Teachers should learn the Peano Axioms to prepare for the common core math. This is almost a single stop answer on how to prepare for common core in math.
Learning the Peano Axioms and the development of arithmetic using them will teach
- 5 Rules for counting by one.
- How counting by one becomes addition by recursion.
- How repeated addition becomes multiplication by recursion.
- How the initial segments or head sets of the natural numbers are formed by set recursion.
- The basic parts of set theory needed for counting, addition and multiplication.
- How to think mathematically.
- How the rules of algebra are the same as the rules for numbers.
- How the logical gaps in Euler’s 1765 Algebra book are filled by 19th century math foundations.
- Real explanations of math as opposed to pseudo explanations.
- The origin of the associative, commutative and distributive laws for numbers and for algebra.
These topics are covered in this blog and in my books.
See links above for more information.
Aim to learn by heart how to prove the associative law of addition for whole numbers. This is within the grasp of teachers and is a significant milestone that Grassmann made in 1861. Grassmann was himself a teacher and he was the one who first proved the associative and commutative laws of addition of whole numbers.