Why Johnny Can't Add: The Failure of the New Math. Paperback – 1 Feb 1974
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Most Helpful Customer Reviews on Amazon.com (beta)
"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
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."
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.]