I don’t hold back from challenging the way mathematics is taught in schools myself, but Keith Devlin’s recent MAAonline column is off base and out of line.
Under the heading It Ain’t No Repeated Addition he criticizes those who identify multiplication as repeated addition.
“Multiplication of natural numbers certainly gives the same result as repeated addition,” he admits, “but that does not make it the same.” Oh yes it does – if by “the same” you mean defining the same binary function, which is a perfectly reasonable and commonly accepted definition. The fact that one basic operation (which actually in fact is not necessarily addition, but rather, in the Peano axiomatization, is incrementation) can be used to define others and then to infer the existence of more elaborate algebraic structures in which the new operations have nicer algebraic properties, is actually one of the most beautiful aspects of mathematics and not something which should be hidden from students by some claim that the larger structure is somehow more fundamental. To me an approach which denies the roots of our subject (as Devlin does when he argues that, when teaching the subject now, we should somehow ignore that fact that 10,000 years ago “the earliest precursor of what is now multiplication was indeed repeated addition”) is off base. But although it is not what I prefer, I can appreciate the appeal of an approach which treats more elaborate mathematical structures as fundamental. So I would not go so far as to call it “out of line”.
Where Devlin is out of line in my book is in the pompous and authoritarian way he presents his case. “Let’s start with the underlying fact. Multiplication simply is not repeated addition” he says baldly – with the implication ‘I am a big pooh-bah in mathematics and what I say goes without question’. To be honest, this tone is not repeated until towards the end of the piece and there it is addressed more to his MAA colleagues rather than the teachers. But “We are the world’s credentialed experts in mathematical structures” is not the approach that I would use to convince teachers of what I considered an error in what they were teaching. A far better tone (and result) might have been achieved by opening a dialogue to discuss the relative merits (with regard to both paedagogy and mathematical integrity) of an evolutionary vs a static approach to the introduction of mathematical operations to our children and students.