The Uncomfortable Truth
I've been coaching math olympiad students for eight years. In that time I've taught hundreds of students, from complete beginners to national team members. The single biggest predictor of how quickly a student improves is not their raw intelligence — it's how precisely they can articulate what they don't understand.
The students who say "I don't get problem 3" plateau quickly. The students who say "I understand why the construction works for n=4, but I don't see why the same argument extends to odd n" — those students improve at a remarkable rate.
Clarity of Thought ≠ Clarity of Expression (But They're Linked)
There's a common assumption that smart people just need to "communicate better" — as if thinking and expressing are separate activities, where you think clearly first and then translate.
I don't believe this. In my experience, vague expression is almost always a symptom of vague thinking. When a student can't explain their solution clearly, 80% of the time they don't actually understand it as well as they think they do.
The fix isn't communication training. The fix is thinking more carefully.
How I Apply This to Software
When I write a code review, I aim to do the same thing. Instead of "this function is complex", I write: "this function has four responsibilities — parsing, validation, transformation, and IO — which makes it hard to test any one of them in isolation."
When I write a technical proposal, I lead with the problem statement, not the solution. This forces me to have the problem statement clear in my head before I start advocating for a particular approach.
A Practical Exercise
Next time you're stuck on a problem — in code, in math, in life — try this: write a one-paragraph explanation of what you've tried and where you're stuck, as if explaining to someone who is intelligent but doesn't have context.
You'll be surprised how often writing that paragraph solves the problem. Rubber duck debugging works because the act of articulating forces you to think precisely.
The Meta-Lesson
Teaching is the most effective form of learning I've found. When you have to explain something clearly enough for a 16-year-old to understand, you discover exactly where your own understanding is fuzzy.
If you want to get better at anything, find someone to teach it to.