Learning Math: What Worked, What Didn't

This post was polished using Claude.

Like everyone, I started learning math in school. I wasn't bad at it, but I was no prodigy either. I decided my path lay in biology, and only returned to math in 2016. You can read about that journey in my previous post. Today I want to write about the how of learning math: what worked, what didn't, and what I wish I'd known earlier.

I think there is something I'd call a mathematical gaze, a style of thinking that some people access easily and others don't. Everything in learning math, at its core, revolves around developing this gaze. Another word for it is intuition.

What is mathematical intuition, exactly? Does it have anything to do with the real world? For a long time I was convinced that solving enough problems would develop it automatically. The truth is more complicated. If you're average, or below average like me, problem-solving alone isn't enough. You need a teacher who already has intuition and can show you how to think like them. That transmission from one mind to another is the only reliable way to generalise beyond the specific equations you've been drilling.

A related insight came to me while thinking about mathematical modelling. People in applied mathematics spend a great deal of time building models of real-world phenomena: abstracting out the parts they care about and expressing relationships between them. Finding the right representations is how they define success. To do that well, you need a solid grasp of the objects mathematics deals with and how they relate, and for that, there is no substitute for practice.

One trick I've found surprisingly powerful: asking which seemingly different problems are actually the same? The right mapping can turn an "impossible" problem into an easy one in seconds.

I also used to binge-watch advanced math lectures on YouTube to "build intuition." You might laugh, but I think it actually works. I'd go in expecting to understand less than 1% of what was being said. Even so, watching people with genuine mathematical intuition think out loud gave me something: a faint but real sense of how math is done, not just the pattern-matching of homework problems, but the branching, exploratory quality of real mathematical thought.

What I've found, in the end, is that for a determined but low-aptitude learner, there is a real path to mathematical understanding, intuition included. It takes only two things: practice, and a teacher who genuinely has the gaze.