If you had fifteen days to learn calculus well enough to pass a comprehensive exam, starting from scratch, how would you do it?
A gut reaction might be to memorize. If you learn everything by rote, you can spit it out on the exam paper, then forget it.
But this only works if your exam questions are narrowly constrained. If you’re going to be exposed to questions you’ve never seen before, memorizing anything other than the most general of procedures is going to be useless.
Maybe the solution is to cram. Get yourself in front of a textbook or notes and review intensely, trying to hold in as much information as you can until after the exam. Even if you know you’ll forget everything shortly after.
This might work for a class which requires you to regurgitate answers, but not if you have to solve problems. Having seen something before is hardly preparation to solve novel problems.
As strange as it may seem, I’m going to argue the opposite—the only possible hope you’d have of learning calculus well enough to pass an exam would be to actually learn it, and learn it deeply.
The Trick is Actually Doing It
There’s a great scene in Lawrence of Arabia where the titular character, played by Peter O’Toole, lights a match and then snuffs it out with the tips of his fingers. Seeing this, another man tries to repeat the feat, yelping in pain. “That hurts!” he then exclaims, “Well, what’s the trick then?”
“The trick is not minding that it hurts.”
The trick to passing a calculus exam in fifteen days, is to actually learn calculus. Not a trick to memorize things or a gimmick to be able to perform well. Instead it’s the opposite, striving to deeply understand the fundamental principles.
The principles are less numerous than the superficial details. It’s only by really learning these, and then aggressively practicing their most common instantiations, that you have any chance of doing well on an exam that has novel problems. Memorizing and cramming, while they seem like they’ll do the trick are actually too slow.
Let me be clear, learning calculus in fifteen days will be less effective than fifty or five-hundred days. But that should be obvious—throwing more time at the problem could only make things worse if you switched to a far less effective method.
Instead, what I’m arguing is that, contrary to the gut instinct of many lazy college students, the only way to get good enough in such a short time is to go deep. This strategy, which I’ll call ultralearning, is the act of flipping that original intuition on its head. Instead of trying to memorize details well enough to pass exams, you try to learn the principles well enough so that your insufficient memorization of the details doesn’t matter quite as much.
This is the approach I used in both the MIT Challenge  and Year Without English . With the former, even if I had through some miracle, gotten through Calculus I by memorizing, I would have been done for with the subsequent Calculus II, Differential Equations, Signals and Systems and dozens of other classes which depended on it. Ultralearning scales. Cramming doesn’t.
Why Learning Faster Matters
Since few people are in a position where they might need to learn calculus in fifteen days, why does this matter?
If the best method used to learn calculus quickly were simply to memorize things, or cram notes, then it would matter little. If this were the case, then learning faster would necessarily mean learning poorly and so those who weren’t under tight time pressures could safely ignore these ideas.
However, if you believe, as I do, that the most effective method is intensive practice combined with trying to form deep intuitions about the principles involved, then this matters a great deal. Because this is also the foundation you can learn anything from.
Ultralearning, is not then just a quick shortcut, but rather the beginning for learning anything. It’s usefulness and necessity are only revealed when passive and lazy strategies stop working because of time constraints.
In the next post, I hope to flesh out this strategy of ultralearning in more detail. However, long-time readers won’t be surprised, it’s a synthesis of the same ideas I’ve advocated for years in the MIT Challenge , Year Without English  and my many articles rethinking self-education.