Mini Blog Post 18: How to teach things well
Introduction
This is a blog post on how to teach things well. I’ll mostly be focusing on forms of teaching that involve preparation and structure, like talks and tutoring, but these ideas transfer pretty broadly. I think teaching and explaining ideas is an incredibly important skill, and one that most people aren’t great at. I’ve spent a lot of time practicing teaching ideas, and I think I’ve found a bunch of important ideas and approaches that work well. I’m giving a talk next week, so I’ll initially focus on how to give good talks, but try to outline the underlying concepts and high-level ideas of teaching. And then talk about how these can transfer to contexts like tutoring, and to teaching specifically maths or applied rationality - the main areas I have actual teaching experience with.
Note: I mostly care about teaching concepts and ideas, and teaching things to people who genuinely want to learn and be there, so my advice will focus accordingly.
I think it’s useful to think about good teaching even if you don’t intend to spend much time teaching - learning and teaching are flip sides of the same process. I’ve found that even when in the role of a student, understanding what good teaching looks like can often fix a lot of the shortcomings of a bad teacher!
Framing
The key insight of this post is that good teaching requires you to be deliberate, and keep the purpose in mind: learning is a process of information compression. When you’re learning something new, you essentially receive a stream of new information. But human cognition doesn’t work by just storing a flood of information as is. The student takes in the information stream, extracts out the important ideas, converts it to concepts, and stores those in their mind. This is a key distinction, because it shows that the job of a teacher is not to give the student information, it’s to get the student to understand the right concepts. Conveying information is only useful as a means to an end to this goal.
In practice, it often works to just give a stream of information! Good students have learned the skill of taking streams of information and converting it to concepts. Often this happens implicitly, they student will absorb and memorise a lot of data, and over time this forms into concepts and ideas in their head automatically. But this is a major amount of cognitive labour. And a good teacher will try to do as much it as possible, to let the student focus their cognitive labour on the important things.
My underlying model here is that we all have a web of concepts in our minds, our knowledge graph. The collection of all the concepts we understand, all of our existing knowledge and intuitions, connected together. And you have learned something when you can convert it to concepts and connect it to your existing understanding. This means not just understanding the concept itself, but understanding where it fits into the bigger picture, where to use it, etc.
The final part there is key - if the student leaves with a good understanding of the ideas in the abstract, but no idea when to think about the ideas again, it’s no better than if they’d learned nothing at all. We call on our knowledge when something related triggers, so in order for a lesson to be useful, you need to build those connections and triggers in the student’s mind.
A key distinction to bear in mind is ideas being legible vs tacit. A legible idea is something concrete that can easily be put into unambiguous words, eg how to do integration by parts. While tacit knowledge is something fuzzier and intuitive, eg recognising the kinds of integrals where you’d use integration by parts in the first place - essentially the intuitions you want the student to have. This is a good distinction to bear in mind, because legible knowledge is much easier to convey, but often your goal is to convey tacit knowledge (at least, it should be!). And there’s a lot of skill to conveying tacit knowledge well, and making it as legible as possible without losing key nuance. And different techniques work better for the two kinds. A lot of my issues with the Cambridge maths course is an extreme focus on legible knowledge over tacit - the underlying intuitions and motivations.
How to teach
There are two key problems when teaching, that any good teaching advice must account for:
Limited ways to convey information
The ideas I’m teaching are stored as implicit concepts in my mind, but in order to convey them, I must translate them into language. This language maps to ideas in my head according to all of my implicit knowledge and worldview, but translates into the student’s head according to their implicit knowledge and worldview. This often creates errors
And converting concepts to language is inherently lossy, ideas have a lot of tacit nuance that is hard to capture
Essentially, words can only convey legible knowledge, and I need to figure out how to hack this to convey tacit knowledge. Or how to find alternate information channels
Alternately, I need to have error checking mechanisms to notice when I’ve failed to convey something well.
A key part of learning is combining knowledge you hear with the ideas already in your head.
But I only have access to what’s in my head, not what’s in the student’s. And by definition this is different - I already understand what I’m teaching!
This is a crippling blow to my ability to teach, and I need to be constantly aware of this and trying to build models of what’s in the student’s heads, and how they receive what I’m saying.
Here are some of the most important tools I have for addressing these problems:
High-level picture
The student’s knowledge graph is big. So the first, and most important part, is identifying which part of the graph they should add these ideas to
Thus you should always highlight where this fits in to the bigger picture. Which questions are we currently trying to answer? Why is this idea interesting at all? Where can the student use this? What are its limitations?
This should always be the first thing when introducing a new idea
Prioritising information
Learning is information compression. This fundamentally means that the student needs to be paying selective attention. When learning something new, the Pareto Principle always applies - 80% of the importance lies in 20% of the ideas.
