Five Principles of Extraordinary Math Teaching: Dan Finkel (Transcript)

Some students noticed that every even number has orange in it, and they were willing to stake a claim. “Orange must mean even.” And then they asked, “Is that right?” This can be a scary place as a teacher.

A student comes to you with an original thought. What if you don’t know the answer?

Well, that is principle three: YOU ARE NOT THE ANSWER KEY.

Teachers, students may ask you questions you don’t know how to answer. And this can feel like a threat. But you are not the answer key.

Students who are inquisitive is a wonderful thing to have in your classroom. And if you can respond by saying, “I don’t know. Let’s find out,” math becomes an adventure.

And parents, this goes for you too. When you sit down to do math with your children, you don’t have to know all the answers. You can ask your child to explain the math to you or try to figure it out together. Teach them that not knowing is not failure. It’s the first step to understanding.

So, when this group of students asked me if orange means even, I don’t have to tell them the answer. I don’t even need to know the answer. I can ask one of them to explain to me why she thinks it’s true. Or we can throw the idea out to the class.

Because they know the answers won’t come from me, they need to convince themselves and argue with each other to determine what’s true.

And so, one student says, “Look, 2, 4, 6, 8, 10, 12. I checked all of the even numbers. They all have orange in them. What more do you want?”

And another student says, “Well, wait a minute, I see what you’re saying, but some of those numbers have one orange piece, some have two or three. Like, look at 48. It’s got four orange pieces. Are you telling me that 48 is four times as even as 46? There must be more to the story.”

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By refusing to be the answer key, you create space for this kind of mathematical conversation and debate. And this draws everyone in because we love to see people disagree.

After all, where else can you see real thinking out loud? Students doubt, affirm, deny, understand. And all you have to do as the teacher is not be the answer key and SAY “YES” TO THEIR IDEAS. And that is principle four.

Now, this one is difficult. What if a student comes to you and says 2 plus 2 equals 12? You’ve got to correct them, right? And it’s true, we want students to understand certain basic facts and how to use them.

But saying “yes” is not the same thing as saying “You’re right.” You can accept ideas, even wrong ideas, into the debate and say “yes” to your students’ right to participate in the act of thinking mathematically.

To have your idea dismissed out of hand is disempowering. To have it accepted, studied, and disproven is a mark of respect. It’s also far more convincing to be shown you’re wrong by your peers than told you’re wrong by the teacher.

But allow me to take this a step further. How do you actually know that 2 plus 2 doesn’t equal 12? What would happen if we said “yes” to that idea? I don’t know. Let’s find out.

So, if 2 plus 2 equaled 12, then 2 plus 1 would be one less, so that would be 11. And that would mean that 2 plus 0, which is just 2, would be 10.

But if 2 is 10, then 1 would be 9, and 0 would be 8. And I have to admit this looks bad. It looks like we broke mathematics.

But I actually understand why this can’t be true now. Just from thinking about it, if we were on a number line, and if I’m at 0, 8 is eight steps that way, and there’s no way I could take eight steps and wind up back where I started.

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Unless… well, what if it wasn’t a number line? What if it was a number circle? Then I could take eight steps and wind back where I started. 8 would be 0. In fact, all of the infinite numbers on the real line would be stacked up in those eight spots.

And we’re in a new world. And we’re just playing here, right? But this is how new math gets invented.

Mathematicians have actually been studying number circles for a long time. They’ve got a fancy name and everything: modular arithmetic. And not only does the math work out, it turns out to be ridiculously useful in fields like cryptography and computer science.

It’s actually no exaggeration to say that your credit card number is safe online because someone was willing to ask, “What if it was a number circle instead of a number line?”

So, yes, we need to teach students that 2 plus 2 equals 4. But also we need to say “yes” to their ideas and their questions and model the courage we want them to have.

It takes courage to say, “What if 2 plus 2 equals 12?” and actually explore the consequences.

It takes courage to say, “What if the angles in a triangle didn’t add up to 180 degrees?” or “What if there were a square root of negative 1?” or “What if there were different sizes of infinity?”

But that courage and those questions led to some of the greatest breakthroughs in history.

ALL IT TAKES IS WILLINGNESS TO PLAY. And that is principle five.

Mathematics is not about following rules. It’s about playing and exploring and fighting and looking for clues and sometimes breaking things.

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