The idea for this book began when I was tutoring undergraduates in calculus, first as an undergraduate at the Rochester Institute of Technology and then as a graduate student at Yale. I was struck by how many students came in with essentially the same question: “I don’t know how to answer this; should I do a derivative or an integral?” Even with very standard applications, the students I tutored often had no idea whether to take a derivative or an integral: they wouldn’t know what to do if they were given a velocity and asked to find a displacement or if they were given an expression representing population and asked to find a rate of growth. If I said “take a derivative” or “take an integral,” they would be off and running: they had memorized all the algebraic rules of calculus and could follow the rules almost perfectly; many knew the unit circle and even some trig identities; many had good algebra skills when it came to solving equations and finding roots. And yet, they had absolutely no idea what it all meant or why they were doing it.
Quite a few of these students, who had attended many different high schools, had already taken calculus in high school and were now taking it for the second time as undergraduates. They had worked hard on memorizing the rules of both algebra and calculus. They lit up with excitement when I laid out the basic ideas of calculus: what rate of change is and how it relates to slope; what accumulation is, and how it relates to area; the meaning of a limit in this context; and the difference between average and instantaneous rate of change. As one student asked when I laid out some very basic principles, “why didn’t anyone tell me this sooner?”
I know that many calculus teachers like to say that students really struggle with the algebra and not the calculus. But what I saw, without a doubt, was students struggling with calculus. I wondered: how could there be such deep and consistent conceptual misunderstandings among students who had invested so much time and energy into calculus, who were conscientious and eager to learn, and who had had a wide range of prior calculus experiences? I found answers to these questions in their calculus texts and syllabi.
Why don’t students learn when to use a derivative and when to use an integral? Because they spend half the class doing derivatives and another half doing integrals. In each of these two halves they narrowly focus on one thing, and the answer to every question involves that thing. Very little of a standard calculus class teaches students to actively think about when and if a derivative or integral is applicable to a given problem.
Why don’t students see the real-world applications of derivatives? Because they go through a unit on limits, a unit on the definition of the derivative, and a unit on derivative rules before getting to a unit involving applications. If a student is confused by limits, they may give up on really trying to understand the class months before they ever see the sort of real-world application that would help them grasp the ideas and purpose of calculus.
Why don’t students appreciate the power and beauty of the Fundamental Theorem of Calculus? Because this theorem is not treated as fundamental at all: it is relegated to a single section and primarily used to transform a list of derivative rules into a list of integral rules. Rather than being a guiding light for the whole class, the Fundamental Theorem becomes little more than another source of rules for diligent students to memorize and follow.
My book, my problems and this website are all an attempt to make this better. I start teaching derivatives and integrals together, from the beginning, so that students learn think about what they mean and when they apply. I start with numerical examples, so that students can gain intuition for many of the big ideas before having to engage with how these ideas work when limits are involved. I teach the Fundamental Theorem early and have students use it throughout the year: it truly is “fundamental” to my class. I want to model the habits of a good mathematician, I want to center the student experience and I want students to think about the delicate dance between theory and application, every day, all year.
If you like this and want a copy of my whole book, please click here