If your students can’t do the proof of the associative law of addition of whole numbers, then they have gaps in their knowledge. Moreover, they don’t know they have them. They will think they know common core math principles but will have hidden gaps in their understanding. Those will catch up with them in fractions, decimals, sets, and algebra.
When you can teach them the associative law of addition, you will have mastery of the math behind common core math. So will they. This will make fractions a snap. It will also get them over using a letter to stand for a variable. This is supposed to happen in first grade in common core. So you better know it well to explain it to first graders.
The younger the student you teach something to the better you need to know it. This way you can come up with the right answers to their questions. They have the same questions you do or math grad students do, and you need to learn the right answers. Those are in the Peano Axioms.
Hermann Grassmann, Dedekind and Peano proved that the only way to fill the logical gaps in Euler’s 1765 Algebra was with the Peano Axioms. The same applies to common core math today.
Handwaving is what Euler did at critical points. Handwaving by you the same way may end up with your students being befuddled for good reasons. The wrong answers don’t teach. That is why Grassmann was not satisfied with Euler’s 1765 Algebra book and found the right answers. Dedekind and Peano finished the job of putting the right answers in a proper structure.
Now you just have to learn it and teach it. There is no other way to teach the concepts of elementary math because these are the concepts of elementary math. Without Peano Axioms you are doing rote learning at critical points without realizing it. This creates conceptual gaps that can open up like sink holes in fractions, sets and using a letter to stand for the unknown.