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Why Johnny Can't Add: The Failure of the New Math. Paperback – 1 Feb 1974


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Amazon.com: HASH(0x9cf1db40) out of 5 stars 10 reviews
35 of 36 people found the following review helpful
HASH(0x9d35530c) out of 5 stars WHY THE MODERN TEACHING OF MATH IS PERVERTED 28 Aug. 2001
By Jos Domingo Moll Vidal - Published on Amazon.com
Format: Paperback
When I was at Hi-School, here in Spain, in the United States of Europe, our teacher used to mention this book to us lots of times. Later on, when at University, I left no stone unturned at several University libraries till I found a copy of the book, a Spanish edition. I read the book just in one session, due to how interesting I found it. Professor Morris Kline (he taught at New York University, I think I remember)shows throughout the book the rare ability to absolutely master every mathematical concept he talks about and at the same time being able to see those concepts with fresh eyes, as if they were new for him, as if he himself were a teenager encountering those ideas for the first time and trying to come to grips with them. True that modern math is far more abstract and powerful than what was the knowledge body of mathematics say in the 17th century for instance. But at the same time, emphasis in detaching ideas from any connections with the physical world, abstracting as an end per se, and letting 'rigorousness' and formalism prevail over intuition, has led some areas of modern math to something which looks like an esoteric exercice consisting of sipping through the symbols in a book. Kline makes for instance the point that too much insistence on 'rigorousness' is equivalent to finding snakes under the jewels, and says that when a mathematician does not care any more or is not sensible to problems such as the movements of planets around a star, the behaviour of density waves (sound) in a cavity, the movement of a mass hanging from a spring, etc., then mathematics are close to over. He says, for instance and amongst many other things, that when mathematics -within a determined mathematical field- go too far away from the physical concepts that may have inspired them and grow out of themselves for too long, then only the fact of that task being in the hands of men with an extremely developed intuition may prevent that mathematical field from finally becoming barren. I found it very interesting how many hypocrisies Morries Kline pointed out in the modern style of teaching mathematics. Some teachers use intuitive, supposedly 'non-rigorous' methods to work out things IN their heads, but try to transmit it to the students in a modern, 'rigorous', and absolutely non-pedagogical way. A double language that usually will have coupled with it an equally twofold moral -hypocrisy-. They think clear in their heads but offer muddy explanations to the students, demoralizing them and making it look as if perfectly assembled and refined mathematical ideas were actually coming out of their heads instantly. It is all vane presumption that does not good to actual teaching. The final victims are the students. This may be one of the most important factors in accounting for the superiority of Japanese students over American ones in several math contests and comparisons held through the years. I too think that intuition is paramount in math and that 'rigorousness' is secondary when confronted to that. Think of Thales from Miletus, Aristarchus, Archimedes, etc. They lacked a good mathematical formalism, they did not even have a positional number system, and even so they worked wonders (Archimedes' laws of statics, mechanics, etc.) and realized by pure intuition things that were only rediscovered more than two thousand years later (say for instance that the planets all revolve round the sun, that stars are other, distant suns, etc.). Nowadays, on the contrary, we seem to have too much formalism, even loads of it, but not that great intuition of the ancient scientists. Sometimes I wonder what Newton or Archimedes would be able to do if they had a period of updating, a Pentium and some other of the resources we have today. Ha ha ha! Kurt Gödel also said once that whereas along the last three centuries abstract mathematics have experienced huge progress, the solution of complicated numerical problems that can be stated in a handful of symbols of elementary arithmetics (consider for example Goldbach's conjecture, or the now 'solved' Fermat's Last theorem)is very backward. This too comes to support the comments made above. I definitely think that Mr Kline's criticisms must be taken very seriously, at least for the sake of giving children a better, or at least decent, mathematical upbringing, instead of killing vocations even before they are born. He speaks along the book with astounding clarity and honesty. Somehow he looks like the character in that telltale, saying that the king is naked whilst almost everybody else -blind people obviously would not see it- too realizes it but does not dare claiming it out 'because no one else does', 'because it goes against tradition', out of cowardice or due to any other obscure reasons. For the good of future generations, for a better education, this book should be read by every physicist and specially by every mathematician, let alone those mathematicians who also have to 'teach' their discipline. I would have given this book six stars if it were possible. Pity that it is currently out of print. I went trough hell to find a copy, but all the pains were worthy by far. Definitely a must for any mathematics library.
8 of 9 people found the following review helpful
HASH(0x9cfc1684) out of 5 stars Sagacious Words 16 July 2007
By James G. Poulos - Published on Amazon.com
Format: Hardcover Verified Purchase
I started to underline what I thought were memorable parts of the book then realized that I would be underlining the entire book. It is a cogent and viciously sharp reading of the problems the author saw in 1972 that has been exacerbated up to today. It is must for teachers and administrators if there is going to be any profound changes in education. The frustration level is so high in education today there might be hope if the people in the position to make changes will listen, unlike in '1972'. It would be irresponsible to the educators profession and the students who are served not to read this book.
6 of 7 people found the following review helpful
HASH(0x9d35536c) out of 5 stars Why Johnny Can't Add 29 Mar. 2011
By Sam Adams - Published on Amazon.com
Format: Paperback
In this short book from 1973, Morris Kline argues against the "new math" curriculum for the teaching of mathematics from elementary school through high school that arose in the late 1950s, became pervasive in the 1960s, and was still current in 1973. Kline believes the approach taken in this curriculum was inappropriately abstract for youth, who cannot have the sophisticated intellectual concerns for axiomatic foundations, structural explication, and rigorous proof that a professional mathematician has. He derides the emphasis on pure mathematics and the demotion or exclusion of teaching how mathematics arises in the service of practical and scientific needs. Research mathematicians, Kline believes, typically don't have a clue how to teach mathematics to elementary and secondary school students, and it was primarily such mathematicians who designed the "new math" curriculum.

