Power Overwhelming - Essays

Words Spent

2025-05-12T13:37:00-04:00

One of my favorite Djikstra programming quotes is about thinking via “lines of code spent” rather than “lines of code produced”. I started using this as a philosophy in my writing too: words spent.

Background

One of the things that’s surprised me about student writing is how poorly words are spent. You’ll have a solution where the trivial boilerplate steps are painfully verbose, and then the actually important parts are missing all the critical details.

I wonder how much of this is because of crummy writing advice. In school essays, even when you have nothing meaningful to say, teachers often impose a minimum word countIn ninth grade, my English teacher preferred the euphemism “develop your ideas” for “write more words”. It wasn’t until halfway through the year I realized why she kept writing that on all my essays. as a “proof of work”. The implied conclusion is that writing many words is the goal. Which is same mistake that counting “lines of code produced” is.

More careful teachers might talk about being concise, and might impose a maximum word count instead (or in addition). Then students get the idea you have this elastic dial that goes from “too little detail” to “too much detail”, and you need to guess the magic “just right” to win your teacher’s approval.

But that’s also mistaken. The “right” numbers of words should depend on what you have to say, just as the number of lines of code should reflect the complexity of the program you’re trying to implement.

Proposal

My proposal is to use “words spent” — just that two-word phrase — as a guideline to steer writers away from bad habits they’ve learned in school. I’ve been using it myself recently and I think it really conveys the right instincts concisely.

Examples:

People instinctively know to avoid costs. If you treat word count as “words spent”, you won’t try to inflate it.
- Orwell’s advice “if a word can be cut out, always cut it out” becomes an obvious corollary. Of course you cut unneeded costs.
People also instinctively know that costs can be necessary. If you buy a car for 5000 dollars, you don’t expect much from it.
Cost prioritization becomes instinctive. You know you should be spending the most words on the most important parts. In principle, the reader can guess what matters by how many words are spent on it.
Cost efficiency becomes natural too. You want good words that do more with less.

Example usage

Individual sentences

You could use “words spent” to guide individual sentences. For example, in a different blog post, I had a draft with the passage:

Olga and I were originally planning to do a variant on Slitherlinks, and we had some prototypes that felt decent by mid-February. I was trying to not be too ambitious because of the tight deadline, so I was trying to keep things simple. We had a verifier up and running too like I did for BWMS, so we felt we could iterate quickly once we got the meta specification.

but on re-reading I deleted a third of the words:

Olga and I were originally planning Slitherlinks variants, and had some decent prototypes by mid-February. I wanted to keep things simple because of the tight deadline. We had a verifier written (like with BWMS), so we could iterate quickly.

(It may be possible to overdo this. If I’m being really aggressive, almost every sentence has some fat, but at some point the sentences start to sound unnatural. I would gladly spend three extra words to make a paragraph flow better, even if those words technically don’t change the meaning of the paragraph.)

Overall picture

But I think it’s more important to use “words spent” at a higher level, rather than micromanaging individual sentences too much. Here are some examples of ways it’s improved my writing style overall:

Sometimes in my writing you’ll see claims whose proof is the single word “Clear” or similar. I’m never terse out of lazinessIf I’m trying to be lazy, it’ll say “to be added later” instead.. It’s always because I’ve decided it’s not worth the words spent to say more.

Conversely, when writing math exposition, I really value good examples. So I know that I should spend a lot of words on examples, and spend fewer words on formal definitions or proofs that I want to de-emphasize.

I know pictures aren’t words, but treat them the same way. (Okay, fine: “a picture is worth 1000 words”.) Point is, good diagrams and pictures are really cost-efficient. They might take a lot of time for me to draw, but the author’s timeTime spent is not words spent! Another bad anti-lesson. Students whose only writing experience is in school might mistakenly conflate words spent with time spent. But if you’re actually trying to write well, your words spent could easily go down over time as you figure out how to be more cost-efficient. Good programmers do this all the time too (refactoring code to be shorter or deleting unnecessary components). is irrelevant to this discussion. You can easily convey something in a half-page diagram that even 1000 words can’t do. If so, do it. (Tables can be really good too.)

Zooming out even further, here’s some thoughts on how the Napkin and LAMV books fit into this framework.

I think I was happy with the Napkin because I was cost-efficient: tons of diagrams, emphasis on intuition over proofs, etc. That’s why Napkin is a good birds-eye view of topics, without getting bogged down by fine or technical details. It’s not a textbook — concisely conveying intuition is the goal of Napkin.

But on the other extreme, LAMV was meant to outright replace the class textbook. That was really freaking expensive, especially for the target audience, who typically really need things spelled out in full. So I think LAMV was successful exactly because it put in writing all the detailed expositionActually it could have been even more verbose if I was careless. If you read the preface of LAMV, I mention that a lot of the solutions to the example questions were initially drafted by ChatGPT, but then for which I would slim down the output like crazy: usually by a factor of 2 or 3. (Actually, now that I think about it, I wonder if humans might tend to spend too few words out of laziness, whereas GPT spends too many words because it doesn’t take time to write words the same way humans do.) that others were simply unable or unwilling to do. In this case the enormous cost paid off.

In ninth grade, my English teacher preferred the euphemism “develop your ideas” for “write more words”. It wasn’t until halfway through the year I realized why she kept writing that on all my essays. ↩
If I’m trying to be lazy, it’ll say “to be added later” instead. ↩
Time spent is not words spent! Another bad anti-lesson. Students whose only writing experience is in school might mistakenly conflate words spent with time spent. But if you’re actually trying to write well, your words spent could easily go down over time as you figure out how to be more cost-efficient. Good programmers do this all the time too (refactoring code to be shorter or deleting unnecessary components). ↩
Actually it could have been even more verbose if I was careless. If you read the preface of LAMV, I mention that a lot of the solutions to the example questions were initially drafted by ChatGPT, but then for which I would slim down the output like crazy: usually by a factor of 2 or 3. (Actually, now that I think about it, I wonder if humans might tend to spend too few words out of laziness, whereas GPT spends too many words because it doesn’t take time to write words the same way humans do.) ↩

A poset of math programs

2025-01-31T13:37:00-05:00

There are a lot of different kinds of math enrichment activities now, ranging from olympiads to math circles to tons of summer programs and so on. I work in the competition sphere, and I used to spend a lot of time worrying about whether I took the right side.

Now that I’m a bit older, I came to the realization that maybe I don’t need to be so intent on comparing my work to others (even though I realize comparing yourself to others is human nature, haha). I eventually told myself: there are lots of people who don’t like olympiad exams; there are also lots of people who do, and it’s just okay for them to co-exist. We don’t need to decide which of the N systems is the best and kill the other N-1, because “best” is so different from person to person anyway.

Instead, we should see the diversity of these activities as a strength: each program brings whatever it can to the table, and we let each student pick whichever one they feel is the best fit for them. Something about free markets being good, y’know? Advanced math education isn’t exactly booming in the United States; it’s small enough I think the biggest leaders are usually both acting in good faith and at least semi-competent. So the quality of the programs seems to me like it should be embedded into a poset rather than a total order. (And improving your program from year N to year N+1 counts as moving up in the poset.)

Richard Rusczyk has a quote I like, where when asked how he’d run US educational policy, he replied:

I typically answer something along the lines of, “I would try to find ways that no one entity or person can run educational policy.” I think education as a whole, like foreign policy and domestic policy for a nation, is far too complex a problem for a single person or single centralized entity to solve. Moreover, the downside of inflicting mistakes on an entire generation are far too large. It’s much better to have the higher variance of allowing localized decisions.

I believe the same thing even in the smaller scope of gifted education.

So I’d like to suggest that everyone in the math enrichment spaces should be treated as teammates. It’s hard to remember this sometimes, especially in the face of serious philosophy disagreements, or when we programs have to compete for the attention of top students (summer is only so long, dates have a tendency to overlap). So that’s one reason I’m writing this post, to remind myself of this. And I’m really happy to see forums like the Summer Mathematics Program Consortium which bring together leaders from different initiatives together.

The story of the AutoCarrot

2024-06-03T13:37:00-04:00

1. Glazed carrots

Okay. Imagine you’re, like, trying to make glazed carrots or something.

Maybe a really simplified recipe looks something like:

Cut your carrots into suitably sized pieces with a knife.
Use a measuring spoon to get the right amount of oil, sugar, salt, etc.
Throw the carrots and other ingredients into a frying pan.
Serve the carrots on a plate.

You’ll notice that there were a bunch of different tools you used. The knife was used to cut the carrots into pieces. The measuring spoon was used to get the right amounts of other ingredients. And the plates are just there for the presentation of your dish. All these tools are things you see in any kitchen, but they do a single, completely unrelated thing.

Now imagine someone asks you:

I’m confused, why do people use a measuring spoon for cooking? Why not just use a frying pan?

If someone asks this question, then it indicates something is really wrong with their understanding of cooking. The English idiom “comparing apples and oranges” would be an understatement. It makes no sense to try to use a frying pan “instead” of a measuring spoon; it’s practically a type-error.

2. Introducing the AutoCarrot

You might be incredulous anyone could ask a question as nonsensical as the example I just gave. But that’s because most people have used a measuring spoon or frying pan sometime in their life, so it’s obvious to them that a measuring spoon and a frying pan are not interchangeable.

Now imagine instead that we lived in a world where you had the AutoCarrot, which was this big box sort of like a laundry machine. The AutoCarrot had a big opening at the top and all you had to do was dump in the carrots from your fridge. It comes pre-installed with a lot of oil, sugar, and salt. The AutoCarrot would automatically cut the carrots for you, add in the right amount of other ingredients, cook them at low heat, and even 3D-print a plate for you to serve the glazed carrots on.

… Actually that sounds kinda nice. Why do four steps when you can do one? NEW STARTUP IDEA LET’S GO.

3. What do you mean you didn’t read the manual?

Imagine the AutoCarrot works great!

At first.

Then one day you decide you want carrots to be less salted. But you’ve never seen a measuring spoon in your life, you’ve only used the AutoCarrot, so you do the only thing you know how to do and send an all-caps email: “HEY EVAN I WANT MY CARROTS TO BE LESS SALTY BUT I CAN’T FIGURE OUT WHAT DIAL ON THIS MACHINE TO USE IS IT THIS FAHRENHEIT ONE????? PLEASE HELP URGENT”.

Or maybe the AutoCarrot ran out of the pre-installed cooking oil, so the carrots aren’t cooking and just burn instead. But you don’t know what cooking oil is, and have no idea what would cause food to get burnt, so you just throw the whole machine out and buy a new AutoCarrot instead of buying a bottle of olive oil for 3 bucks.

And then you wonder why the chefs at actual restaurants are so snobby and keep saying you should be using a measuring spoon and frying pan separately instead of the 350-dollar AutoCarrot you bought on Amazon.com. Geez, don’t they know you have better things to do than learn how to cook? So many extra steps! So much complexity! The chefs are clearly just showing off.

Like, what, you’re telling me I should cut the carrots myself? By hand? And I have to buy my own knife for that? Sheesh! It’s not like you want Michelin-star level carrots, you just want to eat. You’re happy with the AutoCarrot.

4. This is all a metaphor

I work with math students who use LaTeX, and the LaTeX analog of the AutoCarrot has taken over.

The infinitely many stages of grief

2024-04-05T13:37:00-04:00

Where do all the smart, curious, earnest kids go these days?

One of my friends asked me this recently, and I wasn’t sure what to say. In the last ten years, something has changed.

If I had to summarize my concerns in one sentence, I would say this: kids these days no longer feel they’re allowed to work on what they’re interested in or excited about. Instead, they feel obligated to work on whatever happens to be considered the most “important” (or “prestigious”) thing possible.It’s for this reason I consider ambition as a double-edged sword. When ambition isn’t accompanied by excitement, earnestness, curiosity, or interest, it doesn’t usually end well.

But let me do a bit of story-telling.

Hobbies

When I was kid, math contests were seen as a hobby, or sport, or game. Those were the good old days.

Today, that’s no longer true.

For example, at the end of MOP since 2011, we run a selection test called the TSTST, which chooses the finalists for the subsequent year’s IMO team. When I took the TSTST as kid, the feeling in the air was “it’s game day, GLHF!”. I didn’t solve any problems at all my first year, and I still had a blast.

Whereas in 2023, I remember visiting the testing room on the first day of TSTST to help with setting up folders and whatnot, and feeling like I was at a funeral.

The situation has gotten so bad that many students and staff have suggested that we should remove the TSTST altogether from camp. There are some operational reasons for why I don’t think this is feasible, but something about the whole story bothers me:

Honestly, I also find it disappointing that we are trying to run a summer camp for math Olympiad, and apparently are unable to run a math Olympiad during this camp, because the students’ egos are too fragile.

The idea that we should cancel TSTST seems like putting a Band-Aid to treat the symptoms of a much larger underlying issue. Something is really wrong if our top students are unable to handle taking a contest that barely counts.

Induction

It’s easy to point at the contests and say, “haha, see, contests are toxic!”. But when you drill further, I think something much deeper and scarier is going on, and the contests are just one link in a long chain.

For example, one of the reactions to people taking competition results too seriously was extensive propaganda about the uselessness of contests. Everyone is always saying, contest scores are noisy and everyone has bad days. Math contests are cringe and not real math. Competitions are one-dimensional. Contest problems are super uncreative and just solved by a bunch of tricks. There’s more to life than academics. Yada yada yada. The world gives reason after reasonFor the record, I only agree with a proper subset of the reasons that people give. But the correctness is irrelevant to the rest of the post, and this happens to be a sensitive topic, so I won’t delve further. why test scores aren’t a good ruler.

In an ideal world, you would hope the outcome of this messaging would be to transform the contests back into a sport again that people stop treating like their lives depend on it, because it’s just a game, holy crap.

Do you know what happened instead?

It’s like trying to comfort an anorexic by saying their bathroom scale is an inaccurate mechanical scale.There’s a more general lesson here: treating symptoms instead of root causes is misguided at best and actively harmful at worst. This works up until they buy a digital scale and spend every morning calibrating it, and now you’re worse off than you started.

Eight years ago, when I wrote the sentence

Changing the Golden Metric from olympiads to research seems to just make the world more egotistic than it already is.

it was still a hypothetical.I better spell out this implication fully, even though I think it’s implied. The reason I’m so distrustful of the dramatic increase in demand for high school research is that I’m skeptical of the underlying motivation. If we were in a world where there were suddenly an army of smart high schoolers who were super excited about doing long-term math projects, then sure, wonderful, go do the thing you are excited about. However, based on what I actually see, I don’t think this is true (I would love to be wrong about this). Now it’s reality, and it’s not stopping there.

Infinitely many stages of grief.

It’s for this reason I consider ambition as a double-edged sword. When ambition isn’t accompanied by excitement, earnestness, curiosity, or interest, it doesn’t usually end well. ↩
For the record, I only agree with a proper subset of the reasons that people give. But the correctness is irrelevant to the rest of the post, and this happens to be a sensitive topic, so I won’t delve further. ↩
There’s a more general lesson here: treating symptoms instead of root causes is misguided at best and actively harmful at worst. ↩
I better spell out this implication fully, even though I think it’s implied. The reason I’m so distrustful of the dramatic increase in demand for high school research is that I’m skeptical of the underlying motivation. If we were in a world where there were suddenly an army of smart high schoolers who were super excited about doing long-term math projects, then sure, wonderful, go do the thing you are excited about. However, based on what I actually see, I don’t think this is true (I would love to be wrong about this). ↩

The depth of Hanabi

2023-09-19T13:37:00-04:00

This post is a short chrono-logue about my time with the card game Hanabi, which I play with the H-group. Thus, it’s also implicitly an advertisement for why I enjoy the game Hanabi so much.

I think the progression is a bit interesting because it can be divided into almost discrete “stages”, with each stage feeling really different from the last.

0. Casual in-person play: a memory game

Like many other people in my age group, I first met the card game Hanabi in-person at some summer math camp or other (either MOP or SPARC?). The rules are pretty simple to explain, so it’s popular. But we didn’t have much strategy behind it. We had the idea that we played from left to right, a clue means “play all”, and some form of a Finesse-type blind play.

That meant the game felt kind of like a chaotic party game, sort of like the first game of One Night Ultimate Werewolf or something. The game mostly revolved around trying to remember past clues and guesswork at how people would respond. (It’s true that we would allow asking “what do you know about your cards?”, but that was only a subset of the information anyway.) I thought the game was okay, I would play it, but it definitely wasn’t my favorite game, and I’d only play it if others wanted. We were not particularly good at it; we rarely got a perfect score, if ever.

1. Online play: a guesswork game

Around when the pandemic happened, everything got virtualized, and we started playing on the hanab.live website.

We still played with the same crappy convention set (if you can even call it that), but I distinctly remember saying “this is one of the few things that works better virtually than in-person”. That’s because the hanab.live is really feature-rich. The website will completely keep track of the “what do you know about all your cards” situation for you, marking both positive and negative clues, as well as letting you make arbitrary additional notes on each individual card. It also keeps track of the entire history of each game for you, so you can always rewind to see what clues had been given before.

So playing online eliminated the memory component from the game. I’m not saying memory games are always bad (I play Fish, too), but for Hanabi I preferred playing without having memory as a constraint. Nonetheless, because the game was still mostly about guesswork, we still rarely got a perfect score.

2. Starting the H-group convention: learning a new game

Around August 2020 I started learning the H-group conventions, which start to standardize the meaning of “normal” clues. This eliminates most of the guesswork component from the game, which I also thought made the game more interesting. The experience became more about logic and learning strategy, rather than guesswork and memory like it had been at the start.

The H-group plays with a spectrum of experience levels (and they are actually named “levels”, numbered 1-25 in the most recent document). For me, I actually enjoyed the experience of learning a lot of strategies. Despite the large number of conventions, there’s a requirement that strategies the H-group uses should be “intuitive and easy to remember”, so learning each individual strategy usually felt like it made sense, rather than memorizing an arbitrary or contrived rule.

So, for a while, I was slowly moving up through the level system, getting used to the various conventions the group played with.

3. Familiarity: a logic game

Once I started getting more familiar with the basic conventions, I didn’t have to spend as much effort interpreting clues anymore. After a while you develop some instincts and things become more automatic.

Instead, the “getting used to conventions” you start thinking about strategy, in addition to just the logic. If I give a red clue, what do I think the next player will do? And what will the player after that do? Versus if I discard, and let someone else give the red clue, what would happen instead? What would be the next outcome?

I talked about this previously under the name foresight. It’s one example of a way in which logic and strategy mix, in which, given the conventions you know, you want to give clues that are as efficient as possible.

One other thing I should mention: my win-rate skyrocketed. It turns out that even with the most basic level 2 conventions, if you don’t make many mistakes, you can get a perfect score more often than not in standard no-variant games.An earlier version of this post had an exaggerated claim of “well over 80%”. It was pointed out in the comments section this was grossly overconfident, both due to human error or tough deals. The “base” game often felt easy. So we would often play variants which changed the games in ways that made the game harder, like having a suit that was not touched by any rank clues, or suits where there was only one of each card, or playing clue-starved where discarding gives you 0.5 clues back instead of 1.

4. Intermediate play: bending the rules

Once you play variants, Hanabi becomes difficult enough that you cannot simply follow pre-existing conventions too rigidly. There is a great passage in the strategy document that dispels this myth:

Up until now, you may have the impression that the group has a lot of conventions, and that if you just memorize all of the conventions, you will become a really good Hanabi player. Or, you may have the impression that the conventions are like laws and that you must always follow them. Neither of these things are true.

You can read the full paragraph on the conventions website, but the short version is — now that you know the conventions now, break them whenever the situation demands it. This is especially true in really hard variants, where you might always have to take a few risks in order to win because the baseline required efficiency is so high.

5. Advanced play: making trade-offs

When you first learn the H-group conventions, part of the exciting thing is that you can give clues that are really efficient; for example, a single clue could get four or five or more cards to play at once. So, beginners are taught to look for the most efficient clues among the ones they have available to them, and that’s a lot of fun.

However, as you get more advanced, it becomes important to also think about what costs you might incur by trying to set up a really efficient clue. There are a few examples of other parameters to pay attention too:

In serious play, Hanabi also has a parameter called Tempo or Pace, referring to the speed at which the cards are played. A clue can have high efficiency but be slow enough that a less efficient clue would still be better for the team’s win rate. So, it becomes necessary to balance efficiency and tempo.

One related concept that strong players talk about is Bottom Deck Rate; in Hanabi, if you discard a green 3 and the other copy of green 3 was at the bottom of the deck, it’s impossible to win. So there’s a certain notion of “safety”, in which you want to give moves that minimize the number of unsafe discards, and protect cards that might be useful later on in the game. When more efficiency is unnecessary, you often need to trade out some efficiency and give less efficient clues in the hopes of orchestrating safe discards, where you use the foresight I mentioned earlier to try to guide the team down a path where the players whose cards are trash do most of the discarding.

And finally, one more trade-off: the clarity principle. If a player gives a really complicated clue their teammates technically could have figured out, but doesn’t, you might burn the whole game. An “optimal” clue might still be not worth it, even if it’s great on paper, if it’s too confusing for the humans trying to interpret the clue.

During advanced play, it no longer felt like I was just trying to set up the most efficient clue I could. Instead, it felt like I always needed to balance several different metrics and make some judgment calls. That felt really cool to me.

The hanab.live website understands this too. During each game, one of the most important numbers it displays is the future required efficiency, which calculates how many cards you still need to get divided by how many clues you will have over the rest of the game. Good players need to always look at this number when making decisions. If future required efficiency is low, play conservatively and prioritize clear, simple clues that protect cards. When future required efficiency is high, play more aggressively, and be willing to take some risks to try and capture more cards.

6. Competitive play: trust and teamwork

Once you play competitively with expert players, it’s not enough to just be making judgment calls yourself. Because the foresight of thinking several turns ahead is so important, you need to trust that everyone on your team is in sync about judgments. After all, part of picking each move relies heavily on predicting what clues your teammates will give in the future.

That’s why in the H-group, we did post-game reviews after each game, where we go through every move made in the game and talk through any tough decisions that we had to make, or things that in hindsight were mistakes. Because at the highest levels of play, understanding how your teammates play, and the resulting trust and teamwork, become the heart of the game.

An earlier version of this post had an exaggerated claim of “well over 80%”. It was pointed out in the comments section this was grossly overconfident, both due to human error or tough deals. ↩

Pride

2023-06-03T13:37:00-04:00

Sometimes people ask me how many of my students made the IMO, and if I’m in a bad mood I often give the super snarky reply, “I lost track”.The good-mood answer is “a lot”.

That’s actually a white lie. The real answer is “I deliberately don’t keep track”. And in this post I want to explain why.

It’s definitely human nature to be happy when your students succeed, the same way it’s human nature to be happy when your selfies get hearts. In moderation, that seems fine. I think it’s unlikely I ever reach a point where I never brag about OTIS at all.

But there is a fine line between the following two implications:

“I’m super proud of my kids, look what they did.”
“I’m super proud of myself, look what my kids did.”

Without naming anyone in particular, I have seen some instances which I felt were clearly on the wrong side of this line.

When done wrong, this violates a lot of my basic principles. I think it’s important that my students “own” their journeys, and I always remind myself contests are ultimately for the students. I think it’s not cool to use students as pawns for public relations.It goes the other way too: I would be similarly disappointed if students were treating me as a stepping stone to get into MIT.

As an example of a mistake I made in this direction, in late 2022 I floated the idea of having an OTIS alum survey. I rapidly got feedback this felt super dark-artsy and closed the issue. This probably means I’m doomed to having the occasional “wait, you made IMO?” kind of conversation, but so be it.

Status is an Elder God whose power is consistently underestimatedHi Yan! Another hexagon for your collection.. So this post is yet another reminder to myself that wanting to help others is different from wanting to be recognized as a person that helps others, and I’ll be the first to admit this is something I still struggle a lot with.

The good-mood answer is “a lot”. ↩
It goes the other way too: I would be similarly disappointed if students were treating me as a stepping stone to get into MIT. ↩
Hi Yan! Another hexagon for your collection. ↩

A short dissent on USA eligibility

2021-07-15T13:37:00-04:00

Sometime this week the American Math Competitions released the following new policy:

For the IMO, EGMO, RMM, TSTSTs, and TSTs, and MOP, students must be US citizens or US permanent residents. Visas are not a valid substitute. (AMC Policies)

I want to make a rather brief statement on why I was opposed to this change. To do this I want to draw an analogy. In the American Math Competitions, students are asked what gender they identify with, which is used to determine whether they are eligible for the European Girl’s Math Olympiad and also for invitations to MOP.

This means that in theory, you could try to abuse the system by deliberately misrepresenting your self-identified gender. But in practice, nobody has attempted this. So, we continue to allow students to self-identify their gender, in order to make sure to be inclusive to students with gender dysphoria, and trusting our students to have the integrity to report their gender honestly. The other is to be exclusive and follow the sex assigned at birth, which closes a potential loophole that nobody was exploiting.

I think we should have done the same with visa-holding students.

A student with, say, an F-1 visa, is also in between two points, and there is a spectrum of how much time they have spent in the US. Sometimes they have been in the US for only a short time, other times, they have been studying in the USA for all of high school and participated multiple times in AMC/AIME/USAMO track; and then go on to MIT for another four years, and so on. No matter what they do, there will be some eyebrows raised. If they compete with the United States, some people will complain they are not proper citizens; if they compete with their home country, some people will complain they are too American.

Again we have two options. One is to be inclusive and allow anyone legally residing in the United States and feeling they identify as American a chance to represent us. This is what we have done for the last 20-ish years, and I stand behind these teams. The other is to be exclusive and follow legal citizenship, again closing a loophole that nobody was exploiting. Though I understand the rationale, I am still saddened to hear that we have changed to the latter, and I am worried about the message that we send when we tell certain people they are no longer welcome in the USA math community.

(I do believe that one should not compete for spots on more than one IMO country in the same calendar year, even as a dual national. This was the main reason I felt comfortable competing for Taiwan in 2014, because I was no longer eligible to compete for the United States.)

So that is my public statement, but I have one more thing to add. I am hearing lots and lots of rumors going around claiming that specific students (by name) should not have been allowed on the team or even were taking advantage of the USA system in bad faith. To that I have to ask: please don’t spread rumors like this if you don’t know the students in question. It’s not nice to talk about kids you don’t even know behind their backs; but also please take my word for it when I say some of you are really picking the wrong targets.

On a definitely-kinda-related note, the USEMO for this year is gonna be October 30 and October 31, and remains open to anyone who is studying in an American school, even on a visa :) or otherwise identifies as American. If you don’t identify as American you’re instead invited to come to grading or propose problems, or both!

Meritocracy is the worst form of admissions except for all the other ones

2020-01-13T13:37:00-05:00

I’m now going to say something explicitly that I hinted at in June: I don’t think a student deserves to make MOP more because they had a higher score than another student.

I think it’s easy to get this impression because the selection for MOP is done by score cutoffs. So it sure looks that way.

But I don’t think MOP admissions (or contests in general) are meant to be a form of judgment. My primary agenda is to run a summer program that is good for its participants, and we get funding for N of them. For that, it’s not important which N students make it, as long as they are enthusiastic and adequately prepared. (Admittedly, for a camp like MOP, “adequately prepared” is a tall order). If anything, what I would hope to select for is the people who would get the most out of attending. This is correlated with, but not exactly the same as, score.

Two corollaries:

I support the requirement for full attendance at MOP. I know, it sucks for those star students who qualify for two conflicting and then have to choose. You have my apologies (and congratulations). But if you only come for 2 of 3 weeks, you took away a spot from someone who would have attended the whole time.
I am grateful to the European Girl’s MO for giving MOP an opportunity to balance the gender ratio somewhat; empirically, it seems to improve the camp atmosphere if the gender ratio is not 79:1.

Anyways, given my mixed feelings on meritocracy, I sometimes wonder whether MOP should do what every other summer camp does and have an application, or even a lottery. I think the answer is no, but I’m not sure. Some reasons I can think of behind using score only:

MOP does have a (secondary) goal of IMO training, and as a result the program is almost insane in difficulty. For this reason you really do need students with significant existing background and ability. I think very few summer camps should explicitly have this level of achievement as a goal, even secondarily. But I think there should be at least one such camp, and it seems to be MOP.
Selection by score is transparent and fair. There is little risk of favoritism, nepotism, etc. This matters a lot to me because, basically no matter how much I try to convince them otherwise, people will take any admissions decision as some sort of judgment, so better make it impersonal. (More cynically, I honestly think if MOP switched to a less transparent admissions process, we would be dealing with lawsuits within 15 years.)
For better or worse, qualifying for MOP ends up being sort of a reward, so I want to set the incentives right and put the goalpost at “do maximally well on USAMO”. I think we design the USAMO well enough that preparation teaches you valuable lessons (math and otherwise). For an example of how not to set the goalpost, take most college admissions processes.

