Should I gave a make up lecture if some students found what I taught is a bit unclear?

Question

This semester, I am teaching discrete math for computer science students. Today I taught solving linear recurrence equations. The way I did it was not rigors. Instead I used the method of advanced operators described here. I feel this is a bit easier to understand for CS students.

However, there were two top students coming to me after class, suggesting that it is much clearer to do it from a vector space perspective, which they have already learned in some linear algebra courses before.

There are three options for me.

1. Change how I teach this part next time.
2. Give a make-up lecture to explain everything from a vector-space perspective.
3. Do a survey and see what the majority of students wants.

All three options are feasible for me.

I am inclined to choose the last one. Maybe the majority have a different feeling about the lecture. What do you think?

Answer 1

I am inclined to choose the last one. Maybe the majority have a different feeling about the lecture.

If you run a survey and the majority of students "thought the last lecture was too difficult", that won't necessarily be because they already know linear algebra and would have found a vector space perspective easier to understand. It is possible that some of your students don't have a strong understanding of vector spaces, so they probably wouldn't understand what your offer entails (e.g. (re)learning a topic so they can learn a brand new topic).

However, if you think students knowing both perspectives is important, then by all means add another lecture on the topic. If this is mainly because your two top students already know how to do something from their linear algebra course (which it sounds like they do), and you deliberately chose to teach the method from the linked textbook, then I wouldn't burden the rest of the class with a change of direction on a "what if".

If you're really curious about what is a "better" way to teach the topic, experiment with the different approach next term.

Answer 2

My philosophy is schedule über alles. As an academic, time is your most constrained resource. This goes for balancing the hours in your work vs. research, teaching, and service requirements. And in this case it goes for the limited number of class sessions in your semester.

Do you not have a day-by-day schedule for your term? You should. For me, it's the very first thing I map out when preparing a course. What you really don't want (and I've seen it happen to other instructors) is to slide loosely through the term and then realize you need to somehow squeeze 3 chapters of content into the last week of class. The awkwardness you now feel is nothing compared to that.

So keep in mind that you must proceed forward to the following areas of content. If not, be clear-headed: exactly what will get cut from the course if you give a makeup lecture? Make a deliberate choice. And what will get cut from your other work this week as you prepare the new unplanned lecture?

In a college course, students should be expected to fill in gaps outside of class on their own. Actually, that's where the majority of the work should be happening. In particular if it's your top students, and they've found an alternative method that satisfies them, at your option, just let them use either method. But for other students I would lean towards sticking with the book presentation which they can consult and reference more easily.

I recommend the Steven Krantz book How to Teach Mathematics by the AMS as the go-to for general math-teaching philosophy. Section 2.7 on "Handouts" seems relevant here:

If you give a class hour on Stoke's theorem and feel that you have not made matters clear, then you might be inclined to draw up a handout to help students along. You also might suspect that this extra effort on your part will improve your teaching evaluations and, in particular, that students will appreciate all the additional work that you have put in. Well, it won't and they don't...

What I can do is examine my own conscience and tell you what I see. If I give a lesson that is not up to snuff, or if I do a poor job explaining what curvature is, or if I goof up a proof in class, then I can salve my conscience by writing up a handout. It takes about an hour, it is a way of doing penance, and it is a way of working past the guilt of having screwed up in class. In my heart of hearts, however, I know that what I should do is strive to give better classes.

In summary: Of your listed options, (1) is the most on-target. Krantz goes on to suggest that giving a handout online as an optional added resource to students can be okay. I would recommend in the main sticking to the chosen text presentation, regardless if some advanced students have found another method. Spending more class sessions on the topic should be totally off the table, as that way lies a cascade of further problems as other topics get squeezed out of the semester.