"It is our contention that understanding is achieved intuitively and that the logical presentation is at best a subordinate and supplementary aid to learning and at worst a decided obstacle. Hence, instead of presenting mathematics as rigorously as possible one should present it as intuitively as possible. ... So far as understanding is concerned the use of logic in place of intuition amounts, in the words of the philosopher Arthur Schopenhauer to cutting off one's legs in order to walk on crutches." (160)

"Rigor will not refine an intuition that has not been allowed to function freely. The student must experience the gradual passage from what he regards as obvious to the not-so-obvious and to the need for a fuller proof. He will discover the need for rigor rather than have it imposed upon him.

"This approach to rigor is more than a pedagogical concession. If one wished to teach how mathematics developed and how mathematicians think, then the gradual imposition of rigor is precisely what does take place." (163)

"The mind is not a vessel to be filled but a fire to be kindled." - Plutarch
3 of 3 people found the following review helpful
HASH(0x9d12de64) out of 5 stars Answer: Pie in the Sky 30 Jun. 2007
By A Reader - Published on Amazon.com
Format: Hardcover
According to this text, modern academic math movements ignore the authority of nature: mathematics' time-tested problem source from angles (physics from time immemorial) to integrals (newtonian physics) to hypernumbers (mechanics) to spacetime manifolds (relativity). Instead of nature, these movements have, by and large, used "pie in the sky" (p. 8) motivations that appear appropriate only for a small community of graduate mathematics students and their professors.

These are myths created by and for modern mathematical researchers. And these myths, which consist mostly of logic, are those that even intellectual illuminaries such as Newton and Fourier had no need of, never used, and never bothered to develop nor understand.

From page 38: "Apparently the intuitions of great men are more powerful than their logic."

From page 50: "...many perceptive mathematicians have spoken out against the logical approach. Descartes deprecated logic in rather severe language. 'I found that, as for Logic, its syllogisms and the majority of its other precepts are useful rather in the communication of what we already know or...in speaking without judgment about things of which one is ignorant, than in the investigation of the unknown."

From page 55-56: "Henri Poincare [writes]. '[The student] will think that the science of mathematics is only an arbitrary accumulation of useless subtleties; either he will be disgusted with it or he will amuse himself with it as a game and arrive at a state of mind analogous to that of the Greek sophists.'"

From the bottom of page 56: "Many teachers might retort that the student has already learned the intuitive facts about the number system and is now ready for the appreciation of the rigorous version, which exemplifies mathematics. If the student really understands the number system intuitively, the logical development will not only NOT enhance his understanding but will destroy it."

From page 59: "It is easier to incorporate sophistication in trivial matters than to give clear intuitive presentations of the more difficult ideas. Certainly much of the rigor in modern texts was inserted by limited men who sought to conceal their own shallowness by a facade of profundity and by pedants who masked their pedantry under the guise of rigor. One can rightly accuse them of pseudo-sophistication."

From chapter 8: "The symbolic logic does not control or direct the thinking...Not only is symbolic logic not used by most mathematicians but those who do use it do their effective thinking in common language."

From page 99: "Psychologically the teaching of abstractions first is all wrong. Indeed, a thorough understanding of the concrete must precede the abstract."

