Morris Kline

Morris Kline (May 1, 1908 – June 10, 1992) was a Professor of Mathematics, a writer on the history, philosophy, and teaching of mathematics, and also a popularizer of mathematical subjects.

Education and career

Kline was born to a Jewish family in Brooklyn and resided in Jamaica, Queens. After graduating from Boys High School in Brooklyn, he studied mathematics at New York University, earning a bachelor's degree in 1930, a master's degree in 1932, and a doctorate (Ph. D) in 1936. He continued at NYU as an instructor until 1942.

During World War II, Kline was posted to the Signal Corps (United States Army) stationed at Belmar, New Jersey. Designated a physicist, he worked in the engineering lab where radar was developed. After the war he continued investigating electromagnetism, and from 1946 to 1966 was director of the division for electromagnetic research at the Courant Institute of Mathematical Sciences.

Kline resumed his mathematical teaching at NYU, becoming a full professor in 1952. He taught at New York University until 1975, and wrote many papers and more than a dozen books on various aspects of mathematics and particularly teaching of mathematics. He repeatedly stressed the need to teach the applications and usefulness of mathematics rather than expecting students to enjoy it for its own sake. Similarly, he urged that mathematical research concentrate on solving problems posed in other fields rather than building structures of interest only to other mathematicians. One can get a sense of Kline's views on teaching from the following:

I would urge every teacher to become an actor. His classroom technique must be enlivened by every device used in theatre. He can be and should be dramatic where appropriate. He must not only have facts but fire. He can utilize even eccentricities of behavior to stir up human interest. He should not be afraid of humor and should use it freely. Even an irrelevant joke or story perks up the class enormously.[1]

Critique of mathematics education

Morris Kline was a protagonist in the curriculum reform in mathematics education that occurred in the second half of the twentieth century, a period including the programs of the new math. An article by Kline in 1956 in The Mathematics Teacher, the main journal of the National Council of Teachers of Mathematics, was titled "Mathematical texts and teachers: a tirade". Calling out teachers blaming students for failures, he wrote "There is a student problem, but there are also three other factors which are responsible for the present state of mathematical learning, namely, the curricula, the texts, and the teachers." The tirade touched a nerve, and changes started to happen. But then Kline switched to being a critic of some of the changes. In 1958 he wrote "Ancients versus moderns: a new battle of the books". The article was accompanied with a rebuttal by Albert E. Meder Jr. of Rutgers University.[2] He says, "I find objectionable: first, vague generalizations, entirely undocumented, concerning views held by ‘modernists’, and second, the inferences drawn from what has not been said by the ‘modernists’." By 1966 Kline proposed an eight-page high school plan.[3] The rebuttal for this article was by James H. Zant; it asserted that Kline had "a general lack of knowledge of what was going on in schools with reference to textbooks, teaching, and curriculum." Zant criticized Kline’s writing for "vagueness, distortion of facts, undocumented statements and overgeneralization."

In 1966[4] and 1970[5] Kline issued two further criticisms. In 1973 St. Martin’s Press contributed to the dialogue by publishing Kline’s critique, Why Johnny Can’t Add: the Failure of the New Math. Its opening chapter is a parody of instruction as students’ intuitions are challenged by the new jargon. The book recapitulates the debates from Mathematics Teacher, with Kline conceding some progress: He cites Howard Fehr of Columbia University who sought to unify the subject through its general concepts, sets, operations, mappings, relations, and structure in the Secondary School Mathematics Curriculum Improvement Study.

In 1977 Kline turned to undergraduate university education; he took on the academic mathematics establishment with his Why the Professor Can’t Teach: the dilemma of university education. Kline argues that onus to conduct research misdirects the scholarly method that characterizes good teaching. He lauds scholarship as expressed by expository writing or reviews of original work of others. For scholarship he expects critical attitudes to topics, materials and methods. Among the rebuttals are those by D.T. Finkbeiner, Harry Pollard, and Peter Hilton.[6] Pollard conceded, "The society in which learning is admired and pursued for its own sake has disappeared." The Hilton review was more direct: Kline has "placed in the hand of enemies…[a] weapon". Having started in 1956 as an agitator for change in mathematics education, he became a critic of some trends. Skilled expositor that he was, editors frequently felt his expressions were best tempered with rebuttal.

