Five Principles of Extraordinary Math Teaching | Dan Finkel | TEDxRainier
100 Comments


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)

100 thoughts on “Five Principles of Extraordinary Math Teaching | Dan Finkel | TEDxRainier

  1. So glad I incorporate finger use during classroom instruction with my UPK groups. Base Ten in the palms of both hands.

  2. omg. thank u so much for this absolutely inspiring presentation. I think I have the right arguments now to convince my colleagues that this is the kind of math we should be teaching

  3. "Time to struggle?!?" JEEZ. That's an excuse for poor teaching. People struggle plenty without intentionally trying to.

  4. The problem is that most teachers don't know how to teach like that because they were never taught like that themselves. It is difficult thing to employ especially given the constraints of a state,city and school mandated curriculum.

  5. I am a average math student, but I love love the power of math to discover new things like Q-R code using probability, finding patterns. And I knew one day I would become a great mathematician. ……………………..

  6. I see that yes there are every single orange is even but there is also every three has a green, every 5 is blue and every 7 is purple. And the reason why they are multiple colors is bc there is multiple things happening to the number! I dong understand why there are white numbers in the deep orange or what is happening with the deep orange.

  7. The solution to the color ring number chart is below, don’t click read more if you want to figure it out yourself.

    It’s the prime factorization of every number, illustrated using colors.

    1 is grey and is present in every number in the grey border of the rings. Every prime number has its own unique color (except they use orange and a written out number in the orange part for larger primes) and the grey border because it can only be multiplied by 1 and itself. Every composite number it’s prime factors as sections of the shell. For example, the prime factorization of 6 is 2X3. 2 is yellow and 3 is blue, so it has a yellow segment and a blue segment. 16 is 2 multiplied 4 times, 2^4, or 2X2X2X2, so it has a yellow ring separated into 4 segments.

  8. The fact that the kids didn’t immediately recognize what those colour patterns were is a testament to how their education failed them.
    (It’s prime integers)

  9. Incredible presentation! If only Dan could be cloned and placed in every learning institution all over the world!!

  10. Lol, the problem with this is having genius kids who immediately understand what’s going on. No more time for debate. That kid is who I would be.

  11. The 1st five prime numbers have been given unique colours, the rest of the prime numbers are all orange, the composite numbers have colours on the basis of their prime multiples. Eg: 6 is product of 2 and 3 therefore its colour is combination of 2's colour- light orange and 3's colour- green.

  12. One word can change the tone of everything. Posing questions is a less threatening beginning to lessons. And tellikng them I don't know. You tell me. Creates laughter giggles and "noise". Absolutely agree.

  13. I love Math. I played with it. You are ABSOLUTELY RIGHT. Tomorrow I am going to teach 7th graders' Maths and I am an 8th grade. (Well in the Olympics team though). Gonna teach 10th grade stuff tomorrow! Problem: What is the probability of a couple (boyfriend and girlfriend) marrying in their relationship?

  14. "Math, I actually love math," My 11th-grade daughter says. In fact, she's going to talk Calculus 3 in her senior year of high school at the local University.

  15. "I know if I rush to an answer, I would have robbed you of the opportunity to learn" what a beautiful quote. This is exactly what I tell my students. "Figure it out. You can do it." Always give the question, you'll be really surprised with the answers the kids come up with 🙂 Learned a lot of new techniques this way – even from 4th graders. Wonderful wonderful presentation. It’s like you took all the words from my mouth and put it in the most beautiful way – something that I would never have been able to do! Like someone here said – this is the "I have a dream" speech of mathematics

  16. The speech is awesome . In my high school the teachers weren’t like playing math and thinking math instead the were all ways by the rules not let us nourish mathematical thinking. Until today I was trying to love math seeing videos on how to love math and what is math. But because of u mr DAN I have become thinking good of it and I am going to start it right now. U are my true and mind blowing inspiration. HEY! LETS PLAY WITH “ MATH!”

  17. The circular representation is a type of graph which can be used to depict octal bases and, as such, there is an error: have a look at the second ring where 14 is listed twice. [14 base 10 is 16 base 8]

  18. It was entertaining but this was a series of stunts to impress the audience. Which is cool.
    But there is nothing we can learn at a deeper level from this.
    I mean the elephant in the room here is not addressed: TIME.
    We don't have time to do this more than we already do.
    Exploring and playing takes time and all you need to know when you are 18 can't be known this way. You HAVE to take shortcuts.
    Of course, you ask questions, play and explore whenever you can but at some point, rather quickly in fact, you have to just swallow the knowledge in order to know enough in different matters in a certain amount of time.
    Moreover, we don't expect from a child to discover anything in maths that would create a revolution. At least not yet. That's not the goal.
    Learn the stuff and then, when you master it, feel free to just tear it apart and build your way up.
    So yes, it's a feel good conference there and it allows people to remember that playing and exploring should be part of the education but for one reason only: generating interactivity and curiosity in order to keep the children stimulated. That's the real goal. The means to that end is alright but to a certain extent.
    There is a balance to be held between efficiency in terms of knowledge volume and keeping your children interested. And that balance is, more or less, where we already are.
    Some teachers are better than others in maintaining that balance but, in general, that's the guideline already followed by most.
    So, all in all, nice conference, really entertaining, but that's pretty much it.
    What bothers me with the TedX's is that they don't feel like we are going to learn about cool stuff but well that we are going to make stuff look cool using stunts.
    It's all about presentation, very little about substance. In other words and IMHO, it feels fake and quite useless. You don't learn anything, you just feel good about the little you already know about stuff.
    t has a lot to do with communication and not so much with learning.

  19. He made a mistake. Has put two times 14 on the diagram. it should be 14, 15. Then on the outer circle 16 (which is not visible) then 17 (which it is) and so on.

  20. Hi guys, I am a math teacher in a High School. I live in a small village in Vietnam. I want to find some one to talk about teaching math in high school. Thank you.

  21. I love this concept. However, how do I incorporate this philosophy in my classroom when so much emphasis is placed on high stakes, standardized testing & pacing guides?

  22. You like math because you are disabled to like something likeable.Finding paterns in everything, no matter if it does exist or not.

  23. I love math. It is a hobby of mine. I am studying tensor calculus on YT.

    I listened to this ted talk on NPR and smart as this guy is, he spoke only in generalities. All very vague because he did not use any examples to illustrate his point. I wonder if the producers told him to leave math out of the talk because people would tune out? Only half kidding here. But THERE WAS NO MATH in this talk.

    How about an example to illustrate that math is not all about multiplying or dividing or about fractions? That is only arithmetic.

    I am sure this guy is a powerhouse intellect, but his talk fell short in my view.

  24. Start with questions – but what if the students are not interested in questions/answers (this is the case at least sometimes).

    Students need time to struggle – in my experience most people, not just students, want to avoid struggle as much as possible.

    Teacher is not the answer key – as I mentioned before some students are not interested in answers or lack thereof.

    Say yes to your students' ideas – the example 2 + 2 = 12 is a nice one but a student who has problems with regular addition will have very little chance of understanding 'number circle'. that example is appropriate only for the best of students which usually like math already.

    Play – whoever is able to consistently persuade 5 out of 10 students to set aside their phone and play with math deserves a teaching degree, the one who can do it with 7 out of 10 students deserves a doctorate, and the one who can do it with 90% must be some sort of god.

  25. Most people hate Math because modern Education system fails to teach Math in an understanding way. They expect the students to put in most of the work of understanding Math, which as we know, doesn't work, because most students aren't going to put in that much work into understanding Math.
    They just overwhelm us with formulas, equations, theorems, and proofs, and never teach us where those equations and proofs come from. If you just give me a plate of Lasagne, and expect me to make the same dish without giving me the recipe, i will fail at it.
    Same way, if you just give me an equation without explaining me how the equation was derived, i will fail to understand it and fail to make use of it when needed. 
    They expect us to just cram Math formulas and theorems, and spit them out on an Exam, instead of making sure we understand the formula, and why we need it.

  26. What does math stand for?

    M-mental
    A-abuse
    T-to
    H-humans

    Credit : a random comment I found on another vid

  27. Wow…..what a presentation. We are losing so much nowadays in the pursuit of exam grades by cramming kids' heads with 'stuff'. This presentation should be mandatory for teachers 🙂

  28. This is the idea I was trying to figure out! Nice job! The only problem is that there needs to be an explanation for the math. An explanation for hwy it is the way it is. This doesn't mention that.

    Play is absolutely great though. It gives an opportunity for students to discover things on their own.

  29. Of course I agree, though I'm constantly faced with students, parents, or colleagues who say: students should just learn the rules of math, pass the exam and get higher grades. That's all, we want degrees to look for a college or a job. What do you actually do when the teaching and school environment is (almost) all against the 5 principles? Thanks for suggestions! Congratulations mr. Dan.

  30. I feel like these tips only work when you have brilliant students who are interested in the subject. What do you do when your students have really weak foundations and just don't want to touch the subject?

  31. This is all true and fluffy! However, when people are taking college classes or even high school, the whole curriculum is faulty in that the whole educational system is set up to get the students hurry up, finish their semester and get on with their careers. If they allowed students to play, and question and allow them to struggle, nothing would get done!!! The whole system of education must change in order to apply these new principles, otherwise at the moment these ideas are nothing but utopian notions!!! At the moment colleges and universities are nothing but business! They make money off of students. They charge them a lot of money for text books, students take out big loans for education which must pay back, and they are solely pushed to get a grade fast and move on. So how do you expect your five principles to get applied with the current educational system????? How???

  32. The colored circles is quite easy. They are the factors.. That's why the primes only have one color and , say, 8 has the same color as 2 but three of them, 27 has the same color as 3 but 3 of them and 10 has the color of 2 and the color of 5…

  33. every teacher on earth should see this and use this method in there class so many do not and a lot of very intelligent students just get bored and give up because of it when curiosity is shunned by a teacher it has a horrible and lasting impact on a students education and view of the education system as a whole

  34. Lots of people are very obnoxious about having figured out the colour problem. Please remember that he is giving this as an example of a problem he would give to small children.

  35. Math is calculated illusion we created to ward away the fear of uncertainty, using math claim exacts, but when something goes against math we adapt that change into our math creating more illusion

  36. Answer to the color problem is each whole color is a prime number, therefore each non-prime number is multi-colored as it can be written as a product of prime numbers, depending on each color corresponding to its prime multiples.

  37. blue is 5 , yellow is 2, green is 3, red is prime , purple is 7. Just write multiplicative factors and there you have it 🙂

  38. Somehow this video helped me finally figure out the math problem I’ve been fighting with lol I always get my friend’s help and realized I can do this all on my own with some reverse thinking with my equations

  39. There is a connection to the colors and the pictures… the colors correspond to the numbers that multiply to make that number and how many times they are multiplied. for example, 2 is orangeish yellow. 8 has 3 sectors that are this color, so 2 x 2 x 2 (aka 2^3) equals 8. 3 is green. 2 green sectors is 3 x 3. The reddish orange numbers are prime. 4:07

  40. Summary:
    1. Start with a question
    2. Give students time to struggle
    3. You are not the answer key.. try to figure stuff out WITH them.
    4. Say yes to students' ideas and questions: take their ideas and go to conclusions.
    5. Have a play mindset

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