Date: Jan 1, 2011 5:28 PM
Author: Jonathan Groves
Subject: Another Keith Devlin Article on Multiplication
Dear All,

Keith Devlin has written another good article on multiplication

called "What Exactly Is Multiplication?"

In this article, Devlin discusses multiplication from a conceptual

viewpoint: Multiplication is scaling. Actually, multiplication

as scaling and rotating is a better image to use since it extends

to complex number multiplication. The multiplication z_1*z_2

scales the vector z_2 by a factor of |z_1| and rotates it

by arg(z_1).

He also discusses multiplication as an abstract operation in

which we do not define what multiplication exactly is other

than that it is a ring operation with certain properties.

He also mentions in his article that mathematicians rarely think

about what multiplication is since these kinds of questions

do not arise in their work and hence the reason why he has not

addressed to his readers much about what multiplication is in

concrete terms. He does mention that multiplication as a cognitive

process or as a concrete operation is very complex and cites a

414-page book by Harel and Confrey devoted solely to the

complexities of multiplicative reasoning and the development

of multiplicative reasoning in students.

I wholeheartedly agree with his comments that the abstract

mathematical concept of multiplication avoids many of these

numerous complexities associated with multiplication as a

concrete operation; I doubt that I could write 400+ pages

devoted to just multiplication. Perhaps that explains why

so many students and even teachers struggle to understand what

multiplication is in terms of concrete meanings. As Keith

Devlin nicely puts it,

"That is the whole point of abstraction. Though many non-mathematicians

retreat from the mathematicians' level of abstraction, it actually

makes things very simple. Mathematics is the ultimate simplifier."

Of course, the catch is that students need time to reach

this level of abstract reasoning and to learn to appreciate it.

Keith Devlin's latest article can be found at

http://www.maa.org/devlin/devlin_01_11.html.

Jonathan Groves