# An Interesting Juxtaposition

Originally Posted 01-23-2012

Today’s time in the classroom showed an interesting juxtaposition. The “low level’’ class that is focusing on big, deep ideas is on track, but the more advanced class working on “straightforward material” spent a lot of time not going very far.

### Math in Decision Making Success?

My liberal arts course students have made good progress on understanding the idea of a bijection. We haven’t completely formalized it, but I think most of the class basically “gets it.” In fact, today’s four exercises proved to be not enough. They dispatched them quickly and we were done with ten minutes to spare! I’m afraid that means I might need to come up with something meatier to add to Wednesday’s activities. This shouldn’t be a problem—coming up with hard problems is easier than coming up with approachable ones.

### Linear Algebra: The End of the Activity

It took a surprisingly long time to get through the “solve and sort” phase of the activity I started last time. But I am happy with the results. I asked the class to distill some lessons from the experience, and I got this list:
1. The size of the solution set depends on the number of pivots in the matrix.
2. The origin is a solution exactly when the system is homogeneous.
3. If you are working by hand, it can be helpful to look for clever tricks instead of just blindly following the algorithm.
4. If you had to do a lot of these, or even just one of any appreciable size, you want to use a computer. It is not difficult, just tedious.

So, it took a lot longer than I planned, but the main points came across. I used the end of our discussion to introduce the term “rank” and we talked a little about writing the solution set in the standard parametrized form using vector operations.

### Dynamics

If you have bothered to look at these notes, and especially if you are doing something as odd as working through them, know that problem 29 was a total failure.

That is, I am sure the problem is fine, but it is totally misplaced. The students had no idea what to say about it. I’ll have to find the right time to bring that point back up.

Also, one of the more advanced students wrote up a nice presentation of what it means for a sequence to converge along with his proof for task 15. I hope it helps out some for those students who haven’t taken a real analysis course, yet.