The saga continues
In an article dated September 2007 entitled What is conceptual understanding? Keith Devlin mentioned that he wished teachers would stop “saying that multiplication is repeated addition”. This led to enough response email to spawn a dedicated followup article in June 2008: It Ain’t No Repeated Addition. This led to an Internet firestorm and many more articles by Mr. Devlin (one, two, three, four, five) reinforcing the contention that teaching children multiplication as repeated addition is both mathematically wrong and educationally destructive.
I chipped in marginally on the educational question (The “Multiplication is Not Repeated Addition” Research) and solidified my opinion on the mathematical end (“Multiplication is Not Repeated Addition” Revisited) but I had yet to finalize what I thought of the educational issue: does teaching multiplication as repeated addition really hurt students, or is it as Mark Chu-Carroll puts it “a choice between teaching children no intuition, and teaching them a pretty good beginners intuition”?
Since then I have taught several mathematics classes for K-8 teachers as well as helped review most of the recent “mathematics for elementary educators” books, so I’m ready for my final (?) post on the issue. Given Mr. Devlin’s “final” article has been followed by four more, I don’t know how true this will be, but nevertheless–
The short answer
It is fine to expose students to multiplication as repeated addition if great care is taken to avoid the problem of identical mental models.
The long answer
The research I’ve seen does not indicate a significant problem with multiplication-as-repeated-addition at the immediate point of learning; the papers with that goal seem to be of similar type to the paper I already wrote about of not being an honest comparison.
The big issue comes later, with problems like this:
For instance, 12- to 15-year-old pupils were asked how to find the cost of 0.22 gallons of petrol if one gallon costs £1.20. (They were asked only to indicate the operation and not to perform the computation.) The most common answer was 1.20 ÷ 0.22. When the same question was asked with “easy” numbers, such as £2 for the price of a gallon and 5 gallons for the amount of petrol, the pupils answered correctly: 2 × 5. When interviewed, the pupils did not consider it incongruous for the needed operation to change when the numbers changed.
Yes, would could represent the above operation as repeated addition by considering 0.01 to be the unit. The mathematical possibility is irrelevant in terms of the clarity of the student’s mental model; multiplication as repeated addition is inadequate to the task.
Only one of the mathematics for elementary educator books I reviewed explicitly decreed multiplication was repeated addition and left it there; the rest to greater or lesser extents emphasized that students needed to be familiar with different ways of modeling multiplication, like the area model (“the area of a rectangle with side X and Y”) or the correspondence model (“for every X, there are Y things”). What the books fail to mention is it’s quite possible in early grades to “fake” using one model with another model.
The problem of identical models
A student fixed on multiplication as repeated addition can go a long way solely with that model, because with many problems it can be mentally handled in an identical way to other models. By way of example:
You have a brownie pan divided into 3 rows containing 5 brownies each row. How many brownies do you have to serve?
The most obvious model that applies here is an area model, but as given there’s no reason a student couldn’t simply apply repeated addition anyway: 5 for the first row + 5 for the second row + 5 for the third = 15 total, and done. This can proceed for several grades until a student reaches a problem like this:
You have a brownie pan divided into 3 rows containing 5 brownies each row. Unfortunately on the bottom row the bottom half of each brownie was burned and had to be cut off. In total area, how much brownie do you have to serve?
A fairly natural repeated addition can be rescued by considering the columns instead of the rows: . However, add a burn to the right edge
and the student is now lost. Yes, one could consider 1/4 of a brownie to be a “unit” but for a 5th grader this becomes too much, and situations start to arise like the aforementioned petrol problem where whole numbers and decimals used in exactly the same problem result in students providing different operations.
I believe that sole fixation on any mental model is a bad idea, not just multiplication as repeated addition; a student who thinks of multiplication as the area of a rectangle would solve the petrol problem with greater difficulty than a student with a correspondence model. While teachers often (or least we train them to) pay attention to the “inner life” of the mathematical brain, they do not seem to be aware of the identical model problem until it’s too late.
One of my test questions for my K-8 teachers last semester was to write a word problem where multiplication as repeated addition didn’t work naturally. It’s possible with to force students out of their “comfort zone” and make sure they understand multiple mental models. The danger of multiplication as repeated addition is it can serve as a proxy for other models in early grades; as a “default model” the correspondence model is much more flexible and the one that should have greater weight. But it isn’t necessary to avoid multiplication-as-repeated-addition to such an extreme that mulplication tables are memorized without any notion of how they are derived in the first place (as was done in the early history of mathematics education).
A resource to try
Maria Droujkova’s site Natural Math has a page dedicated to different models of multiplication, and they also sell a poster. Here is a sample:
[Thanks to Alexander Bogomolny for sleuthing out the link.]