One reason I believe that our emphasis on the factorabe function in our
entire algebra sequence is wrong is that even the "usable" polynomial
functions are rarely factorable ones that are analyzed well in "standard"
windows. In fact, I would hypothesize that the probability of a factorable
polynomial with small integer coefficients having a real use is close to
zero.
Here is what I call a "usable polynomial." When we collect data and fit
the data to a curve, even if the data does resemble a polynomial, the
polynomial rarely fits in a "standard window" or has the nice coefficients
like the text-book polynomials. The odds are even worse that the
polynomial is factorable. Taylor series is another important use of a
polynomial function. We do not ever expect the partial Taylor series
polynomials to come with nice coefficients or to be factorable. It is
true, that we use the polynomial function to introduce limits,
differentiation and integrations. In the day of technology, however, I am
not convinced that the balance between those simplistic functions and more
usable ones is the correct balance.
Here are some things I think an Intermediate Algebra student should know.
They should know what the slope of a line means in real physical terms--as
the slope of ramps (which are many times given as angles or decimal, not as
the ratio of 2 integers) and they should understand te slope as a rate of
change. We seem to spend so much time on graph paper dealing with slopes
between points with integer values, that students do not recognize 0.04 or
a 4% grade or 20 degrees as slopes. I see no reason not to teach about the
Tangent of an angle as it relates to the slope of a line in Intermediate
Algebra. Students need to recognize that "75.3 miles per hour" or "2.50
per copy" may be a slope on a graph.
Students should be able to determine appropriate scales for showing the
important parts of various kinds of functions.
They should know the relation of the graph to roots to factors, they should
know that a factored and non-factored form of an expression are identities.
Students seem to think that x-2 = (x-1)(x+2) is partly right because
the only error is a sign error. Unfortunately, that opinion comes from
training students that what mathematics is a process. Looking at the
problem more wholistically, the two expressions do not graph out to be the
same function or compute to the same values. The equation is not an
identity, so the factored answer is wrong, not partly right. Students
should know what an identity is.
Students need to know from intermediate algebra the difference between
expressions and equations, linear vs. quadratic (functions, equations,
graphs), have some manipulation skills and know what kind of questions gets
what kind of answer. For example--in general terms, what is the answer to
solving a system of equations? Why is x = 7 obviously a wrong or
incomplete answer to "solve a system of equations"--no matter what the
system is? Why is x = 7 obviously a wrong answer to a problem with the
instruction of "multiply the expressions?"
Students need to know how to analyze quantitave information. I believe
student in every class should in some form collect and analyze data.
Martha
Martha Haehl
Maple Woods Community College
2601 N. E. Barry Rd.
Kansas City, Missouri 64156
(816)437-3147