Sometimes the best advice is no advice

信言不美,美言不信。

I get a lot of questions that are so general that there is no useful answer I can give, e.g., “how do I get better at geometry?”. What do you want from me? Go do more problems, sheesh.

These days, in my instructions for contacting me, I tell people to be as specific as possible e.g. including specific problems they recently tried and couldn’t solve. Unsurpisingly the same kind of people who ask me a question like that are also not the kind of people who read instructions, so it hasn’t helped much. 😛

But it’s occurred to me it’s possible to take this too far. Or maybe more accurately, it’s always better to ask a specific question, but sometimes the best answer will still be “go do more problems, sheesh”.

Here’s a metaphorical example.

Suppose that someone is learning the game of chess, and they just died to scholar’s mate. So they go on the Internet and write something like,

Some cheapskape used scholar’s mate on me, and I died in four turns WTF!? How do I defend this!? We played e4 e5 Bc4 Nc6 Qh5 Nf6 Qxf7.

Okay, it’s a specific question, at least.

If you’re a nice coach, you could give a straight answer to this. Like, “reply g6 if the queen is on h5 and Nf6 if the queen is on f3”. And maybe some nice words about not getting discouraged.

But I wonder if the best advice is really “go play 100 more games”.

Because, well, if someone (a) saw the queen move out, (b) died anyways, (c) can’t figure out what they should have done differently on turn 3, (d) can’t google the answer themself, and (e) is complaining on the Internet about it, then I think two things are clear. First, they have not played many games, and second, they desperately need to learn how to fish.

In some sense, the Scholar’s Mate issue will correct itself automatically after enough games. So I worry that I do a long-term disservice by giving a specific answer over the general answer, and implicitly suggesting that this is how the learning process should work.

Experience is the best teacher of all. No contest. A pupil who doesn’t internalize this, and instead tries to short-circuit the learning process by overfitting their internal models on too few data points, is going to hit a wall really soon.

Unfortunately for me, I don’t play the tough-coach card well, unlike some other people. So for the foreseeable future, I’m still likely to respond with g6/Nf6. Or maybe I will start linking this post when I’m out of patience, we’ll see.

Book pitch

This is a pitch for a new text that I’m thinking of writing. I want to post it here to solicit opinions from the general community before investing a lot of time into the actual writing.

Summary

There are a lot of students who ask me a question isomorphic to:

How do I learn to write proofs?

I’ve got this on my Q&A. For the contest kiddos out there, it basically amounts to saying “read the official solutions to any competition”.

But I think I can do better.

Requirements

Calling into question the obvious, by insisting that it be “rigorously proved”, is to say to a student, “Your feelings and ideas are suspect. You need to think and speak our way.”

Now there is a place for formal proof in mathematics, no question. But that place is not a student’s first introduction to mathematical argument. At least let people get familiar with some mathematical objects, and learn what to expect from them, before you start formalizing everything.

— Paul Lockhart

There was a while I tried to look around to find an introduction-to-proofs textbook that I liked. I specifically wanted to have the following requirements:

  • Pragmaticism: the textbook should not start with foundational issues like logical quantifiers or set theory. I have held a long belief that these are emphatically not the right way to start proofs, because in practice when one really does proofs, one is usually not thinking too much about the axioms of set theory.
  • Substantial: the results one proves as practice should feel interesting. They should have meat. For example, the statement that a tree always has one fewer edge than vertex is not obvious at first, so when one sees the proof it gives an idea. I believe this is important because I want to develop a student’s intuition, rather than try to teach them to work against it.
  • Intuitive: I reject the approach of some other instructors in which students start by proving basic results from first principles like the well-ordering principle, “all right angles are congruent”, etc. I think this is an experience that is worth having, but it should not be the first experience one has. (This is the same reason people’s first programming language is Python and not assembly.)
  • Combinatorial: for competition reasons. My currently recommended combinatorics textbook by Pascal96 is a bit on the difficult side. It would be nice to cover some ground here.

The closest I got was Jospeh Rotman’s Journey Into Mathematics textbook, which satisfies the first three conditions but not the fourth (the book draws from algebra, geometry, and number theory). I adore Rotman’s book and the copy I read at age 12 is tattered from extended use. I’d like to get the combinatorics in, too.

Picking a fight

I should state now this is against common wisdom. Terence Tao for example describes mathematical education in three parts: pre-rigorous, rigorous, post-rigorous. Relevant quotes:

[In the rigorous stage], one is expected to be able to comfortably manipulate abstract mathematical objects without focusing too much on what such objects actually “mean”. … The transition from the first stage to the second is well known to be rather traumatic.

My thesis is that for high school students with an enriched math background, the rigorous and post-rigorous stages should be merged or even inverted. Attending a math circle, going to math camps, or participating in competitions gives you a much better intuition than a typical starting undergraduate would otherwise have access to. I propose that we take advantage of this intuition, rather than ignore or suppress it.

Content

I’m eyeing graph theory as a topic to start off on, if not use wholesale. I think it is an amazing topic for teaching proofs with. Definitions that make sense, proofs that are intuitive but not obvious, lots of pictures that don’t lose rigor, and so on. I imagine I would start there and see where it takes me.

If I go through with it, I think it would take about a year for me to get some initial drafts available to the public.

