Post 34: Learning how to learn
Introduction
The skill of learning things fast and well is phenomenally valuable. It serves as a force multiplier on almost everything you’ll do later in life, and I think it is worth invest a ton of effort into cultivating and developing. Getting better at learning is one of the most valuable things I’ve learned in my degree, and I think it’s a strong contender for my most useful and employable skill. And I think it’s significantly increased the chance that I can do something awesome with my life. I think that anyone who isn’t putting significant effort into improving at this, especially if you’re a student, is systematically screwing up.
This post is an attempt to outline my philosophy of learning, and the tactics/mindset which have worked best for me.
Learning is a useful skill in many different areas of life, with some general skills and some specific skills. I’ll outline many different techniques and approaches, and I recommend reading this post while keeping in mind your goals and contexts, and focusing on the ideas most relevant to you. A lot of my ideas are likely best suited to learning pure maths well, but they’ve seemed to generalise well for me, so I hope this post is of value! And many of my examples will revolve around students, since that’s my context, but the core ideas of effective learning generalise far beyond that. The world is complex and messy, and the process of becoming awesome is a constant process of getting better.
I’d also highly recommend the course Learning How to Learn as a usefully different perspective on good learning! (and from which I shamelessly stole the title of this post)
Iterate!
The most important insight, is that learning is a personal process. There are many possible strategies for how to learn, with different pros and cons. There isn’t a perfect killer app, to find your ideal learning strategy, you just need to try a lot of different stuff! Try something new, review it and see what worked well and didn’t, adapt it and repeat this process until it feels like it works well. This post is essentially a collection of my most successful experiments.
Expect to try many things, and expect most of them to fail. But this is totally fine! This is a textbook example of upside risk! You are going to spend the rest of your life learning new things. If you can marginally improve, that will stick with you for the rest of your life, while a failed experiment has a one off cost. Take a sacrifice in the moment to help your long-term self.
And notice when your current system is broken! If you’re perpetually confused, lose track of what’s going on, and never retain anything, this is a strong sign that it’s worth trying something different. Some things are just intrinsically hard, but you should be able to tell if you’re making progress
Further, a common failure mode is not realising how creative you can be with your learning style! A lot of students I see just absorb a default learning method of going to fast paced lectures and writing down everything they see on the board. And they stick to this style, no matter how confusing or inefficient it feels. But there are so many other ways out there! Ask your friends how they learn things. Ask the smartest people you know how they learn. Try something wild, that you’d never normally do. Don’t be afraid to deviate from the default.
Exercise: Pick something you learned recently, and think about how you learned. Now, set a 5 minute timer, and list other ways you could have learned it. Aim for as many as possible! And don’t confuse the feeling of it being hard to generate more ideas with it being impossible.
Introspection
A significant mistake I see is not respecting the role of emotions, intuitions and introspection in effective learning. Our minds are phenomenally powerful computers, yet our system 2, the conscious verbal part we have access to, is just the tip of the iceberg. Our system 2 alone is utterly incapable of tasks like maintaining a conversation, which computers struggle with, yet our intuitions can manage just fine. A ton of valuable information can be found by paying close attentions to your emotions and intuitions, yet people often dismiss those as “irrational” and not worth listening to.
A few different ways this is helpful:
Notice the feeling of confusion or surprise. This should feel important, and direct you to spend more effort understanding those!
The default state of the world is that you will fail to deeply understand and retain things. A glimmer of confusion or surprise often indicates those weak points, which you know are out there. And you can’t learn effectively until you become able to address those weak points!
If confused, ask somebody for help! Or google “intuitive explanation of …”
Notice the feeling of “this proves too much” - often this shows you’re missing a key limitation of the idea
Notice the feeling of “this feels useless and ungrounded” - often this shows you’ve lost track of the context and bigger picture
Similarly, notice the feeling of following locally but not globally - where you followed each logical step, but feel like there was a trick somewhere you can't put your finger on
Notice what you’re enjoying! Learning is significant cognitive labour, and takes effort. It’s way easier when you feel excited about what you’re working on. Give yourself permission to follow your curiosity, and dive down rabbit holes. Notice the spark of curiosity and nurture it into drive for what you’re trying to do
I find the technique of Noticing to be super useful for getting better at introspection like this, and converting those vague feelings into something useful. I find this emotional awareness significantly drives my actions when learning something new, and triggers several times a day.
The 80/20 Rule
Another key point: there are a lot of things to learn, and a lot of details, and you only have finite time. You need to prioritise. The world is full of wasted motion, and will rarely prioritise for you. A ton of mileage comes from identifying how to get 80% of the value from 20% of the effort, doing that first, and then moving on!
Often the most important parts are getting a handle on the big picture and high-level understanding. Good prompts:
What is going on here? What problems are we trying to solve?
