## Posts Tagged ‘Linear Algebra’

### Inquiry-Oriented Instruction

February 15, 2017

I was part of a grant last semester to implement a set of teaching materials that has been refined over the last decade. The materials use a teaching method called inquiry-oriented instruction, which I would say is a subset of inquiry-based learning (IBL). I used these materials in my abstract algebra class, although there are materials for both linear algebra and differential equations, too.

A very brief description is “intuition comes before definitions.” The materials introduce quotient groups by discussing Even and Odd integers, which students could easily see is a group at that point (using rules like “Even + Odd = Odd”). Once they got familiar with the idea that we could have sets of elements make up a group, we slowly backed our way into the definition of coset. It was pretty impressive to see students very naturally come up with definitions—having the right prompts helped a lot.

As part of the grant, I went to training at North Carolina State to use the materials. I also had funds to have student video my class, which will be used to analyze how well instructors who were not involved with the development of these materials can implement them.

We also used the class video as part of a weekly online working group. The purpose of this group was to prepare us, both in terms of pedagogy and course materials (not everyone was an algebraist), to teach the class. We discussed the purposes of the prompts, talked about what was going well and poorly, and watched video of each others’ classes. I found this immensely helpful.

I would use these materials again (in fact, I am planning on using the linear algebra materials next year). My sense is that my students had an abnormally good grasp of the definitions; previous students have struggled to understand what a coset means, for instance. My focus for the next time I use the abstract algebra materials is to work harder on the technical proofs—I think that my students did better on writing proofs than the previous time I taught the course, but not by a lot. Still, I think that the gains in intuition were worth it.

Links to the abstract algebra, linear algebra, and differential equation materials can be found here in the middle of the page.

### Undergraduate Reseach: Jump Before Looking

May 7, 2014

I talked about my plan for undergraduate research last week. This week, I invited my linear algebra class to join a research team I am forming.

The class is roughly half sophomores and half first-years. They have had calculus and linear algebra. My plan is to come up with a research question based around either finite fields or group actions on cyclic groups. I feel like I have some questions at the appropriate level that have come up in my own research, although I cannot explicitly state them right now. I had better be able to by next fall if any students decide to join the research team.

### Linear Algebra Class Structure

February 21, 2014

I was originally scheduled to teach abstract algebra this semester, but my section was cancelled due to low enrollment. Instead, I am teaching linear algebra, as we had higher-than-expected enrollment there.

The good news is that I can use the same basic course structure for linear algebra that I was planning to use for abstract algebra. The model is this:

1. The semester is divided into two parts. The first part, from January 15th until March 31st, is where we learn the content. The second part is all review and assessment.
2. For the first part, we do IBL-type presentations on Mondays and Fridays. Each day, we can do 4–6 presentations in 55 minutes. On Wednesdays, we review what we learned on the previous Monday and Friday. The reason why I chose Wednesday as the review day was so that students could have at least three nights to prepare for each presentation day.
3. For the second half of the semester, we will alternate between assessment days and review days. Students will be able to choose what they want to review based on what they found most confusing from the first half of the semester AND from the recent assessments.

One advantage of having the Wednesdays saved for review is that I can use it for an emergency presentation day if a Monday or Friday class is cancelled; this has happened twice so far this semester, due to cold and snow (including today).

One problem that I have is that the course notes I wrote for linear algebra have 314 problems in them. Since I am compressing the presentation part into the first part of the semester AND only using Mondays and Fridays for presentations, I only have 20 presentations days for the 314 problems. This means that we need to average 16 problems per presentation day. I accomplish this by designating 6 problems as “Presentation Problems” (which will be presented, naturally), creating video solutions for another (roughly) 6 problems, and then leaving the remaining four-ish problems without solutions (these are mostly computational problems for which the students were given a video “template” on how to do the process).

It took a while to create the videos, but they are pretty much necessary for our course. This course serves as a very gentle “Introduction to Proofs” course, but the level of proof that is expected is of the “figure out how the proof follows directly from the definition” type. Since there are more complicated proofs that need to be done in the course, I would either need to lecture in class, have the students read the proofs from a textbook (which we don’t have), or create video lectures.

