This post contains some thoughts about my fourth session with L. We are working on constructing a proof of the statement 'All primes greater than 3 are of the form 6n + or - 1'.
At the start of today's session, L exclaimed, "I've cracked it!". His proof was as follows: "All numbers of the form 6n+-1 are odd, and all primes are odd. Therefore, all primes are of the form 6n+-1."
I said that there is a hole in this proof. L said he felt "disappointment", a feeling he described as akin to that he used to get at school, along the lines of "I've got it, the teacher's going to be chuffed!" and then finding out that the answer is not correct.
I found it difficult to explain exactly where the hole in the proof is. Among other suggestions, I offered this proof of the (clearly) false statement that all primes are of the form 6n+3: "All numbers of the form 6n+3 are odd, and all primes are odd. Therefore all primes are of the form 6n+3." I think this helped L realise that his proof was not valid, but did not explain why it was not valid, or give any clues as to what a correct proof might look like.
After some discussion about this, I decided to change the direction. I wondered if it would be useful for L to re-state the question in a different form. An important part of the problem solving process is to be able to state the question in a way that makes sense. He came up with the following re-formulation: "All primes are in the 6n+1 column or the 6n-1 column. Why?", which seems more useful, perhaps because it turns attention to the spreadsheet, and the numbers:
Further discussion about L's proof brought attention to the oddness and evenness of the columns. L labelled the columns as follows:
At some point, L said: "Have we been ignoring something important? We've spent a long time focusing on primes and 6n+-1, but not 6n+2, 3, 4... the other columns." It would seem that the wording of the original statement had focussed his attention on primes and 6n+-1, but that re-stating the problem had resulted in him turning his attention to the whole.
There started to be longer pauses in our conversation. L started saying what he could see, talking through the numbers "5, 6, 7, 8, 9, 10,...," and that there were, "No primes in the 6n+2, 6n+3, 6n+4 columns". He said that he, "felt close to something." At this point he asked me to help him further re-phrase the question, but I said that I did not want to say something that might deprive him of the opportunity of arriving at an insight. As a teacher, the amount of guidance one gives depends on the situation.
This was followed by a long period of silence. L said, "I think I can find a proof, but it's not presenting itself at the moment." He described a feeling of trying too hard, similar to that of grasping too hard when trying to unscrew a tight lid from a jar.
He started populating more values in the 6n+2, 6n+3 and 6n+4 columns. The following (abridged) dialogue ensued:
Me: Why are you filling in those columns?
L: There's a comfort in it.
Me: Are there any primes in there?
Me: Would you expect to find any primes in there?
L: No, because they're not in the columns 6n + or - 1.
Me: Are there any other reasons?
L: Why would I not expect to see any primes in +2, +3 or +4...? ... [long pause] ... Do you remember when we were talking about primes having two factors? Well, the numbers in the 6n+2 column have more than two factors, therefore they're not primes.
L: They're all divisible by two.
Me: No primes in there.
L: And 6n+3 are all divisible by 3. And 6n+4 are all divisible by 4... hold on, that's not right... they're all divisible by 2.
L: I'm on the cusp...
We talked for a while, but L was not able to make the final step that would constitute a proof. In a sense, he has done all of the work; all that is needed are a few choice words. How might one arrive at these words?
The session had over-run by around half an hour. I wonder if over-running is counter-productive; I made the difficult decision to end the session. As a final stimulus/clue, I presented him with this image and asked him where the primes are: