Posts Tagged ‘Modeling’

Discrete Mathematical Modeling Postmortem

April 21, 2021

I just got done teaching my Discrete Mathematical Modeling class. See here, here, here, and here for previous posts. Here is my summary of how the class went.

I did not do a good job with the course. I am not down on myself about it, since I don’t think that there was a way for me to do much better: I found out that I was teaching the course a in January, the class started in March (we are on block scheduling this year), and I had two intensive Calculus II blocks between when I found out and when I started teaching the course. I simply didn’t have a lot of time to plan. That said, I am pretty impressed with the level of planning that I was able to do, considering the constraints.

Moreover, I didn’t have a lot of time to adjust once the block started. In particular, I was the chair of a search committee, in charge of registration for my department, advising a student about to defend her thesis, finishing up an independent learning project with three students, and probably a couple of other things during the block. It was kind of a ridiculous confluence of big jobs. Again, I am proud of the way that I managed my time, but I was pretty constrained and I wasn’t able to do everything I wished I could have.

The interesting thing to me is that I wasn’t doing an objectively good job teaching, but I have enough teaching experience so that I could recognize this in real time—I think this was my first subpar teaching effort where I wasn’t subject to the Dunning–Kruger Effect. Here were my main issues.

  • I didn’t have enough time to prepare and adjust, as noted above.
  • In particular, the block schedule hindered my ability to adjust, even if I wouldn’t have had all of the other obligations. I didn’t have a good sense of how well things were working until 1.5 to 2 weeks in, by which time the class was halfway over.
  • My pedagogical content knowledge wasn’t where it needed to be. I hadn’t taught the class before, nor have I thought about modeling deeply before the class began. I simply wasn’t aware of what the students would find easy and what they would struggle with. This made teaching doubly hard—I didn’t know what to look for, and I didn’t always have the fastest way of correcting misconceptions. I didn’t know enough to be able to predict how students would react to the material, so I didn’t have tools available to help them.
  • Related; I mainly stole my modeling problems from trusted sources. However, the problems didn’t do what I wanted of them. In some sense, they weren’t as rich as I wanted them to be. Either I didn’t select the problems well, or my expectations for what problems could be was too high.

I have mainly been teaching the same classes for the past 13 years, and so I have developed a good sense of what to expect students to do. It was an interesting (and unpleasant) experience not be armed with ways to help students. It made me recognize how much progress I have made as a teacher in the last 20 years.

On the plus side, a lot of things went well.

  • Doing a version of ungrading was a really smart move on my part. This gave me a lot of flexibility that I didn’t know that I would need. When the class struggled with one particular project, I had the freedom to walk the class through it. This was really the best thing for their learning, and it didn’t mess up my grading scheme—I just gave them all credit for it. In my original plan, this would have meant that they only would have had four team projects instead of two, and I would have had to wring my hands about how to adjust my grading scheme on the fly. Students also reported that they appreciated the ungrading.
  • The next step in my evolution toward moving my courses toward the CURE end of the spectrum. The projects were open-ended, and the students had a great flexibility to look at problems in the way that they wanted. It was really exciting!
  • I felt like I did justice to the Justice theme. We did some modeling on the Flint water crisis and gerrymandering. Students had to do an individual project where they do modeling on a Justice topic. I thought that it would be difficult for students to find topics, so I supplied them with a default topic (model how to set up a wheelchair program in an airport. How many wheelchairs do you need? How much would it cost? What sort of system can you install to get the wheelchairs to where they need to be?). I was surprised that only two students did the default project, and I was very seriously impressed at the topics students came up with. Some student-generated topics were: how universal childcare would affect the wage gap between men and women in the U.S., how the availability of generic drugs would affect health of people who need them, and how to distribute stuffed animals to children who are detained at the border.
  • I enjoyed working with the students. They were game, and they came up with good ideas. I was a bit surprised at how many of them expressed that they really liked the class, given that I was a bit down on my own teaching.

I will teach this class again. It was really enjoyable, and I would love to do it with the proper amount of prep time.

Ungrading in Discrete Mathematical Modeling

March 24, 2021

My Discrete Mathematical Modeling course has started, and we are off to a good start. Here is what we have done so far:

  • We had a fantasy basketball auction, where the players were bid on only by formulas.
  • We did several Three Act activities.
  • We introduced the ideas of expected values, discrete dynamical systems, and linear programming (OpenSolver crashed, so we didn’t finish the linear programming problem).
  • We talked about Justice.
  • We talked about course policies, including grading.


This course has a Justice theme, and Justice involves power. I wanted to be very aware of the power I held over students, and I wanted to eliminate as many unjust parts of being a teacher as I can.

One thing that I did was to introduce something that is inching closer to ungrading. This is very similar to what I did in my capstone classes these last couple of years. Essentially, I gave them a list of assignments that they need to do. They can keep revising them until they get credit for them. If they do all of the assignments to completion, they get at least a B in the course.

