Not so much.
A new study shows that far from easily grasping mathematical concepts, students who are fed a diet of real-world problems fail to apply their knowledge to new situations. Instead, and against all expectations, they were much more likely to transfer their skills if they were taught with abstract rules and symbols.Chad is quoting from the NYT here, and he follows it up with some nice examples from his own experience as a physics prof. The basic problem is mental compartmentalization. "Many students seem unable or unwilling to take things learned in a class in one department and apply them to subjects studied in a class in another department."
The use of concrete, real-world examples is a deeply ingrained part of the maths classroom. Its advantages have never really been tested properly, for they appear to be straightforward. Maths is difficult because it is a largely abstract field and is both difficult to learn and to apply in new situations. The solution seems obvious: present students with many familiar examples that illustrate the concepts in question and they can make connections between their existing knowledge and the more difficult concepts they are trying to pick up.The train problem is a classic example. Another is the teaching of probability with rolls of a die, or by asking people to pick red marbles from a bag containing both blue and red ones. The idea is that, armed with these examples, students will recognise similar problems and apply what they have learned. It's a technique deeply rooted in common sense, which is probably as good an indicator as any that it might be totally wrong.
What I think this indicates is that teaching is complicated. At some point in my MA year, I came to a realization that shocked me at a fundamental level: I had never and would never receive any formal training in the subject of, well, teaching. All of my training is on the subject: history and philosophy of science.
Somehow, the more specialized the subject matter you mean to teach, the less you're expected to learn about teaching. My friends who are preparing to teach grade school have spent years studying and putting into practice various pedagogical theories. A buddy who teaches high school chemistry effectively has degrees in both teaching and chemistry. A five-minute conversation with any one of them often makes me completely rethink my lesson plan.
Time for the meta: if I apply my suggestion about science to teaching, here's what I get: we should stop teaching students about teaching, and instead start having them do it. Turns out, that's precisely what I'm doing. Oops.
Turns out, it makes as little sense to expect students to automatically pick up abstraction from concrete examples as it does to expect them to pick up skills from facts. There's more than one kind of knowledge. The real trouble seems to be that we only test one kind at once.