The only problem with this dualism is that it really isn't so.
The way mathematics is developed is the same way it should be learned and applied. Your geometry class today does not start with Euclid's axioms, but instead by developing intuition for concepts in the simplest way possible (2D plane, 1D lines), playing around with them, and then formalizing them (which means defining them rigorously to the very last detail, to ensure eg. your concept of a straight line is always a straight line), and generalizing what you learned about them (stating it as theorems). The same is with numbers: I've yet to hear of people starting their arithmetic training with Peano's axioms, yet it's common to teach numbers by pointing at one object and moving another one next to it, and claiming that's now a "two"; adding another one to that gives you a "three" (successor() function, right?).
Sure, most primary education skips this playing around with concepts, and fails to demonstrate failings of not being precise enough, but they are all trivial to demonstrate.
But all of those are what is exactly used in formal/pure math development, and exactly the same processes are used when applying mathematical insights anywhere else. There is no difference at all!
The level of rote memorization and combinatorial explosion rises as you go into more advanced math with more abstract concepts to deal with (which is what I think people really struggle with, since it's a different language, and then intelligence does come into play as well as you start dealing with bigger complexity), but I like to say that math is the simplest form of expression of the human brain, but no simpler. It has well-defined boundaries of our comprehension too (we need to assume some things are true without proving them, aka axioms and base terms).
The biggest problem of all math instruction today is that it's mostly performed by those not understanding the simplicity of it, and thus the only thing they can do is transfer their misunderstanding on to their students. This includes parents who teach their kids basic concepts like basic geometric shapes or numbers.
Another big problem is that people simply fail to realize how much of rote memorization is required for understanding mathematics. Being a different language, you have to learn all the "new" words and grammar to be able to effectively apply it, and people usually discount that as not important, and how only understanding mathematics is hard because of the abstract nature.
The way mathematics is developed is the same way it should be learned and applied. Your geometry class today does not start with Euclid's axioms, but instead by developing intuition for concepts in the simplest way possible (2D plane, 1D lines), playing around with them, and then formalizing them (which means defining them rigorously to the very last detail, to ensure eg. your concept of a straight line is always a straight line), and generalizing what you learned about them (stating it as theorems). The same is with numbers: I've yet to hear of people starting their arithmetic training with Peano's axioms, yet it's common to teach numbers by pointing at one object and moving another one next to it, and claiming that's now a "two"; adding another one to that gives you a "three" (successor() function, right?).
Sure, most primary education skips this playing around with concepts, and fails to demonstrate failings of not being precise enough, but they are all trivial to demonstrate.
But all of those are what is exactly used in formal/pure math development, and exactly the same processes are used when applying mathematical insights anywhere else. There is no difference at all!
The level of rote memorization and combinatorial explosion rises as you go into more advanced math with more abstract concepts to deal with (which is what I think people really struggle with, since it's a different language, and then intelligence does come into play as well as you start dealing with bigger complexity), but I like to say that math is the simplest form of expression of the human brain, but no simpler. It has well-defined boundaries of our comprehension too (we need to assume some things are true without proving them, aka axioms and base terms).
The biggest problem of all math instruction today is that it's mostly performed by those not understanding the simplicity of it, and thus the only thing they can do is transfer their misunderstanding on to their students. This includes parents who teach their kids basic concepts like basic geometric shapes or numbers.
Another big problem is that people simply fail to realize how much of rote memorization is required for understanding mathematics. Being a different language, you have to learn all the "new" words and grammar to be able to effectively apply it, and people usually discount that as not important, and how only understanding mathematics is hard because of the abstract nature.