Who Needs Calculus?
- 2010 4 Jun
I recently explained how parents can put their children on a math track that will lead to at least one year of calculus in high school.1 I ignored the vital question, "Why bother?" In what follows, I'll give you some reasons to bother—reasons, that is, why calculus should be part of your plans for your child.
You may never have studied calculus yourself. You may think of it as a subject only for eggheads, disconnected from real life. I understand why you might think that way and certainly don't mean to suggest that anyone who doesn't study calculus is a failure or a bad person! But I would urge you to consider the following arguments carefully, to make sure you reach an informed decision. The stakes may be higher than you realize.
What in the World Is Calculus, Anyway?
What is calculus?2 In a nutshell, it is the collection of mathematical tools used for studying quantities that change and their rates of change. Where are such quantities in the world? Everywhere. For example:
- The acceleration (rate of change in velocity) of a space shuttle
- The temperature of a pot on a stove—or a nuclear reactor's core or a computer's processor chip—as it heats up or cools down
- The speed of blood flow through a coronary artery
- The electrical current through a light bulb or a circuit board
- Wind speeds in a hurricane
- The loads on a structure—whether house, bridge, or baseball stadium—after a heavy snowfall
- A factory's production rate, defect rate, rate of consumption of raw materials, rate of expenditure on labor, and financing rate on inventory
- Drug absorption rates
- Disease transmission rates
- Rate of curvature of a highway exit or of a letter in a font
Such phenomena are so widespread, in fact, that calculus is easily the most widely applied branch of math. So, why do I think you should include calculus in your child's educational plans?
Reason #1: Options
Many fields are closed to students who can't pass calculus at or near the beginning of their college career. In short, if you're not up to taking calculus, you can rule out any career in engineering, computer science, the natural sciences, economics, finance, or mathematics.
Not only will you have fewer college majors to choose from, but you'll also have fewer colleges to choose from, and possibly less financial aid. The fact that a student is taking calculus in his senior year in high school is a big plus for college admissions and merit-based aid. Having one year of calculus success under your belt and being enrolled in Calculus II when you apply to college in your senior year is an even bigger plus.
Reason #2: Guidance
Calculus is the first taste of real mathematics that most students encounter in their education. As a result, many students who "like math" all the way through grade school and into high school suddenly find themselves disliking calculus. "But this is so different! I thought I liked math." Conversely, some students who didn't "like math" before calculus suddenly find the subject taking on new interest.
This has an important practical consequence. No one should decide on a career in a math-intensive field, or rule out such a career, based on his experience with math before he gets to calculus. And that means that the sooner the student reaches calculus, the better, as it can be such a crucial guidepost concerning his future.
Reason #3: A Head Start
The young person who takes calculus in high school can sometimes place out of having to take the first semester or two of calculus in college. For the student who plans to major in a calculus-intensive field, this can help him get into advanced coursework much more quickly than would otherwise be possible. In such a field, the first semesters of calculus are prerequisites for virtually all higher-level courses.
For many students, a more realistic goal is just to be able to pass calculus when he meets it in college. Why make the often difficult transition to calculus in college, when you could make it in high school, without all the other stresses and distractions of your first year in college? The student who has had a year or two of calculus in high school is much more likely to be able to pass when he has to take it (again) in college.
Reason #4: You Can Do It
It is achievable for the vast majority of students, if you get them started early enough with an incremental, skills-oriented, California Standards-compliant math curriculum.4
Reason #5: God's Glory
God gave me an unforgettable experience once when I was looking out of my window on a flight from Japan to New Jersey. The aircraft was in sunshine, but the sun hadn't yet risen high enough to illuminate the ground. Below me, I could see a few small clouds, brightly lit, and below that, nothing but inky darkness, impenetrable to me up in the bright sunlight.
Then, as I watched, the sun rose on the world below me. We were over the Alaska Range. When the sunlight touched the highest peak, I saw magnificent mountains appear instantaneously out of the dark void. The peaks and ridges were brilliant with golden light, and the valleys, though still in shadow, were dimly visible by the reflected light. In a fraction of a second, the light raced off like fire through dry grass to the northwestern horizon, illuminating more ridges and peaks, bringing a snowy, mountainous world into being out of darkness.
