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During the 60s, people in the US (and also in Europe), school curricula introduces New Math where students began with set theory in the first grade before learning to perform addition or multiplication.

Somehow, this has ended. Can you explain why?

(Of course, in the linked article there is some criticism, but almost everything the government does is criticized and politics don't really care. Why did they care here?)

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    $\begingroup$ One relevant historical artifact that may shed some light on the prevailing attitudes at the time is Tom Lehrer's "New Math" song. It's also quite hilarious! :-) $\endgroup$ – Jim Belk Mar 17 '14 at 19:15
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    $\begingroup$ I think it would be good to have actual published accounts of this. (In addition to the opinions and anecdotes.) $\endgroup$ – Gerald Edgar Mar 18 '14 at 0:50
  • $\begingroup$ I ran into this newly published book [1] in my local library; I can't vouch for it, but it might be of interest (at the very least the bibliographic information should lead to further resources). [1] amazon.com/New-Math-Political-History/dp/022642149X/… $\endgroup$ – GLG Dec 30 '17 at 5:45
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In the first place, the impetus to "reform" math education was motivated by politics, not by any serious observed deficit. By coincidence, there was a "new" style in higher mathematics, reflecting the previous 50-60 years assimilation of set theory and rewriting of many things in terms of set theory. But until the "sputnik scare" no one had incentive to pretend to incorporate set theory or not. The people who "decided" to promote that "New Math" were mostly not mathematicians of any sort whatseover, but, rather, semi-politicized people who needed to be able to "show that they were doing something". After all, set theory was not new, was not what had made any difference in WWII or subsequently, nor was it what made the Moscow school of mathematics what it was.

Many parents objected not on scientific grounds, although it was veiled as such, but because the "new math" was alien to them, and kids who were learning whatever "new math" purported to be were not learning "traditional math". Even though "traditional math" included (I was there...) an enormous amount of repetitive drill, arguably to the point of senselessness, it was orthodox and familiar. Parents could no longer help their kids with their homework...

More substantive than parents' discomfort (though that might have been the dominant political determiner) essentially nothing had changed at the college and university level (or even high school), and kids that knew something about set theory couldn't do the basic algorithmic math for basic chemistry, physics, nor the standard high school math curriculum (whatever the flaws of the latter). Certainly "the new math" did not warm people up to the traditional trigonometry, calculus, etc.

A more insidious problem was that few of the elementary-ed teachers (nor middle-school, nor high-school) had prior preparation in such stuff, ...

In fact, of course, if there were room in the curriculum for it, and if kids were ready for it developmentally, both some sort of "meaning" and "algorithms" could be taught.

But there's neither room in the curriculum, nor (in my observation) are kids ready for more conceptual things at that point. Perhaps it is harder to teach concepts than algorithms, also.

I was in school then, and my parents both taught high-school math, so I heard a lot about this. It is important to note that, for better or for worse, "school boards" (whose qualifications are mostly political) decide textbooks. The "new" books my parents brought home were more interesting to me than doing yet-more elementary arithmetic, which I could already do, but it was also clear that most of the other fifth-graders wouldn't have been in the right state for such stuff, since they were still having trouble "following instructions" about very concrete activities. (I do not have any citations on "development"...)

But, so far as I recall, it's not that there was scientific-grounds objection to set theory itself, but that it displaced indispensable things, ... and was new and scary to non-mathematicians. Also, some of the semi-politicized proponents were blitheringly incompetent math-wise, which made them amusing targets for actual scientists... even though that was not the same thing as a criticism of set theory, etc.

So, with just a nudge from practical issues, the opportunity for political action and argument both created and killed off "new math". (More recently, an attempt to modernize calculus and related material has re-generated "math wars", again with similar extra-mathematical and extra-educational factors dominating the action, ...)

