(It appears to be May 7 — good luck to all the national MathCounts competitors tomorrow!)
1. An 8.044 Problem
Recently I saw a 8.044 physics problem set which contained the problem
Consider a system of
almost independent harmonic oscillators whose energy in a microcanonical ensemble is given by
. Show that this energy can be obtained is
.
Once you remove the physics fluff, it immediately reduces to
Show the number of nonnegative integer solutions to
is
And as anyone who has done lots of math contests knows, this is the famous stars and bars problem (also known as balls and urns).
This made me really upset when I saw it, for two reasons. One, the main difficulty of the question isn’t related to the physics at hand at all. Once you plug in the definition you get a fairly elegant combinatorics problem, not a physics problem. And secondly, although the solution to the (unrelated) combinatorics is nice, it’s very tricky. I don’t think I could have come up with it easily if I hadn’t seen it before. Either you’ve seen the stars-and-bars trick before and the problem is trivial, or you haven’t seen the trick, and you could easily spend a couple hours trying to come up with a solution — and none of that two hours is teaching you any physics.
You can see why a physics instructor might give this as a homework problem. The solution is short and elementary, something that a undergraduate student could understand and write down. But somewhere at MIT, some poor non-mathematician just spent a good chunk of their evening struggling with this one-trick classic and probably not learning much from it.
2. Don’t I Like Hard Problems?
Well, “not learning much from it” is not entirely accurate\dots
Something that bothered me (and which I hope also bothers the reader) was I complained that the problem was “tricky”. That seems off, because as you might already know, I like hard problems; in fact, in high school I was well despised for helping teachers find hard extra credit problems to pose. (“Hard” isn’t quite the same as “tricky”, but that’s a different direction altogether.) After all, hard problems from math contests taught me to think, isn’t that right?
Well, maybe what’s wrong is that there’s no physics in the hard part of the problem; the bonus problems I provided for my teachers were all closely tied to the material at hand. But that doesn’t seem right either. Euclidean geometry might be useless outside of high school, but nonetheless all the time I spent developing barycentric coordinates still made me a smarter person. Similarly, Richard Rusczyk will often tell you that geometry problems trained him for running the business that is now the Art of Problem Solving. For exactly the same reason, thinking about the stars and bars problem is certainly good for the mind, isn’t that right? Why was I upset about it?
Well, I still hold my objection that there’s no physics in the problem. Why? So at this point we’re naturally led to ask: what was the point of the problem set in the first place? And that answer this, you have to ask: what was the point of the class in the first place?
On paper, it’s to learn physics. Is that really all? Maybe the professor thinks it’s important to teach students how to think as well. Does she? And the answer here is I really don’t know, because I have no idea who’s teaching the class. So I’ll instead ask the more idealistic question: should she?
And surprisingly, I think the answer can be very different from place to place.
On one extreme, I think high school math should be mainly about teaching students to think. Virtually none of the students will actually use the specific content being taught in the class. Why does the average high school student need to know what ![{\int_{[0,1]} x^2 \; dx}](https://s0.wp.com/latex.php?latex=%7B%5Cint_%7B%5B0%2C1%5D%7D+x%5E2+%5C%3B+dx%7D&bg=ffffff&fg=000000&s=0&c=20201002)
But on the other extreme, if you have a math major trying to learn the undergraduate curriculum the picture can change entirely, just because there is so much math to cover. It’s kind of ridiculous, honestly: take the average incoming freshman and the average senior math major, and the latter will know so much more than the former. So in this case I would be much more worried about the content of the course; assuming for example that I’m hoping to be a math major, the chance that the (main ideas of) the specific content will be useful later on is far higher.
This is especially true for, say, students who did math contests extensively in high school, because that ability to solve hard problems is already there; it’s not an interesting use of time to be slowly doing challenging exercises in group theory when there’s still modules, rings, fields, categories, algebraic geometry, homological algebra, all untouched (to say nothing of analysis).
What this boils down to is trying to distinguish between the actual content of the given class (something very local) versus the more general skill of problem-solving or thinking. In high school I focused almost exclusively on the latter; as time passes I’ve been shifting my focus farther and farther to the former.
3. 
Now suppose that we are interested in teaching how to think on these problem sets. There’s one other difference between the problem sets and math contests. You’re expected to finish your problem sets and you’re not expected to finish math contests.
I want to complain that there seems to be a stigma that you have to do exercises in order to learn math or physics or whatever, and that people who give up on them are somehow lazy or something. It is true in some sense that you can only learn math by doing. It is probably true that thinking about a hard problem will teach you something. What is not true is that you should always stare at a problem until either it or you cracks.
