5 Reasons to Teach Mathematical Modeling

             5 Reasons to Teach Mathematical Modeling

         To solve modeling problems, mathematicians make assumptions, choose a mathematical approach, get a solution, assess the solution for usefulness and accuracy, and then rework and adjust the model as needed until it provides an accurate and predictive enough understanding of the situation. Communicating the model and its implications in a clear, compelling way can be as critical to a model’s success as the solution itself. 

      Even very young students can engage in mathematical modeling. For example, you could ask students of any age how to decide which food to choose at the cafeteria and then mathematize that decision-making process by choosing what characteristics of the food are important and then rating the foods in the cafeteria by those standards. The National Council of Teachers of Mathematics (NCTM) is providing leadership in communicating to teachers, students, and parents what mathematical modeling looks like in K–12 levels. The 2015 Focus issue of NCTM’s Mathematics Teaching in the Middle School will be about mathematical modeling and the 2016 Annual Perspectives in Mathematics Education will also focus on the topic.

1. Solving rote problems is boring. Modeling involves making genuine choices.

     School mathematics is often presented as a set of steps to be followed in a particular order. Students can follow procedures without understanding (and perhaps not caring to understand) how the steps are connected and why they work. Then, they may rightly see this doing-without-understanding as a useless skill—even though correctly applying the procedures can lead to success on many standardized tests, which are developed to have predictable, standard, easy-to-grade answers. In contrast, mathematical modeling problems are big, messy, real, and open-ended. In modeling, students need to make genuine choices about what is important, decide what mathematics to apply and determine whether their solution is useful. Modeling provides opportunities for students to develop and practice mathematics-related skills, then communicate their understanding and interpretation of the problem.

2. Nobody likes to be wrong. Modeling problems have many possible justifiable answers.

          When people say they dislike math, I imagine them staring at a paper with lots of X-marks all over their work. Think about how we grade school mathematics: Textbook problems provide the correct methodology, and solution manuals contain the correct answers. Problems are designed to facilitate grading, which can deemphasize creativity, elegance, efficiency, and communication. Instead, mathematical modeling problems require answers that not only use valid mathematical arguments but also make sense in context. Good models provide compelling approximations to solutions that can’t be exactly nailed down because the problem is complex, open-ended and messy. You can check out the Moody’s Mega Math Challenge orMathematical Contest in Modeling for examples of mathematical modeling problems. Of course, some models are better than others, and justifying the solution is a critical aspect of the process. 

3. Contrived story problems can be mind numbing. Modeling problems matter to the end-user who needs to understand something or make a decision.

       The way students generally learn mathematics in school does not resemble the way a mathematician does it or the way it is practiced in other fields, such as business, science, computing, and engineering. Problems that ask uninteresting questions about real things, such as apples, are not much of an improvement. Many people have asked me: Is there really any new math to be discovered? The information in textbooks rarely hints at interesting unanswered questions (such as the Millennium problems). In addition to the interesting abstract questions out there, mathematical modeling problems are generally practical. They relate to issues someone needs to understand or decisions someone needs to make, such as when a drug is safe and effective enough to make it available to the public. Modeling makes mathematics relevant to real problems from life.

4. Math is hard and requires practice. Modeling presents problem solving as a creative, iterative process.

        When people tell you that they are bad at mathematics, they will often recount the moment they hit the wall and gave up. They recall a class, a teacher, or a test and perpetuate the idea that if you hit a wall in mathematics you are no good at it. This idea is reinforced by the fact that in school you have to learn particular ideas in a given amount of time or you fail. But here’s the truth: Every mathematician, even one called a genius, hits a wall at some point. Sometimes we get stuck on a problem for years. When we hit a wall, we have to practice harder and longer. We acquire more tools and information. We talk with our colleagues. And like an athlete who misses a shot or loses a game, we only find success if we try new strategies and do not give up. The open-ended nature of mathematical modeling problems can allow students to employ the mathematical tools that they prefer as well as practice skills they need to reinforce. The fact that the process itself involves iteration (evaluation and reworking of the model) clearly communicates that a straight path to success is unlikely.

5. The “mathematician as solitary genius” myth discourages almost everyone. Modeling is inherently a team sport.

I’d just as soon lose the term genius, but if we have to use it, here’s how I see it. A genius is someone whose brain is tickled and delighted by certain ideas. A genius is inclined to think about these ideas far more than most other people, and this perseverance enables them sometimes to think about the ideas in new ways. They are focused, creative, hard-working rule-breakers who put their work ahead of other pursuits. Their work is recognized as exceptional and groundbreaking. My problem with the term comes from the stereotype of a genius (who comes to mind for you?). 

Assumptions about the characteristics of a genius can have a negative impact on those who don’t fit the mold, in terms of recognition of their work, attribution of their success to cleverness (as well as hard work), and their own wrestling with imposter syndrome. Geniuses are seen as people who don’t need any help or collaboration to succeed. But the truth is that we mathematicians often work with other people and have social lives, like anyone else. Those of us who work alone are still reading and building on the ideas of others. Real mathematical modeling problems are so big and messy that in order to solve a problem on a reasonably short deadline, a team approach is almost always valuable. The problems are multi-faceted and open to multiple approaches, so students can contribute their strengths to their team, while learning from the approaches taken by others. Everyone can help make the solution and communication stronger.