Start Making Children Happy With Math
You are here because a story I published in mid-October to Medium went unintentionally viral in some math communities. It was written and titled in haste. Stop Selling Math For Its Usefulness was the name of the story. While I am grateful for the overwhelming response to the story, its tone was tilted towards ranting criticisms of how our children intersect and explore mathematics. And even though the story finished on a whimsical and softer note, I felt that there was some unfinished business here. Sure, let’s “stop” this and that, but what should we “start” doing with our children and the world of mathematics — and I mean, really doing.
If we can have this conversation outside the domain of Hunter S. Thompson’s cheeky commentary on society’s veracity, preferably at a bar, looking at each other squarely in the eyes, then I would hope that the answer would gravitate towards the ridiculously obvious — simple happiness!
I have a theory that the truth is never told during the nine-to-five hours
Hunter S. Thompson
Before you think that seems like some Herculean task or Willy Wonkaesque aspiration, I have a book coming out in 2017 called Pi of Life: The Hidden Happiness of Mathematics. It only took me the better part of a decade to cobble together my book, so I have spent considerable time trying to examine what is the universal truth about math — beyond that it is itself a universal truth:)
Beyond the the steely cold logic and sobering proofs of mathematical might is some fuzzy and warm human stuff that lies in the interstitial medium of Paul Lockhart’s words from his luminary essay/book, A Mathematician’s Lament: How School Cheats Us Out Of Our Most Fascinating and Imaginative Art Form. That wonderfully long subtitle is about as punk as it gets in the world of writing. Seeking out this ineffable goo and presenting it as mathematics’ raison d’etre challenges the status quo with similar defiance:)
It is this reservoir of emotional triumphs and losses that are too often glossed over and that humans, beautifully imperfect, created mathematics — which is beautifully perfect. The story of mathematics is compelling because, while there was lots of punctuating success, it was preceded always by the most natural course of math exploration — repeated failure. Broken paths. Dark corners. Blind alleys. Locked doors. Wrong turns. Dead ends. Every mathematician has not only spent time in this Byzantine world of numbers, but they have done so…quite happily. They were all charmed by the puzzle of mathematics. Curiosity is just not innate, its an emotional precursor to figuring out all the whys that our universe has dumped upon us. In fact, the answers become secondary. And, in math, they often escape generations before they are found.
Let’s just be thankful they were asked to be found.
Currently, in the United States, there is a discussion going on that reflects the wrong direction mathematics is heading towards. It has to do with unnecessarily heated discussions as to where and when students should learn Algebra One. I believe Lockhart nailed it over a decade ago that these kind of discussions were analogous to rearranging deck chairs on the Titanic. But, this is what happens when mathematics is stripped of its free range exploration and organic purpose, put into cans and labeled something like Algebra One. Is this really where students will see unknowns for the first time? Is this really where they will contemplate algebraic thinking — during the hormonal years of being a teenager?
If mathematics is a language, then algebra surely must be the grammar, dictionary, thesaurus and Chicago Manual of Style of this cloaked and coded language. To treat it like some appendage, miraculously extricated from the rest of math, and shown to students like an artifact in a museum or a sad animal in a zoo cage, is — and I am being polite — completely disingenuous to how students should intersect algebra.
Algebra is woven into the tapestry of mathematics. To see all the patterning, colors and weaving requires that we build bridges of understanding and curiosity from arithmetic/number sense. Take for example the inviting picture below of triangle numbers.
The path to algebraic thinking must be inclusive as possible. Everyone must arrive at the chasm that separates arithmetic from algebra. Everyone must believe and try to build this bridge together. So, if we ask children — yes, there is zero need to impart curiosity for algebraic thinking later — how many dots in the next picture, many students might draw the next picture. This is perfectly fine. We get another beautiful model of a triangle number and everyone is still on board. But, what if I asked how many dots in the 11th picture? What this question does is exhaust picture drawing. Naturally exhaust it. Perhaps focusing on the patterning of the numbers 1, 3, 6, 10, etc might be beneficial. Students will gleefully feel that they have “graduated” past drawing cumbersome pictures. You see where this is going…
However, you now offer this question. How many dots in the 1079th picture? Don’t worry. The groaning and screaming is all part of the fun. Pictures? No way. Continue the second difference pattern of +1 for a thousand terms and more? Right…we got better things to do! Well done. We are at the yawning divide between where we are and where we need to get to. Your classroom should be teeming with mini-Gausses by now — show me an easier way to find the dots in this absurd/daunting question. Show me, dammit!
Welcome to the birth of algebraic modeling…or at least the curiosity for it!
No answer…yet. But, hopefully the desire to answer this question has been installed. And, it has been installed with children still smiling and giggling. Last week, I went into my daughter’s Grade 3 class to do some Number Theory. I showed them a structure that was made with 6 cubes. I called it a perfect number. I took cubes of sizes one, two and three, and showed they “went into” this structure six cubes tall “evenly”. I then affixed my “tester cubes” and they were the “same height” as my original structure. Perfect!
I then borrowed the story of Goldilocks and The Three Bears to show that a tower of 8 cubes and a tower of 12 cubes were not perfect. One was too short(1 + 2 + 4 = 7) and the other was too tall(1 + 2 + 3+ 4 + 6 = 16). I then threw bags full of cubes on the tables of each group and asked them to find the next perfect number under 30. About twenty minutes later, two tables screamed in happiness that their skyscrapers 28 cubes high were perfect! Surfing this elation, I trumpeted “do you want to find the next perfect number?!”. Jumping up and down like I was about to throw candy at them, they yelled “YES!” I told them we don’t have enough cubes…:) With my own wide-eyed fascination, I announced that the next perfect numbers were 496, 8128 and 33550336. I wish I had a camera when I said the fifth perfect number…
Children are happy creatures. Well..hopefully. School is too often fraught with alienation and loneliness. Friendship and happiness are assumed or implied. They shouldn’t be. They are crucial to the health and well-being for all of us. Mathematics requires time, patience and guidance. But, once the smiles go away and the eyes grow dim, I am not sure if what we are doing really has any benefit or meaning. If there are no checkpoints to stop and reflect deeply, then we will create sterile and unhealthy discussions where mathematics has the feel of canned tuna and the big question is “do we serve it in Middle School or High School”.
The children at the top. Those are my kids. My book is dedicated to them, Aidan and Raya. I won’t be around forever and I might never have the time to sit down with them and tell them how much I love mathematics and how much happiness it has given me in this life. I want them to always smile when they think of mathematics.
This is why mathematics is important.
There is a famous Kafka quote that is rather long but sums up my feelings on mathematics perfectly. Just replace “Life’s” with “Mathematics”…
“Life’s splendor forever lies in wait about each one of us in all its fullness, but veiled from view, deep down, invisible, far off. It is there, though, not hostile, not reluctant, not deaf. If you summon it by the right word, by its right name, it will come.”
Right word? Happiness:)
