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[Music]
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you
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[Music]
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so we begin here and I'm out for a run
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or a walk and as I often do I look down
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the sidewalk I want to go this way and
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this way and then this way do I go this
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way and then this way no way I cut
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across just like you would and you know
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you would and what's the reason for that
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if you look closely in the shadows you
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can see there's a warn path just like in
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every field in every Park there's a warn
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path where people have cut across so do
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you think to yourself after all your
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years of schooling do you think hey
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that's the Pythagorean shortcut probably
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not okay so why do we do that the answer
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is not because humans are lazy it has
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more to do with as the rapper most def
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put it it's strictly mathematics okay so
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we don't think to ourselves
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hey that side length is shorter than the
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other two side lengths put together or
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we don't say to ourself the square on
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the hypotenuse is equal to the sum of
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the square and the other two sides right
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we just cut across because we know that
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it's faster and it is mathematics here's
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another example right beside my kids
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lovely school an example of the
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Pythagorean shortcut on a hill okay but
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this time is stretched in in three
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dimensions and wouldn't that blow kids
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mind if you were to draw that triangle
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on a balloon for example what happens to
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the angles but this blue kids Minds
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right because they never thought of the
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mathematics this way okay that
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Pythagorean shortcut is in every
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textbook in the last hundred years but
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it's also in our world secondly when we
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let kids play with interesting math this
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sort of thing happens so all I did was a
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kid put these we were playing with the
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squares right so again the sum of the
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two squares equals the sum of the other
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squares but he said okay these three
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squares can't make a right angle
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triangle and truly they can't okay but
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this is an example of the sort of thing
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that kids wonder if we let them because
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you know none of us get out in the world
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and we say I just really really stink at
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reading we don't tend to say that the
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negative messages about mathematics are
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a little more common and we don't want
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that so Stanford education professor Joe
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bowler has been quite influential in
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that she talks about mathematical
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mindsets and we need to have them and we
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need kids to have them so negative
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messages like this don't help us
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so with my own children inspired by my
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own practice I was thinking you know
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what happens when we just play with
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numbers okay and none of us remember
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this but we learned to count we're born
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with something inside us some sense of
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number one few many depending on what
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culture were born into but so I said
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what's a big number and I got five
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because they were both quite little and
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then I got ten so this 10 Ness we can
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count out 10 objects kids learn to do
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that sometimes by rote then they know
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that 10 objects is 10 ok having 10
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cookies is a good thing so this is how
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they get along their way and then
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eventually they learn that numbers get
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bigger and bigger and bigger
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so we were just playing around on a
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piece of paper and I said you give me a
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big number so finally I got million out
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of them right and so I said well okay
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what if you add 1 ok well that's a
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bigger number and then we started
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talking about what happens if you add
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some zeros and they were seeing that
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this you could have this long string of
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numbers
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and that surprised them because they're
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so little and they don't understand
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place value the number system anything
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like that we were just playing
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but then so kalam in his room he emptied
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his piggy bank because he just couldn't
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keep that you know 2 dollars and 42
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cents in so he's he dumped it on his bed
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for some reason I don't know why he was
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naked at the time but and so he says I
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have infinity money I have more money
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than I can count so he's counting it
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he's like like a young Scrooge McDuck or
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like a young Montgomery burns or even
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like a young Donald Trump and he's
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counting it right and so isn't that an
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interesting definition of infinity and
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we don't teach that in school trust me
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we don't so now if you think of infinity
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is more than you could ever count in an
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infinite number of lifetime's this
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lifetime or any other that's a pretty
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interesting definition he has no
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training in math he wasn't in school yet
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and then just from two days ago I got
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this from Twitter because that's how I
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operate I said what do kids say about
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infinity and so Jennifer sent two Osos
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grade sevens in Peel District School
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Board said this and I give Cole Griffin
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and Owen a lot of credit for this it's
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the thought of a number that never ends
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but because it's so indescribable as the
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equation shows it doesn't matter if we
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add one subtract one or add one same
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thing it's still infinity so all of a
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sudden is something gigantic and massive
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that you can wonder about something that
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mathematicians work on but yet twelve
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year olds can grapple with
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just another example of a problem that's
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out there so I give you one penny
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actually I give you the proposition I
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say I'll give you a penny now and then
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I'll give you two more tomorrow do you
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want that for 30 days or do you want a
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million dollars
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and kids are always all like just give
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me the million bucks right now and they
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tell you what they're going to buy but
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it when they start doing the math
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they're surprised right so after ten
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days you have two dollars or something
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but after 30 days you have this pile of
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pennies where it's around ten point
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three or ten point seven million dollars
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and so that's the power of exponential
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growth but they don't know it as that
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but they're surprised and it's an
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interesting result
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just another example of something that's