From page 128: "These men [college mathematics professors and Ph.D.'s in mathematics] do not know even freshman physics nor have they any desire to know it."

From page 130: "The professional purists, representing the spirit of the fragmented research-oriented university, got hold of the curriculum reform and, by their diligence and aggressiveness, created puristic monsters."

From page 133: "Mathematics is likely to attract those who do not feel competent to deal with people, those who shy away from the problems of the world and even consciously recognize their inability to deal with such problems. Mathematics can be a refuge. ...This appraisal of mathematicians is terribly negative, but it seems necessary to discredit the belief that mathematics professors are infallible and a truly superior group."

From page 137: "[The Revolution in School Mathematics] implied that administrators who failed to adopt the reforms were guilty of indeifference or inactivity. But in 1961 this country had had very little experience with the new curricula. A booklet championing and advocating them at that time can be fairly accused of propaganda."
4 of 5 people found the following review helpful
HASH(0x9cfc1a8c) out of 5 stars The Failure of a Flawed Policy 9 Dec. 2012
By Acute Observer - Published on Amazon.com
Format: Paperback
Why Johnny Can't Add, Morris Kline

Professor Morris Kline studied at New York University and received his Ph.D. in mathematics. He was the Director of Electronics Research at NYU's Courant Institute of Mathematical Sciences. This 1973 book is a post mortem on the "New Math" introduced in the 1960s. The `Preface' said the "money, time, energy, and thought expended in this program have been considerable, even enormous". Given the importance of mathematics, has the "New Math" actually improved the teaching of the students? Education is too important a subject to be left to professors of mathematics (paraphrasing Clemenceau). Professor Kline notes the absence of any firm evidence that "New Math" produced genuine improvements. There is a `Table of Contents' for its eleven chapters and 170 pages, a `Bibliography', but no Index.

Chapter 1 illustrates the problems of modern mathematics. Are those questions designed to confuse and stupefy most students? Chapter 2 describes the traditional curriculum. Algebra, geometry, and trigonometry have little practical use for most adults. [So too the plays of Shakespeare?] Perhaps the value of mathematics is in its description of reality, "training the mind" (p.9). The examples about reasoning are best done in a course in Classical Logic. Analyzing claims and statements are part of adult life. Chapter 3 explains the origins of "New Math" (p.15). Mathematics was disliked and dreaded by students, most adults retained little of what they were taught. "Sputnik" launched the funding of "New Math". It assumed that students would learn far more than they had in the past (p.19). Education must be appropriate to the needs of the students (p.22).

Chapter 4 discusses the Deductive Approach, used to teach geometry. It has not been used to teach arithmetic, algebra, and trigonometry (p.24). Does it lead to understanding mathematics (p.33)? Kline says no. The logical foundations of mathematics did not appear until the late 19th century (p.38). Asking students to cite axioms is less important that using them (p.45). Should we think about what we are doing (p.46)? Kline criticizes the use of deductive logic (p.49). Real decisions in life call for judgment. That what is obviously true does not need rigorous proof (Chapter 5). No proof of mathematics was needed in earlier times (p.57). Chapter 6 discusses the words and definitions used in mathematics (p.61). Several hundred terms are used (p.66)! This excessive terminology was criticized (p.68), it is no substitute for substance (p.69). This elaborate notation is never used in the real world (p.72)! Mathematics of mathematics' sake is criticized (Chapter 7). Mathematics exists to understand the physical, social, and economic worlds.

Chapter 8 analyzes the contents of "New Math". Much of it contains what was known before the 18th century. Set theory is of no use in learning elementary mathematics. Kline also criticizes othe topics. "The more general the mathematical concept, the emptier it is." The "New Math" program was adopted without any evidence of its benefits (Chapter 9)! Tests do not measure a student's understanding of concepts but memorization. Chapter 10 discusses the trends in mathematics and its divorce from the physical world. This is the most critical evaluation of "New Math". The mathematics professors who designed "New Math" did not understand teaching in grammar and high schools. Chapter 11 lists the proper directions for teaching mathematics: to help understand the physical world. Can children learn mathematics better when it has a practical benefit? Teaching should meet the needs of the students. Mathematics is a mean to an end. Kline recommends teaching intuitively. Advanced subjects (like set theory) do not belong in elementary or high schools. The real problem is the proper education of teachers. This doesn't exist now, universities do not teach this.

There was an anti-democratic reaction in America in the 1940s. This created policies that went against the interests of the people. This book does not name the powers that forced these changes. [Locally, this was followed by the formation of teacher's unions at the schools.]
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