TEN years ago, Congress adopted the No Child Left Behind legislation, mandating that all students must be proficient in reading or mathematics by 2014 or their school would be punished.

Teachers and principals have been fired and schools that were once fixtures in their community have been closed and replaced. In time, many of the new schools will close, too, unless they avoid enrolling low-performing students, like those who don’t read English or are homeless or have profound disabilities.

Educators know that 100 percent proficiency is impossible, given the enormous variation among students and the impact of family income on academic performance. Nevertheless, some politicians believe that the right combination of incentives and punishments will produce dramatic improvement. Anyone who objects to this utopian mandate, they maintain, is just making an excuse for low expectations and bad teachers.

To prove that poverty doesn’t matter, political leaders point to schools that have achieved stunning results in only a few years despite the poverty around them. But the accounts of miracle schools demand closer scrutiny. Usually, they are the result of statistical legerdemain.

In his State of the Union address in January, President Obama hailed the Bruce Randolph School in Denver, where the first senior class had a graduation rate of 97 percent. At a celebration in February for Teach for America’s 20th anniversary, Education Secretary Arne Duncan sang the praises of an all-male, largely black charter school in the Englewood neighborhood of Chicago, Urban Prep Academy, which replaced a high school deemed a failure. And in March, Mr. Obama and Mr. Duncan joined Jeb Bush, the former governor of Florida, to laud the transformation of Miami Central Senior High School.

But the only miracle at these schools was a triumph of public relations.

Mr. Obama’s praise for Randolph, which he said had been “one of the worst schools in Colorado,” seems misplaced. Noel Hammatt, a former teacher and instructor at Louisiana State University, looked at data from the Web site of the Colorado Department of Education.

True, Randolph (originally a middle school, to which a high school was added) had a high graduation rate, but its ACT scores were far below the state average, indicating that students are not well prepared for college. In its middle school, only 21 percent were proficient or advanced in math, placing Randolph in the fifth percentile in the state (meaning that 95 percent of schools performed better). Only 10 percent met the state science standards. In writing and reading, the school was in the first percentile.

Gary Rubinstein, an education blogger and Teach for America alumnus who has been critical of the program, checked Mr. Duncan’s claims about Urban Prep. Of 166 students who entered as ninth graders, only 107 graduated. Astonishingly, the state Web site showed that only 17 percent passed state tests, compared to 64 percent in the low-performing Chicago public school district.

Miami Central had been “reconstituted,” meaning that the principal and half the staff members were fired. The president said that “performance has skyrocketed by more than 60 percent in math,” and that graduation rates rose to 63 percent, from 36 percent. But in math, it ranks 430th out of 469 high schools in Florida. Only 56 percent of its students meet state math standards, and only 16 percent met state reading standards. The graduation rate rose, but the school still ranks 431st, well below the state median graduation rate of 87 percent. The improvements at Miami Central are too small and too new to conclude that firing principals and teachers works.

To be sure, the hyping of test-score improvements that prove to be fleeting predated the Obama administration.

In 2005, New York’s mayor, Michael R. Bloomberg, held a news conference at Public School 33 in the Bronx to celebrate an astonishing 49-point jump in the proportion of fourth grade students there who met state standards in reading. In 2004, only 34 percent reached proficiency, but in 2005, 83 percent did.

It seemed too good to be true — and it was. A year later, the proportion of fourth-graders at P.S. 33 who passed the state reading test dropped by 41 points. By 2010, the passing rate was 37 percent, nearly the same as before 2005.

What is to be learned from these examples of inflated success? The news media and the public should respond with skepticism to any claims of miraculous transformation. The achievement gap between children from different income levels exists before children enter school.

Families are children’s most important educators. Our society must invest in parental education, prenatal care and preschool. Of course, schools must improve; every one should have a stable, experienced staff, adequate resources and a balanced curriculum including the arts, foreign languages, history and science.

If every child arrived in school well-nourished, healthy and ready to learn, from a family with a stable home and a steady income, many of our educational problems would be solved. And that would be a miracle.