We've already written (twice actually: here and here) about making the jump from "doing problem sets" — which is just a symbol for undergrad and early graduate-school math — to "doing research". In those earlier posts we discussed the phenomenon of being face-to-face with mathematicians who are much, much better than you (and why that's okay), as well as the importance of having "something well-defined to do when you get out of bed in the morning".
In this post we're going to discuss another major experience that will likely be somewhat foreign during this transition — the phenomenon of not making linear progress. Now, once you're not making linear progress, only two choices remain. Either you're going to make no progress at all, or you're going to make non-linear progress. This article assumes that you don't want the former. However, the former and the latter often look and feel very similar. This article will therefore (hopefully) give some insight on how to distinguish between the two, how to obtain the latter, and how to cope with some of the difficulties that the linear-to-non-linear transition introduces.
(Small-Bite) Linear Progress
Prior to college your main experience with "doing math" is "doing homework". This typically looks like learning something very specific in class one day, and then going home that same night and doing 20-40 practice problems on exactly that thing. Every day that goes by, you learn one or two more tangible skills, and you've got 1-3 pages of handwritten practice problems with boxed (and hopefully correct) answers to prove it.
This is linear progress — every day you get a little bit better, and this progress is measurable by things like "number of problems done" or "number of sections in a textbook mastered". Moreover, this progress is (for the most part) pretty enjoyable. You rack your brain a little and then take a little step forward. The topics we learn at this stage in our mathematical careers — and more importantly their pedagogical techniques — are so well understood that we (our species) know pretty well how to dole it out in digestible, bite-sized chunks. Let's therefore call this "small-bite linear progress".
(Big-Bite) Linear Progress
Then comes college and "problem sets" — we've already written (here) about this transition. The way it relates to our current discussion, though, is that this is not a jump from linear to non-linear progress. Rather, it's a jump from "small-bite" linear progress to "big-bite" linear progress.
Instead of sitting in a 45-minute or 1-hour class every day learning the next section in a textbook, you'll sit through 1-3 dense lectures per week. Each lecture will cover many sections of a textbook and maybe even include some non-textbook material. Your homework is not just doing the assigned problem sets, but also spending some hours going back through your lecture notes and decoding the dense material that's been dished out. And then of course there are the problem sets themselves. Instead of 20-40 straightforward practice problems, they'll likely be 2-10 intense problems — each one drawing on several different skills and stretching you in unique ways.
This is a big transition for sure, and one that requires a whole new mindset (hence the article linked above), but it is still linear progress. The material that you're supposed to learn is set for you, and the problems you're supposed to solve are well-defined and assigned. The only difference is that you're supposed to figure more of it out yourself — it's not all put on a silver platter in nice little cut-up bites for you. Hence why we want to call this type of progress "big-bite linear progress".
The Next Transition: Non-Linear, or No Progress
And then comes the next — and final — transition. Research.
This transition requires a quantum leap in mindset shift because not only is progress not going to be linear any more, it won't be guaranteed to exist at all.
At first, research problems will likely be given to you by a research advisor of some kind. But here, the word "problem" is meant to be taken loosely. Much, much more often than not, the problems are not given as: "assume this, that, and this other thing, and then go prove this." And they're certainly not going to be given as "use these two techniques to calculate the following value". If a research problem was that well understood, then the advisor would either just do it themselves or give it to some undergraduate intern over the summer.
Instead, problems will come to you looking something more like this: "I proved this lemma three years ago, here's a link to the paper. Can you try generalizing it a little?". Or maybe something like: "Here's a paper I wrote showing how this one technique can solve this class of problems. It seems to apply to certain subclasses of those problems less efficiently than others. Can you try to make sense of that distinction?"
Notice something new and different about those problems? They're problems that the advisor themselves don't know the answer to, or even how to properly and precisely state them. Now, what makes an advisor a good advisor is that their immense experience gives them an intuition about problems that allows them to be reasonably confident that something interesting will arise from them, but the precise "what" and "how" is for you to figure out.
This is the context in which non-linear progress arises. There isn't a set of textbooks that you know you have to read in order to get to your desired destination. You don't really even know with certainty what your desired destination is!
