Andrei Toom calls this an “extended version” of a talk he gave a few years ago at the Swedish Mathematical Society. At
159 pages [2010 updated version is 98 pages], I would call it a book. Whatever you call it, it’s a must-read for math teachers:
Main Thesis: Word problems are very valuable in teaching mathematics not only to master mathematics, but also for general development. Especially valuable are word problems solved with minimal scolarship, without algebra, even sometimes without arithmetics, just by plain common sense. The more naive and ingenuous is solution, the more it provides the child’s contact with abstract reality and independence from authority, the more independent and creative thinker the child becomes.
When we teach children to solve problems in school, we do not expect them to meet exactly and literally the same problems in later life. Mathematical education would be next to useless if its only use were literal. We want much more, we want to teach children to solve problems in general. In this respect traditional word problems are especially valuable, because to solve a word problem, you have to understand what is said there. This function of word problems is very poorly understood in America.
I suggested that the main educative value of word problems is that they serve as mental manipulatives, paving children’s road to abstract thinking. Pumps and other mechanical appliances are easy to imagine working at a constant rate. Problems involving rate and speed should be (and in Russia are) common already in middle school. Trains, cars and ships are so widely used in textbooks not because all students are expected to go into transportation business, but for another, much more sound reason: these objects are easy to imagine moving at constant speeds.
Also check out Andrei Toom’s articles on math education and humanities.
Hat tip: Mathematics under the Microscope [but I can’t take you back to the original article, because apparently his new blog doesn’t include the old archives.]