It’s back-to-school time again, the time of year when kids get nervous about meeting new classmates and their parents get nervous about poor schooling in math and science. Despite a lot of attention over the past decade, scores for reading comprehension, math, and science on the National Assessment of Educational Progress have either stagnated or gone down. High-school graduation rates have also stagnated or declined.

But some of the measures undertaken to correct these trends might lead you to conclude that school administrators and educational advocates, rather than students, are the ones who could benefit most from better math skills. Consider one of the measures that was supposed to improve the public secondary education: hiring more teachers. Since 1970, labor statistics show that the public school workforce has doubled, with two-thirds of the increase devoted to teachers or teachers’ aides. Meanwhile, secondary-school enrollment rose by just 8.5%. It is fair to ask why no one in the educational infrastructure noted the difference in these two trends long ago and began posing some hard questions.

An even-more interesting case of bad math arose in the late 1990s when the Bill and Melinda Gates Foundation gave grants totaling about $1.7 billion to a movement directed toward breaking up large schools into smaller ones. The idea came from the claim that student achievement improves when schools are smaller. The evidence for this assertion was that there is an unrepresentatively large proportion of smaller schools among high-performing educational institutions.

But there was a problem with the reasoning behind the smaller-is-better movement. It was explained by University of Pennsylvania statistics expert Howard Wainer and McGill University’s Harris Zwerling. Promoters of this idea, said the two statisticians, either didn’t know about or ignored the findings of French mathematician Abraham de Moivre who, in 1730, showed that small samples have high variations. Specifically, he demonstrated that the standard deviation isn’t proportional to the sample size, but instead rises in proportion to the square root of the sample size.

It isn’t surprising that small institutions show up more frequently in lists of better-than-average schools, the two said. The performance of small schools is more variable than that of large schools, so overrepresentation at extremes is to be expected. To prove their point, Wainer and Zwerling looked at the 50 lowestscoring schools in Pennsylvania, where they had academic data to work with. It turned out that smaller schools were overrepresented in this group as well.

There were a lot of complaints about the smaller-school movement. It emerged that “teachers ended up teaching things they don’t really know,” as a Seattle newspaper article concluded. Speaking of the small-school movement, one Michigan State University educator said, “I’m afraid we have done a terrible disservice to kids.” Eventually, the Gates Foundation stopped giving grants to promote the idea.

Wainer has more direct comments on this episode. “Spending more than a billion dollars on a theory based on ignorance of de Moivre’s equation ... suggests just how dangerous that ignorance can be.”

The economics Nobel Prize winner Daniel Kahneman also read about the small-school debacle and said he found it “amusing.” One wonders if, after spending close to $2billion on a misguided idea, Bill and Melinda Gates are as amused by it as Kahneman.

*— Leland Teschler, Editor*