The Board of Education’s District Improvement Plan came under scrutiny Wednesday, with parents saying the plan should have been prioritized and include more dates by which goals would be met.

But the idea of reintroducing “leveling” — where students in a class are grouped according to ability — prompted a lot of discussion though it wasn’t part of the 40-page plan.

The District Improvement Plan for 2015 to 2020 includes “student performance indicators” with goals by 2020 on graduation rates, AP scores and participation, World Language credits by graduation, CMT, CAPT, iReady math tests, attendance and the CT Physical Fitness Test.

But some student performance indicators did not include goals, such as those for in-district assessments of creative and critical thinking, communication and collaboration, individual foreign languages, writing, reading, vocabulary and participation in extracurricular activities.

The plan also includes methods by which student performance goals would be achieved (divided into sections on instructional programs, school improvement plans, leadership capacity and resources) and how to annually review progress toward those goals.

Gerald Kuroghlian, a supervisor of student teachers at Sacred Heart University and longtime educator, said he had heard from Fairfield elementary school teachers that “great pressure” was placed on them from administrators to “move along” slower students so every student was in the same place at the end of the week.

Eileen Liu-McCormack, a school board member, said classrooms include kids who are “very bored,” others who “don’t get it who move along with the tide,” and those who are aligned with the pace of teaching.

“The size of our district should be a competitive advantage for us,” Liu-McCormack said. “We should be able to create more differentiation and more levels. We could probably create a huge amount of differentiation in this district if we decided to go this route.”

Philip Dwyer, the school board chairman, said leveling was a “valid argument,” but he considered its potential reintroduction to be under the purview of Superintendent of Schools David G. Title and the district’s teaching staff.

School board member Donna Karnal said she would “love to see leveling brought back,” but didn’t think the board had that authority. “I personally think leveling is so important,” she said. “There’s no way the school can teach every kid the same way. If we don’t work more individually with kids, instead of grouping them, I think the kids are suffering.”

“I think kids need to be taught the way they’re going to respond the best,” Karnal said.

Board member John Llewellyn said teachers he talked with said leveling would help them instruct students, but that view was in conflict with school administrators. Llewellyn said the school board should have input into whether leveling is reintroduced. “We need to have more say as to what our curriculum is and how it’s delivered,” he said.

Liu-McCormack said teachers also have told her they want leveling brought back. “This could be a board issue if the majority of the board wants to make this an issue,” she said.

Marc Patten, a board member, said the school district had “flip-flopped” on leveling over the past 20 to 25 years and that “no hard-core research” states that leveling is better than teaching to an undifferentiated class. “Until there’s hard-core irrefutable evidence, there’s no reason to turn the Titanic if we don’t have to,” he said.

Board member Jennifer Maxon-Kennelly said research on whether leveling is a more beneficial way to teach is not definitive. She said students who move back and forth between levels could create a “nightmare” for educators.

Carolyn Trabuco, of Sherwood Farm Road, suggested the District Improvement Plan include more metrics for management and comparisons among schools within the district.

Maxon-Kennelly said each school had its own improvement plan and the document discussed Wednesday was an improvement plan for the entire district. Board member Jessica Gerber said results from the schools’ individual improvement plans aren’t consistent because each school doesn’t have the same goals.

Llewellyn said the plan should include interim measurements. “The lack of interim metrics, I think, are problematic,” he said.

Patricia Donovan of Southport questioned how the plan would be prioritized. Dwyer responded that priorities are set by Title and then discussed by the school board during the annual preparation and evaluation of the Board of Education’s budget.

Maxon-Kennelly said she’d like to see an idea of what the first year’s priorities are, and member John Convertito said he would like to see “a pacing structure” in the plan before the board votes on it.

Kelly Jacobson, a parent, said she feels the district does not proactively share information about students’ test results with parents. “Parents in elementary school get absolutely nothing shared with us about the performance of our children,” she said. “Nothing is proactively shared with us.” She said results from three rounds of iReady math tests “are out there but shrouded in secrecy.”

“If we’re going to give our kids these tests, parents need to see the results,” Jacobson said.

Maxon-Kennelly, Liu-McCormack and Convertito said they’d be interested in knowing why the iReady results hadn’t been shared, but Maxon-Kennelly and Convertito added that they didn’t think the District Improvement Plan would be the way to alleviate Jacobson’s concern.

Liu-McCormack said, “Communication is one thing we continually acknowledge we have to do better but actually doing something has been an incredible challenge for us.”

Another parent, Matt Hutzelmann, said the Board of Education should adopt a policy that enables parents to visit their children at lunch.

Hutzelmann, the father of an Osborn Hill School fourth-grader, said the only day he was permitted to visit his child at lunch was on his child’s birthday and that visiting students during lunchtime enables parents to see how their children interact with others.

“It should be more open,” Hutzelmann said. “I’m not the enemy. I’m trying to be involved with my kid’s education.”

“I don’t like the idea for six and and a half hours my kids are in a black box,” he said.

Patten said the idea of preventing parents from visiting their children at school originated with security measures put in place after the massacre at Sandy Hook Elementary School in Newtown in December 2012, based on a recommendation from the Fairfield Police Department. He said police believed that the fewer people who come into a school that don’t belong there on a daily basis, the safer the school would be.

Patten said he didn’t think the policy Hutzelmann favored would be enacted in the near future because the school board defers to the police on safety issues. Maxon-Kennelly said the board could be viewed as irresponsible if it ignored a recommendation from police.

Karnal said children of a parent who never comes to school at lunch could be made to feel uncomfortable by a parent who comes in twice a week. Convertito said some elementary schools have 500 students and if a lot of parents wanted to visit their children at lunch it would create an unmanageable system. He added that some parents may not want strangers sitting next to their child. “The schools are not here for parents to come in and out as they wish,” he said.

Hutzelmann said the board managed having parents visit their children at lunch before and that “a lot has been lost” by preventing them from doing that.

About two dozen parents attended Wednesday’s board meeting, and the board is tentatively scheduled to vote on the District Improvement Plan later this month. Maxon-Kennelly, though, said a vote this month “is probably going to be too hasty.” She said the plan was flexible and could be adjusted.