Also, given that we only have six Presentation Problems each day, I have developed a method of having the students volunteer for the problems that cuts down on the amount of work that I have assigning students to problems. My usual way of doing this is putting one essay quiz on Moodle that asks “Which problems would you like to present?” I still do this for my capstone course, in which we present 15 problems per day. For linear algebra, though, I put one quiz consisting of one multiple choice question for each problem that is to be presented. The students are given three choices: “I want to present this problem,” “I really want to present this problem,” and “I changed my mind—I no longer wish to present this problem” (a student who does not want to present does not need to complete the quiz for that particular question). I assign each question 10 points, 5 points, and 0 points, respectively. These points do not affect a student’s grade, but a there simply so I can look at the quiz summary to see each student’s preference quickly without much clicking. The drawback to this is that there is a lot more to do on Moodle (6 quizzes per day instead just one). However, I created all of the quizzes at the very beginning of the semester, and it didn’t actually take that long to do once I learned about the “duplicate” feature on Moodle.

We are just over halfway through the presentation days, and the class is going really well. I think that I have a remarkably good class, so I cannot really say how this class structure is working; I think that any class structure would work with this particular group of students. On the other hand, this shows that this class structure can work, given the right set of students.

### Again, a new IBL-Peer Instruction Hybrid Model

December 24, 2013

I am continuing to try to figure out a way to effectively use both IBL and Peer Instruction (“clickers”) in my classes.

First, my main constraint: my favorite grading scheme requires students to be given many chances to get questions correct. Ideally, this means that we would finish with new content for the course 1/2 to 2/3 of the way through the semester.

Here is the approach I have been using up until now:

1. First part of the semester: Students get the content from reading the textbook.
2. First part of the semester: Students assimilate the content through Peer Instruction.
3. Second part of the semester: Students do something that resembles (but isn’t actually) IBL.
4. Second part of semester: Assess the students a lot.

Below is the same model I discussed last summer for my abstract algebra class. That abstract algebra class was closed due to low enrollment, and I was assigned linear algebra instead. I am keeping the same model, although I have a lot more exercises/theorems/conjectures in my linear algebra notes than I do for my abstract algebra notes.

Here is the new approach:

1. Mondays and Fridays during first part of the semester: Use IBL and student presentations to introduce the content.
2. Wednesdays during first part of the semester: use Peer Instruction to review and solidify ideas learned on the previous Friday and Monday.
3. Second part of the semester: We review the most difficult material through Peer Instruction and in-class practice.
4. Second part of semester: Assess the students a lot.

Here is the main problem that I am facing: I have 312 exercises in my IBL notes; I basically wrote the notes that I wanted—including many examples to build intuition—and I am now trying to figure out how to shoehorn all of the content into 1/2 to 2/3 of a semester. This works out to an average of about 7 exercises per day if we did IBL work every day of the entire semester, 10 exercises per day if we did IBL work on Mondays and Fridays (and review on Wednesdays) every day of the semester, and 20 exercises per day if we did IBL work on Mondays and Fridays (and review on Wednesdays) every day for half the semester. So I want to see if I can do between 10 to 20 exercises per class IBL class period, which is too much to do without some modifications. Here are the options I can think of to make this happen:

1. Cut some of the content. I don’t want to do this.
2. Provide screencasts of some of the exercises. I want to do this anyway, since part of the goal of our linear algebra class is to introduce students to proofs, and I believe that it is very useful for students to see worked examples. But I don’t want to have to provide 10–15 screencasts each class period.
3. Simply do not cover many of the intuition-building exercises in class; Dana Ernst suggested this to me yesterday, and I think that it is brilliant. There is not reason why I have to do everything in class. Perhaps I could just take questions on any intuition-building exercises after we do the main theorems; I could provide screencasts for some of these if we run out of time.
4. Other ideas?

Right now, my plan is to have students present and thoroughly discuss roughly 5 problems per IBL day, I would do screencasts for roughly 5 problems per day, leaving roughly 10 intuition problems to leave for the students to do.

Do any of you have ideas about how to improve this?

### Student Research in Linear Algebra

December 9, 2010

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.”