By Lucas Castle
Applied mathematicians know the importance of recognizing and promoting mathematical innovation as it relates to industry, medicine, and engineering. They realize that interdisciplinary partnerships are vital to solving the world’s most complex problems. Thousands of mathematicians and computational scientists from around the globe recently convened in Valencia, Spain, for the 9th International Congress on Industrial and Applied Mathematics, which took place this July. Attendees shared recent mathematical advances, collaborated with peers, and facilitated partnerships between academia and industry.
The need for applied mathematics is abundantly clear, and education is a particularly critical component. Presenters at the various minisymposia on mathematics education sought to answer the following question: How are we leveraging our skills as applied mathematicians to truly train and prepare students? Each talk offered a different perspective on this meaningful, complex problem. Some educators spoke of their innovative, problem-focused courses, while others shared their experiences with exposing students—both at the undergraduate and graduate level—to mathematics’ relevance via modeling competitions or industry collaborations. In all cases, students pondered the real-world applications of mathematics both in and out of the classroom.
Inside the Classroom
Undergraduate students often ask me “Why is this useful?” While answering this question is relatively easy, simply listing applications of relevant mathematical concepts still leaves students somewhat unsatisfied. To bypass this issue altogether and make mathematical applications self-evident, many educators are shifting towards problem-centric curriculum design, in which students experience mathematics by tackling relevant and contextualized problems. For example, linear algebra students might develop notions of linear systems by analyzing traffic flow in a series of intersections or studying disease dynamics in epidemic models. Undergraduates in a quantitative reasoning course could design an art gallery—complete with a scale model—to understand the steps involved in the mathematical modeling process. These and other practical experiments effectively demonstrate (rather than simply state) mathematics’ applicability in the real world.
Courses that incorporate projects of this nature also stimulate the development of other desirable, non-mathematical skills. Open-ended problems set the stage for collaborative, multidisciplinary experiences in inquiry-based environments. However, solving a problem is only half the battle. Communication of solutions, both within a group and to intended stakeholders, provides opportunities for students to practice writing and speaking skills via a variety of media. A group performing sensitivity analysis on a model of a stocked lake might convey its results by writing a memo to the U.S. Fish and Wildlife Service with a recommended fish harvesting rate for the lake. Alternatively, a small modeling project on international travel could culminate with oral presentations to a “donor” (in this case, the instructor) that request funding for the hypothetical trip. Using mathematics as the vehicle through which students develop these soft skills reinforces the relevancy of the subject in context.
Of course, teaching and learning in this style is not without challenges. These courses necessitate flexibility and require buy-in from both students and instructors. Teachers must be prepared to generate and facilitate open-ended problems that have no “right” answer. Assessment is also more complicated; grading a particular assignment that yields a variety of solutions, deciding whether traditional hour-long exams fit the material, and balancing rubrics between mathematical concepts and soft skills (such as writing, communicating information or data effectively, etc.) pose difficulties. That said, the rewards far outweigh the obstacles. Contextualized mathematics enhances student attention, information retention, and reflective learning. Students find material in this framework more applicable to their lives and future careers, as the mathematics in question actually matters to them. Furthermore, undergraduates in courses that challenge them with “real problems” tend to outperform their peers in classes that utilize more traditional teaching structures.
Beyond the Classroom
Authentic classroom problems especially benefit students in classes that are taught from a more traditional perspective (like service courses and lower-division math courses). Students taking advanced courses are also likely to encounter at least a few of these types of questions. But no matter what, the classroom will still always be a safe space (facilitated by instructors) that represents only a small taste of work in a mathematical career. How can we provide students with experiences that more rigidly emulate the professional use of mathematics?
The Consortium for Mathematics and its Applications offers two competitive opportunities for students to hone their skills: the Mathematical Contest in Modeling (MCM) and the Interdisciplinary Contest in Modeling (ICM). Over the course of a single weekend, participating teams formulate and communicate a solution to open-ended, broadly-defined problems—such as creating an evacuation plan for the Louvre—for interested stakeholders via a written report. The MathWorks Math Modeling Challenge—sponsored by MathWorks and organized by SIAM—gives high school juniors and seniors 14 hours to tackle a complicated, real-world issue with mathematical modeling and report their results. Participation in such contests enables students to hone their practical mathematical skills and obtain a more realistic picture of an applied mathematics career. Furthermore, faculty are beginning to use these programs as templates to design smaller, local competitions. Such contests provide more chances to engage with mathematics, especially for students who might not normally compete in a national or international competition. Those who do participate receive immediate feedback, which is not possible in large-scale competitions like the MCM and ICM.
Finally, industry itself overflows with problems perfect for student engagement. Many institutions collaborate with laboratories, businesses, and industrial companies that allow both undergraduate and graduate students to conduct research in professional settings. Participants gain firsthand experience using mathematics in business and industry, and companies receive help in solving problems while identifying and training potential future hires.
In conclusion, how do we leverage our skills as applied mathematicians to truly train and coach the future generation? We do so by using our unique experiences, knowledge of mathematics’ utility, and ability to connect with industry partners and immerse them in meaningful, real-world problems. This is the current trend in applied mathematics education for non-majors, graduate students, and everybody in between. Everyone benefits from focusing on problem-solving, and no group is better prepared to provide contextualized experiences in mathematics than applied mathematicians themselves.
Lucas Castle is the Math that Matters Postdoctoral Fellow in the Applied Mathematics Department at Virginia Military Institute. He can be reached at [email protected].