President Obama is campaigning on a promise to recruit 100,000 new science, technology, engineering and mathematics (STEM) teachers by 2022. That works out to roughly seven STEM teachers for each of the 14,561 public school systems in the United States.

Also, you should know that the California State University system reported in May that the Golden State alone will need 33,000 STEM teachers in the next decade — a third of the total Obama proposes to recruit.

Obama’s idea is intuitively plausible: Economic growth depends on a skilled workforce; math and science teachers provide those skills; yet many students get math and science lessons from teachers who lack specific training. Ergo, we need more credentialed STEM teachers.

Is this true? Yes and no. In 2010, 87.8 percent of the nation’s public-school math teachers had either majored in the subject as undergraduates or received certification later, according to the Education Department; for science teachers, the figure was 95.5 percent. Not too shabby.

In majority-black or majority-Hispanic schools, however, the picture was much worse: Uncredentialed instructors made up 19 percent of math teachers in schools where white students were a small minority (10 percent or less of the student body). Yet this is not necessarily a uniquely STEM-related problem. English teachers in predominantly minority schools are also less likely to possess relevant certifications.

In short, training an extra 10,000 STEM teachers a year between now and 2022 might do some good. How much good, and where, and at what ultimate cost, is anyone’s guess.

I don’t necessarily mean to dismiss the president’s idea but to make a different point — one that also applies to Republican challenger Mitt Romney’s promise to set defense spending at 4 percent of gross domestic product.

Governmental inputs — dollars spent, teachers trained — are easy to quantify. So politicians promise more of them and claim success if they hit their targets.

Yet the true measure of any program or policy is how much the public gets out of it. And the “output” of public goods such as education and health — or national security — is notoriously difficult to measure with precision.

Beyond certain broad indicators, such as the literacy rate or infant mortality, the results that we can measure do not necessarily show that more input yields more output.

Consider school spending. Between 1975 and 2007, per-student education outlays doubled, from about $5,000 to about $10,000. Achievement test scores remained flat.

A recent report by the Friedman Foundation for Educational Choice suggests that the increased spending might have financed bureaucratic growth. Between 1992 and 2009, public-school enrollment grew 17 percent, while total school personnel increased 39 percent. Hiring of administrators and other staff, not teachers, accounted for a disproportionate share of the growth.

Of course, it could be objected that test scores might have been even worse without all the additional spending and employees. And I suppose the president’s STEM-teacher target and Romney’s defense-budget goal may help galvanize public opinion on behalf of worthy goals.

Still, I worry that the pursuit of national numerical targets is a bad way to allocate scarce resources. In 1995, President Bill Clinton announced a goal of 8 million new homeowners by 2000. In 2002, President George W. Bush promised 5.5 million new minority homeowners by 2010.

Both men treated homeownership rates as a proxy for non-quantifiable social goods: economic opportunity, community stability, minority inclusion. They were less interested in how many new mortgages might ultimately foreclose.

The federal homeownership push probably contributed to the housing boom and its subsequent crash, which destroyed trillions in nominal household wealth and left many minorities worse off — not to mention the broader economic damage.

In his 1970 essay “Policy vs. Program,” Daniel Patrick Moynihan decried the government efforts of his day, in which “ambitions are repeatedly proclaimed, and just as repeatedly frustrated.” He blamed the tendency to devise separate federal “programs” in response to each perceived social deficit. Moynihan argued instead for “policies” that accounted for government’s own sometimes unintended or counterproductive impact on society — and measured success holistically.

“The test of a program, when this program is part of a policy,” Moynihan wrote, “is not input but output. It is interesting, and at times important, to know how much money is spent on schools in a particular neighborhood or city. But the crucial question is how much do the children learn. Programs are for people, not for bureaucracies.”

Forty-two years later, we are still struggling to put the late senator’s wisdom into practice.