I was working with a learner (A) the other day, on the first of these light-switch problems.
After working on it for a while, A had noticed that the number of switches for each bulb is equivalent to the number of factors for that number, and that an odd number of factors would results in that bulb being left on. A had also developed an efficient way of finding (the number of) factors - in pairs.
When finding the factors of 25, he said: "1 and 25, because 1 and itself are always factors." Then, "five fives are twenty-five," before concluding 25 had 3 factors. He then found the factors of some other numbers, like this:
We talked about previous work we had done on how examples are useful for seeing structure. I asked him: "What's happening here?" but it was not a useful question.
In looking for a useful question, I came to: "What's the same and different about these examples?" but of all the things A noticed, none of them brought him any closer to solving this problem. This, and questions based around 'What do you notice?', are not always as powerful as one might think.
A asked if what he had noticed was significant, and I replied that the process (of looking for what is the same and different) is significant; but he was still no closer to insight about this problem.
I was aware that this was turning into a game of can-you-see-what-I-can-see, or worse: can't-you-see-what-I-can-see?
I then wondered if it would be useful for A to attend to what he was doing whilst creating the examples, rather than creating the examples and then looking at them. I was reminded of a John Mason exercise called 'sandscapes', in which ridge-patterns form when you pour sand on various card-configurations. Whilst doing this exercise, I had found it useful to watch the sand falling, to imagine myself as a piece of sand, in order to understand how/why the ridges were forming as they were.
So, A created a few more examples:
As he created the examples, I asked him "What strikes you?", another un-useful question (in this situation, at least). He continued to notice various things, which were of some interest, but none of which seemed to bring him closer to insight about this problem.
At this point, the idea came to me that A might find it useful to set the examples out differently. I think I suggested he put the examples in order, but it didn't help. He said, "I feel I am missing something obvious." I remarked that someone once said that obvious was the most dangerous word in mathematics. Obvious, ob-viam, in plain sight, but also: in the way, an obstacle...
We were both stuck, me pedagogically. He said he was just "staring at" the examples. I wondered if he could create another example with an odd number of factors, but it is difficult to see how someone might do this. Then I was reminded of a lesson last week with some primary children in which they experienced the usefulness of working systematically through creating aesthetically pleasing patterns.
It was then that I wondered if a way forward here would be to invite A to unlock his creativity, to find other ways of representing the examples that might shed more light on their structure.
I am beginning to understand the importance of creativity in teaching mathematics: for overcoming barriers, breaking out of the ruts we find ourselves in. This was the end of the session; he is going to have a go at this and see if anything comes to mind.