I spend some time working on the second prompt as given, and then the third one, in the hope that they will give me insight. I'm not sure about the notation, so I ignore it. Working on the prompts helps me clarify a conjecture equivalent to what I noticed above: the number of partitions of (say) 6

*including a green (3) as the largest number*is the same as the number of partitions

*into*

*three numbers*. For example:

6 = 1+1+1+3 = 1+2+3 = 3+3 (partitions with 3 as the largest)

6 = 1+1+4 = 1+2+3 = 2+2+2 (partitions into three numbers)

I try out different variants of this, complex enough but not too complex. The statement seems to be true for any rod and any number of partitions! Why? I start thinking about this in terms of 'cuts' of a rod, shown below, as this has been useful in the past for what I vaguely perceive might have been 'similar' problems:

*must*be - a correspondence! I spend quite a long time seeking this correspondence, using a range of examples, but nothing comes of it.

I then have a vague recollection that there may be some way of rearranging the partitions. I get the rods back out:

*What did I do while working on these prompts? I...*

- moved from using rods, as seemed to be suggested by the prompts, to working with numbers as it seemed more efficient (at least, at first - see below!)

- found a systematic way of finding and recording all of the partitions

- became aware of increasing complexity around 6 and 7, so didn't go further

- wondered if a question was to predict the number of partitions for any rod, but wasn't that interested in that question

- worked through each prompt without knowing the relevance but hoping something useful/interesting would appear, thus generating lots of examples

- tried to make sense of the notation but found it confusing so ignored it

- looked for structure across the various examples in various ways (number sequences, ways of counting, ...)

- couldn't find anything recognisable in the number sequences that were being generated

- noticed a surprising similarity between different ways of counting the partitions

- tested this similarity across different examples, which clarified into a conjecture, and a question (why?)

- wanted to convince myself what seemed to be the case

*must*be the case

- was recalling possibly similar problems I had worked on before, which led to different representations / actions

- spent a long time trying to find a correspondence (matching) between the two types of partition, by stressing and ignoring different aspects of various representations, over various examples

- increasingly felt I wasn't going to find what I was looking for using numbers or cuts, and

*eventually*came to a fruitful action - rearranging - through a vague recollection of something I had seen before that might be useful

*What did you do? Can you identify anything else that I did?*

*How might all this inform pedagogy?*