“He knows about different fish, different birds, different species,” Ms. Stewart said. “He’s on it.”

Chase took the city test for the public schools’ gifted and talented kindergarten program, but missed the 90th-percentile cutoff, she said.

Ms. Stewart, a single mom working two jobs, didn’t think the process was fair. She had heard widespread reports of wealthy families preparing their children for the kindergarten gifted test with $90 workbooks, $145-an-hour tutoring and weekend “boot camps.”

The owner of one Manhattan tutoring company, Bright Kids NYC, says the parents of the 120 children her staff tutored spent an average of $1,000 on test prep for their 4-year-olds.

Ms. Stewart used a booklet the city provided and reviewed the 16 sample questions with Chase. “I was online trying to find sample tests,” she said. “But everything was $50 or more. I couldn’t afford that.”

She understands why wealthier families pay for test prep. “They want to help their kids,” she said. “If I could buy it, I would, too.”

Another Bloomingdale Head Start parent, Lawanna Gillespie, a medical aide who said her son Zion also missed the 90th-percentile cutoff, was surprised to hear that prep materials existed for a kindergarten test.

“There are books you can buy with sample questions?” she said. “I never knew that.”

Delores Mims, an education director at the Head Start, said, “Our parents are at a disadvantage.”

Bloomingdale, with headquarters at West 109th Street, is a highly regarded Head Start, and Ms. Mims says she has several 4-year-olds who she feels would do well in a gifted kindergarten program.

Founded as a preschool in 1960 even before the federal Head Start program was established, Bloomingdale became a national prototype. To this day, it’s considered a model, and educators worldwide visit it — recently from Iceland, Indonesia and the Netherlands.

An early 4-year-old graduate, Patrick Gaspard, who grew up to become a White House political adviser, thinks so highly of Bloomingdale that he took one of its founders, Susan Feingold, to meet President Obama.

This week, Bloomingdale marks its 50th year by graduating 100 4-year-olds, 98 percent of them black and Hispanic and all poor (to qualify, a family of three must earn less than $18,300).

Not one of the 100 will be attending one of the city’s gifted kindergarten programs in the fall, according to Bloomingdale officials.

In contrast, in 2007, Ms. Mims says, when she was a teacher, she knew of a half-dozen who were accepted. Back then, under a decentralized selection process, teacher assessment, classroom observation and interviews all played a role.

That approach was criticized as vulnerable to political manipulation and racial favoritism, since districts could take into account increasing diversity in making selections.

“The process was fractured and inconsistent, and programs were too often gifted in name only,” the city education chancellor, Joel I. Klein, said in an e-mail message.

In 2008, Mr. Klein made the score on a citywide standardized test the sole criteria for admission. Mr. Klein is a leading testing proponent for everything from grading schools to rating teachers, and he predicted that a citywide test would be a more equitable solution.

Since then, there have been two major developments, neither looking much more equitable than the old system. Blacks and Hispanics in gifted kindergarten programs dropped to 27 percent this year under the test-only system, from 46 percent under the old system (66 percent of city kindergartners are black or Hispanic).

And a test-prep industry for 4-year-olds has burgeoned. Bige Doruk opened Bright Kids NYC in 2009, and there is so much demand that she says she’s opening a second site this month. She runs a two-month “boot camp” for the gifted test in the fall that includes eight one-on-one 45-minute sessions and two test-prep books for $1,075.

It’s already half-booked, Ms. Doruk said, “and I haven’t even publicly announced it.”

Last year, of 120 children she prepared for the city test, she says 80 percent scored at least 90. “Prepping makes a difference,” she said. “Prep brings anxiety down; children get used to an adult giving them the test and the format.

“A lot of middle- and upper-middle-class families rely on this,” she added.