In considering what motivated Morris Kline to protest, consider Professor Meder’s opinion:[7] I am wondering whether in point of fact, Professor Kline really likes mathematics [...] I think that he is at heart a physicist, or perhaps a ‘natural philosopher’, not a mathematician, and that the reason he does not like the proposals for orienting the secondary school college preparatory mathematics curriculum to the diverse needs of the twentieth century by making use of some concepts developed in mathematics in the last hundred years or so is not that this is bad mathematics, but that it minimizes the importance of physics.

It might appear so, as Kline recalls E. H. Moore’s recommendation to combine science and mathematics at the high school level.[8] But closer reading shows Kline calling mathematics a "part of man’s efforts to understand and master his world", and he sees that role in a broad spectrum of sciences.

Critique of mathematics research

In Mathematics: The Loss of Certainty (ch. XIII: "The Isolation of Mathematics"), Kline deplored the way mathematics research was being conducted, complaining that often mathematicians, not willing to become acquainted with the (sometimes deep) context needed to solve applied problems in sciences, prefer to invent pure mathematics problems that are not necessarily of any consequence. Kline also blamed the publish or perish academic culture for this state of affairs.


  • Introduction to Mathematics (with Irvin W. Kay), Houghton Mifflin, 1937
  • The Theory of Electromagnetic Waves (ed), Inter-science Publishers, 1951
  • Mathematics in Western Culture, Oxford University Press,1953
  • Mathematics and the Physical World, T. Y. Crowell Co., 1959
  • Mathematics, A Cultural Approach, Addison-Wesley, 1962
  • Mathematics for the Nonmathematician, Dover Publications, 1967
  • Electromagnetic Theory and Geometrical Optics (with Irvin W. Kay), John Wiley and Sons, 1965
  • Calculus, An intuitive and Physical Approach, John Wiley and Sons, 1967, 1977, Dover Publications 1998 reprint ISBN 0-486-40453-6
  • Mathematics for Liberal Arts, Addison-Wesley, 1967, (republished as Mathematics for the Nonmathematician, Dover Publications, Inc., 1985) (ISBN 0-486-24823-2)
  • Mathematics in the Modern World (ed), W. H. Freeman and Co., 1968
  • Mathematical Thought From Ancient to Modern Times, Oxford University Press, 1972
  • Why Johnny Can't Add: The Failure of the New Mathematics, St. Martin's Press, 1973
  • Why the professor can't teach: Mathematics and the dilemma of university education, St. Martin's Press, 1977 (ISBN 0-312-87867-2)
  • Mathematics: The Loss of Certainty, Oxford University Press, 1980 (ISBN 0-19-502754-X); OUP Galaxy Books pb. reprint (ISBN 0-19-503085-0)
  • Mathematics: An Introduction to Its Spirit and Use; readings from Scientific American
  • Mathematics in the Modern World; readings from Scientific American
  • The Language of Shapes (with Abraham Wolf Crown)
  • Mathematics and the Search for Knowledge, Oxford University Press, 1985 (ISBN 0-19-503533-X)



  1. ^ Mathematics Teacher 49:171.
  2. ^ Mathematics Teacher 51:428–33
  3. ^ Mathematics Teacher 59:322–330
  4. ^ M. Kline (1966) "Intellectuals and the schools: a case history" in Harvard Educational Review 36:505–11
  5. ^ M. Kline (1970) "Logic versus pedagogy" in American Mathematical Monthly (77:264–82)
  6. ^ D.T. Finkbeiner, Harry Pollard, Peter Hilton (May 1979) American Mathematical Monthly 86:401–412
  7. ^ Mathematics Teacher 51:433
  8. ^ Why Johnny Can’t Add, page 147

External links

A History of Vector Analysis

A History of Vector Analysis (1967) is a book on the history of vector analysis by Michael J. Crowe, originally published by the University of Notre Dame Press.

As a scholarly treatment of a reformation in technical communication, the text is a contribution to the history of science. In 2002, Crowe gave a talk summarizing the book, including an entertaining introduction in which he covered its publication history and related the award of a Jean Scott prize of $4000. Crowe had entered the book in a competition for "a study on the history of complex and hypercomplex numbers" twenty-five years after his book was first published.


Archytas (; Greek: Ἀρχύτας; 428–347 BC) was an Ancient Greek philosopher, mathematician, astronomer, statesman, and strategist. He was a scientist of the Pythagorean school and famous for being the reputed founder of mathematical mechanics, as well as a good friend of Plato.

Boys High School (Brooklyn)

Boys High School is a historic and architecturally notable public school building in the Bedford–Stuyvesant neighborhood of Brooklyn, New York, United States. It is regarded as "one of Brooklyn's finest buildings."


Calculus (from Latin calculus, literally 'small pebble', used for counting and calculations, as on an abacus) is the mathematical study of continuous change, in the same way that geometry is the study of shape and algebra is the study of generalizations of arithmetic operations.

It has two major branches, differential calculus (concerning instantaneous rates of change and slopes of curves), and integral calculus (concerning accumulation of quantities and the areas under and between curves). These two branches are related to each other by the fundamental theorem of calculus. Both branches make use of the fundamental notions of convergence of infinite sequences and infinite series to a well-defined limit.

Modern calculus was developed independently in the late 17th century by Isaac Newton and Gottfried Wilhelm Leibniz. Today, calculus has widespread uses in science, engineering, and economics.Calculus is a part of modern mathematics education. A course in calculus is a gateway to other, more advanced courses in mathematics devoted to the study of functions and limits, broadly called mathematical analysis. Calculus has historically been called "the calculus of infinitesimals", or "infinitesimal calculus". The term calculus (plural calculi) is also used for naming specific methods of calculation or notation as well as some theories, such as propositional calculus, Ricci calculus, calculus of variations, lambda calculus, and process calculus.

Courant Institute of Mathematical Sciences

The Courant Institute of Mathematical Sciences (CIMS) is an independent division of New York University (NYU) under the Faculty of Arts & Science that serves as a center for research and advanced training in computer science and mathematics. It is considered one of the leading and most prestigious mathematics schools and mathematical sciences research centers in the world. It is named after Richard Courant, one of the founders of the Courant Institute and also a mathematics professor at New York University from 1936 to 1972.

It is ranked #1 in applied mathematical research in US, #5 in citation impact worldwide, and #12 in citation worldwide. On the Faculty Scholarly Productivity Index, it is ranked #3 with an index of 1.84. It is also known for its extensive research in pure mathematical areas, such as partial differential equations, probability and geometry, as well as applied mathematical areas, such as computational biology, computational neuroscience, and mathematical finance. The Mathematics Department of the Institute has 18 members of the United States National Academy of Sciences (more than any other mathematics department in the U.S.) and five members of the National Academy of Engineering. Four faculty members have been awarded the National Medal of Science, one was honored with the Kyoto Prize, and nine have received career awards from the National Science Foundation. Courant Institute professors Peter Lax, S. R. Srinivasa Varadhan, Mikhail Gromov, Louis Nirenberg won the 2005, 2007, 2009 and 2015 Abel Prize respectively for their research in partial differential equations, probability and geometry. Louis Nirenberg also received the Chern Medal in 2010, and Subhash Khot won the Nevanlinna Prize in 2014.

The Director of the Courant Institute directly reports to New York University's Provost and President and works closely with deans and directors of other NYU colleges and divisions respectively. The undergraduate programs and graduate programs at the Courant Institute are run independently by the Institute, and formally associated with the NYU College of Arts and Science and NYU Graduate School of Arts and Science respectively.

Edwin E. Moise

Edwin Evariste Moise (; December 22, 1918 – December 18, 1998)

was an American mathematician and mathematics education reformer. After his retirement from mathematics he became a literary critic of 19th-century English poetry and had several notes published in that field.