This is obviously true in math contests too. One of the things I was really bad at was giving up on a problem after hours of no progress. In some sense the time limit of contests is kind of nice; it cuts you off from spending too long on any one problem. You can’t be expected to be able to solve all hard problems, or else they’re not hard.
Problem sets fare much more poorly in this respect. The benefit of thinking about the hard problem diminishes over time (e.g. a typical exercise can teach you more in the first hour than it does in the next six) and sometimes you’re just totally dead in the water after a couple hours of staring. The big guy seem to implicitly tell you that you should keep working because it’s supposed to be hard. Is that really true? It certainly wasn’t true in the math contest world, so I don’t see any reason why it’s true here.
In other words, I don’t think our poor physics student would have lost much by giving up on balls and urns after a few hours. And really, for all the warnings that looking up problems online is immoral, is asking your friend to help really that different?
Power Overwhelming 
Nice blog. I think many parts from your blog posts are on the topic of “applications and learning the actual material” vs “learning how to abstract and learning how to think”.
But this raises the question of what percentage of the population you actually think “knows how to think”, but more importantly, appreciate abstract concepts? Most people would think that the purpose of math is to apply it for engineering. While this may not be bad, (and certainly there is a time to do so), I cringe whenever someone finds out I do math and they talk about Differential Equations as “cool”…
On one hand, not trying to be elitist, but centuries ago, the nobility thought that the general public could never appreciate abstract and beautiful subjects, one of them being Euclid’s Elements (rest being literature, Greek, Latin, etc.) It’s not that different today, where most of the K-12 education system tries to make itself relevant by saying that most of what you learn is “useful in real life” to appeal to the masses. IMO, this is a very bad, but understandable idea. Applied broadly, this in turn basically makes every homework problem you do some trivial exercise in arithmetic (e.g. integrate function, plug into formula, etc.) that’s a huge waste of time for anyone decently intelligent trying to get into college.
On the other hand, what basically happens is that some people already have a foothold in math (through parental encouragement, natural ability, exposure) to break this barrier and see through the “lies”, but what about people who do not have those things? What about people who have an innate appreciation for math beauty, but never had the opportunity to express it, so they lose it? (I think this is more applicable to something like gender gap in math)
They are fed by social outlets everyday; rather than “math is beautiful”, the encouragement is “math is useful.” I didn’t have a good linguistic ability when I was young, so I didn’t appreciate literature and wordplay (but now I do). Being forced to memorize vocabulary words and take memorization tests on “who did what” just reinforced my dislike of English, but now I see it’s not so different for people who only saw school math.
I know that MIT and other related places specifically try to avoid this effect and avoid looking like trade schools, where the latter’s goal is teach practical and “useful” skills (e.g. when you compare CS departments, MIT and Caltech seem to be more theoretical).
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For what it’s worth, I learned algebra by self-studying Topics in Algebra by Herstein. Since the book is ~1/2 group theory and the exercises are all really hard (and I did all the exercises), this meant that I learned algebra by essentially slowly working through a ton of really hard group theory problems.
Interestingly, after doing this, the rest of the algebra book was pretty easy to get through. By spending so long on those problems, I had come to grasp the underlying ideas that pervaded most of algebra. The rest of the book seemed like re-applications of the same meta-principles that I had already learned.
On the other hand, Herstein picks his exercises extremely carefully. Probably many of them are explicitly picked to foreshadow some of these high-level principles. This is certainly quite different from the balls-and-urns problem…
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That’s interesting — this is how I (and I guess most other people) learn subjects like olympiad geometry and math contest in general. Agree that if done correctly, this is about the most effective thing there is. The main reason I haven’t been doing this for my own studies since graduating is that it’s pretty time-consuming.
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I don’t know how to un-physics-ify the problem, but taking your word for it, I agree that it doesn’t seem very helpful or suitable for the course.
On the other hand, do you think that, in practice, many MIT physics students who haven’t seen the trick before are going to spend several hours thinking about it without finding help via a friend or TA? Surely this is another important difference between p-sets and math contests — whether you practice contest problems with others or not, you’re aware that the terminal goal is to perform well on a real math contest, which is almost always taken individually; whereas p-sets are at least somewhat of a goal in and of themselves, and collaboration on them is expected, if not encouraged?
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Fair point. Though my experience is that the most common recourse is to just look up the problems online :)
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Great post, yeah, if you haven’t seen the problem before, then it could be really tough…
Also, you said
“This is especially true for, say, students who did math contests extensively in high school, because that ability to solve hard problems is already there; it’s not an interesting use of time to be slowly doing challenging exercises in group theory when there’s still modules, rings, fields, categories, algebraic geometry, homological algebra, all untouched (to say nothing of analysis).”
Does that mean that nowadays you usually just ignore the practice problems?
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