Two years after the first preschoolers started free pre-kindergarten classes with the proceeds from Philadelphia’s controversial soda tax, Mayor Jim Kenney sat at a tiny table in Southwest Philadelphia and pronounced it a success.

To date, over 4,000 3- and 4-year-old Philadelphians have attended programs, at a cost of $38.4 million to the city. This year, there are 2,250 children in city-funded pre-K seats.

Kenney has long said he sees pre-K as a key item on his agenda, a way to address the problems of the nation’s poorest large city, one with a struggling school system where many children arrive at kindergarten unprepared and struggle to catch up.

Providing high-quality education — city pre-K providers have to meet quality standards many other city early childhood centers cannot — leads people to “becoming contributing members and solid citizens in our city," Kenney said Friday. “They can’t do it without quality early-childhood education.”

The first children who went through PHLpreK programs are now in kindergarten and first grade, but research generally shows that when pre-kindergarten programs are strong, students show lasting academic benefits.

Kenney said the number of children served will eventually rise to 5,500 a year, though that falls far short of the unmet need for high-quality pre-kindergarten programs — in 2015, the last year for which data is available, it was estimated that 17,000 3- and 4-year-olds in the city did not have access to strong pre-kindergarten programs, though conceivably that number has dipped with the advent of the city seats.

Still, the city’s pre-kindergarten program threw itself a birthday party Friday, with Kenney and other officials trooping through Your Child’s World, a center with 42 city-funded pre-kindergarten seats in Southwest Philadelphia.

For Margaret Cobb, city-funded pre-K has been a godsend. She used to send her son, 3-year-old Trent Felder, to a different preschool with state funding when she worked a security job. Then, she got higher-paying work as a SEPTA trolley operator, and made too much to qualify for state aid.

A relative told her about PHLpreK, which has no income requirements, and she quickly got Trent a spot at Your Child’s World, an airy, clean center in a converted roller-skating rink on 71st Street. He’s learning more in city pre-K, Cobb said, and she is better able to pay her family’s bills.

“This took a big weight off my shoulders,” said Cobb.

As Cobb peered inside the classroom where Trent constructed a tall tower with blocks and some of his classmates tackled a counting activity, Kenney eyed the rest of the center, graciously accepting gifts of pretend food from one group of students, insisting he was not President Donald Trump to one skeptic, and confessing his age, “I’m 60 — that’s a six and a zero,” to a little girl in a pink sweatshirt who wanted to shake his hand.

While the city’s pre-kindergarten program has no income requirements, the vast majority of children served come from low-income families; 75 percent are at or below 200 percent of the federal poverty level, less than $50,000 for a family of four. Half of all PHLpreK families earn less than $30,000 annually.

Initially, Kenney pledged to educate 6,500 children annually in city pre-K programs, but expansion plans were put on hold when a lawsuit was filed aimed at stopping the soda tax. The suit was eventually tossed by the state Supreme Court.

Expansion beyond the 5,500 now promised is both a monetary issue and an issue of quality. Philadelphia uses only programs considered “high quality” under the state’s rating system, Keystone Stars, for city seats, and the majority of city centers do not meet that bar.

“Right now, we don’t have enough quality slots and quality centers in Philadelphia,” said Julie Beamon, director of PHLpreK.

With the city’s help, about 2,000 pre-K seats have moved from low- to high-quality under the state’s ranking system, which evaluates centers on teacher preparedness and other measures.

Rachel Sanders, executive director of the program, which has centers throughout the city, has grown her programs because of the city program, which city officials said has created 278 jobs citywide.

“Many of the children," Sanders said, “would not have had the opportunity to participate in high-quality pre-K programs without this.”