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  • $\begingroup$ I understand this Q&A is an old one. But, I just stumbled on it today. Please forgive me and allow me to ask a question here. When did the New Math end in the US and who called it an end? I am very curious because I received New Math Education in high school in 1967-1969 in Taiwan. As far as I know, the New Math was called an end in Taiwan in 1974(5?) because we were told Americans stopped it and we followed suit. I was always wondering what happened. Thanks in advance for any info. $\endgroup$ – scaaahu Apr 25 '18 at 13:07
  • $\begingroup$ I don't really know the details of the school boards' discussions, but from outside it looked like the whole idea gradually became an object of ridicule. The proponents of it ceased supporting it, and, thus, the initial move in that direction stopped and reversed. $\endgroup$ – paul garrett Apr 25 '18 at 16:09
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You might want to read Kline (1973). I haven't read the book, but according to Wikipedia,

In 1973, Morris Kline published his critical book Why Johnny Can't Add: the Failure of the New Math. It explains the desire to be relevant with mathematics representing something more modern than traditional topics. He says certain advocates of the new topics "ignored completely the fact that mathematics is a cumulative development and that it is practically impossible to learn the newer creations if one does not know the older ones" (p. 17). Furthermore, noting the trend to abstraction in New Math, Kline says "abstraction is not the first stage but the last stage in a mathematical development" (p. 98).

You might also want to read Klein (2003).

A substantial number of mathematicians had already expressed serious reservations relatively early in the New Math period. In 1962, a letter entitled On The Mathematics Curriculum Of The High School, signed by 64 prominent mathematicians, was published in the American Mathematical Monthly and The Mathematics Teacher. The letter criticized New Math and offered some general guidelines and principles for future curricula.

By the early 1970s New Math was dead. The National Science Foundation discontinued funding programs of this type, and there was a call to go "back to the basics" in mathematics as well as in other subjects. [...]

References

Klein, David (2003). "A Brief History of American K-12 Mathematics Education in the 20th Century." Mathematical Cognition. Information Age Publishing.

Kline, Morris (1973). Why Johnny Can't Add: The Failure of the New Math. New York: St. Martin's Press.

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    $\begingroup$ A link to the 1962 letter is here. $\endgroup$ – Joel Reyes Noche Mar 18 '14 at 1:43
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    $\begingroup$ The JSTOR link is just a teaser. The complete text can be found here: rationalsys.com/mk_johnny.html (Scroll down to "ON THE MATHEMATICS CURRICULM OF THE HIGH SCHOOL") $\endgroup$ – Ben Crowell Jun 3 '14 at 17:49
  • $\begingroup$ Perhaps related: "On teaching mathematics" by Arnold. $\endgroup$ – Tom Copeland Dec 17 '17 at 18:18
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I think very roughly speaking: "new math" (also known in europe as a fearful period of time, especially for parents) followed a mathematical construction of mathematical knowledge rather than a psychological one.

Mathematically, you would introduce an abstract concept like an equivalence relation first and then introduce concept like terms or fractions as applications. Psychologically, you would go the other way. Obviously, many people can handle fractions without even knowing the word equivalence relation.

So, if a way seems mathematically straight, that doesn't necessarily mean it is psychologically the easiest way or shortest way in time. You may start with set theory to learn the natural numbers; however, simply counting 1, 2, 3,... seems more promising.

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Please refer to Whatever became of the New Math?, a series written by Professor Raimi of the Department of Mathematics of the University of Rochester, and also Whatever Happened To New Math?. Together with PSSC in physics education, SMSG (most popular of the New Math) actually was initiated from professional mathematicians, started to produce high school textbooks (not the elementary school).

I found many articles online (or even a book) details the New Math, but there are not much postmodem on PSSC.

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    $\begingroup$ Links rot (break and lead to nowhere/error) over time. Why not offer a bit of a summary of the series you reference here? $\endgroup$ – JTP - Apologise to Monica Dec 23 '17 at 21:28
  • $\begingroup$ Some excerpts from Whatever Happened To New Math?: The differences in experience and outcomes were as varied as the nation’s geography and, most important, only as good as the teachers. But what new math became is not what it was intended to be. In fact, there never was just one new math. $\endgroup$ – Ming Lei Dec 28 '17 at 1:01
  • $\begingroup$ Popular sentiment was beginning to shift, and with it congressional enthusiasm for the financing of educational reform. Good teaching and good textbooks could not be separated from bad. And for every bright student with a thirst for math, there was one who had trouble figuring the charges on his paper route. New math got no credit for the enthusiasm and all the blame for the ignorance, even in those school districts where it was never seriously adopted. $\endgroup$ – Ming Lei Dec 28 '17 at 1:08
  • $\begingroup$ Where it had been successful, it lingered on in the teaching techniques of individual instructors and in watered-down new-math textbooks, which are still evident in elementary and high schools today. Because it put such stock in creativity and the abstract, new math appealed most to the brighter, college-bound students, who, some have argued, probably would have done well anyway. Average and marginal students could suffer dearly at the hands of an uninspired teacher and a poor textbook, and frequently did. But that spirit-crushing combination could exist independently of new math too. $\endgroup$ – Ming Lei Dec 28 '17 at 1:09
  • $\begingroup$ There is a excellent speech by French mathematician Laurent Lafforgue Why the school. In his speech, he talked about the purpose of schooling and what should be taught in school in general. he stressed the school should teach only knowledge, not thinking, for thinking cannot be taught. The teaching must nourish. Does it mean we still need to teach Euclid's plane geometry? $\endgroup$ – Ming Lei Dec 28 '17 at 1:29
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There is an article by Phillips:

