Puzzles, motivation, and "expert blindness"

I was sitting in my friend's office yesterday, chatting about math and perusing his officemate's extensive book collection. (I was also sitting at said officemate's desk and pretending to be him for the day, not too successfully, but I'm working on that impression.) We both realized we were incredibly hungry and needed to be back on campus in an hour and a half for commitments, so we walked to Oakland to grab a sandwich at Uncle Sam's. I brought along a book for entertainment: Peter Winkler's Mathematical Puzzles: A Connoisseur's Collection. On the walk, my friend reminded me that he hates puzzles.

As an avid puzzle person myself, I couldn't empathize, so I asked him to elaborate. He used, as an example, the following problem we were talking about that I had randomly flipped to in the book:

Prove that every set of ten distinct numbers between 1 and 100 contains two disjoint nonempty subsets with the same sum.
Winkler, P. Mathematical Puzzles: A Connoisseur's Collection. AK Peters: Natick (2004), p. 22.

If you don't know what this means, here's the idea: if you pick any ten numbers between 1 and 100, like {2, 5, 10, 24, 27, 33, 50, 88, 89, 98}, then you should always be able to find two non-overlapping groups of those ten numbers that add up to the same result, like 50+33+5=88 in this case. Of course, 88+10=98 and even 24+27+50=89+10+2 also work, but that's not the point. The idea is to definitively show that you can always do this in at least one way, no matter what the 10 original numbers were!

Now, try working on this problem yourself for a few minutes and see how you feel about it. Does it seem too easy/hard? In between? Do you get frustrated easily, or do you have a vested interest in figuring it out? Do you feel motivated to ultimately solve it? Do you want to just skip to the answer? Do you want a hint or two? (Highlight the text below to see them.)
Hint 1: Use the Pigeonhole Principle.
Hint 2: Consider the number of nonempty subsets of 10 numbers and the number of possible sums those sets can have.

Here is the point that my friend made: there's no motivation to solve this one. Yes, he could probably figure it out if he sat down and thought about it concertedly (and we did, indeed, talk out the solution, with one hint from the book), but why bother? That time is better spent somewhere else, like his research. I couldn't think of a good response to that at the time, but in retrospect, here's what I have: if you're only worried about the end result, then of course the journey isn't going to be much fun! You don't know ahead of time what a puzzle or exercise might teach you, nor do you know what types of skills you'll get to practice. I thought this was a neat (and somewhat surprising) application of a basic mathematical theorem and was glad I had taken the time to think about it.

What does this mean for teaching? Well, our discussion of the above problem was already tainted by the fact that we were both practicing mathematicians and educators, and there wasn't really a debate over whether the problem was interesting or useful in its own right (my friend's position that he just has "better things to do" notwithstanding). Now, imagine a student who might be unsure about whether he/she likes mathematics (or whatever subject you're teaching), whether the pursuit of that subject is in their future; or, even imagine a student who is just taking a course out of necessity and needs to pass to move on to "better" courses. Replace the above problem with some interesting (to you) topic in the course, or some skill that you see as useful for them to learn. How do you make them "like puzzles", in this sense? Like it or not, we have to address situations like this all the time, on a wide variety of scales: How can I make my students see the usefulness of this example from lecture? How can I make my students care enough about the course content to pay attention in class and spend enough time studying outside of it?


These are tough problems, and I'm afraid that there is no generally helpful advice. This is too situational to address. That said, though, I think it's important to keep this in mind all the time. I remember the phrase "expert blindness", or something like that, coming up in a discussion with some folks from the Eberly Center (the teaching center here at CMU). It's a term to describe how people who are already very committed to their subjects tend to implicitly assume that everyone else already has the same enthusiasm and genuine interest, and will just go along for the ride without any questions or doubts. This can be troublesome, of course! You might neglect to properly motivate your students throughout the course, and their commitment will be lackluster; you might fail to point out the usefulness of a certain branch of your topic, thinking that everyone will just find it intrinsically intriguing in its own right; and you might end up teaching to a select handful of top students while leaving everyone else in the dust. I've surely been guilty of this in the past, to some extent, but it's something that I'm consciously trying to eliminate from my teaching ethos. I make myself think hard about every lecture, every example, every homework problem, every segue (both spoken and written), to ensure that a maximum number of students go along with me. This is not easy, but making a concerted effort to address it can go a long way.