New math
Encyclopedia
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 intellectual threat of Soviet engineer
s, reputedly highly skilled mathematician
s, could be met.
through abstract concepts like set theory
and number bases
other than 10. Beginning in the early 1960s the new educational doctrine was installed, not only in the USA, but all over the developed world.
Much of the publicity centered on the focus of this program on set theory
(influenced ultimately by the Bourbaki group and their work), function
s, and diagram
drawings. It was stressed that these subjects should be introduced early. Some of this focus was seen as exaggerated, even dogma
tic. For example, in some cases pupils were taught axiomatic set theory at an early age. The idea behind this was that if the axiomatic foundations of mathematics were introduced to children, they could easily cope with the theorem
s 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
. Most of these topics (except algebraic inequalities) have been greatly de-emphasized or eliminated since the 1960s.
. 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.
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 "New books for new mathematics":
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).
an 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 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 science
s 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
, the New Math was supported by MEXT, but not without problems, leading to child-centred approaches.
as well.
This is the problem Tom Lehrer sings about in his song, “New Math” where he parodies overcomplexity in New Math.
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
was taught in American
United States
The United States of America is a federal constitutional republic comprising fifty states and a federal district...
grade schools, and to a lesser extent in European countries, during the 1960s
1960s
The 1960s was the decade that started on January 1, 1960, and ended on December 31, 1969. It was the seventh decade of the 20th century.The 1960s term also refers to an era more often called The Sixties, denoting the complex of inter-related cultural and political trends across the globe...
. The name is commonly given to a set of teaching practices introduced in the U.S. shortly after the Sputnik crisis
Sputnik crisis
The Sputnik crisis is the name for the American reaction to the success of the Sputnik program. It was a key event during the Cold War that began on October 4, 1957 when the Soviet Union launched Sputnik 1, the first artificial Earth satellite....
in order to boost science education
Science education
Science education is the field concerned with sharing science content and process with individuals not traditionally considered part of the scientific community. The target individuals may be children, college students, or adults within the general public. The field of science education comprises...
and mathematical skill in the population so that the intellectual threat of Soviet engineer
Engineer
An engineer is a professional practitioner of engineering, concerned with applying scientific knowledge, mathematics and ingenuity to develop solutions for technical problems. Engineers design materials, structures, machines and systems while considering the limitations imposed by practicality,...
s, reputedly highly skilled mathematician
Mathematician
A mathematician is a person whose primary area of study is the field of mathematics. Mathematicians are concerned with quantity, structure, space, and change....
s, could be met.
The new mathematical pedagogy
New Math emphasized mathematical structureMathematical structure
In mathematics, a structure on a set, or more generally a type, consists of additional mathematical objects that in some manner attach to the set, making it easier to visualize or work with, or endowing the collection with meaning or significance....
through abstract concepts like set theory
Set theory
Set theory is the branch of mathematics that studies sets, which are collections of objects. Although any type of object can be collected into a set, set theory is applied most often to objects that are relevant to mathematics...
and number bases
Numeral system
A numeral system is a writing system for expressing numbers, that is a mathematical notation for representing numbers of a given set, using graphemes or symbols in a consistent manner....
other than 10. Beginning in the early 1960s the new educational doctrine was installed, not only in the USA, but all over the developed world.
Much of the publicity centered on the focus of this program on set theory
Set theory
Set theory is the branch of mathematics that studies sets, which are collections of objects. Although any type of object can be collected into a set, set theory is applied most often to objects that are relevant to mathematics...
(influenced ultimately by the Bourbaki group and their work), function
Function (mathematics)
In mathematics, a function associates one quantity, the argument of the function, also known as the input, with another quantity, the value of the function, also known as the output. A function assigns exactly one output to each input. The argument and the value may be real numbers, but they can...
s, and diagram
Diagram
A diagram is a two-dimensional geometric symbolic representation of information according to some visualization technique. Sometimes, the technique uses a three-dimensional visualization which is then projected onto the two-dimensional surface...
drawings. It was stressed that these subjects should be introduced early. Some of this focus was seen as exaggerated, even dogma
Dogma
Dogma is the established belief or doctrine held by a religion, or a particular group or organization. It is authoritative and not to be disputed, doubted, or diverged from, by the practitioners or believers...
tic. For example, in some cases pupils were taught axiomatic set theory at an early age. The idea behind this was that if the axiomatic foundations of mathematics were introduced to children, they could easily cope with the theorem
Theorem
In mathematics, a theorem is a statement that has been proven on the basis of previously established statements, such as other theorems, and previously accepted statements, such as axioms...
s of the mathematical system later.
Other topics introduced in the New Math include modular arithmetic
Modular arithmetic
In mathematics, modular arithmetic is a system of arithmetic for integers, where numbers "wrap around" after they reach a certain value—the modulus....
, algebraic inequalities, matrices
Matrix (mathematics)
In mathematics, a matrix is a rectangular array of numbers, symbols, or expressions. The individual items in a matrix are called its elements or entries. An example of a matrix with six elements isMatrices of the same size can be added or subtracted element by element...
, symbolic logic
Mathematical logic
Mathematical logic is a subfield of mathematics with close connections to foundations of mathematics, theoretical computer science and philosophical logic. The field includes both the mathematical study of logic and the applications of formal logic to other areas of mathematics...
, Boolean algebra, and abstract algebra
Abstract algebra
Abstract algebra is the subject area of mathematics that studies algebraic structures, such as groups, rings, fields, modules, vector spaces, and algebras...
. Most of these topics (except algebraic inequalities) have been greatly de-emphasized or eliminated since the 1960s.
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 arithmeticArithmetic
Arithmetic or arithmetics is the oldest and most elementary branch of mathematics, used by almost everyone, for tasks ranging from simple day-to-day counting to advanced science and business calculations. It involves the study of quantity, especially as the result of combining numbers...
. 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.
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
Multiplication table
In mathematics, a multiplication table is a mathematical table used to define a multiplication operation for an algebraic system....
."
In 1965, physicist Richard Feynman
Richard Feynman
Richard Phillips Feynman was an American physicist known for his work in the path integral formulation of quantum mechanics, the theory of quantum electrodynamics and the physics of the superfluidity of supercooled liquid helium, as well as in particle physics...
wrote in "New books for new mathematics":
- "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
Morris Kline
Morris Kline was a Professor of Mathematics, a writer on the history, philosophy, and teaching of mathematics, and also a popularizer of mathematical subjects.Kline grew up in Brooklyn and in Jamaica, Queens...
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).
Across other countries
In the broader context, reform of school mathematics curricula was also pursued in EuropeEurope
Europe is, by convention, one of the world's seven continents. Comprising the westernmost peninsula of Eurasia, Europe is generally 'divided' from Asia to its east by the watershed divides of the Ural and Caucasus Mountains, the Ural River, the Caspian and Black Seas, and the waterways connecting...
an countries such as the United Kingdom
United Kingdom
The United Kingdom of Great Britain and Northern IrelandIn the United Kingdom and Dependencies, other languages have been officially recognised as legitimate autochthonous languages under the European Charter for Regional or Minority Languages...
(particularly by the School Mathematics Project
School Mathematics Project
The School Mathematics Project is a developer of mathematics textbooks for secondary schools, based in Southampton in the UK.Now generally known as SMP, it began as a research project inspired by a 1961 conference chaired by Bryan Thwaites at the University of Southampton, which itself was...
), and France
France
The French Republic , The French Republic , The French Republic , (commonly known as France , is a unitary semi-presidential republic in Western Europe with several overseas territories and islands located on other continents and in the Indian, Pacific, and Atlantic oceans. Metropolitan 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
West Germany
West Germany is the common English, but not official, name for the Federal Republic of Germany or FRG in the period between its creation in May 1949 to German reunification on 3 October 1990....
the changes were seen as part of a larger process of Bildungsreform
Education reform
Education reform is the process of improving public education. Small improvements in education theoretically have large social returns, in health, wealth and well-being. Historically, reforms have taken different forms because the motivations of reformers have differed.A continuing motivation has...
. Beyond the use of set theory and different approach to arithmetic, characteristic changes were transformation geometry
Transformation geometry
In mathematics, transformation geometry is a name for a pedagogic theory for teaching Euclidean geometry, based on the Erlangen programme. Felix Klein, who pioneered this point of view, was himself interested in mathematical education. It took many years, though, for his "modern" point of view to...
in place of the traditional deductive Euclidean geometry
Euclidean geometry
Euclidean geometry is a mathematical system attributed to the Alexandrian Greek mathematician Euclid, which he described in his textbook on geometry: the Elements. Euclid's method consists in assuming a small set of intuitively appealing axioms, and deducing many other propositions from these...
, and an approach to calculus
Calculus
Calculus is a branch of mathematics focused on limits, functions, derivatives, integrals, and infinite series. This subject constitutes a major part of modern mathematics education. It has two major branches, differential calculus and integral calculus, which are related by the fundamental theorem...
that was based on greater insight, rather than emphasis on facility.
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 science
Physical science
Physical science is an encompassing term for the branches of natural science and science that study non-living systems, in contrast to the life sciences...
s and engineering
Engineering
Engineering is the discipline, art, skill and profession of acquiring and applying scientific, mathematical, economic, social, and practical knowledge, in order to design and build structures, machines, devices, systems, materials and processes that safely realize improvements to the lives of...
; and they expected manipulative skill in calculus, rather than more abstract ideas. Some compromises have since been required, given that discrete mathematics
Discrete mathematics
Discrete mathematics is the study of mathematical structures that are fundamentally discrete rather than continuous. In contrast to real numbers that have the property of varying "smoothly", the objects studied in discrete mathematics – such as integers, graphs, and statements in logic – do not...
is the basic language of computing
Computing
Computing is usually defined as the activity of using and improving computer hardware and software. It is the computer-specific part of information technology...
.
