After all, risk-taking is important to Avenues, a giant start-up itself. Avenues World Holdings Inc. raised $85 million through at least two private-equity firms (they are about to close another round of financing), and investors, including pension funds, typically expect returns of 15 to 30 percent on investments. Enrollment will grow to 1,100 students next year, Whittle says, with applications up 10 to 15 percent over last year. (The school, which goes up to 9th grade now, will expand to 12th grade in 2015-16.) Alan Greenberg, Avenues’ president, has said repeatedly that the school could have enrolled with the full 1,635 seats filled but chose to admit students more selectively. He also sent letters to families who declined the school’s offer of admission, explaining to them how they were making a huge mistake.

But for many among a new generation of wealthy New York parents without legacy roots at Horace Mann or Brearley, Avenues, without any legacy of its own, was a welcome option. Trinity accepted only 3.6 percent of nonlegacy, nonsibling kindergarten applicants for entry this fall. (By comparison, freshman acceptance into Harvard was 5.8 percent.) Jacquie Hemmerdinger, from the A.P.A., recalls trying to get her 4-year old twin daughters into Dalton for kindergarten. When they went to the parent interview, the admissions director, Elisabeth (Babby) Krents, held up the essay Hemmerdinger had written about her girls, covered in red ink. Who wrote this? Krents asked. “I got all hot and sweaty,” Hemmerdinger says. “I thought it was a bunch of typos.” Krents then told her that it was the best essay she had read that admissions season. (“I teared up,” Hemmerdinger remembers.) Her girls scored well on the kindergarten entrance exam, and Krents told her that they also fit the school’s need for diversity. She and her husband, a real estate developer, lived in a sprawling Georgian revival house on a tree-lined street that happens to be in Queens. Either way, they never got into Dalton. Avenues was happy to take them.

Avenues’ 215,000-square-foot building, a former grocery wholesale warehouse, is a monument to its ideas about doing things differently. Many Manhattan private schools feel like labyrinthine rabbit warrens; here light streams into the classrooms and the cafeteria and through its stairwells and envelops the gym, where the motto “You Miss 100 Percent of the Shots You Never Take” is emblazoned on the wall. Nearing the end of the first year, teachers and administrators still regularly work 14-hour days in teams, making improvements for next September: immersion in the lower school is being tweaked so that students will learn more math in English, more social-studies classes will be offered in other languages and so on.

But for all of its intentions, Avenues is also starting to feel like a lot of other New York City private schools. It is almost impossible to get into kindergarten — the school received 350 applications for 25 kindergarten slots — and many parents who applied thinking it was a “safety” school were shocked to be rejected. The upper school is considering offering French, a decidedly un-21st-century language, next year because of demand. There are now committees through which parents can formally vent, and 22 new positions being added to the A.P.A. Middle-school students will have a shorter school day next year; despite Avenues’ hopes that students would not overschedule themselves, many were falling asleep on their way to after-school activities. Parent-teacher conferences that included kids may revert to the old format of just parents and teachers.

Maybe there is something to those schools with 100 years of tradition. The staid staples that many Avenues’ parents initially debated — galas, bake sales, book fairs — are now materializing, by parent demand. In late April, a founding-families gala took place at Chelsea Piers. The previous week, the school was buzzing at its first ever book fair. Sam Talbot from “Top Chef” was doing a presentation on the fifth floor, while preschoolers in the black-box theater were making bookmarks with stamps of the Empire State Building and the Statue of Liberty. Face painting and glitter-tattoo stations were packed. A balloon woman was making swords and flowers, but also elaborate wings, pink flamingos and, for one boy, a bodysuit that was built around him. Schulman — ever attentive to the parents — was exuberant. She knew that a book fair, with Avenues panache, would comfort anxious parents. “I knew that books were important and that you turn it into a fun family day,” Schulman said, beaming. “This is the glue.”