Pay-what-you-want model

I want to try this out. I think it would look something like:

  1. You can download the nicely typeset PDF for 20 dollars;
  2. The entire source code is publicly readable on GitHub, so if you can’t pay or don’t want to pay just download the source and compile it. It might not have some formatting polishes or whatever but all the content is going to be there.
  3. If you don’t have a computer to compile things on, email me nicely and I’ll send you a copy.
  4. Pull requests welcome, and if you fix some sufficient number of typos or some major errors I’ll add your name to acknowledgments.

But I’m not sure yet.

Questions for the audience

  1. Is this something people would want to see?
  2. Is there any existing text that already satisfies my requirements?
  3. Is the payment model fair?
  4. Other comments or suggestions?

On choosing exercises

Finally, if you attempt to read this without working through a significant number of exercises (see §0.0.1), I will come to your house and pummel you with [Gr-EGA] until you beg for mercy. It is important to not just have a vague sense of what is true, but to be able to actually get your hands dirty. As Mark Kisin has said, “You can wave your hands all you want, but it still won’t make you fly.”

— Ravi Vakil, The Rising Sea: Foundations of Algebraic Geometry

When people learn new areas in higher math, they are usually required to do some exercises. I think no one really disputes this: you have to actually do math to make any progress.

However, from the teacher’s side, I want to make the case that there is some art to picking exercises, too. In the process of writing my Napkin as well as taking way too many math classes I began to see some patterns in which exercises or problems I tended to add to the Napkin, or which exercises I found helpful when learning myself. So, I want to explicitly record some of these thoughts here.

1. How not to do it

So in my usual cynicism I’ll start by saying what I think people typically do, and why I don’t think it works well. As far as I can tell, the criteria used in most classes is:

  1. The student is reasonably able to (at least in theory) eventually solve it.
  2. A student with a solid understanding of the material should be able to do it.
  3. (Optional) The result itself is worth knowing.

Both of these criteria are good. My problem is that I don’t think they are sufficient.

To explain why, let me give a concrete example of something that is definitely assigned in many measure theory classes.

Okay example (completion of a measure space). Let {(X, \mathcal A, \mu)} be a measure space. Let {\overline{\mathcal A}} denote all subsets of {X} which are the union of a set in {\mathcal A} and a null set. Show that {\overline{\mathcal A}} is a sigma-algebra there is a unique extension of the measure {\mu} to it.

I can see why it’s tempting to give this as an exercise. It is a very fundamental result that the student should know. The proof is not too difficult, and the student will understand it better if they do it themselves than if they passively read it. And, if a student really understands measures well, they should find the exercise quite straightforward. For this reason I think this is an okay choice.

But I think we can do better.

In many classes I’ve taken, nearly all the exercises looked like this one. I think when you do this, there are a couple blind spots that sometimes get missed:

  • There’s a difference between “things you should be able to do after learning Z well” and “things you should be able to do when first learning Z“. I would argue that the above example is the former category, but not the latter one — if a student is learning about measures for the first time, my first priority would be to make sure they get a good conceptual understanding first, and in particular can understand why the statement should be true. Then we can worry about actually proving it.
  • Assigning an exercise which checks if you understand X is not the same as actually teaching it. Okay exercises can verify if you understand something, great exercises will actively help you understand it.

2. An example that I found enlightening

In contrast, this year I was given an exercise which I thought was so instructive that I’ll post it here. It comes from algebraic geometry.

Exercise: The punctured gyrotop is the open subset {U} of {X = \mathrm{Spec} \mathbb C[x,y,z] / (xz, yz)} obtained by deleting the origin {(x,y,z)} from {X}. Compute {\mathcal O_X(U)}.

It was after I did this exercise that I finally felt like I understood why distinguished open sets are so important when defining an affine scheme. For that matter, it finally clicked why sheaves on a base are worth caring about.

I had read lots and lots of words and pushed symbols around all day. I had even proved, on paper already, that {\mathcal O(U \sqcup V) = \mathcal O(U) \times \mathcal O(V)}. But I never really felt it. This exercise changed that for me, because suddenly I had an example in front of me that I could actually see.

3. Some suggested additional criteria

So here are a few suggested guidelines which I think can help pick exercises like that one.

A. They should be as concrete as possible.

This is me yelling at people to use more examples, once again. But I think having students work through examples as an exercise is just as important (if not more) than reading them aloud in lecture.

One other benefit of using concrete examples is that you can avoid the risk of students solving the exercise by “symbol pushing”. I think many of us know the feeling of solving some textbook exercise by just unwinding a definition and doing a manipulation, or black-boxing some theorem and blindly applying it. In this way one ends up with correct but unenlightening proofs. The issue is that nothing written down resonates with System 1, and so the result doesn’t get internalized.

When you give a concrete exercise with a specific group/scheme/whatever, there is much less chance of something like that happening. You almost have to see the example in order to work with it. I really think internalizing theorems and definitions is better done in this concrete way, rather than the more abstract or general manipulations.

B. They should be enjoyable.

Math majors are humans too. If a whole page of exercises looks boring, students are less likely to do them.

This is one place where I think people could really learn from the math contest community. When designing exams like IMO or USAMO, people fight over which problems they think are the prettiest. The nicest and most instructive exam problems are passed down from generation to generation like prized heirlooms. (Conveniently, the problems are even named, e.g. “IMO 2008/3”, which I privately think helps a ton; it gives the problems a name and face. The most enthusiastic students will often be able to recall where a good problem was from if shown the statement again.) Imagine if the average textbook exercises had even a tenth of that enthusiasm put into crafting them.

