Morris Kline (May 1, 1908 – June 10, 1992) was a professor of mathematics who wrote extensively on the history, philosophy, and teaching of that subject. He was also a popularizer of mathematics. He subsequently worked at NYU as an instructor until 1942. Although he was originally trained in topology and worked as an assistant to James Alexander, Kline turned his attention to differential equations and applied mathematics after being convinced by Richard Courant that the greatest contribution a mathematician could make to society was to bring about the understanding of the world.
Writings on the teaching of mathematics
Kline repeatedly stressed the need to teach the applications and usefulness of mathematics rather than expecting students to enjoy it for its own sake. He cautioned, however, that these applications must be carefully selected to suit the level of the course being taught and that at the introductory level, intuition, not rigor, should be the main focus.
He issued multiple objections of how mathematics was taught in 1956, 1966, and 1970, as well as many of the textbooks written during this era. For Kline, an appreciation for rigor took time to be developed and it was ill-advised to teach the abstract before the concrete. Kline criticized the authors of the New Math textbooks, not for their mathematical faculty, but rather their narrow approach to mathematics, and their limited understanding of pedagogy and educational psychology. Many other scholars were also critical of these reforms. Harry Schwartz wrote in his review of the book for The New York Times: "Its significance goes far beyond its immediate topic. It raises the broader issue of how, in field after field in American life, there come to be sudden fixations on supposed panaceas for perceived problems. All too often, however, these panaceas turn out to have unforeseen consequences as bad as or worse than the original difficulties that triggered their adoption." Kline countered by complaining that many of these reviewers did not read the book but only a few excerpts from The Mathematical Intelligencer and noted that he had received many complimentary letters from instructors who shared his opinions and who resented the relentless pressure to undertake research, which came at the cost of good teaching. Kline made frequent use of primary sources, especially in the later chapters.
Ivor Grattan-Guinness indicated that while the book had a number of missed opportunities and technical errors or misinterpretations, this was nevertheless a strong presentation which dedicated much space—about half the book—to developments after 1800, which was unusual at the time of publication. He opined that Kline was at his best when discussing Euclidean geometry, calculus, and the sociology of mathematics over the last three centuries, but faltered in complex variables, linear algebra, and numerical analysis. He omitted probability theory and statistics.
Carl Benjamin Boyer also praised Kline for a detailed discussion of more recent developments in mathematics. Boyer especially liked Kline's handling of non-Euclidean geometry, and for going over topics not commonly found in other books on the history of mathematics, such as the Mathieu functions and the Navier–Stokes equations.
Lester Paldy deemed the book appropriate for teachers of physics as it contained a substantial amount of information on how mathematics had been applied to study physical phenomena.
Writings on mathematical research
Kline urged that mathematical research concentrate on solving problems posed in other fields, such as physics and computer science. In his own words, "the greatest contribution mathematicians had made and should continue to make was to help man understand the world about him." William Barrett of The New York Times praised it as an "immensely readable" account of the decline of mathematics due to conflicting schools of thought.