Notes on Publishing My Textbook

Hmm, so hopefully this will be finished within the next 10 years.

— An email of mine at the beginning of this project

My Euclidean geometry book was published last March or so. I thought I’d take the time to write about what the whole process of publishing this book was like, but I’ll start with the disclaimer that my process was probably not very typical and is unlikely to be representative of what everyone else does.

Writing the Book

The Idea

I’m trying to pin-point exactly when this project changed from “daydream” to “let’s do it”, but I’m not quite sure; here’s the best I can recount.

It was sometimes in the fall of 2013, towards the start of the school year; I think late September. I was a senior in high school, and I was only enrolled in two classes. It was fantastic, because it meant I had lots of time to study math. The superintendent of the school eventually found out, though, and forced me to enroll as an “office assistant” for three periods a day. Nonetheless, office assistant is not a very busy job, and so I had lots of time, all the time, every day.

Anyways, I had written a bit of geometry material for my math club the previous year, which was intended to be a light introduction. But in doing so I realized that there was much, much more I wanted to say, and so somewhere on my mental to-do list I added “flesh these notes out”. So one day, sitting in the office, after having spent another hour playing StarCraft, I finally got down to this item on the list. I hadn’t meant it to be a book; I was just wanted to finish what I had started the previous year. But sometimes your own projects spiral out of your control, and that’s what happened to me.

Really, I hadn’t come up with a brilliant idea that no one had thought of before. To my knowledge, no one had even tried yet. If I hadn’t gone and decided to write this book, someone else would have done it; maybe not right away, but within many years. Indeed, I was honestly surprised that I was the first one to make an attempt. The USAMO has been a serious contest since at least the 1990’s and 2000’s, and the demand for this book certainly existed well before my time. Really, I think this all just goes to illustrate that the Efficient Market Hypothesis is not so true in these kind of domains.

Setting Out

Initially, this text was titled A Voyage in Euclidean Geometry and the filename Voyage.pdf would persist throughout the entire project even though the title itself would change throughout.

The beginning of the writing was actually quite swift. Like everyone else, I started out with an empty LaTeX file. But it was different from blank screens I’ve had to deal with in my life; rather than staring in despair (think English essay mode), I exploded. I was bursting with things I wanted to write. It was the result of having years of competitive geometry bottled up in my head. In fact, I still have the version 0 of the table of contents that came to life as I started putting things together.

  • Angle Chasing (include “Fact 5”)
  • Centers of the Triangle
    • The Medial Triangle
    • The Euler Line
    • The Nine-Point Circle
  • Circles
    • Incircles and Excircles
    • The Power of a Point
    • The Radical Axis
  • Computational Geometry
    • All the Areas (include Extended Sine Law, Ceva/Menelaus)
    • Similar Triangles
    • Homothety
    • Stewart’s Theorem
    • Ptolemy’s Theorem
  • Some More Configurations (include symmedians)
    • Simson lines
    • Incircles and Excenters, Revisited
    • Midpoints of Altitudes
  • Circles Again
    • Inversion
    • Circles Inscribed in Segments
    • The Miquel Point (include Brokard, this could get long)
    • Spiral Similarity
  • Projective Geometry
    • Harmonic Division
    • Brokard’s Theorem
    • Pascal’s Theorem
  • Computational Techniques
    • Complex Numbers
    • Barycentric Coordinates

Of course the table of contents changed drastically over time, but that wasn’t important. The point of the initial skeleton was to provide a bucket sort for all the things that I wanted to cover. Often, I would have three different sections I wanted to write, but like all humans I can only write one thing at a time, so I would have to create section headers for the other two and try to get the first section done as quickly as I could so that I could go and write the other two as well.

