Gathering Thoughts, Part One: Mathematics in Decision Making

[Originally Published 12-19-2012]

Next term, I have three new courses to teach. Well, I have taught one of them (Dynamics) before, but that was before my conversion to Inquiry Based Learning. So, that one will be rewritten from scratch, too. I need to collect thoughts on these and get things ready for next term. So, here it goes…

Mathematics in Decision Making

This is Math 1100 at UNI. The Course Catalog copy reads

Mathematics in Decision Making -- 3 hrs. (MATH 1100) 
Selection of mathematical topics and their applications with an
emphasis on mathematical reasoning. Topics include probability 
and statistics. 

This course typically serves as a terminal mathematics offering for students who only need one course in mathematics to satisfy their Liberal Arts Core requirement on quantitative reasoning. Usually, each section has 65 or 70 students, most of whom would rather be anywhere else. I just checked and I have a section of only 35 at the moment. This might change a bit, but I am pleasantly surprised.

I spoke with one of my colleagues who helps with Liberal Arts Core coordination and Student Outcomes Assessment about what the required topics in probability and statistics are, hoping that they might be minimal. He said that if I viewed my section as experimental, then I could do whatever I wanted. So, it looks like next term I will be teaching an experimental section!

It seems that this is an opportunity. Most of my students will have not had a lot of success in the traditional calculus-prep sequence. Some may have, but they have chosen a program of study that involves minimal mathematics. So, I have just this last chance to show off how interesting mathematics is. This guides my choices.

At the moment here is my plan for the semester. The semester will be divided into three sections, with one major idea for each.

  • Comparing Large Sets.
  • Classifying Triangulated Surfaces
  • Classifying Wallpaper Patterns

Let’s take them in turn.

Comparing Large Sets

We will develop the idea of comparing two sets to see which is bigger, with the goal of seeing that the set of real numbers is bigger than the set of natural numbers in a very meaningful way. My colleague Doug Shaw did this last spring, and I will start by adapting his materials. I recall a conversation with one of my undergraduate roommates (who was studying engineering) about the concepts of countable and uncountable sets that involved him incredulously shouting “But, its infinity!” a lot. Basically I aim to get to this spot and then past it.

Classifying Triangulated Surfaces

I have always felt that the classification of compact surfaces could be (should be) taught to high school students. The really hard part is Rado’s theorem that a compact surface has a triangulation. If we put that in a black box, or just gloss over it, the rest is essentially cut-and-paste. I am going to design a sequence of problems that attacks this. I’ll be adapting the treatments from a couple of grad-level topology books, and maybe John Horton Conway’s “zip proof.”

Classifying Wallpaper Patterns

It is well-known that there are 17 types of wallpaper patterns, if we classify them based on the types of planar symmetries they exhibit. Similarly, there are only 7 types of frieze patterns and only 14 types of spherical patterns. (For the last two, there are some infinite families that I have collapsed into a single type.) These can be classified by discussion their orbifolds, which are decorated surfaces! I think we can make a reasonable unit about symmetry out of this, and rely on our previous work to get the fancy part to work out. This will follow the book The Symmetries of Things by Conway, Burgiel and Goodmann-Strauss in basic outline.

I hope this makes for an achievable and rich semester. I am convinced that the material above is fun, has real depth, but can be made accessible to a general audience.

Tasks left to prepare:

  • Adapt Doug Shaw’s problems about large sets
  • Design problem sequence about triangulated surfaces, including some very hands on days with manipulatives.
  • Design problem sequence about patterns, and find a bunch of examples. Maybe the local Sherwin Williams will part with some old rolls of wallpaper.

Delivery

I’ll be running this as an Inquiry Based Learning experience. To keep things positive, I will have it set up so that students work in small groups (3 students). We will reserve the last fifteen minutes of every meeting for presentations and discussion. Presenters will email me a digest of their work to be bundled together for everyone. [ooh. I have to think about how this goes for the more pictorial parts…] Problems left unfinished become homework.

I anticipated 65 to 70 students, so I convinced my grader to be more of a teaching assistant. He will come to class and help me circulate and lead discussions in productive directions. This is unusual for UNI, but the department head seems OK with this.

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