## Search “counting on” “definition of addition”

Understanding Numbers in Elementary School Mathematics
By Hung-Hsi Wu, Hongxi Wu

ftp://math.stanford.edu/pub/papers/milgram/class-notes-foundations.pdf

https://scholar.vt.edu/access/content/user/wilkins/Public/JECP_2001.pdf

When we search on

we get hits where “counting on” is described as a strategy instead of as a definition.

Counting on is understood widely as a method but not as a definition it would appear.

The hits on the broader search are lower level in abstraction. The first search with definition give us more university type hits.

Cardinality of sets as a definition of addition seems more popular than recursion as in the Peano Axioms.

The cardinality of the union of two sets of disparate elements as the definition of sum has potential pitfalls that we may carry over physical ideas to the definition or proofs of properties.

If we use cardinality as the definition, do we have to use proof by induction? What is done? For example for associativity or commutativity?

What about going from right additive identities to left additive? Or proving the right additive in the first place? Do these require induction in cardinality?

Do we have to prove the recursion theorem first for cardinality? If so how?

Order first with cardinality?

Cardinality seems to be preferred as being closer to physical ideas and manipulatives.