Incidentally, I think being concrete helps a lot with this. Part of the reason I enjoyed the punctured gyrotop so much was that I could immediately draw a picture of it, and I had a sense that I should be able to compute the answer, even though I wasn’t experienced enough yet to see what it was. So it was as if the exercise was leading me on the whole way.

For an example of how not to do it, here’s what I think my geometry book would look like if done wrong.

C. They should not be too tricky.

People are always dumber than you think when they first learn a subject; things which should be obvious often are not. So difficulty should be used in moderation: if you assign a hard exercise, you should assume by default the student will not solve it, so there better be some reason you’re adding some extra frustration.

I should at this point also mention some advice most people won’t be able to take (because it is so time-consuming): I think it’s valuable to write full solutions for students, especially on difficult problems. When someone is learning something for the first time, that is the most important time for the students to be able to read the full details of solutions, precisely because they are not yet able to do it themselves.

In math contests, the ideal feedback cycle is something like: a student works on a problem P, makes some progress (possibly solving it), then they look at the solution and see what they were missing or where they could have cleaned up their solution or what they could have done differently, et cetera. This lets them update their intuition or toolkit before going on. If you cut out this last step by not providing solutions, you lose the only real chance you had to give feedback to the student.

4. Memorability

I have, on more occasions than I’m willing to admit, run into the following situation. I solve some exercise in a textbook. Sometime later, I am reading about some other result, and I need some intermediate result, which looks like it could be true but I don’t how to prove it immediately. So I look it up, and then find out it was the exercise I did (and then have to re-do the exercise again because I didn’t write up the solution).

I think you can argue that if you don’t even recognize the statement later, you didn’t learn anything from it. So I think the following is a good summarizing test: how likely is the student to actually remember it later?

MOHS hardness scale

There’s a new addition to my olympiad problems and solutions archive: I created an index of many past IMO/USAMO/USA TST(ST) problems by what my opinions on their difficulties are. You can grab the direct link to the file below:

https://evanchen.cc/upload/MOHS-hardness.pdf

In short, the scale runs from 0M to 50M in increments of 5M, and every USAMO / IMO problem on my archive now has a rating too.

My hope is that this can be useful in a couple ways. One is that I hope it’s a nice reference for students, so that they can better make choices about what practice problems would be most useful for them to work on. The other is that the hardness scale contains a very long discussion about how I judge the difficulty of problems. While this is my own personal opinion, obviously, I hope it might still be useful for coaches or at least interesting to read about.

As long as I’m here, I should express some concern that it’s possible this document does more harm than good, too. (I held off on posting this for a few months, but eventually decided to at least try it and see for myself, and just learn from it if it turns out to be a mistake.) I think there’s something special about solving your first IMO problem or USAMO problem or whatever and suddenly realizing that these problems are actually doable — I hope it would not be diminished by me rating the problem as 0M. Maybe more information isn’t always a good thing!

Understanding with System 1

Math must be presented for System 1 to absorb and only incidentally for System 2 to verify.

I finally have a sort-of formalizable guideline for teaching and writing math, and what it means to “understand” math. I’ve been unconsciously following this for years and only now managed to write down explicitly what it is that I’ve been doing.

(This post is written from a math-centric perspective, because that’s the domain where my concrete object-level examples from. But I suspect much of it applies to communicating hard ideas in general.)

S1 and S2

The quote above refers to the System 1 and System 2 framework from Thinking, Fast and Slow. Roughly it divides the brain’s thoughts into two categories:

  • S1 is the part of the brain characterized by fast, intuitive, automatic, instinctive, emotional responses, For example, when you read the text “2+2=?”, S1 tells you (without any effort) that this equals 4.
  • S2 is the part of the brain characterized by slow, deliberative, effortful, logical responses; for example, S2 is used to count the number of words in this sentence.

(The link above gives some more examples.)

The premise of this post is that understanding math well is largely about having the concept resonate with your S1, rather than your S2. For example, let’s take groups from abstract algebra. Then I claim that

G = \{ a/b \mid a,b \text{ odd integers} \}

is a group under the usual multiplication. Now, if you have a student who’s learning group theory for the first time, the only way they could see this is a group is to compare it against a list of the group axioms, and have their S2 verify them one by one. But experienced people don’t do this: their S1 automatically tells them that G “feels” like a group (because e.g. it’s closed and doesn’t have division-by-zero issues).

I think this S1-level understanding is what it means to “get it”. Verifying a solution to a hard olympiad problem by having S2 check each individual step is straightforward in principle, albeit time-consuming. The tricky part is to get this solution to resonate with S1. Hence my advice to never read a solution line by line.

Writing for S1

What this means is that if you’re trying to teach someone an idea, then you should be focusing on trying to get their S1 to grasp it, rather than just their S2. For example, in math it’s not enough to just give a sequence of logical steps which implies the result: give it life.

Here are some examples of ways I (try to) do this.

First, giving good concrete examples. S1 reacts well when it “sees” a concrete object like G above, and can see some intuitive properties about it right away. Abstract “symbol-pushing” is usually left to S2 instead.

