By David Bressoud @dbressoud
As of 2024, new Launchings columns appear on the third Tuesday of the month.
FDWK refers to Ross Finney, Frank Demana, Bert Waits, and Dan Kennedy, the team of authors who for many years and multiple editions updated Calculus: Graphical, Numerical, Algebraic, one of the most popular textbooks for the College Board’s Advanced Placement Calculus.
Last month I talked about joining FDWK in 2013. This month I want to talk about changes that I helped initiate for the opening chapters of the 6th edition, with further changes for the 7th edition that will appear next year.
For me, there are two poor ways to begin a first course on calculus. These are either an
extended review of precalculus material or launching into an extended study of limits before
giving students any idea of what calculus is all about. Older editions of FDWK recommended 11
lessons on precalculus topics followed by 11 lessons on limits. Those following these guidelines
would spend four and half weeks before giving students any idea of what calculus is really about
or why they might want to study it.
I agree with the seven principles of teaching described by David Perkins in his book Making Learning Whole. He uses baseball as an analogy which shapes the language he uses. His first principle is Play the Whole Game. I like his description of what this means.
“We can ask ourselves when we begin to learn anything, do we engage some accessible version of the whole game early and often? When we do, we get what might be called a ‘threshold experience,’ a learning experience that gets us past initial disorientation and into the game. From there it’s easier to move forward in a meaningful motivated way.
“Much of formal education is short on threshold experiences. It feels like learning the pieces of a picture puzzle that never gets put together, or learning about the puzzle without being able to touch the pieces. In contrast, getting some version of a whole game close to the beginning makes sense because it gives the enterprise more meaning. You may not do it very well, but at least you know what you’re doing and why you’re doing it.” (Perkins, p. 9)
To be clear, the College Board says nothing about reviewing precalculus before starting to teach calculus, nor do they ask any questions that are purely precalculus on their calculus exams. But as I’ve heard more than one faculty member say, “No one fails calculus because they didn’t know calculus. They failed because they didn’t know precalculus.” This has led to a common misperception among instructors that they must drill precalculus material before launching into calculus.
To correct the impression that time must be spent on precalculus before starting calculus, the 6th edition relabeled the precalculus chapter, which had been Chapter 1, as Chapter 0. We will go one step further in the 7th edition. We suggest that rather than starting with the precalculus chapter, instructors may want to insert relevant topics from precalculus immediately preceding the sections that use those topics. This follows the advice and usage of our new co-author, Mark Kiraly, a high school teacher with many years of experience teaching AP Calculus. We follow his suggestions for where these precalculus topics might be inserted. I know that many teachers already do this. But especially for teachers unaccustomed to teaching calculus, permission and guidance can be helpful.
The next thing that changed in the 6th edition was a new section between Chapters 0 and Chapter 1. We call it the Prologue: Foundations of Calculus. In less than four pages we run a fast overview of the two types of questions that calculus is designed to answer: finding the amount that has accumulated by a certain time given the rate at which it is accumulating and finding the rate at which something is accumulating given the amount that has accumulated by a given time. We then work through two examples: finding distance given velocity and finding velocity given distance as a function of time.
The first of these is integration, the second is differentiation. The order is intentional, harking to the point made in my book Calculus Reordered that integration, the process of accumulating small changes, is both much older and far more intuitive than finding rates of change. I would love to turn our text around and introduce integral calculus first, but that is only my personal preference.
In the 6th edition we downgraded the precalculus chapter and provided an overview of the “whole game.” But that still left a long Chapter 1 on limits. Limits are problematic. In my interviews with students, many expressed their confusion about limits. They know they have to learn techniques for determining limits, but they have little idea of why it is required. Many see it as a rite of passage that can be quickly forgotten once they start learning how to find derivatives and get into the real calculus. I would love to greatly downplay any discussion of limits. Many colleges and universities have done this.
The College Board’s syllabus for AB Calculus prescribes spending 22 to 23 class periods on limits. This is based on their survey of common practice at most universities. While we recommend devoting only half as many class periods, there are still 16 detailed limit topics that students must master and which will be tested on the exam. The AP Calculus syllabus has more topics on limits than in any other unit in the course. Among universities, there has been a general movement to decrease time spent studying limits. One can hope that as more departments of mathematics recognize the pointlessness of extensive treatment of limits, the College Board can ease off on what it prescribes.
FDWK still needs to cover all of the limited topics described in the College Board’s Course and Exam Description. In the 7th edition, we will seek to give more meaning to this chapter by beginning with an extended justification for limits. After a very short intro duction to average velocity, we ask,
“What about instantaneous velocity? Is it possible to talk about a car’s velocity at precisely 3pm? Velocity is change in distance divided by change in time. What happens when there is no change in time?”
To answer this, we turn to the calculus material developed by Thompson. Ashbrook, and Milner, Calculus: Newton, Leibniz, and Robinson meet Technology. We use their idea of looking at a picture of an SUV taken at a shutter speed of 1/1000 second and asking the question “Is it stopped or is it moving?” As we zoom in to a close-up, we can see that there is a little blurring, maybe about 1.5cm.1 The SUV must be moving at a speed of approximately 15 meters per second which is 54 km per hour or around 34 mph. We conclude by observing,
“The streaks are probably not exactly 1.5cm, and even the shutter speed is an approximation, but the more accurately we can bound the lengths of the streaks and the time of exposure, the more accurately we can estimate the velocity.”
Thompson, Ashbrook, and Milner conclude with the observation that “all motion is blurry.” As we then go on to explain, it is only in the idealized world of precise functions that one can determine an instantaneous rate of change. Calculus does not deal with reality. Instead, it provides us with precision tools that enable us to get answers that can come as close as we desire to the actual real-world values. With that glimpse into the big picture, we hope that students will find some meaning in the rest of this chapter.
Another subtle yet significant change to the 6th edition involved the introduction of integration. That will be the topic of my next column.
References
Bressoud, D.M. (2019), Calculus Reordered: A History of the Big Ideas. Princeton University Press. ISBN 978-0-691-18131-8
Perkins, D.N. (2009). Making Learning Whole: How seven principles of teaching can transform education. Jossey-Bass. ISBN 978-0-470-63371-7
Thompson, P.W., Ashbrook, M. and Milner, F. (2019). Calculus: Newton, Leibniz, and Robinson Meet Technology. https://patthompson.net/ThompsonCalc
1 See section 4.3 of Thompson, Ashbrook, and Milner, available at https://patthompson.net/ ThompsonCalc/section_4_3.html
David Bressoud is DeWitt Wallace Professor Emeritus at Macalester College and former Director of the Conference Board of the Mathematical Sciences. Information about him and his publications can be found at davidbressoud.org