# New Math

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New Mathematics or New Math was a brief, dramatic change in the way mathematics was taught in American grade schools, and to a lesser extent in European countries, during the 1960s. The name is commonly given to a set of teaching practices introduced in the U.S. shortly after the Sputnik crisis in order to boost science education and mathematical skill in the population so that the perceived intellectual threat of Soviet engineers, reputedly highly skilled mathematicians, could be met.

## Modern exercises

Mathematicians describe interesting objects with set-builder notation. Under the stress of Russian engineering competition, American schools began to use textbooks based on set theory. For example, the process of solving an algebraic equation required a parallel account of axioms in use for equation transformation. To develop the concept of number, non-standard numeral systems were used in exercises. Binary numbers and duodecimals were new math to the students and their parents. Teachers returning from summer school could introduce students to transformation geometry. If the school had been teaching Cramer's rule for solving linear equations, then new math may include matrix multiplication to introduce linear algebra. In any case, teachers used the function concept as a thread common to the new materials.

Philosopher and mathematician W.V. Quine wrote that the "rarefied air" of Cantorian set theory was not to be associated with the New Math. According to Quine, the New Math involved merely..."the Boolean algebra of classes, hence really the simple logic of general terms."[1]

It was stressed that these subjects should be introduced early. The idea behind this was that if the axiomatic foundations of mathematics were introduced to children, they could easily cope with the theorems of the mathematical system later.

Other topics introduced in the New Math include modular arithmetic, algebraic inequalities, matrices, symbolic logic, Boolean algebra, and abstract algebra.[2] Most of these topics (except algebraic inequalities) have been greatly de-emphasized or eliminated in elementary school and high school since the 1960s.

## Criticisms

Parents and teachers who opposed the New Math in the U.S. complained that the new curriculum was too far outside of students' ordinary experience and was not worth taking time away from more traditional topics, such as arithmetic. The material also put new demands on teachers, many of whom were required to teach material they did not fully understand. Parents were concerned that they did not understand what their children were learning and could not help them with their studies. Many of the parents took time out to try to understand the new math by attending their children's classes. In the end it was concluded that the experiment was not working, and New Math fell out of favor before the end of the decade, though it continued to be taught for years thereafter in some school districts. New Math found some later success in the form of enrichment programs for gifted students from the 1980s onward in Project MEGSSS.[3]

In the Algebra preface of his book Precalculus Mathematics in a Nutshell, Professor George F. Simmons wrote that the New Math produced students who had "heard of the commutative law, but did not know the multiplication table."

In 1965, physicist Richard Feynman wrote in the essay "New Textbooks for the 'New Mathematics":[4]

"If we would like to, we can and do say, 'The answer is a whole number less than 9 and bigger than 6,' but we do not have to say, 'The answer is a member of the set which is the intersection of the set of those numbers which is larger than 6 and the set of numbers which are smaller than 9' ... In the 'new' mathematics, then, first there must be freedom of thought; second, we do not want to teach just words; and third, subjects should not be introduced without explaining the purpose or reason, or without giving any way in which the material could be really used to discover something interesting. I don't think it is worth while teaching such material."

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).

## Other countries

In the broader context, reform of school mathematics curricula was also pursued in European countries such as the United Kingdom (particularly by the School Mathematics Project), and France, where the extremely high prestige of mathematical qualifications was not matched by teaching that connected with contemporary research and university topics.[clarification needed] In West Germany the changes were seen as part of a larger process of Bildungsreform. Beyond the use of set theory and different approach to arithmetic, characteristic changes were transformation geometry in place of the traditional deductive Euclidean geometry, and an approach to calculus that was based on greater insight, rather than emphasis on facility[clarification needed].[citation needed]

Again the changes met with a mixed reception, but for different reasons. For example, the end-users of mathematics studies were at that time mostly in the physical sciences and engineering; and they expected manipulative skill in calculus, rather than more abstract ideas. Some compromises have since been required, given that discrete mathematics is the basic language of computing.[citation needed]

Teaching in the USSR did not experience such extreme upheavals, while being kept in tune both with the applications and academic trends.

Under A. N. Kolmogorov, the mathematics committee declared a reform of the curricula of grades 4-10, at the time when the school system consisted of 10 grades. The committee found the type of reform in progress in Western countries to be unacceptable; for example, no special topic for sets was accepted for inclusion in school textbooks. Transformation approaches were accepted in teaching geometry, but not to such sophisticated level presented in the textbook produced by Boltyansky and Yaglom.[5]

In Japan, the New Math was supported by MEXT, but not without problems, leading to child-centred approaches.[6]

## Popular culture

• Musician and university mathematics lecturer Tom Lehrer wrote a satirical song named "New Math" which revolved around the process of subtracting 173 from 342 in decimal and octal. The song is in the style of a lecture about the general concept of subtraction in arbitrary number systems, illustrated by two simple calculations, and highlights the New Math's emphasis on insight and abstract concepts—as Lehrer put it with an indeterminable amount of seriousness, "In the new approach ... the important thing is to understand what you're doing rather than to get the right answer." At one point in the song, he notes that "you've got [thirteen] and you take away seven, and that leaves five... well, six, actually, but the idea is the important thing."
Lehrer's explanation of the two calculations is entirely correct, but presented in such a way (very rapidly and with many side remarks) as to make it difficult to follow the individually simple steps, thus recreating the bafflement the New Math approach often evoked when apparently simple calculations were presented in a very general manner which, while mathematically correct and arguably trivial for mathematicians, was likely very confusing to absolute beginners and even contemporary adult audiences.[citation needed]
Lehrer states that "Base 8 is just like Base 10, really...if you're missing two fingers."
The tagline of the song is "It's so simple, so very simple, that only a child can do it."
• In 1965, cartoonist Charles Schulz authored a series of Peanuts strips which detailed kindergartener Sally's frustrations with New Math. In the first strip, she is depicted puzzling over "sets, one to one matching, equivalent sets, non-equivalent sets, sets of one, sets of two, renaming two, subsets, joining sets, number sentences, placeholders." Eventually she bursts into tears and exclaims, "All I want to know is, how much is two and two?"[8] This series of strips was later adapted for the 1973 Peanuts animated special There's No Time for Love, Charlie Brown.