Identifying this 20% is significant cognitive labour, because it’s not immediately obvious. They need to pay attention to everything and later filter.
But, the question of “what matters here” is tacit knowledge that I already have! This talk is high labour for the student, but easy for me. Thus, the most important thing a teacher can possibly do is to highlight what matters and what doesn’t, to tell the student where to focus their attention.
In practice, you should always be saying “this is really, really important” or “these are just fiddly details” or “this is a bit of a niche edge case” etc. It is extremely hard to do this too often.
And give more time to the important things. If you say an important point, write it down and put a box around it. Explain why it’s important and how it fits into the bigger picture. Give an alternate explanation, or an example.
This is really easy to miss if you think of teaching as information transfer - where your goal is to tell the student everything and hope they figure out what to pay attention to themselves.
Another tool: Have frequent summaries, highlighting the key points in the previous section.
This is further useful to highlight connections. Some ideas will fit into their knowledge graph more easily than others, and pointing out connections can leverage the easy connections to make hard ones easier
I’m a big of the advice “say what you’re going to tell them, tell them, and say what you’ve just told them”, I think it’s a good way to implement this principle
The fundamental principle behind this is that students retain a tiny fraction of what they hear. If you give a one hour talk, they’ll retain maybe a few minutes worth of content. This is a fundamental fact of the learning process, and the only thing you can do about it is to control what they retain. Focus their attention on the important parts
This mindset helps me identify what’s important. Set a 5 minute timer, and write down everything important that you want people to retain. These are the key points around which your talk should be structured!
Everything else you say should be intended to help these key points stick better - to highlight connections between them and to existing knowledge, to ensure good information transfer, and to convey the tacit knowledge underlying the key points
Another key skill of prioritisation is cutting things. If one part is irrelevant, or it’s boring and fiddly, cut it! It’s painful to not talk about everything cool, but you have limited time - if you don’t actively prioritise, you aren’t avoiding trade-offs, you’re just ceding control to “whatever you leave last”
Anyone who’s gone to a talk by me knows that I have yet to internalise this lesson
If there’s one point you retain from this post, let it be this one - this is incredibly important, and the main mark of a good teacher vs a mediocre one
Understand pre-requisites
A consequence of the Typical Mind Fallacy, is that it’s super easy to forget that your students don’t have all the context you have! This manifests as people teaching ideas without the pre-requisites.
The underlying idea: the ideas you teach are in your knowledge graph, and are built upon existing ideas - these are the pre-requisites. You need to figure out whether the students
A common secondary mistake is to understand pre-requisites, and then try to explain all of them!
A good framing here is inferential distance. The inferential distance of a new idea is the number of steps of new concepts someone must understand before they can understand the new idea. Eg, to teach a young kid about the quadratic formula, they first need to understand the idea of polynomials, for which they need to understand algebra - this is three inferential steps.
A general rule of thumb: never teach things with more than 2 inferential steps. An idea just learned is shaky, and doesn’t yet have good connections built. It’s very, very hard to anchor new ideas onto new ideas.
This is really hard to get right. Pre-requisites require you to have a good model of somebody else’s mind. A useful technique is often to do a practice run on someone from your target demographic, and ask them to flag everything that confuses them.
Further, for groups, this can be an intractable problem, everyone has different prior knowledge. You need to have a clear picture in your head of who the talk is aimed at.
Students should learn actively, not passively
It’s really easy for a student to just passively sit in a stream of information, taking none of it in. This achieves neither of your goals, because to form connections, compress information and connect it to their knowledge graph, they need to be putting in some cognitive labour.
Good ways to encourage this: explicitly telling them that this is important, giving exercises and time to think through them, asking the audience regular questions and giving them some time to think.
Question asking has the failure mode where most people won’t volunteer, or just zone out a bit - I lack great solutions to this problem
This is much easier to handle in smaller settings, I’m bad at handling it in talks. The main solution I have is just to be engaging and keep the pace going well.
Often people will zone out, so having regular breaks, and check-points of “if you weren’t following, we’re changing topic so it doesn’t matter” can help to rectify this
Even short, 30s-120s breaks can be helpful! Encourage people to get up and stretch.
Breaks never feel important, but they really, really help
Give examples
Examples are an insanely powerful tool for teaching things well, and people rarely use them enough. I have never given a class with too many examples (and believe me, I’ve tried)
“You can teach a class with no content, only examples; you can’t teach a class with only content, no examples”
Why examples are awesome:
Examples are an excellent way to resolve lossy information transfer - they’re a completely different channel of communication than normal. If nothing else, they serve as an error check
Examples are a great way to transfer tacit knowledge, without necessarily making it legible - this is what it means to build intuition
Examples can help fit things in to the bigger picture, they can motivate the ideas, and locate where they fit in to the student’s knowledge graph
By giving the student a pool of motivating examples, they can often generate the ideas themselves by generalising from the examples
Examples can bridge the gap from “understanding the knowledge in the abstract” to “understanding where to use these ideas, and where they should come up”
How to use them?
