Devlin's Right Angle, Part II

Let me continue here by addressing some of the counterarguments to Keith Devlin's assertion that multiplication is not repeated addition.

Michael Paul Goldenberg declares that Devlin's complete statement

Multiplication simply is not repeated addition, and telling young pupils it is inevitably leads to problems when they subsequently learn that it is not.

is both right and wrong.

Goldenberg believes that the first clause of Devlin's statement is right because "multiplication simply is not repeated addition" is true for some number systems. And the reason he thinks Devlin is wrong?

So now I need to go out on a limb and suggest that while Devlin isn't wrong, he also isn't quite right, and the problem lies with the nature of school mathematics and its teaching, as well as issues of mathematical maturity.

I agree with the first part, and the second argument is as old as the hills, and it alone is why I started this blog. I have at times referred to it as the Stork Argument, the 170 Hearts Argument, and the Columbus Syndrome. Basically, the argument goes like this:

The truth is unteachable.

I'll come back to this, maybe. Here's Goldenberg again:

One thing I find lacking in his piece is a solid example that would communicate well and clearly to K-5 mathematics teachers (based on the ones I've known and worked with) how multiplication differs in some deep way from addition. I[t] does not suffice merely to assert that the two are, for the most part, not the same.

One has to wonder, given that Goldenberg's essential disagreement with Devlin has to do with Goldenberg's role as shepherd of his local K-5 mathematics flock, why he can't come up with an example "that would communicate well and clearly . . . how multiplication differs in some deep way from addition" himself.

So let's help him out. Below is a cube (I should have used a cone). Each of its six faces is a square. The volume of such a figure can usually be found by employing the formula l × w × h. However, given that multiplication is the same as repeated addition, we should be able to describe the volume of the figure below using only the operation of addition, repeated.

Go.

Part I | Part II | Part III | Part IV | Part V | Part VI | Finale