There Exists An Elegant, Lovely, and Inspiring School Mathematics.
As a newcomer to the poetry and beauty of mathematics, but a dedicated learner, I have often been stunned into submission, struck dumb by its power. Learning mathematics is humbling. I know a tiny slice, and I am but an elementary school teacher, but I want to know more.
Emily Dickinson had God in mind when she wrote, “he stuns me by degrees.” I can honestly paraphrase and say, “mathematics stuns me by degrees.” Formal mathematics may have been the path I chose as a young man, if poetry and literature and philosophy didn’t call first. Well, I would have studied mathematics, or religion, or both. There may not be much of a difference in the end.
One day you find out about the Banach-Tarski paradox. Stunning blow to my understanding of this, our 3D world. Another day, the square root of negative one. Spend some time on the ladder of infinity, leading up to the next tower of infinity, which maybe, possibly, leads to another tower, and your logic, your very senses, your sense of the world- will be disrupted, forever more.
I have seen children blown away by their own insights at times. I have seen creativity and wonder. I have tried to make classrooms places of surprise and wonder. We found pi, because we needed to, and you could do that too, with string and a ruler and some can lids. We broke apart numbers, tore them apart into their atomic bits we call prime numbers, and put them back together again.
We used, as Jordan Ellenberg says of mathematics, our “common sense by other means.” We opened the door a crack into the world of mathematical thinking and reasoning. Some kids walked right through, and some stayed behind.
It is my wish that no kids stay behind. What’s behind the door? Cold hard logic, yes, and numbers, yes, but also shape, space, rates of change, quantity, graphs, chance, and many other things, only some of which are in our curriculum.
Behind the door is play, that most human of desires, as Francis Su would have it, and the thing we so desperately need in our mathematics classrooms. Mathematical objects are meant to be taken down from mental shelves-dusted, examined, and played with. Mathematics is not a museum full of relics, and signs that say “look, but don’t touch.”
For Su, mathematics is, or could be, essential to human flourishing. He tells a story of a man who escapes prison walls by teaching himself mathematics. He walks through a door, in his mind, and enters the mathematical world. It’s a better one, a nicer one, and a more beautiful one, than the world of the prison. He stays there, mentally free, until he is physically free.
But this piece is not about curriculum reform, at least not explicitly. The backwards “E” symbol stands for “there exists…” It may be used to bring into verbal being, into mathematical notation, either a unique object, or a class of objects. I am thinking of one such unique object.
There exists, a school mathematics, that is beautiful, elegant, and true.
There exists a school mathematics that is rigorous and formal, and yet playful.
There exists a school mathematics where kids play with numbers, composing and decomposing with them, like a painter with paint, or a master Lego builder with Lego.
There exists a school mathematics that builds a fundamental conception of arithmetic for kids from the fact that 1+1=2.
There exists a school mathematics where kids play with shape, space, arrangements, quantity and chance.
There exists a school mathematics that coherently ties together big ideas of mathematics, and is not a chopped salad of curriculum expectations.
There exists a school mathematics that contains ideas newer than those of Newton and Liebniz.
There exists a school mathematics that leaves all kids prepared for an increasingly data-driven and mathematical world.
There exists a school mathematics that combines cold hard steely-eyed utility, with starry eyed gazing upon beauty.
There exists a school mathematics that leaves kids begging for more, going home and night to think about interesting tasks and problems, and coming back to school the next day with fresh energy, insights, and ideas.
This school mathematics could, and should, exist. But it doesn’t. Instead we are tied up by politics, standardized testing, and fighting over ideologies and turf. We are bound by cycles of curriculum reform, and periodic scares like “Sputnik”. We are tied up in words by op-eds on either side of the Math Wars.
We fight, and we change things, but we miss what could exist. It is my mission in life to bring that mathematics into being.
Many students might know of mathematics as a closed door, as Bob Dylan sang about:;
“the door is closed for evermore/if indeed there ever was a door.” That door must not be closed. Beauty awaits. Elegance and insight awaits. Confidence and the power of unlocking your own mathematical thinking await.
We need to treat this as a matter of some urgency, so let’s think it, write it, and talk it into being. “There exists a school mathematics…”