Bret Taylor wrote:
>...that is the question.
>
>I guess the answer depends upon whether you want to receive 25 pennies or 1
>quarter for change. :-)
>
>On a slightly more serious note, is there any "mathematical convention" as
>to which of the following are "simplified"?
>
>(x+2)(x+3) or x^2+5x+6
>
>(4+9x)/6 or 2/3 + 3x/2
ahem...
Given the way most math textbooks (and many math instructors) use the verb
"simplify", one might expect them to provide an operational definition of
it, i.e., a definition sufficiently precise that one may use it to
distinguish between cases of "simplifying" and "not simplifying." Have you
ever tried to define precisely what "simplify" means? You can be like my
students and say, "It means to make it [the expression] simpler," but that
only begs the question.
But the textbook says, "Use the distributive property to simplify 2(x+6)"
and gives the result as 2x+12. Now as I'm counting, 2(x+6) contains two
constants, one variable, one addition, and one multiplication. The
"simplified" expression 2x+12 contains two constants, one variable, one
addition, and one multiplication. This is obviously simpler...or did I miss
something?
(And let's not bring up the fact that in the arithmetic of real numbers
there are several valid "distributive properties". Why single one of them
out to bear the honor of the name?)
"But," someone might say, "2(x+6) has parentheses. That's why we want to
simplify it." And it does seem that much of "simplification" deals with
removal of parentheses. Then why do we hate the parentheses so much? I
think they're kind of cute myself...
And has anyone else noticed that after a few months of learning to reduce
fractions in a "knee-jerk" reflex, students learning to add fractions make
the following mistake:
They find the LCD,
They build up the fractions to have the LCD,
Then they notice that the fractions can be reduced,
So they simply MUST reduce them (this is the knee-jerk),
thus they are right back where they started from,
and they can't figure out what went wrong.
And naturally, some teachers respond by teaching, "No, when you're adding
fractions, you must do it THIS way." Wow! That makes everything better,
doesn't it?
Likewise, I've seen students who are comfortable in their memorized
"simplification" skills become very agitated when shown how a NONstandard
form can really "simplify" (in its ordinary english sense) an otherwise
difficult problem. Watch calculus students in a math lab when a tutor is
helping them "rationalize the numerator" to take the derivative of the
square root function _the long way_. "But you're always SUPPOSED to
rationalize the DENOMINATOR!"
Logically, therefore, algebra textbooks responded with "rationalize the
numerator" exercises, and this fixed the problem...or did it?
And here's a related problem: I've seen a textbook that taught a formula,
e.g., A^2 - B^2 = (A-B)(A+B) for factoring the difference of squares, and
subsequently it taught that this formula can be used on "anything of the
same form." So I went looking in that textbook for some explanation of what
"the same form" meant. Zip. Not surprisingly, most students simply replace
"the same form" by "mostly the same operations", using the formula
improperly on (x-3)^2, but never on (x-3)^2 - y^2. And the textbook says
nothing about how they are seeing it wrong.
I wish I had an "exact" solution to these learning problems. If you do,
please mail it to me. Since I don't, here's what I try to do--
1) Help the beginning algebra students (and above) learn what the "form" of
an expression really means. Help them learn to see the "form" of an
expression when they look at it. I wish algebra textbooks had a good source
of exercises on this! None of them seem to recognize that THIS skill needs
improvement.
2) Help them learn the following definition of "simplify": To replace an
expression by an equivalent expression with a more advantageous "form".
One feature that's worth noting: the idea of "more advantageous" is
meaningless unless your work is in context. If you are working with
fractions in an inequality (on paper, not a calculator), it can be very
advantageous to combine everything into a single fraction on one side of the
inequality and a zero on the other. But if you had the same expression
(considered to be a function) and wanted its antiderivative, you might find
it much more advantageous to decompose it into partial fractions. In some
situations it is more advantageous to reduce fractions; in others it is
better to have them built up. Sometimes you want a radical-free
denominator, other times it works to move the radicals INTO the denominator.
And as others have noted, if you want to see if your expression is the same
as someone else's, an agreed-upon "standard form" can be a good thing.
Even with a graphing calculator, I still find it quite useful, as I believe
others have, to be able to easily manipulate expressions into more
convenient forms. And, although I might have difficulty convincing someone
with an opposing viewpoint, I believe that students gain something of value
by becoming facile at expression manipulation, as long as they are also
learning the skill of deciding where to go with it.
[If you are looking for a good place to flame, try inserting a sarcastic
comment immediately following the word "value" in the preceding sentence.]
Oh, and instead of "the distributive property" (a noun phrase applying to
numbers), I have an irrational preference for "distributing" (a verb
applying to any OPERATION that distributes over another).
Well, THIS post took longer than I thought it would. I hope it wasn't a
waste of time.
Kevin Broussard broussard@siskiyous.edu
College of the Siskiyous
800 College Avenue
Weed, CA 96094
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