The current instruction materials on teaching “counting on” and cardinality for grades K to 3 have logical gaps. These in turn come from gaps in teaching preservice teachers. Those in turn are linked to set theory books being focused on counting on for infinite sets instead of for the finite head segments of the natural numbers.

Gaps in teaching

- No notation for successor or count on by one.
- Count on by one as a fundamental function is not a concept taught or understood.
- Defining addition in terms of the counting on by one function is not taught or understood.
- The counting on identities are not known.
- Cardinality is used to define addition, but this is not linked logically to counting on in a strong way.
- Cardinality as a definition of addition defines the addition formula in terms of a mechanical off line process done by a human.
- The relation of order to counting on chains is not brought out enough.
- Ordinal numbers as linked to head sets or tail sets and order by inclusion of these sets is not taught or widely understood for math ed.
- The many little lemmas in Dedekind or in axiomatic set theory books to develop the required set theory, succession, ordinal numbers and cardinal numbers are not listed or articulated for pre-service teachers or students or parents or standards boards.
- The only books containing the math for this are axiomatic set theory books whose orientation is about arithmetic with infinite numbers and not teaching how to build up the basic notions of counting, order and addition for a level below college math majors or even grad students.
- Where the head segments come from is not explained to students, or in materials used by teachers or standard setters or textbook vendors.
- If one imagines taking an axiomatic set theory book like Suppes and taking out everything about arithmetic of infinite numbers, and then removing whatever connecting logic was left leaving a random sequence of independent lessons, and then removing half of those or more, then one gets to what is taught in K-3 or to pre-service teachers or in the materials.
- What the students are learning is not articulated to them or to their teachers or in the standards or in the pre-service courses in college. Only isolated bits are presented and those with very little of the connective structure, the huge number of little lemmas in books like Dedekind or Suppes Axiomatic Set Theory.
- Axiomatic set theory materials typically leave out steps for the student or reader that most readers can’t do all of. This makes them unsuitable for teachers or preservice teachers alike.
- In effect, the math behind elementary math and what math ed is teaching in K-3 is not articulated.
- This means it is not concepts but rote learning, skills and drills.
- Moreover, in practice it is even more that way than in theory looking at the standards materials such as they are, which have some concepts in them.

http://www.doe.virginia.gov/testing/sol/standards_docs/mathematics/index.shtml

http://www.isbe.net/common_core/pdf/math_progression052911.pdf

http://www.charleston.k12.il.us/pdfs/curriculum-maps/math/K_Math_Map.pdf

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