I believe students should see mathematical modeling in school because this way of engaging mathematics can be both inclusive and personally empowering.
Mathematical modeling describes processes used to understand, describe, and predict real-world situations by employing whatever quantitative and computational approaches are useful. Many aspects of applied mathematics, statistics, operations research, algorithms and, data science can fall under this umbrella.
When I say mathematical modeling should have a place in your curriculum, I don’t only mean exposing students to ubiquitous well-known models such as the mass-spring system or predator-prey. I mean the creative process of modeling, like the work students would do in a mathematical modeling competition such as the M3 Challenge or MCM/ICM or in the workplace. I believe this can start as early as kindergarten, though of course here we focus on undergraduate education.
When I say mathematical modeling should have a place in your curriculum, I don’t only mean exposing students to ubiquitous well-known models such as the mass-spring system or predator-prey. I mean the creative process of modeling, like the work students would do in a mathematical modeling competition such as the M3 Challenge or MCM/ICM or in the workplace. I believe this can start as early as kindergarten, though of course here we focus on undergraduate education.
Mathematical modeling is a topic of increased visibility in the education system. First, it appears in the US Common Core Standards for Mathematical Practice (CCSSM-MP4) for students in K-12. Second, job opportunities in business, industry, and government for those trained in the mathematical sciences are increasing while numbers of tenure-track opportunities are flat or decreasing. As someone on a job panel put it: I don’t care what flavor of mathematics you studied in school and bring to the job, once you work for my company, you are an applied mathematician.
Here are 5 characteristics of mathematical modeling that can make it inclusive and personally empowering:
1. Mathematical modeling involves genuine choices. Students can be involved in choosing the situation, the topic, the mathematical and computational tools and, the metric for the success of their model. When students make choices, they can feel a different connection to and ownership of mathematics – mathematical tools and ideas are theirs to choose and use. In this way doing mathematics is more like the creative act of a research mathematician, a writer or an artist – the goal of the work is to create something new and of intellectual value.
2. Students learn how objective functions and assumptions/constraints matter. The general public and the media express some skepticism about mathematical modeling because it seems like maybe you can make the data tell any story you want.
Mathematical modeling helps people learn that by clearly communicating your objective function (goal) and assumptions/constraints you can more honestly and clearly convey why the model makes the prediction that it does. Students also learn that aspects of social justice and equity can be brought into play through these aspects of the model. For example, maybe your business goal is to help the people while staying financially viable, rather than just making the most money.
Mathematical modeling helps people learn that by clearly communicating your objective function (goal) and assumptions/constraints you can more honestly and clearly convey why the model makes the prediction that it does. Students also learn that aspects of social justice and equity can be brought into play through these aspects of the model. For example, maybe your business goal is to help the people while staying financially viable, rather than just making the most money.
3. Mathematical modeling can draw on interdisciplinary ideas and ways of knowing. Because modeling problems are situated in the real world, they almost always have interdisciplinary aspects. Maybe the important information is about how a business is run, or how a machine works, or how humans behave. What are the preferences of the end user? What will make a solution useful and usable? How can visualization help communicate the methods and solutions?
4. Real world problems help students see that mathematics is everywhere. People sometimes think of mathematical modeling as only applied mathematics, but once the problem is extracted and refined to its mathematical form, students can see the structure and beauty of that mathematics. Mathematical modeling can give students ways to practice using familiar tools as well as reach for new mathematical and computational ideas.
Reflective aspects of the modeling assignment can help teachers see how student perspectives on mathematics are changing. After modeling, students often report seeing things in a new way (such as noticing an elevator bouncing as it reaches the bottom of a building) and wondering how to describe this mathematically. Students can also envision career pathways that will call on these skills.
Reflective aspects of the modeling assignment can help teachers see how student perspectives on mathematics are changing. After modeling, students often report seeing things in a new way (such as noticing an elevator bouncing as it reaches the bottom of a building) and wondering how to describe this mathematically. Students can also envision career pathways that will call on these skills.
5. Mathematical Modeling is a team sport that focuses on process as much as product. It may be possible to work on a model alone, but almost always models benefit from multiple perspectives and ways of thinking. Well-structured modeling problems encourage teams to start with a simple common solution and then provide opportunities for teammates to add some complexity and test to see if it improves the model. Teamwork has many challenges, and teams need to be monitored so that students are respectful of all voices.
A rich modeling problem will have multiple solution methodologies and solutions (which depend on the selected objective and constraints). Students can experience mathematics as more than a course with feedback that judges when their answers are right and when they are wrong. This eliminates the phenomenon when every “X” mark on a problem can be like a small cut in a student’s confidence and erode their sense of belonging as a mathematics-doer. Instead, students can discuss how various aspects of solutions make them useful and what ways of communicating solutions are most effective. Students can iterate and improve their answers. Instructors can find ideas of value in every solution.
A rich modeling problem will have multiple solution methodologies and solutions (which depend on the selected objective and constraints). Students can experience mathematics as more than a course with feedback that judges when their answers are right and when they are wrong. This eliminates the phenomenon when every “X” mark on a problem can be like a small cut in a student’s confidence and erode their sense of belonging as a mathematics-doer. Instead, students can discuss how various aspects of solutions make them useful and what ways of communicating solutions are most effective. Students can iterate and improve their answers. Instructors can find ideas of value in every solution.
Here are three ways to make mathematical modeling available to your students:
- Build a final project into a course (could be Calc, Prob/Stats, Lin Al, DEs…) and make sure the project involves some choice. You can still point to a particular tool (such as single value decomposition or compartment modeling) if you need the project to hit a learning objective while leaving the context open to the students.
- Propose a mathematical modeling course for your department. It doesn’t have to be a full number of units – students can take it as an elective. It could be a once a week seminar and it could be pass-fail. Or it could be a full unit, graded course.
- Advise a modeling competition (MCM/ICM) team. Provide time and space for students to do some practice problems and read winning papers. Discuss mathematical and computational approaches. This could be a student club activity or a departmental endeavor.
To learn more about the teaching and learning of mathematical modeling, see these free downloadable reports:
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