If my toilet fills up to the brim because it is clogged, how long do I have to wait/how far down does the water have to go before I can put the plunger in without making it overflow?

(The results of a miscalculation are much more disastrous in this instance.)

That’s why I’m also a little suspicious when someone continually demands context or application. That makes it seem as if students can’t really enjoy the mathematics for its own sake. Now, I’m fine with context and application. I don’t mean to say that we shouldn’t seek them out. Instead I mean to say that for the most part they’re not necessary.

We talked over here about “organizing principles” for a math class. I’d say “make math real world” is as self-defeating an organizing principle as one can find. “Prioritize perplexity,” on the other hand, lets us chase down curious mathematics wherever it lives, either in the world outside the classroom or in the world of numbers and shapes.

While it’s clear to me you prize pure mathematics, it’s less clear to me how you create the experiences around it. Humans like good puzzles, as you say. No disagreement there. But creating those puzzles isn’t a trivial matter.

In the example you chose of deducing congruent opposite angles and sides from a parallelogram’s parallel sides, how do you turn that proof into the sort of satisfying puzzle Lockhart writes about, rather than an exercise assigned by a teacher?