I don't know if this will be helpful or not (bear in mind that musicians only need to be able to count to four!, so I am probably very far from having much to offer on the math front), but a book we've been reading lately sprang to mind as I read through this thread: Moses Richardson's "Fundamentals of Mathematics" (Macmillan, 1941; OOP, but there are editions up through 1973, so lots of copies around). It's a textbook Richardson developed to teach the one-year compulsory math course for humanities majors at his university; while he presupposes previous exposure to elementary algebra and geometry, "almost no accurate recollection of the details of these subjects is prerequisite for this book" (vi). He thinks mathematics has a great deal to offer students who have little need for technical skill in the field, and his objectives here are to give the student:
"(1) An appreciation of the natural origin and evolutionary growth of the basic mathematical ideas from antiquity to the present;
(2) A critical logical attitude, and a wholesome respect for correct reasoning, precise definitions, and clear grasp of underlying assumptions;
(3) An understanding of the role of mathematics as one of the major branches of human endeavor, and its relations with other branches of the accumulated wisdom of the human race;
(4) A discussion of some of the simpler important problems of pure mathematics and its applications, including some which often come to the attention of the educated layman and cause him needless confusion;
(5) An understanding of the nature and practical importance of postulational thinking" (v-vi).
In service of these objectives, he examines several topics (number theory, algebra, logarithms, impossibilities, analytic geometry, functions, limits and the calculus, trigonometric functions, probability and statistics, finite and transfinite cardinal numbers, euclidean and non-euclidean geometry), using the sense of mathematics "as the totality of logical (hypothetico-deductive) systems and their applications" as the organising principle/unifying theme throughout. He points out that while some of the book looks "hard," "the early fundamental paragraphs of a so-called 'advanced' subject, presented at the proper level are often easier to grasp as well as more important and more interesting than the later technically complicated paragraphs of what has traditionally passed for an 'elementary' subject" (ix, vi).
Now obviously, our collective kids are not first-year liberal arts majors, but it seemed to me on reading this thread that this kind of approach might for some people be both very useful and rather intriguing in regard to our kids and their unusual needs. Like many others here, we are bumping up against algebra early, and this struck me as one possible means of slowing things down, as well as laying a pretty thorough grounding in a real understanding of mathematics; especially for homeschoolers operating outside of the system, anyway, I thought I'd toss this out there as an idea. I worry a lot about math, since Harpo seems pretty good at it, and (standard academic dodge) it's not really my area.
I could be completely out to lunch about this book, though! Anyway, so far, both Harpo and I are enjoying this book a lot (I've only had it since Canadian Thanksgiving, though, so we're only a few chapters in at this point).
Hope that helps somebody (or at least wasn't completely irrelevant)--
peace
minnie
