Date: Sep 21, 2004 11:26 AM
Subject: Algebra: How Much is Enough?
Ok, let's back off a little. I'm not advocating not using CAS's in calculus
class. I use them there myself and wouldn't give them up. I am saying we
shouldn't use them as an excuse to give up teaching a lot of the algebra (see
footnote for my working definition) that's now taught.
I like the idea of people learning to think more graphically. Surely that can
complement thinking algebraically, but we don't know that it can replace
I like the idea of people learning more problem solving skills and I use CAS's
to give them opportunities to do so. But these are people with fair algebra
knowledge. I personally doubt that people with a lot less algebra knowledge
would be able to make as good use of the CAS's to solve problems.
Unquestionalby a lot of our citizens don't know much algebra now, but many
(enough?) do know it reasonably well and use it in their work.
I fear some of you are willing to let the level of algebra knowledge among
Americans go down a lot. That could be a big mistake. My concern is that we
may underestimate the importance of educating enough of our citizens in
algebra and fail to teach a generation things they turn out to need.
We have stuffed math knowledge into black boxes like Mathematica, and use the
boxes to do some valuable things. In the work place so far, these black boxes
are used by people who know something about math. How well can people do real
work with them who know very little about math? The jury is out on that one.
We don't yet know how much experience and knowledge needs to be in the head in
order to use the CAS on the desk well.
Algebra: The level one needs depends on the class and or job, but for the
people who take a calculus class (for business calculus take ~75% of what
follows) I'd want them to be able to work with polynomials and rational
expressions (though not as complex as the ones I did as a kid) and have a good
idea of when to do what. For instance they should appreciate factoring as a
step in solving an equation or inequality.
I'd want them to know exponents and logarithms and their properties very
well. I'd want them to understand the idea of one variable determining
another (i.e. functions) and also use of function notation and inverse
I'd want them to know the graphs of several basic functions and be able to
use those graphs in thinking about the solution of problems.