The Calculus: A Genetic Approach

Book Preface

Otto Toeplitz is best known for his contributions to mathematics, but he was also an avid student of its history. He understood how useful this history could be in informing and shaping the pedagogy of mathematics. This book, the first part of an uncompleted manuscript, presents his vision of an historically informed pedagogy for the teaching of calculus. Though written in the 1930s, it has much to tell us today about how we might-even how we should-teach calculus.

We live in an age of a great democratization of calculus. A course once reserved for an elite few is now moving into the standard college preparatory curriculum. This began in the 1950s, but the movement has accelerated in the past few decades as knowledge of calculus has come to be viewed as a prerequisite for admission to the best colleges and universities, almost irrespective of the field that will be studied. The pressures and opportunities created by this popularization have resulted in two significant movements that have shaped our current calculus curriculum, the New Math of the 1950s and ’60s, and the Calculus Reform movement of the 1980s and ’90s. These movements took the curriculum in very different directions.

The New Math was created in response to the explosion in demand for scientists and engineers in the years following World War II. To prepare these students for advanced mathematics, the curriculum shifted to focus on abstraction and rigor. This is the period in which Riemann’s definition of the integral entered the mainstream calculus curriculum, a curriculum that adopted many of the standards of rigor that had been developed in the nineteenth century as mathematicians extricated themselves from the morass of apparent contradictions revealed by the introduction of Fourier series.

One of the more reasoned responses to the New Math was a collective statement by Lipman Bers, Morris Kline, George P6lya, and Max Schiffer, cosigned by many others, that was published in The American Mathematical Monthly in 1962.1 In this letter, they called for the use of the “genetic method:” “The best way to guide the mental development of the individual is to let him retrace the mental development of the race-retrace its great lines, of course, and not the thousand errors of detail.” I cannot believe it was a coincidence that one year later the University of Chicago Press published the first American edition of The Calculus: A Genetic Approach.

The Calculus Reform movement of the 1980s was born from the observation that too many students were confused and overwhelmed by an approach to calculus that was still rooted in the rigor of the 1950s and ’60s. In my experience, most calculus students genuinely want to understand the subject. But as students encounter concepts that do not make sense to them and as they become confused, they fall back on memorization. These students then emerge from the study of calculus with nothing more than a capacity to handle its procedures and algorithms, with little awareness of its ideas or the range of its uses. In the 1980s, departments of mathematics were facing criticism from other departments, especially departments in engineering, that we were failing too many of their students, and those we certified as knowing calculus in fact had no idea how to apply its concepts in other classes. The Calculus Reform movement tried to achieve two goals: to create student awareness of and ability to work directly with the concepts of calculus, and to increase the accessibility of calculus, to make it easier for more students to learn what they would need as they moved into subsequent coursework and careers. It created its own backlash. The argument commonly given against its innovations was that it weakened the teaching of calculus, but much of the resistance came from the fact that it required more effort to teach calculus in ways that improve both accessibility and understanding.