Similarly, drawing pictures, so your S1 can actually see the object. On one extreme end, you can write something like “a point $S$ lies on the polar of $T$ if and only if $T$ lies on the polar of $S$”, but it’s much better to just have a picture:

You can even do this for things that aren’t really geometrical in nature. For example, my Napkin features the following picture of cardinal collapse when forcing.

Third, write like you talk, and share your feelings. S1 is emotional. S1 wants to know that compactness is a good property for a space to have, or that non-Noetherian rings are way too big and “only weirdos care about non-Noetherian rings” (just kidding!), or that ramified primes are the “finitely many edge cases” and aren’t worth worrying about. These S1 reactions you get are the things you want to pass on. In particular, avoid standard formal college-textbook-bleed-your-eyes-dry-in-boredom style. (To be fair, not all textbooks do this; this is one reason why I like Pugh’s book so much, for example.)

Even the mechanics on the page can be made to accommodate S1 in this way. S1 can’t read a wall of text; S2 has to put in effort to do that. But S1 can pick out section headers, or bolded phrases like this one, and so on and so forth. That’s why in Napkin all the examples are in separate red boxes and all the big theorems are in blue boxes, and important philosophical points are typeset in bold centered green text. This way S1 naturally puts its attention there.

But do not force it

On the flip side, if you’re trying to learn something, there’s a common failure mode where you try to keep forcing S2 to do something unnatural (rather than trying to have S1 figure it out). This is the kind of thing when you don’t understand what the Chinese Remainder Theorem is trying to say, so you try to fix this by repeatedly reading the proof line by line, and still not really understanding what is going on. Usually this ends up in S2 getting tired and not actually reading the proof after the third or fourth iteration.

(For the Chinese remainder theorem the right thing to do is ask yourself why any arithmetic progression with common difference 7 must contain multiples of 3: credits to Dominic Yeo again for that. I’m not actually sure what you’re supposed to do when stuck on math in general. Usually I just ask my friends what is going on, or give up for now and come back later.)

Actually, I really like the advice that SSC mentions: “develop instincts, then use them”.

MOP should do a better job of supporting its students in not-June

Up to now I always felt a little saddened when I see people drop out of the IMO or EGMO team selection. But actually, really I should be asking myself what I (as a coach) could do better to make sure the students know we value their effort, even if they ultimately don’t make the team.

Because we sure do an awful job of being supportive of the students, or, well, really doing anything at all. There’s no practice material, no encouragement, or actually no form of contact whatsoever. Just three unreasonably hard problems each month, followed by a score report about a week later, starting in December and dragging in to April.

One of a teacher’s important jobs is to encourage their students. And even though we get the best students in the USA, probably we shouldn’t skip that step entirely, especially given the level of competition we put the students through.

So, what should we do about it? Suggestions welcome.

Undergraduate Math 011: a firsT yeaR coursE in geometrY

tl;dr I parodied my own book, download the new version here.

People often complain to me about how olympiad geometry is just about knowing a bunch of configurations or theorems. But it recently occurred to me that when you actually get down to its core, the amount of specific knowledge that you need to do well in olympiad geometry is very little. In fact I’m going to come out and say: I think all the theory of mainstream IMO geometry would not last even a one-semester college course.

So to stake my claim, and celebrate April Fool’s Day, I decided to actually do it. What would olympiad geometry look like if it was taught at a typical college? To find out, I present to you the course notes for:

Undergrad Math 011: a firsT yeaR coursE in geometrY
handtruck

It’s 36 pages long, title page, preface, and index included. So, there you go. It is also the kind of thing I would never want to read, and the exercises are awful, but what does that matter?

scrshot-tr011ey.png

(I initially wanted to post this file as an April Fool’s gag, but became concerned that one would not have to be too gullible to believe these were actual course notes and then attempt to work through them.)

Math contest platitudes, v3

I think it would be nice if every few years I updated my generic answer to “how do I get better at math contests?”. So here is the 2019 version. Unlike previous instances, I’m going to be a little less olympiad-focused than I usually am, since these days I get a lot of people asking for help on the AMC and AIME too.

(Historical notes: you can see the version from right after I graduated and the version from when I was still in high school. I admit both of them make me cringe slightly when I read them today. I still think everything written there is right, but the style and focus seems off to me now.)

0. Stop looking for the “right” training (or: be yourself)

These days many of the questions I get are clearly most focused on trying to find a perfect plan — questions like “what did YOU do to get to X” or “how EXACTLY do I practice for Y”. (Often these words are in all-caps in the email, too!) When I see these I always feel very hesitant to answer. The reason is that I always feel like there’s some implicit hope that I can give you some recipe that, if you follow it, will guarantee reaching your goals.

I’m sorry, math contests don’t work that way (and can’t work that way). I actually think that if I gave you a list of which chapters of which books I read in 2009-2010 over which weeks, and which problems I did on each day, and you followed it to the letter, it would go horribly.

Why? It’s not just a talent thing, I think. Solving math problems is actually a deeply personal art: despite being what might appear to be a cold and logical discipline, learning math and getting better at it actually requires being human. Different people find different things natural or unnatural, easy or hard, et cetera. If you try to squeeze yourself into some mold or timeline then the results will probably be counterproductive.

On the flip side, this means that you can worry a lot less. I actually think that surprisingly often, you can get a first-order approximation of what’s the “best” thing to do by simply doing whatever feels the most engaging or rewarding (assuming you like math, of course). Of course there are some places where this is not correct (e.g., you might hate geometry, but cannot just ignore it). But the first-order approximation is actually quite decent.

