There Are No Hard Subjects, Only Missing Prerequisites

The common view of learning is that some subjects are clearly harder than others. Quantum mechanics is a lot harder than, say, learning state capitals.

This idea points to some domains of knowledge as being intrinsically harder than others. A related idea, being that if some ideas are intrinsically harder than others, and some people are intrinsically smarter, then maybe some ideas are just too hard for you to learn.

I want to suggest an alternative view. This is that there is no subject that is intrinsically hard. Rather, it’s that some subjects appear hard because they have a lot of prerequisite knowledge, which often students don’t have. Once you obtain the correct background knowledge, most ideas are roughly equally easy to learn.

Recursively Learning Harder Subjects

The reason quantum physics is harder than state capitals is because quantum physics requires more background knowledge, concepts and skills. To learn state capitals, all you really need to know is that states exist and they each have one capital city. To learn quantum mechanics, you need to know differential equations, which requires calculus, which requires algebra, and so on.

Quantum mechanics, compared to state capitals, sits at the top of a much larger pyramid of required knowledge.

At first glance, that might not seem like much of a benefit. If you lack the background in quantum mechanics, what good does it do to say that you’d be able to learn it just as well supposing you had that background?

The reason is that this idea, if true, applies recursively. Meaning that all the prerequisites to quantum mechanics are also easy to learn, provided you have their prerequisites. Keep going down the chain, and eventually you’ll find an idea that’s easy to learn and you can go from there.

Why Do Classes Get Harder?

This theory seems to be betrayed by an experience many people have, which is the progressively increasing difficulty of classes in a particular subject. Students may have aced their high-school math, but then struggle with calculus and need to drop out of a math major in university. What’s going on?

I’m going to suggest there are two effects at work which make it seem like classes are getting harder, when all ideas are roughly equally easy to learn:

  1. Competition is getting stiffer. If you’re doing a PhD in theoretical physics, your peers are now all incredibly smart and driven. You may feel dumb, but that’s mostly because you’re now only surrounded by very smart people. The learning difficulty hasn’t changed, but the competition might have.
  2. Students don’t adequately learn the prerequisites. The other reason ideas may appear to get more and more difficult, is that the student doesn’t actually have the prerequisite knowledge. They may have crammed through an exam and passed without retaining much, or be missing the key concepts and skills which would make a deeper understanding possible.

Limitations to the Theory

This alternative idea, that all knowledge is equally easy to acquire, given the right prerequisites, is probably false in the extreme. Some ideas may be more difficult for some intrinsic reasons, such as being excessively abstract or involving more ideas at the same time.

However, I’d wager that once you remove prerequisite effects, the difference between the difficulty of many ideas is actually quite small. You might even find that some commonly thought-of as “easy” ideas are actually slightly harder than some “hard” ones, simply because the “hard” ones have taller prerequisite knowledge structures.

What this Means For Learning

A common email response I get is someone who is eager to try out the MIT Challenge, then tries to take an “intro” class and gets knocked out by the difficulty.

On the one hand, this shouldn’t be surprising. MIT classes are typically taken by MIT students. MIT students are some of the highest achieving students in the world, often having taking many AP classes and been exposed to ideas that go beyond a normal high-school curriculum.

But what happens after this is that people give up. They imagine they just aren’t cut out for learning hard subjects, so they switch to easier fare. When really, they should be trying to identify what are the prerequisites they are missing and try to learn the topic in more than one step.

I did this for my recent ultralearning challenge. I was reading a book on neurobiology and I couldn’t understand most of it. Instead of admit I’m too dumb for neuroscience, I decided to build my prerequisites by taking a different class to warm me up to the subject.

Recursive Ultralearning

Ultralearning, the deep, intense self-education I’ve been writing about, does depend on smarts. But, perhaps more importantly, it depends on other psychological qualities of grit, determination and the ability to push through difficulty. It also depends on underdeveloped skills of self-education.

One of those skills is the ability to learn things recursively. That, when faced with too difficult an obstacle, the skilled ultralearner will break it down and learn the prerequisites, rather than admit it is an impossible impasse. Sometimes that will be by taking another path in, other times it will be by building a ladder.

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  • William Tarbush

    As an adjunct professor at a small, private University, I would concur. Anything becomes easy when properly prepared for it. Sometimes, I wish I could teach more surveys but other times, I know they are just not as cost-effective as throwing the students to the lions of other courses.

  • William Tarbush

    As an adjunct professor at a small, private University, I would concur. Anything becomes easy when properly prepared for it. Sometimes, I wish I could teach more surveys but other times, I know they are just not as cost-effective as throwing the students to the lions of other courses.

  • Mark Estefanos

    I like the sentiment. The attitude of looking for missing perquisites instead of deciding one isn’t cut out for the subject is definitely a helpful, more accurate, and constructive one.