I did take the time to do some things correctly, mostly LaTeX. Some examples of things I did:

  • Set up proper amsthm environments: earlier versions of the draft had “lemma”, “theorem”, “problem”, “exercise”, “proposition”, all distinct
  • Set up an organized master LaTeX file with \include’s for the chapters, rather than having just one fat file.
  • Set up shortcuts for setting up diagrams and so on.
  • Set up a “hints” system where hints to the problems would be printed in random order at the end of the book.
  • Set up a special command for new terms (\vocab). At the beginning all it did was made the text bold, but I suspected that later I might it do other things (like indexing).

In other words, whenever possible I would pay O(1) cost to get back O(n) returns. Indeed the point of using LaTeX for a long document is so that you can “say what you mean”: you type \begin{theorem} … \end{theorem}, and all the formatting is taken care of for you. Decide you want to change it later, and you only have to change the relevant code in the beginning.

And so, for three hours a day, five days a week, I sat in the main office of Irvington High School, pounding out chapter after chapter. I was essentially typing up what had been four years of competition experience; when you’re 17 years old, that’s a big chunk of your life.

I spent surprisingly little time revising (before first submission). Mostly I just fired away. I have always heard things about how important it is to rewrite things and how first drafts are always terrible, but I’m glad I ignored that advice at least at the beginning. It was immensely helpful to have the skeleton of the book laid out in a tangible form that I could actually see. That’s one thing I really like about writing; helps you collect your thoughts together.

It’s possible that this is part of my writing style; compared to what everyone says I should do, I don’t do very much rewriting. My first and final drafts tend to look pretty similar. I think this is just because when I write something that’s not an English essay, I already have a reasonably good idea what I want to say, and that the process of writing it out does much of the polishing for me. I’m also typically pretty hesitant when I write things: I do a lot of pausing for a few minutes deciding whether this sentence is really what I want before actually writing it down, even in drafts.

Some Encouragement

By late October, I had about 80 or so pages content written. Not that impressive if you think about it; I think it works out to something like 4 pages per day. In fact, looking through my data, I’m pretty sure I had a pretty consistent writing rate of about 30 minutes per page. It didn’t matter, since I had so much time.

At this point, I was beginning to think about possibly publishing the book, so it was coming out reasonably well. It was a bit embarrassing, since as far as I could tell, publishing books was done by people who were actually professionals in some way or another. So I reached out to a couple of teachers of mine (not high school) who I knew had published textbooks in one form or another; I politely asked them what their thoughts were, and if they had any advice. I got some gentle encouragement, but also a pointer to self-publishing: turns out in this day and age, there are services like Lulu or CreateSpace that will just let you publish… whatever you want. This gave me the guts to keep working on this, because it meant that there was a minimal floor: even if I couldn’t get a traditional publisher, the worst I could do was self-publish through Amazon, which was at any rate strictly better than the plan of uploading a PDF somewhere.

So I kept writing. The seasons turned, and by February, the draft was 200 pages strong. In April, I had staked out a whopping 333 pages.

The Review Process

Entering the MAA’s Queue

I was finally beginning to run out of things I wanted to add, after about six months of endless typing. So I decided to reach out again; this time I contacted a professor (henceforth Z) that I knew, whom I knew well from time at the Berkeley Math Circle. After some discussion, Z agreed to look briefly at an early draft of the manuscript to get a feel for what it was like. I must have exceeded his expectations, because Z responded enthusiastically suggesting that I submit it to the Problem Book Series of the MAA. As it turns out, he was on the editorial board, so in just a few days my book was in the official queue.

This was all in April. The review process was scheduled to begin in June, and likely take the entirety of the summer. I was told that if I had a more revised draft before the review that I should also send it in.

It was then I decided I needed to get some feedback. So, I reached out to a few of my close friends asking them if they’d be willing to review drafts of the manuscript. This turned out to not go quite as well as I hoped, since

  • Many people agreed eagerly, but then didn’t actually follow through with going through and reading chapter by chapter.
  • I was stupid enough to send the entire manuscript rather than excerpts, and thus ran myself a huge risk of getting the text leaked. Fortunately, I have good friends, but it nagged at me for quite a while. Learned my lesson there.

That’s not to say it was completely useless; I did get some typos fixed. But just not as many as I hoped.