Why do we care about this?
How could I rederive/generate this idea from scratch if I needed to?
Notice when a decision feels arbitrary and unmotivated, and try to figure out where it comes from
Sit down, and try to write out the key points of a topic from memory. What are the biggest holes in your understanding? What’s missing?
What is currently confusing me? What felt surprising?
Getting the high-level picture is phenomenally valuable - it means that when learning any object level details, I can instantly see where they slot in, whether I care about them or not, what other ideas they connect to, how this could be useful in future, etc.
Another valuable skill is learning when to give up on something and when to move on. What things are not worth caring about? Notice when something is a BS magic trick, or a fiddly detail. I find that maths proofs are often 80% trivial algebra, and 10% magic tricks that are only relevant to that specific problem, and 10% key ideas that are used again and again. And I want to only remember those key ideas, all else is a waste of my time. Some good prompts:
Optimise for what will help you understand other things. Can you imagine this coming up again?
What is the minimal set of ideas I’d need to remember to rederive everything else here?
Ask someone who’s more experienced than you whether this is worth your time
Notice when things feel arbitrary or unmotivated
This is hard because sometimes this shows you haven’t grasped the high-level picture - mentors are useful for resolving this!
This principles also applies to simplifying and compressing. After solving a tricky problem, or understanding a hard concept, compress your understanding to as few words and ideas as possible. Eg, challenge yourself to write down the key ideas of a course in <10 minutes. This pressure often forces me to identify the truly important ideas.
This also applies to the resources you learn from! Some resources are way better than others, and there is significant cognitive labour to learning from a bad resource. And sometimes it’s not even worth bothering! I recommend regularly asking “am I getting value from this?” and every hour writing down what you just learned. If you consistently write down nothing, there’s a decent chance you should change up how you’re doing things!
Two resources I’d highly recommend:
The Napkin by Evan Chen, is far and away the best pure maths book I’ve ever read for getting a feel for the bigger picture
The fast.ai course is an excellent introduction to the practical details and concepts of deep learning, from a very “make things that work” point of view
Teaching
A phenomenally powerful way of learning something deeply is to teach it to somebody else!
One model I have of learning, is that I take in information in the format of language. But in my head, they’re stored in a more abstract, conceptual format. And the key challenge of learning is having a concept translated into language in the first place, and then translate from language into concepts, with as few errors as possible. And this will inevitably leave corruptions and holes, but these aren’t always clear (especially if I struggle to notice confusion or surprise!). Teaching forces me to convert this from concepts back into language, but using different words and framings, so it must go via the conceptual framework in my head. And this both reinforces those concepts, and make the corruptions and holes way more visceral. It’s easy to brush over something confusing when rushing through a textbook, but you can’t brush over it when explaining to somebody else.
Teaching also favours a high-level understanding - it’s really annoying to verbally convey fiddly details to somebody else. My aesthetic sense forces me to favour a coherent, high-level picture, prioritising the key bits and skipping over everything I can justify skipping out. And it’s much more obvious when an explanation is confused, and forces me to go back and understand it more deeply
There are a few different approaches to teaching, and they all have elements of these benefits:
Giving talks
I find giving talks great fun (because I’m a massive extrovert and attention seeker), and it serves as an excellent commitment device. I feel obliged to hold myself to high standards
There are significant positive externalities to doing this! Way more people should give talks
Explaining 1 on 1
This is less formal, and lower stakes/lower pressure
It’s also much lower friction! Find a friend, and offer to explain something cool to them. Or find a precocious student in the year below who wants to learn something advanced
Writing things!
I found it super valuable to write notes trying to teach the intuitions behind my courses
Knowing that somebody else will read the notes held me to a much higher standard
And this was really low friction - it was like normal note writing, but just with a different perspective
Even if you don’t intend to publish things, writing them down is still awesome! It forces you to crystallise things
Ensure it’s in your own words, rather than copying! You need to go via the conceptual language in your head. This forces you to process things
Back when I still went to lectures (rookie mistake), I got much more mileage out of only making notes with summaries of key concepts
When teaching, I find it most useful to focus on the structure, motivation and high-level picture. Thinking about how to efficiently convey ideas into a students mind when they lack context is an excellent way to structure the ideas better in your head.
I think teaching is also excellent, because teaching teaches you better meta-skills for learning well! Teaching and learning are two sides of the same coin - often there is cognitive labour that either the student or the teacher could do. And getting familiar with both points of view makes it clearer what these are, and ways you can fail to learn or fail to teach. I outline this perspective in more detail in my post on good teaching.
Asking Questions
I personally learn really well from other people. And my main tool for this is to ask questions (in both 1-1 or group settings). This is particularly salient to me, because questions are a key part of how I learn, but it feels like most people don’t even conceive of it as a skill worth cultivating or trying at. One of my general life goals is to surround myself with as many smart people as possible, and questions are the process by which I can move knowledge from their head to mine!
