Let's talk about the importance of doing exercises. And if it's not clear, we're not talking about physical exercise (though that's important too, but just not our expertise). Rather, we're talking about mathematical exercises — homework problems, problem set problems, or completely unassigned problems that you do and/or come up with yourself while reading a textbook or paper.
This is (hopefully) not the first time you're hearing this. You've likely heard many teachers or professors describe how important it is do exercises, some more passionately than others. Here, we're going to re-emphasize that yet again but we're also going to give some insight on the matter that we have not seen explicitly mentioned elsewhere. Namely, we're going to clarify exactly what it means to do exercises, and we're also going to describe the awesome phenomenon of how doing more exercises actually makes it so that you have to do fewer exercises.
As is often the case, let's use an analogy to set the stage.
Learning Basketball Plays
We're going to use the act of learning plays in basketball as a powerful analogy throughout this post. If you don't play basketball, no problem, any team sport will suffice. If you don't play any team sport, no problem, any sport will suffice. If you don't play any sport, no problem, having had to learn any physical skill at all (e.g. learning how to eat with a fork or tie your shoes) will suffice. But for concreteness, we'll use basketball.
A "play" in basketball is a specific orchestration of the 5 players on the court to perform a specific set of tasks, in a specific order, with specific fall-back plans based on how things develop. For example, Player 1 passes to Player 2, then P1 goes and sets a screen for P3 (doesn't matter if you don't know what "setting a screen" means) who cuts to the basket. If P3 is open, P2 passes to P3, and if not, P2 looks to see if P4 is open. This is because while this action has been taking place between players P1 through P3, player P5 has been setting a screen for P4. Let's assume this is the end of the play, and let's call this PlayX.
There are a few different ways that you can learn PlayX. One way is to watch your coach draw it up on the blackboard (call this Way1). Another way is to watch 5 players run through it on the court (call this Way2). Another way is to run through it yourself (on the court) from the perspective of, say, Player2 (call this Way3). And another way is to run through it yourself several times, at least once from the perspective of EACH of the players P1 through P5 (call this Way4).
Question for you: if it were the end of the fourth quarter of your state championship game with 5,000 people in the crowd and you were brought into the game to run PlayX in order to hopefully secure a victory for your team, which of the above "ways of learning PlayX" will you wish you had performed?
Doing Exercises
Suppose you're reading a paper or a textbook and an exercise pops up. In a textbook, this might mean you've reached the "problems" portion at the end of each section, or a couple exercises are interspersed in the main thread of the book. In a paper, the authors might be generous enough to set a couple example problems along the way, or the "exercise" might be a more subtle statement like "we then computed that...". The "exercise" in this case would be to redo the calculation (if computationally feasible).
There are a few ways you can "do" the given exercise, which we'll call ProblemX.
Way1: You could pause for a moment, think about it, say to yourself "yeah I see how to do that", and consider yourself done.
We have called this "Way1" because it is exactly analogous to the "Way1" above — watching someone draw up a play on a blackboard. You might get the rough idea of the play, but all the practical details, like how far away players really feel on the court and how the timing will actually play out (really hard to properly reflect "time" on a blackboard) are going to be completely unknown.
Similarly, in the case of math, you might "see how to do it" while actually a) missing some subtle detail that disallows a certain manipulation, b) not fully appreciating how computationally intense a certain aspect of it might be, c) only seeing how to do roughly 95% of the problem, and there's trouble lurking in the last 5%, or d) all of the above.
Way2: You could look up a solution to the problem.
This is again exactly analogous to Way2 in the basketball setting. You've now seen some of those practical details, but you haven't done them yourself. You haven't felt them.
Way3: You do part of the problem, meaning you've actually put pen(cil) to paper and/or coded some stuff up, and then you look up a solution for the rest of it.
Having actually done part of the problem, you've now actually felt what some of those practical difficulties are. However, having looked the rest of it up, you only "know the play" from a limited perspective — hence why this is Way3, analogous to only "knowing the play" from P2's perspective.
Way4: You put pen to paper or fingers to keyboard and you solve the whole damn thing yourself.
This is Way4 because now you know the play. You know the ins and outs, the thorny details. You know why various parts of it need to be the way they are, because of how they interact with other parts. You know that you need to bring in skills that are not specifically crafted for this play, but are necessary regardless.
Question for you: If you were about to walk into a final exam — one that will determine 85% of your grade for the class — and you knew that a problem closely related to ProblemX was going to show up on it, which of the 4 ways above will you wish you had used?
Practicalities
There are some obvious practical issues with wanting to use Way4 for every exercise you ever want to do in your life, and the largest such issue is the finiteness of time that we all have in our lives. To combat this, one has to take a balanced approach. However, early in one's career, we highly recommend a very heavy bias towards Way4, due to the wonderful phenomenon that we'll refer to as the "exponential backoff" of problems...
Exponential Back-Off
Suppose we're back to basketball and trying to teach PlayX to a specific team. We are also afflicted by the curse of finite time, so we want to optimize our approach. What method (i.e., "Way") should we use?
Answer: it depends.
Are we teaching a bunch of 12-year-olds? If so, we're probably stuck with Way4. We'd rather have a team actually know a small handful of plays than a team that "sort of knows" a larger collection of plays. "Sort of knowing" a play is as good as "not knowing" a play when the game comes around, so any method other than Way4 would be a complete waste.
Are we teaching an NBA All-Star team with High-Basketball-IQ players? Imagine a 5-player squad of LeBron James, Chris Paul, Luca Doncic, Nikola Jokic, and Tim Duncan (doesn't matter if you don't know who any of these players are). These are 5 of the smartest basketball players to ever play the game. We could probably use Way1 for 20 different plays — i.e., simply "draw up" 20 different plays on a piece of paper — and all 5 of these guys would remember and execute all of them flawlessly, on their first attempt, likely even adding their own personal flourishes, all perfectly in-sync with each other.
But it's important to understand why this is the case. The 5-man All-Star roster above is comprised of individuals who got that smart because they've used Way4 so friggin' many times in their lives. They have seen and run through so many plays, from so many different perspectives, with so many different added challenges, that they now have a large personal database in their minds of all the different issues, details, timings, and subtleties that might arise. Importantly, they did not get that way by simply staring at blackboards.
Similarly, Terrence Tao or Ed Witten could probably read an advanced graduate textbook on a totally new field and deeply understand all the subtleties and nuances involved while only putting pen to paper once or twice throughout (if at all). The harder challenge would be for us to find a textbook on a field that's actually new to either of those people, much like it'd be hard for us to draw up any plays that LeBron or Jokic haven't already seen and run through 100 times.
So, as you're starting out in your mathematical journey, it's vitally important to use Way4 as often as humanly possible. This is you building your personal database. These are your battle scars, the "experience" that you'll be able to draw on for years.
As an undergraduate you might then need to do your problem set problems as well as several while you're studying, often employing Way4 but sometimes sprinkling in Ways 1-3 just for exposure and/or checking your understanding.
Then, as a graduate student maybe you only have to resort to Way4 for a problem here or there when you can feel yourself losing grip with the main thread. These couple problems will then right the ship for you in many ways and you can read several more sections before having to check back in with pen and paper.
Then, as a researcher you'll likely have to bust out Way4 for very few "exercises", at least in your own field — you'll be on to large-scale research-level problems and you'll be an expert. When you want to explore new fields though, you'll probably be back in the "grad student frequency" of Way4 problems, and that's fine.
And once you're a master in several fields, you'll very rarely have to resort to Way4 for exercises, and will be able to pick up a new field, read a textbook or two, move on to reading papers, and start diving straight into research, almost without ever having to do "exercises" (that said, exercises are always fun!). This is a level that very few people ever get to, including us here at Coho, so please do not rush into this level. Thinking you're at this level before you are can be disastrous, and many people create very successful mathematical careers for themselves without ever getting to this level. This is the Terrence Tao and Ed Witten level — it's the LeBron and Jokic level — and there are plenty of very successful NBA players not named LeBron who still have to actually run through plays from time to time in order to understand them.
Mathematical Sophistication
All of this business with "exponentially backing off" the amount you have to use Way4 throughout your career is almost completely synonymous with the development of one's "mathematical sophistication". This "sophistication" is a term used often but is difficult to define. It's the type of thing you know when you see but you don't know when you try to define it. We'll have much more to say about this elusive quality in the future, but just wanted to drop this nugget here to say that all of what we've discussed above is closely related to this airy substance.