My highschool education did a decent job of this (albeit not quite as strict on explicitly writing out each step), but, as you mention, I was homeschooled and my dad took the time to find textbooks that took this approach. EDIT: I also spent a year studying classical and symbolic logic, which was frustrating at the time but incredibly beneficial in retrospect.

On the other hand, my wife was homeschooled but with a different math curriculum (Saxon math, if you're curious) that heavily focused on memorizing the mechanics (i.e., "this is how you do long division", etc.) but wasn't good at promoting an overall understanding of the material. Instead, it shows math more as a "bag of tricks" and solving a problem means picking the right trick out of the bag and then applying it.

At the beginning of this school year I taught "Algebra in a week"--a review of algebra for students that didn't do well on their math placement exam. I'd sneak in stuff like proofs that (a^b) * (a^c) = a ^(b+c) and comments like "we can rearrange the order of these terms because of the commutative property". At the end, several of my students thanked me for doing this!

(Unfortunately, having TA'd for "Calc II for engineers", I can say that even at the university level not every math professor teaches this way. It's incredibly frustrating to watch students struggle to understand the material because they're being taught a process instead of why that process works. On the other hand, I estimate that most expect math class to be taught that way, and view it as a hurdle that they have to pass in order to get to the classes they want to take. Furthermore, teaching focused on the process does little to discourage the "I can just do this on a computer in the real world" mindset. I could go on for hours on this topic, since it's hardly specific to teaching math...)