### by Peyam Tabrizian, Mathematics

#### Teaching Effectiveness Award Essay, 2013

When I was a freshman at UC Berkeley during Fall 2006 I took Math 54, Linear Algebra and Differential Equations. The main thing that bothered me with this course was the way it was organized: for the first ten weeks, the professor taught the core concepts of linear algebra, and then spent the next five weeks solving differential equations. There were two problems with this: First, by the time we started the differential equations part, I had already forgotten most of what I learned in the linear algebra part, which is a problem because you *need *linear algebra to solve differential equations. Second, it was hard for me to understand the *relationship *between the two subjects. I left this course with the impression that I was taught two completely different topics.

During Fall 2011, when I was a graduate student at UC Berkeley and a GSI for Math 54, I noticed that my students faced exactly the same issue. As a result, their quiz and exam performance on the differential equations part was much worse than on the linear algebra-part. Moreover, some of them told me later on that they were so overwhelmed with the material that they didn’t get anything out of the course, and they consequently suffered in later classes due to a lack of Math 54 knowledge.

Then, during Summer 2012, I had the pleasure to be the instructor for Math 54, which meant that I was able to design my own syllabus and determine the order of the topics to be covered. That’s why, in the spirit of my Math 54 experience as a student and as a GSI, I decided to reorganize things. Instead of teaching the course in two separate chunks, I mixed the topics up in a way that I would *first *teach a linear algebra concept, and then *immediately *apply it to differential equations. For example, right after teaching the students how to diagonalize a matrix (a linear algebra topic), I taught them how to solve a system of differential equations, which essentially involves rewriting the system in terms of a matrix and then *diagonalizing *it. My hope was that the students could see an immediate application of linear algebra to differential equations, which would help them not only solidify the concepts that were taught, but also make them appreciate the beauty and wide applicability of linear algebra.

There were two ways in which I assessed the effectiveness of my strategy. First, on the exams in the summer, I put some questions that were identical to differential equations quiz questions I gave to my students in the fall, as well as some harder questions which required the students to combine their linear algebra and differential equations knowledge. Unlike the fall students, almost all the summer students answered them correctly. Second, a couple of months after the summer course had ended, I asked my students what they thought about my course organization and whether they retained what they learned. They all gave me positive feedback. For example, here is what one of them wrote:

“Learning Fourier Series [a differential equations-topic] while orthogonality [the corresponding linear algebra-topic] was still fresh in my mind made for an effortless transition, allowing me to quickly move on to more advanced techniques and problem-solving skills. I think we learned diagonalization at the ideal time, because it felt very natural and straightforward to apply these techniques to solving various kinds of differential equations soon after. [The teacher’s] seamless transitions between linear algebra and differential equations not only helped me to master the material, but also revealed the underlying beauty and connectedness of mathematics. I [still] frequently use the techniques [the teacher] taught me in Math 54, as his approach led to the type of genuine understanding of the material that becomes permanent in the memory.”

This strategy also unexpectedly opened up some free time in my teaching schedule because when I taught the students a differential equations topic, I didn’t have to constantly remind them about the corresponding linear algebra topic. This also solves a problem many Math 54 summer instructors face: covering such a dense material in only eight weeks. As a result, I was able to give the students more examples and even teach more advanced topics like Markov chains, which they really appreciated.