GUY RAZ, HOST:

When you were a kid, what did you like about math?

DAN FINKEL: I think I understood a sense of, math makes you powerful a little bit. There's this famous story about Gauss - famous mathematician, I think 17th or 18th century - and the teacher gives everyone a problem to add up all the numbers from 1 to 100. And it's something you just do to make people miserable, but Gauss is able to solve the problem very quickly. And the way he's able to do it is to organize the numbers in a way that works for him.

He adds one and a hundred, and that's 101. He adds two and 99. That's 101. Three and 98 - that's 101. And he realizes that he just has a bunch of groups of 101. And it's a multiplication problem way easier to do than just doing it in the order the teacher gave it to him. I think I had something similar where I felt like if I didn't like the problem I was given, I could try changing the numbers and see what would happen.

RAZ: This is Dan Finkel. He's a mathematician.

FINKEL: I can still remember puzzles I did from the time I was 6 or 10 where it was like, draw this shape using just a single line without lifting up your pencil. And I would work on it for so long, and I just was willing to be stuck, I guess. This is sort of the hallmark of being a mathematician - is you don't mind being stuck. But if you're talking about, like, when did I really see mathematics as a beautiful subject, I did go to a math camp the summer after my ninth grade year. And that was the time where I really saw the beautiful math that most people don't get to see until college or graduate school. And my first question was, why has no one shown me this before?

RAZ: Before Dan got his PhD, he was a teacher. And during that time, he noticed a fundamental problem with the way math is taught at school.

FINKEL: We essentially give answers before we give questions. We say, here's how you handle this problem, and I want to make sure you do it so you never get into trouble and never make a mistake. And what we don't do is start with a question and actually allow people to say, oh, that seems interesting. I should be able to do that. There's something strange going on, and now I want to know. Our main problem is we just think of math as this static body of facts rather than an experience.

RAZ: You must've come across kids who said they didn't like it or that they weren't good at it.

FINKEL: Absolutely. Well, a lot of people think they don't like math, so part of my job is to show them that they're wrong. And so what I try to do is give them a different experience of math, something that I would consider an authentic mathematical experience or puzzle or something that invites them to play in a mathematical way. And it sounds like that would be watering it down, but actually, that is an experience of mathematics that is more rigorous and more akin to what mathematicians actually do. And what you find is even the people who think they don't like it start to see that there is something in there that resonates with them.

RAZ: Dan Finkel picks up his idea from the TED stage.

(SOUNDBITE OF TEDx TALK)

FINKEL: So if you are a teacher or a parent or anyone with a stake in education, I offer these five principles to invite thinking into the math we do at home and at school. Principle one - start with a question. The ordinary math class begins with answers and never arrives at a real question. Here are the steps to multiply. You repeat. Here are the steps to divide. You repeat. We've covered the material. We're moving on. What matters in this model is memorizing the steps. There's no room to doubt or imagine or refuse, so there's no real thinking here. If I rush to an answer, I will have robbed you of the opportunity to learn. Thinking happens only when we have time to struggle.

And that is principle two. It's not uncommon for students to graduate from high school believing that every math problem can be solved in 30 seconds or less. And if they don't know the answer, they're just not a math person. This is a failure of education. We need to teach kids to be tenacious, to persevere in the face of difficulty. The only way to teach perseverance is to give students time to think and grapple with real problems.

Everybody needs to work on math to understand it. Everybody is eventually challenged by mathematics. And we build mathematicians by actually giving students time to exercise their curiosity, to try things out, to think things through because without that kind of productive struggle, no real learning takes place.

RAZ: When we come back in just a moment, we'll hear more from Dan on principles three, four and five, and on rethinking our approach to math. Stay with us. I'm Guy Raz, and you're listening to the TED Radio Hour from NPR.

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RAZ: It's the TED Radio Hour from NPR. I'm Guy Raz. And on the show today, ideas about the power of math. So we just heard mathematician Dan Finkel describe 2 out of his 5 principles on how to rethink math education.

So a question's posed, students get time to struggle. And eventually, they come up with something, and they ask, is this right? And then what?

