Stillwell is wrong, or at least not doing his homework, when he says "Everyone concedes that [ideas of number and space] are fundamental, but they have been scandalously neglected..." Back in the early sixties, when Bourbakism was hip and new (as mathematical ideas go at least), everyone agreed that set theory and mathematical logic were fundamental, whence the debactacular attempt to begin teaching first-order logic to seventh graders.
The uncomfortable truth is that math is like a language in that it requires a student to put together a large and messy collection of terms and concepts to express any kind of thought at all. The fact that all of the thousands of entries in this vocabulary can be derived in principle from maybe a dozen axioms is what gives math its power, but nobody learns it that way, proving this or that fact as needed. Real skill comes from practice, which starts out as imitation and then generalizes into problem-solving and proof.
All the emphasis on The Fundamentals of math is really just evidence that most mathematicians are bad at teaching -- it's an attempt to impose mathematical order and rigor on the fundamentally messy and organic process of human learning.
I know John Stillwell, and I think his opinions and beliefs are being mis-quoted by being taken out of context. I think he is right that ideas of number and space are fundamental, and I think he's right that they are being neglected, which is a scandal. I agree that "New Math" was a complete debacle, but I think it's not relevant to this question.
I agree that math is like a language, but I disagree that it requires the student to put together a large and messy collection of terms and concepts. That may be the way you view it, but it's not the way I view it, it's not the way I teach it, and I get pretty good results in masterclasses, and more generic classes, by helping the development of the web of inter-related ideas, without large numbers of apparently unmotivated definitions.
In short, I think Stillwell is right, and I think you're missing his points because you appear not to have read his original works.
Stillwell is wrong, or at least not doing his homework,
Have you read his book Numbers and Geometry, or any of Stillwell's other books?
I actually think he takes the approach you advocate, but you haven't given me quite enough examples of the approach you think best for me to be entirely sure.
http://en.wikipedia.org/wiki/New_Math
Stillwell is wrong, or at least not doing his homework, when he says "Everyone concedes that [ideas of number and space] are fundamental, but they have been scandalously neglected..." Back in the early sixties, when Bourbakism was hip and new (as mathematical ideas go at least), everyone agreed that set theory and mathematical logic were fundamental, whence the debactacular attempt to begin teaching first-order logic to seventh graders.
The uncomfortable truth is that math is like a language in that it requires a student to put together a large and messy collection of terms and concepts to express any kind of thought at all. The fact that all of the thousands of entries in this vocabulary can be derived in principle from maybe a dozen axioms is what gives math its power, but nobody learns it that way, proving this or that fact as needed. Real skill comes from practice, which starts out as imitation and then generalizes into problem-solving and proof.
All the emphasis on The Fundamentals of math is really just evidence that most mathematicians are bad at teaching -- it's an attempt to impose mathematical order and rigor on the fundamentally messy and organic process of human learning.