Efficiently teaching mathematics is an exceedingly challenging task, perhaps even more so than understanding the subject itself. There are numerous approaches to teaching math, but unfortunately, there are just as many ways to do it poorly. What’s striking is that the fundamental competency for effectively conveying mathematical content – and, although probably to a somewhat lesser extent, virtually every type of content – remains consistent, regardless of the audience’s level. Whether you’re instructing secondary school students, conducting masterclasses at the university, or delivering specialized talks to experts at research centers, the key skill is the ability to forget, to set aside, your existing knowledge for a moment.
The problem lies in the natural inclination of lecturers to progress from explaining A to discussing C and finally reaching E, all the while presuming that points B and D are self-evident. However, upon closer scrutiny, it mostly becomes clear that these aspects are transparent primarily to experts who instinctively draw upon their knowledge and experience.
Setting aside your experience requires self-awareness, active listening to your own explanations, and deliberate focus on conveying the material clearly and concisely without assuming prior knowledge. This ensures that your audience fully understands the information before you move on to the next level of complexity. In this sense, teaching involves a cycle of temporarily setting aside your knowledge, carefully evaluating your communication, and understanding the topic anew yourself before moving on.
This technique necessitates thorough preparation though. Your objective is to make the audience wonder why they hadn’t thought of all this themselves. To achieve the goal, you must find a natural, logical, straightforward, and easily accessible path. This path is usually clear to an expert because they have a bird’s-eye view of the subject: having spent years experimenting with the often complex concepts underlying it, they’ve crystallized a simple structure that supports the entire thematic construct. This is what my PhD advisor meant when he challenged my belief that good pedagogical skills were sufficient to teach an undergraduate course, and explained that an active researcher views the topic differently and therefore probably teaches better, assuming of course the researcher has also taken the time to develop their didactic skills.
When a teacher sees the path, or mostly the paths, they first choose their strategy, opting for the “best” of these paths. They then divide the chosen path into smaller segments, which often differ in terms of complexity. While the less demanding segments partly run on their own, the more complex ones require careful reflection, mental experimentation, anticipation of beginners’ reactions, and skillful refinement before the teacher’s efforts develop into a smooth tactical approach that meets the aforementioned requirements of forgetting, listening, and relearning.
While some individuals may inherently possess the skill needed to implement the technique, it is typically developed and perfected through decades of practice. The effort is undoubtedly worthwhile though because the pleasure of hearing yourself speak clearly and witnessing a focused audience is an enduring source of pleasure.