After I resumed breathing, I sat there in awe, looking at those mountains, meditating on God's Word, and thanking Him for letting me see such a sight.
And that was the second most beautiful thing that I have ever seen in God's universe. What was the most beautiful? Calculus. The tools of calculus, at their deepest level, are all aimed at enabling us finite human beings to explore the fringes of infinity, and the views they give us are breathtaking beyond words. I can honestly say that I've never so much felt the greatness of God as when meditating on mathematics, especially calculus. And I've never so clearly felt the reality that we humans are God's image-bearers, made for communion with Him and some measure of knowledge of Him, yet never His equals, always infinitely below Him in wisdom and power.
One example will have to do. The hard parts of calculus are needed because of certain strange characteristics of the real numbers.5 One of these characteristics is that the real numbers are said to be "non-denumerably infinite." What does that mean?
The natural numbers can be listed exhaustively in a sequence, like the obvious one that starts 1, 2, 3 . . . . God, using an infinitely long piece of paper, could list all of them in that order (or various other orders). The same is true of the integers and even the rational numbers, but not the real numbers. There's a proof that looks at the real numbers between 0 and 1 and shows that there cannot exist a sequential list of all of them, in any order whatsoever. That means that not even God, with an infinitely long piece of paper, can list all the real numbers between 0 and 1—much less all of the real numbers. I've gotten scandalized looks from students when I've said that! But then I add the punch line: God doesn't need to put all of them in a sequential list. He can just hold all of them in His mind at once. Wow! And that's just one small example of what I mean.
Now, granted, not everyone is going to be moved by the beauty and power of mathematics to glorify our Creator God the way I am—because they have trouble seeing that beauty and power. But many people can see it. Wouldn't you want to give your child the opportunity to find out if he can?
Norman M. Birkett studied philosophy at Princeton and then spent fifteen years programming computers in the financial industry. For the next nine years, he taught various subjects in a Christian school—including calculus—and helped supervise the math program. Norman and his wife Katharine serve the homeschooling community through self-published textbooks and other resources. See www.ClassicalLegacyPress.com.Copyright 2009. Originally appeared in The Old Schoolhouse Magazine, Winter 2009/10. Used with permission. Visit them at www.TheHomeschoolMagazine.com. For all your homeschool curriculum needs visit the Schoolhouse Store.
1. Norman M. Birkett, "Third Grade? Time to Think about Calculus," The Old Schoolhouse® Magazine, Fall 2009.
2. The word calculus just means a system for calculating, and it is used of systems other than the one we're considering. When we speak of "calculus" as the subject of a math course, however, we mean a particular branch of math—the one called "analysis" by mathematicians.
3. As calculus courses, I included those called "calculus" as well as courses in differential equations, numerical analysis, and real and complex analysis. In one or two cases a course that includes some material from outside calculus but is mostly calculus was counted. On the other hand, I did not include courses that largely consist of using calculus. Those would include a big percentage of all the engineering and physics courses offered.
4. To learn more about California Standards-compliant curricula, see Norman Birkett's article titled "Third Grade? Time to Think About Calculus," found on pages 172-175 of the Fall 2009 issue of The Old Schoolhouse® Magazine. The document found at the following website also may be helpful: www.cde.ca.gov/be/st/ss/documents/mathstandard.pdf.
5. The real numbers can be thought of in various ways. One is that they are the numbers that can be represented by an integer (a positive or negative whole number or zero), followed by a decimal point, followed by an infinite string of digits. It is customary to omit an infinite string of zeroes occurring anywhere after the decimal point. Examples, then, would include 17, which can be written 17.0 (the bar over the zero indicating an infinite string of zeroes); ½, written 0.50; 12 ⅔, written 12.6; and π, which cannot be written out completely (except by God) but can be hinted at as 3.14159 . . . (the dots indicating an infinite string of digits without an infinitely recurring pattern).