Phillips, C. "In accordance with a `more majestic order': the new math and the nature of mathematics at midcentury." Isis 105 (2014), no. 3, 540--563

that presents a thoughtful analysis of this curious episode from the history of math education. The way I understand his argument, the sequence of events was as follows.

  1. Feeding the dissatisfaction with what was felt to be an inadequate situation with math education was the fever characterizing the Sputnik era mentality, when it was felt (at least in the US) that we must catch up with the Russians at any cost.

  2. Certain philosophical assumptions that had originally developed in the community of (mainly pure) mathematicians trickled into the thinking of education decision makers who looked up to the professors.

  3. A significant role here was played by a Piaget--Dieudonne interaction and equivocation on mathematical structures where child psychology was mixed up with mathematical foundations.

  4. A set of measures was proposed that may have been somewhat effective at the highschool level, but at the elementary and middle school level was doomed from the start.

  5. Vast federal moneys were fed into the effort led by the National Science Foundation which was ill-equipped to decide on issues of elementary education.

  6. The proposal was implemented rapidly without proper testing via pilot programs.

  7. By the time people started realizing something was terribly wrong, vast numbers of schools had already been affected, as well an entire generation of schoolers.

Phillips also published a follow-up book

Phillips, Christopher J. The new math. A political history. University of Chicago Press, Chicago, IL, 2015.

I personally found the book a big disappointment. In his 2015 book, he toned down many of the arguments in the 2014 paper and even reversed himself on some issues. One gets the impression that in order to get the book published he had to toe a politically correct line so as to make sure not to be perceived as pouring oil on the conservatives' agenda. In short, I recommend the article but not the book.

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  • $\begingroup$ Do you have the link of the paper? Also, By the time people started realizing something was terribly wrong, did he say in the paper what went wrong? Thanks for paying attention to my inquiry. My comment here can explain why I am so curious. $\endgroup$ – scaaahu May 14 '18 at 7:32
  • $\begingroup$ Notice that the program was never obligatory. There was a set of guidelines and syllabuses and each school in principle decided on its own whether to follow it. On the other hand, Phillips documents that there was a lot of pressure on the publishers to produce books with "new math" or "modern math" in their titles, and corresponding pressure on schools. It never "officially" ended; it doesn't work that way in the US. People just noticed that the teachers weren't prepared, the students weren't learning, and attributed it to the new math. I imagine given enough preparation and teacher... $\endgroup$ – Mikhail Katz May 14 '18 at 8:48
  • $\begingroup$ ... training, this might have actually worked in Taiwan. In Russia I think it worked to a certain extent based on Kolmogorov's prestige. @scaaahu $\endgroup$ – Mikhail Katz May 14 '18 at 8:49
  • $\begingroup$ Yes, it worked in Taiwan to some extent. I myself was kind of benefitted by SMSG. It opened the door of math for me. I have been always wondering why it stopped in the US. I went to the US about ten years after I had the rigorous training in high school and witnesses Americans colleagues in graduate school struggled with elementary set theory stuff, I always wondered what happened their high school education. Thanks for your answer and info. $\endgroup$ – scaaahu May 14 '18 at 9:08
  • $\begingroup$ @scaaahu: I suspect my comment here (especially my suggestion "... you can google their titles ...") will be of interest to you regarding your question. $\endgroup$ – Dave L Renfro May 14 '18 at 10:35
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I would characterize the "New Math" as a "top down" approach to math.

