MITCH STOKES is a Senior Fellow of Philosophy at New St. Andrews College. He received his Ph.D. in philosophy from Notre Dame and an M.A. in religion from Yale. He also holds an M.S. in mechanical engineering and, prior to his teaching career, worked for an international engineering firm where he earned five patents in aeroderivative gas turbine technology. In addition to biographies of Newton and Galileo, his books include A Shot of Faith (to the Head): Be a Confident Believer in an Age of Cranky Atheists (Thomas Nelson), and How to Be an Atheist: Why Many Skeptics Aren’t Skeptical Enough (Crossway).
Format: Hardback (Crown Quartro: 7.44 x 9.68), full-color text.
Mitch Stokes, Calculus for Everyone: Understanding the Mathematics of Change
Text copyright © 2020 by Mitch Stokes.
Illustrations by Summer Stokes, summerstokes.com.
Cover design by Rachel Rosales, orangepealdesign.com.
Published by Roman Roads Press
121 E. 3rd Street, Moscow, Idaho 83843
The only prerequisite for this course is a very basic understanding of Algebra 1. The student should, for example, be familiar with polynomials and be able to manipulate such polynomials (e.g., by combining like terms and manipulating exponents). The focus of the book is on the concepts of calculus proper and not on calculus’s further application to increasingly complicated functions, such as rational, trigonometric, exponential, or logarithmic functions. By keeping to polynomials throughout the course (or, more accurately, to power functions), the student will better identify what is the “calculus part” of calculus. Once the concepts of calculus have been mastered, then students can—if they choose—go on to apply those concepts in ever more complicated ways. By better understanding what calculus is not, the student will better understand what it is.
2 QUIZZES AND EXAMS
I recommend that you use the study questions and exercises in the back of the book as a pool of potential quiz and exam questions. These should be entirely sufficient, though teachers should modify and supplement them as they see fit. But such supplementation isn’t necessary.
3 SCHEDULING OPTIONS
Although the schedule and pace are entirely up to you, the most obvious choice is between either a single-semester or a year-long course. The precise schedule will depend on the specific student(s) and school (which includes homeschools). A teacher may wish to move leisurely through the material over the entire school year to allow the student to digest the material and let it sink in and include time for review. On the other hand, a faster-paced semester-long course may be appropriate for students who have a strong background in mathematics. And in either case, the course would build a strong conceptual foundation for a further, more standard calculus course if you should so choose.
4 READ THE BOOK
Unfortunately, most students do not really read their math textbook. Instead, they merely use it as a kind of reference manual to look up specific concepts and methods when they are stuck on a homework problem. But students should read the text on their own—either before or after the teacher’s presentation of the concepts (or both). One of the goals of this book is to give students practice reading a math book, which is, importantly, unlike reading other types of books.
Of course, on the one hand, a math book should be read like any other book, from front to back, chapter by chapter, line by line. But in other ways, reading math is unique. It will take time for students to adjust to it, and it will take work once they do. A math book should not only be read line by line but word by word and symbol by symbol. Tiny details matter in mathematics, and one small slip—whether it be in writing or reading—can throw everything off course. Remember: Safety First! Also, even if you have read everything, there will be times when you don’t understand a particular sentence or step. When this happens, don’t go on to the next sentence or step. Read it again, slowly turning it over and over in your mind. Indeed, you may have to read the same sentence over and over. Or you may have to return to the beginning of the section or chapter and begin again. In fact, the great Sir Isaac Newton himself sometimes had to do this. For example, when reading through Rene Descartes’ Geometry, Newton found himself getting stymied almost immediately. He then returned to the beginning of the book and reread until he got stuck again, this time a few pages farther on. He repeated this process until he finished the book. You won’t find yourself in such an extreme situation, but if the top mathematician in the world—the man who invented calculus—had to do this, you shouldn’t be surprised if you have to reread sentences, sections, or chapters. And you should probably be surprised if you don’t.
Mathematics obviously has an additional component beyond ordinary English: mathematical symbols. And when it comes to reading mathematical symbols, there are times when you simply have to stare at the formula or step until you recognize what is happening. And of course, there will be times when no amount of staring or reading helps (whether because the author didn’t present it clearly enough, or he made a mistake that the editors didn’t catch, or you simply don’t have the ability to deal with it on your own). This, again, will happen to everyone at some point.
This all may sound horrible, but it’s not nearly so bad if you know ahead of time that you’ll have to do this. And the benefits of reading this closely and carefully are general and go far beyond the math course itself. In addition to careful thinking, it improves overall attention to detail and observational skills, both of which are important human virtues that apply to nearly all of life. Also, once you realize that reading mathematics takes this much work just about everyone, you’ll be less likely to pigeonhole yourself as someone who is “not a math person.”
5 DOING THE PROBLEMS: STUDY QUESTIONS AND EXERCISES
In this book, there are two types of problems for nearly all the chapters: study questions and exercises. The study questions are easy to answer and are intended to help you identify and memorize important facts and concepts. The exercises, on the other hand, are traditional math problems and typically require much more work. The answers to the study questions are in the chapter text itself, while the Solutions to the exercises are provided in the Solutions manual, which is separate from the main text.
– Study Questions
Again, the study questions will draw your attention to some of the facts (or “grammar”) of the chapter. They are to ensure that you notice the highlights in each chapter. Nearly all the answers can be found explicitly in the chapter’s text, and the questions are usually asked in the order the text presents the answers. These are entirely open-book questions (though, teachers and parents, see below). Moreover, the questions sometimes overlap. This overlap is designed to help you understand the same idea from different angles.
Let me point out that we won’t begin doing math problems in the traditional sense until later chapters. (Again, the traditional-style problems are called exercises.) That said, the study questions are, strictly speaking, also math problems since they are ultimately about facts and concepts of mathematics. And if you understand these questions, you’ll understand the material far better than if you had done only the exercises; indeed, they will help you understand the exercises. The distinction between study questions and exercises is entirely artificial.
You should neatly write out (or type) your answers to the study questions and save these answers in a dedicated folder or notebook—a central, easy-to-access location for you to readily review throughout the course.
To teachers and parents: these study questions should also be used as quiz and exam questions. You should also orally ask the student(s) these questions as a means of evaluating their understanding of the chapter and to instigate class discussion. It would also be a great idea to use them as review. You might also want to have students turn in their answers just to keep them accountable, but without carefully checking their answers, since the answers are right there in the text.
Again, neatly write out all your work for these exercises and keep them in your course folder. Having them all in one place will to make it easier to study for quizzes and exams.
By the way, it is entirely appropriate—indeed, necessary—to do the exact same problems more than once, even repeatedly (and even in a single sitting). Repetition is one of the keys to success in mathematics. Review, which is just repetition over a longer period of time, is also crucial, so teachers should assign previous exercises (and study questions) to keep important material fresh in the student’s mind.
Another key to success in mathematics is neatness and organization. As a general rule, you should carefully write out all of the steps. Admittedly, this is a hassle, but doing the steps merely in your head is dangerous. Again, in mathematics our motto is “Safety First!” Also, do not write too small—and don’t cram your work together. There are no points for saving space, and such savings will likely cost you in the form of simple mistakes. And speaking of simple mistakes, everyone makes them in mathematics, even the experts.
– Mitch Stokes, 2020
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