I don’t usually struggle to distinguish toys from tools. Gas stove? That’s a tool. Easy-Bake oven? That’s a toy. Bricks? Tools. LEGO bricks? Toys.
Mathematical tools are similarly distinctive. They harness industrial-strength power—think of Taylor series, or completing the square. Mathematical tools shine floodlights into dark corners. They unlock doors, solve problems, and make attentive students utter, “Whoa, deep.” They often come with complex instruction manuals, requiring weeks (or months (or years!)) of technical training to master.
Mathematical toys… not so much. They’re simple to grasp, fun to handle, and not much substantive good to anyone. Think of Sudoku puzzles, or differentiating cos(cos(cos(cos(x)))). We might get a kick out of poking and prodding such problems, but solving them won’t teach us anything fundamental about the workings of the universe or the necessities of logic. Toy problems aren’t floodlights; they’re more like flashlights dangling off of a keychain.
But just as the Incas mistook the wheel for a mere toy, sometimes mathematicians get it wrong. Sometimes what seems to be a toy is, in fact, a powerful tool.
Sometimes a toy is just a tool in waiting. Continue reading
“Hooray, it’s tax season!” said nobody ever, except for the clinically ill and the clinically sarcastic. But I’m here, in this season of paperwork and low spirits, to offer a hymn of praise to the poor, misunderstood public servants that make income taxes work. No, not IRS agents, although goodness knows those sorry devils could use a defender or two.
I’m talking about tax brackets.
“I’m worried about Joe. He’s been hitting the Klein bottle pretty hard lately.”
“Well, that explains why he’s totally sober. It’s hard to get much liquid in those things.”
Like everyone else, I’m heir to the prejudices of my culture. So even though I know firsthand that teaching is a profession, on some level I see it as an act of self-sacrifice, as a hard path undertaken for the greater good.
This might not sound like a prejudice. Not a damaging one, anyway. It paints a generous portrait of teaching as noble and virtuous—a useful antidote to the too-common caricature of educators as lazy, union-fed bureaucrats. This ethic of “the greater good,” in fact, helped draw many of us into the profession. What we lack in salary and prestige, we make up for in civic-mindedness, in moral conviction, in feeling we’ve taken a high road through our professional life.
But all this high-mindedness has a dark underbelly. In painting the choice to teach as an act of self-sacrifice, I believe that we may harm our schools more than we help them. Continue reading
Every month, I comb through the Google search terms that have led people to my blog. Then I reply to them as if they were letters. It’s a thing.
Q: equation of ugly funny?
Obviously, our senses of humor are as distinct and individual as our handwriting, our Facebook profiles, or the shapes of our noses. But in my view, this is the equation:
Q: how to make gf feel good about bad grades?
A: First, I’m glad you care about your girlfriend’s happiness (or, depending what “gf” stands for, your ghost-friend’s happiness). But rather than helping her feel good about low grades, try this.
Step 1: Help her feel good not about the grades, but about herself. Prove to her (or, in the ghost’s case, “it”) that you value her as an intelligent person, regardless of what her report card suggests.
Step 2: Help her raise the grades. Bad grades really are—as the name implies—bad. They can close off opportunities and create a harder path through your teens and early twenties. Whatever the way to raise her grades—tutoring, getting more sleep, finding a reliably quiet workspace, dropping expendable time commitments—it’s better than just throwing up her hands (or ghostly mist-appendages) and accepting bad marks.
In an email, Bonny Becker asks: How do I go about gaining a better understanding of what my math PhD-seeking son is talking about?
This question haunts those of us with a beloved mathematician in our lives. While most professions have a few dark rooms mysterious to outsiders, the field of mathematics feels like a palace in which every room is dark.