That’s why in the introduction to my geometry book, I explicitly have the line:

Readers are encouraged to not be bureaucratic in their learning and move around as they see fit, e.g., skipping complicated sections and returning to them later, or moving quickly through familiar material.

Put another way: as learning math is quite personal, the advice “be yourself” is well-taken.

1. Some brief recommendations (anyways)

With all that said, probably no serious harm will come from me listing a little bit of references I think are reasonable — so that you have somewhere to start, and can oscillate from there.

For learning theory and fundamentals:

For sources of additional practice problems (other than the particular test you’re preparing for):

  • The collegiate contests HMMT November, PUMaC, CMIMC will typically have decent short-answer problems.
  • HMMT February is by far the hardest short-answer contest I know of.
  • At the olympiad level, there are so many national olympiads and team selection tests that you will never finish. (My website has an archive of USA problems and solutions if you’re interested in those in particular.)
    The IMO Shortlist is also good place to work as it contains proposals of varying difficulty from many countries — and thus is the most culturally diverse. As for other nations, as a rule of thumb, any country that often finishes in the top 20 at the IMO (say) will probably have a good questions on their national olympiad or TST.

For every subject that’s not olympiad geometry, there are actually surprisingly few named theorems.

2. Premature optimization is the root of all evil (so just get your hands dirty)

For some people, the easiest first step to getting better is to double the amount of time you spend practicing. (Unless that amount is zero, in which case, you should just start.)

There is a time and place for spending time thinking about how to practice — one example is if you’ve been working a while and feel like nothing has changed, or you’ve been working on some book and it just doesn’t feel fun, etc. Another common example is if you notice you keep missing all the functional equations on the USAMO: then, maybe it’s time to search up some handouts on functional equations. Put another way, if you feel stuck, then you can start thinking about whether you’re not doing something right.

On the other extreme, if you’re wondering whether you are ready to read book X or do problems from Y contest, my advice is to just try it and see if you like it. There is no commitment: just read Chapter 1, see how you feel. If it works, keep doing it, if not, try something else.

(I can draw an analogy from my own life. Whenever I am learning a new board game or card game, like Catan or Splendor or whatever, I always overthink it. I spend all this time thinking and theorizing and trying to come up with this brilliant strategy — which never works, because it’s my first game, for crying out loud. It turns out that until you start grappling at close range and getting your hands dirty, your internal model of something you’ve never done is probably not that good.)

3. Doing problems just above your level (and a bit on reflecting on them)

There is one pitfall that I do see sometimes, common enough I will point it out. If you mostly (only?) do old practice tests or past problems, then you’re liable to be spending too much time on easy problems. That was the topic of another old post of mine, but the short story is that if you find yourself constantly getting 130ish on AMC10 practice tests, then maybe you should spend most of your time working on problems 21-25 rather than repeatedly grinding 1-20 over and over. (See 28:30-29:00 here to hear Zuming make fun of them.)

The common wisdom is that you should consistently do problems just above your level so that you gradually increase the difficulty of problems you are able to solve. The situation is a little more nuanced at the AMC/AIME level, since for those short-answer contests it’s also important to be able to do routine problems quickly and accurately. However, I think for most people, you really should be spending at least 70% of your time getting smarter, rather than just faster.

I think in this case, I want to give concrete descriptions. Here’s some examples of what can happen after a problem.

  • You looked at the problem and immediately (already?) knew how to do it. Then you probably didn’t learn much from it. (But at least you’ll get faster, if not smarter.)
  • You looked at the problem and didn’t know right away how to start, but after a little while figured it out. That’s a little better.
  • You struggled with the problem and eventually figured out a solution, but maybe not the most elegant one. I think that’s a great situation to be in. You came up with some solution to the problem, so you understand it fairly well, but there’s still more for you to update your instincts on. What can you do in the future to get solutions more like the elegant one?
  • You struggled with the problem and eventually gave up, then when you read the solution you realize quickly what you were missing. I think that’s a great situation to be in, too. You now want to update your instincts by a little bit — how could you make sure you don’t miss something like that again in the future?
  • The official solution quoted some theorem you don’t know. If this was among a batch of problems where the other problems felt about the right level to you, then I think often this is a pretty good time to see if you can learn the statement (better, proof) of the theorem. You have just spent some time working on a situation in which the theorem was useful, so that data is fresh in your mind. And pleasantly often, you will find that ideas you came up with during your attempt on the problem correspond to ideas in the statement or proof of the theorem, which is great!
  • You didn’t solve the problem, and the solution makes sense, but you don’t see how you would have come up with it. It’s possible that this is the fault of the solutions author (many people are actually quite bad at making solutions read naturally). If you have a teacher, this is the right time to ask them about it. But it’s also possible that the problem was too hard. In general, I think it’s better to miss problems “by a little”, whatever that means, so that you can update your intuition correctly.
  • You can’t even understand the solution. Okay, too hard.

You’ll notice how much emphasis I place on the post-problem reflection process. This is actually important — after all the time you spent working on the problem itself, you want to update your instincts as much as possible to get the payoff. In particular, I think it’s usually worth it to read the solutions to problems you worked on, whether or not you solve them. In general, after reading a solution, I think you should be able to state in a couple sentences all the main ideas of the solution, and basically know how to solve the problem from there.

