New Math

After the Sputnik launch in 1957, the U.S. National Science Foundation funded the development of several new curricula in the sciences, such as the PSSC high school physics curriculum, BSCS in biology, and CHEM Study in chemistry. Several mathematics curriculum development efforts were also funded as part of the same initiative, such as the Madison Project, SMSG, and UICSM.

These curricula were quite different from one another, yet shared the idea children's learning of arithmetic algorithms would last past the exam only if memorization and practice were paired with teaching for understanding. More specifically, elementary school arithmetic beyond single digits makes sense only on the basis of understanding place value. This goal was the reason for teaching arithmetic in bases other than ten in the New Math, despite critics' derision: In that unfamiliar context, students couldn't just mindlessly follow an algorithm, but had to think why the place value of the "hundreds" digit in base seven is 49. Keeping track of non-decimal notation also explains the need to distinguish numbers (values) from the numerals that represent them[1], a distinction some critics considered fetishistic.

Topics introduced in the New Math include set theory, modular arithmetic, algebraic inequalities, bases other than 10, matrices, symbolic logic, Boolean algebra, and abstract algebra.[2]

But the New Math wasn't just a list of topics. All of the New Math projects emphasized some form of discovery learning. Students worked in groups to invent theories about problems posed in the textbooks. Materials for teachers described the classroom as "noisy." Part of the job of the teacher was to move from table to table assessing the theory that each group of students had developed and "torpedoing" wrong theories by providing counterexamples. For that style of teaching to be tolerable for students, they had to experience the teacher as a colleague rather than as an adversary or as someone concerned mainly with grading. New Math workshops for teachers, therefore, spent as much effort on the pedagogy as on the mathematics.

The New Math is often described as a short-lived movement with no lasting influence on current teaching practice. For better and for worse, that's not the case. One example concerns the introduction of the field axioms. Understanding the ways in which an arithmetic formula can be reordered helps students solve mental arithmetic problems such as 3+18+7 by recognizing that the problem can be reordered as 18+(3+7) and recognizing numbers whose sum is ten. (If the problem were 3−18+7, the grouping of operations would have to be considered more carefully before reordering, because subtraction isn't commutative.) On the other hand, the words "commutative" and "associative" are hard to remember, hard to spell, and therefore intimidating. But they are still sometimes taught, in part because the use of New Math textbooks was much more common than the provision of New Math teacher preparation workshops. Traditional teachers can add "commutative" to the weekly spelling list without actually giving students challenges in which the Commutative Law is helpful.

The style of work in modern elementary mathematics lessons is very heavily influenced by the New Math. Organizing the classroom space into table groups of (generally) four to six students facing each other rather than facing front was a New Math innovation. The use of manipulatives (physical blocks of wood or plastic that can be combined to illustrate ideas about quantity and shape, such as Cuisenaire rods) didn't start with the New Math, but it wasn't until the New Math popularized their use that they became universal. Publishers of current curricula provide manipulatives in table kits.[3] [4]

Another enduring result of the New Math has been the willingness of teachers and curriculum developers to use arithmetic algorithms other than the ones used in the 19th century. Those algorithms were designed to minimize the number of steps needed for an experienced adult to carry out a calculation. But in the 21st century all but the simplest computations are done by machine. Educators still have children do arithmetic, because some practical experience is necessary to understand the mathematical ideas. But they invent algorithms that are easier to understand, rather than faster to carry out.

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. In an effort to learn the material, many parents attended 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.

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."[5]

In 1965, physicist Richard Feynman wrote in the essay New Textbooks for the "New" Mathematics:[6]

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 are 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 worthwhile teaching such material.

In his book Why Johnny Can't Add: the Failure of the New Math, Morris Kline says that 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."[2]^: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."[2]^:98

As a result of this controversy, and despite the ongoing influence of the New Math, the phrase "new math" is often used now to describe any short-lived fad that quickly becomes discredited.

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. 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.

Again, the changes were 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.

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 [sic] presented in the textbook produced by Vladimir Boltyansky and Isaak Yaglom."[7]

In Japan, New Math was supported by the Ministry of Education, Culture, Sports, Science and Technology (MEXT), but not without encountering problems, leading to student-centred approaches.[8]

• Musician and university mathematics lecturer Tom Lehrer wrote a satirical song named "New Math" (from his 1965 album That Was the Year That Was), 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." The chorus pokes fun at parents' frustration and confusion over the entire method: "Hooray for New Math, New Math / It won't do you a bit of good to review math / It's so simple, so very simple / That only a child can do it."[9]
• New Math was the name of a 1970s punk rock band from Rochester, New York.[10]
• 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?"[11] This series of strips was later adapted for the 1973 Peanuts animated special There's No Time for Love, Charlie Brown. Schulz also drew a one panel illustration of Charlie Brown at his school desk exclaiming, "How can you do 'New Math' problems with an 'Old Math' mind?"[12]
• In 1999, Time placed New Math on a list of the 100 worst ideas of the 20th century.[13]
• In the Pixar animated film Incredibles 2 (which is set approximately in the 1960s), Bob Parr (Mr. Incredible) can be seen wrangling with New Math in his son's homework, and promptly describes it as making no sense at all, and then to ridicule "I don't know that way! Why would they change Math? Math is Math! MATH IS MATH!"[14]

• André Lichnerowicz – Created 1967 French Lichnerowicz Commission
• Comprehensive School Mathematics Program (CSMP)
• Secondary School Mathematics Curriculum Improvement Study (SSMCIS)
• List of abandoned education methods
• School Mathematics Study Group (SMSG)
• New New Math - a satirical term for the Math Wars of the 1990s

• Adler, Irving (1972). The New Mathematics (revised ed.). New York: John Day Company. ISBN 0-381-98002-2.
• Mashaal, Maurice (2006). "New Math in the Classroom". Bourbaki: A Secret Society of Mathematicians. American Mathematical Society. pp. 134–145. ISBN 9780821839676. This work was originally published as Bourbaki: une société secrète de mathématiciens (2002, ISBN 2842450469, in French) and the 2006 English-language version was translated by Anna Pierrehumbert.
• Phillips, Christopher J. (2014). The New Math: A Political History. University of Chicago Press. ISBN 9780226185019.

• Tom Lehrer Deposit #10