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