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.

  1. 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.
  2. 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.
  3. 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.