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Re: A statement on what is wrong with standard calculus
Posted:
Feb 25, 1996 6:36 PM
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I confess, that when I started this, my working definition of algebra
was pretty much along the lines of Neil Stahl's. I understand from
Cornelio Hopmann that he feels that this definition is too inclusive;
he wishes to distinguish between arithmetic, algebra, and the ability
to handle basic functions. Fine.
A further confession. When Cornelio Hopmann first raised the question
"what is algebra?" I did not understand his difficulty. Upon further
thought, I realize that this is actually a key question.
Some background:
I am teaching at the College of General Studies at Boston Univ.
CGS is a two year program for incoming freshmen who barely meet
admission requirements. If they do well in our program, they transfer
into the college of their choice for their junior year.
The point is, most of my students do not have strong mathematics
backgrounds. This year, we are using a preliminary edition of a reform
calculus text being prepared at Clemson University. Students are
required to use a graphing calculator. Most have Texas Instruments'
TI-85.
The authors of this text frequently assert that students do not require
"algebra" to successfully complete this course, and further, that
requiring "algebra" would make calculus concepts less clear.
There are several things I like about the text, but they are not
relevant to the present discussion. Here are some of my problems:
1. The authors note in their calculator guide that the TI-85 will not
compose functions. If you have formulas for f(x) and g(x) and you
want to enter f(g(x)) into the calculator, you have to work out the
formula by hand, and then enter it. To my mind, that requires
"algebra".
2. When first finding derivatives as the limit of a sequence, the
constants in the functions in the first group of problems all
contain many significant figures. The claim is that this will
make the concepts clearer. I personally do not see how working with
f(x) = 2.334965 x^3 - 13,9960.6 x^2 + 4.5967 x - 101.283
will make concepts any clearer than working with
g(x) = x^3 - 3 x^2 + 2 x - 1
will. (The problems are my own invention, but the first will give you
a flavor of the problems in the text.)
3. There are not many problems for calculating derivatives by hand.
Here is problem 12:
12x
y = 100,000(1 + .05/12)
The students are expected to figure out that
x
(12x) ( 12 ) x
(1 + .05/12) = ((1 + .05/12) ) = 1.05116189788
I chose to begin the year with a review of basic exponent and
logarithm rules (not in the text, I made my own handout). This sort
of calculation is still beyond the level of most of my students.
When I took calculus (the honors course, at the University of Chicago,
with a computer supplement) my instructor said that whatever else we
might learn, by the end of the year we would at least know algebra.
And he was right. We needed to use algebra in our calculations, and
so we learned it. We also learned a lot of calculus.
In my experience, the skills necessary to cope with (at least the
first and third) of the above problems are first taught in a course
labeled "algebra" which is why I have used that term. I would like to
know what types of calculations fall under the heading of the
"algebra" that my students allegedly don't need.
If anyone can shed some light on this, I would appreciate it.
Diana Watson
Boston University
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