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TEDx Talks
Published: 2016-02-17
Source: https://www.youtube.com/watch?v=ytVneQUA5-c
a friend of mine told me recently that her six year old son had come home from school and said he hated math 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
%HESITATION and exhilarating journey of discovery or descents into tedium frustration and despair mathematical mis education is so calm and we can hardly see it we practically expect math class to be repetition and memorization of disjointed technical facts you were not surprised when students are 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 that a anyone really who wants to dazzle them with a statistic do you know that if you insert a single statistic into an assertion people are ninety two percent more likely to accept it without question yeah I totally made that up and ninety two percent is you have to wait 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 are squandering our chance touch life after life with the beauty and power of mathematical thinking I lead a workshop on this topic recently and at the end woman raised her hand and
said the experience made her feel and this is a quote like a gods as maybe the best description I've ever heard from 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 remade a cart who famously proclaimed I think therefore I am but the car to look 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 concedes that affirms and denies wheels and refuses then imagine is 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 any one of the
state 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 the steps to multiply you repeat here the steps to divide you repeat we've covered 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 what would it look like if we start with a question for example here the numbers from one to twenty now there's a question lurking in this picture %HESITATION hiding in plain sight what's going on with the colors not intuitively it feels like there's some connection between the numbers in the colors
I maybe it's even possible to extend the coloring the more numbers the same time the meaning of the colors is not clear it's a real mystery 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 if course I'm not going to tell you what it is I don't think of myself
as the mean person but I am willing to deny you what you want because I know if I rush to any answer I would have robbed you of the opportunity to learn thinking happens only when we have time to struggle and that is principle to it's not uncommon for students to graduate from high school believing that every math problem can be solved in thirty seconds or
less and if they don't know the answer there's not a math person this is the failure of education we need to teach kids to be tenacious and courageous persevere in the face of difficulty only way to teach perseverance is to give students time think and grapple with real problems I brought this image into a classroom recently and we took the time to struggle in the longer
we spent the more the class came alive with Victor students made observations they had questions like why do the numbers that last column always have one's in blue in them it doesn't mean anything that the green spots are always going diagnose and what's going on with those little weight numbers in the red segments it's important that those are always awed numbers struggling with a genuine question
students deep in their curiosity in their powers of observation they also develop the if you believe he does take a risk some students notice that every even number has orange in it and they were willing to stake a claim Ornish must mean even and then they asked is that right this can be a scary place as a teacher a student comes to you do you have
an original thought what if you don't know the answer well that this principle three you are not the answer key teachers students may ask you questions you don't know how to answer in this can feel like a threat but you are not the answer key students who were inquisitive it is a wonderful thing to have in your classroom if you can respond by saying I don't
know let's find out math becomes an adventure the parents viscose for you to 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 do you or try to figure it out together teach them that not knowing is not failure a 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 how to the class because the you know the answers will come from me they need to convince themselves and argue with each other to determine
what's true and so one thing says look two four six eight ten twelve I checked all the even numbers they all have words in them what more do you want and another student as well wait a minute I see what you're saying but some of those numbers have one wants peace some of two or three but look at forty eight stop for lunch pieces are you
telling me that forty eight is four times as even as forty six there must be more to the story by refusing to be the answer key 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 affirmed denied understands and all you have
to do is the teacher it's not be the answer key and say yes to their ideas I'm not responsible for now this one is difficult what if a student comes to if it's to plus two equals twelve you've got to correct them right 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 dis empowering yeah accepted studied and disproven is a mark of respect it's also far more convincing to be showing your wrong by your peers been told you're wrong by the teacher
allow me to take this a step further how do you actually know that two plus two doesn't equal twelfth what would happen if we said yes to that idea I don't know let's find out so if two plus two equals twelve then two plus one would be one less so that would be eleven and I don't mean to put zero which is just too would be
ten but if two is ten then one would be nine and zero would be eight and I have to admit this looks bass it looks like we broke mathematics but I actually understand why this can't be true now just the thing about it if we were on a number line if I'm at zero eight is it steps that way and there's no way I could take
it steps and wind up back where I started unless what if it wasn't a number line what if it was the number circle well then I could take eight steps and why background thirty eight would be zero in fact all of the different members on the real line would be stacked up in those its spots and where the 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 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 the two plus two equals four but we also 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 two plus two equals twelve and actually explore the consequences it takes courage to
say what if the angle in a triangle didn't add up to a hundred eighty degrees or what if there were a square root of negative one or what if there were different sizes of infinity but that courage in those questions led to some of the greatest breakthroughs in history all it takes is a willingness play and that is principle five mathematics is not about following rules
it's about playing in exploring and fighting looking for clues and sometimes breaking things Einstein Coldplay the highest form of research and a math teacher who lets their students play with math gives them gift of ownership playing with Matthew feel like running through the woods when you were a kid and even if you were on a path it felt like it all belong to you parents if
you want to know how to nurture the mathematical instincts of your children play is the answer what books are you reading a play is to mathematics in a home filled with blocks and puzzles and games and play it's a home where mathematical thinking can flourish I believe we have the power to help mathematical thinking flourish everywhere we can't afford to miss use 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 marriage thank you
