At the Jeremiah E. Burke High School, daily attendance is 90 percent this year and the school has partnerships with UMass Boston and City Year, a civic-focused nonprofit. And once a year, 60 or so teachers and some 600 students head to New Hampshire and climb Mt. Monadnock as part of a community-building exercise that is symbolic, the headmaster says, “of reaching new heights.” Students have also taken trips to Cape Verde and to a black engineers’ conference in Pittsburgh.

But the Washington Street school, which reopened in 2008 after undergoing a $49.5 million renovation, is barely on the radar screen for middle school students looking at high schools, according to figures from the school district.

This year, 11 students picked it as their first choice, 12 students picked it as their second choice, and 24 students picked it third.

The total number of students citywide who requested the high school in the first round this year was 125. The Snowden in Copley Square received about 2,000 requests, Fenway High some 1,200, and Tech Boston Academy on Dorchester’s Peacevale Road around 900.

The low figures for Burke High illustrate the challenge that Mayor Thomas Menino and school officials have talked about with increasing frequency as they seek to overhaul the school assignment process: Convincing parents that the school nearby is as good as the one miles away, or at least good enough.

“Historically, it has not been perceived as a viable opportunity,” said Burke High School Headmaster Lindsa McIntyre. “And it has been plagued with some challenges. We can’t ignore that. I think that reputation still exists in the minds of many,” she said, while stressing that the Burke is “one of the safest schools in the district,” with just 45 suspensions this academic year.

McIntyre was appointed to the job in 2009, coming from Community Academy, an alternative Boston high school. She had also worked as the King Middle School, the City on a Hill Charter School, and at Burke High in 1996.

The 1990s were a rough period for the Burke, which had its accreditation taken away at one point. It currently is tagged as a “turnaround,” or underperforming, school.

One anecdote about the Burke has been told and retold: A piece of the ceiling fell on Menino when he was touring the school in 1995, an event that spurred him to pour money into improving the building.

Four years ago, the school re-opened with renovated classrooms, computer labs, a new gymnasium, a performing arts wing, and a new cafeteria. The freshmen and sophomore classes have been broken down into so-called “academies” in order to “home in on the individual needs of students,” McIntyre said.

The daily schedule has been re-designed so that the core topics – English, math, history and science – are covered in classes 80 minutes long. Teachers sometimes stay at the school until 6:30 or 7 p.m.

Still, the doubts persist. “It’s a perception,” said McIntyre. “Part of the work we need to do is marketing our school and changing the perception. The reputation precedes us and we need some support in turning that around.” Some of that remedial work includes reaching out to middle schools for open houses, she said.

Jorge Martinez, head of Project R.I.G.H.T., a neighborhood stabilization group, said that schools like the Burke need more resources. He expressed frustration at the loss of a $9 million federal teachers grant because of the continuing bickering between the administration and the Boston Teachers Union over the teachers’ contract. Some of those funds were meant for the Burke, Martinez said, adding that “the reality is the schools in the communities of color have been underperforming and under-subsidized.”

The numbers at the Burke tell their own story: 94 percent of students are black and Latino; 23 percent fall into the special education category; and 76 percent are considered low-income. The drop-out rate for the 2010-2011 school year was 14.8 percent, down from the nearly 16 percent rate the school experienced in 2006.

“We think we’re effecting valuable change,” McIntyre said. “Are we there yet? Not according to the data. But we’re on our way.”