Honestly, the core issue might really be cultural, rather than an admissions problem. I wish there was a way we could do the MOP selection as we do now without also implicitly sending the (unintentional and undesirable) message that we value students based on how highly they scored.

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

2019-08-25T13:37:00-04:00

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.

An opening speech for MOP

2019-06-01T13:37:00-04:00

While making preparations for this year’s MOP, I imagined to myself what I would say on orientation night if I was director of the camp, and came up with the following speech. I thought it might be nice to share on this blog. Of course, it represents my own views, not the actual views of MOP or MAA. And since I am not actually director of MOP, the speech was never given.

People sometimes ask me, why do we have international students at MOP? Doesn’t that mean we’re training teams from other countries? So I want to make this clear now: the purpose of MOP is not to train and select future IMO teams.

I know it might seem that way, because we invite by score and grade. But I really think the purpose of MOP is to give each one of you the experience of working hard and meeting new people, among other things. Learn math, face challenges, make friends, the usual good stuff, right? And that’s something you can get no matter what your final rank is, or whether you make IMO or EGMO or even next year’s MOP. The MOP community is an extended family, and you are all part of it now.

What I mean to say is, the camp is designed with all 80 of you in mind. It made me sad back in 2012 when one of my friends realized he had little chance of making it back next year, and told me that MAA shouldn’t have invited him to begin with. Even if I can only take six students to the IMO each year, I never forget the other 74 of you are part of MOP too.

This means one important thing: everyone who puts in their best shot deserves to be here. (And unfortunately this also means there are many other people who deserve to be here tonight too, and are not. Maybe they solved one or two fewer problems than you did; or maybe they even solved the same number of problems, but they are in 11th grade and you are in 10th grade.)

Therefore, I hope to see all of you put in your best effort. And I should say this is not easy to do, because MOP is brutal in many ways. The classes are mandatory, we have a 4.5-hour test every two days, and you will be constantly graded. You will likely miss problems that others claim are easy. You might find out you know less than you thought you did, and this can be discouraging. Especially in the last week, when we run the TSTST, many of you will suddenly realize just how strong Team USA is.

So I want to tell you now, stay determined in the face of adversity. This struggle is your own, and we promise it’s worth it, no matter the outcome. We are rooting for you, and your friends sitting around you are too. (And if the people around you aren’t your friends yet, change that asap.)

117(d): Please don't tax PhD tuition waivers

2017-11-08T13:37:00-05:00

This is a rare politics post; I’ll try to keep this short and emotion-free. If parts of this are wrong, please correct me. More verbose explanations here, here, here, here, longer discussion here.

Suppose you are a math PhD student at MIT. Officially, this “costs” $50K a year in tuition. Fortunately this number is meaningless, because math PhD students serve time as teaching assistants in exchange for having the nominal sticker price waived. MIT then provides a stipend of about $25K a year for these PhD student’s living expenses. This stipend is taxable, but it’s small and you’d pay only $1K-$2K in federal taxes (about 6%).

The new GOP tax proposal strikes 26 U.S. Code 117(d) which would cause the $50K tuition waiver to also become taxable income: the PhD student would pay taxes on an “income” of $75K, at tax brackets of 12% and 25%. If I haven’t messed up the calculation, for our single PhD student this means paying $10K in federal taxes out of the same $25K stipend (about 40%).

I think a 40% tax rate for a PhD student is a bit unreasonable; the remaining $15K a year is not too far from the poverty line.

(The relevant sentence is page 96, line 20 of the GOP tax bill.)

Against the "Research vs. Olympiads" Mantra

2016-08-13T13:37:00-04:00

There’s a Mantra that you often hear in math contest discussions: “math olympiads are very different from math research”. (For known instances, see O’Neil, Tao, and others. More neutral stances: Monks, Xu.)

It’s true. And I wish people would stop saying it.

Every time I’ve heard the Mantra, it set off a little red siren in my head: something felt wrong. And I could never figure out quite why until last July. There was some (silly) forum discussion about how Allen Liu had done extraordinarily on math contests over the past year. Then someone says:

A: Darn, what math problem can he not do?!

B: I’ll go out on a limb and say that the answer to this is “most of the problems worth asking.” We’ll see where this stands in two years, at which point the answer will almost certainly change, but research ≠ Olympiads.

Then it hit me.

Ping-pong vs. Tennis

Let’s try the following thought experiment. Consider a world-class ping-pong player, call her Sarah. She has a fan-base talking about her pr0 ping-pong skills. Then someone comes along as says:

Well, table tennis isn’t the same as tennis.

To which I and everyone else reasonable would say, “uh, so what?”. It’s true, but totally irrelevant; ping-pong and tennis are just not related. Maybe Sarah will be better than average at tennis, but there’s no reason to expect her to be world-class in that too.

And yet we say exactly the same thing for olympiads versus research. Someone wins the IMO, out pops the Mantra. Even if the Mantra is true when taken literally, it’s implicitly sending the message there’s something wrong with being good at contests and not good at research.

So now I ask: just what is wrong with that? To answer this question, I first need to answer: “what is math?”.

There’s been a trick played with this debate, and you can’t see it unless you taboo the word “math”. The word “math” can refer to a bunch of things, like:

Training for contest problems like USAMO/IMO, or
Learning undergraduate/graduate materials like algebra and analysis, or
Working on open problems and conjectures (“research”).

So here’s the trick. The research community managed to claim the name “math”, leaving only “math contests” for the olympiad community. Now the sentence

“Math contests should be relevant to math”

seems totally innocuous. But taboo the world “math”, and you get

“Olympiads should be relevant to research”

and then you notice something’s wrong. In other words, since “math” is a substring of “math contests”, it suddenly seems like the olympiads are subordinate to research. All because of an accident in naming.

Since when? Everyone agrees that olympiads and research are different things, but it does not then follow that “olympiads are useless”. Even if ping-pong is called “table tennis”, that doesn’t mean the top ping-pong players are somehow inferior to top tennis players. (And the scary thing is that in a world without the name “ping-pong”, I can imagine some people actually thinking so.)

I think for many students, olympiads do a lot of good, independent of any value to future math research. Math olympiads give high school students something interesting to work on, and even the training process for a contest such that the IMO carries valuable life lessons: it teaches you how to work hard even in the face of possible failure, and what it’s like to be competitive at an international level (i.e. what it’s like to become really good at something after years of hard work). The peer group that math contests give is also wonderful, and quite similar to the kind of people you’d meet at a top-tier university (and in some cases, they’re more or less the same people). And the problem solving ability you gain from math contests is indisputably helpful elsewhere in life. Consequently, I’m well on record as saying the biggest benefits of math contests have nothing to do with math.

There are also more mundane (but valid) reasons (they help get students out of the classroom, and other standard blurbs about STEM and so on). And as a matter of taste I also think contest problems are interesting and beautiful in their own right. You could even try to make more direct comparisons (for example, I’d guess the average arXiv paper in algebraic geometry gets less attention than the average IMO geometry problem), but that’s a point for another blog post entirely.

