We’re arrived at the eleventh installment of this series, and it would be worth asking whether we have arrived at any conclusions yet about what multiplication really is. We’ve seen multiple manifestations of the idea of multiplication, and it would be understandable if you are left with the question: “will the real multiplication please stand up?”
Though I think the question is useful, I think it equally useful to defer answering it straight on. At least, by deferring answering the question this far, we have collected a good set of scenarios, manifestations, embodiments, contexts, structures, in which the numerical relationships match some or all of our notions about multiplication. In each of them, we can put together an instance where we can point to a two of something, point to a three of something, and then point to a six of something, and see their relationship.
Rather than looking for the single ‘real’ multiplication, I would prefer to look for candidates, or exemplars, around which the other manifestations can rally. In other words, look for ways to slice and dice the data we’ve collected here and see if we can see something coherent emerge.
I’ll show you how I slice the data gathered in the previous posts, and I won’t suggest that it is the right way or the only way – but I trust that it is an interesting way.
One key notion I find in the slicing is the notion of scaling. If you get twice as many vacation days as I do, that would be an example of scaling. If you make three times the salary I make, that would be an example of scaling. If you are half my age, ditto. The comparisons are based on ratios, and all comparisons are between similar things: we compared vacation days to vacation days, we compared salary to salary and we compared age to age. If I get a 20% raise, my new salary is 1.20 times my current salary. It scaled up by 1.20. If a realtor sells my house and gets a commission of 6% of the sales price of the house, that’s an example of scaling. The scale factor is .06; we start with an amount in dollars (the sales price) and multiply by .06 and end up with an amount in dollars (the commission). If I draw an arrow and then ask a child to draw an arrow that is twice as long, that’s an example of scaling. We start with a length, of the original arrow, and we end up with a length of the new arrow – twice as long.
I suggest the name scalar multiplication for this.
In science, we would talk about this in terms of dimensions or units. One of the things being multiplied has a unit or dimension, such as dollars or inches or pounds or degrees. The other thing being multiplied is a “plain” number, no units of dimensions. When you multiply a number of pounds times 2, the result is still in pounds. The number 2 is not in pounds, it is not in ‘anything’. It is simply the number 2. We double the number of pounds.
Completely different from this notion is the thing we encountered when we looked at man hours, in the landscaping example. You don’t get man hours by scaling up men and you don’t get man hours by scaling up hours. You get man hours by having a crew of men work for a number of hours. The resulting unit is neither men nor hours but something new, something distinct from both. Let’s see if we can illustrate this. A picture for an hour is reasonably easy to find and reasonably easy to appreciate:
A picture for a man, a landscaper, is easy to locate too:
Much harder, not surprisingly, is coming up with a picture for a man hour. If you look in Google images, you will find a dearth of images that evoke a man hour. Instead, to match the landscaping example, let’s use a picture that evokes an amount of work, something that stands for the size of the job. Here’s something:
I’m not suggesting that this amount of leaves is in any way realistic as standing for the amount of cutting/raking/collecting/bagging etc that a professional landscaper would handle in an hour. It will have to do here. The point is that a man hour represents an amount of work done.
Below is our picture, then, for a crew of 4 men working for 3 hours.
Note that I’m proposing this picture very explicitly as a picture for , not simply as a picture for .
That the units, or dimensions, play an important role in science is well-understood. That they may play a central role in mathematics isn’t usually appreciated. What I want to suggest is that they play an equally central role in characterizing what multiplication is. Some of our earlier examples already hinted at this .
The same logic and structure is present in that example, but it is even harder to picture fully, since the notion of mints per roll, as clear as it is to kids and adults alike, is not a unit that particularly allows for a clear visualization. And the same is true for many of our other example units, like miles per hour, miles per gallon, oranges per row, and more.
I suggest the name unit multiplication for this.
So, one way to slice multiplication. Another one in a subsequent post.