A Pound of Gold or a Pound of Hay?

Jonah Lehrer of The New Yorker claims that psychological research shows that "[t]he vast majority of people respond quickly and confidently" -- with the same wrong answer -- to the following simple problem:

A bat and ball cost a dollar and ten cents. The bat costs a dollar more than the ball. How much does the ball cost?

Upon reading the above sentences, I immediately saw a simple algebra problem. In mathematical terms, the first two sentences could be expressed as:

a + b = 1.10 (1)

a - b = 1.00 (2)

In the above equations, a represents the bat and b the ball. I didn't bother with the equations, and didn't need to, anyway, to see that the "popular" answer of ten cents (which the problem statement might suggest to someone in a hurry) was wrong. Such a price would lead to a contradiction between (1) the total price of the two items and (2) the stated difference in the price between the two. A price of one dime for a ball would mean that the bat would have to be worth (1) $1.00 and (2) $1.10 at the same time.

The article goes on to note the following:

And here's the upsetting punch line: intelligence seems to make things worse. The scientists gave the students four measures of "cognitive sophistication." As they report in the paper, all four of the measures showed positive correlations, "indicating that more cognitively sophisticated participants showed larger bias blind spots." This trend held for many of the specific biases, indicating that smarter people (at least as measured by S.A.T. scores) and those more likely to engage in deliberation were slightly more vulnerable to common mental mistakes. Education also isn't a savior; as Kahneman and Shane Frederick first noted many years ago, more than fifty per cent of students at Harvard, Princeton, and M.I.T. gave the incorrect answer to the bat-and-ball question.

Over half of these students failed to spot a simple contradiction! Assuming the people surveyed were not told to hurry to give an answer, this suggests that an incredible number of people out there do not, as a matter of course, integrate one item of knowledge with another. This is astounding to me on one level, although the fact that people who fail to do this are also more prone to cognitive bias is not.

An explanation I would have liked to see considered is what kind of thinking methods people have been taught or exposed to over the course of their educations. "Progressive" education, informed by the anti-integrative philosophy of Pragmatism, has profoundly affected much of American education for decades. I suspect that if such an explanation were considered, the fact that education doesn't generally save people from such slips would seem much less surprising. Furthermore, the fact that intelligent people are more prone to cognitive bias would also make sense, given the fact that most spend more time than average with the comprachicos.

-- CAV