    That being said, I don’t think you can say all ideas are the same difficulty. Some ideas are assembled of many smaller ideas, and some even require you to understand implications of the interactions of those smaller ideas. Some ideas are more abstract, and we know that people vary in their ability to manipulate abstract concepts since that’s what an IQ test measures. Some ideas are just far more complex than others. Some ideas are far more unintuitive than others, making them harder to grasp.

    Another thing to note is that learning the prerequisites of a subject also teaches you the skills you need to take on harder tasks of that nature. By learning algebra, you’re learning a prerequisite for calculus, but you’re also cultivating the skill of thinking abstractly, which will make you more capable of doing calculus. You can see this in how learning algebra or proofs will make you a better programmer, even if they’re not directly related. Think along the lines of how lifting weights will make you better able to wrestle.

    EDIT: I’d just like to reiterate that I think your model is better than common wisdom, and a much more beneficial one to have, I just disagree with the small point of “all ideas are the same difficulty.”

  • Mark Estefanos

    I like the sentiment. The attitude of looking for missing perquisites instead of deciding one isn’t cut out for the subject is definitely a helpful, more accurate, and constructive one.

    That being said, I don’t think you can say all ideas are the same difficulty. Some ideas are assembled of many smaller ideas, and some even require you to understand implications of the interactions of those smaller ideas. Some ideas are more abstract, and we know that people vary in their ability to manipulate abstract concepts since that’s what an IQ test measures. Some ideas are just far more complex than others. Some ideas are far more unintuitive than others, making them harder to grasp.

    Another thing to note is that learning the prerequisites of a subject also teaches you the skills you need to take on harder tasks of that nature. By learning algebra, you’re learning a prerequisite for calculus, but you’re also cultivating the skill of thinking abstractly, which will make you more capable of doing calculus. You can see this in how learning algebra or proofs will make you a better programmer, even if they’re not directly related. Think along the lines of how lifting weights will make you better able to wrestle.

    EDIT: I’d just like to reiterate that I think your model is better than common wisdom, and a much more beneficial one to have, I just disagree with the small point of “all ideas are the same difficulty.”

  • evamotch

    I fount this really helpful. I have to teach myself Quickbooks- you made me realize I’m finding it so hard because I know nothing about accounting– I should read a basic accounting book first.

  • evamotch

    I fount this really helpful. I have to teach myself Quickbooks- you made me realize I’m finding it so hard because I know nothing about accounting– I should read a basic accounting book first.

  • Scott Young

    True, but I think the difference you mention is exaggerated.

    What does it mean for an idea to be complex? That means it has a lot of component parts. Those component parts can be learned separately, thus making the final composite idea less difficult.

    Here’s a line of reasoning I think is compelling:

    Human working memory is incredibly limited, and while there is some natural variation, you don’t see human beings with orders of magnitude more general working memory than others. The way around this limitation is by creating cognitive “chunks” of information that can be recalled directly from long-term memory. Having chunks = having prerequisites. Since human beings’ working memory doesn’t vary by more than an order of magnitude, then the difficulty of ideas can’t vary by more than this.

  • Scott Young

    True, but I think the difference you mention is exaggerated.

    What does it mean for an idea to be complex? That means it has a lot of component parts. Those component parts can be learned separately, thus making the final composite idea less difficult.

    Here’s a line of reasoning I think is compelling:

    Human working memory is incredibly limited, and while there is some natural variation, you don’t see human beings with orders of magnitude more general working memory than others. The way around this limitation is by creating cognitive “chunks” of information that can be recalled directly from long-term memory. Having chunks = having prerequisites. Since human beings’ working memory doesn’t vary by more than an order of magnitude, then the difficulty of ideas can’t vary by more than this.

  • Mark Estefanos

    Well put! I agree, ideas don’t differ by much more than an order of magnitude. What an empowering idea.

  • Mark Estefanos

    Well put! I agree, ideas don’t differ by much more than an order of magnitude. What an empowering idea.

  • Bobby

    Your idea is interesting, Scott, and reminded me of website: https://www.metacademy.org/. If you want to learn gradient descent, metacademy will list its prerequisites, which will in turn have their own prerequisites, etc. I think your idea might be slightly oversimplified but I think you’re on to something. In any case, being able to break down a subject into its individual components is an important skill to have, especially when tackling more massive undertakings. Combining this idea with Cal Newport’s “deep work” concept I imagine could be quite a duo.

  • Bobby

    Your idea is interesting, Scott, and reminded me of website: https://www.metacademy.org/. If you want to learn gradient descent, metacademy will list its prerequisites, which will in turn have their own prerequisites, etc. I think your idea might be slightly oversimplified but I think you’re on to something. In any case, being able to break down a subject into its individual components is an important skill to have, especially when tackling more massive undertakings. Combining this idea with Cal Newport’s “deep work” concept I imagine could be quite a duo.

