I think I've mentioned this before in my earlier blog entries about the need for narrative-based or pedagogical methods that follow roughly the historical developments of subjects of study. This is especially true for mathematics curriculum whereby problems and puzzles (and their possible solutions) are examined not as prescribed-solutions-perfected-from-on-high but rather as story-telling demonstrations of how people have struggled mentally and intellectually in formulating questions/issues at hand and in seeking their solutions.
Reading a book on Newton, for eg, makes one realize that often geometrical insights help arithmetic understanding and vice versa. James Gleick's book, Isaac Newton (2003), talks about how Newton came to realize that the quantity of time is not just the passage of time, per se, as most of us "understand" it to be, but may be captured as a length in a line: time equals distance; or that "colour" may be said to be a degree of refraction - ie, that we may describe spectrometry of light (the different colours) by "their different refrangibility to sort them out." (Gleick, p.82)
Gleick goes on to say that
Newton's letter was itself an experiment, his first communication of scientific results in a form intended for publication... ...He had no template for such communication, so he invented one: an autobiographical narrative, step by step, actions wedded to a sequence of reasoning. (p. 82)
The passive approach to rote teaching and learning is really mind-numbing and literally kills curiosity; a historical narrative based pedagogy not only inspires one to dig further (mostly on one's own initiative) but informs one of multifarious aspects of what education asks of us (teacher and student alike) - to discuss and interact with each other to draw out insight and context of the problem naturally (ie, as opposed to dogmatic recitation of supernatural "facts"). The cultivation of humanist scientific mindset offers infintie possibilities without having to resort to apparently divinely-inspired laws perfected without human participation.
One other insight that I've come to realize is what ties elementary school maths to higher level maths are the four fundamental operations of arthmetic (ie, one need only addition, subtraction, division, multiplication to negotiate one's way thru this wonderous field of human activity); the differences and degrees of difficulty come about as a result that in lower maths one only concerns oneself with counting numbers, integers and rationals (for simplicity: both positive and negative numbers, say), while in higher maths the elements of arithmetic have come to include the broader irrationals, algebraic, transcendentals, and imaginary numbers for solutions. Nothing of the grammar of arithmetic has changed, only the elemental terms that may be included in the language have been broadened.
But all of this comes about naturally as one explores the historical developments of mathematics. A number problem is realized, seek its solution. The story line is simple but it is a human story nonetheless where the developments in human thought are humanized - ie, opened up to not only English-speaking peoples but to all. Folk etymology of the rational vs irrational numbers in English may be by-passed completely; same for trying to fit latin- and greek- terms to the English grammar from which much math phobia arises; the pitfalls of word-games and deeper philosophical issues are much better clarified/formulated when one is not forced to rely solely on just one or similar language structures to generate insights but open to a broader and richer linguistic base to draw upon or from.
To wit: Indian maths created the negative numbers and zero, for eg. more than a thousand years ago (practically if not actually) while Europe struggled dogmatically/theologically with them right up to the modern age - in most certain, definitive terms as only hubris affords.
Always start from first principles, let them carry you where they may. Even the venerable axioms of Euclid are yet productive, still able to generate new insights never before thought of or realized though two thousand years old they are. And yet forever young. Even the realization that something is "missing" in the logical structure opens up new vistas. Linguistics and maths - the new bones, skins and stone (as Balikci said of the Natsilingmiut).