If You Had 15 Days to Learn Calculus, How Would You Do It?

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.

Ultralearning

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.

  • Jonh

    I’d start with Khan Academy’s Calculus: https://www.khanacademy.org/math/differential-calculus

  • Jonh

    I’d start with Khan Academy’s Calculus: https://www.khanacademy.org/ma

  • Great article! Looking forward to the fleshed out approaches.

  • Till H. Groß

    Great article! Looking forward to the fleshed out approaches.

  • Srikant Mahapatra

    I think an even better course to get started would be Dr. Jim Fowler’s (Ohio State University) brilliant Calculus One on Coursera: https://www.coursera.org/learn/calculus1.
    Then one could follow it up with its sequel Calculus Two: Sequences and Series.

  • Srikant Mahapatra

    I think an even better course to get started would be Dr. Jim Fowler’s (Ohio State University) brilliant Calculus One on Coursera: https://www.coursera.org/learn….
    Then one could follow it up with its sequel Calculus Two: Sequences and Series.

  • Fernando Belmonte Archetti

    I find Khan’s method to be much too slow.

  • Fernando Belmonte Archetti

    I find Khan’s method to be much too slow.

  • Alekx

    the best calculus site is 17calculus.com

  • Alekx

    the best calculus site is 17calculus.com

  • Alexander Ladroma

    Not minding the hurt… : (

  • Alexander Ladroma

    Not minding the hurt… : (

  • Jason Bailey

    Scott, didn’t you use MIT’s “Open CourseWare” on the university’s website for the MIT Challenge? It seems like a good idea to start there by taking the Calculus 1 course and working through to the next and so on. That is my plan. My goal is to work through the Calculus courses there and then take the actual Calc classes at my local university. This way, I’ll already know and understand the material, making it much easier to get A’s.

  • Jason Bailey

    Scott, didn’t you use MIT’s “Open CourseWare” on the university’s website for the MIT Challenge? It seems like a good idea to start there by taking the Calculus 1 course and working through to the next and so on. That is my plan. My goal is to work through the Calculus courses there and then take the actual Calc classes at my local university. This way, I’ll already know and understand the material, making it much easier to get A’s.

  • bobango

    I am very confident that you are correct, Scott. It’s been years since I formally took several calculus courses in college. I was good at it, but have not really used it much in my practical life. On the other hand, I would have every confidence of becoming proficient in the material again in a few weeks or a month for the very reason you indicated. When I learned it the first time around, I learned the deep principles and I practiced applying them religiously.

  • bobango

    I am very confident that you are correct, Scott. It’s been years since I formally took several calculus courses in college. I was good at it, but have not really used it much in my practical life. On the other hand, I would have every confidence of becoming proficient in the material again in a few weeks or a month for the very reason you indicated. When I learned it the first time around, I learned the deep principles and I practiced applying them religiously.

  • Peter Finan

    Fully agree – a few years ago I sat the New York Bar exam using this approach. I’m European, had only three months to prepare, was working full time (until the last four weeks), and had never sat an exam in the US in my life. I thought very hard about the approach I needed to to take to have the best chance to pass in those circumstances. I ended up preparing and following a personalised plan which matched what you outline above – striving to understand the issues/principals together with really intense practice of sample/past exam questions. Stuck to it day after day for three months. Got the exam and have no doubt the approach nailed it. End justifies the means 🙂

  • Peter Finan

    Fully agree – a few years ago I sat the New York Bar exam using this approach. I’m European, had only three months to prepare, was working full time (until the last four weeks), and had never sat an exam in the US in my life. I thought very hard about the approach I needed to to take to have the best chance to pass in those circumstances. I ended up preparing and following a personalised plan which matched what you outline above – striving to understand the issues/principals together with really intense practice of sample/past exam questions. Stuck to it day after day for three months. Got the exam and have no doubt the approach nailed it. End justifies the means 🙂

  • Scott Young

    I agree. Khan’s advantage is that he is so prolific. I don’t personally find him to be either the best teacher, nor the most concise one.

    Not a whole course, but for good intuitions about calculus, I recommend BetterExplained.com

  • Scott Young

    Yeah–if you can successfully pass the MIT exams, you’ll probably need very little extra practice to do your university ones.

  • Scott Young

    I agree. Khan’s advantage is that he is so prolific. I don’t personally find him to be either the best teacher, nor the most concise one.

    Not a whole course, but for good intuitions about calculus, I recommend BetterExplained.com

  • Scott Young

    Yeah–if you can successfully pass the MIT exams, you’ll probably need very little extra practice to do your university ones.

  • Thanks! Learning is a never ending process. Calculus is a part of Mathematics, which can be easily done by different techniques to pass it in quick time.

  • Daniaazi

    Thanks! Learning is a never ending process. Calculus is a part of Mathematics, which can be easily done by different techniques to pass it in quick time.

  • Ankit Gaurav

    Will try implementing things you pointed out.
    Thanks

  • Ankit Gaurav

    Will try implementing things you pointed out.
    Thanks

  • Anwar

    Hi Scott,
    I was wondering if there is any good online mathematics degree with reasonable price . Could you suggest me.
    Thanks in advance

  • Anwar

    Hi Scott,
    I was wondering if there is any good online mathematics degree with reasonable price . Could you suggest me.
    Thanks in advance

  • This is actually very useful for students preparing for competitive examinations, where the questions asked are rarely repeated (over a span of close to 40 years)–it is always better to understand a concept fundamentally than just superficially.

  • Vruta Gupte

    This is actually very useful for students preparing for competitive examinations, where the questions asked are rarely repeated (over a span of close to 40 years)–it is always better to understand a concept fundamentally than just superficially.

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