The last chapter of my book, Teaching Mathematics Through Problem-Solving in K-12 Classrooms is titled “Mathematics Can Be Playful. The book is available at Amazon here.
The intended audience is teachers. Teaching is itself a creative act. I use the “Exist Quantifier” to speak of teaching, as teaching is also a “bringing into being” of ideas, concepts, knowledge, and relationships, inside a classroom.
Teaching is an aspirational act. We have tremendous power to make things “exist” in our classrooms. The book is about the type of mathematics education I think should exist, all over the world.
This piece discusses that idea that math can be more playful than it often is, in our classrooms.
The limits of the human imagination are that there are no limits, unless we accept, like Hamlet, that our skulls are cages that lock us in, like nuts inside of nutshells. I prefer to think we are the kinds and queens of our own “infinite spaces”. Infinite spaces of thought.
Our imaginations are huge, and our imaginations are playful, and we can bring our thoughts into being in the world.
So I had this idea that “math is play”, or at the very least, it can be more play-ful than it often is, in the world, and in our classrooms. Mathematics doesn’t really get enough credit for its playfulness, and its true mischievous spirit.
Mathematics does have its reputation for its cold and austere beauty, as G.H. Hardy said, but I picture it taken out of its formal wear, and put into some more comfortable clothes. Mathematics should be the life of the party.
The premise is simple. Humans are born to play. Play is a basic human desire. Mathematics can be about play, or at least it can be more playful, than we usually make it in our classrooms.
Play builds hope, as Francis Su says. If we define math as play, then math builds hope. Math play builds hope. Mathematics builds hope in children, or at least, it can.
Science gets all the “hands-on” love in schools. Kids get to learn science by playing, doing, and touching. They can fly paper airplanes to learn about flight. They can mix oil and water to see what happens. Mathematical ideas need more hands-on love. They need to be translated from the perfect world in which they live (if you, perhaps, take the Platonist view of things), into our world.
Mathematical objects shouldn’t sit on dusty shelves, or in museum cases. They are to be admired, for they are beautiful. But they are meant to be played with.
Here are some axioms for a playful mathematics. If you are a teacher, these principles could serve you well to create your own “pedagogy of playful mathematics.” If you just want to play with some mathematics, they will also serve you well.
- Accept that humans are born to play. Play is irrepressibly human.
2. We can “play to learn.” Not all learning is playful, but much of it can be.
3. Mathematical ideas are meant to be played with.
4. Playing and thinking are not at odds with each other. A “playing state” can be a “thinking state”.
5. All classrooms are thinking classrooms: get kids talking, thinking, reasoning and wondering as they learn mathematics. Even simple things like “2+2” can be thought about.
6. Mathematics is full of surprises. Surprises are interesting.
7. A pedagogy of playful mathematics is a pedagogy of hopeful mathematics. All mathematics classrooms should be playful classrooms.
7b) Dr. Cathy Bruce envisions a playful pedagogy as a kind of guided play for children, with teachers guiding kids through powerful and big ideas in mathematics. Let it be so!
8. Embrace the mess of the problem-solving process- listen and talk to your students as they work. Listen to your own self-talk, while you work!
8b) Thinking is messy. We don’t always get things right the first time. Course corrections, revisions, and even total transformations of the work we are doing is needed. Thinking is a messy process, but we are thinking beings, and our classrooms are thinking classrooms.
Kids know how to play. It’s us who have forgotten. Their worlds are imaginative ones, full of ninjas and secret spaces and places. Our worlds are all “fake news” and constant pings and notifications on smartphones and politics and mortgages and taxes and “who’s going to pick up the kids from swimming?”
Kids know about play. I get yelled at if my kids don’t get to play outside as long as they want. They don’t stop playing to eat. They literally forget to eat. Kids at play are in a deep, deep, state of what Csikszentmihalyi calls “flow”. My children can build Lego worlds and cities for hours. They invent games with their toys, and go deep inside the play world they have created.
“Workflow” is not real flow. Maybe you are lucky, and you have a job that lets you get into a deep flow state. For most of us, our time is chopped into little bits. Emails that appear in bold, oh-so-important, and moving from meeting to meeting, and task to task, race to the bottom of a neverending to-do list, just to get to the end of the day, just to get up and do it again.
Kids know about play. It’s us adults who have forgotten.
Kids typically have to leave their imaginative worlds at the schoolhouse door. School is rules and assignments and bells and endless curriculum standards. You must stop thinking, and working when that bell goes, because the bell means: “get up and switch from doing math to doing art.” You must produce art on demand. You must produce math when it’s time. Don’t stop “producing.”
