Math professor Alexandre Borovik has a post on understanding why arithmetic is difficult to learn. It is because it has a greater conceptual structure than is realized. He points out this complexity goes from hidden to explicit by use of the Peano Axioms and proofs by induction.
Borovik points out that to learn the definition of addition and then the proofs that addition is associative and commutative are hard work. So children have to do this to learn addition.
However, this is harder for them precisely because the teacher and parent do not understand the Peano Axioms and proof by mathematical induction.
The book Pre-Algebra New Math Done Right Peano Axioms is easier to learn from than any other source on the Peano Axioms and addition and the proofs. It is approximately 391 pages long on just the Peano Axioms, order and addition. It also discusses set theory basics and some pedagogy similar to these web pages.
The book has 178 Examples, 463 Exercises, 98 definitions, 45 Lemmas with proofs, references to over 96 webpages, 58 chapters, and is 391 pages when formatted as pdf with Latex. By going so slowly it can build up to the definitions of addition and the proofs of the addition laws. By going so slowly it is suitable for below college level. In particular, it is suitable for parents, self study for teachers, and for the more curious students. It is ideal for home schooling, because home schooling is a greater risk to miss out on the structure and logic of math with fewer people involved.
Professor Borovik’s web page discusses the right zero identity and right shift identity. The latter could be called the right successor identity. The text Pre-Algebra New Math Done Right Peano Axioms covers names for identities and shows how to prove from the right identities the left zero identity and left shift identity. This is simpler than full addition. The shift (or successor) identities are a place to practice proofs and learn proof by induction at a lower level of difficulty than full addition.
We will see in a later post that pre-service teachers are taught Peano Axioms in Germany.
At 6 minutes Prof Spannagel starts to write the Peano Axioms on the blackboard. He has an entire series of lectures on Youtube on the Peano Axioms for his pre-service teacher students.
“In der Arithmetikvorlesung aus dem Wintersemester 2010/11 an der PH Heidelberg spricht Prof. Dr. Christian Spannagel über die Folge der natürlichen Zahlen.”