Problem Solving with Joe Kyle
Problem-solving - "Oh, that's when it's in words."
These days, we hear talk of "problem-solving" so often, in so many walks of life, that there's a real danger that it will become a term devoid of real meaning, in my view. Within the realm of the mathematical sciences, though, it is the beating heart of our discipline and there are authors who take a scholarly and serious-minded approach to the topic. But all too often it is reduced to such banal strategies as "read the problem twice" (why not thrice?) and "underline the question".
To be fair, I was never convinced that the greatly admired Pólya (1945) offers much substance: 'formulate a plan and carry it out' seems a little lacking in depth, to my eyes. When I am working on a problem, I play around with it, change it a bit, go for a walk, do another problem while the other 'ferments' in the back of my mind, etc, etc - it all seems much more messy and chaotic and miles away from the rather clinical instruction to devise and implement a plan.
Given all this, I don't blame the young teacher I recently talked to who, when asked what she understood by problem-solving replied: "oh, that's when it's in words." At least she had produced something pragmatic and practical that could be taken into the classroom and used.
My worry, now that the term has found a place in the National Curriculum, is that problem-solving will lose any true meaning and degenerate to little more than an advertising slogan.
A real issue for those tasked with 'teaching' problem-solving is the fact that - to put it in the guise of a tautology - a problem is only a problem for as long as it's a problem! I refer here to the fact that once you've seen the trick that unlocks a problem, then - for you - it loses the status of being a problem; it becomes a fact, a known technique or whatever.
For example, the venerable "nine dots" problem that asks for nine dots in a three-by-three array to be covered by four consecutive straight lines, without the pen leaving the paper. Many of you will have seen this before and will know the trick. For those who may not have seen it before, I won't spoil the fun - save to add that it is widely believed that the phrase "thinking out of the box" has its origins in this little problem.
Here's an example of what I think is a good problem: find the biggest product possible from whole numbers that sum to 100.
Why do I like it? Well right at the outset, it penalises lazy thinking! I have seen many (quite bright minds!) assume that the question is asking for the biggest product possible with two numbers that sum to 100. That's a simple exercise in quadratics, but is not what the question asks - a timely reminder to always read the question carefully.
Also, I like it because you can play with the ideas in a simpler context: experimenting with whole numbers that sum to 10, rather than 100, is feasible by hand and begins to indicate where the answer might lie. It also offers rewards for "thinking out of the box" in the sense that extra insight comes from dropping the restriction that the numbers have to be integers. Some intriguing calculus emerges that leads to a walk-on cameo role for the constant e = 2.718...
I have not provided answers to the problems discussed; this is intentional. Problem-solving is not a spectator sport; you have to do it yourself. But I am very happy to respond to any enquiries by email (j.kyle@bham.ac.uk)
Reference
Pólya, G. (1945). How to Solve It. New Jersey, USA: Princeton University Press.