DiBeos

Today, I will share with you my favorite strategy for problem-solving.

Just think about it: When we sit down to solve a mathematical problem, especially one that is difficult we usually tend to find ourselves looking through our huge, mental world of logic, symmetries, definitions, theorems, and so on. But our brains have two ways of finding the solution to this problem. It’s two ways of thinking, actually: divergent thinking and convergent thinking. When you understand how they work, how they’re different, and how they cooperate with each other, it will really help you get insights when studying any area of mathematics. 

But before actually defining each of these ways of thinking, I want to tell you about a beautiful illustration of this duality so that you can feel how they work, and it’s actually the way many mathematical proofs are discovered.

When we look at a final version of a proof, it looks like an elegant piece of work. None of it is messy. But, if you ever have the chance to ask the mathematician how they came up with it, their story will probably contain a lot of messy notes and crazy ideas. These ideas usually come from leaps of speculation and beliefs in conjectures which turn out to be false.

The point is that the initial steps in any proof are full of painful constructions, and basically, all kinds of different mistakes. And then finally after sorting all of that mess out, mathematicians come up with a clean proof.

This kind of process is true not only for high level research, but pretty much in any learning process. My personal favorite strategy when I am learning something new is to explore all kinds of possible and random explanations. And only then do I start to eliminate theories that my judgment tells me are wrong, or are imprecise, or are just not very clear. And finally I create my own explanation for myself.

I hope you noticed a pattern here. First, in this whole process we diverge, or collect the data. And then, we converge, we clean up the data that we collected.

Divergent thinking is something broad. Think about an explosion of possible explanations for some kind of problem. When you’re thinking divergently, you need to get rid of the critical filters in your brain to take a look at ideas that may seem crazy, without any constraints. We also call it “brainstorming”. When mathematicians and physicists are at this stage, they might ask “What if the symmetry group were extended?” , “Can I perturb this structure?” , “What if I reframe this problem geometrically instead of algebraically?” , “What happens if I drop continuity?”. At this stage, you are not trying to be right, you’re trying to find new routes to explore, and study various interpretations and look at the same object through many kinds of lenses. So at this point, your goal should be to just explore, and be creative, and find new connections, with your mind all over the place. 

After that, you need to go in the direction of convergent thinking. It’s when you need to reduce and eliminate all of those unfruitful ideas from the brainstorming phase. Here you need to start defining your terms precisely, and maybe use  already known (and well-established) theorems. Now you have to check boundary conditions, eliminate contradictions, and turn your broad set of options into a single path that will take you forward. So essentially, convergent thinking is about deciding what stays and what has to be removed because it’s irrelevant. 

If you want to see a practical example of this technique in practice, watch the video above.

In mathematics education (although, this is especially true for modern schools), teachers overemphasize convergent thinking. So, students are trained to look for the “right” method quickly, to be able to spot it right away. They have to memorize formulas, and they have to eliminate “bad” ideas early on. But this approach discourages them from exploring and just makes them have a shallow understanding of the subject. 

True mathematical insight comes from following a bad idea far enough to see what exactly is wrong with it. And when you do so, you learn more than you would from just applying a set of techniques without ever questioning whether this is the best approach.