I’ll return to the planned series after this one side post. Keith Devlin at Devlin’s Angle has written many times about how multiplication is not repeated addition and how he thinks it is a detriment to teach children this falsehood. I strongly suggest going and reading all of his articles before reading this. He makes the analogy that he can take a car or a bike to work. The end result happens to be the same, but we shouldn’t pretend like the process is the same. I agree. Just because two processes happen to produce the same results does not mean we can conclude that they are the same process. It is a false analogy as I will show. Repeated addition and multiplication don’t just happen to come out with the same answer. They come out with the same answer because they are the same process.
Before we begin I’ll say a few words about the research done on this topic. I haven’t read it yet, so I can’t comment on whether or not I think the results are valid (it is amazing how much bias … yes, even among professional research scientists … happens in experiments involving observing and quantifying how well children understand things). If teaching children in a certain way increases understanding, then by all means go ahead. I’m no expert on the subject. The whole idea of how to teach a concept seems to have nothing to do with whether or not multiplication is repeated addition and that is what I’m addressing.
First, I agree with Devlin on one point. We can abstractly define multiplication to be a basic operation that is separate from addition. He never really makes this precise, so let’s do that first so we see exactly why they are different operations and why he is saying we confuse the two.
Let’s say we have a set that we’ll call . The sole purpose of this is because our notation is causing the problem. We should just think that is the number . We can define an addition and a multiplication as abstract arbitrary rules that satisfy certain axioms. Note that the multiplication a priori has absolutely nothing to do with the addition (and even a posteriori in most cases). We’ll say the addition is and the multiplication is . Yes, we’re going to use that large clumpy symbol to try to avoid future confusion.
Despite the multiplication being completely arbitrary, notice that we can still define something that we’ll call “additive multiplication”. We’ll denote this with a dot. In other words, . No matter what integer we pick we can define this additive multiplication. There is no reason to stop there. We can define to be some element of the set with the property that . We can do one at a time just means or maybe even better it is adding two and a half times: . We can extend this to the negatives. I won’t worry about the interpretation of this because Devlin admits negatives pose problems in any interpretation. In algebra we’d call this giving our set with the addition a -module structure since we’ve defined a multiplication . It doesn’t ever even mention the multiplication, so indeed it is completely separate.
Let’s cautiously reintroduce old notation. We have these two arbitrary operations on the integers: addition and multiplication . As was pointed out already we also have this other thing and it is very good of Devlin to point out that this other thing need not agree with the multiplication rule we’ve defined. The reason we don’t notice that they are two different things is because when we do the dot and think of repeated addition we specify with the symbol that we want to repeat addition three times. When we do multiplication we’ve reused the same symbol in a different way. Namely, that is just an element of the set on which we defined multiplication which we could be notating as say to keep things separate. I applaud Devlin at this point for pointing out the conflation of these two separate things (although he never told us this was the real reason for the confusion).
Let’s start pointing out some issues with what Devlin says. First off, it is mentioned several times in every single one of his articles that the concept of multiplying by repeated addition only works for positive whole numbers. Hold on just a second. Now that we’ve figured out what is really going on we see this is not true at all. Our set could be anything, say is the real numbers. We still have this -module structure on it. So it makes perfect sense in the repeated addition paradigm to do say . This is a far cry from only positive integers. We just used rational numbers, irrational numbers, and negatives and didn’t run into any problems. In fact, even the alternative proposed by Devlin will have issues with real numbers and the way it is overcome is exactly the same way it can be overcome in this situation, but we won’t get into that here.
The second major point is what Devlin wants to replace the paradigm with. He thinks it should be replaced with the idea of “scaling”. Let’s look at the example that he thinks proves his point. I’ll just quote verbatim the relevant section:
Take a piece of elastic, and tie two knots in it, one near each end. Ask the child to measure the distance between the knots. Suppose it turns out to be 5 inches. Now, as the child watches, slowly stretch the elastic until the distance between the two knots is 10 inches. Get the child to measure it again. Now ask the child to write down a mathematical description of how the new length depends on (is related to) the original length. I would hope the child writes down
10 = 2 x 5
In more general terms, what you did was double the length, or, as an equation:
new length = 2 x old length
I would be very surprised if the child wrote down
10 = 5 + 5
new length = old length + old length
and if he or she did, you would have your work cut out trying to put them right before they have serious trouble in the math class. Sure, these addition equations are numerically correct. But so what? What you have just shown the child when you stretched the elastic has absolutely nothing to do with addition and everything to do with multiplication.
What horrifies me here is that this little example actually shows exactly the opposite of what he claims. He thinks if the child says the new length is the old length plus the old length they are in serious trouble, but I’ll rephrase the example to show that if they think in ANY other terms they will be in big trouble and this way is actually the correct way of thinking.
What exactly does Devlin mean when he uses the word “double”? Really. Seriously. I want to know. Here is what I mean when I use it. The number is causing problems again because it gives us something tangible to apply our arbitrary multiplication rule to. Take a piece of elastic. Take a stick of unknown length. Tie knots at where the ends of the stick are. Stretch the elastic. When using the stick to see how far the elastic is stretched you find that you use the stick exactly twice. In other words, the new length is the old length plus the old length.
This isn’t some strange rhetorical trick. Now that we can’t accidentally apply our multiplication rule we see that when we use the word double the only possible meaning it can have is (length). The dot operation is what we mean. Now we’ve come to the amazing part of all of this. Devlin wants us to define the multiplication to mean “scaling” which is a priori completely separate from the “additive multiplication” of the dot. But when we go to define what we even mean by scaling we find that we are taking as the definition to be repeated addition. This is why the two agree! They are actually the same! If the child tries to tell you that the elastic is (length) in which he means the length is an arbitrary multiplication rule, then obviously the child is going to be in way more trouble because he has completely missed the point.
Devlin seems to have worked out that in abstract math land the “additive multiplication” and the times multiplication are very often different. From this he seems to be concluding that because of this they are different for the number systems we teach children. This is not correct. In principle, they could be different, but in reality the dot defines a perfectly valid multiplication and we take that to be the definition (even abstractly). They don’t just happen to give the same answer. They are the same. Even if we take multiplication to mean “scaling” when we go to figure out what we mean by that word we come back to repeated addition.