Eudoxus of Cnidus

Eudoxus of Cnidus (; Ancient Greek: Εὔδοξος ὁ Κνίδιος, Eúdoxos ho Knídios; c. 390 – c. 337 BC) was an ancient Greek astronomer, mathematician, scholar, and student of Archytas and Plato. All of his works are lost, though some fragments are preserved in Hipparchus' commentary on Aratus's poem on astronomy. Sphaerics by Theodosius of Bithynia may be based on a work by Eudoxus.


Hippasus of Metapontum (; Greek: Ἵππασος ὁ Μεταποντῖνος, Híppasos; fl. 5th century BC), was a Pythagorean philosopher. Little is known about his life or his beliefs, but he is sometimes credited with the discovery of the existence of irrational numbers. The discovery of irrational numbers is said to have been shocking to the Pythagoreans, and Hippasus is supposed to have drowned at sea, apparently as a punishment from the gods for divulging this. However, the few ancient sources which describe this story either do not mention Hippasus by name (e.g. Pappus) or alternatively tell that Hippasus drowned because he revealed how to construct a dodecahedron inside a sphere. The discovery of irrationality is not specifically ascribed to Hippasus by any ancient writer. Some modern scholars though have suggested that he discovered the irrationality of √2, which is believed to have been discovered around the time that he lived.

Kline (surname)

Kline is a surname. Notable people with the surname include:

Adam W. Kline (1818–1898), New York politician

Boštjan Kline (b. 1991), Slovenian alpine ski racer

Branden Kline (b. 1991), American baseball player

Brittani Kline (b. 1991), American fashion model

Charles H. Kline (1870–1933), American politician, mayor of Pittsburgh, Pennsylvania

Christopher Paul Kline (b. 1979), American musician known as "Vertexguy" or "the Vertex Guy"

Franz Kline (1910–1962), American artist

Jerry Kline (born 1951), American businessman

John Kline (Harlem Globetrotter) (fl. 1950s), American professional basketball player

John Kline (politician) (b. 1947), United States Representative from Minnesota

Kevin Kline (b. 1947), American actor

Lindsay Kline (1934–2015), Australian cricketer

Mark Kline, American physician

Morris Kline (1908–1992), American professor of mathematics and writer

Otis Adelbert Kline (1891–1946), American adventure novelist

Paul Kline (1937–1999), British academic psychologist

Phill Kline (born 1959), American lawyer and politician from Kansas

Richard Kline (b. 1944), American actor who played Larry on the sitcom Three's Company

Scott Richard Kline (1967-2015), given name of Scott Weiland, American singer and founding member of the alt-rock group Stone Temple Pilots

Stanley F. Kline (1901–1942), United States Navy sailor and Silver Star recipient

Virginia Harriett Kline (1910–1959), American geologist, stratigrapher, and librarian

Mathematics and the Search for Knowledge

Mathematics and the Search for Knowledge is a book by Morris Kline on the developing mathematics ideas, which are partially overlap with his previous book Mathematics: The Loss of Certainty, as a source of human knowledge about the physical world, starting from astronomical theories of Ancient Greek to the modern theories.In contrast to the numerous theories, that have appeared since the ancient times up to Newton's theory of gravitation, which are describe different physical phenomena and were often intuitive and can be mechanically explained, but all modern theories, such as electromagnetism, theory of relativity, quantum mechanics, are the mathematical description of reality, which could not be granted with a clear interpretation, which would be available to human senses.

About the concepts that appear and are used in these theories to describe the physical world, we are the only known - mathematical relationships that they are satisfy (for example, an electromagnetic radiation, wave-particle duality, spacetime, or an electron).

Due to the limitations of our senses capability (for example, from the whole spectrum of electromagnetic radiation the human eye perceives only a small part) and the ability to mislead us (for example, optical illusion), human is forced to use the mathematics as a tool that allows us to not only to compensate the imperfection of our senses, but also to obtain new knowledge, which are not available to our sensory perception.

The author brings us to the idea that the physical world, is not what available to us in our sensation, but rather what human-made mathematical theories say.

