Those results are achieved, the companies say, by repurposing cognitive tasks initially developed by psychologists as tests of mental abilities. With technical names like the antisaccade, the N-back and the complex working memory span task, the exercises are dressed up as games that become increasingly difficult as students gain mastery.

Conceived to appeal to adults, especially baby boomers looking to stanch the effects of aging, Lumosity now draws one-quarter of its audience from students between the ages of 11 and 21, according to Michael Scanlon, the company’s scientific director. “I was taken aback that so much of our user base is so young,” he said. “The particular audience I had in mind at the earliest stages of the company was my mother.” In response to requests from schoolteachers, the fee is now waived — $15 a month — for students in their classrooms. More than 1,000 teachers and 10,000 students have enrolled this year, Mr. Scanlon said.

For the one-on-one training at LearningRx, fees are decidedly higher, from about $80 to $90 an hour in Upper Montclair. The students come with learning disabilities, with grades they want to improve in a competitive academic environment, all with hopes of just being sharper.

TAYLOR WEBSTER, 16, trains daily for lacrosse with a personal coach. “She has natural athletic ability,” said her mother, Samantha Newman-Webster. “But it’s really through her training that she has been able to achieve to the point where she’s being sought out by college recruiters.” Would brain training, the family wondered, do for her grades what physical training did for her lacrosse game?

Ms. Newman-Webster enrolled Taylor, already a B student at the private Montclair Kimberley Academy, at LearningRx in February. “I felt like I wanted to do whatever I could to make her learning easier, faster, deeper,” she said. “I knew she was going to be taking the SATs, and they say it will improve whatever you’re trying to do.”

Speaking by cellphone on the way to a lacrosse game, Taylor explained, with a laugh, what it’s like: “In the beginning your head is sore. Honestly, I had headaches after going there the first few times. It was kind of tedious and made my brain hurt. But I started getting better and better at it. It kind of became a competition for me to do better each time.”

She’s now studying for the SAT. “It used to take me an hour to memorize 20 words. Now I can learn, like, 40 new words in 20 minutes.”