ON command, Eze Schupfer reads aloud the numbers on a worksheet in front of her: “42, 43, 12, 13.” Then she begins to trace them.

“Is that how we write a 12?” her instructor, Maria Rivas, asks. “Erase it.”

“This is a sloppy 12, Eze,” she says. “Go ahead: a one and a two. Smaller. Much better.”

Eze moves to 13.

“Neater,” Ms. Rivas insists. “Come on, you can do it.” Finally, she resorts to the kind of incentive that Eze, her pink glitter sneaker barely grazing the ground, can appreciate: “You’ll get an extra sticker if you can do a perfect 13.”

Eze is 3. She is neither problem child nor prodigy. And her mother, Gina Goldman, who watches through a glass window from the waiting room, says drilling numbers and letters into the head of a 3-year-old defies all the warmth and coziness of her parenting philosophy — as well as the ethos of Eze’s progressive preschool. But she began bringing Eze and her older brother to these tutoring sessions nearly a year ago on the advice of a friend, and has since become the kind of believer who is fueling a rapid expansion of Junior Kumon preschool enrichment programs like this one, a block from the toddler-swollen playgrounds of Battery Park City.

As competition in education has spread down, the tutoring industry has followed.

Research suggests that there is little benefit from this kind of tutoring; that young children learn just as much about math, if not more, fitting mixing bowls together on the kitchen floor. But programs like Kumon are gaining from, and generating, parents’ anxiety about what kind of preparation their children will need — and whether parents themselves have what it takes to provide it. For those whose idea of enrichment is introducing “Buenas Noches, Luna” into their toddlers’ bedtime reading ritual, this is yet another reminder that no matter how much you do, there is always some other program that — who knows? — just might mean a difference.

“The best you can say is that they’re useless,” said Alison Gopnik, a professor of psychology at the University of California, Berkeley, who compared the escalation of supplemental education with Irish elk competing to see which had the biggest antlers. “The result is that they go around tottering, unable to walk, under the enormous weight of these antlers they’ve developed,” she said. “I think it’s true of American parents from high school all the way down to preschool.”

Other tutoring companies like Sylvan have also moved into the prekindergarten market. But Kumon, a Japanese import that calls itself the world’s largest math and reading enrichment program, has pushed most aggressively, admitting students as young as 2. Those young students have become an increasingly important part of its business: Kumon grew by about 12 percent last year, to 250,000 students nationwide; Junior Kumon grew by more than 30 percent. In New York, where the company is colonizing storefronts like so many Starbucks, enrollment in Junior Kumon has tripled since it began opening centers in 2007.

“Age 3 is the sweet spot,” said Joseph Nativo, chief financial officer for Kumon North America. “But if they’re out of a diaper and can sit still with a Kumon instructor for 15 minutes, we will take them.”

While worksheets are considered soul sapping in the schools where many Junior Kumon parents aspire to send their children — the kind of affluent places holding screenings of “Race to Nowhere” — at Kumon, they are the essence of the experience. Repetition, derided elsewhere as drill and kill, is considered the key to developing concentration.

PARENTS pay $200 to $300 a month for their 2-, 3-, 4- or 5-year-old to spend up to an hour twice weekly being tutored at a Junior Kumon center — 20 to 30 minutes each on reading and math. Children are then expected to do 20 minutes of homework on each subject every day, with their parents guiding and grading them. Recommended reading lists start in preschool with “Goodnight Moon” and “Each Peach Pear Plum.”

Ms. Goldman, who works at an agency representing artists, took her son, Huxton, to Kumon last September, shortly after he turned 5, in an effort to make sure, she said, “that he wouldn’t be behind the curve” when he started kindergarten the next year.

If she was hesitant, her children’s preschool was “disgusted,” she said. But she saw immediate results — Huxton began reading, adding and subtracting — and Eze, newly 3, started soon after. “These results translated into a self-esteem boost that I didn’t anticipate,” Ms. Goldman said. “They’ve gotten that there’s a thrill in achieving something. I care more about that than I care about them reading.”

She liked that it required her to participate in her children’s homework. “I treat them both with more respect now, because I see what they’re capable of intellectually,” she said. And she liked that it was reward-based, even as she recognizes that some experts decry using prizes as incentives to learn. “That’s life,” she said. “If you do something, you get something.”

Now, as she considers what extracurricular activities to keep on their schedule (her children also take swimming, karate, music, art and German classes), Ms. Goldman said she would not think of giving up Kumon.

“It used to seem like a horrible thing to do to your kid — why would you force this?” she said. “I don’t feel that way anymore.”

Officials at Kumon say it is often thought of as an “Asian” program — a description they reject. There is no competition or pressure, they say; everyone gets a perfect score on the worksheets because they are given countless opportunities to correct themselves. But Junior Kumon has proven popular among many accomplished immigrant parents.

Estee Bauernebel, who grew up in Australia of Korean, Japanese, Chinese and Malaysian heritage, took Kumon classes herself as a child. Seeing the education her son was getting in a public elementary school “was heartbreaking,” she said. “In Australia they’re doing basic multiplication in first grade.”

She enrolled him first, at 7, and then her daughter, Mia, at 3, after she became jealous of her brother’s homework — a not uncommon story at Junior Kumon centers. “She used to cry at night because she couldn’t read,” Ms. Bauernebel said. “It was so traumatic for all of us.”