For the olympiad level, I have a whole different post dedicated to reading solutions, and interested readers can read more there. (One point from that post I do want to emphasize since it wasn’t covered explicitly in any of the above examples: by USA(J)MO level it becomes important to begin building intuition that you can’t explicitly formalize. You may start having vague feelings and notions that you can’t quite put your finger on, but you can feel it. These non-formalizable feelings are valuable, take note of them.)

4. Leave your ego out (e.g. be willing to give up on problems)

This is easy advice to give, but it’s hard advice to follow. For concreteness, here are examples of things I think can be explained this way.

Sometimes people will ask me whether they need to solve every problem in each chapter of EGMO, or do every past practice test, or so on. The answer is: of course not, and why would you even think that? There’s nothing magical about doing 80% of the problems versus 100% of them. (If there was, then EGMO is secretly a terrible book, because I commented out some problems, and so OH NO YOU SKIPPED SOME AAAHHHHH.) And so it’s okay to start Chapter 5 even though you didn’t finish that last challenge problem at the end. Otherwise you let one problem prevent you from working on the next several.

Or, sometimes I learn about people who, if they do not solve an olympiad problem, will refuse to look at the solution; instead they will mark it in a spreadsheet and to come back to later. In short, they never give up on a problem: which I think is a bad idea, since reflecting on missed problems is so important. (It is not as if you can realistically run out of olympiad problems to do.) And while this is still better than giving up too early, I mean, all things in moderation, right?

I think if somehow people were able to completely leave your ego out, and not worry at all about how good you are and rather just maximize learning, then mistakes like these two would be a lot rarer. Of course, this is impossible to do in practice (we’re all human), but it’s good to keep in mind at least that this is an ideal we can strive for.

5. Enjoy it

Which leads me to the one bit that everyone already knows, but that no platitude-filled post would be complete without: to do well at math contests (or anything hard) you probably have to enjoy the process of getting better. Not just the end result. You have to enjoy the work itself.

Which is not to say you have to do it all the time or for hours a day. Doing math is hard, so you get tired eventually, and beyond that forcing yourself to work is not productive. Thus when I see people talk about how they plan to do every shortlist problem, or they will work N hours per day over M time, I always feel a little uneasy, because it always seems too results-oriented.

In particular, I actually think it’s quite hard to spend more than two or three good hours per day on a regular basis. I certainly never did — back in high school (and even now), if I solved one problem that took me more than an hour, that was considered a good day. (But I should also note that the work ethic of my best students consistently amazes me; it far surpasses mine.) In that sense, the learning process can’t be forced or rushed.

There is one sense in which you can get more hours a day, that I am on record saying quite often: if you think about math in the shower, then you know you’re doing it right.

 

Make training non zero-sum

Some thoughts about some modern trends in mathematical olympiads that may be concerning.

I. The story of the barycentric coordinates

I worry about my geometry book. To explain why, let me tell you a story.

When I was in high school about six years ago, barycentric coordinates were nearly unknown as an olympiad technique. I only heard about it from whispers in the wind from friends who had heard of the technique and thought it might be usable. But at the time, there were nowhere where everything was written down explicitly. I had a handful of formulas online, a few helpful friends I can reach out to, and a couple example posts littered across some forums.

Seduced by the possibility of arcane power, I didn’t let this stop me. Over the spring of 2012, spring break settled in, and I spent that entire week developing the entire theory of barycentric coordinates from scratch. There were no proofs I could find online, so I had to personally reconstruct all of them. In addition, I set out to finding as many example problems as I could, but since no one had written barycentric solutions yet, I had to not only identify which problems like they might be good examples but also solve them myself to see if my guesses were correct. I even managed to prove a “new” theorem about perpendicular displacement vectors (which I did not get to name after myself).

I continued working all the way up through the summer, adding several new problems that came my way from MOP 2012. Finally, I posted a rough article with all my notes, examples, and proofs, which you can still find online. I still remember this as a sort of magnus opus from the first half of high school; it was an immensely rewarding learning experience.

Today, all this and much more can be yours for just $60, with any major credit or debit card.


Alas, my geometry book is just one example of ways in which the math contest scene is looking more and more like an industry. Over the years, more and more programs dedicated to training for competitions are springing up, and these programs can be quite costly. I myself run a training program now, which is even more expensive (in my defense, it’s one-on-one teaching, rather than a residential camp or group lesson).

It’s possible to imagine a situation in which the contest problems become more and more routine. In that world, math contests become an arms race. It becomes mandatory to have training in increasingly obscure techniques: everything from Popoviciu to Vieta jumping to rectangular circumhyperbolas. Students from less well-off families, or even countries without access to competition resources, become unable to compete, and are pushed to the bottom of the IMO scoreboard.

(Fortunately for me, I found out at the 2017 IMO that my geometry book actually helped level the international playing field, contrary to my initial expectations. It’s unfortunate that it’s not free, but it turned out that many students in other countries had until then found it nearly impossible to find suitable geometry materials. So now many more people have access to a reasonable geometry reference, rather than just the top countries with well-established training.)

II. Another dark future

The first approximation you might have now is that training is bad. But I think that’s the wrong conclusion, since, well, I have an entire previous post dedicated to explaining what I perceive as the benefits of the math contest experience. So I think the conclusion is not that training is intrinsically bad, but rather than training must be meaningful. That is, the students have to gain something from the experience that’s not just a +7 bonus on their next olympiad contest.

