I have talked briefly about Paul Zeitz's book The Art and Craft of Problem Solving and have just read Learning and Doing by John Mason. Both of these books give the reader strategies for getting started on problems; I am going to summarise what they have to say, then give some examples of these strategies in action.
Zeitz's suggests you start by getting oriented. He talks about some obvious steps such as reading the question, identifying the hypothesis, comparing to to previous problems, etc. He also talks about classifying the problem (is it a problem to solve, or a problem to find?). He also suggests that restating the problem in your own words might be useful.
So far, so obvious. But how exactly do I get oriented? Zeitz suggests three or four strategies:
- Get Your Hands Dirty involves experimenting, such as plugging in some numbers and seeing what happens. For me, this is actually how I get oriented.
- Wishful Thinking and Make it Easier are connected strategies; they are about making the problem simpler in some way, perhaps by making the numbers or algebra easier to deal with, or drawing a picture, perhaps solving a simpler version or just part of the problem. Again, this is part of getting oriented for me.
- Penultimate Step is interesting. If you know what the desired conclusion is, ask yourself what might reach that conclusion in single step, and aim for this. This is not something I do very often, but would probably make me a better problem solver!
As a mathematician you may do some or all of these automatically, but how many of your students do?
An interesting idea that Zeitz also talks about is the 'Fantasy Answer' strategy. When faced with a complex problem, just guess an answer, in the interests of just getting something down on paper and making yourself feel a bit better. Although the answer will almost certainly be wrong, rereading the question in light of the fantasy answer may shed some more light on steps forward.
Mason has a similar view of solving problems, but used the language of specializing. More precisely, he talks of specializing appropriately. Specializing appropriately is about making an abstract problem more concrete to you, which means to describe the problem using objects that you are familiar with and that you can confidently manipulate. (This is connected to the idea that what constitutes a problem or an exercise being in the eye of the beholder.)
He gives the examples of using physical objects, numbers, fewer dimensions - basically, the same as Zeitz's get your hands dirty and make it easier strategies above. He makes the valuable point that, as your mathematical expertise grows, the 'objects' which give you confidence, and with which you specialise, become increasingly abstract.
These strategies are clearly very similar, and match with my experience of solving problems. When writing the scaffolded problem sets for this site, the first step I suggest in nearly all cases is to make it easier, solve a smaller version of the problem, or put in some numbers. Here are a couple of examples:
These strategies for getting started are crucial to improving as a problem solver - they help you get something down on paper and give you an idea of what is going on. All this increases confidence, which increases our perceived chances of solving a problem, and make us more likely to keep trying!
Now, if all this has made you feel confident you could tackle any problem, here is an Olympiad problem that you might like to try: