When my child was in middle/high school, I insisted that he learn the math in an “old school way”: When he learned geometry, I hunted down materials that covers the “two columns” proof and insisted he learn it (I do not care whether the school teaches it in this way or not.) When he learned the limit, infinite sequences/series topics in calculus, I also hunted down materials that teach the epsilon-delta (or episilon-N in the case of infinite sequences/series) proof.
In the stone age when I was in high school, the school did not teach calculus. But I found some math book which teaches this concept by myself. (I think I first learned the begining calculus from two books. I only remember the name of one books: “What is mathematics” by Courant (sp?) The other is a book translated from some Russian book. I also got some “frame-by-frame”-based book (somewhat like a “workbook”) which teaches limits, mainly its epsilon-delta proof.
Calculus was not in my high school course sequence and I could still manage to self-study it in some semi-official way (i.e., with some reasonably rigorous, convincing proof.) My child’s high school math sequence includes calculus; so I think he has no excuse to not learning it properly.
When my child learned physics, I also insisted he could derive most of the formula he used. When he learned special relativity, I also handed him a small book (like 150-200 pages ONLY?) by Resnick (He is the same author who coauthored a popular college physics textbook by Halliday and Resnick) which covers the axiomatic treatment of special relativity, using algebra only. Prior to that special relativity book, I handed him two other more “introductory” special relativity book.
Did I “abuse” my child by asking him to learn “too much” additional topics? He sometimes complain; but being a “good” child, I think he has at least studied like 50+% of these extra learning materials that I passed to him.
On the other hand, we rarely paid attention to the scores he got in various tests (esp. SAT.) In 7th grade, while many students particiated in SAT tests for the purpose of getting into some “Duke talent search” program, the day happened to have a conflict with his recital. We would rather want him to particitate the recital than that “stupid” test. (This actually gave him some small headache when he wanted to take some class at a local community college diring the summer after sophomore. Many other students who had taken the SAT could just take the course; he needed to take and pass some “academic screening” test required by the community college before he could take the test.)
After he had graduated from college, at one time, he mentioned to me that he could have majored in physics (even though he did not.) I consider his statement as a confirmation/compliment to my long time efforts of insisting that he learn such topics in the “old school” way.