I’m being perfectly honest when I say that calculus itself is not very hard. “What?!! Did he just say that?!” Yes, I said it. And it’s true. “But what about the horror stories? The sheer panic induced when even the word calculus is mentioned?” Well, the horror stories are also true. I can’t deny it. Calculus classes do indeed cause much pain and suffering, even tears. “But you just said …” I know. It looks like a contradiction. But it isn’t.

The answer lies entirely in the student’s preparation. With proper training and preparation (and a good work ethic), calculus becomes a beautiful, rewarding experience. In fact, it convinced me to become a mathematician. But without solid preparation, calculus can be a miserable, meaningless exercise in endurance.

Now, what preparation is needed for a genuinely successful study of calculus? (“Successful” means a true mastery of the art, not just a good grade.) Well, just like any craft, you first need good tools. Your tools for calculus are arithmetic, algebra, Cartesian geometry, and trigonometry. How sharp are they? How comfortable do they feel in your hand?

In general, students considering a calculus course are reasonably proficient in basic arithmetic. But I have seen exceptions. I will only say that it’s impossible to teach calculus to someone who struggles adding fractions. Such a student is better off practicing arithmetic anyway. Arithmetic is much more useful. It’s like the toolbox, without which all the other tools are lost.

Regarding algebra, a physicist friend of mine once commented that calculus is where true competency in algebra is solidified, because algebra is used in every phase. It’s the measuring tape. Misuse it, and nothing squares up. Remember logarithms, exponents, factoring polynomials, finding asymptotes, etc.? Yeah, all that stuff. Well, calculus is where all that stuff really shines. Obviously, the algebra measuring tape had better be calibrated correctly.

Think of Cartesian geometry as the hammer in the toolbox. It’s important, and well-used, but it doesn’t really require much maintenance. There’s no need to take it out for regular sharpening. You just use it when needed. Honestly, I never had any issue with my students’ knowledge of Cartesian geometry. That’s probably because the concepts needed are pretty basic: slopes, areas, equations of lines, etc. The trickiness in calculus involves their application. The concepts of slope and area are applied in ways the student has not seen. But that technique is part of calculus itself, not part of the preparation.

And now for trigonometry. Sigh. Sadly, this tool resembles Grandpa’s old handsaw hanging from a nail in a darkened corner of the garage. Draped in spider webs, covered with rust, the teeth that remain are rounded and dull. Powered by enough muscle and self-hatred, it can still cut … in a rough, crooked way. But abuse and neglect have rendered it incapable of fine craftsmanship. Too often, this precisely describes a calculus student’s knowledge of trigonometry.

Why is this the case? I think it’s because trig is merely viewed as a prerequisite for calculus, rather than a worthy end itself. Bright students are pushed through trig to get to calculus so they can try for college credit. If a student truly masters trigonometry, then that’s a great plan. If not, you’re doing them no favors by rushing into calculus. The result is a second-rate knowledge of both subjects.

And believe me when I say that trigonometry is far more useful to most folks than calculus. I have used trigonometry to answer questions posed by woodworkers, machinists, tarp-makers, and roofers. Any guesses on how many calculus questions they’ve asked me? Don’t slight trigonometry just because it isn’t calculus.

Okay, that covers the tools. But that’s not enough. Mere ownership of tools doesn’t produce fine furniture. Skill in their use is also needed. In our mathematical analogy, skill is the difference between knowing facts, and using them. For instance, I could ask a trigonometry student to graph y = 4cot(3×+1), thus demonstrating factual knowledge. Or instead, I could ask him to find the height of the telephone pole across the street without crossing the street … using only a protractor, level, tape measure, and paper towel roll. That solution will demonstrate skill.

How is this skill developed? Word problems. Logic problems. Hard Problems. Anything requiring multi-step reasoning and strategy. Nothing elicits greater fear and trepidation from a student than a complex word problem. But no solution generates greater satisfaction. The best way to conquer a fear of word problems is to become good at them. And the best way to become good at them, is to work more of them. Start the development of reason at a young age; build fluency of translation between verbal and mathematical language; embrace challenges, and the fear will never gain a foothold. Then, with that kind of preparation, I tell you once again that calculus itself is not very hard.

Copyright 2017, The Old Schoolhouse®. Used with permission. All rights reserved by the Author. Originally appeared in the Fall 2017 issue of The Old Schoolhouse® Magazine, the trade publication for homeschool moms.

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