calisthenics

n. pl. - 柔軟體操, 運動
n. - 柔軟體操, 運動

Brain Calisthenics for Abstract Ideas

Like any other high school junior, Wynn Haimer has a few holes in his academic game. Graphs and equations, for instance: He gets the idea, fine — one is a linear representation of the other — but making those conversions is often a headache.

Or at least it was. For about a month now, Wynn, 17, has been practicing at home using an unusual online program that prompts him to match graphs to equations, dozens upon dozens of them, and fast, often before he has time to work out the correct answer. An equation appears on the screen, and below it three graphs (or vice versa, a graph with three equations). He clicks on one and the screen flashes to tell him whether he’s right or wrong and jumps to the next problem.

“I’m much better at it,” he said, in a phone interview from his school, New Roads in Santa Monica, Calif. “In the beginning it was difficult, having to work so quickly; but you sort of get used to it, and in the end it’s more intuitive. It becomes more effortless.”

For years school curriculums have emphasized top-down instruction, especially for topics like math and science. Learn the rules first — the theorems, the order of operations, Newton’s laws — then make a run at the problem list at the end of the chapter. Yet recent research has found that true experts have something at least as valuable as a mastery of the rules: gut instinct, an instantaneous grasp of the type of problem they’re up against. Like the ballplayer who can “read” pitches early, or the chess master who “sees” the best move, they’ve developed a great eye.

Now, a small group of cognitive scientists is arguing that schools and students could take far more advantage of this same bottom-up ability, called perceptual learning. The brain is a pattern-recognition machine, after all, and when focused properly, it can quickly deepen a person’s grasp of a principle, new studies suggest. Better yet, perceptual knowledge builds automatically: There’s no reason someone with a good eye for fashion or wordplay cannot develop an intuition for classifying rocks or mammals or algebraic equations, given a little interest or motivation.

“When facing problems in real-life situations, the first question is always, ‘What am I looking at? What kind of problem is this?’ ” said Philip J. Kellman, a psychologist at the University of California, Los Angeles. “Any theory of how we learn presupposes perceptual knowledge — that we know which facts are relevant, that we know what to look for.”

The challenge for education, Dr. Kellman added, “is what do we need to do to make this happen efficiently?”

Scientists have long known that the brain registers subtle patterns subconsciously, well before a person knows he or she is learning. In a landmark 1997 experiment, researchers at the University of Iowa found that people playing a simple gambling game with decks of cards reported “liking” some decks better than others long before they realized that those decks had cards that caused greater losses.. Some participants picked up the differences among decks after just 10 cards.

Experts develop such sensitive perceptual radar the old-fashioned way, of course, through years of study and practice. Yet there is growing evidence that a certain kind of training — visual, fast-paced, often focused on classifying problems rather then solving them — can build intuition quickly. In one recent experiment, for example, researchers found that people were better able to distinguish the painting styles of 12 unfamiliar artists after viewing mixed collections of works from all 12 than after viewing a dozen works from one artist, then moving on to the next painter. The participants’ brains began to pick up on differences before they could fully articulate them.

“Once the brain has a goal in mind, it tunes the perceptual system to search the environment” for relevant clues, said Steven Sloman, a cognitive scientist at Brown University. In time the eyes, ears and nose learn to isolate those signs and dismiss irrelevant information, in turn sharpening thinking.

Good teachers at all levels already have their own techniques to speed up this process — multiplication flash cards, tips to break down word problems, heuristic rhymes — but scientists are working to tune students’ eyes more systematically and to build understanding of very abstract concepts.

Fractions, for one. Most American middle school students, though they understand what fractions represent, don’t do so well when tested on their ability to change one fraction, like 4/3, to another, like 7/3, by adding or subtracting (many high school students bomb these tests, too).

Enlarge This Image

Michal Czerwonka for The New York Times

PHYSICS IN ACTION From left, Christopher Allen, Andrea Leal and Gabe Boros conduct an experiment at New Roads. Gabe said he uses "tricks" to eliminate wrong answers.

Perceptual Learning

How does a student learn from gut insinct? Try these samples to find out.

Basic Math

Measurements and Graphing: Match the equation to the graph and learn to perceive basic measurement concepts.

Positive and Negative Feeback

Extreme Ball: Time a fan to blow and push a ball attached to rubber bands.

Extreme Population: Help your city reach a population of one million citizens.

Stabilize Ball: Time a fan to blow and stabilize a ball attached to rubber bands.

