Somewhere in the mid-1990s, my professor of neurolinguistics, Slavko Milekic, showed something to the class: he threw a piece of chalk into the trash can from across the room. “How did I do that?” he asked in his always-delighted, full-of-wonder tone. “I could not possibly have done that calculus problem in the time it took me to turn, see the trash can, raise my hand, accurately assess the mass of the chalk, and predict the correct direction and force to communicate to my arm.”
It wouldn’t surprise me if Slavko knows calculus. But he definitely didn’t use it to throw the chalk.
We do math all the time. We are really good at it. You’re not bad at math. Our math notation is bad at math. Our math teachers are bad at helping us understand it because we’re trapped in using notation that is convenient to use for upper-class Europeans who are aspiring to have their work typeset with a Gutenberg press.
Nicky Case has had plenty of journal papers typeset with the descendants of the Gutenberg press. But at least the ones I’ve read are about explicitly not using those technologies to teach, instead using structured toys that give the player exploratory opportunities. They’re clever, often funny, sometimes melancholy. They are presented with sincere emotion because that’s how we understand where to focus our intentions.
When teaching with these tools, I find that it’s really important not to tell the kids what to discover. They play with these ideas, building them out according to their curiosity, discovering things that I wouldn’t have thought to tell them to discover.
I first learned how to draw a network diagram in college, probably to conceptualize neural networks. It was considered, in the early 1990s, a pretty advanced way of looking at problems. But Nicky Case turned it into a toy, and now my students are playing with it to represent the flow of heat in their computers, the flow of calories in an ecosystem, the flow of resources in economies, and other dynamic systems.
This way of thinking about problems — where it’s reduced not to static Latinate and Greek characters in black and white written from left to right on a flat piece of paper, but instead in as many dimensions as necessary, using symbols designed for the purpose and described in colors that are appropriate to the conversation — is having a profound effect already. My students are learning ways to talk about systems like winning strategies for Rock, Paper Scissors, like how increasing populations of fish can cause a crash in the number of birds, like how one can find beauty in the harmonies of patterns that move at the same rate but different distances.
When we restrict the entire range of thought processes to those who already share the most in common with the inventors of the processes, we waste the ideas when our society wants — needs — to change its perspective on the world. If we can’t think in terms of interdependent systems; if we can’t recognize that beauty and fun are indicators that a process finds the (admittedly local) minimum of energy expenditure, then we literally cannot solve the challenges of climate change. We will lack the perspective, we will fail to adapt, our creativity will falter, and we will continue shoveling coal into the sacrificial burner of infinite growth until the harm finally consumes those who are causing it.
As we pass through this first generation of simulation-aware students, I expect that new forms of description will arise that don’t imagine themselves to be written in black ink on a piece of ground wood pulp. I want my students to be able to gesture, build, virtualize those concepts for their intellectual and creative descendants.
We have some real opportunities here to understand the world and change our relationship to it. And if we don’t — if, instead, we refuse math to students who need it because Isaac Newton’s brilliance came with an extremely uncommon set of circumstances — then not only do we risk the world around us, but we also risk never finding all the Isaac Newtons in it.