Day Two, Now I know what you don’t know!

Originally Posted 01-11-2012

One of the best part of teaching in an inquiry-based style is the immediacy of communication between student and teacher. Since every class meeting is filled with communcation between everyone in the the room, an active and engaged instructor learns quite quickly what the students’ real misconceptions are. This is in contrast to a more “traditional” lecture format class, where a teacher likely doesn’t see that the students don’t understand something until an exam, and even then it is not always clear what they don’t understand. Today, this problem was brought home in each of my three classes.

Dynamics

The issue here was about the following problem:

Task 7. In medieval Italy, a man named Leonardo Bigollo posed a problem about
rabbits. Suppose that it takes one month for a pair of rabbits to reach maturity,
and another month for a male-female pair of adult rabbits to, uh, “generate” a
male-female pair of newborn rabbits. Let’s keep track of pairs of baby rabbits
and pairs of adult rabbits. His original question was this:

If you begin with a pair of newborn rabbits, how many rabbits of each type do
you have after one year?

Make a table showing the number of pairs of rabbits of each type for the
first year. Then use your table to design a dynamical system modeling this
situation. What is your phase space? What is the dynamics? Can your phase
space be extended?

This led to a really rich discussion of how to model a physical problem as a dynamical system. In the end we finally came around to see that this is best handled as a model on a two-dimensional space. Many students tried to set up a model with the recursion that defines the Fibonacci sequence, but that doesn’t fit the mold of a dynamical system set-up (X,f), where X is the phase space, and the function f:X \rightarrow X is some unchangeable rule defining how you get from one state to the next. But a few did propose X = \mathbb{N}\times\mathbb{N} and f(n,a) = (a, n+a). I think we got there by the end of the hour.

Linear Algebra

We spent our time today on the geometry of linear combinations, and the geometry of intersecting lines in the plane. This is a much smaller deal than before, but a few groups were having trouble because the idea of making a set as the intersection of other sets, or as a union of other sets, was unfamiliar.

Math in Decision Making

I gave my students more problems about constructing bijections between large sets. One particular problem (that I stole from Doug Shaw, a combinatorialist) gave them fits.

Question 6. Let T be the set of 10 letter words, where the first two letters are
repeated, such as MMAQRESEDQ. Let J be the set of 9 letter words. Which is
larger |T| or |J|? Or are they the same?

It turns out that the basic problem was in figuring out what counted as an element of T. We talked about it for ten minutes. I think some students clued in to how it works, but a significant fraction are still confused. By the way, these students are refreshingly honest about when they don’t get it. Whatever the reason is, they are much more comfortable about sharing that they don’t yet “get it.” The math and math-ed majors only say this with really pained looks on their faces, as if it hurts them to admit it.

Summary

So I got to see what parts of my basic assumptions are not so automatic for my students. I will try to smooth the process out in the next iteration of each class, but I prefer not to try to completely “fix it.” Using IBL means letting the students struggle with difficult ideassorry, it really means supporting them through a struggle with ideas they find difficult. Now, the concepts above can be confusing for a newcomer to mathematics. I can’t really change that, but I can anticipate it and make sure it comes up in a way that allows the students to handle the question and move on.

I had a lot of fun in the classroom today, and I feel very tired. It is easy to forget what an active classroom can do to you.

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