This looks like something straight out of SMSG (meep meeb -- is that Sputnik?).
As an enrollee in the 1960s, we were always being made to rewrite goofy integers by spelling them out in longhand, e.g. 1578 = 1000 + 500 + 70 + 8. This was the "duh language" (the low level stuff) of our 1st and 2nd grades.
Then yes, "distributive property" -- so fun to say that -- and little demos such as Paul's above.
But hey, where there's one "property" there's gotta be more (I like this recent Supreme Court joke: "just *one* congressional district in Texas must be redrawn" (assuming it's a tiling and neighboring states are off limits)).
So what are the other properties? Here came a failure:
Reflexive: If a == b, then b == a. Associative: Given operation @, (a@b)@c == a@(b@c).
Why failure? Because there was no framework for counter-culture, by which I mean, no coherent mathematical systems that *break* the above rules were ever shared (of course we have many such systems, just they hadn't gotten to any examples in our sequence yet, perhaps because it seemed criminal to teach "breaking the law and getting away with it" to impressionable youngsters).
Yes, these "laws" work for the integers, but why make a law out of such things? Where's the rhyme to this reason?
So the whole "properties/laws" business now seems suspect, undermined. What real *work* do they do? If none useful, then why are we wasting my/our time on 'em? Don't kids *always* want to know this in math, and don't they have a *right* to a good answer (better than "because we need it on tests" (what's called a stupid tautology)).
There's an easy fix of course. Delve into number theory just deep enough to hit polyhedra and permutations. Show where the associative property might fail, without destroying the sense of the game (matrix algebra a good example, and not surprisingly a language for rotating polyhedra as well). And as for reflexive violations, maybe have differences in the __eq__ of left and right side arguments (more on this elsewhen).
Paul's insistence that a proof be "symbolic" is of course a red herring. What's a Chinese ideogram, a symbol or a picture? How about square root of two, in surd notation. What's that? Pictures, symbols... what's the difference? But now we're into philosophy and the foundations of mathematics. Mathematicians don't usually make good philosophers -- Wittgenstein pointed this out, and proved his case pretty well, I thought.