The math in K-6 is actually the most difficult in terms of foundations of mathematics. Ordinary whole numbers were not put into the foundations of math until the end of the 19th century and after rigor was put into calculus.
We need continuing education in math for K-6 teachers so they can get a firmer grasp of the foundations of numbers. This would emphasize both deep learning but also useful phrases for teaching concepts correctly.
The distinction between signed numbers, plus two and minus three, and unsigned numbers, magnitude two or size two has to be given greater emphasis in continuing education of teachers and of students.
Unsigned numbers need to be first class objects in math education. Too much of the role of unsigned numbers is given to positive numbers. The absolute value of plus two should be taught as unsigned two not plus two. This corresponds to the history of math in which negative numbers were allowed in in the last few centuries.
Language is built around unsigned numbers. Onto that foundation is grafted signed numbers. This will work better if we keep unsigned numbers as first class objects in math education. So part of the time, we are just teaching unsigned numbers.
Signed numbers are conceptually more difficult. The history of human use of math shows that. They came later not earlier because of their extra load of abstraction. This extra load is a burden for children and causes many to drop out of math.
Continuing education in math of K-6 teachers will emphasize treating unsigned numbers as first class numbers. So that teachers and students have words and phrases to indicate they are talking about unsigned numbers not signed numbers. Then signed numbers are a composite object of a sign and a size, i.e. a sign and an unsigned number.
This continuing education will also help teachers supplement their income with math tutoring.
