```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 multiplicationcalled "What Exactly Is Multiplication?"In this article, Devlin discusses multiplication from a conceptualviewpoint: Multiplication is scaling.  Actually, multiplicationas scaling and rotating is a better image to use since it extendsto complex number multiplication.  The multiplication z_1*z_2scales the vector z_2 by a factor of |z_1| and rotates itby arg(z_1).  He also discusses multiplication as an abstract operation inwhich we do not define what multiplication exactly is otherthan that it is a ring operation with certain properties.He also mentions in his article that mathematicians rarely thinkabout what multiplication is since these kinds of questionsdo not arise in their work and hence the reason why he has notaddressed to his readers much about what multiplication is inconcrete 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 abstractmathematical concept of multiplication avoids many of thesenumerous complexities associated with multiplication as aconcrete operation; I doubt that I could write 400+ pagesdevoted to just multiplication.  Perhaps that explains whyso many students and even teachers struggle to understand whatmultiplication is in terms of concrete meanings.  As KeithDevlin 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 reachthis 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
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