I think the message “training is bad” might be even more dangerous.

Imagine that the fashion swings the other way. The IMO jury become alarmed at the trend of train-able problems, and in response, the problems become designed specifically to antagonize trained students. The entire Geometry section of the IMO shortlist ceases to exist, because some Asian kid wrote this book that gives you too much of an advantage if you’ve read it, and besides who does geometry after high school anyways? The IMO 2014 used to be notable for having three combinatorics problems, but by 2040 the norm is to have four or five, because everyone knows combinatorics is harder to train for.

Gradually, the IMO is redesigned to become an IQ test.

The changes then begin to permeate down. The USAMO committee is overthrown, and USAMO 2050 features six linguistics questions “so that we can find out who can actually think”. Math contests as a whole become a system for identifying the best genetic talent, explicitly aimed at weeding out the students who have “just been trained”. It doesn’t matter how hard you’ve worked; we want “creativity”.

This might be great at identifying the best mathematicians each generation, but I think an IMO of this shape would be actively destructive towards the contestants and community as well. You thought math contests were bad because they’re discouraging to the kids who don’t win? What if they become redesigned to make sure that you can’t improve your score no matter how hard you work?

III. Now

What this means is that we have a balancing act to maintain. We do not want to eliminate the role of training entirely, because the whole point of math contests is to have a learning experience that lasts longer than the two-day contest every year. But at the same time, we need to ensure the training is interesting, that it is deep and teaches skills like the ones I described before.

Paying $60 to buy a 300-page PDF is not meaningful. But spending many hours to work through the problems in that PDF might be.

In many ways this is not a novel idea. If I am trying to teach a student, and I give them a problem which is too easy, they will not learn anything from it. Conversely, if I give them a problem which is too difficult, they will get discouraged and are unlikely to learn much from their trouble. The situation with olympiad training feels the same.

This applies to the way I think about my teaching as well. I am always upset when I hear (as I have) things like “X only did well on USAMO because of Evan Chen’s class”. If that is true, then all I am doing is taking money as input and changing the results of a zero-sum game as output, which is in my opinion rather pointless (and maybe unethical).

But I really think that’s not what’s happening. Maybe I’m a good teacher, but at the end of the day I am just a guide. If my students do well, or even if they don’t do well, it is because they spent many hours on the challenges that I designed, and have learned a lot from the whole experience. The credit for any success thus lies solely through the student’s effort. And that experience, I think, is certainly not zero-sum.

I switched to point-based problem sets

It’s not uncommon for technical books to include an admonition from the author that readers must do the exercises and problems. I always feel a little peculiar when I read such warnings. Will something bad happen to me if I don’t do the exercises and problems? Of course not. I’ll gain some time, but at the expense of depth of understanding. Sometimes that’s worth it. Sometimes it’s not.

— Michael Nielsen, Neural Networks and Deep Learning

1. Synopsis

I spent the first few days of my recent winter vacation transitioning all the problem sets for my students from a “traditional” format to a “point-based” format. Here’s a before and after.

Technical specification:

  • The traditional problem sets used to consist of a list of 6-9 olympiad problems of varying difficulty, for which you were expected to solve all problems over the course of two weeks.
  • The new point-based problem sets consist of 10-15 olympiad problems, each weighted either 2, 3, 5, or 9 points, and an explicit target goal for that problem set. There’s a spectrum of how many of the problems you need to solve depending on the topic and the version (I have multiple difficulty versions of many sets), but as a rough estimate the goal is maybe 60%-75% of the total possible points on the problem set. Usually, on each problem set there are 2-4 problems which I think are especially nice or important, and I signal this by coloring the problem weight in red.

In this post I want to talk a little bit about what motivated this change.

2. The old days

I guess for historical context I’ll start by talking about why I used to have a traditional format, although I’m mildly embarrassed at now, in hindsight.

When I first started out with designing my materials, I was actually basically always short on problems. Once you really get into designing olympiad materials, good problems begin to feel like tangible goods. Most problems I put on a handout are ones I’ve done personally, because otherwise, how are you supposed to know what the problem is like? This means I have to actually solve the problem, type up solution notes, and then decide how hard it is and what that problem teaches. This might take anywhere from 30 minutes to the entire afternoon, per problem. Now imagine you need 150 such problems to run a year’s curriculum, and you can see why the first year was so stressful. (I was very fortunate to have paid much of this cost in high school; I still remember many of the problems I did back as a student.)

So it seemed like a waste if I spent a lot of time vetting a problem and then my students didn’t do it, and as practical matter I didn’t have enough materials yet to have much leeway anyways. I told myself this would be fine: after all, if you couldn’t do a problem, all you had to do was tell me what you’ve tried, and then I’d walk you through the rest of it. So there’s no reason why you couldn’t finish the problem sets, right? (Ha. Ha. Ha.)

Now my problem bank has gotten much deeper, so I don’t have that excuse anymore. [1]

3. Agonizing over problem eight

But I’ll tell you now that even before I decided to switch to points, one of the biggest headaches was always whether to add in that an eighth problem that was really nice but also difficult. (When I first started teaching, my problem sets were typically seven problems long.) If you looked at the TeX source for some of my old handouts, you’ll see lots of problems commented out with a line saying “too long already”.

