I want to discuss some misconceptions most people have about math, or math education. When I graduated from high school, I had the same misconception that most people have. The idea is that you have these math exercises, and then there's this black-and-white moment where you suddenly understand math. Or that there are these exercises, you follow a procedure, you get the correct answer, and then someone grades your solution and says, "Yes, this is correct."

But when you observe how a research mathematician actually practices math, it is so different from how you learn math at school. Or even how you do math in Russian math circles. They don't just give you exercises—they give you problems that require investigation. You may not solve them in an hour. You may need days, weeks, or even years to investigate a mathematical problem.

Problems, by contrast, do not come with algorithms attached. By their very nature, they require investigation, which is both an art and a science, demanding technical skill along with focus, tenacity, and inventiveness. Math circles teach students these skills, not with formal instruction, but by having them do math and observe others doing math. Students learn that a problem worth solving may require not minutes but possibly hours, days, or even years of effort. They work on some of the classic folklore problems and discover how these problems can help them investigate other problems. They learn how not to give up and how to turn errors or failures into opportunities. ~ Mathematics via Problems by Skopenkov

The interesting part is that there's always a new angle you can take on some mathematical issue. What math researchers really learn is how to make up definitions. They look at a mathematical structure and ask, "Okay, what's important about this structure, and what can I ignore?" By forming those definitions, they're able to come up with interesting questions. So it's much more a process of sense-making, and sense-making is open-ended. You're never done constructing frames. There are always new anchors you can use to construct, modify, or replace your frames.

That also reminds me of the paper When is a Symbol Symbolic? It's about the relationship students have to symbols, and how they have to go through this transition from enactive, to iconic, to symbolic in order to make sense of them. Whenever you encounter a new symbolic equation, you don't yet know what it means. You have to infer its meaning from context. You have to translate the symbol into something more concrete—a context where you can actually develop an image or an intuition for what the equation means. You make guesses, and then you gradually deepen your understanding of those concepts until you're able to use them in more complicated questions or higher levels of abstraction.

So this is a never-ending process. It's very different from, "Okay, follow these steps, get the right answer, and you're done." It's an open-ended investigation. An open-ended conversation. You're never done.

It took me a really long time to understand that. I had to spend a long time outside of school learning what math actually is—what mathematical research actually is. And I'm really glad I have an internet connection, because it gives me access to mathematicians who openly share their process. I'm really grateful that I get to see how they actually think.

And that's also why I'm not frustrated anymore when I don't understand some mathematical structure one hundred percent. Especially after reading data-frame theory, I was just like, "Duh." Frame fixation is what holds me back. You're never done understanding mathematics. Fixating on one frame only limits you. You have to build your understanding gradually.

It's totally normal that things don't make sense immediately. This isn't school math. That's just how the process works.