The First Review

Not very much happened for the rest of the summer while I waited impatiently; it was a long four month wait for me. Finally, in the end of August 2014, I got the comments from the board; I remember I was practicing the piano at Harvard when I saw the email.

There had been six reviews. While I won’t quote the exact reviews, I’ll briefly summarize them.

  1. There is too much idiosyncratic terminology.
  2. This is pretty impressive, but will need careful editing.
  3. This project is fantastic; the author should be encouraged to continue.
  4. This is well developed; may need some editing of contents since some topics are very advanced.
  5. Overall I like this project. That said, it could benefit from some reading and editing. For example, here are some passages in particular that aren’t clear.
  6. This manuscript reads well, written at a fairly high level. The motivation provided are especially good. It would be nice if there were some solutions or at least longer hints for the (many) problems in the text. Overall the author should be encouraged to continue.

The most surprising thing was how short the comments were. I had expected that, given the review had consumed the entire summer, the reviewers would at least have read the manuscript in detail. But it turns out that mostly all that had been obtained were cursory impressions from the board members: the first four reviews were only a few sentences long! The fifth review was more detailed, but it was essentially a “spot check”.

I admit, I was really at a loss for how I should proceed. The comments were not terribly specific, and the only real action-able item were to use less extravagant terms in response to 1 (I originally had “configuration”, “exercise” vs “problem”, etc.) and to add solutions (in response to 5). When I showed he comments to Z, he commented that while they were positive, they seemed to suggest that the publication may not be anytime soon. So I decided to try submitting a second draft to the MAA, but if that didn’t work I would fall back on the self-publishing route.

The reviewers had commented about finding a few typos, so I again enlisted the help of some friends of mine to eliminate them. This time I was a lot smarter. First, I only sent the relevant excerpts that I wanted them to read, and watermarked the PDF’s with the names of the recipients. Secondly, this time I paid them as well: specifically, I gave 40 + \min(40, 0.1n^2) dollars for each chapter read, where n was the number of errors found. I also gave a much clearer “I need this done by X” deadline. This worked significantly better than my first round of edits. Note to self: people feel more obliged to do a good job if you pay them!

All in all my friends probably eliminated about 500 errors.

I worked as rapidly as I could, and within four weeks I had the new version. The changes that I made were:

  • In response to the first board comment, I eliminated some of the most extravagant terminology (“demonstration”, “configuration”, etc.) in favor of more conventional terms (“example”, “lemma”).
  • I picked about 5-10 problems from each chapter and added full solutions for them. This inflated the manuscript by another 70 pages, for a new total of 400 pages.
  • Many typos and revisions were corrected, thanks to my team of readers.
  • Some formatting changes; most notably, I got the idea to put theorems and lemmas in boxes using mdframed (most of my recent olympiad handouts have the same boxes).
  • Added several references.

I sent this out and sat back.

The Second Review

What followed was another long waiting process for what again were ended up being cursory comments The delay between the first and second review was definitely the most frustrating part — there seemed to be nothing I could do other than sit and wait. I seriously considered dropping the MAA and self-publishing during this time.

I had been told to expect comments back in the spring. Finally, in early April I poked the editorial board again asking whether there had been any progress, and was horrified to find out that the process hadn’t even started out due to a miscommunication. Fortunately, the editor was apologetic enough about the error that she asked the board to try to expedite the process a little. The comments then arrived in mid-May, six weeks afterwards.

There were eight reviewers this time. In addition to some stylistic changes suggested (e.g. avoid contractions), here were some of the main comments.

  • The main complaint was that I had been a bit too informal. They were right on all accounts here: in the draft I had sent, the chapters had opened with some quotes from years of MOP (which confused the board, for obvious reasons) and I had some snarky comments about high school geometry (since I happen to despise the way Euclidean geometry is taught in high school.) I found it amusing that no one had brought it up yet, and happily obliged to fix them.
  • Some reviewers had pointed out that some of the topics were very advanced. In fact, one of the reviewers actually recommend against the publication of the book on the account that no one would want to buy it. Fortunately, the book ended up getting accepted anyways.
  • In that vein, there were some remarks that this book, although it serves its target audience well, is written at a fairly advanced level.

