He starts by comparing exercises with problems. He calls an exercise a mathematical question that you know how to answer immediately, such as 237 x 68. A problem is a mathematical question that you do not know how to answer immediately (although it may be an exercise for someone else).
He goes on to define three 'levels' of problem solving: strategy, tactics and tools. Strategies are mostly non-mathematical, often psychological, general ideas for starting and pursuing problems - more on these below. Tactics are mathematical methods that work for a number of different problems, such as symmetry or parity. Tools are less important - they are techniques and 'tricks' that work in specific situations.
I like this idea of splitting problem solving into different aspects. It feels similar to playing chess, where a strategy is an overall scheme you follow to win a game, upon which all your moves are based, whereas a tactic is a short sequences of moves you use time and again, perhaps to capture a piece, such as a pin or skewer.
Zeitz discusses a (quite large) number of strategies and tactics in his work. I will talk about many of these in future posts, but for now let's look at a few of the key strategies in action: 'wishful thinking', 'make it easier' and 'get your hands dirty'. Consider this problem from the non-calculator US AMC10 test, designed for KS4 students:
So how to begin? Of course, you should try it for yourself before reading further...
Here's one way of going about it. Wishful thinking and make it easier say to me that I would be able to solve this problem easily if the numbers were smaller, and get your hands dirty suggests I should try calculating some of these smaller numbers and see what happens. So here goes:
So should we end here? No, there is reflection to be done. Firstly, our strategies of wishful thinking, make it easier and get your hands dirty have served us well for this problem, so let's remember them for future use. But was there an easier way to solve this problem? And what else have we learnt? What have we learnt about factorials and square numbers?
As a teacher I like to think about how we can extend this question further. We could think about the nature of square numbers. For example, in this question we used the fact that a square multiplied by a square is a square - is this always true? Why? Why not investigate this idea further by posing a question to students like this: