The numbers game

I remember two things about Miss May’s class at Mount Eden Normal Primary School in 1969. First was listening to Neil Armstrong setting foot on the surface of the moon, broadcast over the school’s classroom intercom system. I was six, sitting cross legged on the mat… a man of the world, because I had travelled with my family from Scotland by boat, venturing through the bleak streets of Naples, my mum’s hand firmly gripped in one hand and my die-cast Thunderbird 2 (with fully operational Thunderbird 4 pod) in the other for safety; and skirting through the war in the Suez Canal prevented from stopping in Aden because of the shooting. I had even integrated with the strange sounding natives of New Zealand, with their weird accents, and suddenly it all paled with the words “One small…crackle…step…crackle…for man…”.

But even that shock didn’t prepare me for what was to follow.

Mathematics traumatised me like nothing I had ever known. When introduced to those little coloured blocks (whose name I have deleted from my memory bank) I realised for the first time that, however strange the world is, it would never be capable of terrorising me as much as maths. I am not to proud to say that I ran away. When the time came for maths I hid. There was an especially excellent mature oak in the playground, not far from my beloved monkey bars. When it was wet the hollow in the branches would fill with water and once I shared the space with an exceptionally large weta. But these discomforts I could tolerate. Arithmetic I could not.

In time I taught myself to sit quietly, eyes glazed in a trance like state while the wonders of numbers were absorbed by my peers. By my teens I had developed elaborate doodling systems and mildly comical ways of distracting my more diligent colleagues.
Some things should remain a mystery I rationalised (in that irrational way that teenagers have). I managed to graduate with no qualification in maths.

So it seems curious to me that I have become rather interested in the subject and its relationship with problem solving and creativity. Who would have thought?

I picked up a book in a garage sale, quite randomly, called How to Solve It: A New Aspect of Mathematical Method . Where was this when I needed it? Still, when the pupil is willing, the master will appear. In this case the master is a Hungarian mathematician, Polya, whose book was first published in 1945. The truly fascinating thing about the book is the methodology for problem solving. And there is not a number in sight.

1. First, you have to understand the problem.
2. After understanding, then make a plan.
3. Carry out the plan.
4. Look back on your work. How could it be better?

If you are still stumped Polya says "If you can't solve a problem, then there is an easier problem you can solve: find it." Or: "If you cannot solve the proposed problem try to solve first some related problem. Could you imagine a more accessible related problem?"

Impressively simple really. Common sense you say. But where it gets really interesting is his introduction of heuristic methods for simplifying the problem.

Analogy:
Can you find a problem analogous to your problem and solve that?

Generalization:
Can you find a problem more general than your problem...?

Induction:
Can you solve your problem by deriving a generalization from some examples?

Variation of the Problem:
Can you vary or change your problem to create a new problem (or set of problems) whose solution(s) will help you solve your original problem?

Auxiliary Problem:
Can you find a subproblem or side problem whose solution will help you solve your problem?

Is there a problem related to yours and solved before?:
Can you find a problem related to yours that has already been solved and use that to solve your problem?

Specialization:
Can you find a problem more specialised?

Decomposing and Recombining:
Can you decompose the problem and "recombine its elements in some new manner"?

Working backward:
Can you start with the goal and work backwards to something you already know?

Draw a Figure:
Can you draw a picture of the problem?

Auxiliary Elements:
Can you add some new element to your problem to get closer to a solution?

They might seem arcane when listed out that way, but I’d encourage you to pick a couple of techniques that might work for you when you are stuck.

I scratched the list out in my notebook while I was in a café in the city, somewhere along the way I have misplaced my copy of the book so have ordered a handful from Amazon.

Boy are some of my friends going to surprised by their Christmas gifts this year. Both of them.