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interesting
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numbers themselves ok so 100 squares
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pretty common in school we teach kids to
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count by twos count by threes factors of
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24 but what's pictured in yellow are the
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prime numbers so those indivisible
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little atoms of numbers you know and so
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there's no real visible pattern okay if
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we did multiples of 2 there'd be a
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pattern there would be columns it'd be
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nice and neat kids are really interested
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in that in my experience they seem to
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get more spread out as you get along the
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number system so interesting and so
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intriguing
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and so what do kids think when they get
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to school
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one kid in grade six said math is a way
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of describing this world and she wanted
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to go to things like money but how much
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and then baking but how long in these
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things important well they're
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quantifiable right and do we want kids
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to come out of come out of school and
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come home and say I had math today we
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learn how to describe our world or do we
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want to leave grade 12 or college and
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university we leave all our schooling
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and we say hey I learned math I learned
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how to describe the world I think that's
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a pretty good outcome
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but what are my own children say so
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column on the top I'm just admitting it
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right now I said what is math he just
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repeated that back from a couple weeks
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before he doesn't know what multiplying
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is infinity times infinity is still
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infinity but that's what he said math is
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which is fun but so aleck in the bottom
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left he said 1 plus 1 is 2 try it if you
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know a 3/4 five-year-old say what is
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math kindergarten age students they'll
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sit they'll give you an example of
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arithmetic and math surely is arithmetic
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but it's a whole lot more of that than
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that
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arithmetic is interesting kids will say
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that they'll just give you that one
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example of the fact they know
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just one of those mic drop moments so I
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put it out to Twitter and so Malka
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Rosenfeld's daughter lzd said space and
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time any distance and numbers ok so
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again do we want kids to come home from
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school and say I use math today it's all
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about space time distance in any numbers
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I think we'd like that I'd be pretty
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amazing but sometimes what does happen
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ok and so Jamie Mitchell great teacher
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in Burlington his daughter Stella and I
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hope you don't feel this way he says
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math is numbers she says math is numbers
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and annoyingness and hatred I want to
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start the hashtag save Stella and all
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the Stella's out there because math is
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more surprising wonderful and
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interesting than that somehow she's got
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there in her school experiences and her
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math teacher father's quite heartbroken
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about it but I love this quote because
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no not all kids feel that way but some
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do okay and that's a problem so
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what if every kid came out of school and
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they said math is surprise and Wonder
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and playing with number and a whole lot
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more than that but what if they said
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that that would be a pretty great
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outcome
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and
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so taking you through where I sit as a
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teacher of mathematics in the school
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system so this is school board materials
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from 1991 and what they're basically
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saying is teachers aren't going to teach
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like you were taught and that's true and
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sometimes parents want to come in and
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get mad about us and we've all seen
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maybe those like Common Core math things
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that go viral and Facebook parents get
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mad okay so back up to 1957 Russia
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launches a satellite called Sputnik the
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space race has started America says
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we're not good enough we're not as good
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at the Soviets as math good at math as
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the Soviets and so they invent something
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called the new math 1957 so almost 60
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years later we're still fighting this is
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right after 1989 the details don't
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matter but continent-wide Reform
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National Council of teachers of
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mathematics in the United States we
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decide we want to change it we decide
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the old ways won't do
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1991 okay we still have these same
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conversations so there's new math
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there's old math there's new old math
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there's old old math and we're still
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fight okay sometimes we do call it the
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math wars and so what happens how are we
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portrayed in the newspaper you know I
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track this stuff people say don't read
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the comments I say always do under every
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article about math test scores math
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teaching it could hold me to this kids
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these days they can't make change
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it always appears and people get mad
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about it so you know math is about a
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whole lot more than just making change
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and yes kids should be able to make
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change accurately but we have machines
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called cash registers just to do that
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and we're better at thinking and cash
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registers are good at carrying out
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routines
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so what do we need in our classrooms we
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need kids to be skilled with number
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concepts we need them to be skilled with
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arithmetic but we need them exposed to
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the big powerful and interesting
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concepts make no mistake about it and we
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need our classrooms to be wide open
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spaces for students thinking okay
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because we're born into it that's our
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human birthright an uninterrupted stream
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of consciousness that were born into
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from birth until death
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we have that we have our power of our
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thinking and so do we teach primarily
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through worksheets or wonderings
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probably a little bit of both kids want
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to know if pie ever ends I know because
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we had a Wonder wall and some kid just
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stuck that to it I didn't write that
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that's what kids want to know they
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wonder and I want for our classrooms
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that they're talking spaces kids should
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talk to kids so math just doesn't happen
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between me and my textbook or me and my
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teacher I talked to other kids talking
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changes everything in my experience
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they should be wide open spaces for
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thinking as I said and kids should get
