How a Mathematician Solves a Problem

One of my colleagues at a previous school emailed round this problem and it caught my attention. Possibly just the bright colours, but it has a simple beauty to it, reminding me of the sort of problems that Ed Southall often shares.

I got stuck into it straight away, first putting together the triangles in a rather naive way that wasn’t any use at all. Realising that I wasn’t getting anywhere, I changed tack and went for the ‘brute force’ approach of applying trigonometry to get the answer.

Key point: if one approach isn’t working, don’t get stuck in a rut. Ask yourself what other mathematical concepts you know that may be related to the problem, and try those instead.

Getting an answer of 180 degrees suggested to me that there would be a neater approach., probably involving putting all three of these triangles into one large triangle. I started trying to do so.

Key point: Get as many ideas on paper as possible, don’t give up if your first couple of attempts don’t work out.

This is a bit of a mess, but in retrospect it’s my favourite page. Trying out a load of options quickly can give you a feel for what might work. I eventually got the idea that two of the angles needed to be on the ‘base’ of the triangle, and it was then a matter of varying the two angles and the distance between them until the third angle appeared at the top.

Key Point: Sometimes drawing a geometry problem more neatly will help you more than you might expect.

Then I decided to start trying to draw this out a bit more neatly; this helped me notice that I could create a key isosceles triangle.

Finally, of course you should write your work up neatly in order to present it to your readers (in my case, a triumphal email to my colleague!)

Key point: Rough working is great, but make sure you write up your solutions neatly at the end. Of course, your teacher will be even happier if they can see your rough thought process too.

I looked at my work at the end of the process.
a) I felt that sense of satisfaction that reminded me why I like maths.
b) I thought it would make a good lesson for my pupils.

In summary, here are generalised versions of my strategies:

  • If one approach isn’t working, don’t get stuck in a rut. Ask yourself what other mathematical concepts you know that may be related to the problem, and try those instead.
  • When you’re confident you’re going in the right direction, get as many ideas on paper as possible, don’t give up quickly.
  • Sometimes drawing a geometry problem more neatly will help you more than you might expect.

I know that the idea of general problem-solving skills (even within the domain of maths) is controversial. Of course, I wouldn’t have been able to solve this problem if I didn’t have a secure understanding of properties of triangles (obviously), proportional reasoning (to scale up and down the triangles) and trigonometry (to give me faith that it was worth continuing!)

However, I do feel that some of the general strategies suggested in  Thinking Mathematically have improved my ability to solve this kind of problem. It was interesting to listen to Dylan William’s thoughts on this topic: Craig Barton suggested that he was moving away from the idea that general problem-solving skills are useful and Dylan was less convinced that he learned general skills from George Polya’s classic book. He did admit, however, that ‘trying a simpler case’ (in the terms of Thinking Mathematically: ‘specialising’) is a pretty useful general principle. This, alongside writing down what you know and want and drawing a bigger, clearer diagram, are the main general strategies which I’ve seen maths students use effectively.

Personally, I’m fairly confident that I wouldn’t have been so effective at solving this problem 10 years ago, when I’d just graduated with a first class maths degree. If nothing else, teaching makes you a better mathematician.

Want to teach your pupils general problem-solving skills? Take a look at my resources here: http://lukepearce.eu/problem-solving/

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