You cannot understand modern education policy without a grasp of the achievement gap. On average, low-income students have lower academic achievement than affluent students. Black or Hispanic students similarly score lower on standardized tests, on average, than white or Asian students.

School leaders say they want to reduce those gaps but are uncertain about how to do it. They should read a new book by Arlington County educators who mounted one of the most sustained assaults on the achievement gap ever seen in this area.

The book is “Gaining on the Gap: Changing Hearts, Minds, and Practice,” by Robert G. Smith, Alvin L. Crawley, Cheryl Robinson, Timothy Cotman Jr., Marty Swaim and Palma Strand. The main man is Smith, the Arlington school superintendent from 1997 to 2009.

When Smith took over Arlington schools, I thought he was headed for an embarrassing failure. He said he intended to do everything possible to reduce the achievement gap and report his progress every year, letting his reputation rise or fall on the results. I wrote that he was taking a big risk. Reducing the gap was too difficult to make it the prime issue.

He did it anyway and made me look bad: From 1998 to 2009, the portion of black students passing Virginia Standards of Learning tests in Arlington rose from 37 to 77 percent. For Hispanic, students the jump was from 47 to 84 percent. The gap between non-Hispanic white and black passing rates dropped from 45 percentage points to 19. Between Hispanics and non-Hispanic whites, the gap shrank from 35 points to 12.

The power of the book is not so much in the numbers but how Smith and his well-chosen team of administrators and teachers did this. They made sure all staff knew how kids at all levels were doing and how important it was that they improve. There were startling differences between what Smith asked them to do and what the federal No Child Left Behind law mandated for all districts in the country.

Smith went far beyond the law’s focus on reading and math scores in grades three through eight and in high school. He insisted on measuring each major ethnic group, plus low-income students, students with disabilities and students learning English, on: the percentage passing first-year algebra with a C or better by the end of eighth grade; the percentage enrolled and passing advanced courses in grades six through 12; the percentage completing the third year of a foreign language by the end of grade 11; the percentage of sixth- through eighth-graders taking electives in art, music and theater; the percentage meeting or exceeding criterion levels on the Virginia Wellness-Related Fitness Tests; and several other measures.

No Child Left Behind put all districts on an improvement schedule that made few adjustments to reality. If a school had a bad year, it acquired an unattractive label and had to get moving or risk more sanctions. Smith, on the other hand, required detailed reporting of results at all levels, but then, he says in the book, the “emphasis shifted from results to recommended revisions to goals, objectives, indicators, or targets, depending on the results achieved. For example, if targets had been exceeded in the previous year and targets previously set for the next year had also been surpassed, the targets would be revised upward. In other instances, the data may have gone in the opposite direction, occasioning a need to adjust targets downward to keep their achievement within the realm of the possible.”

How shockingly reasonable. Smith didn’t insist, as the federal government did, that all schools strive toward 100 percent proficiency by 2014. He just wanted them to improve. I think there are problems with measuring schools by these gaps, but Smith and his splendid team produced a better school system.

That is worth copying if Congress ever gets around to replacing No Child Left Behind.