Five Principles of Extraordinary Math Teaching | Dan Finkel | TEDxRainier

Summary

In this talk, the speaker emphasizes the beauty and power of mathematical thinking, lamenting how often math is misrepresented as tedious memorization rather than an exhilarating journey of discovery. They advocate for a transformative approach to teaching math, rooted in curiosity and exploration, and outline five principles: start with a question, allow time to struggle, embrace not knowing the answers, say 'yes' to students' ideas, and encourage play. By fostering these principles, we can help students experience the joy and creativity of math, ultimately nurturing a lifelong love for the subject.

Takeaways

• 😟 The script starts with a personal anecdote about a child disliking math, highlighting the common issue of math aversion.
• 🧐 The speaker emphasizes the beauty and power of mathematical thinking and its life-changing potential, contrasting with the negative experiences many have with math education.
• 📚 The script describes 'mathematical miseducation' as a widespread problem that leads to a lack of motivation and understanding among students.
• 💡 It points out the dangers of lacking mathematical literacy, such as being easily manipulated by misleading statistics and limited career opportunities.
• 😌 The speaker humorously illustrates the persuasive power of made-up statistics, showing how people are less critical when numbers are involved.
• 🌟 The potential of mathematical thinking is underscored through a workshop experience where it was described as 'like a God', indicating its transformative nature.
• 🤔 The script introduces five principles for incorporating thinking into math education, starting with asking questions rather than providing answers.
• 🕒 Principle two stresses the importance of giving students time to struggle with problems, fostering perseverance and genuine understanding.
• 🙅‍♂️ Principle three asserts that teachers and parents should not act as 'answer keys', encouraging students to explore and find answers themselves.
• 💬 Principle four encourages saying 'yes' to students' ideas, even incorrect ones, to promote active participation and respect for their thinking process.
• 🎲 Principle five promotes the idea that math is about exploration and play, not just following rules, which can lead to new mathematical inventions and understanding.
• 🏠 The speaker suggests that a home environment filled with play and games can nurture children's mathematical instincts and thinking.

Q & A

What is the main issue the speaker addresses regarding math education?

The speaker addresses the issue of 'mathematical miseducation', where math is often taught through repetition and memorization, leading to a lack of motivation and even a lifelong aversion to math among students.

Why does the speaker mention the importance of mathematical literacy?

The speaker emphasizes the importance of mathematical literacy because without it, individuals' career opportunities are limited, and they become vulnerable to manipulation by entities that use statistics to deceive.

What is the fabricated statistic mentioned by the speaker to illustrate a point?

The speaker mentions a fabricated statistic that inserting a single statistic into an assertion makes people 92 percent more likely to accept it without question, to demonstrate how easily people can be misled when they are not comfortable with math.

What does the speaker suggest as the starting point for a good math class?

The speaker suggests that a good math class should start with a question, rather than beginning with answers, to promote authentic thinking and curiosity among students.

What is the significance of the color-coded numbers example used by the speaker?

The color-coded numbers example is used to illustrate how starting with a question can lead to an engaging and mysterious exploration, which is more likely to stimulate real thinking and learning compared to simply memorizing steps.

Why is it important for students to struggle with a question according to the speaker?

Struggling with a question is important because it allows students to develop perseverance, curiosity, and observational skills, and it encourages them to take risks and engage in the learning process more deeply.

What is the role of the teacher according to the speaker's third principle?

According to the speaker's third principle, the teacher should not act as the 'answer key' but rather facilitate an environment where students can explore, question, and find answers together, turning math into an adventure.

What does the speaker mean by saying 'yes' to students' ideas and questions?

The speaker means that teachers should validate students' participation in the mathematical thinking process by accepting their ideas and questions, even if they are incorrect, to foster a respectful and empowering learning environment.

How does the speaker connect the idea of a number circle to the concept of modular arithmetic?

The speaker connects the idea of a number circle to modular arithmetic by explaining how accepting the notion that 2 plus 2 could equal 12 on a number circle leads to a valid mathematical system with practical applications, such as in cryptography.

