In my last newsletter, I received the following thoughtful comment from Tivadar Danka, who writes The Palindrome:
One reason I like written technical content is precisely what you consider a con: you are forced to research the things you don't understand. Otherwise, you can't proceed. I am convinced that it is the best way to learn, as it makes you actively do things, not just ride along.
Don't get me wrong, I love video content, but it can often give you the false illusion of understanding something. In mathematics at least, real understanding comes from solving problems and using the available objects/tools so solve problems.
I wanted to disagree, but I couldn't.
After thinking about it for a while, I realized we're solving different (but related) problems. Each problem has its own ecosystem of approaches and trade-offs.
I haven't seen a comparison of these problems and their ecosystems yet, so I figured I'd write one up.
In this 13th issue of The Pole, I'll cover the benefits of, the approaches to, and the problems with learning technical subjects. Math will be the primary example, but I believe most of this writing applies to learning in general.
A bullet summary:
The benefits of learning math
What Makes People Engage With Math: a TED talk
The main problem people encounter when learning math
Solution 1: The academic route
The problems with the academic route
Solution 2: The hacker route
The problems with the hacker route
if you have problems at the top, it can be extra frustrating to solve them
you end up learning things that aren’t useful
you’re betting big that you identified the right problem
it can be boring from an artistic perspective
Comparing both routes side-by-side
learning for fun
learning to create something
learning to solve a problem
The solution in the end
The next issue (#14) will directly address video vs written content!
The benefits of learning math
Why do people learn math? Or anything?
In the end, there are two types of motivation: extrinsic and intrinsic motivation.
Some people learn math as a means to an end. Knowing math can help you solve problems and make money. This is extrinsic motivation.
For others, the means are the end. The fulfillment and satisfaction comes from the learning. This is intrinsic motivation.
From personal experience, the latter is a far greater motivator. I get a lot of joy from learning math. That joy comes from two places: asking questions and getting answers to them.
I make that distinction because, at least to me, they're two different kinds of joy.
Getting answers to questions is satisfying in a specific way. Like unclogging a drain and watching the water flow. Or untangling a knot. It feels like a release.
But the greater joy, at least for me, is the asking part.
There are few greater feelings than when I'm going about my day, and, out of nowhere, a JUICY question pops into my head.
...what happens if we combine A with B and a little bit of C?
...what if X?
...what if we dropped assumption Y?
...why isn't Z possible again?
Immediately my brain starts running simulations and scenarios. A movie starts playing in my head. I feel my limbs wanting to start tinkering.
It feels like an entire world of possibility opened up me, and me alone. Like a gold rush, but I'm the only one that can see the gold.
It calls to me, but not in an over-riding sense. It's not an instruction that I must set aside what I was doing and make myself follow. It's more like seeing the light of a lighthouse that I had a burning desire to sail towards all along.
It's like I'm channeling the spirit of an idea. It feels supernatural, sometimes. The better I understand a subject, the higher resolution those scenarios, the juicer the questions become, and the more possessed I get.
(Hopefully it's the closest I ever come to actually being possessed.)
What Makes People Engage With Math: a TED talk
In fact,
gave a TED talk a few years ago about what makes people engage with math.He showed a chart of his 4 most popular videos at the time:
He remarked that the appeal of the 4th and 2nd videos were the relevance of the topics. They're about subjects with good answers to the question, when will I ever use this? They're about subjects with practical reasons to learn about them.
He then contrasted that with the 3rd and 1st most popular videos. These videos had no obvious practical application. Instead, their appeal was emotional. Non-extrinsic.
The 3rd most popular video was about how number of collisions between 2 blocks turns out to have the same digits as pi.
From the video:
The total number of number of digits in the collision are the same as pi. 3.141592.. at this point, it does not matter if the physics is idealized.. if you have a soul, you have to know why!
It's a one dimensional situation. There's no circle. I don't see a circle. And pi's digits are counting something. That's a very weird thing for pi to do, that's not what it does..
It's not [popular] because it's useful, it's because the story has drawn you in.
The number 1 most popular video was about solving a hard problem on a math test.
From the video:
The video is not about the problem... it's a story about you, dear viewer, whoever you are, whatever your background in math, you're not actually that different from the top students.
...in the same way that people watching Star Wars can get a little buzz by thinking,
Hm, what if I had The Force?
I like to think that people watching [this video] get that same little buzz by thinking,
Hm, what if I were to solve the hardest problem on a Putnam test?
Yeah, it's a fiction, but that's exactly what pulls you in!
So, yeah. Learning math can be fun and also useful. Why don't more people learn math?
