Reflection on Chris' students visiting

I cannot exactly recall how positive and negative integers were taught to me, even after a while of brain-squeezing. If anything, I found that they "made sense" to me, although I could not exactly say why.

A big theme for the students would be their discovering of advanced mathematical concepts and pedagogical techniques.

Fractions can be readily visualized and demonstrated, and one pair of enterprising boys who run a shoe painting business used that to great effect, using a gridiron football field, which conveniently comes with yard lines in multiples of 10. 

When it comes to students taking on signed arithmetic, by far the most prevalent was that of a "yin/yang" analogy- "chocolate/milk"; "pepper/milk"; "fire/water"; et cetera. On that note, one pair of girls, who used chocolates and milks along with flow charts, particularly impressed me because they were using Boolean logic by themselves without being taught about it. Using "chocolate" for "positive" and "milk" for "negative", they had four flow charts depending on the signs of the numbers:

Chocolate -> chocolate -> chocolate

Chocolate -> milk -> Not chocolate

Milk -> chocolate -> Not chocolate

Milk -> milk -> Not milk

One pair of boys had apparently discovered the concepts of "additive identity" and "subtraction as the inverse of addition" without using such words. They said that 3 can be like 3 + 1 - 1; 4 spots of fire, but one of them cancelled out by 1 bucket of water. It is one thing to be like "zero plus anything is zero"; rewrites like this are what fuels algebraic proofs, and they showed themselves capable of grasping that.

While using the coordinate axes to plot all solutions to a linear Diophantine equation is a classic approach, these boys had found a way to turn it into a game by throwing some Sponge Bob and treasure hunting into it; after finding the "right" pair of coordinates, you discover where you are supposed to "go to next", and the problem goes on. Have they been frequenting teacherspayteachers.com or something, seriously!?