This makes sense to me as far as the course goes, too. Modeling is similar to research, in that the students won’t end up at a predictable place (and they shouldn’t).

Discrete Math Modeling Planning

February 3, 2021

I suddenly have to teach a modeling course. I am excited about this, but it is a lot of work—particular for someone who doesn’t know a lot about modeling.

There were two decisions I had to make right away.

  • Will this be a class on models, or will it be a class on modeling?
  • How active will this class be?

These were answered simultaneously, in some sense. First, some background.

A course on models teaches students a bunch of models (linear exponential, etc), and then has the students apply the models to different problems. A course on modeling starts with messy problems, and works from there to figure out how to make sense of it. Grading in a course on models is straightforward: you have a good idea what to expect from a student, since they will most likely be applying a linear model when you are working on the chapter on linear models. Grading in a course on modeling is surprising. The students might apply a variety of different models, and their assumptions might be very different.

Both courses have their place, but I think that a course on modeling is more appropriate in my liberal arts setting. So I am looking for messy, real-world problems that students can approach. One problem, which I am already using in Calculus II, is “Use mathematics to predict how many people in [insert geographic region here] will have Covid on June 30, 2021.” I also ask about cumulative cases. Questions like this will be at the heart of my course.

This also requires students to be very active, since this course is about the modeling process rather than mastering known modeling techniques. This aligns with my goal to up the levels of active learning in my class (which have been good for about a decade, but I can still fall into lecturing too much sometimes).

There is some trickiness here. How are students suppose to learn about what models are possible? What will the assignments look like? What will grading look like?

I will start to answer the first here: I am hoping to introduce models by some combination of

  • Working through some models together, with me making suggestions (and giving mini-lectures where appropriate) when students get stuck.
  • Having the students work in different teams to create models, and then having them share the results. They will likely do different things, which will help the different teams learn about new techniques (I will be coaching the individual teams to help them refine their ideas, of course).
  • Introducing models as they would be done in a course on models (I am not a purist). I will likely introduce several models in the first week of course without practicing them a lot. This will introduce them to some tools.

As always, I am looking for advice for this course—it you know of messy, real-world problems that would be good for a non-calculus-based modeling course, I would love to hear them!

Discrete Mathematical Modeling

January 27, 2021

Our semester has started now, and I am teaching on the block schedule again (three hours of class on MTRF, and I have one hour of office hours each of those days, too).

After years of being good, I have over-committed myself this semester. This is only partially my fault. In addition to my usual teaching and chair duties, I volunteered to lead two student research projects, which I love doing. I am also teaching one extra class that was canceled due to low enrollments—four students would have graduation issues if we didn’t do this, so I am happy to do this, too. We are also hiring, and I am chair of that committee; I love that work, too.

All of this would be manageable. However, I recently found out that I also need to teach a third course this semester, and that is what is putting me over the top. The teaching isn’t so much a problem, but rather I don’t have much time to plan it.

The course I am teaching is brand new (no one has taught it before), and it is called discrete mathematical modeling. While I am a bit stressed about finding time to plan it, I am really looking forward to it (I also know that it will get done. Because it has to get done.). We are going to talk about optimization, discrete dynamical systems, and modeling using probability, and the only prereqs are a good high school math background (in particular, calculus is not a prerequisite).

I am just starting to plan this. Here are my initial ideas prior to starting the proper way.

  • The focus will truly be on “modeling” rather than “models.” The latter is showing students a bunch of models other people have made (linear models, exponential models, dynamical systems, Markov chains, etc), and then having them apply them to situations. While there will be some of that, I want the course to be more about the process of creating the models. This is half-baked (the execution, that is, not the idea), but my hope is to provide students with semi-messy situations, have them come up with some sort of mathematical model, and then perhaps show them a standard model for that situation.
  • In-class will be 90% having students create models and compare the results. The remaining 10% are showing them “my” models (a standard way of modeling the situation).
  • Specifications grading seems appropriate here. I need to figure out a set of specifications for modeling reports.
  • Several places have suggested that it is a good idea to quiz students on the basics of the standard modeling techniques, since students otherwise will tend to use the same modeling method repeatedly.
  • I want to incorporate some game theory (model the Cold War!), but I am going to try to stop myself; we are in a condensed block schedule, and I don’t have much time to plan it. I need to keep this simple.
  • The main technology will be spreadsheets.

One last cool thing is that this course has a Justice theme, so roughly 25% of the class should be about Justice. This is a helpful constraint to narrow down my choices, and I am also happy to learn more about Justice. I have already benefited from the MAA’s book on social justice (paid for by my work on the AMS Math Reviews).

My main sources that I am going to draw from are:

More than usual, I welcome suggestions on anything related to modeling.