Your exercise analogy is perfect. I think your concept of Math is in direct opposition to the author's (and mine).
The author is arguing that math is about logically solving problems based on a small set of simple rules. That students memorizing a formula are learning a formula but not learning Math. Like you said, Math is like exercise, where you challenge yourself with harder problems and over time you become a better problem solver. The shortcoming with American public schools, or whatever we're talking about here, is that the teacher's incentive is to get their student to achieve a competency in repeating memorized instructions, something that can and is tested. This leads to numerous students who don't enjoy or understand Math for what it really is, and fail to develop the part of their brain that solves problems logically. Instead, they learn a bunch of formulas that are totally pointless unless they happen to need that exact formula in the future. So, evaluating teachers based on the performance of their students at applying different formulas is part of the problem, not the solution.
I'm not sure we disagree too much. My view is that you need to teach strong fundamentals so you don't need to think (ie, worry) about them later. This reduces degrees of freedom, and frees up your mind. The reason you do this is precisely that it lets you focus on higher level abstractions. It might just be me (probably is) but I found that math got more intuitive this way. Also, if your aim is to make the basic building blocks both reliable and intuitive, it will get used more in daily life. If you need to reference a text, it will ~never happen. [1,2]
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[1] What are <legos>, but simple geometric objects?
[2] Life is an open book test, there is just not time or the bandwidth. And I'm not talking about memorizing a calculus text, etc just for fun. But more basic - the bulding blocks. But if you want to play for high stakes later on this is the Ante, IMHO. That probaly wasn't clear in the initial comment I made (as several people have pointed out logic~abstraction, etc). So I'm not saying this should be the limit or ceiling of an education. Its just the opposite. But its shocking the kinds of things that happen otherwise. And its not fair to the kids who never get a shot at doing serious work, because their fundamentals are unsound. "Subtle bigotry of low expectations" and all that.
The author is arguing that math is about logically solving problems based on a small set of simple rules. That students memorizing a formula are learning a formula but not learning Math. Like you said, Math is like exercise, where you challenge yourself with harder problems and over time you become a better problem solver. The shortcoming with American public schools, or whatever we're talking about here, is that the teacher's incentive is to get their student to achieve a competency in repeating memorized instructions, something that can and is tested. This leads to numerous students who don't enjoy or understand Math for what it really is, and fail to develop the part of their brain that solves problems logically. Instead, they learn a bunch of formulas that are totally pointless unless they happen to need that exact formula in the future. So, evaluating teachers based on the performance of their students at applying different formulas is part of the problem, not the solution.