As of a month ago, more than a third of the nation's school districts were not completely ready for any problems they may encounter with their computer systems when the new year arrives, according to a survey released yesterday by the Education Department.

The late September survey showed that 27 percent of public school districts had not yet debugged administrative systems in their central offices, such as those containing payroll and student records, and a full 44 percent had not fixed Y2K problems in school buildings where heating systems, security alarms and refrigerators in cafeterias could be disabled.

And nearly 15 percent of the 1,000 districts surveyed indicated their school buildings would still not be ready Jan. 1, raising a worst-case scenario of unheated classrooms, frozen pipes and flooded corridors in an estimated 2,000 to 15,000 schools nationwide.

"A lot of school districts are cutting it very close, and some of them are not going to make it," predicted John A. Koskinen, director of the White House's Council on Year 2000 Conversion.

Schools will be closed on New Year's Day, a Saturday, and most are scheduled to reopen the following Monday or Tuesday, giving officials a couple of days to remedy any Y2K-related problems. But the public could be affected in the meantime because many communities have designated schools as emergency shelters in case of broad failures in essential, computer-dependent systems.

"The schools may not be ready to serve this other emergency function," warned Marshall Smith, deputy secretary of education.

Smith said he would not be surprised if 1,000 to 1,500 of the nation's 90,000 schools remained closed unexpectedly beyond the Christmas break, forcing parents to scramble to make day-care arrangements.

School officials in the District as well as in Detroit, Dallas, Oakland and some other big cities have already decided to delay reopening schools by a few days to allay public fears and ensure Y2K problems can be resolved before students return, according to Henry Duvall, spokesman for the Council of Great City Schools. In the District, schools are scheduled to reopen Jan. 5.

The Education Department survey did not identify which school districts were prepared and which were not. But a survey by the General Accounting Office last month described the preparedness of the nation's 25 largest districts, including three in the Washington area.

The District and Fairfax County schools told the research arm of Congress that their Y2K problems would be solved by the end of this month; Prince George's County said it would not be ready until after Nov. 30.

Overall, representatives of school districts played down the likelihood of major problems, saying noncompliance was likely concentrated in small rural districts that do not depend heavily on computers.

"The largest ones--they're ready," said Duvall, whose council represents 58 large urban districts. "We might be out in front of some of the suburban and rural school districts."

Joseph Villani of the National School Boards Association said small rural districts--such as 80 in Montana administered from a single room--were likely to be unprepared but probably would not experience major disruptions. He said some small districts have a limited capacity to address anticipated computer problems, while others have had trouble securing Y2K funding from state or local governments.

The Education Department survey also found that 39 percent of the 1,350 colleges surveyed did not yet have their computers ready for Year 2000. Department officials said colleges could experience delays in processing student aid as well as losing research data and building heat. A spokesman for higher education, though, doubted those problems would be widespread.

"We ultimately think Y2K will be a nonevent at most campuses. Colleges are pretty sophisticated users of technology," said Terry Hartle, senior vice president of the American Council on Education.