[wcarey]

Well, I'd be curious what your first ten sentences to a group of, say, 11th graders would be about Newtonian gravitation, and how those would compare to what you'd say about the same subject to 8th graders.

It seems elsewhere in the thread that you don't consider it lying if one gives a general disclaimer that models aren't perfect, but I wonder whether or not that general disclaimer wouldn't inoculate the idea that multiplication is repeated addition? It's an imperfect model that super useful for all the numbers available to students when they learn what multiplication is. And I wonder further whether there's not an implicit disclaimer that what you're learning isn't the final word inherent in the structure of the educational system?