This post has been prompted by Maria Droujkova's Multiplication Models poster which summaries situations in the world where multiplication happens, showing the great variety of possible situations.
I think it's fair to say, however, that in the individual develops only a small number of structures for multiplication in their brain which they use to support calculation in all of these situations. There is a point at which they perceive multiplication is going on and their brain starts up its multiplication systems.
We are consciously aware of some of the mental models for multiplication and division we have but not of others. For example in division we deliberately teach quotitioning (chunking) and partitioning (splitting/sharing) but even if partitioning is not taught, research shows it develops anyway. We naturally flip between these two structural methods without being aware of what we are doing. We also rely on known facts and derived algorithms.
In multiplication we are taught repeated addition (subdivided in different ways for different algorithms), but I suspect most of us also have a scaling structure which supports our thinking. This is not taught in most schools and even I, as an experienced maths teacher, find it difficult to describe.
I recently worked on a PhD proposal about exploring the consequences of deliberately and explicitly teaching students a scaling model for multiplication (which also applies very easily to division).
You simply draw a line which is 1 (0 is at the beginning of it and 1 is at the end) then ask the student to put a dot where the number we are multiplying BY is. They are effectively picturing a number line.
Then they rescale their number line to make the original line (which was 1) the number we are multiplying and estimate the value of the dot on this rescaled number line.
So for example 150 x 7
0_______1 .(student might put the dot here)
0_______150 .1000? student guess
We only get an estimate for the value of the dot (although students are very likely to start using and developing methods to make their answers more accurate).
My intention was to use this at an intervention with students whereby when they are given a multiplication problem they do this first before calculating the exact answer in whatever way they have already been taught.
Some of you may spot that this challenges not just the way we 'see' multiplication but also the way we see number - bringing out the dual nature of number as being not just about counting but also about scaling.
Anyway I'm not going to be able to take this forward at present so it's here for anyone to take anything they want from it.
I'd absolutely love feedback!
I'll post a more detailed version of the thinking behind it and its place in current academic research on the next post.