“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).

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.