“Joyce Irvine versus millions,” Ms. Irvine said. “You can buy a lot of help for children with that money.”

Burlington faced the difficult choice because performance evaluations for teachers and principals based on test results, as much as on local officials’ judgment, are a hallmark of the two main competitive grant programs the Obama administration developed to spur its initiatives: the stimulus and Race to the Top.

“I was distraught,” said Ms. Irvine, 57, who was removed July 1. “I loved being principal  I put my heart and soul into that school for six years.” Still, she counts herself lucky that the superintendent moved her to an administrative job  even if it will pay considerably less.

“I didn’t want to lose her, she’s too good,” Ms. Collins said, adding that the school’s low scores were the result of a testing system that’s “totally inappropriate” for Wheeler’s children.

Justin Hamilton, a spokesman for the United States Department of Education, noted that districts don’t have to apply for the grants, that the rules are clear and that federal officials do not remove principals. But Burlington officials say that not applying in such hard times would have shortchanged students.

At the heart of things is whether the testing system under the federal No Child Left Behind Act of 2001 can fairly assess schools full of middle-class children, as well as a school like Wheeler, with a 97 percent poverty rate and large numbers of refugees, many with little previous education.

President Obama’s Blueprint for Reform says that “instead of a single snapshot, we will recognize progress and growth.” Ms. Collins says if a year’s progress for each student were the standard, Wheeler would score well. However, the reality is that measuring every student’s yearly growth statewide is complex, and virtually all states, including Vermont, rely on a school’s annual test scores.