The way I often say it is that most of people's mathematical education - even most of the math that people take in college, up to about junior year - is mechanical and recipe driven. It's usually about "in this situation, do this to solve the problem." You don't need much actual insight into what's going on, you just need to be able to recognize the patterns and know when to apply what. This makes sense, since the vast majority of people who take even differential equations are not people who need that kind of deep knowledge - they're often engineers or scientists who need to know how to solve a particular kind of problem when they encounter it.

In undergrad, I thought I was good at math. Then I got to grad school and took grad level courses in algorithms and operations research. That was when I realized that I had learned - necessarily - mostly recipes, not fundamental mathematics.

Memorizing the periodic table does not make you a chemist. It won't even suffice to pass chemistry 101. High school math amounts to barely more than memorizing the periodic table.

I took 10 semesters of 'math' in college, was on a 'math' team in high school, and got a 780 on the 'math' SATs. Yet by the OP's definition, I don't know any Math because I'm not doing his definition of Math.

That's like arguing that you can't swim unless you are an Olympic swimmer. I can't do the back stroke like Michael Phelps, but I sure can swim.

High school math - at least for most - includes Calculus 1. Yes, that involves memorizing integrals and using them later. But if you are going to be that flippant about it, basically what you can marginalize everything. Organic chemistry? Meh, that's just a bunch of memorizing compounds, it is not more difficult than 1st grade memorization of multiplication tables. Anatomy? Ha, that's child's play, it's just like memorizing the names of colors.

I think the poster is arguing against the conflation. Organic chemistry is not memorizing compounds; mathematics is not memorizing integrals; anatomy is not like memorizing colors. There is a great difference between understanding and memorizing.

I had a very bad history teacher. On most tests, he required the class to learn about 20 different dates (only) and some facts like where a peace treaty was signed. Nobody cared what those events meant, what were connections between the dates etc. Now, many years later, I don't recall anything. I learned nothing. Was I really learning history or just arbitrary numbers? You could claim that it was real history, and probably be technically correct, but I think that's silly. In my mind, the semantics of what I knew did not resemble history at all. In contrast, my chemistry teacher visibly required understanding, and till now I can discuss basic chemistry, even though I did not exercise my skills for a long time.

It's the same with mathematics, learning algorithms means essentially you are programmed to do mechanical work, just I like I was acting like a lookup table during history classes. "Real" mathematics starts when the problem you are doing is not directly amenable to a mechanical approach.

Exactly. In the case of mathematics, if you aren't doing proofs yet, you don't even know what math is. Most high school math curricula are completely devoid of proofs or abstract reasoning with the exception of at most a few months during geometry. How many college students have wrongly expected "Modern Algebra" to be a rehash of what they learned in high school? The distinction between mathematics and what gets taught in high school is very similar to the distinction between computer science and computer programming.

Your analogy doesn't quite work because the difference between a high school swimmer and an Olympic swimmer is a difference of degree. That its, they both do the same thing, one is just (much) better.

But the difference between what people do when they take most math courses up to about their third year in college, and what mathematicians do, is a difference of kind, not degree. You can see my other comments in this thread to see what I think that difference in kind actually is.

Okay. so your statement here, in a very convoluted way, says that high school maths alone does not make you a mathemtician, and implies that I am saying the opposite.

Of course high school maths alone will not make you a mathematician, but its still maths, and its silly to say otherwise! This is my only point here.

The problem is that the same word has to pull double duty. Most math courses even in college teach you how to use math. They do not teach you how to do math. Science is similar. During an undergraduate education in physics, you don't really do physics. You learn about physics. You don't actually do physics until you get to grad school and start doing research.

Math is a lot easier to do independently than Physics though. Maybe it is different for pure mathematics, but applied mathematics can be used independently and in class without going to grad school. It is like someone going to school for programming and learning different programming languages, but never really doing to much with it. It would be up to that person to do some independent projects for themselves and the same with applied mathematics.

I don't even think a degree in physics or pure mathematics is even worth much by themselves without grad school.

And your point is right because I would not have been able to do Calc I if I did not know Trig and Algebra. The same way that I would not have been able to do Algebra without without basic math. I hate memorization and prefer immersion, but it was important to remember how to do that stuff because it was needed later on.

I love the more logical side of math that does not even deal with that many numbers (which is why I am majoring in), but the stuff in school is definitely maths. The way it is taught is questionable, but it should not be dismissed.