Practice problem #1 for M3 Challenge 2021 focused on businesses that have been affected by COVID-19. “Pandemic Winners and Losers” garnered great questions and conversation via live text chat on February 10. Dr. Neil Nicholson, problem author and professor of math and actuarial science at North Central College, gave great advice for Challengers. We have highlighted some of it here. You can see the whole conversation on the Practice Problems page.
Dr. Nicholson: As our time here is winding down, I just want to pass along some thoughts about the math modeling journey.
- It is indeed a journey. You can only get better at math modeling by doing math modeling. Don’t be discouraged if the problems are hard at the beginning. Keep going!
- Hopefully this practice problem reiterates the importance of all the “parts” of modeling. Don’t just breeze over those initial steps. Set a firm foundation and you’re setting yourself up for success!
- HAVE FUN. You’re doing this competition because you enjoy it. Don’t stress, don’t worry, don’t get worked up. Do your best and enjoy the time you get to spend working on the problem. When it’s done, be proud of the work you’ve done.
Q & A
Student: I have no idea where to start or what I am doing.
Dr. Nicholson: Being new to modeling simply means you’ll get better every time you work another problem. Don’t be discouraged! The first time I tried one of these I had no clue where to begin. Years later and many problems practiced, I start to see themes, know what questions to ask, and am able to jump to potential mathematical approaches much more quickly.
Another tip I have (and this is a MAJOR perk of participating in M3): “I have no idea…I am new…” reads as if this is a solo effort. Keep in mind that M3 is a team competition. Working with your peers will spark points of view you may not come up with on your own. Discussion breeds creativity (or as my 4 year-old says, “Teamwork makes the dream work.”)
Student: “Where did this problem come from? What inspired it?”
Dr. Nicholson: The inspiration for this one is fairly obvious: life has changed so much over the past year. A natural thing to ask is, “How? Why? And what’s next?”
I wrote this the way I did (asking you to think more about the initial “parts” of math modeling than is typically asked in the practice questions) because, as an M3 judge for many years, I’ve seen how teams that set a strong foundation early on in their solutions tend to be more successful when it comes to doing the actual math modeling. It’s so much easier to attack a specific, clearly worded question than something open-ended with multiple interpretations. Setting proper assumptions helps in building the models, often by introducing parameters or inspiring certain variables to include.
Student: How do you redefine the problem? And how important is HIGH LEVEL math?
Dr. Nicholson: I was just talking about the different math approaches to a specific question in my Senior Seminar course (college-level senior math majors) yesterday. We looked at 4 solutions to one of the M3 questions from a few years ago, each a Top Finalist team. What we saw was that the math used was so different from team-to-team. There was a Markov process/linear algebra approach, a differential equation approach, a probability approach, and one that involved little more than algebra.
What WAS uniform across the winning solutions, however, was all the “little” things: well written expository language, strong assumptions and justifications, good use of sources, clear explanations of the work, a demonstration that the team really understood what their math was saying, self-reflection on the strengths and weaknesses of their model, a fabulous executive summary, etc. All of these things didn’t depend on the level of math used, meaning that a team (and most competing teams do NOT have that “high level” math ability) has the ability to be successful regardless of their advanced skills.
Student: We are stuck on the steps to actually create a model – how do we do it?
Dr. Nicholson: There are a lot of approaches to creating models–they need not use software, but that’s one option. I’d suggest looking at solutions from previous years (the top 6 from each year are posted on the M3 Challenge website) and looking at the math approaches they use. Common approaches that come to mind that I see often:
2) Statistics and/or probability
A few GREAT resources to help with this are the guides created by SIAM:
Student: How do you break down or group all the variables related to a specific problem?
Dr. Nicholson: There are lots of ways to attack the problem–and I wouldn’t say any one of them is stronger than another. What I would say is to, as a team, clearly identify what it is you want to accomplish with your solution, and as your modeling process proceeds, always stay focused on that topic.
For this problem, assume the data you need is there and then build a model accordingly.
Student (Specific example): Regarding the building of a model to depict the success of our chosen industry (the college industry), we have multiple variables to consider, i.e: annual profit, proportion of students who can afford tuition, proportion of faculty trained in online teaching, etc. Could you suggest any methods on how we should go about implementing our variables into the solution model, or rather, how we could group together certain considered variables in order to best shape our model? Other than that, should we generally tend to prioritize creating a solution with a singular, all-encompassing model, or multiple smaller models that relate a few variables at a time?
Dr. Nicholson: Depending on the question, one vs many models often answers itself. I’d anticipate the more variables you use, the more likely it is you’ll have multiple smaller models that can be created and then those results fed into another model to answer “the” question.
Suggestions for creating a model for “college success?” “Success” is very open and can be defined how you choose it to be. Then, how you build a model depends on that definition. In this case, you’ve gotten a lot of variables that aren’t related working into “success.” I’d get the sense for many of these variables that looking at simple change before-during-projected after gives you somewhere to start building a model. For future projections, probabilistic models (ie, the likelihood that those changes stick or revert back to previous levels) might help.
One note: “annual profit” isn’t necessarily defined for colleges. Most colleges & universities are non-profit entities. Perhaps “growth in endowment principle” (ie, “money in” to the college from places other than tuition)