a fable about the origins of those helpful counting thingies
Once there weren’t numbers,
and life was cold and sad.
You might say “I’ve got lots of stuff!”
but not how much you had.
You could gather flowers,
but you couldn’t count them up.
You could ask for chocolate milk,
but not a “second” cup.
And though their eyes could see just fine,
the people still were blind.
They held things in their arms and hands,
but never in their minds.
or, How to Call Out Mistakes without Trampling the Mistaken
It was the end of our first day on limits—a deep and slippery concept, the engine of calculus—when Melanie exclaimed, “Wait. Shouldn’t that limit be 4, not 6?”
Nope—it was 6. Melanie’s error suggested that she’d missed the lesson’s most basic truth, an idea that the class had spent the day paraphrasing, analyzing, and shouting in chorus. Talking one-on-one, I could have coached her through the misconception. But hers was a public declaration, in front of the whole room.
Even before the words had left Melanie’s mouth, I could hear the groan welling up among the students, murmured ridicule and the slapping of foreheads soon to follow. They all knew it. She didn’t. From Melanie’s blushing, you could read her self-esteem falling like a mercury thermometer.
And so I found myself confronting one of the teacher’s daily puzzles: what do you say when a student is wrong?
Stop leering and asking me “What’s your sine?” I’ve already told you, it’s ½. Why don’t you ask me about my hobbies or something?
Ooh, this isn’t totally related, but what you just said reminds me of a great story…
Now and then, an article lands in my inbox, promising that some technology will remake the classroom. Our schools, apparently, are as outdated as car-phones or medical leeches. It’s time to welcome the flipped classroom, the MOOC, the data-driven world.
It’s not all wrong, I’m sure. But it makes me wonder: How well do we actually know the classroom? Before we start drastic renovations, we should make sure we’ve got a clear view of the structure that’s already in place. And I’m not sure we do.
The story of the classroom is devilishly hard to tell.
We’re always lumping them together, scientists and mathematicians. They’re “STEM” professionals: bespectacled, smart, pleasantly soft-spoken until you conflate Star Trek and Star Wars, after which their wrath is visited upon you.
But the fact is that, aside from being the butt of cheap jokes, mathematicians and scientists don’t share all that much in common.
And you can tell that from the way they look at each other’s fields.
When it comes to research, scientists view mathematics the way a handyman views a toolkit. To a scientist, math is a way of solving problems, as practical as a step-ladder or a roll of duct tape.
…when you run into a college classmate who dropped out after suffering from health issues. You always meant to write a nice, sympathetic letter of support, but it never crested to the top of your to-do list, and now your long silence seems callous. The classmate sees you, looks away, then marches right up to you and asks, point-blank, the question you’ve dreaded for years:
“Why,” your classmate spits, “does raising both sides of an algebraic equation to an even power potentially introduce extraneous solutions?”
…you’ve already skimmed every worthwhile article in the newspaper. You completed the crossword, the Sudoku, even the word jumble. Grudgingly, you turn to the paper’s last remaining puzzle: the Partial Differential Equation of the Day.