Some of the reviews were cursory like before, but some of them were line-by-line readings of a random chapter, and so this time I had something more tangible to work with.

So I proceeded to make the changes. For the first time, I finally had the brains to start using git to track the changes I made to the book. This was an enormously good idea, and I wish I had done so earlier.

Here are some selected changes that were made (the full list of changes is quite long).

  • Eliminate a bunch of snarky comments about high school, and the MOP quotes.
  • Eliminate about 250 contractions.
  • Eliminate about 50 instances of unnecessary future tense.
  • Eliminate the real product from the text.
  • Added in about seven new problems.
  • Added and improved significantly on the index of the book, making it far more complete.
  • Fix more references.
  • Change the title to “Euclidean Geometry in Mathematical Olympiads” (it was originally “Geometra Galactica”).
  • Change the name of Part II from “Dark Arts” to “Analytic Techniques”. (Hehe.)
  • Added people to the acknowledgments.
  • Changes in formatting: most notably I change the font size from 11pt to 10pt to decrease the page count, since my book was already twice as long as many of the other books in the series. This dropped me from about 400 pages back to about 350 pages.
  • Fix about 200 more typos. Thanks to those of you who found them!

I sent out the third draft just as June started, about three weeks after I had received the comments. (I like to work fast.)

The Last Revisions

There were another two rounds afterwards. In late June, I got a small set of about three pages of additional typos and clarifying suggestions. I sent back the third draft one day later.

Six days later, I got back a list of four remaining edits to make. I sent an updated fourth draft 17 minutes after receiving those comments. Unfortunately, it then took another five weeks for the four changes I made to be acknowledged. Finally, in early August, the changes were approved and the editorial board forwarded an official recommendation to MAA to publish the book.

Summary of Review Timeline

In summary, the timeline of the review process was

  • First draft submitted: April 6, 2014
  • Feedback received: August 28, 2014
    Second draft submitted: November 5, 2014
  • Feedback received: May 19, 2015
    Third draft submitted: June 23, 2015
  • Feedback received: June 29, 2015
    Fourth draft submitted: June 29, 2015
  • Official recommendation to MAA made: August 2015

I think with traditional publishers there is a lot of waiting; my understanding is that the editorial board largely consists of volunteers, so this seems inevitable.

Approval and Onwards

On September 3, 2015, I got the long-awaited message:

It is a pleasure to inform you that the MAA Council on Books has approved the recommendation of the MAA Problem Books editorial board to publish your manuscript, Euclidean Geometry in Mathematical Olympiads.

I got a fairly standard royalty contract from the publisher, which I signed off without much thought.

Editing

I had a total of zero math editors and one copy editor provided. It shows through on the enormous list of errors (and this is after all the mistakes my friends helped me find).

Fortunately, my copy editor was quite good (and I have a lot of sympathy for this poor soul, who had to read every word of the entire manuscript). My Git history indicates that approximately 1000 corrections were made; on average, this is about 2 per page, which sounds about right. I got the corrections on hard copy in the mail; the entire printout of my book, except well marked with red ink.

Many of the changes fell into general shapes:

  • Capitalization. I was unwittingly inconsistent with “Law of Cosines” versus “Law of cosines” versus “law of cosines”, etc and my copy editor noticed every one of these. Similarly, cases of section and chapter titles were often not consistent; should I use “Angle Chasing” or “Angle chasing”? The main point is to pick one convention and stick with it.
  • My copy editor pointed out every time I used “Problems for this section” and had only one problem.
  • Several unnecessary “quotes” and italics were deleted.
  • Oxford commas. My god, so many Oxford commas. You just don’t notice when the IMO Shortlist says “the circle through the points E, G, and H” but the European Girls’ Olympiad says “show that KH, EM and BC are concurrent”. I swear there were at least 100 of these in the boko. I tried to write a regular expression to find such mistakes, but there were lots of edge cases that came up, and I still had to do many of these manually.
  • Inconsistency of em dashes and en dashes. This one worked better with regular expressions.

