I love this discussion and want to throw in my two cents. Why not teach beginning students group theory as the foundation to addition and subtraction? While i

I love this discussion and want to throw in my two cents. Why not teach beginning students group theory as the foundation to addition and subtraction? While it wouldn't solve this particular problem since fractions require fields, it would give teachers the tools to explain precisely why addition of two tables is not a group or field operation; they are from different sets and require measurement with units to define a new inclusive set. My own understanding of arithmetic and algebra was enhanced immensely by a book outlining the derivation from Peano axioms. Ultimately the concepts are very simple and kids intuitively do the steps without having the names for the mental processes that they're doing. The educational hurdles to me seem to be 1) identifying the memorized names of the sequence of natural numbers with a procedure for generating them; 2) distinguishing between procedures, objects, numerals, and numbers; 3) basics of set theory; 4) natural induction.