Morris Kline’s book, Why Johnny Can’t Add, is an incisive critique of the pedagogy of the New Math movement. New Math emphasized imposing mathematics upon children in an abstract, axiomatic style. It was widely regarded as a failure and has been dead for almost half a century.
It is now widely accepted that students should learn their multiplication tables. For all the complaints about Common Core mathematics, the Common Core has required such memorization from its inception in 2010.
Kline’s criticisms of the New Math are still frequently mentioned today. Yet the second chapter of Why Johnny Can’t Add is a fundamental attack on the traditional math curriculum from pre-algebra onward. (By “traditional math curriculum”, Kline refers to mathematics education in the United States prior to the 1950s.) It has largely been forgotten. I summarize its contents here without evaluation.
Arithmetic
After students learn to add, multiply, and divide manageable natural numbers, they turn to more complex operations. Kline illustrates this transition with the addition of fractions:
One can understand why simply adding the numerators and denominators will not work, how adding fractions differs from multiplying them, and how to derive the answer from first principles; alternatively, one can merely learn a technique that produces the correct answer.
A good teacher would no doubt do his best to help students grasp the rationale of the process, but on the whole the traditional curriculum does not pay much attention to understanding. It relies on drill to get the students to do the process readily. (p. 5)
Algebra
As the mathematics becomes more advanced, it increasingly matters whether students merely become competent mechanical calculators and information-retrieval systems or understand what they are doing. When students reach polynomials, they again perform addition, multiplication, and division. But because students memorize techniques without understanding the underlying ideas, they come to see mathematics as a grab bag of unrelated methods.
They are like pages torn from a hundred different books, no one of which conveys the life, meaning and spirit of mathematics. The presentation of algebra begins nowhere and ends nowhere. (p. 6)
Geometry
When the time comes for geometry, students abruptly encounter a new practice: constructing proofs. Proofs are central to mathematics:
The concept of proof is fundamental in mathematics, and so in geometry the students have the opportunity to learn one of the great features of the subject. (p. 6)
Unfortunately, the genuine practice of proving theorems receives little attention. Textbooks give students little sense of what it is like to struggle with a problem or discover a solution. Instead, they present only the end product—the written proof—and teach a few rote techniques for passing exams.
Hence the student cannot see the rationale and he does the same thing in geometry he does in algebra. He memorizes the proof. (p. 6)
As they move from geometry to intermediate algebra, bright students may wonder why proofs appear in the former but not the latter. Yet this line of thought—which threatens to lead to an authentic insight into mathematics—is quickly extinguished when “proof is again abandoned in favor of technique” (p. 6).
The Gravest Defect
Kline has a number of other criticisms. For example, students are often taught falsehoods such as
- cannot be factored. (It can be factored over the real numbers using irrational coefficients.)
- cannot be factored. (It can be factored over the complex numbers.)
But the “gravest defect” of the traditional mathematics curriculum is its “lack of motivation” (p. 7). Every teacher has heard the question, “Why do we need to learn this?” Kline thinks this is the essential question, and that the traditional mathematics curriculum has no good answer.
Motivation means more than a psychological stimulus. Genuine motivation also supplies insight into the very meaning of mathematics. (p. 12)
It is not enough to say that mathematics will be useful later in life. Unless one teaches mathematics or works as a scientist or engineer, one is unlikely to use most of what one learns in mathematics class. Nor can one defend mathematics on the ground that it trains the mind when “the learning is almost always memorization” (p. 6).
For Kline, the traditional mathematics curriculum is a failure and an absurdity. It does not teach mathematics at all. It is a sort of pretense or play-acting.
The failure to present the meaning of mathematics is analogous to teaching students how to read musical notation without allowing them to play the music. (p. 23)
Kline concludes that reform is needed, but that the New Math was not an adequate solution.
All citations to Why Johnny Can’t Add are from the 1973 edition. The featured image is a painting by Nikolai Petrovich Bogdanov-Belsky in the State Tretyakov Gallery, Moscow.
