A lot of students who excel at math in high school end up pursuing computer science or engineering in college. Only a few continue down the road of pure mathematics. It’s not because they aren’t capable—many of them are brilliant. So what causes this drop-off? Why does advancing in pure math feel so much harder than expected?
To understand that, we need to think about how we learn, not just what we learn.
Math as a Game: The Defense Analogy
Imagine that learning is like playing a game. Each new concept is an enemy you need to defeat. In high school, the “defense stats” of math problems are relatively low. You can attack them with the standard moves—memorized formulas, pattern recognition, step-by-step procedures—and make steady progress.
But in pure math, the material gets denser, more abstract, and harder to penetrate. It’s like facing enemies with high armor: your usual attacks (rote learning or surface-level understanding) don’t do much damage anymore. At some point, unless you use special effects, you can’t even scratch the surface.
That’s the real shift. It’s not just that math gets harder—it demands a completely different style of play.
Special Effects: How to Deal Damage to Dense Concepts
These “special effects” aren’t magical powers—they’re skills and habits that help you make progress when the usual tools no longer work. Here are a few that matter most:
- Accepting temporary confusion
In pure math, it’s normal to feel lost at first. You can reread a definition three times and still feel like it’s written in another language. That’s okay. Being confused doesn’t mean you’re bad at it—it means you’re in the process of learning deeply. The trick is not to panic. Sit with the confusion. Wrestle with it. Often, understanding only arrives after you’ve spent time feeling unsure. - Rephrasing and teaching
If you can explain an idea in your own words—or better yet, teach it to someone else—you’re solidifying your understanding. Talking through a concept out loud or trying to write it clearly can expose gaps in your thinking and help you close them. It’s like testing your upgraded move in the game to see if it really works. - Making things concrete
Even the most abstract ideas are built from simpler pieces. Draw pictures. Play with toy examples. Trace back where definitions come from. One very underrated trick is writing your own notes—not just copying the textbook, but rephrasing the ideas in a way that makes sense to you. When you can translate dense material into your own internal language, you start to build real ownership over the ideas. - Being patient and persistent
In pure math, breakthroughs don’t come instantly. Some ideas take days—or weeks—to really click. You might feel like you’re wandering in the fog, but each step is building something. Sometimes, even making loose associations—like linking a theorem’s name to the name of a concept or an author—can act as scaffolding. It may not feel like much, but these small hooks help structure your understanding. Be patient. Stay engaged. Things do eventually fall into place.
Final Thoughts
Pure math isn’t just “harder math”—it’s math with higher armor, denser defenses, and fewer obvious entry points. It asks you to level up your approach. You can’t brute-force your way through with memorized techniques. But if you’re willing to develop new strategies—to sit with confusion, talk through ideas, make the abstract concrete—you’ll find that even the most intimidating ideas can be cracked.
And when they do crack, when the armor breaks and the idea finally lands—that’s when the game becomes really fun.
Stability of Learning 