## Student Research in Linear Algebra

One huge thing that I learned this semester: listen when Derek Bruff speaks. I have taken two things from him this semester:

1. Clickers are an extremely useful tool in teaching (and students love them), and
2. Poster presentations are a good idea.

I am going to focus on the latter for now. I am teaching a linear algebra course, which is the first upper-level course that (most of) our mathematics majors take. The class is mostly sophomores, although there was a large number of freshmen in my class this semester.

I had my students do research projects this semester. They were not required for everyone, although you needed to complete on if you were to get an A for the course. Also, I would very subjectively take your project into consideration for students who will not get an A. In all, 16 student out of 24 students opted to do a project.

The students had to write up a paper (in $\LaTeX$) and do a poster presentation. I suggested several topics for them to research:

1. Describe how a real world application works. For instance, describe how linear algebra is used when you Google something.
2. Given an $n \times n$ matrix with entries $1,..,n^2$, what is the largest possible determinant?
3. Given an $n \times n$ matrix with entries $1,..,n^2$, what is the largest possible eigenvalue?
4. Given an $n \times n$ matrix with all entries equal to $0$ or $1$, what is the largest possible determinant?
5. Given an $n \times n$ matrix with all entries equal to $0$ or $1$, what is the largest possible eigenvalue?
6. Given an $n$ and an eigenvector $\vec{v}$, can you determine an $n \times n$ matrix that has $\vec{v}$ as one of its eigenvectors?
7. Suppose that Player A always puts a $1$ in an $n \times n$ matrix, and Player $B$ always puts a $0$ in the matrix. Player $A$ goes first, and then they alternate turns. Suppose that Player $B$ wants the matrix to have determinant zero, and player $A$ wants the determinant to be anything but zero. Who can always win the game, what should the player do to win, and why will it work?
8. Write a computer program that solves systems of equations, finds kernels of matrices, etc.
9. Create your own project. If there is some question or application that interests you, let me know. I will help you determine if it is at the right level for Math 239.

I explicitly told the students that they were not expected to solve the problem. Rather, they had to be able to make progress on it. For instance, they did not need to find the exact largest determinant, but they should be able to find a lower bound for the largest determinant by constructing a family of matrices that achieve their lower bound.

My project format was very similar to Derek’s (I even used the same three award categories), but there were some differences. First, I did not have my students turn in a draft. This is largely because I did not have my act together, not because I am opposed to it.

Second, I had the students work individually. They had been working in teams all semester, and I wanted them to have something they could definitely create on their own. They were, however, allowed to confer with each other about projects. At most, I would have had 24 projects, so this was doable (in part because of the next paragraph).

Finally, the major difference between Derek’s format and mine was in grading. My grading was essentially a 0/1 system: either you did the project, or you didn’t. This made grading a little easier, and I am guessing (based on the psychology literature) made the project more enjoyable for the students. There were a couple of projects where I suspect the student did not put in much work, but only a couple. Those who did the project wanted to do it. (I did not really grade the projects, but I did read all of them to provide comments and feedback).

I was happy with the results. This is the first course where students see proofs in any sort of serious way, and the proofs they do see typically require them to just move one step beyond a definition. Thus, I did not expect sophisticated proofs. But the students worked hard and made interested conjectures (and a couple proved a theorem). The worst part is that I forgot to bring my camera, so there were no pictures.

The students also enjoyed it. I did a brief survey. Here are the results for the rating scale questions—”1″ means “Not much” and “5” means “A lot” (I averaged the numbers together for convenience, not for correctness).

1. How much did you learn from doing the projects? 3.86
2. How much did you enjoy and/or get a sense of accomplishment from doing the project? 4.43
3. How worthwhile was it to see posters of other people’s projects? 4.00

A couple of interesting points about the data:

1. One student said that he/she did not spend enough time, and this got in the way of learning and enjoyment. If this student’s numbers are omitted, the averages for the first two questions become 4.00 and 4.62, respectively.
2. I surveyed everyone in the course on the last question—even those who did not do projects. The people who did not do projects averaged 3.00 on the last question; people who did projects averaged 4.36.

I will end with some selected comments, the first three of which really warm a linear algebra teacher’s heart.

1. “Learned that proofs are important and nothing can be assumed about patterns and their continuation.”
2. “I learned the importance of a proof!”
3. “I think it was a very good idea to have a project applicable to linear algebra…I did learn that without a solid proof of something, it can be easy for someone to prove you wrong :)”
4. “I learned a new thinking style of how to observe from the inside and outside perspective of problems.”
5. “…I mostly learned how to work by myself/teach myself something new.”
6. “I learned a lot about working in $\LaTeX$ and a lot about discovering things on my own about mathematical concepts.”
7. “It was just enjoyable to ‘nerd out’ over a math project and to explore different topics w/o the aid of a professor (no offense…)”
8. “I found myself losing track of time, quite enjoyable.”
9. “Well actually it drove me nuts but it was fun.”
10. “I enjoyed doing the project because it made me think more deeply on the question.”
11. “It solidified a lot of the concepts we learned in class + helped a lot in learning to organize a long math document.”
12. “It was painful to go through it first, but playing the ‘game’ and argue it with friends was such fun during that time.”
13. “I know that I would have really enjoyed it if I had put more time + effort into it. It was fun to have something to work towards.”
14. “I learned that procrastinating is a terrible idea and that I need to be patient when working stuff out because I will figure it out eventually.”
15. “I liked the feeling when it was all done and I knew all the work I put into it was worthwhile.”
16. “The math was OK until I figured it out then it was great.”