Last week, I gave Josiah a Montessori presentation showing that a number is divisible by nine if the sum of its digits is divisible by nine. Part way through the presentation, Josiah started talking about using other bases to check divisibility by other numbers than nine. Instead of exploring Josiah's ideas, as all teachers know we should in these situations, I ignored the interruption and pushed on with My Plan. Fortunately Josiah was not put off that easily and once we had finished with the presentation, he spent some time on his own exploring divisibility using different bases. What had occurred to him was that just as a number is divisible by nine if the sum of its digits in base ten is divisible by nine, a number is divisible by, say, "a", if the sum of its digits in base a+1 is divisible by a.
Josiah tried an example, checking whether 764 is divisible by seven by converting 764 to base eight and then adding the digits. If their sum is divisible by seven then 764 is divisible by seven.
I thought the conclusion Josiah reached after his investigation was interesting: he commented that knowing that a number is divisible by seven if the sum of its digits in base eight is divisible by seven is not in fact all that useful seeing as it's quite time-consuming to convert a number to an unfamiliar base. This is borne out by the fact that he didn't actually do the conversion to base eight correctly. He came up with 1214; the correct figure is 1374.
Sadly, looking at the page on which Josiah carried out the calculation, it is not possible to follow what he did. I have tried to instill in the kids that their maths working should be clear and easy for someone else to follow because I remember at university that in tests and exams, points would be given for correct steps in a problem even if the final answer was wrong ... but you didn't get the points for correct steps if the steps you took were not clear!
I think it's great that to Josiah, maths is something that you explore and analyse. When I was his age, maths was something that I enjoyed (and did well at) but I simply followed the steps I had been taught - it didn't occur to me to explore or even to think about what I was doing.