The main problem people encounter when learning math
If you ask most people what their biggest issue is learning math, their answer is something like:
it's hard
If you probe further and ask why it's hard, you'll get something like:
it's complicated or it's confusing
Which, if you think about it, is ironic. Why? Because math is, by definition, the most straight-forward subject in existence.
How could anyone be unclear about something so transparent?
Well, let's take a look at some math. Here’s part of a scientific paper with a lot of math:
Unless you're part of a specific handful of people on the planet, this is total gibberish to you.
For fun, let's count some of the words and symbols we would need to know to understand this.
On a small part of a single page, there are already at least 16 things we need to learn to make sense of this image.
And it doesn't stop there. For example, if you look up SDE, that stands for Stochastic Differential Equation.
Oh, great, what does stochastic mean? What's a differential equation?
If you look those words up, no doubt you'll find even more words and symbols you don't know.
Down the rabbit hole you go!
Here's the good news: because it's math, those words have specific meanings. The way they relate to each other is (in most cases) well-defined.
There is a place for it in your mind. It's knowable. You can learn it, if you wanted to.
It's a language. If you manage to get the words (and their relationships between each other) to stick in your head, you'll be able to speak the language.
That, I argue, is the real problem: getting the ideas to stick in one's head. Not the complexity of the ideas themselves.
To be clear, I do believe explaining things is hard.
But not compared to overcoming the biological bottleneck from short-term to long-term memory.
Ok, so how do people get ideas shoved into their long-term memories?
The academic route
How have most people learned math in the past?
Colleges and universities.
Here's a summary of the process:
start with the basics (e.g. Arithmetic)
take the next class in the sequence (e.g. Algebra)
keep going until you hit the advanced classes (e.g. Topology)
pick your specialization
There's an assumption here that it's worth making obvious:
You're a newbie, so you have no sense of what the the problem space and solution space look like. You have no reason to be looking at scientific papers because you have no idea
if a paper is worth your time to digest, and
if it is, to what extent you should invest time into understanding it.
Instead, you should start with the principles of the field and build up from there. Towards the end, you'll know
which problems you're interested in solving
what the existing solutions are to those problems, and
what you could try to do better.
At some point, you'll be able to read that scientific paper and any other paper you want.
In fact, you'll be able to come up with original ideas and write your own papers.
The problems with the academic route
Great, so we have an established route for learning math. What's the problem with it?
Well, you don't need to look far to find criticism of colleges. I'll spare you the trouble with a quote from the fast.ai website:
Typically, math courses first introduce all the separate components you will be using, and then you gradually build them into more complex structures. The problems with this are that students often lose motivation, don’t have a sense of the “big picture”, and don’t know which pieces they’ll even end up needing.
Let's assume satisfaction and motivation come from understanding.
The academic route is un-motivating because most of the understanding comes at the end. We spend most of the time memorizing things. The ratio of cognitive load spent to understanding gained is abysmal.
But then, once the ideas are in our head, a whole new world opens up to us:
The hacker route
But let's say you don't have that kind of time (or patience). Let's say you have a more refined idea of what you want to learn and why (or, at least, you think you do).
In that case, the bottom-up academic route probably isn't for you. In fact, it might be better to start at the top.
With the academic route, you start with the basics and build from there. It's more of a just-in-case philosophy.
With the hacker route, you start with the problem and learn what you need to solve it. A just-in-time philosophy.
Get started, learn as you go. Often this means building off an existing solution.
You will often find the entrepreneurial community embodying this philosophy.
fast.ai elaborates on why this is useful:
We have been inspired by Harvard professor David Perkin’s baseball analogy. We don’t require kids to memorize all the rules of baseball and understand all the technical details before we let them have fun and play the game.
Rather, they start playing with a just general sense of it, and then gradually learn more rules/details as time goes on.
All that to say, don’t worry if you don’t understand everything at first! You’re not supposed to. We will start using some “black boxes” or matrix decompositions that haven’t yet been explained, and then we’ll dig into the lower level details later.
The problems with the hacker route
So the hacker route sounds pretty nice. What's the catch? Why not charge in, guns blazing?
Let’s take that paper from earlier as an example. What's wrong with following the rabbit hole?
Some weaknesses:
if you have problems at the top, it can be extra frustrating to solve them
Why? Because often the problem is below the surface, which means it’s a black box to you. Since you didn’t start from the bottom, it can be hard to understand the relationships between each component. Without that context, the problem can be overwhelming.
you end up learning things that aren’t useful
Think back to the jobs you’ve been hired for. Have you ever had the person training you show you how to do something, but the way you were taught to do it is better?
