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The Six Pillars of Calculus: Biology Edition

Lorenzo Sadun
Publication Date: 
Number of Pages: 
Pure and Applied Undergraduate Texts
[Reviewed by
Russell Jay Hendel
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This book, a first year undergraduate text for Calculus I, is recommended for instructors who wish to remedy poor student understanding, performance, and enjoyment of calculus, provided they are willing to make changes in their curriculum and the way they deliver calculus instruction. After stating the sources of the book, the rest of the review will detail the innovations and changes introduced by it.
A primary source for this book is the 1994 textbook Calculus in Context (by J. Callahan, D. Cox, K. Hoffman, D. O’Shea, H. Pollatsek, L. Senechal) connected witth the Five Colleges Calculus Project. The book and Project stress the role of calculus as a ”language and a tool for exploring the whole fabric of science with differential equations as a unifying theme.” This project is based on a National Science Foundation grant which required that the results be made available for others to adapt. Six Pillars further benefited from Eric Stade, who based on this NSF project, created a calculus course at the University of Colorado aimed at Biology students.
A good example of the flavor of the book is contained in the second chapter which presents the SIR (Susceptible, Infected, Recovered) model for spread of disease and particularly the spread of COVID-19. The model is approached without calculus. Instead, equations are developed that model the spread of disease.  In attempting to solve these equations, it is natural to take limits, and develop the idea of derivatives and differential equations. The approach is ”problem → equations → calculus definitions and theorems” rather than the traditional ”calculus definitions and theorems → illustrative problems-→ equations and formulas.” The formal definition of limit and derivative are in fact delayed to the next 2 chapters which revisit the SIR
model and provide computer approaches to the problem using MATLAB. Finally, in Chapter 5 a ”traditional” Calculus chapter is presented presenting the formulas for derivatives of polynomials and trigonometric functions and their applications to graphing. Throughout these chapters real-world problems like SIR are presented.
This is about half the book. Chapter 6 introduces the transcendental calclulus fucntions (exponential and logarithm) by regarding them as functions that help solve differential equations (again: The book theme of ”real-world problems → models → formal calculus.”)  Chapters 7-9 do for integration what was previously done for differentiation. It is the last chapter, 9, that
is the traditional calculus chapter on techniques of integration. Chapter 7 simply introduces a variety of natural sum processes which we are told in Chapter 8 can be done by the integral.
Chapters 10 and 11 introduce functions of several variables and series. 
Since the book spends considerable time on real world problems and natural attempts to solve them without yet knowing calculus, in order to adequately cover this material in one semester, the book is light on continuity, differentiability, and omits l’Hoptial’s rule, implicit differentiation, logarithmic differentiation, partial fractions, and trigonometric substitution.
Should you then as in instructor use this book? If you want students to enjoy the calculus experience, understand and appreciate calculus as a tool to describe and model the real world, if the other departments and upper-level courses you service don’t mind certain omissions, and finally if your chair and dean are supportive, I would recommend going for it. The book is written for students who intend to go into biology.  Other books are also being developed by the author for students interested in other fields of science.
Russell Jay Hendel, holds a Ph.D. in theoretical mathematics and an Associateship from the Society of Actuaries. He teaches at Towson University. His interests include discrete number
theory, graph theory, applications of technology to education, problem writing, theory of pedagogy, actuarial science, and the interaction between mathematics, art, and poetry, and biblical literary exegesis.