“How much do students learn in school?”  The question is harder than it seems.  You get one answer if you measure their knowledge at the end of the school year or right before graduation.  You’ll probably get a very different answer, however, if you measure their knowledge a year, five years, or twenty years after graduation. 

The latter measure is clearly more important.  What good is “knowledge” that melts in your mouth like cotton candy?  As you’d expect, however, we rarely measure long-term learning.  Instead, we look for our keys under the streetlight because it’s brighter there.  Our obsession with student achievement ends with graduation.

Fortunately, there are a few noble exceptions.  My favorite: Bahrick and Hall’s “Lifetime Maintenance of High School Mathematics Content” (Journal of Experimental Psychology, 1991).  The authors assembled a large sample of current students and adults (ages 19-84).  They collected detailed information on their mathematics coursework, IQ, and other variables.  And they reached some remarkable discoveries.

Here’s how knowledge of algebra decays over a lifetime.  The lines (top to bottom) show the fitted scores for (1) people who studied more than calculus, (2) people who studied calculus, (3) people who didn’t study calculus but took another algebra course, and (4) people who didn’t study calculus and only took one algebra course. 


If you’re a cheerleader for education, you’ll fix your gaze on the top two lines.  People who go beyond calculus don’t just master algebra; they know algebra almost perfectly for the rest of their lives.  People who stop with calculus do almost as well, their average score slowly declining from 90% to 75% over the course of fifty years.  Wow!

If you’re someone like me, however, you’ll fix your gaze on the bottom two lines.  After all, most students never take calculus.  What benefit does this vast majority get out of higher mathematics?  Not only is their proficiency low, but it decays fairly rapidly.  Ten years roughly halves their edge over pre-algebra controls.

Non-economists will probably interpret this as an argument for making everyone take calculus.  But what about the cost?  Does it really make sense to torture everyone with four years of advanced mathematics to ensure that they don’t forget their first year?

Before you answer, consider one more finding from the paper.  Bahrick and Hall constructed a measure of how much subjects “rehearsed” – i.e., used – algebra in their daily lives:

Those who reported no rehearsals were assigned zero on the scale; those who reported only activities or occupations of low or intermediate relevance were assigned a 1; those who reported highly relevant activities were assigned a 2 if they engaged in these activities 5 or fewer hours per year, a 3 for 6 to 100 hours per year, and a 4 for more than 100 hours per year.  Those whose profession involved the continuous performance of highly relevant activities were assigned a 5.

Survey says:


The vast majority of subjects rarely use algebra no matter how proficient they are.  Look at the fractions with rehearsal scores of 0 or 1: 89% for people who studied less than calculus, 91% for people who studied calculus, and 70% for people who studied more than calculus.  If the students who already take calculus don’t use it in real life, why on earth should we push weaker students to match their achievement?

P.S. Today, October 17, GMU will be hosting a “flash debate” on government support for higher education between me and Pulitzer-prize winner Steve Pearlstein.   The time: 1:30 PM.  The place: GMU Fairfax campus, North Plaza, by the big clock tower.