What is the final principle the speaker offers for nurturing mathematical thinking?

The final principle the speaker offers is the importance of play in mathematics, suggesting that play is to mathematics what books are to reading, and that a home filled with mathematical play encourages the development of mathematical thinking.

Outlines

00:00

📚 The Power and Miseducation of Math

The speaker begins by sharing a personal story about a child's aversion to math, which contrasts with her own love for the subject. She emphasizes the transformative power of mathematical thinking and laments the common miseducation in math that leads to aversion and avoidance. The speaker criticizes the expectation of math classes as mere memorization and repetition, which fails to inspire students. She highlights the consequences of lacking mathematical literacy, including limited career opportunities and susceptibility to manipulation by misleading statistics. The speaker humorously illustrates the power of numbers by making up a statistic that, despite being fabricated, carries weight due to people's discomfort with questioning numbers. She calls for a reevaluation of how math is taught, advocating for an approach that starts with questions and fosters genuine thinking.

05:05

🤔 Encouraging Persistence and Curiosity in Math

In this paragraph, the speaker addresses the misconception that math problems should be solved quickly and the negative impact of this belief on students' self-perception as 'not a math person.' She advocates for teaching tenacity and courage in the face of difficult problems by allowing students ample time to think and engage with these problems. The speaker shares an experience of using an image in a classroom to stimulate thinking and discussion, emphasizing the importance of struggling with a question to deepen curiosity and observation. She also discusses the role of the teacher and parent as facilitators of this process, encouraging them to embrace not knowing the answer as an opportunity for exploration and learning, rather than a threat.

10:05

🔍 The Role of the Teacher Beyond Being the 'Answer Key'

The speaker continues the discussion on the role of educators, emphasizing that they should not be seen as the sole source of answers. She encourages teachers to embrace questions they cannot answer as opportunities for collective exploration and learning. The speaker highlights the importance of allowing students to take risks with their ideas and to engage in mathematical conversations and debates. She stresses the value of students convincing themselves and each other of the truth, rather than relying solely on the teacher's authority. The speaker also touches on the concept of saying 'yes' to students' ideas, even incorrect ones, as a form of respect and a way to foster a more convincing learning experience through peer interaction.

🎲 The Courage to Explore and the Playfulness of Mathematics

In the final paragraph, the speaker delves into the idea of play and exploration in mathematics, using the hypothetical scenario of '2 plus 2 equals 12' to illustrate the process of questioning and discovery. She points out that by entertaining such ideas, students can engage in a deeper understanding of mathematical principles and even lead to innovative concepts like modular arithmetic. The speaker argues that mathematics is not just about following rules but about creativity, exploration, and sometimes challenging established norms. She concludes by advocating for a mathematical education that encourages play, curiosity, and courage, which can lead to a love for math and a preparedness to meet future challenges with creativity.

Keywords

💡Mathematical thinking

Mathematical thinking refers to the cognitive processes used in mathematical reasoning, problem-solving, and proof. In the video, it is presented as a powerful and transformative way of thinking that can change lives and is central to the speaker's argument for a more engaging approach to teaching math. The speaker emphasizes the beauty and power of mathematical thinking, contrasting it with the common perception of math as a tedious subject.

💡Miseducation

Miseducation in the script refers to the flawed educational practices that lead to students developing a negative view of mathematics. It is related to the theme as the speaker discusses how traditional math education, which often focuses on memorization and repetition, can result in students disliking math and lacking mathematical literacy, which in turn limits their career opportunities and makes them vulnerable to manipulation by various entities.

💡Literacy

In the context of the video, literacy is not just about reading and writing but also about having the ability to understand and engage with mathematical concepts. Mathematical literacy is crucial as it empowers individuals to make informed decisions and not be easily swayed by misleading statistics or financial schemes, which the speaker illustrates with humor by making up a statistic about the persuasive power of numbers.

💡Authentic questions

Authentic questions are those that arise naturally from a situation and have a compelling need for an answer. The speaker uses the term to emphasize the importance of starting math education with genuine curiosity and inquiry rather than just providing answers. An example from the script is the question about the relationship between numbers and colors, which is not immediately obvious and thus stimulates real thinking.

