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.