Those closely (or even not so closely) following the mathematics portion of the Internet may have run across a fuss last year over Keith Devlin’s article It Ain’t No Repeated Addition, contending that teaching children multiplication as repeated addition is both mathematically wrong and educationally destructive. Numerous furores arose (for example on this thread) and Mr. Devlin himself followed with two more columns, It’s Still Not Repeated Addition, and Multiplication and Those Pesky British Spellings. I have been reading every blog followup and planning an “ultimate” post judiciously examining all the evidence and delivering a final verdict.
This is not that post. There were subtle misunderstandings going on and I have concluded everyone is right (in some senses) and everyone is wrong (in some senses), which is a recipe for a complicated essay — don’t expect the final result for a while.
However, I have run across something that warrants sharing now. Keith Devlin cited a particular research paper, The development of the concept of multiplication. I read it, and was startled when I came across this:
The picture in the booklet was shown to the child. The experimenter said: “Amy’s Mum is making 2 pots of tomato soup. She wants to put 3 tomatoes in each pot of soup. How many tomatoes does she need?”
Repeated Addition Group
The picture in the booklet was shown to the child. The experimenter said: “Tom has three toy cars. Ann has three dolls. How many tyos do they have together?”
Stop for a moment before I give my take: I want to know if you see what I see.
Two different objects for two different children? How does this give a natural intuition of multiplication?
I can guarantee without even giving the experiment that the first model would be more successful than the second, but it’s more a matter of a badly-formed example than a true comparison of the concept they’re aiming at.
More importantly, it doesn’t match how I’ve seen multiplication depicted in education. Repeated addition tends to be more like:
Frank bought a $3 sandwich every day for four days. How much money has he spent total?
Repeated addition intuitively yet — one can think of the exact same problem with the “correspondence” model given above. If given without commentary, one would not know which model it would fit.
Yesterday, in the supermarket, a customer dumped a whole basket full of power bars on the counter. The checkout counter person started to scan them in, one by one. After seeing about seven of them scanned, with a lot more to go, the customer piped up: “they are all the same!” The checkout person laughed and started to count them.
(This was part of a thirteen part series on multiplication by Bert Speelpenning.)
That is, if Frank has bought 10 power bars worth $2 each, how much money has he spent? The motivation here is again entirely by repeated addition, but there’s no reason 10 power bars can’t match with pots and $2 can’t match with tomatoes. Furthermore, in terms of a mental arrangement, it’s much more common to have a geometric sort of layout:
That is, verbally one might describe the situation as repeated addition (due to the natural situation) but cognitively perhaps something else is going on. (Which consequently suggests some awareness in teaching the topic is warranted … but that’s a topic for my longer essay.)
I’m reminded of that article from Nature from last year which tried to assert abstract examples were better than concrete ones by making a badly-formed concrete example and then surprise — of course people are confused! I have gathered educational researchers occasionally make the questions of their experiments in a “value neutral” sense, that is, as long as the letter of the definition of the concept is followed, it’s valid. This ignores the other intrinsic qualities that make one example clearer than another. Even something as small as using boats as an example in a landlocked area vs. one next to the ocean can have a cognitive influence.
Something else I found curious about the paper is it doesn’t match with the recent efforts (spawned by the original Devlin essay) to rid multiplication of repeated addition. Joshua Fisher tried a model based on the geometry of a grid, while Maria Miller tried to follow Mr. Devlin’s suggestion of multiplication as a scaling factor. However, if the “correspondence” model given in the research paper truly avoids repeated addition, then (from my personal textbook knowledge) teachers don’t need to change their examples at all. Is the “addition” model given in the research is perhaps one that used to pop up in British textbooks? Were we all even arguing about the same thing?
If you’re curious: while it wasn’t up at the time of the controversy, someone has since uploaded the research paper to the Internet for all to read.