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I am hardly an advocate for having anyone waste time memorizing trivia
(unless that's his/her free choice). But I am very happy that there are
useful ways to make memorizing things easier and wish I'd known them as
The tool I described is not broken, however: it's rusty, but not beyond
polishing. If I needed it, restoring the periodic table (the
presidents' list I have NOT forgotten, just to be accurate) would be
far easier for my having memorized it in the first place.
Further, please don't confuse any particular example I might offer with
a generalization about the real focus of this issue: are mnemonics
useful tools for mathematics students? My answer would be a qualified
"yes." At the same time, I'll just suggest that for some people, it
appears that memorizing is either everything or nothing. I take a
middle ground. I learned SOH-CAH-TOA from a high school student I
tutored for the SAT's back in the early 1980's, when I was NOT a
mathematics person, but was starting to get interested in becoming one.
It seemed a harmless and useful device at the time and still does.
Because the NAMES of the trig functions are a convention, and so are
their definitions. The conceptual issues are not to be confused with
the official terminology (though there are interesting things to be
gleaned from the derivations of those terms, and as a long-time English
teacher with a fondness for roots, prefixes, and the like, I'm always
curious about where terminology comes from, regardless of the field).
Knowing the many meanings and uses of the sine function is far more
important than knowing the name, but it's rather useful to KNOW the
name and the fact that it can be thought of, from one viewpoint, as the
ratio of the opposite side and hypotenuse of an angle of interest in a
right triangle. If that's ALL you know, you're not very far along, but
if you don't know that, you may not be able to do as much with trig as
you'd like. It depends on you and the context.
That's why calling something "trivia" or "useless" is a dangerous game:
for WHOM and under what conditions is something trivial or useless? Can
any of us presume to know that for everyone in all situations? I, for
one, cannot. So as a teacher, I'm happy to share mnemonic systems with
students. But I offer the system, if the student is interested, under
the belief from personal experience that once you have a method, you
get the most mileage out of creating your own connections. Would that
be a constructivist approach to memorization? ;^)
Now, I have observed far too many math classrooms and teachers to think
that there aren't lots of dull things going on out there. But that
doesn't lead me to the conclusion that I should NEVER teach facts.
Realistically, facts are precisely the kinds of things that call for
direct instruction. But since I want students to gain much more than
factual or procedural knowledge of mathematics, direct instruction is
only a component of my teaching repertoire, one that teachers should
know. Unfortunately, for many, that's all they know. But it would be
unfortunate, too, if a teacher completely ignored the practical need
for knowing some facts: there are things that discovery learning was
not meant to address, and effective presentation of key facts and
terms, at the appropriate time and in useful ways, is part of good
teaching. Of course, I know where the balance lies for MOST teachers
these days, but that's changing. It won't help the process, however, to
throw babies out with bath water, at least from where I sit. Your
mileage may vary, and probably does.
Michael Paul Goldenberg, Director
1810 Fair St
Ann Arbor, MI 48103
734 644-0975 (c)
"Until this moment, Senator, I think I never really gauged your cruelty
or your recklessness ... If it were in my power to forgive you for your
reckless cruelty, I would do so. I like to think I am a gentle man, but
your forgiveness will have to come from someone other than me ... Sir,
at long last, have you no sense of decency?” (Joseph Welch, during the
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