In our last session, me and L considered the question 'How many examples do you work through?'
Today we continued work on the problem "Prove that all primes >3 are of the form 6n+-1."
During the week, L sent me a video of this thinking, from which this is a screenshot:
From this he concluded that not all primes are of the form 6n+-1. It is interesting that he substituted primes into the form 6n+-1, something that he later questioned.
What is most interesting, however, is that the example 6 x 11 - 1 = 65 presented, for him, a counter example to the statement "All primes >3 are of the form 6n+-1." [I'll call this statement A] How, as a teacher, can I help him see that finding a non-prime of the form 6n+-1 is not a counterexample to the statement?
[I have since given him this example: I make the statement "All cows have four legs." I go out and find a non-cow that has four legs. Is this a counterexample to the statement?]
We talked about our previous work in which we proved the statement "All primes >3 are odd." [let's call this statement B] We talked about how the logical argument we came up with was produced by looking at examples, but did not require an example. The crux of this, and much mathematical reasoning, is the transition from 'What seems to be the case?' to 'What must be the case?'
L articulated a logical argument for the truth of statement B: "All evens are not prime (because they are all divisible by 2...), therefore all primes are odd." I wanted to provide an image, so showed him this table. I also invited him to draw two circles labelled 'odd' and 'even', in which we started populating with whole numbers from 4 upwards. Inside the odd circle came another circle, the primes. Just as we started doing this, he stated that the converse of statement B was false: "... but not all odds are prime," of which 9 and 15 were examples.
In order to work towards a similar proof of statement A, I presented another statement: "All primes >3 are one more or one less than a number in the 3 times table." [call this statement C]
He decided to draw a table with three columns with heading 'primes', 'one more or one less', and '3 times table'. He went through all of the primes up to 23, then said that statement C "looks like it is true". I asked him what the mathematician in him wanted to do next, he said that he would like to explore primes more until he could find one that wasn't one more or one less. He invited me to pick some larger primes. I chose 101 and 331. He showed these were both one more than the three times table.
L was still thinking in terms of collecting evidence, which is what is required to convince ourselves about what seems to be the case. But how do learners of mathematics discover when and how to make that leap from what seems to be the case to what must be the case?
I asked L to articulate what he thought at this point. He came up with this logical argument: "All primes are one more or one less than a number in the 3 times table... because all numbers in the three times table are divisible by 3." However, he felt that there was something missing from this argument, that he was "blagging it". This is often how it feels when trying to form a mathematical proof!
L decided to try creating some circles and populating them with numbers as we had done before in order to provide a more rigorous argument.
I invited him to try to remember exactly what we did with the circles. However, I was aware there was a 'pedagogical gap' here. It was me who had created the circles, and it could not be clear to him how I done this. The essence of the circle creation, and the essence of the argument - was that the two circles contained all possible numbers, and that one of the circles (odd) contained all the primes.
He made four iterations of circle creation, shown below:
L tested each iteration by populating them with numbers, starting at 4. We scrapped each iteration when certain numbers couldn't find a home. In iteration #4, each number had a home, but sometimes two. We continued to populate it with numbers up to 13. At one point I asked him to cover up the prime circle as I felt it might help him see that all of the numbers were in the other two circles, and that all of the primes were inside the +-1 circle. He said "We don't need a prime circle!" and then said "We are going about this the wrong way, the starting point should be primes." I felt we were close to something, but that this final comment might take us further away from a proof. But this is about the process, not necessarily arriving at a solution.
L then drew two circles labelled '3x table' and '+-1" and started populating them with primes, up to 41, which were all in the +-1 circle. The reason I felt this was taking us away from a proof is that it is just more evidence for what seems to be the case. At 41, he said "Shall we keep going?". I replied "Do you want to?". He said "Not really..." which may have had a double meaning - that he could see that more examples were not needed, but also that he was finding it hard to keep going. I asked what is a common question for me: "What do you want to try?"
L was at that critical mathematical stage. We are convinced that something is true, it almost seems obvious that it is true, but cannot find that missing piece of the argument to create a proof why it must be true. At some point previously he had said "Are you going to put me out of my misery?" I had registered this, but had resisted up to this point. Now, however, I felt it would be beneficial to show him this:
I asked L what he was attending to. He said that his attention was drawn to the 3n column in the middle, and then that it was "a way of writing all of the numbers from 4 upwards." We started highlighting the primes. The conversation then went as follows:
L: I get that.
L: It's a way of articulating a proof.
Me: Do you see how that's a proof?
L: You can be sure the primes will be in the 3n-1 or 3n+1 column.
Me: How can you be sure?
L: Because none of them are in the three times table... that's the thing! All primes are of the form 3n-1 or 3n+1 because no primes are in the three times table.
At this point, we decided to stop. I invited L to return to the 6n+-1 problem using some of the ideas we talked about in this session. He liked the table representations, and is going to work on using them.
Before finishing, I drew L's attention to what was the same and different about the 'odd/even' and '3n' tables, and the importance of his observation that this was "a way of writing all of the numbers from 4 upwards". It will be interesting to see if he can generalise to the 6n case.