## Math Facts are not a Democracy

Not all math facts are created equal.   Math facts are not a chaos of random statements all equally important.

If we push math facts at children they do not form the structure of math from Peano Axioms to fractions as functions and then fractions as a type of number as an independent creation in effect.

This is what the Rips, Bloomfield, Asmuth paper mentioned earlier says.  Just pushing math facts on children does not give them number concepts as encapsulated in the Peano Axioms.  For example, the idea that the natural numbers don’t circle back on themselves is not something that comes just from learning a list of numbers from 0 to 9 or even up to 19.   They write that this does not come until much later.

It really should be taught explicitly and taught why.  However, you wouldn’t think to say it if you didn’t study the Peano Axioms.

Looking at the Number Line does not lead you to articulate explicitly that the natural numbers don’t circle back.  Nor does it lead one to use a term like cycle for a chain of natural numbers that did lead back to its starting point.  The Peano Axioms imply that the natural numbers do not contain any cycles.  This is an important concept.

If starting from m, we got by a chain of succession back to m, then m would have two predecessors.  So this is ruled out, although within the system of Peano Axioms, the proof requires the right structure of propositions to demonstrate it.