The Right and Virtuous Path

Which now leads me to what I think is a culture issue.

MOP alumni prior to maybe 2010 or so were classified into two groups. They would either go on to math research, which was somehow seen as the “right and virtuous path”, or they would defect to software/finance/applied math/etc. Somehow there is always this implicit, unspoken message that the smart MOPpers do math research and the dumb MOPpers drop out.

I’ll tell you how I realized why I didn’t like the Mantra: it’s because the only time I hear the Mantra is when someone is belittling olympiad medalists.

The Mantra says that the USA winning the IMO is no big deal. The Mantra says Allen Liu isn’t part of the “smart club” until he succeeds in research too. The Mantra says that the countless time and energy put into running each year’s MOP are a waste of time. The Mantra says that the students who eventually drop out of math research are “not actually good at math” and “just good at taking tests”. The Mantra even tells outsiders that they, too, can be great researchers, because olympiads are useless anyways.

The Mantra is math research’s recruiting slogan.

And I think this is harmful. The purpose of olympiads was never to produce more math researchers. If it’s really the case that olympiads and research are totally different, then we should expect relatively few olympiad students to go into research; yet in practice, a lot of them do. I think one could make a case that a lot of the past olympiad students are going into math research without realizing that they’re getting into something totally unrelated, just because the sign at the door said “math”. One could also make a case that it’s very harmful for those that don’t do research, or try research and then decide they don’t like it: suddenly these students don’t think they’re “good at math” any more, they’re not smart enough be a mathematician, etc.

But we need this kind of problem-solving skill and talent too much for it to all be spent on computing R(6,6). Richard Rusczyk’s take from Math Prize for Girls 2014 is:

When people ask me, am I disappointed when my students don’t go off and be mathematicians, my answer is I’d be very disappointed if they all did. We need people who can think about these complex problems and solve really hard problems they haven’t seen before everywhere. It’s not just in math, it’s not just in the sciences, it’s not just in medicine — I mean, what we’d give to get some of them in Congress!

Academia is a fine career, but there’s tons of other options out there: the research community may denounce those who switch out as failures, but I’m sure society will take them with open arms.

To close, I really like this (sarcastic) comment from Steven Karp (near bottom):

Contest math is inaccessible to over 90% of people as it is, and then we’re supposed to tell those that get it that even that isn’t real math? While we’re at it, let’s tell Vi Hart to stop making videos because they don’t accurately represent math research.

Addendums (response to comments)

Thanks first of all for the many long and thoughtful comments from everyone (both here, on Facebook, in private, and so on). It’s given me a lot to think about.

Here’s my responses to some of the points that were raised, which is necessarily incomplete because of the volume of discussion.

To start off, it was suggested I should explicitly clarify: I do not mean to imply that people who didn’t do well on contests cannot do well in math research. So let me say that now.
My favorite comment that I got was that in fact this whole post pattern matches with bravery debates.

On one hand you have lots of olympiad students who actually FEEL BAD about winning medals because they “weren’t doing real math”. But on the other hand there are students whose parents tell them to not pursue math as a major or career because of low contest scores. These students (and their parents) would benefit a lot from the Mantra; so I concede that there are indeed good use cases of the Mantra (such as those that Anonymous Chicken, betaveros describe below) and in particular the Mantra is not intrinsically bad.

Which of these use is the “common use” probably depends on which tribes you are part of (guess which one I see more?). It’s interesting in that in this case, the two sides actually agree on the basic fact (that contests and research are not so correlated).
Some people point out that research is a career while contests aren’t. I am not convinced by this; I don’t think “is a career” is a good metric for measuring value to society, and can think of several examples of actual jobs that I think really should not exist (not saying any names). In addition, I think that if the general public understood what mathematicians actually do for a career, they just might be a little less willing to pay us.

I think there’s an interesting discussion about whether contests / research are “valuable” or not, but I don’t think the answer is one-sided; this would warrant a whole different debate (and would derail the entire post if I tried to address it).
Some people point out that training for olympiads yields diminishing returns (e.g. learning Muirhead and Schur is probably not useful for anything else). I guess this is true, but isn’t it true of almost anything? Maybe the point is supposed to be “olympiads aren’t everything”, which is agreeable (see below).
The other favorite comment I got was from Another Chicken, who points out below that the olympiad tribe itself is elitist: they tend to wall themselves off from outsiders (I certainly do this), and undervalue anything that isn’t hard technical problems.

I concede these are real problems with the olympiad community. Again, this could be a whole different blog post.

But I think this comment missed the point of this post. It is probably fine (albeit patronizing) to encourage olympiad students to expand; but I have a big problem with framing it as “spend time on not-contests because research”. That’s the real issue with the Mantra: it is often used as a recruitment slogan, telling students that research is the next true test after the IMO has been conquered.

Changing the Golden Metric from olympiads to research seems to just make the world more egotistic than it already is.

Things SPARC

2016-02-13T13:37:00-05:00

[EDIT 2018/03/05: This description seems significantly less accurate to me now than it did a few years ago, both because my views/values have changed substantially, and because SPARC has changed direction substantially since I attended as a junior counselor in 2015. I’ll leave it here as a reference, but should be taken with a grain of salt.]

I often get asked about what I learned from the SPARC summer camp. This is hard to describe and I never manage to give as a good of an answer as I want, so I want to take the time to write down something concrete now. For context: I attended SPARC in 2013 and 2014 and again as a counselor in 2015, so this post is long overdue (but better late than never).

(For those of you still in high school: applications for 2016 are now open, due March 1, 2016. The program is completely free including room/board and you don’t need rec letters, so there is no reason to not apply.)

The short version is that maybe 1/4 of the life skills I use on a regular basis are things I picked up from SPARC. (The rest came from some combination of math contests and living in college dorms.) On paper SPARC seems like a math or CS camp, but there is a strong emphasis on practicality in the sense that the instructors specifically want to teach you things that you can apply in life. So in addition to technical classes on Bayes’ theorem and the like, you’ll have classes on much “softer” topics like

Posture (literally about having good body posture)
Aversion factoring (e.g. understanding why I’m not exercising and fixing it)
Expanding comfort zones (with hands-on practice; my year I learned to climb trees)

and so on. This makes it hard to compare to other math camps like MOP (though if you insist on drawing a comparison, I think many MOP+SPARC students agree they learned more from SPARC).

Some more testimonials:

What benefits do students get from attending SPARC?
What is like to attend SPARC?
The SPARC video gives a good sense of the atmosphere at the program.

