How We Learn

by Benedict Carey

  Not this time. The mixed-study group got nearly 65 percent of the artists correct, and the blocked group only 50 percent. In the world of science, that’s a healthy difference, so the researchers ran another trial in a separate group of undergraduates to double-check it. Once again, each student got equal doses of blocked and mixed study: blocked for six of the artists, mixed for the other six. The result was the same: 65 percent correct for those studied in mixed sets, and 50 percent for those studied in blocks. “A common way to teach students about an artist is to show, in succession, a number of paintings by that artist,” Kornell and Bjork wrote. “Counterintuitive as it may be to art history teachers—and our participants—we found that interleaving paintings by different artists was more effective than massing all of an artist’s paintings together.”

  Interleaving. That’s a cognitive science word, and it simply means mixing related but distinct material during study. Music teachers have long favored a variation on this technique, switching from scales, to theory, to pieces all in one sitting. So have coaches and athletic trainers, alternating endurance and strength exercises to ensure recovery periods for certain muscles. These philosophies are largely rooted in tradition, in a person’s individual experience, or in concerns about overuse. Kornell and Bjork’s painting study put interleaving on the map as a general principle of learning, one that could sharpen the imprint of virtually any studied material. It’s far too early to call their study a landmark—that’s for a better historian than I to say—but it has inspired a series of interleaving studies among amateurs and experts in a variety of fields. Piano playing. Bird-watching. Baseball hitting. Geometry.

  What could account for such a big difference? Why any difference at all? Were the distinctions between styles somehow clearer when they were mixed?

  In their experiment, Kornell and Bjork decided to consult the participants. In a questionnaire given after the final test, they asked the students which study method, blocked or interleaved, helped them learn best. Nearly 80 percent rated blocked study as good or better than the mixed kind. They had no sense that mixed study was helping them—and this was after the final test, which showed that mixing provided a significant edge.

  “That may be the most astounding thing about this technique,” said John Dunlosky, a psychologist at Kent State University, who has shown that interleaving accelerates our ability to distinguish between bird species. “People don’t believe it, even after you show them they’ve done better.”

  This much is clear: The mixing of items, skills, or concepts during practice, over the longer term, seems to help us not only see the distinctions between them but also to achieve a clearer grasp of each one individually. The hardest part is abandoning our primal faith in repetition.

  Math scores, however, don’t lie.

  • • •

  Despite its leadership in technical innovation and discovery, the United States has long lagged in math education, usually ranking around ninth or tenth in the world—as measured by performance in eighth graders—far behind countries like South Korea and Finland. Experts and officials are perpetually debating how to close that gap, and in the late 1980s the nation’s premier organization of math teachers—the National Council of Teachers of Mathematics—convened a meeting of leading educators to review and reshape how the subject was taught. It was a gargantuan job and, like so many grand-scale efforts, became bitterly contentious. The central disagreement was over teaching philosophy: Do students learn most efficiently in classes that emphasize the learning of specific problem-solving techniques, like factoring and calculating slope? Or do they benefit more from classes that focus on abstract skills, like reasoning and number sense—knowing, for example, that 2/3 + 3/5 is greater than 1, without having to find a common denominator? The former approach is bottom-up; the latter is top-down.

  This being education, the debate was quickly politicized. The top-down camp became “progressives” who wanted children to think independently rather than practice procedures by rote. (This group included many younger teachers and university professors with doctorates in education.) The bottom-up camp became “conservatives” who saw value in the old ways, in using drills as building blocks. (Its core was made up of older teachers and professors of math and engineering.) The math wars, as they were known, caused confusion among many teachers. Math education was virtually devoid of decent research at the time, so neither side had the ammunition to win the argument. The typical experiment involved academics or outside experts descending on a class or school with a novel math, history, or writing curriculum and announcing “improvements” that were hard to interpret, given that the measures (the tests) were often new themselves, and few experiments tracked the teachers’ commitment to the program.

  Teachers, then as now, see enough new approaches come and go over time that many become constitutionally skeptical. Plus, this clash over math was (and is) about philosophies, and in math of all subjects it is results that matter, not theories. “One of the things you see that’s so baffling, when you’re a new teacher, is that kids who do great on unit tests—the weekly, or biweekly reviews—often do terribly on cumulative exams on the same material,” Doug Rohrer, who was a high school math teacher in Palo Alto, California, in the late 1980s, told me. “The kids would often blame the test or even blame me explicitly, saying I gave them trick questions.” What made those questions so tricky, explained Rohrer, was that “math students must be able to choose a strategy—not just know how to use it—and choosing a strategy is harder when an exam covers many kinds of problems.” For practical teaching issues like this one, the math wars debate was irrelevant.

  Rohrer toyed with the idea of developing a different curriculum, one that rejected the idea of teaching in blocks (two weeks on proportions, say, then two weeks on graphs) and instead mixed problems from previously studied topics into daily homework to force students to learn how to choose appropriate solution strategies rather than blindly apply them. To solve a problem, you first have to identify what kind of problem it is. Rohrer was lying on his futon in his studio apartment one day, staring at the ceiling, and thought, Okay, maybe it’s time to write a textbook of mixed problems. He soon found out that someone already had.

  That someone was a retired Air Force officer turned math teacher in Oklahoma City. In the 1970s, John H. Saxon was teaching math at Rose State College and growing increasingly exasperated with the textbooks the college used. The books’ approach left students fuzzy on the basics, and quick to forget what they’d just studied. So one day Saxon decided to write out some problem sets of his own, with the goal of building algebra skills differently—i.e., more incrementally—than the standard curriculum. His students improved fast, and soon he was developing entire lesson plans. Between 1980 and 1990, Saxon authored or coauthored twelve math textbooks for kindergarten through high school, plus a couple of college texts. His central innovation was a process of “mixed review.” Each homework assignment included some new technique—solving simultaneous equations, for example—along with a number of problems from previous lessons, say, solving equations for x. Saxon believed that we grasp a new technique more clearly when using it alongside other, familiar ones, gradually building an understanding of more abstract concepts along the way. His books built a following, mostly among private schools, homeschoolers, and some public districts, and he soon became a lightning rod in the math debate. Saxon was a bottom-up man. He thought the reformers were dangerous and they returned the compliment.

  Rohrer wasn’t sure what he thought about the math wars or, for that matter, about Saxon. He does remember picking up the Saxon books and looking at the chapters. They were different, all right. The lessons, in Rohrer’s view, were not in logical order. Yet the problems were mixed, from all sorts of different lessons—precisely the approach he thought would help his own students.

  He let it drop. Rohrer was ready to walk away from math teaching altogether, and entered graduate school in experimental psychology.
It was in 2002—eight years after he finished his degree—that he again began to think about learning. For one thing, he’d read the 1992 Schmidt-Bjork paper on motor and verbal learning. And he returned to the central problem he’d had while teaching high schoolers. His students didn’t need to remember more. Their weakness was distinguishing between problem types—and choosing the appropriate strategy. Mixing problem types (he had not yet heard the term interleaving) looked like it might address just this weakness.

  We’ve done well so far to avoid doing any real math in this book, but I think it’s time to break the seal. In the past decade, Rohrer and others have shown in a variety of experiments that interleaving can improve math comprehension across the board, no matter our age. Let’s take a look at one of those studies, just to show how this technique works. We’ll keep it light. This is fourth grade geometry, and a little review never hurt anyone. In 2007, Rohrer and Kelli Taylor, both at the University of South Florida, recruited twenty-four fourth graders and gave each a tutorial on how to calculate the number of faces, edges, corners, and angles in a prism—given the number of base sides. The tutorial is self-explanatory and perfectly doable, even for people with math allergies. In the diagrams below, b is the number of base sides:

