Teaching math as if there’s only one correct way to solve a problem makes us think that we’re problem-solving, but instead, we’re “answer-getting.” I’ve seen it so many times, but none bothers me more than watching elementary and middle school students solve word problems in this way.
Consider this typical middle school prompt: A store is selling 6 bags of marbles for $18. What is the unit price for a bag of marbles? When I read this problem, I picture a child looking up at me and asking, “Does ‘of’ mean multiplication?” It has happened to me so many times when I visit math classrooms.
There’s no secret code. Of could mean “multiply,” but it might not. These are the highly counterproductive questions that children ask when they have been presented with a “single way” to solve word problems, such as looking for keywords.
In this example, the students may immediately multiply 6 × 18. If you then ask them why the unit price of a single bag of marbles would cost $108 — and be so much more than the price of 6 bags of marbles — they will look at you with uncertainty. This is the end result of answer-getting.
Problem-solving is a distinct cognitive experience. Instead, we ask, what is happening in the problem? It is not mindlessly following a single prescriptive set of steps. The way to solve this problem, and every problem, is to understand what is happening. But that means there will be many paths to the answer. How I understand the problem might be quite unlike how you understand it.
The right way is the wrong way
When we are taught to rely on a singular, step-by-step process as the true way to solve a math problem, we turn off our problem-solving brain. These skills require continuous work to keep them sharp, and the constant reliance on someone else’s “exact right” method dulls them. Over the years, we may even lose at least some of our problem-solving acumen by not using it.
This reliance also discourages courage — we need to take chances to solve problems, and insistence on following a singular method prevents us from risking wrong answers via experimentation.
We can solve problems in many distinct ways. In fact, trying different approaches is fun as well as instructive, and it is necessary when problem-solving gets hard — which is often when the problems are most worth solving. Engineers who write software code or build bridges make a conscious attempt to solve problems in more than one way, even when a solution is readily available to them.
Why not solve it and move on?
First of all, if you dig deeper to find more than one solution, you can decide which one among them is less expensive, more durable, or more elegant — whichever outcome matters most to you. Second, and perhaps more significant, when problem-solving gets really hard and the way ahead isn’t clear, you need to be ready to try anything. And the first step of the “try anything” approach is to back up and examine a problem from every angle, or at least from more angles than you initially see.
In the real world, of course, we often resort to looking at problems from fresh angles out of desperation. “Try anything” is the motto. As one member of a two-working-parents household with elementary school twins during the COVID-19 pandemic, we often were compelled to try anything to solve problems regarding work, social distancing protocols, on-and-off remote school, and limited childcare.
To expose the myth of a single correct method for the sham that it is, we need to understand the consequences of answer-getting versus problem-solving. Because we’ve been brainwashed into believing that answer-getting is good and because most of us spent years in answer-getting math curricula, we don’t realize the negative effects it has on us.
Here are typical ways we respond in an answer-getting environment:
- The mind goes blank. For a moment, nothing occurs to us because we’re not allowed to use our minds creatively.
- Racing heart. We react anxiously as we try to remember how the teacher did the math on the board. What was her first step again?
- Negative self-talk. For a moment, we have the germ of an idea, an instinct about how to start solving a challenging math problem, but because we’ve been conditioned to seek the answer only one way, we chastise ourselves for thinking we know better than what we’ve been taught, and we revert to standard operating procedure.
- Reluctance to talk through questions and concerns. We’re embarrassed to bring up these issues with others, assuming they are “right way” adherents. This reluctance to involve others is an obstacle to a creative, collaborative process.
The overarching effect of an answer-getting system is disempowerment. We feel defeated before even attempting to work on a problem.
Here are recent conversations I have had with children and adults on what this sort of math feels like:
“I want to use decimals. The teacher wants me to use fractions for no reason. I just have to do what he says. There is no freedom to do the math the way you want to do it, even if my way is easier for me. No one listens to me.”
“I actually remember getting dinged on a high school math test even when I had the right answer, but I had solved it my own way. As a teenager, that made me furious. Now looking back as an adult, I think about it like tennis. If you are drilling me so I learn or improve a new skill like backhand volleys, then I can understand the reasoning for forcing a specific approach. But if you have no reason whatsoever for forcing your way on me, it still steams me to think about it.”
A problem-solving approach conjures significantly contrasting responses — responses that reflect a sense of empowerment and courage. Ideally, schools would teach math with problem-solving as its driving principle rather than the myth of a single right way. To approach this ideal, however, we need to understand what problem-solving is all about.
Math should be taught as a collaborative process, much as other subjects are taught.
How to counteract the myth
We make math a performative rather than a learning experience. When the teacher asks the class, “What is the answer to 63 plus 37?” he turns math into an individual sport.
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Add the myth of speed, and each student is scrambling to come up with the answer first and win the game. The answer becomes the only thing that matters, and both understanding and collaboration fall by the wayside.
No doubt, some of you might wonder if I’ve lost my math mind. After all, we need to get the answers right so that we can purchase the right amount of carpeting to cover a room’s floor or make sure that our rocket makes it to the moon.
Again, this is an issue of integrative complexity. Of course, we need to know what 63 + 37 equals. But if that’s all we know, then we’re missing out on a lot of what math offers.
Fortunately, we can learn in a way that we obtain precision as well as other benefits. Consider again 63 + 37. What if the teacher framed the question this way: “Don’t tell me the answer. It’s 100. How would you start calculating 63 plus 37 in your head? What is your first step?” Now math is a process.
I have gotten the chance to hear second graders’ brains working at this moment many times. Each time is a joy. One might say, “I broke this up as 60 plus 30 plus 3 plus 7. And the next thing in my head I saw was that it was 93 plus 7. And then I knew that was 100.” Another second grader might offer another option: “I looked at this for a moment, and I saw that 3 and 7 make 10. So I knew I had a 60 plus 30 plus 10. And I know that’s 100.”
This is the math people need in their lives. This is what is needed to build bridges. It’s also how you build a deep math sense.
Math should be taught as a collaborative process, much as other subjects are taught. We often view math as distinct from other subjects in K–8, as something that must be taught as an individual sport where everyone is on his or her own to come up with the right answer first. Other subjects are taught as team sports, ones where process matters, where students don’t rely on tricks, where students are encouraged to work together, and where a variety of ways to answer may be acceptable. But when it comes to math, collaboration and process work are subordinated or eliminated.
