I have recently found a curious phenomenon in which teaching math can actually lead to new mathematical ideas, and as a result possibly new research. In this post, I’ll discuss some ways this can happen.
I write about and develop my mathematical ideas in the parts of this blog where I chronicle my learning journey in math and science. While I am not sure whether some of these ideas may be suitable for original research, the manners in which some of these ideas arise can nevertheless inspire a similar possibility for practicing researchers at the cutting edge of the field.
In general, teaching someone a subject helps to increase your own understanding of it. Part of the pathway towards that additional understanding can involve asking new questions about the subject matter, which can lead to new investigations and ideas with it. There are a number of concrete example ways this could arise:
- The search for more “intuitive” or “teachable” definitions can lead to questions about equivalencies to alternative characterizations. In the case that equivalency turns out to be true, it offers an exciting new approach to the concept; in the case that it doesn’t, it carries implications for education where instructors need to remember to distinguish things properly, and it opens up avenues to study different, non-equivalent concepts that may nevertheless be related and/or motivated similarly. A number of my ideas have arisen from this, a recent example being alternative “Boolean algebra-like” characterizations of common math structures like rings (as discussed in this post.)
- Relatedly, statements that can be made as “informal approximations” in the midst of motivating a concept or result can inspire new research if we ask whether we can actually formalize and prove some aspect of this approximability, or possibly even prove equivalency up to some level. Again, further avenues of research can result here depending on whether or not such a formalization turns out to be true. A number of my ideas have arisen from this — for example, my “topology as an algebraic structure” exploration started with me claiming that topology is fundamentally different from algebra, which then led to the follow-up question of whether that was based on the way the topology axioms are standardly written or actually fundamentally true.
- Instructors teaching students about their research, especially when such research is not up until that point well-known by others, can spur greater developments on it — exposure of different ideas and intuitions to each other can result in fruitful new work. As an example, in a famous anecdote, Richard Feynman used to carry five or so different problem-solving approaches in his head that he would apply to any new problem he encountered, which possibly enabled him to see connections and try non-standard approaches that people up until that point hadn’t considered simply because they weren’t in the headspace to think about such approaches. This is also a good reason why many math classes, even for undergraduates, try to expose students to unsolved problems.
- Relatedly, structuring new classes in non-standard ways can lead to fruitful new approaches and connections. For example, consider a class that combines and weaves two subjects that aren’t traditionally discussed together. Such a course could yield new research insights especially when presented to students who can develop that point of view further.
There are probably other similar ways in which research questions can arise from teaching scenarios, which I will add to this post later on once I think of them.
In fact, many of my ideas that I have posted about (or will post about) have arisen from these kinds of questions (especially from the first two points, since I didn’t major in math so I wasn’t exposed too much to alternative-style courses. However, the easy access of course materials online, as with for example MIT OpenCourseWare, makes even the other points accessible to non-math majors too.) In this way, teaching can enable researchers at the cutting edge to come up with new ideas, spurring them to develop new theories which in turn can help advance humanity’s overall understanding of math.
