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.
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
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.