Solving problems we don't know how to solve
Ideally, we stay in the Goldilocks zone when we’re learning: not too easy, not too hard, but juuust right. (Image by Michal Jarmoluk)
Some of my students recently faced an interesting problem on a placement test for the brilliant Beast Academy math curriculum.
It went like this:
Ella begins at 9 and skip-counts by 7’s. Jack begins at 9 and skip-counts by 8’s. What is the next number that both Ella and Jack will say?
The thing about this problem, which several students struggled with enough to involve their parents (some of whom also struggled with it), is that it it’s more of a puzzle than a problem. “I haven’t been taught to solve these,” said one, as though this is was just some procedure to be memorized.
But the point isn’t to learn how to solve a bunch of problems like this one. Ideally, in our mathematics education, we learn to apply the skills and knowledge we have to any context that requires them: tipping someone in a restaurant, determining how much you can pay in rent based on your income, or calculating the correct amount of pressure in the steam generator of a nuclear submarine. We need to be able to think, not just carry out a procedure.
Unfortunately, much of the typical American mathematics education consists of learning to carry out increasingly complex procedures with little understanding of them. We’re teaching fifth graders stuff that used to be learned in seventh grade, but don’t be fooled — the kids aren’t any better at doing the work. I see kids from all over the country who are struggling with the basics: fractions, multiplication, perimeter, et cetera. And when it comes time to apply what they do know, they can’t. After all, the Jack and Ella problem above really just requires the ability to add and multiply…but if you’re not used to thinking, or you figure that all of math is a maze that you’ll never find your way out of, you will fail.
In order for our students to get better at math, they need to begin to see math problems as puzzles to be solved. Theoretically, they can use what they know to learn to solve puzzles of increasing complexity instead of being explicitly taught every step. This is more interesting, more efficient, and better for students’ developing brains.
After all, nobody really cares what the next number is that Jack and Ella will say. It doesn’t matter. It’s not real. The benefit of problems like these is not in finding the answer — it’s in the process of solving. It’s in the way each one stimulates the mind and allows the student to add depth to their understanding of how different concepts (for instance, addition, multiplication, sequences, and prime factorization) relate to each other.
If we want students who can succeed at higher level mathematics, let’s give them hundreds of problems to solve that don’t overtax their computation skills but instead, challenge their problem-solving skills. How many eggs are in 364 dozen? Boring — I’ll use a calculator for that every time. But how about “Dara has nine U. S. coins for a total of 68 cents. How many nickels does she have?” (another Beast Academy problem). Now you’ve got something interesting to play with that can’t be solved by simply punching numbers into a calculator.
Even as I say this, I feel like a weirdo. I wonder why the vast majority of math programs aren’t set up this way in the first place. I feel the pressure to push kids through the same old math sequence that isn’t working in order to get them ready for Algebra I in ninth grade — even if they’re coming into sixth grade with a shaky foundation of basic arithmetic. But I’ve been working on that for eight years, and I know it doesn’t work. The only thing that truly works is to help a student to strengthen their mathematical foundation, no matter how long it takes. And the best way to do that is not by drilling, but by problem solving.
The good news is that, in the long run, this approach saves time and frustration. It creates independent learners who don’t run to their parents when they encounter problems they don’t already know how to solve. Students who are well-versed in solving puzzles understand that discovering one you don’t already know how to solve is the whole game. These students are eager for what comes next, ready to flex their math chops as the problems get harder and the stakes get higher.
Best of all, when people learn how to face a challenge and prevail in mathematics, they can transfer that confidence to other domains. Such students become strong, flexible thinkers who relish a chance to grow and prove themselves. What more can you ask for out of an education?
