This year, as part of my duties, I am a College Fellow for the undergraduate Algebra (Hons) sequence at the University of Chicago. I conduct a weekly problem session for the students, which serves as an opportunity to review class material, go over tricky points in homework problems, and introduce students to more examples.
On occasion, I’ve come across the problem of wanting to make a subtle point. Often, my desire to make that particular subtle point seems to simply be an ego issue. I’m not referring to the kind of subtle points that there is universal agreement need to be made — the kind that every book introducing the topic makes explicitly. I’m referring, instead, to subtle points that may be considered pointless both by beginners and experienced people. The beginners consider them pointless because they don’t comprehend the point, while experienced people may consider them pointless because they understand it and don’t see what the fuss is about.
One example is the two definitions of group — one definition that defines a group as a set with one binary operation satisfying a number of conditions, and the other defining a group in terms of three operations (a binary operation, a unary operation, and a constant function) satisfying some universally quantified equations. The latter definition is termed the universal algebraic definition, and its importance lies in the fact that it shows that groups form a variety of algebras, which allows for a lot of general constructions. The former definition is important because it indicates that essentially just one group operation controls the entire group structure. This kind of differentiation may seem pointless, or “yeah, so what?” to a lot of mathematicians.
Another example, that actually occurred in a recent problem session, is that of differentiating between a polynomial and a function. if is a ring, a polynomial in gives rise to a function from to . However, polynomials carry more information than simply the functions associated with them: in other words, different polynomials can give rise to the same functions (this happens, for instance, over finite rings, though it does not happen over infinite integral domains). I’d made this point a few times in earlier problem sessions, and there was even a homework problem that essentially did this stuff.
So, during a problem session review of the notion of characteristic polynomial, I decided to make this subtle point distinguishing the characteristic polynomial from its associated induced function. I made the point that in order to compute the characteristic polynomial, we need to do a formal computation of over a polynomial ring rather than simply think of a function that sends to . This is a rather subtle, and in many ways, an apparently useless point (in fact, I don’t know of too many places that make this distinction very carefully at an introductory stage). However, I wanted to make it in order to rub in the difference between a polynomial and its associated function.
I hovered over this point for quite some time, so I guess a reasonable fraction of the students did get it, but at the beginning, one girl simply didn’t see the distinction between the two things, and was honest enough to admit it. So it took a couple of minutes to spell the distinction out.
In this blog post, I want to explore some of the arguments for making subtle points, and some effective ways of doing so without stealing too much attention away from the not-so-subtle-but-more-important point.
Make sure that people are already exposed to situations where the subtle points matter, or where naive intuition fails
In the case of the characteristic polynomial, the students had already done a homework problem that showed that over a finite integral domain, different polynomials could define the same function, and over an infinite integral domain, different polynomials always defined different functions. (A third part of the problem also showed that for infinite rings that are not integral domains, different polynomials may define the same function).
I don’t think doing this problem made people immediately appreciate the subtle difference between a polynomial and its associated function. However, having done the problem, they had a backdrop of reference when I pointed out the subtle distinction. In other words, I could say, “As you’ve already seen in this problem you did, different polynomials may define the same function.” This creates a back-stitch between stuff they did and stuff they’re seeing.
Make sure that the subtle point doesn’t overwhelm the main point
In general, I venture into the subtle point only if it either reinforces the main point being made, or if people have already understood the main point and need further understanding of the subtle point in order to round up their understanding.
Moreover, I also try to make the subtle point in such a way that if people don’t get the subtle point, this does not interrupt their flow of understanding the remaining material.
The case for subtle points: segmentation, replay, and subconscious influence
What if a large fraction of the audience ignores or does not understand the subtle point? I think there are still reasons for making subtle points.
The first is segmentation of the audience. Even if a lot of the people in the audience do not get the subtle point, there may be a few (who have an ear for subtlety and have understood the rest of the material well enough to focus on the subtlety) who can appreciate the subtle point. For these people, not having the subtle points makes the session a waste of their time, so the subtle point is a real value-add for them without being a significant detraction for anybody else. This is similar to all the features that software developers may add that help accomplished software users without distracting the simple users.
The second is replay. This is probably not applicable much to the problem sessions, because students are unlikely to review the problem session material too many times. Besides, they cannot replay a video of the problem session. But replay plays a greater role for online course videos, course texts, and other things that can easily be reviewed and replayed. In the case of replay, a subtle point that is missed by the audience the first time may be picked up during replay time.
The third is subconscious influence. The argument here is that people in the audience, even if they do not understand the point completely, hear the arguments and these make a subconscious impact — for instance, they might understand the argument more quickly if it appears in the future, or spot similar ideas if they see them elsewhere.