What doesn't work
We all agree that this is unsatisfying:
You aren't allowed to divide by zero, because it's a rule.And many of us (read: me, yesterday) think that this is better:
What's 10 divided by 2? It's 5, because 10 split up evenly into two groups has 5 in each. 10 divided by 1? 10 in 1 group has 10. But 10 in 0 groups? What would that even mean?And, then, having clearly and elegantly explained why dividing by zero would be a very, very silly thing to do, we go back to the day's main topic:
So 2 to the negative 3rd power is 1/8...Wait, hold on.
I get it: if you think of division as evenly grouping items, then dividing by zero makes no sense. But that's just the normal wear and tear of a mathematical model. We ask kids to believe that exponentiation is like repeated multiplication, and then we ask them to forget that when we introduce negative powers. Multiplication is repeated addition until you throw in "2.3 times 5.1" and then everything goes to hell. We freaking give quadratics imaginary solutions, and our failure to imagine what "0 groups" looks like is stopping us from dividing by zero? Yeah, right.
(Also, wouldn't zero groups have no items in them?)
And another thing: we tell kids that division by zero is undefined. Skeptically, they take out their calculators and punch some keys and get an error message. "Wait! He's right. It gives you an error."
When it comes to dividing by zero, there is a lot wrong with the standard teachery maneuvers:
- Any sort of "wtf would 0 groups mean" argument does not show that division by 0 is non-sensible. All that it shows is that this particular model of division -- the grouping model -- breaks down for non-integers. That's normal in math. Kids should be regularly creating and discarding conceptual models.
- "Undefined" is the best we can do? Language matters, and saying that 5/0 is undefined makes it sound like, shoot, well, we were going to get around to it but we just chose to let it slide.
- The kids are checking their calculators to see if division by zero makes sense. For crying out loud, that's not math. They're wondering whether to believe you or not, because what you're saying doesn't make sense. Hell, everyone knows that 5 divided by 0 is 0. It just makes so much sense...
This works better
March in front of the classroom. "What's 3 divided by 0? Someone tell me NOW," you say.
If your students are a bunch of sissies and nerds they'll shout "You can't divide by zero!"
"Oh don't give me that math teacher stuff. Who says that I can't divide by zero? Give me a real answer."
That's all it takes. Really. They've been waiting in every math class since they were 8 to get this off their chests.
"3 divided by 0 is 0."
OK, cool. Now we've got something to work with. Ask the class, agree or disagree?
But if 3 divided by 0 is 0, and 5 divided by 0 is 0, then wouldn't this follow?
So 5 = 3, right?
They squirm. They try something else. Maybe 5 divided by 0 is 5? You can handle that too. The point is to lead them to contradiction, and let them grapple with that tension. There are other ways to tug out the contradictions.
Here's why this is better:
- We shouldn't be telling kids that the reason that we don't divide by zero is because an intuitively pleasing model fails. That should never stop a good mathematician.
- Telling kids that division by zero is "undefined" sounds lazy. It's more accurate and informative to say that division by zero leads to contradiction.
- How do you help kids see that it leads to contradiction? Take suggestions from the kids of what division by zero should mean, and then let them see the implications. Let them try to make things consistent. Make the choices clear. Sure, we can think of division that way. We'd just have to refine our rule for multiplying fractions. So, what's your new rule for multiplying fractions?
All of this is more authentic than what I used to do. ("What I used to do"? I'm talking about yesterday. Things move fast around here.) The biggest change in what I'm doing this year is slowing down and having kids make arguments in class. Proof and argument is how I'm helping my kids make sense of this stuff. There's no shortcut or substitute.