December 1 is the deadline for families to send in their applications for gifted and talented kindergarten classes — a program that has long been criticized as a bastion of the white and privileged. This year could be different.

This month, children will be assessed under a new citywide process the Department of Education says is more fair and accessible than the old system.

Among the applications will be two from the Jordans and the Pallazhcos, families whose starkly different backgrounds show some progress has been made. Their experiences navigating the application process this year also show the many obstacles the education department faces as it attempts to make gifted and talented programs more equitable.

Mary Beth Jordan, a social worker who lives on the Upper West Side, has been researching the choice public schools around the city and the process to get into them since her 4-year-old daughter, Tess, was born. At first, she and her husband, a hedge fund analyst, fought over the idea of moving to the suburbs. Her husband wanted a bigger house and thought the schools would be better, but they decided to stay after Ms. Jordan convinced him they could find quality schools in the city.

They discovered that Tess might be gifted after she took a privately administered IQ test.

“I’ve kind of been tracking this, and I’m also a social worker, so I’m used to navigating a big sprawling bureaucracy,” Ms. Jordan said. “I definitely have an advantage.”

Although neither parent believes in prepping toddlers for tests, they made sure to take Tess to museums often, do puzzles, and read her books to help stimulate her mind.

Luz Pallazhco, an Ecuadorian immigrant whose family shares a cramped one-bedroom apartment in Greenpoint with her two brother-in-laws, said she guessed her son, Adrian, might be above average when at age 4 he began helping his older sister Jessica with her first-grade homework. She and her husband, a construction worker, didn’t know the English words to describe his capabilities until a teacher at her son’s Head-Start program suggested he apply to a school for gifted and talented children.

“She says that the boy has needs that are not equal to a normal school,” Ms. Pallazhco said in Spanish. “There, they’re going to give him homework that will bore him. I had never heard of it, until she explained.”

The new application process uses the same two citywide assessments for every child who applies, which is meant to level the playing field between families like the Pallazhcos and the Jordans. Before, individual schools used a medley of different deadlines and tests, some of them given by expensive private doctors.

The Otis-Lennon School Ability Test is read out loud to students, meaning the test can be given in any language. A child’s preschool teacher fills out the other assessment, the Gifted Rating Scale.

“Giftedness is not about language,” the executive director of Assessment and Accountability for the schools, Jennifer Bell-Ellwanger, said. The executive director of gifted and talented programs, Anna Commitante, added that it’s not about social class, either.

One of the dangers for the administration as it tinkers with the system is the wrath of voters who view the gifted and talented programs as a way to keep middle-class families in the public school system. Some of those parents have said they worry that a wider pool could make it harder for their children to get into the programs of their choice, especially the more competitive citywide programs.

The city doesn’t have information on the incomes of parents of students in gifted programs, but according to statistics compiled from schools’ self-reported data, 25% of gifted and talented program students are white, 33% are black, and 20% are Hispanic.

In the school system as a whole, whites make up about 15% of students, blacks 34%, and Hispanics nearly 40%, according to education department figures. Department officials say they will use the data from the new citywide assessment to compile their own demographic data and to look for areas in the city with too many or too few programs.

The standardization of assessments won’t solve the inequities in the system by themselves, education experts say. The most daunting task faced by the education department is making sure parents of potentially gifted children in all corners of the city are aware that the testing process exists. Then it must help them navigate through a system that even well-informed parents can find confusing.

“The process takes place before kids enter school, so there needs to be some way of informing parents who may not be as savvy about the citywide program,” the director of the National Research Center on the Gifted and Talented, Joseph Renzulli, who has also advised the administration on its gifted and talented programs, said. “If there’s going to be a party, we need to get an invitation to the party if we’re going”to attend.

Along with the new citywide assessment, Ms. Commitante said the city has also done more aggressive outreach than in years past.

Without realizing it, the Jordans and Pallazhcos recently crossed paths at an open house for the Anderson School, a highly competitive program on the Upper West Side that accepts applications from across the city. Holding a bilingual flyer from the Head-Start program, Ms. Pallazhco sat in the back row of the auditorium with her husband and Adrian and Jessica, who squirmed in their seats and munched on cookies from a table of hors d’oeuvres. They had traveled for an hour to get there. She looked on quizzically as school administrators and teachers stood up one by one to talk about the school in English.

Ms. Jordan and her husband had come with a couple of friends from the neighborhood. Following instructions posted on the school’s Web site, they didn’t bring Tess and Derek, their 3-year-old.

Afterwards, Ms. Pallazhco chatted with a Spanish-speaking PTA member who answered some of her questions.

“I don’t know anybody who knows about this type of school,” she said later. “The people who are from here, they’re the ones who know about it.”

At several points in the process, both families hesitated about applying, though their worries were very different.

Ms. Pallazcho dropped out of school after fifth grade. Her husband dropped out after sixth.

“I’m really worried if it’s a school where they’ll send him with homework that I won’t be able to help him with because I just went to elementary school,” she said. “If people don’t help at home, students can’t do it alone.”

Ms. Jordan isn’t worried about homework, but she is worried about some of the lessons Tess may take home with her if she attends a city gifted and talented program. Having grown up in a racially homogenous town in central Massachusetts, Ms. Jordan said one of the main reasons she wanted to stay in New York City is because she wants her children to be exposed to diversity.

On a tour of another gifted and talented programs on the Upper West Side recently, she said she was disappointed.

“The general education classrooms were 80 to 90% kids of color. It was the opposite in the gifted and talented,” she said. “They wouldn’t have to tell you what kind of classroom it was. You could tell by the skin color.”

She was about to ask what the school was doing about the disparity when one of the other parents on the tour, Cynthia Nixon, an actress on the HBO series “Sex and the City,” asked first.

“They responded by saying they’re trying to do more outreach. They say they’re trying,” Ms. Jordan said. “I want my kid to have the best education possible, but do I want my kid growing up and thinking only white kids are smart? I don’t know.”