A lot of discussion in education today focuses on how to make certain concepts more palatable to students and more easily understandable. However, it seems that there isn’t enough emphasis on teaching people how to learn those concepts, especially when they come from suboptimal explanations. I conceptualize this as a difference between teaching content vs teaching process, and in general I’d argue both are incredibly important for students’ future success. In this post, I elaborate further on some aspects of teaching process that I think we should incorporate more deliberately in education.
Teaching Process in Math
Say for example that we want to teach a subject like math. There is a lot of emphasis today on producing visualizations of math concepts via video formats, and this offers possibilities for education that simply weren’t feasible before digital technology. I’m super excited by what these formats and methods can achieve, both today and in the future.
However, at the same time, ultimately a lot of research and graduate school communications are still conducted today in written formats, via papers and textbooks. It is an interesting debate as to whether video and other digital-specific formats will eventually replace these even in the furthest reaches of research, or whether there are fundamental attributes to written content that make them always preferable in research settings. (For example, at least today, I find written content to be much quicker to produce; as long as this is the case, it would then offer a faster turnaround, making it preferred for research communications.) Regardless, for the time being, research is still often conducted with the frequent use of written content.
Thus, if you’re preparing students to go further in math, say for graduate school and/or eventual research positions, it’s not enough to teach the math content well; you also have to instruct the students in how to read written math effectively themselves. For example, graduate school professors seem to often assign their students entire textbooks to study, where people who come in with an ability to parse these resources effectively will be at a significant advantage. This is why many undergraduate math programs contain courses on exactly this, often as part of something like “Mathematical Communication” or “Mathematical Exposition.”
Teaching Process Beyond Math
Another example of teaching process, with more general applicability beyond math, connects to an often-quoted maxim that says something like, “you only truly understand a subject when you can summarize it in two sentences.” Certainly, it is awesome to keep searching for better and simpler explanations of concepts (without losing their essential meaning or oversimplifying them), and it is also great to keep finding better approximations to concepts that can require fewer prerequisites and enable more people to understand them. However, I disagree a lot with this maxim. In general, people should not expect a constant limit on the size of such explanations, especially one that can work with the same effectiveness across all subjects, topics, and contexts. Instead, if they really want to learn the concept, then it is in their best interest to develop a more scalable process for learning from a wider variety of explanations, even when such explanations may not be the most optimal or simple.
In general, there is a difference between aspiration and expectation. We can always aspire to teach content better and more simply, but ultimately, we should set expectations appropriately with students that such higher-quality presentations may not always be available. (For example, this can be the case for research-scope topics where people still haven’t even figured out much of the actual content, let alone how to effectively present it.) Generally, many valuable concepts can require more elaboration and more prerequisites, and for maximum flexibility, it is better for students to adopt the framework of “what prerequisites do I need for this and what is the expected length of the explanation” as opposed to “explain it to me in two sentences.”
Parting Thoughts
There are many other aspects of teaching process that I think are important to consider, including some subject-specific ones. I’ll probably elaborate on more of these in future posts. But in general, educators should have the goal of serving not just as instructors, but also as mentors: explain the content more effectively, and in parallel enhance students’ abilities to derive from wider varieties of explanations and presentations that they may encounter in the future.
