When I was a child, I always found math easy. Fractions, calculus, formulas—they were always visual. Fractions were pizzas; equations were deformations of the p

When I was a child, I always found math easy. Fractions, calculus, formulas—they were always visual. Fractions were pizzas; equations were deformations of the plane; integration was tracing out a shape using a slope. So I'm sympathetic to the author's desire for better visualization and teaching tools. But when I reached college, I became frustrated with math. It just wasn't easy anymore, the way programming was: I could pick up a programming book, read it in a weekend, and understand it. But when I tried to read an advanced math text, I became lost after 10 pages. Eventually, I figured out what had happened: The information density of college-level math texts is insane. Even if you're bright and talented, it may take you a day to understand a single page. And there's no substitute for working carefully, finding concrete examples, and slowly building a deep understanding. Here's an example that involves programming. Once upon a time, I needed to understand monads, in hope of finding a better way to represent Bayesian probabilities. I started with the monad laws, a handful of equations relating unit, map, and join. I read countless monad tutorials, and dozens of papers. I read every silly example of how monads are like containers, space suits, C++ templates, and who knows what else. I wrote little libraries. I learned category theory. I wrote a monad tutorial. I eventually wrote a paper explaining a whole family of probability monads: http://www.randomhacks.net/darcs/probability-monads/probability-monads.pdf http://www.randomhacks.net/darcs/probability-monads/probabil... And then one day, I thought about the monad laws again. I realized, "Hey, that's it. That's all. Just unit, map, join, and a handful of equations. Anything which quacks like a monad, is a monad. How did I ever think this was complicated?" But when I look at the monad laws today, there's this huge structure of connections in my head. All that work, just to grasp something so simple, and so easy. So I'm all for building better visualizations, and for helping people to understand math intuitively. That's an important step along the path. But math doesn't stop at an intuitive understanding. When you really understand it, the equations will suddenly be easy, and everything will fit together. And then you'll encounter the miracle of math: Your deep understanding will become the raw material for the next level. Counting prepares you for addition, addition prepares you for multiplication, basic arithmetic for algebra, algebra for calculus, and so on. And someday, I hope that my rudimentary understanding of category theory will prepare me to understand why adjoint functors are interesting.