Further thoughts about multiplication. It is great that Keith is making this work more widely known, but I have recently been realising that even the book he refers to underplays the difficulty children have with division. As mathematicians we often assume that it is enough to know about reciprocals, and to see division not as a separate thing but as the inverse of some other multiplication - I am talking here about multiplication on real numbers of course.
But for school students, even those in upper secondary school, the meaning of division is still tied up with algorithms they learn to carry it out - algorithms mainly concerned with chunking numbers and then performing some kind of repeated subtraction. The meaning of scaling gets totally lost in the algorithm. Furthermore, the meaning of division is also very strongly modelled by partition, rather than scaling, and the everyday language used in school to trigger division enforces this - "how many of these go into this?" As educators we should not underestimate how strongly these meanings linger - meanings that are associated with repeated addition but not with scaling.
I have been wondering what school students' understanding of division would be like if instead of learning algorithms for dividing they always constructed rational numbers to represent multiplicative inverses, and the 'answer' to a division is then expressed as an ordered pair rather than a single decimal number. Then the dominant mathematical action when dealing with 'division' would be finding common factors to express the multiplicative relation in its simplest form.
Happy New Year
Anne Watson Professor of Mathematics Education, Department of Education, University of Oxford (Linacre College) ________________________________________ From: Post-calculus mathematics education [MATHEDU@JISCMAIL.AC.UK] On Behalf Of Jonathan Groves [JGroves@KAPLAN.EDU] Sent: 01 January 2011 22:28 To: MATHEDU@JISCMAIL.AC.UK 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 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.