Teaching OTIS made me appreciate the amount of power I have on the other side of a mentor-student relationship. Basically, when I design a problem set, I am making decisions on behalf of the student: “these are the problems that I think you should work on”. Since my kids are all great students that respect me a lot, they will basically do whatever I tell them to.

That means I used to spend many hours agonizing over that eighth problem or whether to punt it. Yes, they’ll learn a lot if they solve (or don’t solve) it, but it will also take them another two or three hours on top of everything else they’re already doing (OTIS, school, trumpet, track, dance, social, blah blah blah). Is it worth those extra hours? Is it not? I’ve lost sleep over whether I made the right choice on the nights I ended up adding that last hard problem.

But in hindsight the right answer all along was to just let the students decide for themselves, because unlike your average high-school math teacher in a room of decked-out slackers, I have the best students in the world.

4. The morning I changed my mind

As I got a deeper database this year and commented more problems out, I started thinking about point-based problem sets. But I can tell you the exact moment when I decided to switch.

On the morning of Sunday November 5, I had a traditional problem set on my desk next to a point-based one. In both cases I had figured out how to do about half the problems required. I noticed that the way the half-full glass of water looked was quite different between them. In the first case, I was freaking out about the other half of the problems I hadn’t solved yet. In the second case, I was trying to decide which of the problems would be the most fun to do next.

Then I realized that OTIS was running on the traditional system, and what I had been doing to my students all semester! So instead of doing either problem set I began the first prototypes of the points system.

5. Count up

I’m worried I’ll get misinterpreted as arguing that students shouldn’t work hard. This is not really the point. If you read the specification at the beginning carefully, the number of problems the students are solving is actually roughly the same in both systems.

It might be more psychological than anything else: I want my kids to count how many problems they’ve solved, not how many problems they haven’t solved. Every problem you solve makes you better. Every problem you try and don’t solve makes you better, too. But a problem you didn’t have time to try doesn’t make you worse.

I’ll admit to being mildly pissed off at high school for having built this particular mindset into all my kids. The straight-A students sitting in calculus BC aren’t counting how many questions they’ve answered correctly when checking grades. They’re counting how many points they lost. The implicit message is that if you don’t do nearly all the questions, you’re a bad person because you didn’t try hard enough and you won’t learn anything this way and shame on you and…

That can’t possibly be correct. Imagine two calculus teachers A and B using the same textbook. Teacher A assigns 15 questions of homework a week, teacher B assigns 25 questions. All of teacher A’s students are failing by B’s standards. Fortunately, that’s not actually how the world works.

For this reason I’m glad that all the olympiad kids report their performance as “I solved problems 1,2,4,5” rather than “I missed problems 3,6”.

6. There are no stupid or lazy questions

The other wrong assumption I had about traditional problem sets was the bit about asking for help on problems you can’t solve. It turns out getting students to ask for help is a struggle. So one other hope is that with the point-based system is that if a student tries a problem, can’t solve it, and is too shy to ask, then they can switch to a different problem and read the solution later on. No need to get me involved with every single missed problem any more.

But anyways I have a hypothesis why asking for help seems so hard (though there are probably other reasons too).

You’ve all heard the teachers who remind students to always ask questions during lectures [2], because it means someone else has the same question. In other words: don’t be afraid to ask questions just because you’re afraid you’ll look dumb, because “there are no stupid questions“.

But I’ve rarely heard anyone say the same thing about problem sets.

As I’m writing this, I realize that this is actually the reason I’ve never been willing to go to office hours to ask my math professors for help on homework problems I’m stuck on. It’s not because I’m worried my professors will think I’m dumb. It’s because I’m worried they’ll think I didn’t try hard enough before I gave up and came to them for help, or even worse, that I just care about my grade. You’ve all heard the freshman biology TA’s complain about those kids that just come and ask them to check all their pset answers one by one, or that come to argue about points they got docked, or what-have-you. I didn’t want to be that guy.

Maybe this shaming is intentional if the class you’re teaching is full of slackers that don’t work unless you crack the whip. [3] But if you are teaching a math class that’s half MOPpers, I seriously don’t think we need guilt-trips for these kids whenever they can’t solve a USAMO3.

So for all my students, here’s my version of the message: there are no stupid questions, and there are no lazy questions.

Footnotes

  1. The other reason I used traditional problem sets at first was that I wanted to force the students to at least try the harder problems. This is actually my main remaining concern about switching to point-based problem sets: you could in principle always ignore the 9-point problems at the end. I tried to compensate for this by either marking some 9’s in red, or else making it difficult to reach the goal without solving at least one 9. I’m not sure this is enough.
  2. But if my question is “I zoned out for the last five minutes because I was responding to my friends on snapchat, what just happened?”, I don’t think most professors would take too kindly. So it’s not true literally all questions are welcome in lectures.
  3. As an example, the 3.091 class policies document includes FAQ such as “that sounds like a lot of work, is there a shortcut?”, “but what do I need to learn to pass the tests?”, and “but I just want to pass the tests…”. Also an entire paragraph explaining why skipping the final exam makes you a terrible person, including reasons such as “how do you anything is how you do everything”, “students earning A’s are invited to apply as tutors/graders”, and “in college it’s up to you to take responsibility for your academic career”, and so on ad nauseum.