Finally, if you attempt to read this without working through a significant
number of exercises (see §0.0.1), I will come to your house and pummel you
with [Gr-EGA] until you beg for mercy. It is important to not just have a
vague sense of what is true, but to be able to actually get your hands dirty.
As Mark Kisin has said, “You can wave your hands all you want, but it still
won’t make you fly.”
— Ravi Vakil, The Rising Sea:
Foundations of Algebraic Geometry
When people learn new areas in higher math,
they are usually required to do some exercises.
I think no one really disputes this: you have to actually do math to make any progress.
However, from the teacher’s side,
I want to make the case that there is some art to picking exercises, too.
In the process of writing my Napkin …
Math must be presented for System 1 to absorb and only incidentally for System 2 to verify.
I finally have a sort-of formalizable guideline for teaching and writing math,
and what it means to “understand” math.
I’ve been unconsciously following this for years
and only now managed to write down explicitly what it is that I’ve been doing.
(This post is written from a math-centric perspective,
because that’s the domain where my concrete object-level examples from.
But I suspect much of it applies to communicating hard ideas in general.)
S1 and S2
The quote above refers to the System 1 and System 2 framework from Thinking,
Fast and Slow.
Roughly it divides the brain’s thoughts into two categories:
- S1 is the part of the brain characterized by fast, intuitive, automatic,
instinctive, emotional responses, For example, when you read the text “2+2=?”,
S1 tells you (without …
In yet another contest-based post,
I want to distinguish between two types of thinking:
things that could help you solve a problem,
and things that could help you understand the problem better.
Then I’ll talk a little about how you can use the latter.
(I’ve talked about this in my own classes for a while by now,
but only recently realized I’ve never gotten the whole thing in writing. So here goes.)
1. More silly terminology
As usual, to make these things easier to talk about, I’m going to introduce some words to describe these two.
Taking a page from martial arts, I’m going to run with hard and soft techniques.
A hard technique is something you try in the hopes it will prove something
— ideally, solve the problem, but at least give you some intermediate lemma.
Perhaps a better definition is “things that will …
I think it would be nice if every few years I updated my generic answer to “how
do I get better at math contests?”. So here is the 2019 version.
Unlike previous instances, I’m going to be a little less olympiad-focused than I usually am,
since these days I get a lot of people asking for help on the AMC and AIME too.
(Historical notes: you can see the version from right after I
graduated and the version
from when I was still in high school.
I admit both of them make me cringe slightly when I read them today.
I still think everything written there is right, but the style and focus seems off to me now.)
0. Stop looking for the “right” training (or: be yourself)
These days many of the questions I get are clearly most focused on trying to
find a perfect plan — questions like …
One of the pieces of advice I constantly give to young students preparing for
math contests is that they should probably do harder problems.
But perhaps I don’t preach this zealously enough for them to listen,
so here’s a concrete reason (with actual math!) why I give this advice.
1. The AIME and USAMO
In the USA many students who seriously prepare for math contests eventually
qualify for an exam called the
AIME (American Invitational Math
Exam). This is a 3-hour exam with 15 short-answer problems; the median score is maybe about 5 problems.
Correctly solving maybe 10 of the problems qualifies for the much more difficult
USAMO.
This national olympiad is much more daunting, with six proof-based problems given over nine hours.
It is not uncommon for olympiad contestants to not solve a single problem (this
certainly happened to me a fair share of times!).
You’ll …
I know some friends who are fantastic at synthetic geometry.
I can give them any problem and they’ll come up with an incredibly impressive synthetic solution.
I also have some friends who are very bad at synthetic geometry,
but have such good fortitude at computations that they can get away with using
Cartesian coordinates for everything.
I don’t consider myself either of these types; I don’t have much ingenuity when it comes to my solutions,
and I’m actually quite clumsy when it comes to long calculations.
But nonetheless I have a high success rate with olympiad geometry problems.
Not only that, but my solutions are often very algorithmic,
in the sense that any well-trained student should be able to come up with this solution.
In this article I try to describe how I come up which such solutions.
1. The Three Reductions
Very roughly, there are …
