Solving systems of equations are very important, but why do the 2x2 systems
always fit in a standard window? Why do the 3x3 systems always have nice
coefficients and the answers generally come out just as nice? The problems
need to change to make good use of calculators.
If we teach factoring, why not teach a simplified verstion of the rational
root theorem at the intermediate algebra/beginning algebra levels so that
students can estimate roots of the graph and use the roots to help them
make factoring choices? At very least, students can use the graph and
concept of identity to check whether factoring, simplification,
multiplication, etc. is correct. When comparing the graph of the factored
vs. non-factored forms of the expression, there is no partially correct
answer--either the original expression and the answer graph out to be the
same, or the answer is wrong. If the expression has more than one
variable, the students can use the straight-up calculations feature to test
points a few values to see whether the question expression and answer
expression seem to be identities. We can use the calculator to train the
student to build in checks and balances--let's ween them off of the back of
the book.
We do a disservice to the few students who do go on to higher levels of
mathematics. At the lower levels, they also are allowed to think that
mathemtics is the study of manipulation skills and 10 minute applications.
The graphing calculator allows us and them to focus on mathematical
concepts. A graphing calculator can help broaden the concepts that we
already teach as well as bring higher level concepts to lower level
classes. However, overlaying a graphing calculator over a very traditional
"focus on the procedures" course can have the effect of dummying down the
class.
Where does data collection and curve fitting or real-life problem solving
that involves REAL-life numbers and functions fit into the curriculum or
these courses? To point to a recent discussion, why should Johnny factor
at the expense of understanding percent change, looking at raw data and
producing a function that approximates the data? Why would anyone in their
right mind study curve fitting and real-life problems and real-life very
large and very small real numbers without using technology? Why would
anyone in their right mind study behaviour of polynomial and rational
functions and limit their study to the nice functions that can be analyzed
easily with pencil and paper techniques?
Even students who are struggling with basic arithmetic skills can study
mathematical concepts of the real world along with adding, subtracting,
multiplying and dividing. We too many times assume that a student cannot
think mathematically or apply mathematics before they become skilled at a
myriad of procedures. The truth is, when students learn mathematics from
in-context use which often requires the use of technology, students can
then learn to think mathematically and make sense of the procedures.
By refusing to "let" students use technology, we short change them by
focusing mathematics so much on abstract procedures and contrived
substitutes for applications that they miss out on a lot of usefull as well
as theoretical mathematics.
Martha
>In a message dated 97-04-30 00:49:16 EDT, you write:
>
><< The WAMATYC conference was also this last weekend, 212 participants seems
> to ring a bell. One of the topics discussed was the extent to which
> graphing technology should be integrated into intermediate algebra. The
> panel that discussed the issue seemed comfortable with graphing
> calculators in calculus and pre-calculus but questionable in
> intermediate. What do you all think? >>
>
>I think that they should be used in Intermediate Algebra for *demonstrations
>and simulation* purposes only. Actually letting the students use them at
>this point in their young math careers will cause them to become too
>dependent. They students really need to learn to do stuff by hand first.
>
>I think the calculators should be first introduced to the students (for use
>on exams and homework) in some of the freshman level courses: Pre-calc,
>business calc, stats, etc.
>
>I teach a pre-calc course using the TI-85....... but I test my students with
>and WITHOUT their calculators! My students have told me that if I hadn't
>done this (ie tested them without), they would have become addicted to the
>calculator and wouldn't have learned to do the stuff themselves.
>
>Karen
>Orange Coast College
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Martha Haehl
Maple Woods Community College
2601 N. E. Barry Rd.
Kansas City, Missouri 64156
(816)437-3147
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