I’ve posted before about the teaching of mathematics, and the question of teaching “problem solving” versus teaching basic mathematical skills, like how to add numbers, add fractions, or solve quadratic equations.
It’s all part of teaching mathematics, so we must teach both, and we must find a balance. On the one hand, students cannot solve interesting problems without having the basic skills needed to solve them. Skills are learnt by practice, by drill, by doing lots of similar exercises. On the other hand, learning skills by themselves is a bit dry and becomes more interesting when students can see a use for them.
Many years ago we used to teach lots of skills, with no problem solving. Now we are in danger of a complete switch the other way around. There was a comment on this by mathematician and Fields medallist Vaughan Jones in a New Zealand newspaper recently.
I believe there is a difference between asking a child “what is 3+2” and asking “if I have three apples and my friend gives me two apples, how many apples do I have?” The latter is known as a word problem, albeit a simple example. A word problem has to be converted into an equation, which then has to be solved. In this example, the child must realize that the solution to the word problem is the solution to the sum “3+2”. That conversion, from a word problem to an equation, is a crucial part of problem solving. When the equation is found, it must then be solved, using the skills (in this case, addition) previously learnt by drill.
Word problems can be contrived, but there are enough good examples to use. If you haven’t already seen it, there is a great article A Mathematician’s Lament by Paul Lockhart about teaching mathematics and solving problems.