But of course there were plenty of other mistakes like missing spaces, missing degree spaces, punctuation errors, etc.

Cover Art

This was handled for me by the publisher: they gave me a choice of five or so designs and I picked one I liked.

(If you are self-publishing, this is actually one of the hardest parts of the publishing logistics; you need to design the cover on your own.)

Proofs

It turns out that after all the hard work I spent on formatting the draft, the MAA has a standard template and had the production team re-typeset the entire book using this format. Fortunately, the publisher’s format is pretty similar to mine, and so there were no huge cosmetic changes.

At this point I got the proofs, which are essentially the penultimate drafts of the book as they will be sent to the printers.

Affiliation and Miscellani

There was a bit more back-and-forth with the publisher towards the end. For example, they asked me if I would like my affiliation to be listed as MIT or to not have an affiliation. I chose the latter. I also send them a bio and photograph, and an author questionaire, asking me for some standard details.

Marketing was handled by the publisher based on these details.

The End

Without warning, I got an email on March 25 announcing that the PDF versions of my book were now available on MAA website. The hard copies followed a few months afterwards. That marked the end of my publication process.

If I were to do this sort of thing again, I guess the main decision would be whether to self-publish or go through a formal publisher. The main disadvantage seems to be the time delay, and possibly also that the royalties are lesser than in self-publishing. On the flip side, the advantages of a formal publisher were:

  • Having a real copy editor read through the entire manuscript.
  • Having a committee of outsiders knock some common sense into me (e.g. not calling the book “Geometra Galactica”).
  • Having cover art and marketing completely done for me.
  • It’s more prestigious; having a real published book is (for whatever reason) a very nice CV item.

Overall I think publishing formally was the right thing to do for this book, but your mileage may vary.

Other advice I would give to my past self, mentioned above already: keep paying O(1) for O(n), use git to keep track of all versions, and be conscious about which grammatical conventions to use (in particular, stay consistent).

Here’s a better concluding question: what surprised me about the process, i.e, what was different than what I expected? Here’s a partial list of answers:

  • It took even longer than I was expecting. Large committees are inherently slow; this is no slight to the MAA, it is just how these sorts of things work.
  • I was surprised that at no point did anyone really check the manuscript for mathematical accuracy. In hindsight this should have been obvious; I expect reading the entire book properly takes at least 1-2 years.
  • I was astounded by how many errors there were in the text, be it math or grammatical or so on. During the entire process something like 2000 errors were corrected (admittedly several were minor, like Oxford commas). Yet even as I published the book, I knew that there had to be errors left. But it was still irritating to hear about them post-publication.

All in all, the entire process started in September 2013 and ended in March 2016, which is 30 months. The time was roughly 30% writing, 50% review, and 20% production.

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Against the “Research vs. Olympiads” Mantra

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 more. 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 \neq 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.

  1. 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.

  2. 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).

  3. 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).

  4. 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).

  5. 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.

___ Students Have to Suffer

This will be old news to most of the readership of this blog, but I realize I’ve never written it down, so time to fix that.

Fill in the blank

Let’s begin by playing a game of “fill in the blank”. Suppose that today, the director of secondary education at your high school says:

“___ students just have to suffer.”

This is not a pleasant sentence. Fill in that blank with a gender, and you’d be fired tomorrow morning. Fill in that blank with an ethnic group, and you’d be fired in an hour. Fill in that blank with “special needs”, and you’d be be sued. Heck, forget “___ students”, replace that with “You”. Can you see someone’s career flashing before their eyes? How could you possibly get away with saying that about any group of students?

Those 500 hours

“Smart students just have to suffer.”
Director of Secondary Education at Fremont Unified School District

This happened to me. I haven’t told this story enough, so I will tell it some more.

