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 trigo-
nometry 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 trigonom-
etry 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 mathemati-
cal analogy, skill is the difference between
knowing facts, and using them. For in-
stance, I could ask a trigonometry student
to graph y = 4cot(3x+ 1), thus demonstrat-
ing 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 solu-
tion will demonstrate skill.
I have used trigonometry to answer questions posed by woodworkers,
machinists, tarp-makers, and roofers.
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NO ONE KNOWS WHAT IT’LL TAKE TO
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