“Brain calisthenics” — such as computer-based exercises in quickly linking graphs to equations — help students internalize abstract ideas and see patterns intuitively, say cognitive science researchers in a *New York Times* story.

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.

In a 2010 study, UCLA and Penn researchers used perception training to teach fractions to sixth graders in a Philadelphia public school.

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.

Notice how few students understand fractions.

Reading the comments reminded me of the parable of the six blind men and the elephant. Every reader seems to think the research proves their theory: Kids need more practice; kids need to construct knowledge, kids need real-world examples, kids need visuals.

I’m not doing well with abstract ideas this week, due to a horrible cold and a racking cough, but here’s UCLA’s graphs ‘n equations module.

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