Often when giving a point, I’ll give a micro-example to give context to it - eg “sometimes straight lines aren’t enough to model data, eg with a quadratic”. This should be quick and effortless, the example should make immediate sense to the students, with 0 inferential steps
(I can’t believe it’s taken me 18 posts to get to my first nested example :( )
Examples can be good at the start, to motivate things and show the questions we’re trying to answer. It can be good to give an example, and then constantly refer back to it as we generalise the example into a concept
After introducing a complex idea, go through a long example and illustrate which parts of the example embody the complex idea
Use examples to illustrate the importance and relateability of an idea - eg if explaining how to think about good planning to students, give an example of a student with a deadline crisis that they missed - everyone relates to this
Examples contain a lot of information, so the idea of information prioritisation applies strongly here - tell the students what to pay attention to in the example, and why it’s interesting
Visual information and diagrams
Often tacit knowledge manifests as a literal picture in my head - draw this!
This is another good alternate communication channel
Our visual memory and processing is often much better than our abilities with language - this can work well for clarified confusing and complex parts
Pacing
An easy mistake that I often fall into is to give a section the amount of time it takes me to say it. I convert the ideas into words, and just read through what I’ve come up with.
This is the fallacy of viewing teaching as giving an information stream! Time should be allocated for people to process and compress information, and they need more time for hard parts and less time for easy parts
This is hard to get right intuitively - when you understand an idea, it feels easy!
A good trick: make a high-level summary, and rate each section out of 5 for difficulty. Then go and explicitly give more time to those sections - eg add more intuitions, say the key ideas more, give more examples
Note - pacing doesn’t mean the speed at which you speak, it’s about the time you give to different ideas!
Note that difficulty =/= importance. If one part is hard, but unimportant, cut it. Or give a brief overview, explain the important idea, and say “don’t sweat the details”
Another trick - explicitly tell students which ideas are important and worth paying attention to, and which aren’t
If doing a practice run (which you totally should), regularly check in with the test audience about pacing - the default state of the world is that you get pacing wrong
It's key to get pacing right - people zone out in slow, easy sections, and get lost in fast, hard sections. Your job as the teacher is to keep as many people as possible absorbing information at the optimum rate.
It’s very hard to give accurate time estimates for things - my trick is to have a bonus section at the end which I’ll cut if need be, and to pace in the moment according to my existing notes and my intuitions
Understand the mindset behind a question
When somebody asks a question, the default response is to answer it. This is a failure to be deliberate! The student asks a question because they’re confused about something, and your goal is to resolve that confusion - answering the question directly is just a means to an end.
This is an important distinction, because often questions are weird. They’re confused, or don’t quite make sense, or are asking about unimportant things. This manifests, for me, as the student’s mind not making sense. And it’s easy to get frustrated, or just to answer the dumb question directly. But this is ineffective.
A related effect - somebody asks a question that isn’t the real question they want to ask. Eg, a student at a university open day who asks “how many A-Levels did you do?”, when what they really care about is “how many A-Levels do I need to do to get in”
Your goal should be to understand the state of mind from which that question made sense - once you’ve done this, you can often resolve the confusion directly, or answer the question they really care about.
Do this by asking clarifying questions, trying to answer and saying “did that answer your question?”, giving them several interpretations of what they’re really asking and asking whether any resonate, paraphrasing the question back to them, etc.
The main trigger to look for here is a note of confusion - the question feeling a bit off, or out of nowhere, something isn’t quite making sense.
It’s a delicate balance between doing this and moving on with the talk - try to gauge whether many people share the same confusion, if not, just move on
Often doing this can uncover Pedagogical Content-Knowledge, common ways that people misunderstand the ideas you’re teaching. It’s super valuable to collect these, because then you can recognise them in future and dissolve them directly.
Illusion of transparency
A consequence of the typical mind fallacy - it’s easy to think you’ve clearly communicated knowledge when you really, really haven’t. As a consequence, you need to put a lot of effort into being grounded and calibrated - because often a confused audience won’t feel like a confused audience to you.
Ask questions! Especially ones that highlight the key ideas, eg “how to do easy thing X” or “what was the key idea in here?”
Do hand-polls - ask people to indicate their understanding by putting their hand up high if it’s clear, and low if it’s less clear. This is a good technique, because most people will actually do it, unlike “does anyone have any questions?” or “is this making sense to people?”
