Writing in Calculus
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Making Group Projects in Calculus Manageable and Creative
UME Trends, September,1995
Tommy Ratliff
Group projects can be a very time-consuming and stressful part
of a calculus course. In the past when I assigned projects, I
was generally happy with the process and the results, but the
students were somewhat nervous about writing papers for a math
class, and I was nervous about administering and grading
papers.
However, I made a few adjustments this spring when I assigned
three group projects in my Calculus II course and in the
second semester of a combined Pre-Calculus/Calculus I sequence.
I was so pleased with the entire experience that I plan to assign
group projects in all of my calculus courses from now on.
I have three main reasons for assigning projects:
- I want
the students to work on more difficult and open-ended problems than
I could assign in regular homework sets.
- I want the students to
improve their mathematical communication skills by working in groups
and by writing their results in self-contained papers.
- I want to keep the course fresh and interesting by varying the
structure of the class meetings and by giving the students the opportunity
to be creative in their papers.
One of the primary reasons why the projects worked so well this spring
was that I used a checklist developed by Annalisa
Crannell at Franklin & Marshall College to grade the papers.
She has an excellent article
in Primus [Vol. 4, 1994, pp. 193-201)]
explaining the checklist, which
contains eleven items such
as ``Does this paper: Clearly restate the problem to be solved? Define
all variables, terminology, and notation used? Use correct spelling,
grammar, and punctuation?'' I gave the students a copy of the checklist
to use as a guide when writing their papers, which not only eased
their apprehension about writing for a math class, but also improved the
overall quality of the papers.
I felt much more comfortable with my grading
since I had a well-defined set of criteria to apply
to every paper. In addition to helping me be consistent in my grading,
this also made the grading surprisingly efficient: I could grade a set
of twelve papers from my Calculus II class in under three hours.
The second reason for the success of the projects this spring was that
each project was written as a letter from a fictitious character
to the students asking for their advice on some problem. This clearly
defined the target audience for the paper
and gave the students an idea of the mathematical background that they
should assume of the reader.
The plot lines in the projects were
a little bit goofy, although not imprecise, which helped relax the
students and gave them the opportunity to
be creative when writing
their papers.
For example,
in a Calculus II project on infinite series, Wile E. Coyote writes to
the students that he has
a recurring nightmare that he and the Roadrunner are standing at
opposite ends of a road that is 1 kilometer long. He can move toward
the Roadrunner at a speed of 1 meter/second, but after each second the
road stretches uniformly and instantaneously by 1 kilometer. He wants
the students to tell him if he ever catches the Roadrunner, and if he
does, how long it will take. With some help, the students made a few
specific calculations and recognized the pattern that the distance
between the Coyote and Roadrunner after n seconds is
n(999-(1/2 + 1/3 + 1/4 + . . . + 1/n) meters.
Since the harmonic series diverges, they
determined that the Coyote will eventually catch the Roadrunner, and
they used the integral test to approximate that it will take
2e^999-1 seconds.
Some of the best papers I received all semester tried
to place
this length of time in perspective.
One compared to the time that the earth had been in
existence (although the group members disagreed on whether this
was 4 billion or 10,000 years), and another explained the length of
time in a poem called ``Ode to Coyote'' that concluded that it was
``Just too many millennium to comprehend.''
I used several projects from Student Research Projects in Calculus
[Cohen, Gaughan, Knoebel, Kurtz, MAA, 1991]
(although I took a few liberties with
the plot lines), and I wrote several
on my own. (If you are interested in seeing
copies of the projects or the checklist, I have placed them on my World Wide
Web homepage http://www.stolaf.edu/people/ratliff/, now
http://www3.wheatoncollege.edu/tratliff.)
The classes met three times
a week for 55 minutes, and I handed out the projects at the end of one
meeting and gave the students the next meeting to work on the projects
in groups of two or three. Each group turned in one paper about a week
later that usually ranged from four to seven pages and counted for 10\%
of their final grade. I allowed the
students to pick their own groups, and while some students did shift
groups after the first project, I did not have any serious problems
with any of the groups. I will need to keep a closer eye on the
dynamics of the groups in the future, probably by
asking the students to evaluate the contributions
made by each member of the group.
I did not assign homework on the days that they worked on the projects,
but I did not reduce their homework in any other way. I gave three exams
and a comprehensive final as I usually do, and I was
pleasantly surprised that the students did not feel overwhelmed.
On their end-of-semester evaluations, only
5 of the 33 students who responded
said that the course required much more effort than their
other courses, while 24 said that the effort was somewhat more or about
the same.
A number of the students said that the projects
helped them understand the concepts of the course. As one student put it,
``The group projects, although somewhat time-consuming, were a great
learning experience for relating what we were learning in the book
to real life problems.'' In retrospect,
the projects were a very rewarding, and very manageable, addition to the
course, for both the students and me.
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