Does Algebra Help You Think Better?

A couple of days ago Andrew Hacker wrote a New York Times op-ed asking “Is Algebra Necessary?” It was prompted by the growing trend to require a passing grade in algebra as a condition for graduating from high school, and Hacker argues that this trend is doing a lot of damage, helping to make dropouts out of kids who are perfectly adequate in every way except their ability to manipulate abstract symbols. Eugene Volokh, however, suggests that Hacker is wrong:

Though I’m not certain of this, I suspect that algebraic problem-solving teaches useful mental habits that both open up possible future careers and also help train people’s general problem-solving abilities.

I’m not picking on Eugene here. I just had his post handy as an example of an argument that I’ve seen frequently in response to Hacker’s piece. So I’m curious: is there any evidence at all that knowledge of algebra (not arithmetic, algebra) teaches useful mental habits or improves people’s general problem-solving abilities? Obviously algebra is useful if you plan to learn more math in order to pursue a science or engineering career. But for your garden variety high-school grad, does knowledge of algebra truly instill an ability to reason better? I have to say that my personal experience is that it doesn’t: people with a strong math background don’t seem to reason any better than anyone else. I suspect that those of us who are good at algebra tend to vastly overestimate its impact on our mental habits.

Just to be absolutely clear here: General numeracy is useful, and it’s especially useful for understanding numerical problems. (Duh.) That’s not what I’m asking about. What I’m asking is whether mere knowledge of algebra produces better mental habits in other areas of life. It might! But is there any actual evidence to back this up?