The American approach is to build conceptual understanding through time-consuming student-centered discovery of multiple solutions and explanations of relatively simple problems. An internationally more successful approach is to build conceptual understanding through teacher-directed instruction and individualized practice in challenging math problems.I got a little flack for my sweeping statement about an “American approach” so I followed up with:
I should clarify what I mean by “American approach”: the approach inspired by national movements like the Common Core and the NCTM standards.The various objections fell into several categories:
1. The pedagogy I’m calling “American” is rare throughout the U.S.: most classrooms still follow a traditional model.
But even if most students are still sitting in rows with the teacher in front, more and more are using Reform Math textbooks like Everyday Math and Investigations, which solicit multiple solutions and verbal explanations for relatively simple math problems. Even if teachers matter more than textbooks, textbooks can place a ceiling on how challenging the material is. That's why traditional texts that date back to the 1960s and earlier are so much better than today's textbooks: they don't place such a low a ceiling on mathematical challenge. Instead they provide math-expanding opportunities for those who can handle them.
2. International comparisons based on test scores are unfair because Europe is “white” and Chinese students cheat. (Yes, one commenter actually said this, repeatedly).
But being white doesn’t make you good at math; China is only one of several Asian countries I discuss; and the many Chinese (and other Asian) nationals who disproportionately populate the top PhD programs and math-intensive careers here in the U.S. probably didn’t get where they are by cheating on math tests.
3. International comparisons based on performance on the PISA test are unfair because other countries track out their lowest performing students prior to age 15-16, the age range of students taking the PISA.
I’d be curious to see statistics on how large this effect is; I’ve looked around a bit and found nothing. Presumably our scores, too, are affected by dropouts and no-shows.
4. International comparisons based on the relative mathematical difficulty of high school exit exams are unfair because these don’t tell us how most students actually did on the various problems.
I’d argue that the predominance on some of these exams of much more challenging problems than American high school students ever see on any standardized test or graduation test tells us something about what kinds of mathematical opportunities students from other countries are getting that their American counterparts may not be.
5. In addition to international comparisons being unfair, a comparison within a province of one country of student performance before and after a student-centered discovery-oriented curriculum was introduced is also unfair. Why? Because it ignores what was going on concurrently in the rest of the country at large.
Then what kind of comparison is fair?
6. The Finnish exam and the Chinese Gao Kao are no more difficult than our Common Core-inspired exams.
My impression is that people who believe this haven’t looked closely at the mathematical demands of these tests, and/or believe that applying math to real-world situations and “proving” things using graphs (common in America's Reform Math and Common Core-inspired exams) to be of a mathematical challenge equal to or greater than the “mere” manipulation of abstract symbols. People with this impression should take a look at the research produced by professional mathematicians and check out the ratio of graphs and “real-world” situations to sequences of abstract symbols.
7. Students at an elite private high school do really well with a discovery-based curriculum.
If I were forced to enact a student-group-centered, discovery-based curriculum somewhere, I’d do it at a highly selective high school whose students were admitted, in part, based on their aptitude for (and therefore their solid foundational knowledge in) math. Such students stand the greatest chance of learning additional math independently, and from one another, and without too much loss in efficiency compared to what’s possible in more teacher-directed, individualized-problem-solving classrooms.