It’s easy to conceive of learning as a passive process, where knowledge is dispensed by a mentor and I just absorb it, but I think this is a terrible approach. Learning should always be active, as you compress the ideas into the conceptual language in your head, and notice errors, holes and confusions. And when learning from talk or a mentor, asking questions is the main way to actively engage! And if you are not constantly on top of things and processing information, I think you’re significantly and systematically missing out.
There are a few different mindsets for asking questions:
When learning, you want to fit concepts into your knowledge graph, the web of ideas in your mind and how they fit together. If you’re not sure
When learning, you’re always trying to fit the new ideas into your head. But sometimes this doesn’t quite fit, and you feel confused. In my head, I try to ensure this feels like an error, and feels important. If this ever happens, ask about it!
It’s easy to be insecure about this and fear looking stupid. I think this is generally an unhelpful mindset. If you’re confused, likely other people are too! I try to conceive of question asking as a public service, and people generally appreciate my questions!
It’s easy to hold yourself to high standards with questions, but I think this is also silly. An excellent question is “when you said … I felt confused. Can you elaborate?”
Teaching is hard! It’s hard to tell what the audience most wants to hear! Asking questions is an excellent way to create a dialogue and make yourself easier to teach
A useful concept for learning is surface area - getting context, connections and associated ideas to the object-level thing you care about. And questions can be optimised for getting surface area!
Open ended questions, that rely on their intuitions:
What are common misconceptions in your field
What did you find most surprising when learning about this?
What are the most important problems/ideas?
What are you currently working on, and why?
The key here is that context & surface area lets you ask questions. But the teacher has lots already. And good questions can use their context to find the right questions, without taking much work from you
Ask for examples! Especially for typical examples, and for counter-examples.
Volunteer your own examples and ask whether they’re suitable
Try to paraphrase back the key ides in your own words, and ask whether it’s a good summary
Paraphrasing is a phenomenally valuable learning technique. If the only thing you retain from this post is paraphrasing, I am happy!
Paraphrasing forces you to convert the ideas into conceptual language and back immediately. It creates a tight feedback loop that highlights errors and confusions immediately
And remember, the default state of the world is that you fail to learn things well! Paraphrasing can reveal this and fix it immediately. You’re definitely wrong about something, and want to learn what
Paraphrasing forces you to be constantly active and on your toes! It’s hard to zone out if you know you’re going to be paraphrasing back.
It’s useful in the same way that teaching is great!
In a 1-1 conversation, I paraphrase something at least every 5 minutes, and this is a key part of how I learn
Paraphrasing is great for the teacher - it shows you’re paying attention, and that they’ve communicated well
In a group setting, it’s often helpful for other people to hear a different framing and perspective, or to highlight something they were confused about but hadn’t realised yet
It’s much easier to correct than to communicate well in the first place, so this can identify misconceptions!
Another failure mode is focusing on someone’s conclusions, not their model. But the model, the way they generate conclusions, is way more interesting, and far more educational. If passive, it’s easy to see the conclusions and take them at face values. But paraphrasing makes a missing model far more visceral.
On a more meta-level, another way to use questions is by seeking feedback. As I outline in more detail here, seeking feedback is really important. It can identify your weak points, your misunderstanding, and your strengths. All of life is a trade-off, and you need to allocate your ability to understand improve accordingly. Feedback gives valuable information for doing this.
Feedback is also a great way to mine a mentor or teacher for information! You use their context to identify what you should most care about. And again, it’s much easier to correct than to teach!
Some tips:
Be open to criticism! Feedback is useful and valuable, and you likely have a reflexive to be defensive. This is a bias, and systematically deviates you from the truth, and should be solved with a counter-bias.
Giving good feedback can be hard, and takes cognitive and emotional labour. Clearly signalling you’ll take it well is both polite, and makes you more likely to get it
Paraphrase back the feedback, and ensure you’ve understand it well. Try to reveal the model beneath it, and see what you can learn from that
Remember, feedback is awesome! Problems are for fixing, but you can’t fix a problem until you identify it. Feedback identifies your weak points, and you can iterate and improve them, get more feedback, and steadily become more awesome. High-quality feedback is a sign that somebody is an ally, not an enemy - they’re helping you become stronger.
Ask for feedback for the most important things, you can prioritise even here!
Eg, ask for feedback on how you learn, how you prioritise, etc. All of the most important meta skills!
Ask for feedback for what you’re uncertain about! Information is most valuable on the things you understand the least
Ask for feedback on the weak points you’re procrastinating on doing anything about, if it might serve as useful motivation!
