Time for some wrap-up and some pulling together, after twelve prior installments in this series. This doesn’t mean that I think I have arrived at some kind of eternal truth about multiplication. Instead, the whole premise of this work is that there is always deeper to go, and that digging deeper will unearth more riches. Yet none of that prevents me from summarizing where we’ve been and where we’ve been going, and where we’ve ended up thus far.

When I started this series, I didn’t know where it would lead. I had some sense that tracing different ideas of multiplication would be fruitful, and I had a definite intention of side-stepping some controversies that are raging in official educational circles, mostly the back-and-forth whether multiplication really is repeated addition. Mostly, I wanted to get away from the – in my mind, limiting – notion that multiplication is something you do with numbers. I wanted to see how multiplication as an idea could arise in steps – sometimes easy and effortless, sometimes haltingly, sometimes requiring going back on earlier steps – out of a process of a person trying to make sense out of a number of different patterns and structures that you see in the world. These patterns arise around grouping, around scaling, around arrays, around units.

In the prior two posts, in my own attempt at making sense of the diverse worlds of multiplicative structure, I’ve divvied up those worlds according to two different rubrics. In one, I distinguished between *scalar multiplication* and *unit multiplication*; in the other, I distinguished between *unary multiplication* and *binary multiplication*. Neither way of divvying things up is common in the math world, though the distinction between scalar multiplication and unit multiplication is quite familiar and natural to physics, and the distinction between unary multiplication and binary multiplication is quite familiar and natural to computer science – if not necessarily under those names. (In physics, it might be more common to talk about dimensions and dimensionless constants; in computer science it might be more common to talk about unary operators and binary operators.)

I think it is important to underline that both these ways of divvying up multiplicative structure are squarely targeted at *ideas* of multiplication. It is a way to sort notions and concepts and models of multiplication. I’m not suggesting that “unit” exists as something out in the world. In fact, we’ve seen that a certain arrangement of oranges can be seen as 3 rows with 2 oranges per row, and from a slightly different perspective as 2 columns with 3 oranges per column. In the first way, “row” becomes a unit, and “oranges per row” another unit; in the second way, “column” becomes a unit, and “oranges per column” another unit. This wouldn’t necessarily match the way a physicist thinks of units. And yet, all the dimensional analysis the physicist is familiar with would all apply to the units the way I have distinguished them here.

Similarly, to find out if somebody thinks of converting the number of rolls of Mentos to the number of candies as a relationship between two numbers (with the “14” being thought as constant, and implicit) or as a relationship between three numbers (with the “14” thought as another independent variable, and explicit), you would have to ask that person or observe them carefully. It’s in how they are making sense of the situation, not something you can tell by looking at the rolls of Mentos long enough.

I think it is also important to underline that I’m not proposing this taxonomy as in some way directing a new curriculum for teaching multiplication, whether in elementary school or when revisiting it in a different setting in middle school. I do propose that it is very fruitful for teachers to be looking at ideas of multiplication, exploring the way that kids and adults have made sense of the structures they encounter, and the way that they deal with dislocation and discomfort when their current ideas don’t fit new challenges.

Most importantly, I think it useful when teachers look underneath their own familiarity in thinking of multiplication in a particular way, and can appreciate the rich landscape of ideas that their students bring in. The only way a student has to learn something new is to start from where he/she is – simply because that’s the only starting point they’ve got. It is this observation that makes teaching such a challenging activity, sometimes frustrating but always rich.

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