You all know the score. A train leaves one city travelling at 35 miles per hour and another races toward it at 25 miles an hour from a city 60 miles away. How long do they take to meet in the middle? Leaving aside the actual answer of 4 hours (factoring in signalling problems, leaves on the line and a pile-up outside Clapham Junction), these sorts of real-world scenarios are often used as teaching tools to make dreary maths “come alive” in the classroom.
Except they don’t really work. 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.
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.