Notes On Learning Hard Things

Reflections on the art of studying physics

The number one most read page on my site is a guide to learning physics.

This is a curious thing. Physics has a reputation as something hard to learn. I think – and in truth, this is only speculation – that people see the equations and numbers and weird diagrams of multidimensional space and conclude that a great deal of effort must be invested to make any kind of sense from them.

They may be right. But clearly there is at least some appetite amongst my readers to undertake this difficult task anyway. I thought, therefore, that I’d try to give some advice to those who are thinking about doing so. And though what follows is mostly about physics, I believe it generalises to many hard topics.

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Foundations

1. You don’t run a marathon by getting up one day and running twenty-six miles. No one can do that. And you can’t learn quantum theory by sitting down one day with a copy of Advanced Quantum Mechanics and working through it page by page, either.

2. The order in which you do things matters. Get up and run a mile before you start thinking about twenty-six. Build a base, so that you are running twenty to thirty miles a week. Start going for longer runs – five miles, eight, ten, and so on, until you are running fifteen or twenty at a time. If you do this, you’ll find your sense of distance subtly shifts. In the beginning a ten mile run seems like a lot – and, in fairness, it is – but once you are able to run six or seven miles, ten seems much more possible. And once you get beyond it, and start running fifteen or twenty, suddenly ten miles feels rather easy after all.

3. The order in which you do things matters. Build a foundation of mathematics before you leap into the deeper theories of physics. You’ll need to understand calculus before you can grasp electromagnetism; you’ll need to know algebra before you can do calculus.

4. Part of the challenge is figuring out the right order. As a beginner, how do you know what you need to study quantum mechanics? How much algebra do you need before you can move on to calculus? Fortunately, you are not the first person to tackle this problem. There are guides out there. Universities structure learning to progressively lay down the knowledge you need to go further, and many make this structure publicly available. Textbooks will often tell you the level of knowledge you need to take them on – they might not always be right in this assessment, but if a book says you should have taken a course in classical mechanics before reading it, then find a textbook that covers classical mechanics. And if all else fails, and you run into material you simply don’t understand, then admit this, step back, and seek out the foundation you are missing. Conversely, if things are too easy, then move on to something harder.

5. Breadth is often better than depth. Mathematics is the foundation of physics, but you don’t need to be an expert mathematician to study physics. Instead you need to know enough to know where the tools are, and where to look up more when you need it. You need to be aware that (e.g.) matrices are a thing, even if you don’t remember every detail about how they work. And you need to know where to go to find that detail if you need it.

6. Don’t waste time memorising things. Physics is not a subject where you need a great memory or need to recall facts and equations in an instant. Most things can be looked up. You need to build something like a map in your head – you need to know how one thing relates to another, even if you don’t recall the precise relationship. Again, you need to know enough to know where to find more.

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Study

7. Textbooks are the best and quickest way to build theory. Yes, you can (and probably should) watch lectures on YouTube or read blog posts and articles, but these are a complement to good textbooks. When done well, a book takes the reader through a subject in a logical way, presents information clearly, and is more complete than even the most thorough lecture course. Good textbooks, of course, can be hard to find. But there are resources out there that aim to solve that problem too. Never fear abandoning a textbook that simply isn’t good or doesn’t work for you. There are plenty more.
   

8. Read, read, then read again. The first time you read a chapter of a textbook you will not understand everything. That is okay. Read until you reach the end or you reach a point at which you don’t understand anything any more. Then read it again. This time pay attention to the places you get stuck. Make an effort to understand what is happening. Look things up; go back to your foundations if necessary. The first pass through gives you an outline. The second gives you understanding. The third puts that understanding in context.

9. Theory by itself is not enough. It is one thing to read and understand, it is another to put that understanding into practice. So you need to solve problems. You need to use your knowledge in the real world, and put it to work without the guidelines of a textbook. This can be hard. It can feel uncomfortable. If it does, that is a good sign – learning means pushing beyond the boundary of what you already know into the region you don’t. And most of us feel a natural resistance to stepping away from safety and comfort.

10. Where to find problems? A good textbook will contain problem sets related to the subject you are reading about. If it doesn’t, try related textbooks or university courses. Try re-deriving the proofs shown in the book without looking at them – that is, by working them out on your own.

11. Work alone at first. When I first started studying I used to do the opposite: the physics students would gather and work together as one big group on a problem set. An answer would emerge, and everyone would present more or less the same solution to the professors. But this was not a serious way to learn. Later, I started working alone. Instead of drawing on the wisdom of the group, I would lock my door and force myself to think about the problem and the solution. This was hard – it took me much longer to find the answers – and it was sometimes frustrating. But I learned more this way. It forced me to go back and really understand the root of the problem. And sometimes I would find a solution that no one else had.

12. Don’t be tempted to ask ChatGPT for help, or to Google your way around a tough problem. Stick with it. And don’t look at the answer key for “hints”.

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Feedback and Shortcuts

13. It is often tempting to think a list of textbooks and access to the Internet is enough to learn something. But it isn’t. Think about a language. You study the book and do the exercises. But how do you speak in practice? If you go to a bakery in France and start speaking textbook French, then two things will happen. One, your pronunciation will certainly be wrong, or at least mark you out as someone who learned from a book or a tape. And two, you won’t speak like a native, and you’ll get confused when the baker throws back some perfectly common but technically incorrect expression at you. So the book alone is not enough. You have to take your skill into the real world and use it among other people.

14. After study, you need feedback. People who purely work alone tend to drift into building elaborate theories and models that fall apart as soon as they are exposed to an outside view. To avoid this, you need to expose your work to others, and you need to do this often.

15. In practice, this means finding others to study with you. Universities offer many ways to do this. Students attend lectures as a group and move through the material at the same pace. Seminars offer a way to talk and discuss ideas. Tutorials give feedback in small groups. If you are studying by yourself this will be harder to do. The best way is to find a committed friend to study with you. But there are also many online communities. Tools like ChatGPT may be able to help, but they are often sycophantic and unwilling to give you the hard feedback you will sometimes need.

16. You need feedback to catch errors. But you also need it to gather tacit knowledge, the things that everyone knows, but no one writes down. The purpose of a degree in physics is not merely to teach you a set of equations and laws that give you the power to predict the future, but also to teach you how to think like a physicist. And this is not something obvious – there is no course on this. It is instead something you pick up by osmosis, a kind of tacit understanding that spreads by human interaction.

17. Studies of chess grandmasters have found they rely on intuition more than deliberate reasoning. How do you build intuition? In chess you do it by playing a lot of games and studying a lot of past games. Essentially, you need experience, but you need the kind of experience in which you are repeatedly exposed to similar problems and learn the common patterns that link them. Same in physics. You need to do enough exercises that you start to learn the landscape and build intuition. In time you’ll realise that many problems are similar to one another or can be tackled with a similar framework, even when they initially look completely different. The only way to build this is by solving a lot of problems.

18. Look for shortcuts. Physics is full of them. Some are mathematical tricks; others are about framing problems in the right way. Indeed, a lot of problems that at first look almost impossible can be rendered simple by switching to a different frame of reference. For some reason, though, I have found few good lists of all the shortcuts and tricks out there. So keep track of the shortcuts you encounter along the way, and never underestimate how useful they can prove.