Low Math Scores and Growth Theory

I have always felt uncomfortable being pushed to worry about low standardized test scores in the U.S. A new round of these math test results have just come out. What to think?

I'm 40 now. U.S. math test scores were low when I was in school in the 1970s. The world did not end. This was not for lack of warnings that this was a sign that the sky was falling.

Even so, I was one of the best in my high school at math, and I went off to college to get a B.A., M.A., and Ph.D. where my math skills were routinely trounced by foreign students.

I'm older now, and I discount these test scores, but I wasn't sure why until today. My sanguine view about them had little justification. But it was sort of like the deficit - I knew it had to be overblown because there's no way that a statistic that out of whack can be meaningful and not have really obvious effects.

So, today I read an article entitled "The Last Time You Used Algebra Was ..." in the New York Times. This point was what I needed:

In math, as in chess, countries that produce the most grandmasters per capita - like Hungary and Iceland - not only don't rule the world, they don't even rule chess. Sheer power counts, as it did in chess for the Soviets. America may lose math literacy surveys, but it dominates number-crunching in every sphere from corporate profits to supercomputers to Nobel prizes.

This makes a lot of sense to me when taken through the lens of growth theory and new growth theory. In growth theory it is critical to differentiate between level effects and growth rate effects. In new growth theory, non-excludable, non-rivalrous ideas are critical for growth. Now some explanations.

A level effect is a result that is determined by the size of an aggregated variable. A growth rate effect is a result that is determined by the growth rate of a per capita (disaggregated) variable. In combination they can lead to some unusual implications. One is that a person's income is determined by both the level of technology that surrounds them and the growth rate of that technology - but that the former has a positive effect while the latter has a negative effect. The positive effect is easy to understand: more technology means more income. But the negative effect is harder. It occurs because technological growth needs to be supported. Think about it: a major upgrade in Microsoft Windows might necessitate that you skip a vacation because you've got to spend money to support your use of that new technology. It may be good for you in the future because you have more technology, but that's the level effect. The growth is a pain in the neck because you have to pay for it.

Non-excludable goods are ones that you can't keep other people from consuming just because you are. Non-rivalrous goods are ones that your use of does not detract from its use by others. A good example with both features is calculus: just because you learn it doesn't keep someone else from learning it, and just because you use it doesn't mean that other can't. A pear just doesn't have either of those features.

So, what on Earth does this have to do with standardized math test scores? Math test scores are an average across all students within a country. It is great if these scores grow, but in the short run that's expensive. What's important is the overall level of math knowledge in a country. The U.S. has tons of that - far more than anyone else.

And why is that? It's a combination of three factors: 1) we're already large, 2) we are not very interested in excluding immigrants, and 3) we are not very interested in playing off one immigrant against another.

So, the U.S. is a big, non-excludable, non-rivalrous sponge for international math skills. This is one more levels variable in our favor that helps make us rich.

Comments

[Jim Mitchell]

What a fancy of way of saying that we depend on immigrants for our technical base.

I'd suggest that we ought to worry because the world is changing in a manner (e.g. India) that will allow them to return/stay home. For my kids that could be a big problem.

[Dave Tufte]

I'm not sure why it would be a problem to rely on immigrants for our technical base. It would seem to me that if a skilled immigrant wants to emmigrate from (say) Jordan to the U.S., that this is our gain and Jordan's loss. But, my guess is that this would be viewed negatively by a layperson in either country - although the Jordanian layperson is more likely to be correct.

Having said that, you've hit a nail on the head in your second paragraph (my guess is that you think it is a different nail than the one I hit, but I'm not so sure). What I was driving at is that it is important to be both big and smart. The U.S. is in an OK position because we are big and kinda' smart on average.

I know, not as smart as we could be, but better off on average than most countries around the world - remember that the standardized tests cover only a subset of countries, and then only the kids in the school in those places.

But apparently our bigness can compensate for our lack of smarts. Japanese kids score better than Americans, but that isn't enough to compensate for the fact that our country is twice as big. So, bigness must be quite important.

Fortunately, we are likely to retain our place as the third biggest country for some time, perhaps forever. India and China, on the other hand, are currently big and not as smart on average. But as they get richer, and get and keep more kids in school for longer, they will get smarter on average. We are in the same page in that India will win this race - they will soon overtake China in size, their educational system is better top to bottom, and they have more committment to it than the Chinese.

So if we are worried about the very long run (sometime after I'm dead and buried), then we can envision India, China, and the U.S. (in that order) being the major powers on the globe, and India and China being rather far ahead of the U.S.

So our destiny is to be something like Japan or Germany is now. Is that such a bad place to be? If it is, we can invest a lot more in technical education, but has that made the countries that scored higher than us on the PISA tests better off than they would have been otherwise? I'm not so sure.