Tricia Pietravalle remembers her days in preschool: Playing in the sandbox, doing some painting, listening to stories. So when she met with her son’s preschool teachers, she was taken aback when they showed her “this whole grid of how they’re evaluating him, academically, socially, emotionally.”

Joseph was learning letters and numbers in full-day classes and making connections she didn’t expect from a 3-year-old. “It was overwhelming — in a good way,” she said.

For generations, children have started school with kindergarten, when they’re 5. Not in the District. For most families in the city, school now starts at 3.

In his State of the Union address, President Barack Obama called for a dramatic shift in early childhood education: free public preschool for all low- and moderate-income 4-year-olds. Opponents voiced concern about increased spending and ineffective government programs. Supporters argued that the initiative could jump-start learning, using the latest brain research to erase some of the disadvantages faced by children from low-income families.

The District is already doing something more ambitious. Nearly 13,000 of the city’s roughly 15,000 3- and 4-year-olds are attending public preschool.

“Any 3-year-old in the District is guaranteed a spot to be at a full-day preschool program. That’s basically unheard-of,” said Daphna Bassok, an assistant professor at the Curry School of Education at the University of Virginia.

So as national and state leaders consider a major expansion of public education, the city offers an example of how that that can play out on the ground.

Inside Amidon-Bowen Elementary School, where an AppleTree Early Learning public charter school has taken over a couple of classrooms, 3-year-olds were buzzing about happily one recent morning. Some were playing with blocks; some had puppets and were making shadows. Pietravalle’s daughter Lucy and several other children sat at a small table, and teacher Briana Bourne asked, “What do you want to know about outer space?”

When Lucy said she wanted to learn about planets, Bourne asked, “What sounds do you hear in the word ‘planet’?”

Lucy wrote a P on her paper, and Bourne gave her a high-five. “What other sounds do you hear in PLLAANNETT?”

It’s a rare opportunity for the city, said Jack McCarthy of the AppleTree Institute for Education Innovation, which runs public preschools at seven sites in the District, including the one in Southwest Washington that two of Pietravalle’s children attend.

“Here is a place where funding is in place, universal preschool is policy,” McCarthy said. If the quality could be improved and ensured for all students, he said, “we could close the achievement gap here in five years.”

The president’s call — outlined last week without any funding plan — reflects a growing national movement to enroll children in public school before kindergarten. Federal Head Start programs have been available for low-income children for decades, and many special education students start school before kindergarten. But more states are adding classes for 4-year-olds.

In the Washington area, when and how a public-school education begins is influenced by where you live. In Loudoun County, most children start with half-day kindergarten when they’re about 5. In Arlington County, parents can pay, on a sliding scale, for Montessori programs for 3- and 4-year-olds, or they may qualify for a free program for low-income 4-year-olds. Charles County offers full-day classes for 3-year-olds at six schools with high percentages of low-income children.

In the District, officials created a program about three years ago, leveraging federal Head Start funding to offer seats to 3- and 4-year-olds in public schools with many children from low-income families. The program has a research-based curriculum, improved professional development and ongoing assessments of the children, according to Danielle Ewen, director of the office of early childhood education for D.C. public schools.

About 6,000 students are enrolled in D.C. school system’s preschool program, and 7,000 or so more attend charter schools and other programs that offer free preschool. The National Institute for Early Education Research at Rutgers University estimated the annual cost, across the board, at $122 million in 2010-11.

The institute gives the traditional school system’s preschool standards a favorable rating, in part, because they require teachers to have a college degree and ensure class sizes stay small.

But Andrew Coulson of the Cato Institute questioned whether the school system is the best provider of early education. According to his analysis, traditional D.C. public schools spend, on average, $29,000 per student on educating students from kindergarten through 12th grade. “I don’t think anyone would say children in D.C. are getting $29,000 worth of education.”

Michela English, of the nonprofit Fight for Children, which is investing $10 million in early education programs in the District, said the city has been a leader in recognizing the value of early education. “But the coverage is still spotty, and the quality is not uniform, so outcomes are not uniform,” English said.

Fight for Children rates AppleTree an excellent program, and it is one to which the group has given private money to match a large federal innovation grant.

AppleTree’s McCarthy said teaching social skills is an important part of school readiness; children need to learn not only letter sounds and counting but also how to listen, focus, and overcome frustration. AppleTree also recognizes the importance of play — and of naps.

In the classroom, 3-year-olds are still 3-year-old. Most children played quietly or learned to write letters with help from teachers. But one little boy whispered, “I want my mommy.” Another threw sand. A third howled in despair, tears running down his face, after being reminded to pay attention.

A little boy chased a girl around a table trying to get a card she was holding with the letter “U” on it, both of them giggling, until a teacher’s aide sat down with them and showed them how to match the letter to a picture of Uranus, then to match an “M” to Mars.

Pietravalle said she was apprehensive before her son, Joseph, started, wondering whether 3 was too young for school. But he adapted quickly. In his second year of preschool, Joseph, who just turned 5, is reading simple books. “I had no idea how good it was going to be for him,” she said. “It was really eye-opening.”