Devlin's Right Angle, Part III

So I'll start to wrap up here—just one more post after this, I promise—by making a long story short and a short story a bit longer.

Devlin has the right angle on this topic. Multiplication is not repeated addition, and we should really stop telling kids that it is.

However, as I tried to relate in my first post on the topic, Devlin is wrong to ask classroom teachers (or homeschoolers) to make this adjustment. Even if teachers are not, in effect, forced by state standards or their current mathematics curricula to introduce multiplication as repeated addition, separating the two operations from each other requires a fundamental change in how most elementary mathematics teachers go about the business of teaching multiplication. For evidence of that fact, you need go no further than Homeschool Math Blog, run by Maria Miller, who has an actual financial stake in these kinds of arguments. So I agree with Goldenberg when he says that "i[t] does not suffice merely to assert that the two [multiplication and repeated addition] are, for the most part, not the same."

Yet, as I tried to get at in my second post, it does not similarly suffice for teachers—or textbook publishers or curriculum experts—to simply throw up their hands and declare that, well, the nice NPR math guy might be right, but he didn't spell out for me word for word how I am to change my practice, so I'll leave well-enough alone.

Of course, that doesn't mean that educators should just agree with Devlin and move on to implementation. I thought this, from Maria, for example, was a pretty good rejoinder:

Even our word "multiply" refers to multiple copies of the same... people and animals "multiply", we talk about multiples, etc. We use the word "times" referring to doing the same thing over and over, such as "I opened the door three times."

And consider these grafs from a book by Jan Gullberg:

Multiplication was thus defined by Robert Recorde (c. 1542) [spelling edited slightly for clarity]:

Multiplication is such an operacion that by two sumes producyth the thyrde, whiche thyrde sume so manye times shall containe the fyrst, as there are unites in the second.

[. . . .]

The earliest form of multiplication known is the Egyptian method of duplation, which reduces multiplication to a form of continued addition. The method . . . was frequently copied by other peoples, and is commonly found in textbooks from the Renaissance.

But it also doesn't mean that educators should simply reject Devlin's idea out of hand. After all, we're not ancient Egyptians. And it's never going to be 1542 again.

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