I was just thinking about this the other day. Personally, I think that math falls into two categories, though I think I would distinguish them differently from you (If I'm understanding you correctly). Its kind of like the difference b/t the hammer maker and the carpenter, the producer and the consumer. For me, mathematics (the kind you research and which is abstract and theoretical) is largely in the hammer maker camp. We'll call this math X, these guys are creating and polishing tools (aka in analysis providing proofs and arguments for why the real numbers can be considered complete or that a derivative actually can be taken on a given class of functions).
Then there is Math "Y". This is all the guys who use those things the X guys are selling, the proverbial hammers they have produced. They assume the X guys did their work correctly and that when they use the products they've bought i.e. the rules, theorems and strategies developed by the X guys, to solve a particular equation or problem, the answer is correct. For example, they assume the limit of the sum of two polynomial functions on the reals is equivalent to the sum of the limits of those functions - they don't care about all the nitty gritty details and justifications - the X guys figured all that out for them. They Y guys are much more concerned with figuring out how to get the rocket into space or ensure the skyscraper is soundly built.
I would say from my experience, very little of mathematics education is in the X camp, I'm not saying this is a bad thing though, perhaps it is similar to the fact that most programmers are not compiler programmers or programming language creators :)
Then there is Math "Y". This is all the guys who use those things the X guys are selling, the proverbial hammers they have produced. They assume the X guys did their work correctly and that when they use the products they've bought i.e. the rules, theorems and strategies developed by the X guys, to solve a particular equation or problem, the answer is correct. For example, they assume the limit of the sum of two polynomial functions on the reals is equivalent to the sum of the limits of those functions - they don't care about all the nitty gritty details and justifications - the X guys figured all that out for them. They Y guys are much more concerned with figuring out how to get the rocket into space or ensure the skyscraper is soundly built.
I would say from my experience, very little of mathematics education is in the X camp, I'm not saying this is a bad thing though, perhaps it is similar to the fact that most programmers are not compiler programmers or programming language creators :)