This is an account of the 5th session with L. We have been working on creating a proof that all primes are of the form 6n+-1. L started by saying he had reached a "tipping point", and wanted to know the proof. He marked his desire to "have a solution".
I obliged. After I guided him through a proof, he expressed that he was "a bit disappointed!" L was eager to try another problem, but I insisted that we spend some time thinking about what we had learned from solving this one. In order to understand what we talked about, it is necessary to outline the proof, and L's thinking about it. Firstly, he had realised that all numbers can be written as one of 6 forms/columns like this:
Now, L was aware that each of the columns 6n, 6n+2 and 6n+4 did not contain primes, as these numbers are all divisible by 2. At one point in our work, he had also noted that the column 6n+3 does not contain any primes because all these numbers are divisible by 3, but this was more slippery.
All that was left was to put this all together, and then state something like "Therefore all primes must be in the other two columns, 6n + and -1." When I said this to L, he wanted to add a because after this final sentence. He wondered whether this was due to the structure of language, or the presence of the 'why' question at the top of the image above.
What I think was most significant, however, is that L (by his own description) was trying to find a proof by scrutinising the two columns 6n+1 and 6n-1, and the primes therein. The subtlety of this proof is that we need instead to consider the absence of primes in the other columns! I likened this to the idea of figure and ground in Gestalt psychology, and this famous image:
Here, the 6n+-1 columns are the candlestick that is left when we exclude the two faces, or vice versa. This also brings to my mind the power of stressing and ignoring, as described by Gattegno. I reminded L of the moment when working on this problem when he turned his attention to the whole, and how I described it as significant at the time. This is a valuable experience that we can refer back to.
Another important experience was exploring of the role of examples in forming a mathematical conjecture. L is now aware that examples are vital in allowing us to get a sense of structure that may lead to a conjecture, but that no number of examples can constitute a proof, and that ultimately something else is needed.
He also discovered that it is useful to set out examples in a way that may allow structure to become more visible, such as a table, or some other diagram. This setting out of examples into a table also allowed us to re-state the question in a way that made it simpler to work with.
I added that we had also gained familiarity with some important mathematical content, including primes, divisibility and using algebraic forms such as 6n+1.
Here are L's initial notes on this reflection/discussion:
He is going to write up them up in a more useful form; I will attach this to the end of this blog.
Imagine a row of light switches, all turned off. Someone comes along and switches each one (on). Then a second person comes and switches each second one off. Then the third person comes along and switches every third one either on or off, depending on whether it is currently off or on. Then a fourth person, and so on... Which light switches are left on at the end of this process?
L chose to work on the problem with 10 light-switches and 10 people. This is a good example of starting with something manageable. Worked on a spreadsheet, he identified that the first, fourth and ninth light-switches were left on. I asked him what questions he had next. This is what he came up with, following some discussion: