the “Critical Issues in Math Education” workshop, which took place in Berkeley, Calif., this past March and was sponsored by the Mathematical Sciences Research Institute. The overarching theme pertained to the following question: How can we individually and collectively advance the teaching and learning of mathematical modeling in K-16? The panelists and presenters represented a diverse set of stakeholders, including university professors and industry leaders. Many universities now offer graduate degrees in mathematical modeling, and a quick internet search for jobs involving “math modeling” returns hundreds—if not thousands—of positions in data science, analytics, and more. Our biggest stakeholders in the future of math modeling are the students themselves, who risk being left behind in this next wave of technological and industrial advancement without a foundation in modeling techniques.
Introducing Modeling: First Steps
In my experience, one of the biggest obstacles in transitioning a conventional classroom into one that supports a modeling mindset is the resulting tension between a more traditional, textbook-dependent curriculum and the open-ended nature of modeling scenarios. This friction can create significant anxiety for teachers who genuinely want to get a taste of modeling but do not know where to begin. Introducing modeling problems also requires ceding some classroom control; students are not authentically invested in the problems if they do not possess a certain level of autonomy in the process.
For teachers who are not quite ready to dive into the deep end of modeling, breaking down the process and scaffolding the activities into “bite-sized” explorations can ease them into modeling while acclimating students to a more collaborative, inquiry-based form of mathematics instruction. The Math Modeling Hub and the Math Modeling Faculty Mentoring Network both provide online resources and support for instructors at the K-16, graduate, and industry levels. In addition to GAIMME, numerous resources offer guidance and motivation for teachers, including SIAM’s Math Modeling: Getting Started and Getting Solutions and Math Modeling: Computing and Communicating.
Opportunities also exist for students and teachers to grapple with a modeling problem without sacrificing content or class time. High school modeling contests have increased in number and scope after I introduced modeling to my curriculum in 2002. Since 2004, I have had teams participate in the High School Mathematical Contest in Modeling (HiMCM), sponsored by the Consortium for Mathematics and Its Applications. Several teams have placed at or near the top over the years.
Students in Greta Mills’ honors seminar on math modeling at Oxbridge Academy participate in the 2017 High School Mathematical Contest in Modeling. The students designed a mountain resort that could serve as the location for a future Winter Olympic Games. Photo courtesy of Oxbridge Communications.
In 2011, my students first partook in the MathWorks Math Modeling (M3) Challenge, a competition (formerly known as Moody’s Mega Math Challenge) that SIAM has organized since 2006. In the spring of 2012, I served as the first high school judge for the M3 Challenge. I was incredibly nervous at the prospect of judging alongside some of the leading experts in mathematical modeling, but the judging process actually calmed my nerves. I could see that my students’ efforts were in line with what other teams produced, and have enjoyed serving as a judge every year since.
HiMCM and the M3 Challenge have grown exponentially in recent years, and both contests provide students of all levels with the chance to work through open-ended, important, and real-world problems. Interestingly, one year I had a team of eager precalculus students participate in HiMCM alongside my high-level modeling students; they outperformed half of the modeling class, proving that math level is not a barrier to the modeling process.
Student Reflections on Modeling
A student enrolled in my modeling course this year after experiencing a great deal of math anxiety in previous math classes. He has become so passionate about modeling that he applied for—and received—a college scholarship for computational modeling. He is currently participating in a school trip to China focused on computational modeling and artificial intelligence.
High school modeling experience has motivated a number of strong math students to pursue computational mathematics. Even those who have followed other paths voiced their appreciation of exposure to modeling’s collaborative and iterative nature. “When minds can collaborate, we realize possibilities that we wouldn’t have thought of before,” said one student. “I think it’s good to learn to work together towards solving a problem and present the solution in a polished manner.”
1 Both handbooks are freely available online.
 Guidelines for Assessment and Instruction in Mathematical Modeling Education. (2019, February). (2nd ed.). Philadelphia, PA: Society for Industrial and Applied Mathematics and Consortium for Mathematics and Its Applications.
||Greta Mills is a mathematics teacher and chair of the Mathematics Department at Oxbridge Academy in West Palm Beach, Fla.