Now, here is my own list of concrete things which SPARC has taught me, in no particular order:

Being significantly more introspective / reflective about life. Example: realizing that some class/activity/etc. are not adding much value to life and dropping them.
Being interested in optimizing life in general; I now find it fun to think about how to be more productive and live life well the same way I like to think about hard math problems.
Thinking about thinking: things like mental models, cognitive biases, emotions, aversions, System 1 vs System 2,
Becoming very aggressive at conserving time. I’m much more willing to trade money for time, and actively asking whether I really need to do something, or if I can just axe it.
Using game theory concepts to think about the world. College tuition is expensive because this is the Nash equilibrium. Recognizing real-life situations which are well understood as games, like prisoner’s dilemma, chicken, etc.
Applying Bayes’ theorem and expected value to real life. Trying out X activity has constant cost but potentially large payoffs, hence large positive EV.
Being able to use probabilities in a meaningful way. Being able to tell the difference between being 90% confident and 70% confident in an event happening.
Actively buying O(n) returns for O(1) cost.
Being more willing to take less conventional paths, like essentially dropping out of high school to train for the IMO (I describe this in the first FAQ here). Another good example I haven’t done myself (yet) is taking a gap year.
Using Workflowy. It’s a big part of why I can think as clearly as I do. In context of SPARC, this is a special case of understanding the idea of working memory, which is also an idea I picked up from camp.
Writing a lot more. This is probably actually a consequence of the things above rather than something that I directly learned from SPARC; some combination of understanding working memory well, and being much more reflective.
Peer group and culture. This is a bigger one than people realize. In the same way that math contests establish a group of people where it’s cool to think about hard math problems, the SPARC network establishes a group of people with a culture of encouraging people to think about rationality. It’s very hard to be good at reflection in an isolated environment! SPARC lets you see how other people go about thinking about how to live life well and gives you other people to bounce ideas off of.

I’m sure there’s other things, but it’s hard for me to notice since it’s been so long since I had to live life pre-SPARC. And there’s some things that other people learned from SPARC that never stuck with me (lots of the social skills, for example). Much like your first time attending MOP, there will be more things to learn than you’ll actually be able to absorb. So the list above is only the things that I myself learned, and in fact I think the set of things you acquire from SPARC more or less molds to whichever particular things matter to you most.

Writing

2015-03-14T13:37:00-04:00

In high school, I hated English class and thought it was a waste of time. Now I’m in college, and I still hate English class and think it’s a waste of time. (Nothing on my teachers, they were all nice people, and I hope they’re not reading this.)

However, I no longer think writing itself is a waste of time. Otherwise, I wouldn’t be blogging, even about math. This post explains why I changed my mind.

1. Guts

My impression is that teachers in high school got it all wrong.

In high school, students are told to learn algebra because “we all use math every day”. This is obviously false, and somehow the students eventually are led to believe it.

You can’t actually be serious. Do people really think that knowing the Pythagorean Theorem will help in your daily life? I sure don’t, and I’m an aspiring mathematician. (Tip: Even real mathematicians stopped doing Euclidean geometry ages go.) It’s hilarious when you think about it. We’ve convinced millions of kids all over the country that they’re learning math because it’s useful in their lives, and they grudgingly believe it.

The actual answer of why we teach math in schools is that it is supposed to teach students how to think. But even the teachers have lost sight of this. Most high school math teachers are now just interested in making sure their students can “do” certain classes of problems in a short time, where “do” here doesn’t refer to solving the problem but regurgitating the solution that’s already been presented. The process is so repetitive and artificial that in high school I wrote computer programs to do my homework for me, because all the “problems” were just the same thing with numbers changed. If you’re interested in just how far off math is, I encourage you to read Lockhart’s Lament.

How can this happen? I think the answer is that many high schoolers don’t really have the guts to think, “my math teachers don’t have a clue”, even though they like to joke about it. I have the guts to say this now because I know lots of math. And it’s amazing to know that millions and millions of people are just plain wrong about something I believe in.

But on to the topic of this post…

2. The world lied to me

I was always told that the purpose of English class was to learn to write. Why is this important? Because it was important to be able to communicate my ideas.

Dead wrong. Somehow the skill of being able to argue on the nature of love in Romeo and Juliet was going to help me when I was writing a paper on Evan’s Theorem years down the road? That’s what my parents said. It sounds absurd when I put it this way, but people believe it. (And let’s not forget the fact that theorems are named by last name…)

I claim that the situation is just like math. People are just being boneheads. As it turns out, the standard structure of an English essay is nothing more than a historical accident. Even the fact that essays are about literature is a historical accident. But that’s beyond the scope of what I have to say.

So what is the purpose of writing? It turns out that there is one, and that it has nothing to do with communication. It’s that writing clarifies thinking.

3. Writing lets you see everything

“I sometimes find, and I am sure you know the feeling, that I simply have too many thoughts and memories crammed into my mind… At these times… I use the Pensieve. One simply siphons the excess thoughts from one’s mind, pours them into the basin, and examines them at one’s leisure.”

– Harry Potter and the Goblet of Fire

Here’s some advice to all of you still in doing math contests – start keeping track of the problems you solve.

There’s superficial reasons for doing this. A few days ago I was trying to write a handout on polynomials, and I was looking for some problems on irreducibility. I knew I had seen and done a bunch of these problems in the past, but of course like most people I hadn’t bothered to keep track of every problem I did, so I could only remember a few off my head. So I had to go through the painful process of looking through my old posts on the Art of Problem Solving forums, searching through old databases, mucking through pages of garbage looking for problems that I did ages ago that I could use for my handout. And all the time I was thinking, “man, I should have kept track of all the problems I did”.

But there are deeper reasons for this. As I started collating the problems and solutions into a list, I started noticing some themes in the solutions that I never noticed before. For example, basically every solution started with the line “Assume for contradiction that $f$ is not irreducible and write $f = g \cdot h$ ”. And then from there, one of three things happened.

The problem would take the coefficients modulo some prime or prime power, and then deduce some things about $g$ and $h$ . Obviously this only worked on the problems with integer coefficients.
The problem would start looking at absolute values of the coefficients and try to achieve some bound that showed the polynomial had to reduce in a certain way.
If the problem had multiple variables, the solution would reduce to a case with just one-variable. This was always the case with problems that had complex coefficients as well.

You can’t really be serious – I’m only noticing this now? Here I was, already a retired contestant, looking at problems I had done long long ago and only realizing now there was a common theme. I had already done all the work by having done all the problems. The only difference was that I didn’t write anything down; as a result I could only look at one problem at a time.

Needless to say, I was very angry for the rest of the day.

4. External and Working Memory

Why does this happen? More profoundly, it turns out that humans have a finite working memory. You can only keep so many things in your head at once. That’s why it’s a stupid idea to not write down problems and (sketches of) solutions after you solve them and keep them somewhere you can look at.

I probably did at least 1000 olympiad problems over the course of my life. Did I manage to keep all the solutions in my head? Of course not. That’s why at the IMO in 2014, I didn’t try a maximality argument despite the $\sqrt n$ in the problem. I think if I had kept better records I wouldn’t have missed this. How else do you get exactly $\sqrt n$ in the lower bound? It’s not even an integer! Poof. There goes my neat 42.

I didn’t realize this wasn’t just a math thing until much later. I was talking about something along these lines during my interview for Harvard College; my interviewer was an artist. When I was talking about writing things down because I couldn’t keep them all in my head, he said something that surprised me – his easel was covered with sticky notes where he wrote down any ideas that occurred to him. He called it “external memory”, a term I still use now.

It’s actually obvious when you think about it. Why do people have to-do lists and calendars and reminders? Because you can’t keep track of everything in your head. You can try and might even get good at it, but you’ll never do as well as the old-fashioned pen and paper.