So what do you do? The way we see it, there's three things you need to do (in various combinations, at various times). First, you have to keep learning things. Second, you have to keep trying things. And third, you have to keep talking to your advisor (and other experienced researchers). All of these have their own versions of non-linear progress, but we'll focus on the first two in the remainder of this post.
Non-Linear Progress Type 1: Learning
You've found, or been given, a research project and you're just starting out on it. The first thing to do is learn as much as you can about it, of course. But what that looks like in practice is very different from the types of learning you've likely done before.
You might be tempted to start reading every paper in the bibliography of your advisor's paper, and then every paper in the bibliographies of all of those papers, and so on, until eventually you're rebuilding the whole field from Euclid's "Elements". We recommend against that.
You might also be tempted to read every textbook written in the last 20 years about any subject remotely close to the subjects that arise in the paper. We also recommend against that. For some concrete tips on what to do — and not just what not to do — take a look at some of those articles we linked to above.
What we want to emphasize in this article is the fact that this process of figuring out what it is you need to learn, and then actually learning it, can take weeks or months. And during many of those days, you're going to feel like you're not making progress at all. This, friend, is what non-linear progress feels like — it feels like no progress at all 90-98% of the time.
But then you find the right paper or the right section of a textbook that gives you a whole new perspective on your problem. Or maybe that paper or textbook section simply points you in a well-defined direction, so that your next 2-3 weeks of learning are rapid and exciting.
At this point, there are two vitally important things (in our opinion) to do to help you both stay sane, as well as continue to make non-linear progress in the future. First, for sanity's sake, you need to take some time to appreciate the fact that those initial weeks and months of floundering in the dark were not wasted days. Rather, they were bouncing you around closer and closer to the thing you needed. The only way to find a needle in a haystack is to look through the entire haystack.
Well, unless the needle is metallic and you have a magnet. That is to say, the second thing you need to do is reflect on that journey and honestly assess whether you could have found that needle a little faster. Did you spend too much time reading things "at a high level" and not really learning much from them? Did you not spend enough time asking other people questions?
Needles in haystacks will always be hard to find, but the difference between "no progress" and "non-linear" progress lies in staying sane and honestly assessing what you're doing and if/how you could be doing it better. And this brings us to the second type of non-linear progress:
Non-Linear Progress Type 2: Doing
You've learned enough about your problem now and you've got yourself a nice, concrete goal that you're trying to achieve. You've turned a vague wisp of a whisper of an idea into a precise statement of something to prove or calculate, and now the fun really begins. Namely, you have to actually figure out how to do it. This, too, will be plagued with non-linear progress (and hopefully not no-progress).
We haven't done any systematic studies on this, but we'd be willing to bet that 95-99.9% of all things tried in research, fail. This means several things. First and foremost, it means that you're going to spend a lot of time failing, which means you're going to spend a lot of time feeling like you're not making any progress. Second, it means that when progress does come, it often comes fast. We (at Coho) still marvel at how it seems like all that "pent up failure" almost always leads to an explosion of productivity. And third, it's both vitally important and almost entirely your own responsibility to constantly try finding ways of decreasing your failure rate from, say 98% to even just 97.5%. Those tiny adjustments often mean the difference between no progress and non-linear progress.
One of our favorite concrete techniques for assessing whether we're making no progress or non-linear progress is the following. When you're about to try something, can you identify what you would learn even if the thing you're about to try fails? If you can't answer that question, then it's probably a no-progress thing that you're about to do. Most of the time, in trying to answer that question, a no-progress thing can turn into a non-linear progress thing with some surprisingly small tweaks. Which brings us to...
The Takeaway
We've written a lot here, but hopefully have drawn a somewhat simple picture, so long as we can accept a couple basic truths. They are:
- In research, linear progress does not exist.
- This means we're forced to accept either no progress, or non-linear progress.
- We probably want the latter.
- To obtain the latter, we must
- First accept the fact that most things fail in research
- Develop systems that ensure we're learning from our failures.
With these systems in place, combined (obviously) with hard work and discipline, one can continue making progress in research and actually enjoy all the (brief but dramatic) ups and the (long, drawn-out) downs.
If you have other ways of optimizing your hit rate for "non-linear progress", please reach out and let us know, we'd love to hear about them!