FINKEL: Well, so the third principle is you are not the answer key. And this is tricky because rightness and wrongness does matter in mathematics, but the current kind of problematic state of mathematics education is partly one that has come about because we are obsessed with right answers. If all you care about is if the answer is right or not, then it's very easy to not worry if the students understand. All you care about is, could we get them to write the right number down on the paper?

And students are geniuses at this because they've learned that their teachers care about answers. Their teachers know the answers. All they need to do is read the teacher and see whether they show them that the answer is correct. And we need to actually just get away from this obsessive compulsion to say, all that matters is the answer and whether it's right or not, and get into a situation where we say, more important to me is how you are thinking about it.

There are sometimes moments where a student has a wrong answer but a brilliant idea, an idea that is worth exploring and developing. And there are also times when a student has a right answer, and yet maybe they arrived at it in a way that didn't show us anything new.

(SOUNDBITE OF TED TALK)

FINKEL: By refusing to be the answer key, you create space for this kind of mathematical conversation and debate. And this draws everyone in. 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 two plus two 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 student's right to participate in the act of thinking mathematically.

So what we're really trying to do in all this is give students ownership over their own mathematical thinking and make sure they have a place at the table of mathematical thought. We want to make sure that they are welcome there and that they know that you value their right to participate in this process.

RAZ: I mean, the idea, I guess, is math shouldn't feel intimidating.

FINKEL: Right. I mean, it shouldn't feel like it belongs to somebody else. Like, when you do math, you shouldn't feel like you're going into a room where all the furniture is covered with plastic sheets and you're not allowed to be in there. It should feel like this is your place, that you're allowed to break things. You're allowed to question. You're allowed to put your own mark on things, and there's a place for you to do it.

(SOUNDBITE OF TED TALK)

FINKEL: But allow me to take this a step further. How do you actually know that two plus two doesn't equal 12? What would happen if we said yes to that idea? This is how new math gets invented. It takes courage to say, what if two plus two equals 12, and actually explore the consequences. That courage led to some of the greatest breakthroughs in history. All it takes is a willingness to play.

And that is principle five. Mathematics is not about following rules. Parents, if you want to know how to nurture the mathematical instincts of your children, play is the answer.

(SOUNDBITE OF MUSIC)

RAZ: I mean, could you make the argument that we're, you know, wired to be open and interested in math from an early age?

FINKEL: Yeah, I would, and I do make that argument. I think that human beings are fundamentally pattern-seeking creatures, and mathematics is all about pattern and structure and finding order and placing order on a sometimes disorderly world. If you spend time with 2 and 3 and 4 and 5 and 6-year-olds, they are constantly organizing, sorting, counting, structuring, doing this work that is fundamentally mathematical work because they feel it empowers them. This is, I think, very much a natural human instinct.

RAZ: I mean, I have seen that with my own children - right? - from when they were little kids counting to 100 and then, you know, counting to 500, just sitting there for like forever counting to 500 or, you know, or asking questions about what's bigger, this or that. It is true that numbers - they're a prism through which they see the world, certainly early in life.

FINKEL: Yeah. I think that it's just such a rich world to explore. And it's, I think, such a pleasure that young kids are just drawn to that. We don't need to teach that, we just need to nurture it so it's still there because the people who end up becoming the great mathematical thinkers and scientists and engineers really have that same sense of wonder and same sense of kind of creative delight. It's not that it came out of nowhere, it's that they just didn't lose it and it got nurtured and grew along the way.

RAZ: OK. Aside from obviously wanting kids to love numbers and math, you also argue in your talk that there are real-world consequences when, you know, when people don't have math literacy.

FINKEL: Yeah. What's funny is there are so many places to look for where not knowing mathematics or not having a kind of numerical literacy can really undermine us. There is a certain amount of financial literacy, just numerical understanding you need to make it through the world. You need to be able to look at the newspaper. You see a graph. It is telling you something about the way the world is. It's nice to be able to actually understand what it's telling you. Or take out a loan. It's nice to know what that means about how much money you're going to be paying. And it's so easy to be tricked if you don't understand those things, that it really puts you in danger, I think, if you are afraid of math.

(SOUNDBITE OF MUSIC)

RAZ: That's mathematician Dan Finkel. He's the founder of the consulting business Math For Love. You can see his full talk at ted.com.

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