A new article in the journal Science describes a controlled trial conducted by Ohio State University researchers testing modern approaches to math education. Though the subject of debate for many years, most math teachers use a variety of concrete examples to demonstrate mathematical concepts and techniques.
A more traditional approach is to head straight for the abstract algebraic approach. A New York Times summary of the article gives the following example:
One train leaves Station A at 6 p.m. traveling at 40 miles per hour toward Station B. A second train leaves Station B at 7 p.m. traveling on parallel tracks at 50 m.p.h. toward Station A. The stations are 400 miles apart. When do the trains pass each other?...An experiment by the researchers suggests that it might be better...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.
Sitting in college lectures, we still see this traditional approach very frequently and the flexibility and generalizability afforded by the abstract rather than specific real world examples seems an obvious choice to university professors. They obviously don't care which train gets to Topeka first.
As the researchers at Ohio State concluded, most people, when presented with the abstract tools, are better-prepared to solve a variety of applied problems (including problems like the train question above) than those who were given real-world examples:
The problem with the real-world examples, Dr. Kaminski [one of the OSU researchers] said, was that they obscured the underlying math, and students were not able to transfer their knowledge to new problems.
“They tend to remember the superficial, the two trains passing in the night,” Dr. Kaminski said. “It’s really a problem of our attention getting pulled to superficial information.”
While the researchers conducted their trial on college volunteers, they noted that evidence exists that similar problems occur for younger students. This suggests both that students in both primary and secondary schools should see math presented abstractly and that primary school teachers really need a fundamentally better understanding of mathematics to be able to present it in such a way. Even the use of manipulatives was discouraged, except to jumpstart some students.