Why Can't Johnny Take a Derivative?

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John Hood at The Corner cites a recent Brookings report in support of the familiar conclusion that American students are far more confident in their math abilities than students in other countries who do better on average. Indeed, across countries, there seems to be an inverse correlation between average confidence in math ability and average scores. One read on that data is that focusing on boosting student self-esteem has created a nation of Lake Woebegone classrooms, full of mostly mediocre kids convinced they're all above average. But that seems like a hasty inference for a couple reasons.

First, as Brookings notes, though confidence and achievement are inversely correlated between countries, they're positively related within each country. This is not, pace Brookings, a paradox, because when you think about it, there's no reason at all to expect confidence to track achievement across countries. To paraphrase Robert Nozick, very few of us sit around thinking: "Hey, I've got two opposable thumbs and I've mastered a natural language—I'm pretty good for a primate!" When we want to know how successful we are at something, we compare ourselves to our own social groups, not to all other Americans, and certainly not to all human beings. So even in the absence of any kind of unusual national egotism, you'd expect average national confidence to track, not average national score, but the distribution of scores.

Imagine two countries. One has a familiar bell curve distribution of math scores, with a steep drop-off from the median around which most students are clustered. Most kids are going to look around and conclude they're about average. The second has a double-peaked or bimodal distribution, with lots of kids clustered at the bottom, a dip toward the median, and another big cluster a bit above average. You'll see something like half the kids in that country concluding they're better than average. Note that I haven't said anything about the average scores between countries here; the average in the first country could be much higher, and nothing changes. The effect could be magnified if you've got, say, clusters of crappy schools in which nobody's very good at math, but someone's got to be valedictorian—at least, for certain ways of breaking the math-talent distribution out into discrete schools.

One additional factor is that some of the lowest levels of math confidence come out of high-scoring Asian countries where, at the risk of overgeneralizing, there tend to be norms against seeming boastful. The low reported levels of confidence there might very well just be a sign of higher rigor and more stringent expectations in schools there. But it might also reflect the desire of kids who are good at math (even by local standards) and know it to avoid seeming arrogant.

The Lake Woebegone explanation is actually plausible enough on face: Kids who're made to think they're very good at something, unless they have a special passion for the subject, probably won't feel much pressure to improve. But it'd be interesting to see someone check the alternative hypothesis by looking at a breakdown of the score distributions within each country.