repeated addition redux

Some minds are changed a bit, and I think for the better, by this LinkedIn discussion, but it also includes a good example of exactly what I was concerned about in my comment on Keith Devlin's original piece on the topic.

Here's the link:

Should kids be told that multiplication is repeated addition? | LinkedIn.

First the good.

The question posed is a real paedagogical question to which I don't think we know the answer, and the discussion led to some interesting examples of ways that the scaling interpretation of real number multiplication might be conveyed to small children . I don't think it's yet clear how early they can truly identify and compare different geometric magnitudes  (viz experiments comparing volumes for example) but this is certainly a worthwhile area for research.

But now the bad.

Part of what turns people against mathematics is the certainty of its results which means that one's mistakes are hard to hide.  But what's even worse than being found indisputably wrong when wrong is being told with a claimed mathematical level of authority that one is wrong when one cannot understand the argument.  And I think Devlin's campaign against "repeated addition" makes it more likely for children to be exposed to that kind of dogmatic smackdown.

My objection to Devlin is that he claims as mathematical fact positions which are essentially philosophical. He makes no attempt to show any inconsistency in the  definition of multiplication of natural numbers by a recursive algorithm of repeated addition, but just claims in his first piece that this doesn't capture what might be called the "essence" or "real meaning" of the corresponding operation on the reals and in his most recent effort he makes the claim that recursion is not a case of repetition.  While not mathematically provable, either of these may indeed be a reasonable philosophical position (though I don't subscribe to either of them). And they may have valid paedagogical implications (which I might agree with despite not agreeing with his philosophy).  But I have always suspected that the way he said it would encourage people to dogmatically insist that repeated addition is  "not multiplication. Not in any sense; not even on the naturals". And unfortunately there was an instance demonstrating just that in the course of the LinkedIn discussion.

My fear is that teachers who are a bit insecure about their understanding of mathematics will be led by such attitudes to undermine rather than support the understanding of a child who, on being introduced to multiplication says "Hey, I get it. This is just repeated addition!"

When that happens, I have no doubt that the teacher's response should be an enthusiastic "Yes!",  and it should come with no anxious look or subsequent undermining "but..."

It is perfectly fine to follow the "Yes!" with some thing like "and here are some things we can use it for.."  (see the wonderful poster by Maria Droujkova for some nice examples) . And of course these may include, as Maria does, examples which help to prime the student for seeing things another way. But it is up to the student to eventually see the next step, not for the teacher to immediately suggest that she has somehow fallen short.




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