Mathematics classrooms aren’t usually seen as imaginative spaces, which is a pity, since the creation of mathematics is one of the most imaginative leaps ever taken by humans. Mathematics allows us to create other worlds, whole new imaginary worlds. Kids should build “castles” out of mathematics, in their minds.
Dr. Eugenia Cheng speaks of how “if you have a big imagination, you can do mathematics.” We all have big imaginations, we just don’t use them enough. I want a “no limits mathematics”, bounded only by our imaginations, and if our imaginations have no limits, then there are no limits.
Mathematics can be a creative endeavour. If we don’t see math as a generative process, a creative process, then we will not find creative thinking. The range and variety of the student work, their thinking, is where creativity is to found, and celebrated. Subtle divergences in solution paths. New insights into common problems. New lines of attack on problems using familiar tools like adding, subtracting, multiplying, and dividing. Interesting representations using drawings, or concrete materials.
Thinking classrooms are creative classrooms. Get kids talking and thinking about interesting mathematics, and they will show you that mathematics is a creative subject.
I was waiting to be bowled over by stunningly divergent solution paths, sudden and stunning “eureka” moments. Those do happen, but they are not common.
Don’t wait to be bowled over by sudden and stunning eureka moments. We should watch for more subtle evidence of creativity. For example, students using new thinking tools, or subtly tweaking a solution path or process they may have got from talking with their classmates. Creativity is there to be found in the math classroom.
It may just be a subtle variation on the math you expected. It could be a question that arises from a student, and spurs them to create more math. Sometimes, only sometimes, it’s a lightning flash of insight- an amazing and new solution path that you have never seen before.
Mathematical play builds virtues that enable us to flourish in every area of our lives. For instance, math play builds hopefulness: when you sit with a puzzle long enough you are exercising hope that you will eventually solve it. Math play builds community — when you share in the delight of working on a problem with another human being. And math play builds perseverance — just as weekly soccer practices build up the muscles that make us stronger for the next game, math investigations make us more fit for the next problem whatever that is, even if we don’t solve the current problem.
Young children play to learn about the world. Many educators have in the past tended to discount the mathematical experiences children bring to school with them. Rather, we assume children don’t have much in the way of mathematical experiences to bring into the classroom. This view reinforces the idea of math as esoteric and strange, a secret world that only teachers can give kids access to. From birth, though, children are learning about shape, quantity, and space, by exploring the world around them through play.
It’s often said, something like, “play is the highest form of learning.” A pedagogy of playful mathematics might even be the highest form of mathematics pedagogy. Kindergarten teachers in play-based programs are highly trained at helping kids learn through play. Play, in general, drops off, as kids go through the grades. What if we played more, in every grade?
We hear versions of this argument now and again. It goes something like this: “they just live in the apartment complex over there, they don’t have much life experience. Perhaps we are looking for more formal “prior knowledge” about things like mathematics, and we miss informal or tacit knowledge that we can use in our classrooms.
There is a tendency, I think, to “hide” mathematical concepts from young children. I don’t think we should. I think we should talk about infinity with three year olds. I think five year olds should twist doughnuts into toruses. I think eight year olds should break apart the standard multiplication algorithm to see how it works. I think nine year olds should break apart numbers into their atomic prime bits, and put them back together again.
All children should play with big and powerful mathematical ideas.
You might hear a conversation like this between two five year olds:
“How high can you count?”
“I can count to infinity!”
“Oh yeah? I can count to infinity plus one…”
Humans are born to wonder about number. We are born to think. I want schools for all our children where they are encouraged to play with mathematics. Math is play, or at least it should be. Math in schools is about play, or at least it could be.
Let kids play with the big and powerful ideas and tools of mathematics, and let them surprise you with the power of their thinking.
Coda: Curiosity is a State of Being
I am reading the da Vinci biography right now. Above all, he was curious. He took notes about things he wanted to ask about, to get advice, or learn about. His mind ranged across disciplines, across mathematics, science, art, engineering. His mind never stopped being curious. Curiosity was his lived state of being for his entire life.
To live is to be curious. Kids now about curiosity. If you know any young children, you probably have noticed they are very curious. They need to know “what”, “why”, “when”, “where”, and “how”. If they need to know how to find fossils, or how to make a paper airplane, they REALLY need to know, and won’t be satisfied until they do know.
As we enter the working world, it’s very possible that our curiosity becomes muted, or even disappears. Curiosity goes from something gnawing at our brains, to something that we barely even indulge, because we are busy, and because we just got 22 emails, and because our iPhone just pinged.
I keep saying that life is learning, or it should be, and being curious leads to learning. A state of curiosity is an endless state of needing to learn.
Originally published at medium.com on December 5, 2017.
Originally published at medium.com on December 5, 2017.