Teaching in the USSR
Soviet Union
The Soviet Union , officially the Union of Soviet Socialist Republics , was a constitutionally socialist state that existed in Eurasia between 1922 and 1991....
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 BoltyanskyVladimir BoltyanskyVladimir Grigorevich Boltyansky , also transliterated as Boltyanski, Boltyanskii, or Boltjansky, is a Soviet and Russian mathematician, educator and author of popular mathematical books and articles. He is best known for his books on topology, combinatorial geometry and Hilbert's third problem.-...
and YaglomIsaak YaglomIsaak Moiseevich Yaglom was a Soviet mathematician and author of popular mathematics books, some with his twin Akiva Yaglom.Yaglom received a Ph.D. from Moscow State University in 1945 as student of Veniamin Kagan. As the author of several books, translated into English, that have become academic...
.
In Japan
Japanese mathematics
denotes a distinct kind of mathematics which was developed in Japan during the Edo Period . The term wasan, from wa and san , was coined in the 1870s and employed to distinguish native Japanese mathematics theory from Western mathematics .In the history of mathematics, the development of wasan...
, the New Math was supported by MEXT, but not without problems, leading to child-centred approaches.
Example
A "New Math Way" is to not only learn how to accomplish subtraction by regrouping in the normal decimal system, but learn it in base 8Octal
The octal numeral system, or oct for short, is the base-8 number system, and uses the digits 0 to 7. Numerals can be made from binary numerals by grouping consecutive binary digits into groups of three...
as well.
- Base 8: regrouping the eights column means adding eight to the ones column and subtracting one from the eights column
-
- i.e. 342 − 173 = 147.
This is the problem Tom Lehrer sings about in his song, “New Math” where he parodies overcomplexity in New Math.
Popular culture
- Tom LehrerTom LehrerThomas Andrew "Tom" Lehrer is an American singer-songwriter, satirist, pianist, mathematician and polymath. He has lectured on mathematics and musical theater...
wrote a satirical song named "New Math" which centered around the process of subtracting 173 from 342 in decimalDecimalThe decimal numeral system has ten as its base. It is the numerical base most widely used by modern civilizations....
and octalOctalThe octal numeral system, or oct for short, is the base-8 number system, and uses the digits 0 to 7. Numerals can be made from binary numerals by grouping consecutive binary digits into groups of three...
. 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 emphasis on insight and abstract concepts of the New Math approach. Lehrer's explanation of the two calculations is entirely correct, but presented in such a way (at rapid speed, with minimal visual aids, and with side remarks thrown in) as to make it difficult for most audience members to follow the rather simple calculations being performed. This is intended to poke fun at the kind of 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. Summing up his opinion of New Math is the final sentence from his introductory remarks to the song: "...in the new approach, as you know, the important thing is to understand what you're doing, rather than to get the right answer;" at one point in the song, he inserts the assertion that 13 – 7 = 5, which is (deliberately) incorrect.
- Lehrer stated that "Base 8 is just like Base 10, if you're missing 2 Fingers."
- Lehrer, at the end of the song, said that he often wanted to write a mathematics textbook someday that would be a million seller, entitled "Tropic of CalculusTropic of Cancer (novel)Tropic of Cancer is a novel by Henry Miller which has been described as "notorious for its candid sexuality" and as responsible for the "free speech that we now take for granted in literature." It was first published in 1934 by the Obelisk Press in Paris, France, but this edition was banned in the...
".
- "New Math" was also the name of a 1970s punk rock band from Rochester, NY.
- In the Simpsons episode "Dog of Death," Principal Skinner refers to the New Math:
Kent: But there's already one big winner: Our state school system, which gets fully half the profits from the lottery.
Skinner: [talking with his teachers] Just think what we can buy with that money... History books that know how the Korean War came out. Math books that don't have that base six crap in them!
See also
- André LichnerowiczAndré LichnerowiczAndré Lichnerowicz was a noted French differential geometer and mathematical physicist of Polish descent.-Biography:...
– Created 1967 French Lichnerowicz Commission - Comprehensive School Mathematics ProgramComprehensive School Mathematics ProgramComprehensive School Mathematics Program stands for both the name of a curriculum and the name of the project that was responsible for developing curriculum materials....
(CSMP) - List of abandoned education methods
Further reading
- Adler, IrvingIrving AdlerIrving Adler is an author, mathematician, scientist, and educator. He is the author of 56 books about mathematics, science, and education, and the co-author of 30 more, for both children and adults. His books have been published in 31 countries in 19 different languages...
. The New Mathematics. New York: John Day and Co, 1972 (revised edition). ISBN 0-381-98002-2 - Maurice Mashaal (2006), Bourbaki: A Secret Society of Mathematicians, American Mathematical SocietyAmerican Mathematical SocietyThe American Mathematical Society is an association of professional mathematicians dedicated to the interests of mathematical research and scholarship, which it does with various publications and conferences as well as annual monetary awards and prizes to mathematicians.The society is one of the...
, ISBN 0-8218-3967-5, Chapter 10: New Math in the Classroom, pp 134–45.