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....