When I was a senior in high school, I was enrolled in two classes and would thereafter run off to take graduate math at UC Berkeley. (Notes here.) This was fantastic and worked for a few weeks, so I got to learn real analysis and algebraic combinatorics from some nice professors.

Then the school district found out, and called me in for a meeting. The big guy shows up, and gives me this golden quote. I was then required to enroll in five classes a day, the minimum number of classes required for me to count towards the average daily attendance funding for my school district.

And that is why, for three periods a day, five days a week, I was forced to sit in the front office, saying “Hi, how may I help you?”.

(I didn’t even get paid! Could’ve asked for a cut of that ADA funding. It didn’t all go to waste though; I spent the time writing a book.)

Everywhere Else

Since I’ve had fun picking on my school district, I will now pick on the Department of Education.

“While challenging and improving the mathematical problem-solving skills of high-performing students are surely every-day objectives of those who teach such students, it is not a problem, relatively speaking, of major import in American education.”
Department of Education Reviewer

Oh boy.

The point is that the problem of neglecting gifted students isn’t at the level of individual teachers. It’s not a problem at the level of individual schools, or individual cities. This is a problem with national culture. The problem is that as a culture we think it’s okay to say a sentence like that.

Replace “high-performing” with any adjective you want. Any gender, any social class, any ethnic group, whatever, and you will get a backlash. But we’ve decided that it’s okay to mistreat the gifted students, because no one complains at that.

Maybe it’s too much to ask that schools do something special for top students. Can you at least not get in their way? Like not forcing students to be an office assistant for 500 hours to obtain ADA funding? Or more generally, how about just not forcing students to take classes which are clearly a waste of time for them?

Next Actions

So what can you do to change the national culture? As far as I can tell, this is mostly a lost clause. I wouldn’t bother trying.

The reason I wrote this post because I went through most of high school not really being aware of just how badly I was being mistreated. I’m really writing this for myself four years ago to point out that, man, us nerds really got the ugly end of the deal.

What you can do (and should) is make small local changes. You can persuade individual schools to make exceptions for a kid, and frequently individual teachers will do what they can to help a gifted student as well. Each individual student has good chance of finding a way around the big bureaucrats that rule the wastelands.

Ask a lot of people: if one administrator says no, ignore them and ask another one. Be prepared to hear “no” a lot, but keep waiting for the one or two crucial “yes” moments. If push comes to shove, switch schools, apply to college early, etc. Take the effort to get this one right. (See 56:30-60:00 of this for more on that.)

Dear past self, yell a little harder at the big guy when he comes, maybe you can save yourself 500 hours as an office aid.

Addendum: A Happy Story

In the comments, someone wrote the following:

Did your mistreatment as a gifted student hinder you in any significant way? … Where would you be today had the system not failed you?

I think it’s impossible to know. But here’s another story.

  • When I was in 7th grade, my school tried to force me to take pre-algebra. My mom begged the school teachers until they finally relented and let me take Algebra I. At the time, my 12-year-old self couldn’t have cared less: both classes were too easy for me, and I spent most of Algebra I playing Tetris on my TI-89.
  • Two years later this happened again: the school wanted to force me to take Algebra II. This time, my mom begged the teachers to let me take precalculus instead, which they eventually did. My 14-year-old self also couldn’t care less; both classes were too easy anyways, and I spent most of precalculus playing osu on my iPhone.
  • Two years later I was in Calculus BC, again bored to tears and in the last HS math class offered. That’s when my parents were able to persuade the school to let me take classes at UC Berkeley instead, since I had exhausted the HS math curriculum. I did very well in my first undergraduate classes, which then allowed me to take graduate classes for the rest of high school.

These professors were the ones that wrote my reference letters for college applications, which got me into all the top schools in the country (Berkeley, UCLA, MIT, Princeton, Stanford, Harvard). Without these reference letters I would certainly not have had as many options; winning the USAMO and making the IMO didn’t happen for me until the end of senior year.

But it wasn’t until I met the guy quoted above that I found out that I had unwittingly “broken district rules”, and technically shouldn’t have happened. (Belated thanks to those individuals who stuck out their necks for me!)

So here’s a surprisingly clear example of a near miss. Suppose that my mom had been more polite, or my school had been a little more firm, and any of the three events above didn’t occur. Not only would I have lost some college choices (potentially including MIT), I wouldn’t even know that this was the key event I could have changed.

[Bonus question: I estimate about 2% of USA high school students take the AMC. How would my life have changed if I had been in the other 98%?]

By analogy, if you ask me now what ways I’ve been affected, how am I to tell you? Without an Earth simulator I can’t point to which of the other 100 times I was mistreated hurt me the most. All I can do is point out that I (and many others) are being mistreated, which really should not be okay in the first place.

Against Hook-Length on USAMO 2016/2

A recent USAMO problem asked the contestant to prove that

\displaystyle  (k^2)! \cdot \prod_{j=0}^{k-1} \frac{j!}{(j+k)!}

is an integer for every {k \in \mathbb N}. Unfortunately, it appears that this is a special case of the so-called hook-length formula, applied to a {k \times k} Young tableau, and several students appealed to this fact without proof to produce one-line solutions lacking any substance. This has led to a major controversy about how such solutions should be graded, in particular whether they should receive the {7^-} treatment for “essentially correct solutions”, or the {0^+} treatment for “essentially not solved”.

In this post I want to argue that I think that these solutions deserve a score of {1}.

1. Disclaimers

However, before I do so, I would like to make some disclaimers:

  • This issue is apparently extremely polarized: everyone seems to strongly believe one side or the other.
  • This was an extremely poor choice of a USAMO problem, and so there is no “good” way to grade the HL solutions, only a least bad way. The correct solution to the dilemma is to not have used the problem at all. Yet here’s the bloodied patient, and here we are in the emergency room.
  • While I am a grader for the USAMO, I am one of many graders, and what I say in this post does not necessarily reflect the viewpoints of other USAMO graders or what score the problem will actually receive. In other words, this is my own view and not official in any way.

One last remark is that I do not consider the hook-length formula to be a “well-known” result like so many contestants seem to want to pretend it is. However, this results in the danger of what constitutes “well-known” or not. So in what follows I’ll pretend that the HL formula is about as well-known as, say, the Pascal or Zsigmondy theorem, even though I personally don’t think that this is the case.

One final disclaimer: I am unlikely to respond to further comments about an issue this polarized, since I have already spent many hours doing so, and I’ve done enough of my duty. So if I don’t respond to a comment of yours, please don’t take it personally.

2. Rule for citations

Here is the policy I use for citations when grading:

  • You can cite any named result as long as it does not trivialize the problem.
  • If the result trivializes the problem, then you are required to prove the result (or otherwise demonstrate you understand the proof) in order to use it.

This is what I’ve heard every time I have asked or answered this question; I have never heard anything to the contrary.

Some people apparently want to nit-pick about how “trivialize” is not objective. I think this is silly. If you really want a definition of “trivialize”, you can take “equivalent to the problem or a generalization of the problem” as a rule of thumb.

Clearly it follows from my rule above that the hook-length formula deserves {0^+} grading, so the remainder of the post is dedicated to justifying why I think this is the correct rule.

3. Subjective grading

I would rather have an accurate subjective criteria than a poor objective one.

In an ideal world, grading would be completely objective: a solution which solves the problem earns {7^-} points and a solution which does not solve the problem earns {0^+} points. But in practice this is of course not possible, unless we expect our contestants to write their solutions in a formal language like Coq. Since this is totally infeasible, we instead use informal proofs: students write their solutions in English and human graders assign a score based on whether the solution could in principle be compiled into a formal language proof.