  Half the children performed blocked study. They worked eight “face” problems (FFFFFFFF), then eight “edge” problems (EEEEEEEE), eight “corner” problems, and eight “angle” problems in a row, with a thirty-second break in between, all in the same day. The other half worked the same number of each type of problem, only in randomly mixed sets of eight: FCEAECFA, for example, followed by CAAEFECF. The tutorials were identical for each group, and so were the problems. The only difference was the order: sequential in one group and mixed in the other. The next day the children took a test, which included one of each type of problem. Sure enough, those in the mixed-study—interleaved—group did better, and it wasn’t close: 77 to 38 percent.

  One fairly obvious reason that interleaving accelerates math learning in particular is that tests themselves—the cumulative exams, that is—are mixed sets of problems. If the test is a potpourri, it helps to make homework the same. There’s much more going on than that, however. Mixing problems during study forces us to identify each type of problem and match it to the appropriate kind of solution. We are not only discriminating between the locks to be cracked; we are connecting each lock with the right key. “The difficulty of pairing a problem with the appropriate procedure or concept is ubiquitous in mathematics,” Rohrer and Taylor concluded. “For example, the notorious difficulty of word problems is due partly to the fact that few word problems explicitly indicate which procedure or concept is appropriate. The word problem, ‘If a bug crawls eastward for 8 inches and then crawls northward for 15 inches, how far is it from its starting point?’ requires students to infer the need for the Pythagorean theorem. However, no such inference is required if the word problem appears immediately after a block of problems that explicitly indicate the need for the Pythagorean theorem. Thus, blocked practice can largely reduce the pedagogical value of the word problem.”

  Rohrer puts it this way: “If the homework says ‘The Quadratic Formula’ at the top of the page, you just use that blindly. There’s no need to ask whether it’s appropriate. You know it is before doing the problem.”

  The evidence so far suggests that interleaving is likely applicable not just to math, but to almost any topic or skill. Badminton. History (mix concepts from related periods). Basketball (practice around the free throw line, not repeatedly from the line). Biology. Piano. Chemistry. Skateboarding. Blindfolded beanbag throwing, for heaven’s sake. Certainly any material taught in a single semester, in any single course, is a ripe target for interleaving. You have to review the material anyway at some point. You have to learn to distinguish between a holy ton of terms, names, events, concepts, and formulas at exam time, or execute a fantastic number of perfect bow movements at recital. Why not practice the necessary discrimination skills incrementally, every time you sit down, rather than all at once when ramping up for a final test? As mentioned earlier, many musicians already do a version of mixed practice, splitting their sessions between, say, thirty minutes of scales, thirty minutes of reading new music, and thirty minutes of practicing familiar pieces. That’s the right idea. Chopping that time into even smaller pieces, however—of fifteen minutes, or ten—can produce better results. Remember: Interleaving is not just about review but also discriminating between types of problems, moves, or concepts.

  For example, I still take classes when I can in Spanish and Spanish guitar. Every time I look at a list of new vocabulary words, I take that list and combine it with a list of at least as many older words. I do more kinds of mixing with the guitar (maybe because there’s more to mix than words and reading). I do one scale, two or three times, then switch to a piece I know. Then I go back and try again the portions of that just played piece—let’s say it’s Granados’s Spanish Dance Number 5—that I messed up. Play those two times, slowly. Then I’m on to a (different) scale, followed by a few bars of a totally new piece I’m working on. Enough for one pass. I take a break and play a few riffs from the first tune I ever learned, “Stairway to Heaven” (somehow it never gets old), and after that I’m ready to dive into Spanish Classical.

  That is interleaving. And it’s sure to be highly individual, far more effective for some subjects or skills than for others. The important thing to know is that you’re essentially surrounding the new material or new skill set with older stuff, stuff you already know but haven’t revisited in a while, whether it’s a Jimmy Page solo or a painting by Georges Braque.

