For those who have read and heard Keith Devlin, he is pretty close with Jo Boaler who you may also have heard about. Keith Devlin, you will recall, writes a column called Devlin’s Angle in MAA and also is known as “that math guy” at NPR.
He made a big name for himself some years ago when he claimed that multiplication “Ain’t no repeated addition”.
Well, yes, in formal, higher level mathematics, there is a general definition of multiplication that must meet several conditions. Technically, it is a function which maps two objects (numbers, functions, even shapes) from a set, into one object, (e.g., f(2,4) = 8 ) and the function is commutative, associative, distributive, and has an identify function called “1” in which a*1 = a.
What he seems to miss is that this general function does in fact include repeated addition as a means of informing the particular relations between numbers. For example, the times table chart showing the various facts is formally taken as an axiomatic definition; i.e., 4 x 2 = 8. That is, it equals 8 because we define the function that way. The inconvenient fact that Devlin likes to dodge is that the definition is informed by repeated addition.
Devlin and others of like mind think that teaching multiplication as repeated addition results in confusion when we teach fractional multiplication. Actually it isn’t that confusing, and using an area model incorporates the ‘repeated addition’ form of multiplication to get the end result.
Devlin and others who repeat his refrain got to their higher understanding by starting with repeated addition, which they now disdain. Remove the ladder much, Keith?
It is a throwback to the 60’s new math in which multiplication was defined formally. The joke was that kids knew that 5 x 4 = 4 x 5, but didn’t know that it also equals 20.
Funny that the top performing nations like Singapore manage to teach multiplication as repeated addition.
Devlin came under a lot of criticism for his series of articles, yet he stood by them and defended his stance vigorously in subsequent articles.
Boaler doesn’t go quite so far, but she is more for “fluently deriving” the math facts than straight memorization. So 9 x 8, should be looked at as 9×9 – 9. That way, kids who don’t know all their facts can derive them. And also satisfy her idea of what “deeper understanding” is. Really, folks, multiplication isn’t that hard. Nothing against the strategy she talks about, but the disdain for memorization (because it supposedly eclipses understanding) is just more money-making nonsense for Boaler and her ilk.
In my opinion, of course.