**Olga Goulko** (Department of Physics)

STEM EDX Fellow AY 20-21

### Overview

The idea behind Just in Time Teaching (JiTT) is that students solve an exercise (or answer questions) shortly before class. After the submission deadline, which is typically an hour or so before class, the instructor reviews the results and then adjusts the class material accordingly. For example, the instructor might spend more or less time on certain topics, depending on how well the students seem to understand them. The instructor may also read out certain answers (anonymously) and use them as basis for discussion to introduce the topic.

### JiTT Resources

- Novak, Gregor, E. Patterson, A. Gavrin, W. Christian, and Just-in-Time Teaching. “Blending Active Learning with Web Technology.” Upper Saddle River, NJ: Prentice Hall (1999).
- JiTT Digital Library Resources
- Wikipedia entry about JiTT

### Description of Activity, Assessment, and Reflection

I have used many different warm-up activities in my PHYSIC 616 Mathematical Methods graduate course. Some were more successful than others. I have included some successful ones along with a few unsuccessful ones to help others design their own.

For each warm-up, I will describe the activity, explain the main topics to be covered, give some sample student responses, and discuss how effective each was.

### General Structure of the Activities

In my implementation, the submissions were graded based on honest participation only (not correctness) and contributed 5% to the final grade. I assign only one warm-up per week, due two hours before the first lecture of the week, even if there are several lectures during each week.

The students always gave positive feedback about this technique and I believe it helps me teach more effectively. The extra work associated with designing and grading the warm-up exercises is not too great and is offset, in my opinion, by the benefits. Participation in the warm-ups could be better, especially as the semester progresses. According the student surveys that I conducted, the main reasons are that students tend to forget to submit the warm-up or are too busy with other things.

### Successful Warm-Ups

**Exercise 1. Classification of differential equations**

Please classify the differential equations below (order, linear/non-linear, ODE/PDE, homogeneous/inhomogeneous)

*Comments*: The goal of this exercise was to highlight and dispel common misconceptions regarding differential equations. Typically, students easily distinguish between PDE and ODE and can identify the order of the equation. The more difficult part is to realize that a linear equation must be linear in the dependent variable (the function one has to solve for) and not in the independent variable (which the function depends on) and that the inhomogeneous part of the equation does not need to be a constant but may depend on the independent variable.

Common mistakes thus include classifying equation a) as linear and equation d) as non-linear but homogeneous. Equation c) has also been wrongly classified as inhomogeneous on occasion, due to the way it is written. Most students made at least one such mistake. Another misconception is that linear is the same as constant coefficients, while the coefficients actually may depend on the independent variable. In this case b) would be wrongly classified as non-linear. I discussed these concepts in the lecture following the warm-up, relating them to the warm-up exercise.

**Sample Student Response to Exercise 1**

Example student response to the exercise on differential equations. I highlighted the incorrect responses. The student also indicated which aspects were most difficult.

I hope that letting the students think about these properties first and then explaining the background improves understanding and retention. I also like this question because it is simple to state and does not take long for the students to answer.

**Exercise 2: Exam Review**

Please formulate one question that you would like to have discussed during the exam review on Monday. As always, you can also post questions to the discussion board.

*Comments*: Some students are reluctant to ask questions during class or to reach out with questions. Forcing them to formulate a question as part of a low-stakes assignment allows me to get a better picture of where the difficulties lie. The exam review session is then built on the requests of the students. This allows me to prepare more detailed answers and examples in advance.

**Sample Student Response to Exercise 2**

Another more complicated Maclaurin series centered somewhere non-zero!

**Exercise 3. Wronskian**

Please read section 5.1.4 of the lecture notes. Compute the Wronskian of the functions *f*_{1}(*t*) = *t* and *f*_{2}(*t*) = *t*^{3}.

*Comments*: This is a very simple computational exercise, which is trivial to complete if the student read the lecture notes. I find that having a specific goal makes students more likely to read and internalize the lecture notes before class. At the same time this exercise does not take much time to complete. All students solved this exercise correctly. When I discuss the Wronskian during the lecture I mention that they have already computed one example during the warm-up.

**Sample Student Response to Exercise 3**

### Less Successful Warm-Ups

**Exercise 4. Green’s functions**

Give an example of an application of Green’s functions in physics. Have you encountered Green’s functions in other physics courses and in what context?

*Comments*: The answers the students gave were typically very dispersed and do not provide a good basis for class discussion. I still make some statements about them during the lecture and mention some specific examples that the students submitted. For continuity it is important not to skip any warm-ups, so in this sense this serve the purpose.

In the future, I would try making the question more concrete, for example: “Green’s functions play an important role in electromagnetism. Discuss an example.”

**Exercise 5. Fourier transforms**

Prove the following properties of the Fourier transform stated in the notes

• constant shift in time,

• scaling in time,

• nth derivative,

• and another property of your choice.

*Comments*: The drawbacks of this exercise are that it is too long and too complex, given that the students would need to complete it before the class, without having heard the explanation of the instructor. Because I knew I would not have time to discuss all of the proofs during the lecture, I delegated several to the warm-up, and this is not a good reason. I was also hoping that some of the proofs would be simple enough and could be skipped. Instead, the students were confused and discouraged. During the lecture I demonstrated most of the proofs to the students explicitly.

In the future, I think that just shortening the question would do a lot of good. For example to just ask to prove the first property and maybe have an example of another similar proof in the notes. Then the students could read the example proof and do a simple analogous proof. More complicated proofs would be delegated to the lecture and homework.