💡Tenacity

Tenacity in the video is portrayed as the quality of being persistent in the face of difficulty, which is a key aspect of the learning process in mathematics. The speaker argues against the notion that math problems should be quickly solvable and instead encourages students to persevere, reflecting on the genuine struggle involved in solving complex problems.

💡Inquisitive

Being inquisitive is highlighted as a positive trait for students in the script, as it signifies a natural curiosity and desire to learn. The speaker encourages teachers to embrace students' inquisitiveness and to engage with their questions, even if it means admitting 'I don't know,' turning math into an adventure of discovery.

💡Number line

A number line is a fundamental mathematical concept used to represent numbers in a sequential and ordered manner. In the video, the speaker uses the idea of a number line to explore the concept of modular arithmetic, or number circles, where numbers wrap around in a circular pattern, challenging traditional notions of addition and leading to new mathematical insights.

💡Modular arithmetic

Modular arithmetic, also known as clock arithmetic or number circles, is a system where numbers 'wrap around' after they reach a certain value, similar to how the numbers on a clock work. The speaker uses this concept to illustrate the idea of exploring unconventional mathematical ideas and how such exploration can lead to practical applications, such as in cryptography.

💡Courage

Courage in the context of the video is the willingness to challenge established norms and to explore ideas that may seem absurd at first, like the idea that '2 plus 2 could equal 12.' The speaker emphasizes the importance of courage in mathematical thinking, as it allows for the exploration of new concepts and the potential for groundbreaking discoveries.

💡Play

Play is described as a crucial element in learning mathematics, as it encourages exploration, creativity, and a sense of ownership over the learning process. The speaker likens the act of playing with math to the freedom and discovery experienced during childhood play, suggesting that a playful approach can foster a love for math and a deeper understanding of its concepts.

💡Curiosity

Curiosity is presented as a driving force behind the learning process in mathematics. The speaker advocates for an educational approach that stimulates students' curiosity, leading them to ask questions, explore possibilities, and engage deeply with mathematical concepts, as opposed to merely following rules or memorizing facts.

Highlights

The speaker expresses concern over children disliking math due to its common miseducation.

Mathematical illiteracy can limit career opportunities and make people vulnerable to misleading statistics.

A fabricated statistic can be persuasive due to people's discomfort with questioning numbers.

The potential of mathematical thinking to transform lives is being squandered.

A workshop participant felt empowered by mathematical thinking, likening the experience to 'feeling like a God'.

René Descartes' philosophical inquiry into the nature of thinking is suggested as a foundation for mathematical education.

Five principles are proposed to enhance mathematical thinking in education.

Starting with a question rather than answers can stimulate authentic mathematical thinking.

Students need time to struggle with problems to foster perseverance and thinking skills.

Teachers and parents should not act as the sole source of answers, encouraging exploration instead.

Accepting all ideas, even incorrect ones, into mathematical discussions can promote respect and learning.

Exploring the consequences of incorrect ideas can lead to deeper understanding and creativity.

The concept of number circles or modular arithmetic demonstrates the value of exploring unconventional ideas.

Mathematics should be about play and exploration rather than strict adherence to rules.

Playing with math can nurture a sense of ownership and foster mathematical instincts.

The speaker advocates for a shift in how math is taught to inspire courage, curiosity, and creativity in students.

The goal is to make mathematical thinking an enjoyable and empowering experience for all students.

Transcripts

Translator: Spring Han Reviewer: Mirjana Čutura

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,

(Laughter)

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 percent more likely to accept it without question?

(Laughter)

Yeah, I totally made that up.

(Laughter)

And 92 percent 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."

(Laughter)

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.

A good place to start

is with the words of the philosopher and mathematician René Descartes,

who famously proclaimed, "I think, therefore I am."

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 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 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.

(Laughter)

I don't think of myself as a mean person,

but I am willing to deny you what you want.

(Laughter)

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?"

(Laughter)

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 ...

(Laughter)

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.

(Applause)