  • Phuong VU

    Hi Scott,
    That’s a very useful approach. What would you do in case of steep learning, says, you have limited time (e.g: need to complete that intensive course within a month). When I look up on the “stepping stone”, I also got overwhelmed by how many information/extra courses there is, and, as someone who basically has no idea yet about the subject yet (says I’m a right brainer sitting through some very technically coding course), it is hard for me to “filter out” what to prioritize. What would you do? Go ask an expert on which steps to take? Sometimes they know, other times they are just in the field for so long that they don’t really know where to start from scratch, for someone who has a completely different background.

  • Phuong VU

    Hi Scott,
    That’s a very useful approach. What would you do in case of steep learning, says, you have limited time (e.g: need to complete that intensive course within a month). When I look up on the “stepping stone”, I also got overwhelmed by how many information/extra courses there is, and, as someone who basically has no idea yet about the subject yet (says I’m a right brainer sitting through some very technically coding course), it is hard for me to “filter out” what to prioritize. What would you do? Go ask an expert on which steps to take? Sometimes they know, other times they are just in the field for so long that they don’t really know where to start from scratch, for someone who has a completely different background.

  • bobango

    I’m a professional trained cognitive psychologist, so good luck with the learning there. I think you’ll find the most demanding part is the advanced stat, if you aren’t prepared for it. The problem there is that most college stat courses are cookbook courses. They teach formulas, with little insight into the underlying ideas. A graduate stat course in a strong cognitive psychology program is going to require you to think about what these concepts mean and where to apply them. In the end, it is immensely gratifying to go through. Having offered these comments, I otherwise endorse your perspective on why some topics may seem “harder” than others.

  • bobango

    I’m a professional trained cognitive psychologist, so good luck with the learning there. I think you’ll find the most demanding part is the advanced stat, if you aren’t prepared for it. The problem there is that most college stat courses are cookbook courses. They teach formulas, with little insight into the underlying ideas. A graduate stat course in a strong cognitive psychology program is going to require you to think about what these concepts mean and where to apply them. In the end, it is immensely gratifying to go through. Having offered these comments, I otherwise endorse your perspective on why some topics may seem “harder” than others.

  • Akira1643

    This is an excellent post that all students of all ages need to hear. Instead of giving up on math, that 10th grade student will understand that what lies ahead of them isn’t some immovable wall. It is actually the fact that he has to backtrack to get the ladder to climb over it.

  • Akira1643

    This is an excellent post that all students of all ages need to hear. Instead of giving up on math, that 10th grade student will understand that what lies ahead of them isn’t some immovable wall. It is actually the fact that he has to backtrack to get the ladder to climb over it.

  • Damias Mcdonald

    One thing I found useful is to find someone who just finished the course. Often they can tell you what you need to focus on in the pre-reqs. Most of the time it’s really only a few things you need to be well versed in, the rest you just need to have passing knowledge of.

    If you don’t have access to one, I go over the course quickly to see what it’s like and what I understand. Whatever I don’t understand I google it and try to read it’s simple.wikipedia version then see if that’s enough. If not I try to find online resource, mostly video, that’ll break it down for me.

  • Damias Mcdonald

    One thing I found useful is to find someone who just finished the course. Often they can tell you what you need to focus on in the pre-reqs. Most of the time it’s really only a few things you need to be well versed in, the rest you just need to have passing knowledge of.

    If you don’t have access to one, I go over the course quickly to see what it’s like and what I understand. Whatever I don’t understand I google it and try to read it’s simple.wikipedia version then see if that’s enough. If not I try to find online resource, mostly video, that’ll break it down for me.

  • Alexander Ladroma

    Thanks a lot for this Scott, really. I was about to shift to real estate rather than engineering this morning. How I wish you wrote this a week earlier. There’s no stopping now and actually thinking of doing PhD, ; )

  • Alexander Ladroma

    Thanks a lot for this Scott, really. I was about to shift to real estate rather than engineering this morning. How I wish you wrote this a week earlier. There’s no stopping now and actually thinking of doing PhD, ; )

  • Michelle Johnson

    I taught myself Quickbooks with a book that my predecessor got in a hands on Quickbooks class. It includes lots of practice exercises with each lesson. A book like this may be helpful. You don’t need to understand everything about accounting to use Quickbooks, maybe a few key concepts. Once you identify those, you can just Google them.

    I think the trickiest part of learning on your own is figuring out what the the prerequisites are. It requires thinking differently than we usually do during the day.

  • Michelle Johnson

    I taught myself Quickbooks with a book that my predecessor got in a hands on Quickbooks class. It includes lots of practice exercises with each lesson. A book like this may be helpful. You don’t need to understand everything about accounting to use Quickbooks, maybe a few key concepts. Once you identify those, you can just Google them.

    I think the trickiest part of learning on your own is figuring out what the the prerequisites are. It requires thinking differently than we usually do during the day.

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