I swear I’ll go back to math after this post, but I honestly wanted to better understand what the fuss over multiplication being repeated addition was all about. I looked up the research that Devlin had quoted when he was talking about how multiplication is not repeated addition. I read chapter 7 of Nunes and Bryant Children Doing Mathematics.
Honestly, I can breathe a sigh of relief. I now understand what all the fuss is about and no one actually seems to believe that multiplication is not repeated addition (which is good since it is!). When they say that they are actually referring to a few other things which I’ll soon elaborate on. The misunderstanding comes from people not being precise with their language and saying something they don’t actually mean.
Let me clear up some of the things I’m saying versus some of the things I’m not saying. Recall that all of this came about because of questioning whether or not we should teach children that multiplication is repeated addition or if we should use some other concept like “scaling”. First off, multiplication is repeated addition (and I think most people actually agree with this regardless of what they say), so under no circumstances should we ever remove that from our education of children. What I’m not saying is that this is the only concept we should teach. Obviously when teaching anything you should present as many different viewpoints as possible to try to get the best understanding. The idea of scaling is one such other viewpoint (which I’ve pointed out is by definition repeated addition).
After very carefully reading what Nunes and Bryant have to say I think they are perfectly fine with multiplication being repeated addition. There seems to be two different ways they use the concept that “multiplication is not repeated addition.” First, they use it to mean that certain real world situations for which using multiplication is quickest often have more complicated structure than where addition is quick and easy. They continually use the term “multiplicative situation”. It seems to me that they aren’t at all concerned with what multiplication is but how to get children to convert real world situations into symbols for which they can then do the multiplication.
If you have a group of children and a pile of candy that you distribute evenly among the children, then complicated ideas happen like increasing the amount of candy increases the amount that each child will get while increasing the number of children decreases the amount each get. But again, saying that this shows that multiplication is not repeated addition is just misusing that phrase. It doesn’t show that at all (because it’s not true). The complexity of a real world situation has no bearing on the actual operation of multiplication. It seems to me that this is the easy to detect misuse of that phrase.
The slightly more subtle way it is misused is when talking about things like “scaling”. I quote from the book, “In multiplicative situations where the relationship between two variables is concerned, a new number meaning emerges, a factor, function, or an intensive quantity connecting the two variables.” Can I just emphasize something here? Notice the phrase new number meaning. Here it is folks. I seem to have pinpointed the whole misunderstanding. The problem isn’t at all with multiplication being confusing. The confusing thing is that numbers are absurdly abstract entities (even for adults).
Before this point in their education children haven’t had to deal with numbers as abstract entities. It was always 3 apples or 5 pieces of candy. A scaling factor is unitless. The hard conceptual part about this stage of their education is that these abstract entities can take on many different meanings all in the same problem. This has nothing to do with multiplication (because as an operation it is just repeated addition). It has to do with the concept of what a number is. It only looks like multiplication is causing the confusion because this is the first place the concept of a number as something other than a concrete number of objects appears. If our curriculum was slightly different and some other math problem introduced a new number meaning before multiplication it would look like that was the cause as well when it is really the concept of number itself that is the cause.
Now that we’ve pinned this down it seems that when people say “multiplication is not repeated addition” what they really mean has nothing to do with the true meaning of that phrase. What they mean is that when they give children a word problem associated to the operation of multiplication it involves more complicated ideas than the previous word problems dealing only with addition. The other thing they seem to mean is that confusion arises when children have to deal with a more abstract notion of number. All of this makes me feel better that it seems that no one is making the false statement that “multiplication is not repeated addition” and intending for the actual meaning of that statement to be true. The thing that worries me is that this strange stock phrase is being used when much better more precise phrases could be used that convey the actual problem. This phrase seems to make people believe the problem is with multiplication itself rather than with the true sources of difficulty.