This isn’t just about “I need to remember to do $X$ in exactly $Y$ time”. There’s a reason we use blackboards during math lectures instead of just talking. The ideas in math are really, really hard, because math is only about ideas, and nothing else. If the professors didn’t write the steps on the board, no one would be able to keep more than two or three steps in their head at once. The difficulty is only compounded by the fact that math has its own notation. We didn’t develop this notation because we were bored. We developed notation because the ideas we’re trying to express are so complex that the English language can’t even express them. In other words, mathematicians were forced to create a whole new set of symbols just to write down their ideas.

5. An Imperfect Analogy to Teaching

But so far I haven’t really argued anything other than “if you want to remember something you better write it down”. There’s a difference between a to-do list and an exposition. One is just a collection of disconnected bullet points. The other needs to do more, it needs to explain.

The following quote is excerpted from Richard Rusczyk’s article “Learning Through Teaching” ).

You can’t just “kind of get it” or know it just well enough to get by on a test; teaching calls for complete understanding of the concept.

How do you know that?

When would you use that?

How could you come up with that in the first place?

If you can’t answer these questions for something you “know”, then you can’t teach it.

I knew this was true from my own experiences teaching, but it took me more time to realize that writing well is a similar skill. The difference is the medium: when you’re teaching in person, you get real-time feedback on whether what you said makes sense. You don’t get this live feedback when you’re writing, and so you need to be much more careful. Yet all the nuances of teaching are still there – distinguishing between details, main ideas, hardest steps; deciding what can be worked out from what other things, even deciding which things are worth including and which things should be omitted.

This all really started to become obvious to me when I started my olympiad geometry textbook. In senior year of high school, I decided that I had a good enough understanding of olympiad geometry to write a textbook on it. I felt like I could probably do better than all the existing resources; not as hard as it sounds, since to my knowledge there aren’t any dedicated books for olympiad geometry.

After I had around 200 pages written, I realized that I had gotten a lot better at geometry. There were lots of things that happened in the process of thinking about the best way to teach geometry.

Most basically, I did in fact fill in gaps in my knowledge. For example, I studied projective transformations for the first time in order to write the corresponding section in my book. The ideas definitely clicked much faster when I was thinking about how to teach it.
I made new connections. I realized for the first time that symmedians and harmonic quadrilaterals are actually the same concept; I discovered a lemma about directed angles that I wished I had known before; I found a new proof to Menelaus using an elegant strategy I had used on Monge’s Theorem. None of this would have happened from just doing problems.
Most profoundly, I got a much better understanding for when to apply certain techniques. One of the main goals of my book was to make solutions natural – a reader should be able to understand where a solution came from. That meant that at every page I was constantly fighting to try and explain how I had thought up of something. This unending reflection was exhausting and reduced me to a rate of about one page written per hour\footnote{But conveniently, this process is something that just requires a laptop, not even paper and pencil. So I got a lot of pages written during office assistant.}. But it improved my own ability significantly.

Ultimately what this exemplifies is that trying to explain something lets you understand it better. And that’s in part because you can only manage so many things in your head at once. If you think keeping track of your appointments in your head is hard, try doing that with a complex argument. Can’t do it. Writing solves this problem.

6. Finding the Truth

But that’s not a perfect analogy. What I’ve presented above is a model where you have ideas in your head and you output them onto paper. This isn’t totally accurate, because as you write, something else can happen: the ideas can change.

I’ll draw an analogy from painting, again courtesy of Paul Graham.

The model of painting I used to have is that you would have something you want to draw, and then you sit down and draw it, then polish up the details. (That’s how I did all my high school art projects, anyways.) But this turns out to not be true: Countless paintings, when you look at them in x-rays, turn out to have limbs that have been moved or facial features that have been readjusted. I was surprised when I first read this. But it makes sense if you can think about it: how you can be sure what’s in your head is what you want if you can’t even see it yet?

I propose that writing does the same thing. I don’t start by thinking “these are the ideas and I will now write them down”. Rather, I just write my thoughts down, not sure where they’re going to end up. That’s how my geometry textbook actually got written. I didn’t start with a table of contents. I started by putting down ideas, finding the connections between them, noticing new things I hadn’t before. I created new sections on the fly as the need arose, added new things as I thought of them, and let the whole thing sort itself out with a simple \verb+\tableofcontents+. You can even think of the table of contents as a natural bucket sort – put down related ideas near each others, add section headers as needed, and bam, you have an outline of the main ideas. And I never know what this outline will look like until it’s actually been written.

By the same token, revising shouldn’t be the art of modifying the presentation of an idea to be more convincing. It should be the art of changing the idea itself to be closer to the truth, which will automatically make it more convincing. This is consistent with the Latin: the word “revise” literally means “see again”.

This is where high school and college essays get it really wrong. In a college essay, the goal is to “sell an idea” to the reader. If something in the essay looks unconvincing, you fix it by trickery: re-writing it in a way that it sounds more convincing without changing the underlying idea. The way you say something goes a long way in selling it. That’s what English class should have taught you. Sure, some teachers tell you to make concessions or counterarguments, but you’re doing this to try and pretend to be “honest”. You only write such things with an agenda in mind.

But since when are you always right? That’s absurd. The English class model is “I have a thesis that I know is right, and now I’m going to explain to the reader why”. But how can you know you’re right about a thesis before you’ve written it down? If the thesis and its accompanying argument is even remotely complex, it wouldn’t have been possible to sort through the whole thing in your head. Worse still, if the thesis is nontrivial, odds are that someone who is about as smart as you will disagree with you. And as Yan Zhang often reminds the SPARC attendees, you should really only expect to be right about half the time when you disagree with someone about as smart as you. If an essay is supposed to move you closer to the truth, and your original thesis is wrong half the time, do you scrap half your essays? Unfortunately, I don’t think you’d ever pass English class that way.

The culture that’s been instilled, where the goal of writing is to convince, is intellectually dishonest. I might even go to say it’s dangerous; I’ll have to think about that for a while. There are times when you do want to write to convince others (grant proposals, anyone?) but it seems highly unfortunate that this type of writing has become synonymous with writing as a whole.

7. Conclusion

So this post has a few main ideas. The main purpose of writing is not in fact communication, at least not if you’re interested in thinking well. Rather, the benefits (at least the ones I perceive) are

Writing serves as an external memory, letting you see all your ideas and their connections at once, rather than trying to keep them in your head.
Explaining the ideas forces you to think well about them, the same way that teaching something is only possible with a full understanding of the concept.
Writing is a way to move closer to the truth, rather than to convince someone what the truth is.

So now I’ll tell you how I actually wrote my geometry book, or this blog post, or any of my various olympiad articles. It starts because I have an idea – just a passing thought, like “this would be a good way to explain Masckhe’s Theorem”. Some time later I’ll another such thought which is related to the first. Then a third. My memory is especially bad, so pretty soon it bothers me so much that I have to write it down, because I’m starting to lose track. And as I write the first ideas down, I start noticing new ideas, so I add in these ideas, and then more new ideas start flooding in. There are so many things I want to say and I just keep writing them down. That’s how I ended up with a 400-page textbook written from what originally was just meant to be a short article. There were too many things to say that other people hadn’t said yet, and I just had to write them all down. The miraculous things is that these ideas naturally sorted themselves out. The bulleted main ideas I listed above weren’t things I realized until I looked at the resulting table of contents.

I’m sometimes told by people I respect that they like my writing. But I think this actually just translates to “I like the ideas in your writing”, and so I take it as a big compliment.