On the surface it seems like a good idea. Hell, if it were introduced to me in Elementary school I think I’d be better off. I had just learned about this topic recently and I am still intrigued about all the reasoning behind its introduction.
The New Math, like curriculum reforms proposed before it and many certainly since, was unpopular with parents who, because they had been raised with a different approach to thinking about math, found it unintuitive. It is not clear that the people who designed the curricula quite anticipated how much parental assistance played in formal education, but in any event, if the parents feel bewildered by their children's mathematics homework, this makes it pretty unpopular.
It became a punchline, and then a political punching bag: conservative groups used it as a sign of how out of touch educators and intellectuals were with the common man, not only arguing against the specifics of the "New Math" itself, but against the idea that anyone other than Ivory Tower mathematicians really needed to understand mathematics in anything more than a practical fashion, one gained through rote memorization. By the 1970s had included it as part of a "back to basics" call that was a rejection of liberal governance. You can contextualize this within the rightward drift and identity/grievance politics that were stoked during Nixon's rise to power, and the "New Math" became yet another battleground on which "culture wars" were fought.
It should also be note that it never really got the foothold in American curricula that either its boosters or detractors implied it did; many states and districts did not adopt it, or did not update their textbooks often enough that their students would have been exposed to it.
Lastly, along with all of the above, the National Science Foundation cut the funding for the group behind the New Math curriculum dramatically (by 50%) as part of general budgetary contractions caused by the financial pressures that flowed from the Vietnam War. So there ceased to be a formal organization promoting and managing the curriculum by 1972, and this was right when the "back to basics" campaigns heated up. Even Nixon got in on it in: "How much in Federal dollars went to further this innovation? Also, how many other, less well known, ‘innovations’ have actually hurt schoolchildren?" The (contested) appearance of declining test scores in mathematics generated a large amount of newspaper coverage played into a growing discontent.
All of this added up to New Mail "failing," in that the term became essentially negatively associated in the eyes of huge numbers of parents, associated with a progressive politics (which is somewhat ironic, given that it was part of a Cold War-based STEM reform meant to keep Americans competitive with the Soviets in a race for "scientific manpower"), and generally considered as a disaster. But even its failure was not complete: many of its approaches and concepts stayed on in mathematics curriculum and textbooks, but they dropped the "new" language that became so associated with it.
A colleague of mine who is a political psychologist frequently points out that people who don't have an opinion on something are pretty easy to convince one way or the other, and once they are convinced or have an opinion, they are almost impossible to "convert" to another viewpoint. One of the difficulties with math curriculum reform at the primary level is, again, that parents often are called upon to assist with it, and parents have a very strong sense of what the "right way" to do math is, since they had to learn a way (right or wrong) themselves. It is indeed bewildering to see alternative mathematical approaches to things we feel are intuitive because we had to learn them (see the endless memes and articles about the impossibility of understanding basic Common Core math today), even though the conventions we learned are no less arbitrary (and sometimes harmful in their own ways). If you combine this with a political environment that was rapidly honing in on such frustrations as markers of political identity, during an increasingly polarized, partisan period of American politics (one we still live in), it is not all that surprising that New Math failed so famously.
The above is derived largely from Christopher Phillips, The New Math: A Political History (University of Chicago Press, 2015), chapter 6 and epilogue. Phillips' book is the best (really only) history of New Math out there. For more on Cold War STEM reforms in general, see John L. Rudolph, Scientists in the Classroom: The Cold War Reconstruction of American Science (Palgrave, 2002). For more on the changing American political headways in the late 1960s and early 1970s, I am partial to Rick Perlstein's Nixonland.
To add a bit more to /u/restricteddata's answer, here's a bit of "New Math" history from an older question that references the Tom Lehrer song:
First to the big picture, the difference between American public and private school is more a matter of marketing and perception than one of pedagogy and practice. By the mid-60's when "New Math" happened, private school was coded as wealthier, whiter, and more child friendly than public schools, which were now filled with children of all genders, all classes, races, religions, and in many places, children with disabilities. By 1960, the seeds of the "common school" experience for all American children planted in the 1840s had come to bloom. At the same time, the dramatic increase in the size of the student population after World War II, known as the Baby Boom, meant many public school classrooms were packed and poorly resourced, especially in cities. This was especially true in New York City, where Lehrer grew up. And especially, especially true on the Upper East side where he lived.
Because of the crowded classrooms and growing sentiment of public education as non-exclusive, parents with the means and desire to leave crowded, less prestigious city schools, did so. Some of those parents, mostly white, moved their children to private schools that best matched their vision of what school should look like. Or their children attended the same private schools they attended as a young person. While some picked progressive, child-centered schools (Montessori was hitting its stride in the states around that time), many chose private schools with deep roots and traditions. Their private status and history meant they were likely to follow a classical liberal arts curriculum, which stressed Latin, Greek, sciences, the English canon, rhetoric, math, and history. Typically in such schools, math focused on traditional algorithms, often using pedagogical approaches that were more about the school's traditions than modern pedagogical practices. (More recitation and route memorization than games and manipulatives.) Private schools weren't required to hired licensed teachers and often hired content area experts. So, a private school was more likely to have mathematician teaching whereas public schools were more likely to have trained teachers with math content expertise.
The nature of private schools in the era was that, baring rules related to safety and hygiene, local and state governments were pretty hands off. This meant that when the news of Sputnik hit and it was all hands on deck in the public sector, private schools could keep doing what they were doing without much interference. My hunch is that's what he was referring to with that particular line. There simply wasn't the same level of pressure on public schools to adapt as there was on private schools. Likewise, private schools were often barred from receiving the federal funding for teacher training and curriculum writing that was awarded following Sputnik.
Now to "New Math." The phrase has come to refer to any widespread changes in math instruction - which is why the phrase on a textbook in The Incredibles 2 and the father's exasperated "math is math!" line gets a laugh by modern audiences. (We can talk about Common Core in 2029.) The work of reshaping math ed, though, started before the space race. Sputnik itself didn't change math education, rather it loosened the purse strings at the state and federal level. Following World War II, the National Counsel for Teachers of Mathematics put out a call for a shift how math was being taught and the call was echoed across the system. (To be clear, virtually every discipline was going through this process. Each generation of educators looks for ways to improve upon education but the changes to math were the most obvious and clearly observable in children's homework, which is why it's more likely to be noticed by parents and the media.)
Lehrer was an mathematician and as such, was likely part of, or at last privy to, conversations about changes to mathematical pedagogy post World War II. I found an utterly delightful article called "The Original New Math: Storytelling versus History" by David Roberts and Angela Walmsley (The Mathematics Teacher, October 2003) that does a deep dive into the song and stresses a few important points about "New Math":
Meanwhile, teachers were generally in support of changing math practices - teacher prep has always struggled to get it right and that was the case in the 1950s. Too much of the focus was on teaching, and not on making sure students were learning. From a teacher quoted in the article:
I am completely sold on SMSG. I would never want to go back to our former traditional program. SMSG is written in such an interesting manner that mathematics now is able to take its place as one of the more interesting subjects in the junior high curriculum.