Kumon officials say they had always allowed preschoolers into the programs. But about 10 years ago, they repackaged and began marketing it as Junior Kumon. “People sometimes believed ‘my kids are too young,’ ” said Mr. Nativo, the chief financial officer. “We highlighted the fact that the earlier you start, the better.”

The company created additional flash cards aimed at younger children, and altered the setup of the centers. While students in older Kumon programs work independently at tables set up in rows, Junior Kumon students sit around horseshoe-shaped tables, overseen by “assistants” — they are never called teachers. The company installed windows between the parents’ waiting rooms and the Junior Kumon rooms, to ease any separation anxiety.

“The Junior Kumon program is the anchor of our center,” said Amy DeBock, who owns a franchise in South Grafton, Mass., that serves the affluent communities surrounding it. “The students who do well in Junior Kumon, they will continue and continue and continue with the program.”

She was concerned when the economy collapsed that parents would pull their children out. But she has seen no dropoff. “I really think parents will get rid of everything else but education for their children,” she said. “With all the information coming out about the competition and the push on education, especially in our area, parents are just wanting their children to have the best, to be ahead.”

Most Kumon centers are franchises. But recognizing the prohibitive costs of rent in New York City, the parent company began opening centers itself in New York City a little over three years ago. It now has 36 in the five boroughs — 13 in Manhattan — with 14 more expected to open this year.

At the Battery Park City center, students walk in with a sense of purpose, carrying light-blue pouches with their homework. They check in on an attendance sheet and search out their folders with new worksheets, then sit down and take out their pencils.

For the uninitiated — not to mention anyone with small children — a visit to Junior Kumon can be unnerving.

Little Kate Wattenberg, a 6-year-old who recently graduated from Junior Kumon, has just finished 90 multiplication problems in six minutes. As she prepares to check out for the day, she shows the center’s director, Diana Sutowski, the entry in her reading journal for the last book she read, “Knights in Shining Armor.” “I liked that he did become a knight because he practiced a lot,” she had written.

“Can you tell me what the future tense of practice is?” Ms. Sutowski asks.

“I will practice,” Kate says.

“The past tense?”

“Past is it already happened,” Kate thinks aloud. “I practiced.”

Before she leaves, Ms. Sutowski congratulates her with a high five.

Kate’s mother, Jessica, brought her to Junior Kumon because, she said: “I do try to follow education, and I am scared for the future of our country. These children are going to be central to our social security, to our political decisions.” Like most students at Kumon, Kate comes from a background where she is likely to get plenty of educational stimulation and sophistication — when Ms. Wattenberg tells her daughters that they are going home to make pizza with a friend that evening, Kate’s younger sister, 4, cheers, “I want pesto!”

Many who study child development say that that kind of exposure — to parents who use big vocabulary words and bring books into the house — is the best preparation for a young child, and question how much programs like Junior Kumon really help.

Parents and policy makers may be ambivalent, Professor Gopnik said, but the popularity of programs like Kumon is “pushing very young children’s lives and curriculum in preschool programs more and more in this direction.”

“Part of them are saying, ‘This isn’t right, 3-year-olds should be playing in the sandbox and putting together mixing bowls,’ but then they’re thinking that maybe if the kid next door is doing it, it’ll be time to go to Harvard and my child won’t have the same advantage,” she said. “We are in a culture where education is the path to success, and it’s hard for people to recognize how deep and profound learning is when children are just playing.”

“When you’re putting blocks together, you’re learning how to be a physicist,” agreed Kathy Hirsh-Pasek, a psychologist at Temple University and an author of “Einstein Never Used Flash Cards.” “When you’re learning how to balance things and calculate how tall you can make your building, you’re learning how to be a physicist. Having your kid drill and kill and fill in worksheets at 2 and 3 and 4 to the best of our knowledge so far does not give your child a leg up on anything.”

“Yes, your child might know more of his letters than the child who spent Saturday in the sandbox,” she said. “But the people who are team players, who are creative innovators, they are the ones who are going to invent the next iPad. The kids who are just memorizing are going to be outsourced to the kids in India who have memorized the same stuff.”

Programs like Junior Kumon may not do harm, she said. But they do help push a consensus that young children need more and more structured curriculum.

At the Battery Park City Kumon, Maverick Scott, a financial analyst, pulled a book from his backpack: “Every Child an Achiever: A Parent’s Guide to Kumon.” It had helped persuade him to bring his son, Cyrus, to Junior Kumon at age 4.

“This is math and reading,” he said. “They don’t do that in preschool.”

Cyrus sometimes resisted doing the homework, he said. This was obvious when Cyrus had already counted how many pages he had to complete before they even started. Now Kumon work has become routine, Mr. Scott said, even on vacation. “It’s very natural.”

Now 5, Cyrus has advanced into the classes for older children.

Back in the Junior Kumon room, Ella Shrotri was completing her third day. Her parents brought her here shortly after her older brother started Kumon. She was more developed than he was socially, they said, but they wanted to help her develop academically.

Ms. Rivas, the instructor, helped guide Ella’s tiny hand as she traced a lower-case “a.” “A circle then, down,” she said gently. “We’ll try again next time,” she said finally.

There would be time. Ella was still two weeks shy of 3.