Standard 1 of the Standards for Mathematical Practice begins with the words, “make sense of problems,” which makes sense to me, since you can’t solve problems--or at least solve them thoughtfully and efficiently--if you don’t understand them.
Often, however, students skip sense-making, and instead just reach for their pencils and calculators in pursuit of the correct answer. It doesn’t matter how they get the answer as long as it’s right. But what’s more significant to me than students’ propensity for answer-getting is math teachers’ propensity for answer-getting. Not just in how they teach math, but in how they do math.
To illustrate, here’s a problem a student gave me when I was a new teacher:
Place the numbers 1 - 8 in the grid, using each number once, such that no consecutive numbers are in boxes that touch vertically, horizontally, or diagonally.
It took me around 20 minutes to solve this problem, which is about average among the hundreds of students and teachers I’ve given it to in my classroom and workshops. But a handful of teachers and students, most notably a fourth-grader who was an average math student, have nailed this problem in a minute or two. What did they do that the rest of us didn’t do? Simple: they thought about the problem before trying to solve it. They made sense of the problem, and the solution became obvious to them. The rest of us, on the other hand, just grabbed a pen or pencil and plugged in numbers.
On a positive note, most students and teachers I’ve given this problem to have eventually solved it. And in doing so, they’ve embodied the words that follow “make sense of problems...” in Practice Standard 1: "... and persevere in solving them.” But while I’m all for productive struggle, I’m also for strategic problem solving. (I’ll post a comment in a few days with a strategic solution to this problem. Please share yours too.)
My point here doesn’t just apply to outside-the-box problems like the one above. A lack of sense-making is also evident for straightforward problems like this one:
Solve: x/5 + 18 = 19
I’ve given this problem at workshops too, and most teachers have attacked it just as most students would: promptly and procedurally--i.e., subtract 18 from both sides, then multiply both sides by 5. And they’ve gotten the correct answer, 5, as a result. But they’ve also laughed at themselves when a colleague has shared a more intuitive approach: “x/5 has to equal one in order for the statement to be true, so x must be 5.”
The issue again is teachers’ knee-jerk tendency to solve problems before making sense of them. And if teachers have this tendency, it stands to reason that students will too. How, then, can you shift the emphasis from answer-getting to sense-making (and thoughtful problem-solving) in your classroom? Here are a couple of suggestions to get you started:
- Replace the age-old math mantra, “show your work,” with “explain your thinking.”
- Make “pencils down, minds on” a new mantra for the start of the problem-solving process, since sense-making begins with thinking about math, not doing math.
Image provided by GECC, LLC with permission.