This morning I did something that took me quite a long way out of my comfort zone, and I’m very glad I did it.

We have a study school here in Cambridge at the moment, forty Year 10 students from London who are spending four days with us becoming better problem solvers. As part of this, I agreed to model solving a problem I hadn’t seen before, in front of the students. My colleague Charlie had found a suitable problem he didn’t think I was familar with:

Find all positive integers m, n with n odd such that 1/m + 4/n = 1/12

My first reaction was panic. I didn’t immediately see a complete route to a solution so I was very worried that I wouldn’t be able to solve it. Eventually though, I made a start and tried some things out, worked out some conditions that m and n had to satisfy, and found one solution. But the question had asked for all solutions, so I knew I needed to work more generally.

As I continued to work on the problem, people in the room started to help me out. The Year 10 students made a couple of suggestions, as did the student ambassadors sitting at the back of the room. Of most help to me was Charlie who knew one possible route to a solution and made suggestions that he thought might help me to get there.

Eventually, I realised that I had found an equation the solutions to which would give me all possible values for m and n. At this point, I was very happy to stop but the students wanted me to carry on and find at least one solution using my method so I continued, and once I’d shown them that my method worked I went and hid in the corner to recover my wits while Charlie drew out the key points of what I had done.

Charlie talked very little about the maths I’d done, but rather identified ways in which I’d used my problem-solving toolkit to help me. Having read Polya’s “How to Solve It” quite recently, the idea of problem-solving heuristics was on my mind so the narrative I gave while thinking out loud about the problem had phrases such as “This has helped me in the past so I’ll try it here” and “This isn’t going to work, what else can I try?”

The best part of the morning came quite a bit later. The students were working on the problem solving booklet we’ve put together for them, and one was working on Hidden Dimensions. He wrote down some algebraic expressions, and then used exactly the same technique I’d used on the problem earlier to solve his problem. Making that connection for himself, without any prompting, is exactly the sort of thing we were hoping this week would foster in these students!