Diminishing Returns

Originally published April 6, 2020.

I immediately become apprehensive when people state they are going to do every problem in X book/Y chapter or the past Z years of the AMC 10/12. There's a lot of reasons why this is bad - in the former case, this is very restrictive and prevents you from moving on when you need to, and in the latter case, this seems fairly boring/stressful and is a surefire recipe for apathy. This is not to say a plan or outline for how to train is bad. Knowing what you're going to do, even if it's just a rough idea in your head, is better than not knowing. [1] If nothing else, you'll know how to spend your time training and won't have to waste any of it thinking "what should I do?" while being unproductive. But I recommend against having detailed plans for training because you don't really know what you should be doing until you start getting good at it - at which point, you have more than enough of an idea of how to do it and probably should be doing something else anyways.

With this being said, two pitfalls that students commonly fall into while making/following plans of study are:

1) Being a completionist.

2) Strict adherence to "the plan."

Let me share an anecdote. When I was in 6th grade I was just starting math competitions, was barely aware what MATHCOUNTS was, thought the AMC 8 was "the competition" of the year, and had no clue what the AIME was despite nearly qualifying for it until February or March. My preparation mostly consisted of doing random Alcumus problems with my friend during English/History class, and we occasionally did past problems from the AMCs and AIMEs and marveled at the prospect of making AIME. [2] My lack of knowledge about math competitions and my having no idea of what training was supposed to look like [3] meant it made some degree of sense for me to do Alcumus back then. As I learned more about the math competition scene and got my values and philosophy aligned, I gradually transitioned to using better resources.

It would make no sense and be terribly inefficient for me to do Alcumus now on the scale I did it before. I wasn't naive enough back then to think I'd have an idea of what I'd be doing in three years, and I am not arrogant enough to think that I'll know what I'll be doing in three years from now. If I had to guess three years ago, my answer would've been embarrassingly wrong. I don't doubt the same will happen if I guess now what I'll be doing in three years. So the worst thing you can do is let past you dictate future you's actions.

And it's not just (younger) math competition kids who do this! I frequently see "regular math kids" do this when studying for a test [4] and it usually doesn't turn out well. What they fail to consider is that, to put it bluntly, the review problems suck and the book they are reading from is boring and poorly written, and they need to get better problems from better books. The main driver of your studying should be you: the standard is "How much does this help?" In competition math books are "naturally selected out." While not everything good gets the exposure it does, nearly everything that gets the exposure it does is good. This is because you cannot distort reality in competition math or in learning - you either write something that helps kids learn math or you get ignored. In school, this is not the case; standards are arbitrarily selected, and the people who pick your study material couldn't be more detached from your situation. But it is very easy to pretend to do work when you know how to do real work; this is why I am very happy to see "normal kids" learn and be interested in math.

Footnotes

[1] The latter is still okay.

[2] Specifically, "There's no way we'll be getting into AIME."

[3] "Spam past tests" is probably not the right answer.

[4] Usually takes the form of "I need to do all of the review problems in Chapter X!"