Often formal education is formal for a reason: the battle-tested techniques become standardized. Sometimes, when you go off tinkering on your own, you end up discovering an improvement, but most of the time, best practices are best.
you’re betting big that you identified the right problem
Jumping straight in relies heavily on you knowing exactly what you're looking for. If you're not clear on what problem you're solving, you could end up wasting a lot of time solving the wrong problem (or reinventing the wheel for the right problem).
If your vision of the problem isn't clear, it's easier to get distracted by solutions to problems similar to yours but ultimately aren’t your problem.
it can be boring from an artistic perspective
Let’s say you’re learning to draw. You’re not trying to go to art school. You just want to learn to draw portraits. Specifically, you want to learn to draw a portrait of your crush.
You go to the art store, you buy some supplies, you head home, you whip out a picture of your crush, and you start drawing.
It is at that moment that you realize: you SUCK at drawing.
Which is unfortunate, because you wanted it to be good. You wanted to post it on Instagram, hoping they’d see it and swoon.
Which leaves you with two choices: learn how to draw good, or copy the picture exactly.
If you’re going all-in on the hacker route, that might mean copying the picture as the first step and then modifying it from there.
But WOW is that boring. That would drain your work of its soul and make it a product. You’d rather make a bad drawing - at least it would be yours.
Comparing both routes side-by-side
Okay, so both the academic and hacker routes have their weaknesses. Can we combine them somehow and get the best of both worlds?
(I’m resisting the urge to add a Miley picture.)
To answer that question, let’s say you’re learning something for one of three reasons:
learning for fun
learning to create something
learning to solve a problem
It seems to me that the trade-offs of each approach take different forms, depending on the reason.
Learning for fun
When you’re learning something for fun, the academic route focuses on the details. The big picture comes later, if at all.
The hacker route focuses on solving the problem right in front of you - the details come later, if at all.
Learning to create something
When you’re learning something to create (i.e. for artistic purposes), the academic route focuses on originality. It’s detached from solving an actual problem. You’re learning what good art is so you can adapt it to your unique style. Because it’s divorced from reality, it’s hit or miss whether or not it becomes good.
The hacker route focuses on quality. It’s solving a specific problem, so it should work as soon as possible, ideally from the beginning. It’ll be good, originality be damned.
Learning to solve a problem
When you’re learning to solve a problem, the academic route focuses on risk mitigation. You’re an empty cup. Innocent and naive. So take your time, start from the beginning, learn the ropes. Don’t go on wild goose chases, don’t get distracted by the noise of society, don’t do dangerous things.
The hacker route focuses on saving time. You’re not a sheep. You have ideas. You can clearly see what needs to be done. Why can’t we just do it this way? Why isn’t this way better? Come on now, let’s get to it!
The solution in the end
So what approach should I take? You might ask.
In my experience, it’s helpful to have a holistic approach. Don’t be loyal to a particular philosophy or medium.
You might take an academic approach to solving a problem because you want to become a domain expert. Or maybe you should take a hacker approach because the house is on fire and it needs to be put out NOW.
You might take a hacker approach to painting because you want to step into the shoes of another artist. Or maybe you’re just trying to doodle and relax.
Maybe you learn to play boardgames better by hearing all the instructions up front - a more academic approach. Or maybe you like to learn as you go. The hacker approach.
Or, maybe you like to hack for a little bit and then academic for a little bit. Flip flop between them! Whatever works for you!
The comparisons to learning math really hit home for me here. I loved math as a kid and was good at it for a while. I think I enjoyed it, even when it became harder. Reading your essay, I see where I started to lose excitement for it and I believe it is when I was learning foundational pieces that I didn't have context for thus the payoff came way later in the semester. Great read Josh!
I'm definitely more of a hacker. Often times when learning in an academic environment I feel like the pace is so slow, whereas I can cover 2-3 subjects in the time it would take me to finish 1 class in school, and I get the instant gratification of solving a problem in real-time.
I also find it especially hard to learn from others, as I have a much different learning style than most. If I take notes I never find them useful, and if it is important I will remember it anyways so it feels like a waste of time. Being fully engaged in listening requires that your language processor be focused on listening. Writing uses the same part of your brain as listening so you can only do one or the other effectively.
Now, on the subject of video content, I would say it depends on the video. If it is a scholastic video intended to cover a specific subject without any back and forth, then I would agree.
Where I would disagree, is when you can have two professionals going back and forth, asking questions of each-other and filling in the gaps that a normal "lesson" would leave out. I find that a dialogue between two experts is more revealing than anything one professor could squeeze into a linear lesson plan.
Great article.