What this means is that in grading, there are subjective decisions made all of the time. A good example of this is omitting steps. Suppose a student writes “case {B} is similar [to case {A}]”. Then the grader has to decide whether or not the student actually knows that the other case is similar or not, or is just bluffing. One one extreme, if {A} and {B} really are identical, then the grader would probably accept the claim. On the other extreme, if {A} and {B} have some substantial differences, then the grader would almost certainly reject the claim. This implies that there is some intermediate “gray area” which cannot be objectively defined.

Fortunately, most of these decisions are clear, thus USAMO grading appears externally to be mostly objective. In other words, accurate grading and objective grading are correlated (yay math!), but they are not exactly the same. All scores are ultimately discretion by human graders, and not some set of rigid guidelines.

Citations are a special case of this. By citing a theorem, a student is taking advantage of convention that a well-known proof to both the student and grader can be omitted from the write-up.

4. Citing the problem

In light of this I don’t think you should get points for citing the problem. I think we all agree that if a student writes “this is a special case of IMO Shortlist 1999 G8”, they shouldn’t get very many points.

The issue with citing HL in lieu of solving the problem is that the hook-length formula is very hard to prove, and it is not reasonable to do so in an olympiad setting. Consequently, it is near certain that these students have essentially zero understanding of the solution to the problem; they have not solved it, they have only named it.

Similarly, I think you should not earn points for trivializing a problem using Dirichlet, Prime Number Theorem, Zsigmondy, etc. This is also historically consistent with the way that grading has been done in the past (hi Palmer!).

5. Citing intermediate steps

Now consider the usage of difficult theorems such as Dirichlet, Prime Number Theorem, etc. on a solution in which they are merely an intermediate step rather than the entire problem. The consensus here is that citing these results is okay (though it is not unanimous; some super harsh graders also want to take off points here, but they are very few in my experience).

I think this is acceptable, because in this case the contestant has done 90% of the solution, and cannot do the remaining 10% but recognize it as well-known. In my eyes this is considered as essentially solving the problem, because the last missing bit is a standard fact. But somehow if you cannot do 100% of the problem, I don’t think that counts as solving the problem.

What I mean is there is a subjective dependence both on how much of the solution the student actually understands, and how accessible the result is. Unfortunately, the HL solutions earn the worst possible rank in both of the categories: the student understands 0% of the solution and moreover the result is completely inaccessible.

6. Common complaints

Here are the various complaints that people have made to me.

  • “HL is well-known.”
    Well, I don’t think it is, but in any case that’s not the point; you cannot cite a result which is (a) more general than the problem, and (b) to which you don’t understand the proof.

  • “Your criteria is subjective!”
    So what? I would rather have an accurate subjective criteria than a poor objective one.

  • “It’s the problem writer’s fault, so students should get {7}.”
    This is not an ethics issue. It is not appropriate to award points of sympathy on a serious contest such as the USAMO; score inflation is not going to help anyone.

  • “It’s elitist for the graders to decide what counts as trivialized.”
    That’s the grader’s job. Again I would rather have an accurate subjective criteria than a poor objective one. In practice I think the graders are very reasonable about decisions.

  • “I don’t think anyone disputes that the HL solution is a correct one, so certainly the math dictates a {7^-}.”
    I dispute it: I don’t think citing HL is a solution at all.

  • “Why do we let students use Pascal / Cauchy / etc?”
    Because these results are much more reasonable to prove, and the “one-line” solutions using Pascal and Cauchy are not completely trivial; the length of a solution is not the same thing as its difficulty. Of course this is ultimately subjective, but I would rather have an accurate subjective criteria than a poor objective one.

  • “HL solutions have substance, because we made an observation that the quantity is actually the number of ways to do something.”
    That’s why I wish to award {1} instead of {0}.

  • “Your rule isn’t written anywhere.”
    Unfortunately, none of the rules governing USAMO are written anywhere. I agree this is bad and the USAMO should publish some more clear description of the grading mechanics.

  • “The proof of the HLF isn’t even that complicated.”
    Are you joking me?

In summary, I don’t think it is appropriate to give full marks to a student who cannot solve a problem just because they can name it.