Spaced Repetition
An underrated specific technique for learning well is spaced repetition. Education research has robustly shown that the best way to retain information is to be repeatedly and actively tested on it. And, the more times you’ve been tested on it, the longer it takes to forget it. So, if you review at the optimal time, the intervals increase exponentially, and your retention is good that entire time. So the total effort to remember something for the rest of my life just isn’t that high. And there is now software like Anki that can do this for you!
Rote memorisation tends to get a bad rap. I think it’s a valuable part of learning, because it ensures I have important knowledge at my fingertips. The difference between immediately knowing a fact and having to look it up is significant, because this removes friction. And friction is a significant hindrance on my ability to do things, be creative and make progress. Though, importantly, spaced repetition shouldn’t be used to achieve understanding. The ideal use case is taking something you already have a high-level picture for, and to ensure it sticks
A few tips for doing this well:
Anki works excellently for simple facts. Remembering statistics, facts about the world, programming syntax, learning a foreign alphabet, etc
In large part, this is because it’s designed for flashcard style, question and answer format. But this is surprisingly versatile!
I’ve also found it useful for intuitions, motivations and getting a high-level picture of things! Eg, to make maths cards, schema like “we care about … because …”, “the key move to prove … is …”, “the definition of … is sensible because …”
This takes an understanding I currently have, and ensures it sticks
Spaced repetition really shines for long-term gain. It’s really depressing to learn something awesome and to forget it a few months later
The structure of school and university sucks for this, you’re incentivised to cram for the exam, and to forget it all immediately after
I’m currently in the process of converting the useful 20% of my degree into Anki cards, while I still have it in my head
If you’re a student and think what you’re learning in your degree is intrinsically worthwhile and don’t use spaced repetition, I think you’re systematically missing out!
A good litmus test: Would I like to retain this 5 years from now?
This differentiates the trivial bullshit details, from the key points and understanding that I actually care about
As with all things, a good approach to Anki is systematised! I try to always spend 10-15 minutes in the morning on it, while brushing my teeth and getting ready, and have locked everything else on my phone during this time
Generally, I think more people don’t use spaced repetition because it’s slow, unrewarding and only pays off in the long-term. But building systems can reduce these costs while preserving the long-term rewards
Cloze deletions are awesome - they let me write the idea down in prose and to delete the key sections, making cards much lower overhead to generate!
A cloze deletion takes a sentence like “Golden retrievers are a species of [dog]” and makes the front “Golden retrievers are a species of […]”, and the back “Golden retrievers are a species of [dog]”
The main failure mode of early Anki users is making cards that are long and effortful. I try to make all cards answerable in <10s if I remember things. If you’re just starting out, aim to make many cards, and to power through quickly
It isn’t always obvious what to use Anki for. I’m trying to cultivate the habit of noticing all cool ideas I read or learn that feel worth retaining, and making cards for those
Anki isn’t for everyone, but if you haven’t tried it, I think there is significant value of information to trying it out! As Michael Nielson puts it, “Anki makes long-term memory a choice”. Don’t let long-term memory be something that just happens to you.
Conclusion
Overall, I think learning well is a phenomenally useful skill, and cultivating it should be a major life priority! The ability to learn well is a key force multiplier on anything else I might want to do. And it makes sense that most people underinvest in it - the short-term costs of doing anything different can be high, and the biggest payoffs are over the rest of your life. We have a systematic bias against this kind of longterm thinking!
Hopefully the ideas I’ve outlined in this post can serve as some useful starting points and prompts. I’d love to hear anybody else’s perspective on effective learning, if your methods and philosophy differ from mine!
To summarise some of the key high-level framework I’ve outlined in this post:
Learning is an iterative process. I receive ideas as language, convert them into concepts in my head.
The default state of the world is that this conversion process will go wrong. This is fine, and to be expected! I will have errors, and want to identify and iterate on those errors ASAP
The feelings of confusion and surprise are the first warning signs of this, and I want to be as sensitive as possible to these
Paraphrasing and teaching is a great way to further
To fit them in well, I need to know where to add them in. This requires having a clear picture of the high-level context. This is of fundamental importance, and should be prioritised
This is an active process. I need to be doing things, actively participating. If I’m being passive, I’m systematically screwing something up.
This is fundamentally a process of information compression. Most details are easy, a few are crucial. I want to become sensitive to what is and is not worth prioritising, and learning when to cut my losses.
Often I need to look past the surface level details to identify the key important ideas beneath the surface
Learning takes significant cognitive labour. This is something I want to allocate well, and so I should figure out what is and is not worth my time - most things aren’t! Prioritisation for topics is key!
And remember, learning is a personal, iterative process! Try things, experiment, review how well it worked, and iterate! Your life should be a steady process of getting better at learning. And if you’ve been stuck in a rut learning wise, are you happy with that? Or could things be better?
