I was recently thinking about defining things vs explaining them. When you're trying to understand something, I think it's easier to use a story like approach. "If you take the line making a circle (or circumference if you already know that one) and measure its length. Then you take the diameter…"

Once you know something, it's easier to shift that knowledge into a definition. "the ratio of a circle's diameter to its circumference." It's more concise and is more usable to tell subsequent stories and make subsequent definitions.

I think this might be a common mistake when explaining things. When we understand something, it's kind of stored as a stack of definitions. "Ratio between diameter and circumference" assumes the learner knows ratio, diameter and circumference. If they don't, the tendency is to build up their definitions. "Ratio is __. circumference is __. pi is__…" But definitions take a bit more effort to plug into your brain.

I think there's a programming paradigm analogy though I'm not quite fluent enough in the concepts to be confident about it. To venture.. a declarative programming's "Case Where" expression is more difficult to grasp than procedural "if then" if you're not familiar with conditionals in general.

I think this is why encyclopedias can be hard to understand. They kind of define things.

The best way I've seen pi explained visually is pi-unrolled.gif: https://commons.m.wikimedia.org/wiki/File:Pi-unrolled-720.gi...

That is a great visualization but one something that bothered me about it was that you're constrained to watch it unravel at it's own playback speed.

Another approach: http://betterexplained.com/articles/prehistoric-calculus-dis...

This would be a great resource for an advanced second grader.

There are just definitons. Not interesting at all.

It would indeed be more interesting to explain why pi is that odd number and not just say 3. But that is probably not so easy.