New Math


New Mathematics or New Math was a dramatic change in the way mathematics was taught in American grade schools, and to a lesser extent in European countries, during the 1950s-1970s. Curriculum topics and teaching practices were changed in the U.S. shortly after the Sputnik crisis. The goal was to boost students' science education and mathematical skill to meet the technological threat of Soviet engineers, reputedly highly skilled mathematicians.

Overview

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 in chemistry. Several mathematics curriculum development efforts were also funded as part of the same initiative, such as the , SMSG, and .
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 from the numerals that represent them, 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.
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.

Legacy

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+ and recognizing numbers whose sum is ten. 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 four to six students facing each other rather than facing front was a New Math innovation. The use of manipulatives 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.
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.

Criticism

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."
In 1965, physicist Richard Feynman wrote in the essay New Textbooks for the "New" Mathematics:
In his book , 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." 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."
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.

Other countries

In the broader context, reform of school mathematics curricula was also pursued in European countries, such as the United Kingdom, 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:
In Japan, New Math was supported by the Ministry of Education, Culture, Sports, Science and Technology, but not without encountering problems, leading to student-centred approaches.

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