Natural number

In mathematics, the natural numbers are those used for counting (as in "there are six coins on the table") and ordering (as in "this is the third largest city in the country"). In common mathematical terminology, words colloquially used for counting are "cardinal numbers" and words connected to ordering represent "ordinal numbers".

Some definitions, including the standard ISO 80000-2, begin the natural numbers with 0, corresponding to the non-negative integers 0, 1, 2, 3, …, whereas others start with 1, corresponding to the positive integers 1, 2, 3, …. Texts that exclude zero from the natural numbers sometimes refer to the natural numbers together with zero as the whole numbers, but in other writings, that term is used instead for the integers (including negative integers).The natural numbers are a basis from which many other number sets may be built by extension: the integers (Grothendieck group), by including (if not yet in) the neutral element 0 and an additive inverse (−n) for each nonzero natural number n; the rational numbers, by including a multiplicative inverse (1/n) for each nonzero integer n (and also the product of these inverses by integers); the real numbers by including with the rationals the limits of (converging) Cauchy sequences of rationals; the complex numbers, by including with the real numbers the unresolved square root of minus one (and also the sums and products thereof); and so on. These chains of extensions make the natural numbers canonically embedded (identified) in the other number systems.

Properties of the natural numbers, such as divisibility and the distribution of prime numbers, are studied in number theory. Problems concerning counting and ordering, such as partitioning and enumerations, are studied in combinatorics.

In common language, for example in primary school, natural numbers may be called counting numbers both to intuitively exclude the negative integers and zero, and also to contrast the discreteness of counting to the continuity of measurement, established by the real numbers.

The natural numbers can, at times, appear as a convenient set of names (labels), that is, as what linguists call nominal numbers, foregoing many or all of the properties of being a number in a mathematical sense.

New Math

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 change involved new curriculum topics and 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 technological threat of Soviet engineers, reputedly highly skilled mathematicians, could be met.

The phrase is often used now to describe any short-lived fad which quickly becomes highly discredited.


The quadrivium (plural: quadrivia) is the four subjects, or arts (namely arithmetic, geometry, music and astronomy), taught after teaching the trivium. The word is Latin, meaning four ways, and its use for the four subjects has been attributed to Boethius or Cassiodorus in the 6th century. Together, the trivium and the quadrivium comprised the seven liberal arts (based on thinking skills), as distinguished from the practical arts (such as medicine and architecture).

The quadrivium consisted of arithmetic, geometry, music, and astronomy. These followed the preparatory work of the trivium, consisting of grammar, logic, and rhetoric. In turn, the quadrivium was considered the foundation for the study of philosophy (sometimes called the "liberal art par excellence") and theology.The quadrivium was the upper division of the medieval education in the liberal arts, which comprised arithmetic (number), geometry (number in space), music (number in time), and astronomy (number in space and time). Educationally, the trivium and the quadrivium imparted to the student the seven liberal arts (essential thinking skills) of classical antiquity.

Rafael Bombelli

Rafael Bombelli (baptised on 20 January 1526; died 1572) was an Italian mathematician. Born in Bologna, he is the author of a treatise on algebra and is a central figure in the understanding of imaginary numbers.

He was the one who finally managed to address the problem with imaginary numbers. In his 1572 book, L'Algebra, Bombelli solved equations using the method of del Ferro/Tartaglia. He introduced the rhetoric that preceded the representative symbols +i and -i and described how they both worked.

Social gadfly

A gadfly is a person who interferes with the status quo of a society or community by posing novel, potently upsetting questions, usually directed at authorities. The term is originally associated with the ancient Greek philosopher Socrates, in his defense when on trial for his life.

Why Johnny Can't Add

Why Johnny Can't Add: The Failure of the New Math is a book written by Morris Kline, first published in 1973. In it, Kline severely criticized the teaching practices characteristic of the "New Math" fashion for school teaching, which were based on Bourbaki's approach to mathematical research, and were being pushed into schools in the United States. Reactions were immediate, and the book became a best seller in its genre and translated into many languages.

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