el

Algebra With a Critical Thinking Approach

Kor six months, I have participated in a Thinking
Skills Project at the Community College of Aurora. I
was among the first dozen instructors who were
trained in the recognition and use of 22 thinking skills
and then asked to train students in one or more of
these skills—not as separate course content but as
content integrated into our regular curriculum.

The Process

Algebra is definitely not the preferred language in
the world, even though more people speak and write it
than any other language; and if there is any one topic
in algebra that would win the award for least-liked, it
would probably be exponentials and logarithms. And
just to make sure that we were taking ona real task, I
decided to attack graphing by function analysis (as
opposed to plotting of points). The following events
occurred in my College Algebra class.

The students first analyzed a large selection of ex-
ponential functions to determine how many different
components there were. We were then able to extrapo-
late the following general form: f(x)=aB°*“+d

We checked out each of the five constants individu-
ally to see which values were acceptable/ unacceptable;
we also discarded the trivial cases such as B = 1 or 0.
The bulk of the classroom time was spent testing to see
how each constant affected the general graph. Next
came the most difficult task of all—determining the
order in which to look at the constants for graphing
purposes (Bbacd).

Then came the fun part. We created a mnemonic to
lock that sequence in our memories: Big boys are
cuddly dudes. (The wording may be due in part to the
fact that the class was 70% female.)

The entire process took 3/4 of a two-hour class
session, and we spent the remainder of the class prac-
ticing a variety of exponential functions.

The Outcomes

The process was magical! The students forgot their
fears while they hunted down the offending creatures
and put together a general “mug shot” of exponential
functions. Their fears then dissipated as we “poked” at
cach part and watched as the graph wiggled.

Some students, for the first time in their lives, were
doing some real power thinking. They found them-
selves “in control”; they set a goal and accomplished it.
They were working together as a team, sharing partial
thoughts to trigger others’ thoughts. We were in-

volved so deeply with concept development and a
attainment and pattern recognition that breaktime
came and went unnoticed.

Three of the students whose academic and/or math
skills were judged marginal for this class blossomed
into mathematical fireballs for this class period. After
20 minutes of class, I was doing little more than facili-
tating. Two weeks later when tested on this material,
ALL of the students did A-level work.

For the remainder of the term, | was dealing with a
new group of people. They WANTED TO KNOW so
much. They were also WILLING TO DO so much. Out
of the original class of 34, 18 earned the grade of “A.”
For this course I have had my “standard” syllabus and
my “ambitious syllabus.” After seven weeks, I
switched to the latter, only to find that it was too short
on aspirations.

The results of this activity in my classroom was
exciting for the student—and the instructor—though
one aspect caused a minor alarm. Enrollment for the
class was allowed to exceed not only the preferred
instructional size but the classroom capacity, and the
usual withdrawals did not take place. In all twelve
classrooms in this project, retention was up! [One
word of warning for schedulers: Most students love
being active participants in the teaching/learning
process.]

An additional spin-off of this process was the
students’ introduction to the scientific method, how-
ever abbreviated it might be. Many of these students
might never be exposed to this method again. So it
was important that they experienced gathering raw
data, synthesizing a general statement that fits all the
data, and then testing the statement as a valid (and
centuri¢s-old) method of increasing knowledge.

This approach on graphing exponentials may be a
reinvention of the wheel. However, combined with the
“thinking skills” approach to general classroom deliv-
ery, it confirmed my love of teaching and my students’
love of learning.

 

 

Frank A. Neckel, Division Chair, Mathematics

For further information, contact the author at Commu-
nity College of Aurora, 791 Chambers Road, Aurora,
CO 80011.

 

 

Suanne D. Roueche, Edjtor
ER

December 9, 1988, Vol. X, No. 30

Tho University of Texas at Austin, 1988
Further duplication is permitted by MEMBER
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