The question of how age links to ability to understand a proof, e.g. not leave steps out, is obviously important to transition to college math.
At Harvard they have the course Harvard Math 55.
Harvard Math 55
If students in high school have difficulty identifying left out steps in proofs because of age, then Harvard Math 55 for intro freshman would be a mistake.
This then raises the issue of what is going on in the mind at age levels that makes it hard to learn proofs or complicated proofs.
Does this mean at younger ages, drill and procedural math should be the focus and only at later age levels should critical thinking be stressed? Or does it mean that critical thinking has to be introduced at younger ages to create the development in critical thinking needed at later age levels?
This hypothesis or way of thinking would also suggest the use of procedural techniques as much as possible to teach concepts as opposed to the other way around.
Successor prime notation can be used to teach natural number concepts. 1960s New Math did not use prime notation for successor. Dedekind 1888 did. So Dedekind may have been ahead pedagogically.
n+0 = n
n+m’ = (n+m)’
This notation helps teach that addition as a concept comes from successor as a concept. Here the operator nature is stressed.
When we use ordered pairs, we stress the function.
(n,0) -> n
If one looks at the Devlin blog article on Mathematical Thinking, one will see that it is not focused on age level development issues and whether we can teach not leaving out proof steps at younger ages, or we should try anyhow, or anything of the kind.
Truth tables and propositional calculus with their symbols are a way for younger people to learn proofs if that is something they are particularly challenged on because of lower critical thinking skills at lower age levels. If this is correct, New Math in the 1960s was right to try to introduce such methods. They are suited to teaching concepts in a procedural and mechanical way suited to lower age levels. If we wait until people are in their 20s to teach proofs, that may not work either. We may have missed pushing and stimulating the needed development for critical math thinking in the math focus.