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to conjecture which is to map as
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hypotheses to science right just
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advancing these powerful conjectures
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with their own thinking and finally
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wondering and grappling with really
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really big ideas
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like what makes a circle a circle and
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one kid wondered that and it's actually
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a really profound question so we can say
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it's the most perfect expression of
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circumference compared to diameter and
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that ratio is PI but I don't know if
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that's satisfactory to a kid or not or
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what happens if you slightly deform it
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you know we all know what happens to our
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car tires if it starts going flat for
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example kids want to know kids want to
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know things like this how many people
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have lived on this planet we can apply
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mathematics and reasoning to this so we
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can come up with an estimation tool we
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can look up a few facts and it's
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checking for that reasonableness and
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kids could come up with an interesting
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number really interesting question this
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one you probably can't read but it says
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if you divide a number up so much that
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it equals zero is that number actually
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infinity okay so you can't actually
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divide by zero but you imagine if if I
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divide any number like eight and I
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divide that number by the number next to
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zero for all intents and purposes I
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would have an infinite number of parts
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and this kid was reasoning that through
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but that something happens at zero right
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you just can't do it and kids want to
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know and it's not in our curriculum but
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it probably should be and so I say to
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you let kids lead with the power of
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their thinking because they have it and
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they must okay our classrooms must be
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thinking spaces for children for all our
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students
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and let's student thinking lead is kind
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of my philosophy and it's the reason why
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there's so many seemingly messy photos
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of student work because when you get in
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that moment and you get talking with
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them and you see that work develop it's
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just amazing okay they start to find a
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pattern as it says at the bottom they
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lead with the power and strength of
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their thinking this was a project we did
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for the Ministry of Education so loving
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the math living the math that was called
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and this picture I always loved because
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it's all these different solutions to
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the same problem they look different
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they are different they're reasoned out
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differently there's different
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mathematics in a lot of them so we had
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some teachers in a room looking up and
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we said what's all the different ways
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this question could be answered and
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there was still more okay and that's
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surprise on the teacher end we thought
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we knew what they were going to do and
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they did something even better than that
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astonishingly diverse and then you get
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kids like this one his insight so we
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have this triangle pattern he saw it as
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groups of three being stuck on the side
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that's what he's circled but he had to
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explain it to his classmates it was
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about to burst out of him okay and other
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students saw the pattern differently
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maybe growing up or down and that's how
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he saw it and he needed to share it but
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these sudden startling insights these
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Eureka moments where they're surprised
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and stunned even by the power of their
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own thinking we need that this is one of
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my favorites so if I offer you $20 off
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or if I offer you 20% off and you have
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to pick well he reasoned out that $100
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was the tipping point for that so where
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the discount becomes better basically
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and this was his hypothesis but if the
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story is a little better okay the
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teacher in that classroom said he's he's
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on special education support he never
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talks this kid hates math but he was the
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one and he was working with a couple
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other kids he needed to share this so
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badly because his thinking was activated
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okay he understood the context I guess
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about shopping he understood he had this
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insight and he needed to share it and
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that's what we want those reluctant kids
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that are just it's bursting out of them
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to share their thinking and they
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surprised us the teacher was surprised
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and this happens over and over and over
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again when you let kids talk to each
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other and we want all of us saying this
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I know when I'm being ripped off and so
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would we want our sons and daughters
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nieces nephews and friends we want from
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ourselves that we come home from school
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and say I learn how to use mathematics I
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know when I'm being ripped off I think
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that's pretty good I just had to do with
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soft drinks pop so but they said they
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reasoned ever they said I can't take
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this it's a ripoff and that leads to
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this next one is one of my favorites I
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walk down the hall and I said a buck 54
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32 grams of baked potato chips are you
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kidding me nobody even likes those and
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so I came up with this problem right
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what's a fair price and so kid one kid
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said 27 cents and they looked up a bunch
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of chips on the internet and some flyers
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and 27 cents was what they came up with
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and then they knew from the power of
00:16:59
their own mathematics suit I'm assuming
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they never bought these again and so
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great book how not to be wrong by Jordan
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elenberg there's no equations in it I
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highly recommend it but he called
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mathematic mathematics the extension of
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common sense by other means and I would
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love for all of us if we say hey I went
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to school I learned math I learned how
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to extend my common sense by other means
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and those means are mathematics I would
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love that for all of us and for all the
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children out there I would like it very
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much if we saw mathematics as a source
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of surprise and Wonder and not just
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something that happens in textbooks are
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not just something that happens in
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schools and I'll give the last word to
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kids as I always like to do because they
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often say it best so one kid in grade
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six said math is the most powerful force
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in the universe use it wisely thank you
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very much
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you
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[Applause]
