Junior lecturer

In the first half of 2024, I lectured first year mathematics at Stellenbosch University, covering mainly precalculus and calculus. I taught a class of approximately 220 students.

My duties

• Teaching classes (5h of lectures each week)
• Compiling tutorials & memoranda
• Compiling & marking exams
• Corresponding & meeting with students as needed

My feedback

I set up an anonymous feedback system for my classes whereby students could review my lectures and submit comments. My feedback was generally positive, with the main criticisms usually pertaining to the pace of the module, my vocabulary being inaccessible, or my tendency to speak too quickly at times.

Several students expressed appreciation of my enthusiasm toward the subject. Some comments along these lines are given below (selected with some obvious bias):

Love the Lectures, I never liked math in High school but your enthusiasm and knowledge make this module enjoyable.

I appreciate your style of lecturing. Your focus on developing a deeper understanding of the concepts reflects what I believe to be the benefit of an academic university environment.

Overall your class is interesting and you make maths (that is a headache for some) to be fun! Your intelligence is inspiring and your excitement about the subject is refreshing.

I really enjoy your enthusiasm for math, it is highly contagious. Thank you for your positivity.

My mathematics lecture is what I look forward to quite often in the week. It is a conducive and learning friendly environment. My lecturer’s passion is what inspires me to be as passionate about mathematics.

My lecturing style

My lectures tend to be highly structured, with sections, theorems, comments, etc all numbered and cross-referenced.

My pedagogy is generally to begin by formulating definitions / theorems in their full abstract generality before launching into concrete examples. I appreciate that there is some contention about whether this is the most effective pedagogy, so I wish to briefly motivate this decision.

The alternative approach is to structure lessons in such a way that students build towards abstract ideas by generalizing / extending known concrete examples. The thinking is that this makes the abstract notions, when first encountered, feel more natural and familiar, and that it more closely mirrors the process of doing mathematical research. Generally, I agree with these advantages. However, I feel that this approach has some major downsides, when applied to a large introductory course:

1. It buries the lead. Leaving important abstract results for late in the course gives students too little time to digest / memorize / play with / apply them. On the other hand, leading with the result allows time to process it, and it contextualizes the examples that follow. In short, a difficult concept encountered early is far more likely to stick.
2. It alienates weaker students. Students enter a freshman course with a wide range of mathematical exposure and ability. Few students are yet mathematically mature enough to identify and abstract patterns in some given set of concrete examples. As such, the feeling of self-discovery that comes from a concrete-to-abstract pedagogy is lost on many.
3. It takes too long. A freshman course has a lot of ground to cover, spanning many topics. For an abstract idea to feel well-motivated, one needs to first see a lot of examples. There simply isn’t time to do this effectively during the lectures. Instead, lecture time is more economically spent in parsing the abstract notion and showing a few examples, with the remainder of examples being relegated to tutorials / exercises.

That being said, I am not dogmatic about this approach. I have limited teaching experience, and I am certainly open to the idea that alternative pedagogies can be more effective in some / all cases.