The Importance of Learning Your Prerequisites

Mastery learning -- one of the most reliable, largest-effect-size techniques for elevating student learning outcomes -- centers on learning prerequisites. In fact, the famous Two-Sigma Problem is centered around the effectiveness of mastery learning.

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In the scientific literature, one of the most reliable, largest-effect-size techniques for elevating student learning outcomes centers on learning prerequisites. It’s called mastery learning – students demonstrate proficiency on prerequisites before being asked to learn material that depends on those prerequisites.

Just to put this in perspective: the famous “Two-Sigma Problem” comes from Benjamin Bloom’s 1984 paper on the effectiveness of mastery learning. (There are many other studies demonstrating the effectiveness of mastery learning, but I want to highlight this particular one because… well, it’s the frickin’ Two-Sigma Problem! Everybody who knows anything about education knows about the Two-Sigma Problem. It’s like the Riemann Hypothesis in the field of education.)

The paper was called “The 2 Sigma Problem: The Search for Methods of Group Instruction as Effective as One-on-One Tutoring,” and the whole idea was that when you take one of the most effective learning techniques leveraged by 1-on-1 tutors (namely, fully individualized mastery learning), and you do a loose approximation of mastery learning across a classroom of 30 students, you still get a very large effect size – large enough that it inspired Bloom to search for other techniques that could similarly be loosely approximated in a classroom setting to elevate learning outcomes even further.

Don’t get me wrong, I get the allure of the top-down thinking, and I don’t mean to claim that it’s never useful. If you want to learn ML, then you can think top-down to figure out what fields of math you need to learn in order for ML to become accessible to you. You’ll find that you absolutely need to learn calculus, linear algebra, and prob/stats, and you can skip stuff like abstract algebra, number theory, etc.

But ultimately, the learning has to occur bottom-up. Are you really going master computing neural net weight gradients via backpropagation by asking “what does that squiggly ‘d’ mean,” “why do you have to chain-multiply the derivatives like that,” “how do you calculate the derivative of any activation function,” etc., all the way down to the depths of whatever is the last piece of math you’ve mastered?

No, all you’re going to do with those questions is create a roadmap of what you need to learn. Which is essentially a calculus course. Except your roadmap will be terrible because you don’t actually know the subject yourself – it will be be missing all sorts of gaps that you don’t even realize are missing because, which is to be expected given that you don’t actually know the subject.

You’ll try to climb back up the skill tree implied by your incomplete roadmap and you’ll repeatedly get stuck trying to climb up to the next branch that you can’t reach because there are prerequisites that you don’t realize you’re missing.

Most people in this situation will eventually just give up due to all the friction. Only those who have extremely outsized perseverance and generalization ability have any chance of fighting through and making it to the other side. And even then, it will take longer (and they’ll likely end up with more holes in their knowledge) than if they just sucked it up and worked through a well-sequenced calculus course.

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