a) an attempt in the mid-50's through the 70's to update the K-12 and undergraduate math curriculum by incorporating ideas from set theory, abstract algebra, topology and other topics from "modern" (that is, < 200 years ago) mathematics including modulo arithmetic, the binary system, linear algebra and the like....
b) an attempt to totally swindle innocent schoolchildren out of a grasp of ordinary arithmetic and undergraduates out of the good old fashioned calculus by clouding the issue with such superfluities as Venn diagrams, Möbius strips, alternate-base arithmetic, symbolic logic, signs like < and > and other trivia.
The reasoning was sound. Mathematics, as taught at anything but a graduate level had become a fossilized and sterile discipline, in which almost no discovery had been made in at least 200 years. Math books tended to use the same examples and problems from edition to edition, and some, the infamous A, B, and C problems, had been knocking around for at least a century. Math, and math teaching, therefore, tended to be a rather staid collection of unchanging rules, facts, and processes that were often given piecemeal with little explanation: innovation in math education was limited to such minutiae as audio vs. visual memorization methods for addition and multiplication tables (my private nemesis -- before I learned to put them on a grid, it made no sense), and the like. Even worse, very little of this curriculum was all that applicable to mathematics, as such: proofs, of the sort used in papers on the subject, were hardly seen until the last year or two of study. However, it was easy to teach, and easy to grade, and if anyone had problems with it, well...math is hard.
At the same time, mathematics was rapidly expanding, and concepts that were once considered hopelessly esoteric, such as linear and abstract algebra, topology, and computer programming (considered, for a time, to be a wholly mathematical exercise), were rapidly becoming important in industry, business administration, economics, and defense, as well as the sciences. Moreover, at least parts of many of these fields did not call for much in the way of skill in calculation, but were more akin to problems in logic, the humanities, and even ordinary common sense than the usual counting-to-the-calculus round.
With this in mind, it seemed logical to make at least the rudiments of higher math part of everyday parlance, to reveal a bit of the structure underlying arithmetic and algebra, and to thoroughly overhaul math teaching methods. Many textbooks were given color and more pictures and diagrams, age-appropriate word and story problems replaced the creaky legacy time/distance and steam-engineering conundrums, and various props, such as Cuisenaire rods and abacii came to the fore, as well as games like Wff'n'Proof and Tuf. The prime example, and still the gold standard of this kind of teaching is/was the freshman-level Introduction to Finite Mathematics, which moves seamlessly from compound expressions to game theory in one semester, punctuated by various speculations on such things as the intelligence of blondes and jocks, baseball, and Lewis Carroll.
When taught well, the New Math program was a resounding success -- I went from being inept in the times tables to multiplying matrices in a year's time, and would have gone much further if I'd stayed in similar programs. As it was, I went back to Old Math next year (taught by an uninspired fossil) and crashed and burned, consoling myself with rumors of the DoD of the day practically kidnapping able "young" minds...heheheheheheh....to Points Unknown out West. Hey it was the Seventies.
The truth was, most elementary- and high school teachers didn't want to have to learn a lot of weird concepts just to be able to get kids to deal with long division and trigonometry. Parents hated not being able to help kids with their homework, and although kids loved it, there was little follow-up on the really good parts. Consequently, after 1971 or so, many school programs again became "traditional with a gloss of New Math", retaining the new, glossy textbooks, some props and "enrichment activities", and token coverage of Venn diagrams. This has led to a dismal situation that has not been helped by time...