Modern fads in math teaching tend to lean heavily on so called manipulatives - the theory being, apparently, that the hands are the window to the math brain. Kenneth Chang reports in The New York Times that Ohio State U reseachers have cast doubt on this theory. An experiment by the researchers suggests that it might be better to let the apples, oranges and locomotives stay in the real world and, in the classroom, to focus on abstract equations, in this case 40 (t + 1) = 400 - 50t, where t is the travel time in hours of the second train. (The answer is below.) “The motivation behind this research was to examine a very widespread belief about the teaching of mathematics, namely that teaching students multiple concrete examples will benefit learning,” said Jennifer A. Kaminski, a research scientist at the Center for Cognitive Science at Ohio State. “It was really just that, a belief.” . . . In the experiment, the college students learned a simple but unfamiliar mathematical system, ess...