There comes a time in everyone's life (well, everyone who does STEM things beyond the required high school courses) where you go from "homework" to "problem sets". This is a big transition in one's life, and it's one that many students are not properly prepared for. We therefore want to give a little insight into what this looks like.
What's The Difference?
For much of one's early years in math, you get assigned "homework". Homework looks like the following — you learned how to factor polynomials in class today, and now you have to go home and factor 20 polynomials before tomorrow's class. For example, a homework assignment might be something like "problems 1-35 odd" or "problems 1-75 every other odd". Each problem is another polynomial that needs factoring, the only difference between them being the coefficients. You know what you need to do for each, and now you just need to do that over and over again.
If you understood the factoring lesson and are reasonably good at your multiplication tables, each polynomial will take a minute or two to factor, and so you'll be done with your math homework in around 30 minutes. Nice.
But then comes the day where you start getting assigned "problem sets". This might be in an advanced high school class (for me, it was AP Physics in high school), or a college course. Instead of having to solve 30 problems a day, you now have to solve 5-10 problems per week. Sounds nice, right?
Be Careful What You Wish For
As anyone who has been faced with "problem sets" will tell you, the world of problem sets is not easier than that of homework. Your total number of problems might have gone down, but the size — and nature — of those problems have changed, and in a big way.
The main goal of "homework" is to help you build muscle memory, where the "muscle" is your brain. Homework hones in on a very specific skill — like factoring a polynomial — and has you repeat exactly that process many times over. This is incredibly important. You need these skills to be ingrained in your subconscious so that when you're out in "the wild," you can call upon those skills without much thought.
If you're riding a bicycle through the crazy streets of New York City, you need to be thinking about all of the unique challenges that the current state of the world is sending you (cabs, potholes, pedestrians, etc.) These challenges are unique to each bike ride. What you can't be wasting any precious brain power on are things that are not unique to the current problem — how to switch gears, maintain balance, and brake. These skills need to be pure muscle memory before you hit the streets of a major city.
While homework is there to help you build muscle memory, problem sets are there to help us...think.
You learn concepts — not recipes — in class and then you're given a number of problems that can be solved using those concepts. It is often not clear how those concepts can be applied to solve those problems, and this requires you to think — a lot. And...
Thinking Is Hard
When one first transitions out of "homework" and into "problem sets," the first thing they typically notice is just how difficult and deeply unnerving thinking is. Staring at a problem that you don't know how to solve is very unsettling. It's not just the technical challenge of "how do I solve this?" — there's the emotional challenge of having thoughts like "can I do this?", "am I smart enough?", "am I good enough?" constantly popping up in your head.
This is why you only get, like, 10 problems a week ;)
All too often this is the point at which people start disliking math. Often, the transition from homework to problem sets is synonymous with the transition into "proof-based math," and so people tend to think that they "don't like proofs." Of course, that's sometimes the case, but I'd be willing to bet that it's also a lack of preparation for the abrupt shift in what "math homework" looks like.
And this is why we wanted to write this article — to give (some of) that preparation. Simply knowing that this transition will occur, and that we all face the same challenges with it, can help us handle it.
What Not To Do
We have witnessed many students make this transition and by far the most common reaction to it is to look up solutions. A typical workflow looks like the following: look at a problem, decide within 30 seconds that it's too hard, google a solution, see that the solution makes sense (sometimes people don't even do this, which is even worse!), write down that solution, and repeat until the problem set is done.
We will have much, much more to write about why this workflow is deceptively terrible, and we'll offer some better approaches, but for now we'll just say the following: try not to work like this.
These 10 problems per week that you have to solve are supposed to be hard. They're supposed to make you tremble in your boots. You're going to have doubts about your ability to solve these problems, or to solve any problems for that matter. These are not fun feelings to have, and they'll make you want to immediately reach for your phone to google a solution. RESIST THAT TEMPTATION FOR AS LONG AS POSSIBLE.
Allocate at least one hour for each problem, even if that's a whole hour of staring at a blank piece of paper. One rule: no phones allowed! Sit and think. Meditate for all I care (though you might as well think about the problem).
Sit with the discomfort, get to know it, make friends with it. Give it a name if you have to. Realize that it is precisely this discomfort that is going to make you better, stronger, and smarter. Try something. Anything. Write down something — anything — that you know about the problem. Try to turn the problem into an easier problem.
Even if you eventually do look up a solution, the longer you've wrestled with the problem, the more you'll appreciate why that solution is the way it is, and the longer that solution will be burned into your memory.
Welcome To The Streets
We'll write many more articles about how to actually make this transition and some specific problem-solving approaches that we've found helpful, but this article is not about that. This article is meant to simply expose you to the fact that this transition will occur. For many reading this, that transition may have already occurred without you explicitly knowing it.
Once you've made this transition, you're officially in the crazy streets of New York City. The training wheels are off and, like any coming-of-age story, there will be moments (many of them, at first) where you wish you could go back. Solving 30 problems a day seems like heaven compared to having to think so friggin' much.
However, after a period of pain, struggle, and doubt, something amazing happens. You become friends with frustration. You embrace the journey from ignorance to understanding, and you become addicted to it. You no longer fear the next problem set, but rather look forward to seeing which problems you'll be banishing to the Realm Of The Solved this week.
That said, you'll never stop loving that moment where you finish that f***ing problem set every week.