Seek feedback and iterate
Teaching is hard. You’re fundamentally trying to convey tacit knowledge, via lossy and low bandwidth communication channels, into an alien mind that you have very limited access to. The default state of the world is that you suck at this
The solution is to regularly ask for specific, actionable feedback and calibration, and to actually put meaningful effort into updating on this! Feedback is one of the main ways you can better understand the mind of a student.
Teaching 1-on-1
Practicing tutoring and explaining things one on one can often be more valuable! I think a great use of time for most students is to do tutoring - it’s pretty fun, you get paid decently, and you get way better at explaining ideas. And the ability to explain an idea clearly in a conversation is an amazingly applicable skill - I use this all the time in daily life.
The main difference is that it’s a lot easier to get them to be active, and it’s much easier to adapt the pace and difficulty well. Essentially, invert all of the ideas in my post on how to learn from conversation
The key technique is asking the student to paraphrase what you’ve said back to you
This forces them to be active, and to process information
It identifies errors, and helps you to correct them
Often you can then recognise th
It helps build a model of what’s going on in their mind
It helps you calibrate the difficulty and pacing
If you’re a tutor who doesn’t do get the student to this, I think you’re missing out on a major free win
This is also super effective when explaining ideas to friends, though can seem a bit rude
Get the student to tell you the key points/ideas in what you’ve said
Get the student to generate examples, especially typical examples
Here, understanding the mindset behind the question is even more important. You should always do this when they ask a question, especially if they seem dissatisfied with your answer.
Teaching Maths
It’s easy to neglect the tacit information - the intuitions, underlying concepts, motivations. This is terrible. One of the most important parts of teaching maths well is to convey this high-level overview
Every proof can be heavily compressed. Most proofs have some key ideas, followed by repeatedly doing the obvious thing. “Repeatedly doing the obvious thing” will inevitably be compressed in the student’s mind, so you should skip saying it at all, and just give the key ideas
Examples, especially motivating examples, are incredibly important. It’s really, really hard to learn a new concept without having a clear motivating example in mind.
Examples teach tacit knowledge well - they illustrate what you can and can’t do, and why you care about ideas
After learning rigour for a while, you’ll end up with post-formal intuitions, where you mostly ignore rigour and think intuitively, but can drop into rigour if need be. Most of the cognitive labour in maths is reaching this point, and a good teacher will try to give as much of the post-formal intuition as possible
Maths, especially pure maths, is often formalising an intuition. Probability is the formal study of uncertainty. Groups are the formal study of symmetries. Topology is the formal study of continuous deformations (things which don’t rip or glue). Pointing this out is vital
A good way to find these is to notice which questions the topic newly lets you answer. This is a great way to motivate things!
Diagrams are awesome
Often you begin being able to answer a type of question with a lot of tacit knowledge, and are expected to pick all this up with examples. Often 80% of this can be captured in an explicit algorithm - this is a great way to make tacit knowledge legible.
Teaching Applied Rationality/Life Advoce
These are much more about tacit knowledge than explicit, so these should be done in a workshop format with a big focus on exercises
Emphasise that everything is highly personal and subjective - all ideas should be adapted to your mind and your circumstances
Pairing people up works well for getting them to actually do the exercises!
Ideally, boil down the tacit knowledge to a rough algorithm, and alternate explaining steps and getting the students to practice them
The impact of the class is mostly students retaining key insights and mental habits - what the “time when I should apply this idea” feels like from the inside. This is the small fraction of information they’ll retain. Thus it’s your job to boil down the idea to these habits, say this explicitly, and structure the class to reinforce them
Much of the impact comes from the students taking action after the class - this is hard! You want to emphasise this, and minimise barriers
Give time for the students to generates lists of ways to apply the ideas, and how they’re relevant in their day to day life
Give time for them to set reminders for actions taken after the class
Make it feel actionable - it’s easy to think an idea is important, but for it to feel abstract. Eg, to think that prioritisation is a good idea, but to never get round to it.
Emphasise how the idea fits into everyday life and everyday problems
Give a lot of examples of how to use it - this can form connections like “oh! I never thought of using it for that”
This conveys the tacit knowledge of when to use the idea!
Emphasise relatability and importance - give examples of a bad situation where the technique wasn’t applied, and make it feel visceral and relatable
Conclusion
If you’re planning on teaching something in the future, I hope these thoughts were useful! But even if not, I think these skills transfer excellently to explaining things in everyday life. And that thinking about teaching can make you a much more effective learner.
I find that often, as a student, I can help the teacher be more effective by asking the right questions - asking them which information is the most important, checking that my understanding is correct by paraphrasing back, asking them for the motivations and higher-level picture. The feeling of “something not fitting into my knowledge graph well” can be made into a pretty visceral one. And realising the habits of students that hinder them from learning, like being passive instead of active, and not trying to do information compression themselves, can help me recognise when I fail to do those things!
