Recently, a friend contacted me wanting to learn more about maths, possibly with a view to sitting an A-level exam in the future, but really just to learn about mathematics.
We set ourselves to have a Skype conversation each week, starting last week. Before our first conversation, I sent him a set of problems and invited him to choose one. Looking at the problems made him feel nervous; he did not choose anything to work on. It turns out that he had not had a good experience of learning maths at school.
During our first conversation he chose the following problem: Prove that all prime numbers are of the form 6n+1 or 6n-1. After discussing the definition of prime numbers, he identified a few primes (2, 3, 5, 7), a few non-primes (6, 8, 10), and then made a couple of conjectures:
(1) "Even primes between 0 and 100 are rare."
(2) "All odd numbers (apart from 1) are prime."
Adding to the first conjecture, I suggested he might like to explore (even) primes between 0 and n for some n > 100.
Following the session I wrote an email summarising what we had talked about, and included some thoughts about working on mathematics generally, including this (the 'MGA cycle') from John Mason (see more about this below):
In the bottom left hand corner is a question that intrigued me: "How many examples do you work through? What's reliable? x of n?"
During our conversation, it became clear that these examples were not enough to convince him that there were no even primes (>2) between 0 and 20,000. He could not be certain there would not be some even prime in there somewhere. He was wondering how many examples one needed to create to become convinced, thinking about this in a statistical sense, hence the question: "What's reliable? x of n?".
I had not considered this interpretation of the role of examples in learning mathematics. Do we (teachers) expect learners to understand how (single) examples in mathematics can represent generality, in a way that instances of everyday phenomenon do not?
I asked him why he had moved on from exploring primes between 0 to 100, to 0 to n, after only seven examples. He said that the examples he had created were enough to convince him "in a lazy way" that there would not be any even primes in that range. I then asked (L for learner):
Me: "Do you think there might be an even prime that is not 2?"
L: [pause] "No!"
Me: "Why not?"
L: "The moment you create an even number, you always get a third stem [factor]... so on that logic..."
Me: "How many examples do you need to convince you that that logic will hold?"
L: "I was thinking you could create (say) three examples for every interval of 1000, so here we would need 60 examples for the range 0 to 20000, but this is still dealing with probability, and is not reliable... statistically it doesn't feel OK."
I drew his attention back to his logic about the 'third stem', and asked: "Does the number of examples matter? Does it convince you more, the more examples I give you?" At this point something seemed to dawn in him. He could see that it was precisely because all even numbers are divisible by two meant that none of them are prime (apart from 2).
This dialogue highlights the complex role of examples in mathematics. We need them to identify structure, without which we are unlikely to be able to make a logical argument, but the logical argument does not make use of an example. This said, I noted out that I felt that his 'wildcard' example 12,864 was a representative example of the generality of the problem, in the sense that it is an even number about which there is nothing special, through which one can determine the structure of the 'proof'.
L started by trying to create what he called a "syllogistic argument", which I took to mean that he was attempting to form a logical argument similar to the one above, but bypassing the creation of examples. I am not sure exactly what this looked like, but after a while of this he said: "I've got myself in a tizz!" and recognised that this approach was not helpful - what was needed were examples, in the first instance. I referred back to the MGA cycle (above), that the manipulation of examples leads to getting-a-sense-of the problem, from which something might be articulated, and the cycle starts again with expanded awareness of structure.
He quickly realised that 9 was not prime (as it was divisible by 3), followed by 33. He could see how it was connected to the first conjecture (being divisible by 3 means there is a third stem), which led to a new conjecture: "All odd numbers are prime except those that are divisible by 3." He then chose a "clumsy number" (similar to his 'wildcard') 27937, and stated that it was prime as it was not divisible by 3. We talked about numbers that were divisible by three for a while, and then something happened.
He was working on something on his calculator. I asked him what he was doing, and he said he was dividing 27937 by other odd numbers (5, 7, 9, 11, 13... ) to see if it was divisible by any of them (it turns out that 27937 is not prime, being divisible by 13). It was not clear how this shift of attention had occurred, he said he just felt a "niggle" and "just decided to start doing it". Manipulating, getting-a-sense-of and articulating had led him to the seemingly spontaneous action of testing for prime-ness.
I suggested an answer to his question "How many examples do you work through?" might be: As many as you need to determine the structure of the problem. He then created this diagram, his version of the MGA cycle:
For next week, he is going to work on the original 6n+-1 problem that stimulated all of this activity.