I'm concerned about the double-sided chip explanation offered by Judy's textbook. Representing multiplication as "putting in" and "taking out" seems as contrived as the "rule," or at least I think a student would see it that way. Students often have very distinct interpretations of the "-" sign; my students typically and explicitly figure out whether this dash signifies subtraction or negativeness, and their interpretations of an expression depend largely on their interpretations of this symbol.
I'm therefore leery of introducing -2 * -3 as taking away two sets of negative three each, simply because this asks us to read one "-" in one way and the other "-" in another way. Also, the notion of taking away is, in my mind, reserved for subtraction.
Unfortunately, I associate multiplication with the idea of quantities of quantities: 6 sets of 8, for example. This does not lend itself well to the shift into negative numbers, and I'm eager to know of how other teachers make this shift. I suspect that it *can't* be done smoothly, i.e., that the shift requires thinking about numbers in a totally different way than we (students) have thought about them when our binary operations were on positive numbers. So perhaps instead of stretching (and breaking) conceptualizations of positive numbers to make them fit the "rules" of negatives, we should develop ideas about why these "rules" make sense with respect to themselves. Perhaps the history buffs on the list can help us here.
Kreg A. Sherbine | To doubt everything or to believe Apollo Middle School | everything are two equally convenient Nashville, Tennessee | solutions; both dispense with the email@example.com | necessity of reflection. -H. Poincare