Stabilize Population: Help your city's population stabilize at 500,000 citizens.

In a 2010 study, researchers at UCLA and the University of Pennsylvania had sixth graders in a Philadelphia public school use a perception-training program to practice just this On the computer module, a fraction appeared as a block. The students used a “slicer” to cut that block into fractions and a “cloner” to copy those slices. They used these pieces to build a new block from the original one — for example, cutting a block that represented the fraction 4/3 into four equal slices, then making three more copies to produce a block that represented 7/3. The program immediately displayed an ‘X’ next to wrong answers and “Correct!” next to correct ones, then moved to the next problem. It automatically adjusted to each student’s ability, advancing slowly for some and quickly for others. The students worked with the modules individually, for 15- to 30-minute intervals during the spring term, until they could perform most of the fraction exercises correctly.

In a test on the skills given afterward, on problems the students hadn’t seen before, the group got 73 percent correct. A comparison group of seventh graders, who’d been taught how to solve such problems as part of regular classes, scored just 25 percent on the test.

“The impressive thing for me was that we went back five months later, after the summer, and the gains had held up,” said Christine Massey, director of the University of Pennsylvania Institute for Research in Cognitive Science and a study co-author. When the younger students returned as seventh graders in the fall, they scored just as high as they had the previous spring on tests of fractions that they had not seen. Knowing what a fraction represents is one thing, the authors say, but repeatedly seeing and manipulating all those fractions by slicing and cloning drives the concept home once and for all.

The research team found similar results in high school sophomores who practiced with the software that Wynn Haimer used, working to match algebraic equations with graphs.

“I find that often students will try to solve problems by doing only what they’ve been told to do, and if that doesn’t work they give up,” said Joe Wise, a physics instructor at New Roads School, where the study was done. “Here they’re forced to try what makes sense to them and to keep trying. The brain is very good at sorting out patterns if you give it the chance and the right feedback.”

The modules are less demanding than problem sets, but they’re not video games — they’re homework. “To be honest, I’ve got so much to wrap up this year that I haven’t really used the program much,” said Gabe Boros, one of Mr. Wise’s students. “I did try it a couple of times and improved a little, but often I have to guess or use tricks to eliminate the wrong answers.”

Which is the whole idea: Subtle shortcuts are the very stuff of perceptual intuition. With practice, neurons in the visual cortex and elsewhere specialize to identify these signature patterns, and finding them frees up mental resources for deductive reasoning, to check answers or to move on to harder problems. Such perceptual intuition isn’t cheating — it’s what the big-shot experts do. In the case of graphs and equations, it includes making quick judgments about where lines should intercept the axes and about their slope, even when that is not at all obvious.

On the surface at least, this may sound like the approaches that SAT or LSAT prep courses take, using time-saving strategies and informed guessing. But there is a difference, researchers say. The prep courses teach to the test, but perceptual training tools are aimed at the underlying skills — manipulating fractions, graphing equations. “It’s not how well you do, but how well you learn,” as Mr. Wise put it.

Ideally, perceptual training does more than breathe life into abstract principles, the same way that repairing engines instills a lived experience of internal combustion mechanics. It also primes students to apply the principles in other contexts. This ability to transfer, as it’s known, is fundamental to scientific reasoning and is among the highest goals of teachers at all levels.

Here, too, perceptual learning may help. In a series of experiments, researchers at Indiana University have had students practice on software that models scientific principles, like positive feedback loops. In one, middle school students use a mouse to add “slime mold” to a slide and watch as it spreads faster the more they add. The process fuels itself.

“The kids who have seen this situation will transfer it to other positive feedback loops, like global warming,” said Rob Goldstone, director of the cognitive science program at Indiana University. “The more ice that melts, the more heat that’s absorbed into the earth, the warmer it gets, which melts more ice, and so on.”

“Once they have the concept, I can refer back to it,” said Nancy Martin, a science teacher at Jackson Creek Middle School in Bloomington, Ind., who has worked with Dr. Goldstone. “I can say, ‘Remember how the ants worked, or the slime; does that have anything to do with what we’re discussing today?’ ”

In an education system awash with computerized learning tools and pilot programs of all kinds, the future of such perceptual learning efforts is far from certain. Scientists still don’t know the best way to train perceptual intuition, or which specific principles it’s best suited for. And such tools, if they are incorporated into curriculums in any real way, will be subject to the judgment of teachers.

But researchers are convinced that if millions of children can develop a trained eye for video combat games and doctored Facebook photos, they can surely do the same for graphs and equations.