The nicest formal definition of multiplication is "the operation that distributes over addition". When defining operations on abelian groups that might be called "multiplication", it's the distributive property that makes the term useful, and brings us into the categories of ring, semiring, [weird math prefix here]-ring, etc.
But the thing is: repeated addition does distribute over addition! So there is really a very natural extension from "multiplication is repeated addition" to "multiplication is any operation that preserves the nicest property of repeated addition, which is distribution".
I hasten to add that as a GTA I have had untold numbers of students who apparently did not learn the distributive property correctly -- likewise, the biggest difficulty my students seem to have with dimensional analysis in practice is that they have trouble dividing fractions symbolically. Also, substitution (replacing an expression with a letter) continues to trip students up: e.g. when pointing out that, say, newtons per coulomb is the same as volts per meter. (Students are no doubt tired of hearing me yak about how math expressions are a form of communication...)
Sometimes, I do think we need to teach students to "manipulate expressions" rather than just "solve problems", but then again, don't we do that already?
But the thing is: repeated addition does distribute over addition! So there is really a very natural extension from "multiplication is repeated addition" to "multiplication is any operation that preserves the nicest property of repeated addition, which is distribution".
I hasten to add that as a GTA I have had untold numbers of students who apparently did not learn the distributive property correctly -- likewise, the biggest difficulty my students seem to have with dimensional analysis in practice is that they have trouble dividing fractions symbolically. Also, substitution (replacing an expression with a letter) continues to trip students up: e.g. when pointing out that, say, newtons per coulomb is the same as volts per meter. (Students are no doubt tired of hearing me yak about how math expressions are a form of communication...)
Sometimes, I do think we need to teach students to "manipulate expressions" rather than just "solve problems", but then again, don't we do that already?