“What if I say ‘six less than a number’? Michelle?” the teacher asks.
“Six minus N,” Michelle answers. Incorrect.
Aubrey guesses the only other possibility: “N minus six.” Great.
The kids repeat this form of platoon multiple choice. Watched in real time, it can give the impression that they understand.
“What if I gave you 15 minus B?” the teacher asks the class, telling them to transform that back into words. Multiple-choice time. “Fifteen less than B?” Patrick offers. The teacher does not respond immediately, so he tries something else. “B less than 15.” This time the response is immediate; he nailed it. The pattern repeats. Kim is six inches shorter than her mother. “N minus negative six,” Steve offers. No. “N minus six.” Good. Mike is three years older than Jill. Ryan? “Three X,” he says. No, that would be multiply, wouldn’t it? “Three plus X.” Great.
Marcus has now figured out the surefire way to get to the right answer. His hand shoots up for the next question. Three divided by W. Marcus? “W over three, or three over W,” he answers, covering his bases. Good, three over W, got it.
Despite the teacher’s clever vignettes, it is clear that students do not understand how these numbers and letters might be useful anywhere but on a school worksheet. When she asks where variable expressions might be used in the world, Patrick answers: when you’re trying to figure out math problems. Still, the students have figured out how to get the right answers on their worksheets: shrewdly interrogating their teacher.
She mistakes the multiple-choice game they are mastering for productive exploration. Sometimes, the students team up. In staccato succession: “K over eight,” one offers, “K into eight,” another says, “K of eight,” a third tries. The teacher is kind and encouraging even if they don’t manage to toss out the right answer. “It’s okay,” she says, “you’re thinking.” The problem, though, is the way in which they are thinking.
* * *
• • •
That was one American class period out of hundreds in the United States, Asia, and Europe that were filmed and analyzed in an effort to understand effective math teaching. Needless to say, classrooms were very different. In the Netherlands, students regularly trickled into class late, and spent a lot of class time working on their own. In Hong Kong, class looked pretty similar to the United States: lectures rather than individual work filled most of the time. Some countries used a lot of problems in real-world contexts, others relied more on symbolic math. Some classes kept kids in their seats, others had them approach the blackboard. Some teachers were very energetic, others staid. The litany of differences was long, but not one of those features was associated with differences in student achievement across countries. There were similarities too. In every classroom in every country, teachers relied on two main types of questions.
The more common were “using procedures” questions: basically, practice at something that was just learned. For instance, take the formula for the sum of the interior angles of a polygon (180 × (number of polygon sides − 2)), and apply it to polygons on a worksheet. The other common variety was “making connections” questions, which connected students to a broader concept, rather than just a procedure. That was more like when the teacher asked students why the formula works, or made them try to figure out if it works for absolutely any polygon from a triangle to an octagon. Both types of questions are useful and both were posed by teachers in every classroom in every country studied. But an important difference emerged in what teachers did after they asked a making-connections problem.
Rather than letting students grapple with some confusion, teachers often responded to their solicitations with hint-giving that morphed a making-connections problem into a using-procedures one. That is exactly what the charismatic teacher in the American classroom was doing. Lindsey Richland, a University of Chicago professor who studies learning, watched that video with me, and told me that when the students were playing multiple choice with the teacher, “what they’re actually doing is seeking rules.” They were trying to turn a conceptual problem they didn’t understand into a procedural one they could just execute. “We’re very good, humans are, at trying to do the least amount of work that we have to in order to accomplish a task,” Richland told me. Soliciting hints toward a solution is both clever and expedient. The problem is that when it comes to learning concepts that can be broadly wielded, expedience can backfire.
In the United States, about one-fifth of questions posed to students began as making-connections problems. But by the time the students were done soliciting hints from the teacher and solving the problems, a grand total of zero percent remained making-connections problems. Making-connections problems did not survive the teacher-student interactions.
Teachers in every country fell into the same trap at times, but in the higher-performing countries plenty of making-connections problems remained that way as the class struggled to figure them out. In Japan, a little more than half of all problems were making-connections problems, and half of those stayed that way through the solving. An entire class period could be just one problem with many parts. When a student offered an idea for how to approach a problem, rather than engaging in multiple choice, the teacher had them come to the board and put a magnet with their name on it next to the idea. By the end of class, one problem on a blackboard the size of an entire wall served as a captain’s log of the class’s collective intellectual voyage, dead ends and all. Richland originally tried to label the videotaped lessons with a single topic of the day, “but we couldn’t do it with Japan,” she said, “because you could engage with these problems using so much different content.” (There is a specific Japanese word to describe chalkboard writing that tracks conceptual connections over the course of collective problem solving: bansho.)
Just as it is in golf, procedure practice is important in math. But when it comprises the entire math training strategy, it’s a problem. “Students do not view mathematics as a system,” Richland and her colleagues wrote. They view it as just a set of procedures. Like when Patrick was asked how variable expressions connected to the world, and answered that they were good for answering questions in math class.
In their research, Richland and her collaborators highlighted the stunning degree of reliance community college students—41 percent of all undergraduate students in the United States—have on memorized algorithms. Asked whether a/5 or a/8 is greater, 53 percent of students answered correctly, barely better than guessing. Asked to explain their answers, students frequently pointed to some algorithm. Students remembered that they should focus on the bottom number, but a lot of them recalled that a larger denominator meant a/8 was bigger than a/5. Others remembered that they should try to get a common denominator, but weren’t sure why. There were students who reflexively cross-multiplied, because they knew that’s what you do when you see fractions, even though it had no relevance to the problem at hand. Only 15 percent of the students began with broad, conceptual reasoning that if you divide something into five parts, each piece will be larger than if you divide the same thing into eight parts. Every single one of those students got the correct answer.