Here is the full text and summary of Dan Finkel’s talk titled “Five Principles of Extraordinary Math Teaching” at TEDxRainier conference. In this enjoyable talk, he invites us to approach learning and teaching math with courage, curiosity, and a sense of play.
TRANSCRIPT:
A friend of mine told me recently that her six-year-old son had come from school and said he hated math. And this is hard for me to hear because I actually love math.
The beauty and power of mathematical thinking have changed my life. But I know that many people lived a very different story.
Math can be the best of times or the worst of times, an exhilarating journey of discovery or descent into tedium, frustration, and despair. Mathematical miseducation is so common we can hardly see it.
We practically expect math class to be repetition and memorization of disjointed technical facts. And we’re not surprised when students aren’t motivated, when they leave school disliking math, even committed to avoiding it for the rest of their lives.
Without mathematical literacy, their career opportunities shrink. And they become easy prey for credit card companies, payday lenders, the lottery, and anyone, really, who wants to dazzle them with a statistic.
Did you know that if you insert a single statistic into an assertion, people are 92% more likely to accept it without question? Yeah, I totally made that up.
And 92% is — it has weight even though it’s completely fabricated. And that’s how it works.
When we’re not comfortable with math, we don’t question the authority of numbers. But what’s happening with mathematical alienation is only half the story.
Right now, we’re squandering our chance to touch life after life with the beauty and power of mathematical thinking. I led a workshop on this topic recently, and at the end, a woman raised her hand and said that the experience made her feel — and this is a quote — “like a God.”
That’s maybe the best description I’ve ever heard for what mathematical thinking can feel like, so we should examine what it looks like.
But Descartes looked deeper into the nature of thinking. Once he established himself as a thing that thinks, he continued, “What is a thinking thing?”
It is the thing that doubts, understands, conceives, that affirms and denies, wills and refuses, that imagines also, and perceives. This is the kind of thinking we need in every math class every day.
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 1: 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 the model is memorizing the steps. There’s no room to doubt or imagine or refuse, so there’s no real thinking here.
What would it look like if we started with a question? For example, here are the numbers from 1 to 20. Now, there’s a question lurking in this picture, hiding in plain sight. What’s going on with the colors?
Now, intuitively it feels like there’s some connection between the numbers and the colors. I mean, maybe it’s even possible to extend the coloring to more numbers. At the same time, the meaning of the colors is not clear. It’s a real mystery.
And so, the question feels authentic and compelling. And like so many authentic mathematical questions, this one has an answer that is both beautiful and profoundly satisfying.
And of course, I’m not going to tell you what it is. I don’t think of myself as a mean person, but I am willing to deny you what you want.
Because I know if I rush to an answer, I would’ve 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 and courageous, 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.
I brought this image into a classroom recently, and we took the time to struggle. And the longer we spent, the more the class came alive with thinking.
The students made observations. They had questions. Like, “Why do the numbers in that last column always have orange and blue in them?” and “Does it mean anything that the green spots are always going diagonally?” and “What’s going on with those little white numbers in the red segments? Is it important that those are always odd numbers?”
Struggling with a genuine question, students deepen their curiosity and their powers of observation. They also develop the ability to take a risk.
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.”
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.
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.
Einstein called play the highest form of research. And a math teacher who lets their students play with math gives them the gift of ownership. Playing with math can feel like running through the woods when you were a kid. And even if you were on a path, it felt like it all belonged to you.
Parents, if you want to know how to nurture the mathematical instincts of your children, play is the answer. What books are to reading, play is to mathematics.
And a home filled with blocks and puzzles and games and play is a home where mathematical thinking can flourish. I believe we have the power to help mathematical thinking flourish everywhere. We can’t afford to misuse math to create passive rule-followers.
Math has the potential to be our greatest asset in teaching the next generation to meet the future with courage, curiosity, and creativity.
And if all students get a chance to experience the beauty and power of authentic mathematical thinking, maybe it won’t sound so strange when they say, “Math? I actually love math.”
Thank you.
SUMMARY OF THIS TALK:
In Dan Finkel’s talk titled “Five Principles of Extraordinary Math Teaching,” he emphasizes the importance of transforming the way we teach and think about mathematics. Finkel’s passion for math is evident as he shares his insights on how to make math an engaging and empowering experience for students. Here are the key takeaway points from his talk:
1. Math Education Crisis: Finkel begins by acknowledging that many students dislike math, and mathematical miseducation is pervasive. Students often perceive math as a repetitive and memorization-driven subject, which leads to a lack of motivation and a dislike for math.
2. The Power of Mathematical Thinking: Finkel believes in the beauty and power of mathematical thinking, which has the potential to change lives. However, he emphasizes that we are squandering the opportunity to inspire students with this way of thinking.
3. Five Principles for Extraordinary Math Teaching:
a. Start with a Question: Finkel suggests that instead of starting with answers, math education should begin with questions. Genuine questions spark curiosity and invite students to engage deeply with mathematical concepts.
b. Thinking Happens Only When We Have Time to Struggle: Finkel highlights the importance of giving students the time to grapple with real problems. Perseverance and tenacity are vital skills that can only be developed when students are allowed to struggle with challenging questions.
c. You Are Not the Answer Key: Teachers should embrace the idea that they don’t have to know all the answers. By saying “I don’t know, let’s find out,” educators can create an environment where students explore and discover solutions collaboratively.
d. Say “Yes” to Their Ideas: Finkel encourages educators to accept and explore students’ ideas, even if they are incorrect. This approach fosters a culture of respect and empowers students to participate in mathematical thinking.
e. Willingness to Play: Mathematics is not just about following rules; it’s about playing, exploring, and being creative. Finkel believes that math should be approached with a playful attitude, which encourages students to take ownership of their learning.
4. The Power of Play: Finkel emphasizes that mathematics is about more than just following rules; it’s about playing, exploring, and sometimes breaking things. He likens play in mathematics to the highest form of research, as it encourages creativity and ownership of the subject.
5. Nurturing Mathematical Instincts: Finkel advises parents to foster their children’s mathematical instincts through play. He suggests creating a home environment filled with mathematical toys, puzzles, and games to encourage mathematical thinking.
6. The Potential of Math Education: Finkel believes that math has the potential to teach the next generation courage, curiosity, and creativity, preparing them to face the future with confidence. By embracing these principles, math education can inspire a love for math in all students.
In summary, Dan Finkel’s talk outlines five principles for extraordinary math teaching: starting with questions, allowing time for struggle, not being the answer key, saying “yes” to students’ ideas, and promoting a playful approach to mathematics. He believes that by adopting these principles, we can transform the way math is taught and empower students to appreciate the beauty and power of mathematical thinking.
Resources for Further Reading:
Can You Solve the Troll’s Paradox Riddle? – Dan Finkel (Transcript)
The Beauty and Power of Mathematics: William Tavernetti (Transcript)
A New Type of Mathematics: David Dalrymple at TEDxMontreal (Full Transcript)
Related Posts
- Transcript of Why Do Our Brains Love Music? – Dr. John Rehner Iversen
- Transcript of Pope Leo XIV Remarks To U.S. Audience For First Time In Chicago
- Transcript of Angela Duckworth’s Commencement Speech to 2025 Penn GSE Graduates
- Transcript of Jay Shetty’s Princeton University Class Day Commencement Speech
- Transcript of Maggie Rogers’ 2025 Commencement Speech At NYU Tisch School of the Arts