Many moons ago, I used to book and promote concerts, so I don’t pose that question with disdain towards people who put shows together. Scientists, on the other hand…. (heh)
In large part due to the media blitz surrounding Amanda Ripley’s book on education, The Smartest Kids in the World, in which she dwells almost exclusively on PISA scores for international comparisons, the results of the PISA international comparisons in math and reading are back in the news–tests on which the U.S. is middling and which are also very problematic. It also doesn’t help that Bill Keller of the NY Times is pimping this book, given his basic lack of understanding of educational statistics–he could worse than read Mikoto Rich’s excellent reporting in his very own newspaper (and let’s not forget Bill Keller’s cheerleading for the Iraq War, so I have complete faith in his ability to get the policy right…).
Anyway, back to the PISA evaluations. Ripley refers to PISA as “the most respected test of teenagers in the world.” It’s essentially a constructivist test. In a recent response to critics, Ripley described PISA thusly:
Unlike TIMSS, PISA was designed to test students’ abilities to apply knowledge to solve real-world problems and think for themselves. (TIMSS is a test of school curriculum.) I was most interested in those higher-order thinking skills, since they are increasingly valuable in the modern economy. To see if the hype on PISA was true, I took the test myself, and I found it to be a remarkably sophisticated test.
Here’s a sample question–I’m going to put the answer below the fold (no cheating!):
For a rock concert a rectangular field of size 100 m by 50 m was reserved for the audience. The concert was completely sold out and the field was full with all the fans standing. Which one of the following is likely to be the best estimate of the total number of people attending the concert?
20,000. I, like a lot of people, got this wrong because, given a choice of one or four people per square meter (the field is 5,000 square meters), I thought one person was the better answer–because there are things called fire codes (promoted and booked concerts, remember?). Had it been fill-in-the-blank, I would have probably gone with 10,000 or 15,000 thousand.
This type of question highlights two problems with supposedly common sense, real world questions. One problem is that this is a typical culturally variant question. People used to crowds would probably answer 20,000, while those from Big Sky country would probably feel claustrophobic at one person per square meter. As we’ve discussed before, this sort of cultural variance is a massive problem when comparing countries.
But there is a more significant problem, and it gets back to what the fundamental goal of a mathematics education should be. If the goal is to enable students to tackle real-world problems, PISA isn’t bad. But the problem is that college mathematics–and the sciences–require formalism and abstraction. You need to be able think about the question posed above as “4xy“. This affects many fields. I could, if I choose to do so, explain the very basics of the neutral theory of evolution without recourse to math. But to do anything even slightly more difficult, you need to be familiar with formal math. That’s before you get to the harder stuff. Thinking creatively is great, but, at some point, you have to know things and have a facile use of abstraction. How you test–that is, what is important for students to learn–matters. Do we want a testing regime that would lead to smart, but mathematically untrained, college students. Will we offer them remedial mathematics in college if they suddenly decide to be STEM majors? Given the cost of college, is that an ethical thing to do?
Lest you think I’m overstating the problem, it’s worth noting that two hundred Finnish college math professors don’t like Finland’s curriculum for similar reasons (pdf, p. 11; boldface mine)):
When PISA results showed Finland to be the top country in the world in math, a group of more than two hundred university mathematicians in Finland petitioned the Finnish education ministry to complain that, regardless of what PISA was indicating, students increasingly were arriving in their classrooms unprepared in mathematics. Knowledge of fractions and algebra were singled out as particularly weak areas. Two signers of the petition posed the question, “[A]re the Finnish basic schools stressing too much numerical problems of the type emphasized in the PISA study, and are other countries, instead, stressing algebra, thus guaranteeing a better foundation for mathematical studies in upper secondary schools and in universities and polytechnics.” One Finnish researcher, analyzing national data, compared the math skills of 15- and 16-yearolds on tests given in 1981 and 2003. Sharp declines were registered on calculations involving whole numbers, fractions, and exponents. The explanation: “‘Problem Solving’ and putting emphasis on calculators have taken time from explaining the basic principles and ideas in mathematics.”
I dunno, but fractions and exponents are kinda important. Algebra is pretty key too. But I’m a scientist so what do I know?
Obviously, most people aren’t going to use high level mathematics (though many skilled trades do). But high school should enable most, if not all, students to take college mathematics. Because concert planners are cool (even if I was one!), but we also need people who use formal math.