  As I read it, the science suggests that interleaving is, essentially, about preparing the brain for the unexpected. Serious climbers and hikers have a favorite phrase: It’s not an adventure until something goes wrong. By wrong they mean wrong wrong. A rope snaps; the food supply flies overboard; a bear crawls into the tent. I think interleaving prepares us for a milder form of wrong. Every exam, every tournament, every match, every recital—there’s always some wrinkle, some misplaced calculator or sudden headache, a glaring sun or an unexpected essay question. At bottom, interleaving is a way of building into our daily practice not only a dose of review but also an element of surprise. “The brain is exquisitely tuned to pick up incongruities, all of our work tells us that,” said Michael Inzlicht, a neuroscientist at the University of Toronto. “Seeing something that’s out of order or out of place wakes the brain up, in effect, and prompts the subconscious to process the information more deeply: ‘Why is this here?’ ”

  Mixed-up practice doesn’t just build overall dexterity and prompt active discrimination. It helps prepare us for life’s curveballs, literal and figurative.

  Part Four

  Tapping the Subconscious

  Chapter Nine

  Learning Without Thinking

  Harnessing Perceptual Discrimination

  What’s a good eye?

  You probably know someone who has one, for fashion, for photography, for antiques, for seeing a baseball. All of those skills are real, and they’re special. But what are they? What’s the eye doing in any one of those examples that makes it good? What’s it reading, exactly?

  Take hitting a baseball. Players with a “good eye” are those who seem to have a sixth sense for the strike zone, who are somehow able to lay off pitches that come in a little too high or low, inside or outside, and swing only at those in the zone. Players, coaches, and scientists have all broken this ability down endlessly, so we can describe some of the crucial elements. Let’s begin with the basics of hitting. A major league fastball comes in at upward of 90 mph, from 60 feet, 6 inches away. The ball arrives at the plate in roughly 4/10 of a second, or 400 milliseconds. The brain needs about two thirds of that time—250 milliseconds—to make the decision whether to swing or not. In that time it needs to read the pitch: where it’s going, how fast, whether it’s going to sink or curve or rise as it approaches (most
pitchers have a variety of pitches, all of which break across different planes). Research shows that the batter himself isn’t even aware whether he’s swinging or not until the ball is about 10 feet away—and by that point, it’s too late to make major adjustments, other than to hold up (maybe). A batter with a good eye makes an instantaneous—and almost always accurate—read.

  What’s this snap judgment based on? Velocity is one variable, of course. The (trained) brain can make a rough estimate of that using the tiny change in the ball’s image over that first 250 milliseconds; stereoscopic vision evolved to compute, at incredible speed, all sorts of trajectories and certainly one coming toward our body. Still, how does the eye account for the spin of the ball, which alters the trajectory of the pitch? Hitters with a good eye have trouble describing that in any detail. Some talk about seeing a red dot, signaling a breaking ball, or a grayish blur, for a fastball; they say they focus only on the little patch in their field of vision where the pitcher’s hand releases the ball, which helps them judge its probable trajectory. Yet that release point can vary, too. “They may get a snapshot of the ball, plus something about the pitcher’s body language,” Steven Sloman, a cognitive scientist at Brown University, told me. “But we don’t entirely understand it.”

  A batting coach can tinker with a player’s swing and mechanics, but no one can tell him how to see pitches better. That’s one reason major league baseball players get paid like major league baseball players. And it’s why we think of their visual acuity more as a gift than an expertise. We tell ourselves it’s all about reflexes, all in the fast-twitch fibers and brain synapses. They’re “naturals.” We make a clear distinction between this kind of ability and expertise of the academic kind. Expertise is a matter of learning—of accumulating knowledge, of studying and careful thinking, of creating. It’s built, not born. The culture itself makes the same distinction, too, between gifted athletes and productive scholars. Yet this distinction is also flawed in a fundamental way. And it blinds us to an aspect of learning that even scientists don’t yet entirely understand.