The idea was to have young students see the "big picture." The danger is that they fail to "see the trees for the forest" (the reverse of the usual malady).

I was a grade/junior high school student during the "new math" debate of the late 1960s.

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    $\begingroup$ Why would you say that the "New Math" ended, in answer to the OP's question? As your answer currently stands, it does not appear to address this. $\endgroup$ – J W Jun 15 '14 at 6:42
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    $\begingroup$ One example was "carrying," in subtraction discussed in the Tom Lehrer clip. Students in the 1950s and early 1960s were taught to do it "mechanically," from right to left. As a grade school student in the 1960s, I was exposed to the "new math" (left to right) which led to greater understanding, but my teachers warned about a greater chance of error. $\endgroup$ – Tom Au Jun 15 '14 at 19:07
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They failed to deliver the punchline: the uniqueness of the reals, as a [complete] ordered field.

Once numbers were under­stood as forming an ordered field, and the positive integers among them as a certain inductive subset, and once the language of sets became standard, so that statements with quantifiers made sense, then and only then would students see... (http://web.math.rochester.edu/people/faculty/rarm/igno.html)

Most of the alterations to the curriculum narrated by Kline can be seen from this perspective. One introduces congruences to show examples of other fields that fail to meet some of the axioms of the reals; one introduces basis different of ten in order to show other ways to express the reals that actually are the same, isomorphic if you wish, field (Note particularly how binary notation was introduced but not related to set theory operations, contrary to its use in modern computer applications). In some presentations even equations were subordinated to the reinforcement of the idea of expanding, or completing, a numerical set.

Discussions such as Is it advisable to avoid teaching "multiplication as repeated addition"? , asking if multiplication and addition should be considered independent, are a remmant of this perception; what is discussed at the end is how fast can students be presented to the concept of field, and then how to argue that the combination of multiplication and sum in the reals is the most natural or unique.

Surely New Math had other intermediate goals, but the real line was its final act of fireworks. Very much as the final fireworks bouquet of Euclid "classical curriculum" is the existence of five and only five regular solids.

EDIT: The unpublished Mathématique moderne 4 of Papy, with notes on the real field, is avalaible online at http://www.rkennes.be/Articles%20de%20Papy/ListesArticles.htm (for free, but lot of popups)

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  • $\begingroup$ Your use of Raimi's quote seems misplaced --- he was talking about the natural numbers and he was using this as an example of instilling logical deduction in mathematics teaching. Also, the uniqueness of the reals is a very narrow and limited topic, probably only suitable for honors level in the 12th school year, if at all. See the references in Where can I find primary sources from the New Math movement in the 60s?. $\endgroup$ – Dave L Renfro Apr 25 '18 at 7:31
  • $\begingroup$ @DaveLRenfro Perhaps I suffer some confirmation bias, as I found Raimi's quote after writing the answer. Still I think it supports that they were working towards this goal. That the topic was unsuitable for K-8 is probably the reason why they failed to deliver it. $\endgroup$ – arivero Apr 25 '18 at 12:19
  • $\begingroup$ Disclaimer: I studied under the European Spanish version of New Math, perhaps not exactly the same flavour. $\endgroup$ – arivero Apr 25 '18 at 12:23
  • $\begingroup$ I second @DaveLRenfro's comment: the advanced mathematical properties of the real field are largely irrelevant to the success or failure of the new math movement. The field in question is not constructed in high school and not even in freshman calculus in the vast majority of cases. Besides, its uniqueness is a bit of a mirage because it hides specific commitments in set-theoretic foundations. These foundations were specifically developed in the 1920s to ensure the uniqueness of the reals, so there's a bit of a back-scratching exercise here. Moreover, the real line in Edward Nelson's... $\endgroup$ – Mikhail Katz May 14 '18 at 11:25
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    $\begingroup$ Actually this construction is usually attributed to Cantor (even though he did not have the notion of an equivalence relation), not Cauchy. Cauchy had the notion of a "Cauchy sequence" but wasn't worried about constructing the reals: to him unending decimals were enough :-) $\endgroup$ – Mikhail Katz May 15 '18 at 11:11

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