[pinealservo]

The Peano Axioms are not about defining what 'addition' and 'multiplication' are; they're about presenting a model of the natural numbers along with the operations of addition and multiplication in first-order logic. This makes them great fodder for worksheets in proof-writing courses (and I did glance through your worksheet; it looks like a great resource!), but doesn't necessarily expose the standard mathematical notion of what 'addition' and 'multiplication' are! If you ask a random mathematician out of the blue what the axioms of arithmetic are, my guess is that you won't often get the Peano axioms as an answer, but rather the standard algebraic ring or field axioms.

Although it seems very common to define multiplication as repeated addition in dictionaries and materials for kids, it is in fact only a valid definition for a rather narrow conception of numbers, i.e. the natural numbers. It doesn't work without exceptions for the Integers, the Rationals, or the Reals. Considering that we want students to eventually be able to deal with the Real numbers, I think it would be better to avoid defining multiplication to be something that doesn't work outside of the Naturals! We would be in quite a pickle trying to explain the calculation of the area of a circle in terms of repeated addition...

By calling what they're teaching the 'repeated addition strategy' it seems like they've thought about this; it's indeed a strategy for calculating a product of two natural numbers. But that makes the marking off of a point all the more perplexing, because both repeated addition schemes are equally valid strategies for computing the same product, by virtue of the commutative property of multiplication! Which is indeed generally an axiom and not a derived theorem in the more general case of multiplication, because multiplication is not generally defined in terms of repeated addition. In general, the axioms only say that multiplication distributes over addition: https://en.wikipedia.org/wiki/Field_(mathematics)

My kids are actually going through this phase of their curriculum right now, and I know that here, at least, they do teach the commutative property of multiplication fairly quickly after multiplication is introduced. So I'm not really sure what pedagogical point of the grading of this assignment would be, but perhaps there is some point to it. Fortunately my kids have not run afoul of this kind of thing.