How to Remember Concavity

My Calculus students struggled to remember the difference between the two types of concavity, so I made up a poem for them:

My fiancee, who also teaches Calculus, invented a different mnemonic… Continue reading


Following Recipes

Another pep talk with the class, another metaphor hijacked. This is a theme in my life.

Me: Remember, guys. Math isn’t just about following a recipe.

Student #1: It isn’t?

Me: No! Math is about logical reasoning. If I only wanted to teach you how to be detail-oriented and copy an example, we’d have a cooking class. You’d still learn to follow directions, and at the end we’d have a tasty cake, instead of some silly graph. Continue reading

The Quadratic Formula Must Die! (or, Long Live the Quadratic Formula!)

Algebra students are often compelled to memorize the following jumble of symbols:

Every adult I meet seems to remember this equation. Some quote it proudly. Others recall it grudgingly, fists clenched. Some people sing it (to the tune of “Pop Goes the Weasel,” though the rhythm’s a little forced). Many believe it captures the essence of mathematics: mystical formulas, taught to everyone, comprehensible to few.

I’m ambivalent. I can’t deny the quadratic formula’s usefulness, but I can’t let its tyranny go unchallenged, either. This formula has planted itself, like a virus, in the minds of otherwise healthy adults. Our society demands that its members know this equation, but makes no mention of its context, its uses, its colorful history.

I come not to praise the quadratic formula, nor to bury it, but merely to shed a few rays of light. What is it for? What is its story? And where on Earth does it come from? Continue reading

A Ray of Light

There are moments of teaching I like to remember – episodes of cleverness, compassion, success. And then there are the other moments, the ones that my thoughts tend to flee, the ones I prefer not to think about. This is a story about both.

One Friday after school, a student came to me with questions. As a 12th-grade transfer, she found herself struggling to catch up with the students who had already spent years in the crucible of our intense charter school. To graduate that year, she needed to take my Statistics class concurrently with Algebra 2. She was failing them both. Continue reading

The Graph Factory

I want my students to see graphing as a subtle, meaningful craft. But when I mess up and assign too many graphs for homework, they just sprint through them, cranking them out like cheap factory products. It goes something like this…

Me: How’s that graphing going?

Student: No time, man! I’ve got sixty logarithms that need to ship to customers tonight, and the assembly line’s been down for hours. I’m cranking out asymptotes by hand over here – I’ve got no time for your funny business!

Me: But why? What’s the point of these graphs?

Student: Hey, not my place to ask questions. I just hit my graph quotas, and try to make it home for dinner with the wife and kids.

Me: But you’re making mistakes. Sine curves don’t have sharp corners.

Student: So slap a warning label on ‘em, for all I care! Continue reading

Why can’t you divide by zero?

Part 1: Dividing By Smaller and Smaller Numbers
by a High School Math Teacher 

Suppose you’ve got a pizza. A nice charcoal-cooked one from New Haven, or an oven-hot Chicago deep-dish, or even one of those organic San Francisco artisan pies that somehow make artichoke hearts seem like they belong on a pizza. And, generous soul that you are, you’ve decided to share.

How many people can you feed if everybody gets half of a pizza (a hearty helping)?

Well, it’s 1 pizza ÷ ½ pizzas per person = 2 people.

And how many can you feed if everybody gets 1/10 of a pizza (a cheesy snack)? Continue reading