My wife loves her vegetable garden. Michele will spend hours with her fingers in the rich soil, nurturing her tomatoes, zucchini, lettuce, green beans, jalapenos, watermelon, and sweet potatoes.
I love Michele’s garden too: I get to eat the produce it generates. The problem is, so do all of the local wildlife which didn’t get the memo from the township that we own this land now and they need to move along.
So over the past several years, I’ve built three fences for my wife’s garden. It’s not that they were bad fences, or that they didn’t stay up. It’s that the problem we were solving kept changing.
My first fence was awesome: nice, straight metal posts hammered into the ground at equal intervals around the garden’s perimeter; welded wire fencing tightly stretched between the posts and anchored into the soil to prevent forward-thinking groundhogs from digging underneath. And it worked exactly as intended. Those groundhogs stayed out all summer, as did the foxes, cats, squirrels, and other small mammals.
The deer? Not so much. To them, the fence merely defined the boundaries of a conveniently-located salad bar. We didn’t get much of a harvest that first year.
Year two brought fence number two. Taller posts. A second tier of fencing above the first one. This fence kept out the deer, and a lot more of the food made it into the house.
I thought I was done with fences, until this year when Michele said, “I want to expand my garden.” So with the help of my son, we added more fence around the new section, and the result is the verdant miracle you see in the picture above. And in case you were wondering: ratatouille made with fresh-picked zucchini is absolutely spectacular.
Math as an Innovation Incubator
What’s all of this have to do with math? you ask. Well, the fence was built to solve a particular problem, and it wasn’t a solution I could look up in the answer key to a book. It wasn’t a nice, neat problem created by a textbook-writer either. It was messy (both literally and figuratively) and complicated, and the parameters kept changing as the situation evolved.
There was certainly computation involved in solving the problem: how much square footage we needed to have in the garden, how much fencing to buy, how far apart we’d put the posts. But it was more than that. Every version of the fence needed to innovate–granted, on a very small scale–in order to work in our unique situation.
I learned how to innovate by solving problems throughout my life. It’s not something I learned in school, at least not directly. But our modern world needs more innovators, and if we’re going to build these skills in our students, the mathematics classroom is a fantastic place to start. There are two habits of mind you can build into your math instruction to begin creating more innovators. I call them Conjecture and Chaos.
Conjecture: Capitalizing on Curiosity
The human brain is naturally curious. We are wired to wonder. John Medina, molecular biologist, explains in his book Brain Rules that it comes from our need to explore our environment.
Babies are born with a deep desire to understand the world around them, and an incessant curiosity that compels them to aggressively explore it. This need for explanation is so powerfully stitched into their experience that some scientists describe it as a drive, just as hunger and thirst and sex are drives. (Medina, 2014, p. 247, emphasis mine)
Most math classrooms never capitalize on this curiosity. We hand students predefined and preprocessed problems for which there is one, single correct answer and then walk them step-by-step through the procedure for finding that answer. Worse, we wait until after they’ve drilled the skills to even introduce them to that problem.
Instead, try sparking their curiosity by introducing your students to an intriguing conundrum and then using it to introduce the skills they’ll need to solve it. You may be surprised at how many of your students figure the skills out on their own in the course of solving the problem.
When students struggle to get an answer, encourage them to guess and explain their partial reasoning. Never end with the answer to a question, either. Always ask your students follow up questions like these:
- Why do you think so?
- How do you know?
- Why did you solve it that way?
- What was hard about solving that problem?
- How did you overcome the difficulty?
Professional mathematicians exhibit this kind of curiosity all the time. It’s what propels them to explore new ways of thinking about the world and math. James H. Simons, founder of the National Museum of Math, turned his incessant curiosity into a billion dollar empire. And he did it by studying math.
Chaos: Embrace the Mess
Math class needs to be a place with a healthy dose of chaos. I’m not talking about the kind of chaos my students experienced during my first year as a teacher, when I struggled with even the most basic classroom management tasks. I’m talking about the messiness of real problem solving. Thomas Edison famously said, about the path to finding the right filament for his light bulb, “I have not failed. I’ve just found 10,000 ways that won’t work.”
One of the biggest problems with math instruction is that we show students the one way that someone else figured out would work, and then ask them to rehearse it to perfection. We never let kids experience the ten thousand ways that won’t work. I’m not talking about aimless, endless anarchy here, but we can’t expect kids to learn to cook without making a mess in the kitchen.
Terry Tao is another world-class mathematician, one who embraces the Chaos of real math work. When he tackles a complex problem, he never takes a straight line from problem to solution. The work usually involves weeks or even years of false starts, failed attempts, and flawed logic.
If one of the world’s greatest mathematicians is a messy problem solver, why wouldn’t we let our students be messy? When students struggle to solve a challenging problem, don’t be too quick to come to their rescue and provide them with the next step. Focus on the process.
One way to do this is to give students the correct answer to the problem up front. Now there’s no pressure to find an answer. Instead, the challenge is to figure out exactly how someone got to that answer. Another strategy is to use lots of non-routine problems. Consider this one:
If Citizen’s Bank Park were completely filled with popped popcorn, how long would it take the Phillies to eat it all?
Given the Phillies’ level of success during the 2015 season, perhaps we ought to seriously consider filling the ballpark with popcorn. But I digress.
The point here is that there is no way to be certain about the answer. Students need to defend and explain their thinking, moving them towards deeper thinking practices that will serve them well and help them be more innovative.
Tomorrow’s Innovators in Today’s Classroom
With these two ways of thinking embedded in the culture of your classroom, you can begin to nurture students who can think creatively and find innovative solutions to complex problems. What other ways do you use math instruction to promote innovation and reasoning? How do you get your students to think outside the box? Share your ideas in the comments below.
5 Principles of the Modern Mathematics Classroom
Want to learn more? You can explore Conjecture and Chaos more deeply, as well as three more habits of an innovative mind, Communication, Collaboration, and Celebration, in my upcoming book, 5 Principles of the Modern Mathematics Classroom.
Medina, J. (2014). Brain rules: 12 principles for surviving and thriving at work, home and school (2nd ed.). Seattle, WA: Pear Press.