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?

L: Nah!

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?

[long pause]

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.

Me: Why?

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.

Me: Right.

[pause]

L: I'm on the cusp...

L: There's a comfort in it.

Me: Are there any primes in there?

L: Nah!

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?

[long pause]

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.

Me: Why?

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.

Me: Right.

[pause]

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: