Spring 1998
Vol 6 Issue 2
IN THIS ISSUE...
New Alignments in Calculus Instruction
Change: Do We Really Have a Choice?
Change, Learning, and the Future.
Using a Student's Fund of Knowledge to Guide Discovery
You Say You Want and Evolution?
SEE ALSO...
The Labyrinth
Maricopa Center for Learning and Instruction
New Alignments in Calculus Instruction
Contributed by Connie Carruthers, Ed Chandler, Alan Jacobs,
Judy Lambert, John Losse, Jim Vicich, and Keith Worth on Behalf of the
SCC Mathematics Department.
Ten years ago mathematics faculty across the United States were challenged to
improve the three-semester calculus sequence -- challenged to make it a pump, not
a filter (Steen 1988). At Scottsdale Community College we came to the slow realization
that despite our earnest intentions our calculus served as a filter, not a pump.
We had such a dense filter that only 38% of Calculus I students were passing with
a grade of C or better. And as reported by their instructors from other disciplines
even those students could not apply calculus concepts to science and engineering
problems.
Though less than 5% of SCC math students take calculus, the calculus course sequence
stands at the center of the department's mission. It's a capstone, the culmination
of the algebra course sequence; but it's also the starting block to begin the formal
study of mathematics, science and engineering. We wanted the calculus sequence to
serve more students and serve them better. It needed to draw them up through algebra
and on to more advanced studies. Therefore, starting in 1994, we set about to align
four important factors: a change in the curriculum emphasis, different approaches
to teaching, a new vision of the classroom setting, and improved access to computer
technology.
We removed the existing individual desks from our newly-designated calculus classroom
and redesigned it to align with our pedagogical vision which would put learning,
not teaching, at the center. We oriented nine custom-designed four-person worktables
so there was no "front" to the room; we positioned white boards on all
four walls to facilitate group and individual presentations; we put computers and
monitors under the table, so its surface was unobstructed for group work; we invested
in comfortable student chairs. This environment would support our goal that students
take increasing responsibility for their own learning.
We selected a textbook, Calculus by Hughes-Hallett, Gleason, et. al., which placed
an increased emphasis on understanding calculus concepts through a variety of representations:
numerical, graphical, and verbal, as well as the traditional symbolic forms. It's
a readable textbook which supported our use of technology and student groups for
investigations of mathematical concepts. This text, referred to as Harvard Calculus,
not only aligned with our instructional goals, but it was also used at the three
state universities.
To make all these changes work, we developed a team-teaching rotation plan. We began
with four faculty members team-teaching calculus sections during the 1994-95 school
year. Eventually all members of the department would team-teach calculus with a veteran
of the new environment. We wanted to develop new uses of the environment together.
After all, if we expected our students to collaborate on projects, we should do it
ourselves!
Each faculty member experimented with various pedagogical changes. This new setting
gave us encouragement to try a number of ideas, such as collaboration and student
presentations. It allowed us to find a new balance in our teaching. According to
Keith Worth:
The format and classroom have allowed daily assessments that are so effective that the old idea of unit testing is almost moot. I get a really clear idea of each student's understandings and misconceptions by observing their presentations of problems. Each student refines their understanding of the fundamental ideas involved in calculus on a daily basis. In presenting a problem, they have to extend their thinking beyond just the answer; they become more interested in the process underlying the problem solution. Over time I see them refining and improving their problem-solving strategies. Many of my former students have returned to tell me that they continue to use this process improvement model in their other classes with great success.
One of John Losse's changes was to expect students to read the textbook before class. He used class time to build upon the book rather than to explain the book. One of his students returned a year or so later, after transferring to a four-year school. She described a math professor who was difficult to follow in class. John expected this to be a prelude to her saying she was not doing well in that class but instead she said "but it's OK, because now I know I can understand math from reading the book, and I'm doing fine."
The experience of team teaching has been a career highlight for several of us. Connie Carruthers especially enjoyed the interaction with two different faculty members, and two different teaching styles. Students observed how the faculty talked to each other and seemed to appreciate the teamwork involved. Exposure to that kind of faculty discourse in every class meeting was a tremendous addition to their learning.
Judy Lambert values learning and modeled Ed Chandler's style of thoroughly investigating a topic using numerical, analytical and graphical methods. She characterizes this style as the best way to gently lead without rushing to answers.
This experience has affected the way we teach the rest of our classes. John Losse speaks for all of us when he states that "most of my classes are recognizably different than they were five years ago, and not just because of changes in technology." This experience has served to improve our classroom teaching, because, as Jim Vicich says, "we are developing the ability to provide a variety of learning experiences for our students. We don't all teach the same; far from it, we are developing an even greater diversity of styles."
The positive benefits of this innovation have been two-fold: for students and for faculty. Not only have students improved their oral presentation skills, their collaborative skills, and their confidence in their own abilities, but more students have achieved greater learning in calculus than before. The student success rate (grade of "C" or better) improved dramatically. In the ten semesters prior to Fall 1994, Calculus I students had an average success rate of 38%. Since Fall 1994, the success rate improved to an average of 53%. Likewise, Calculus II students had an average success rate of 46% in the 10 semesters prior to Spring 1995. After our changes, their success rate improved to an average of 64%. Of course, grades are somewhat subjective, so we invoked another quality measure. We continued to give Calculus I students the same common final exam that we had given in the years before. We found that the achievement of post-innovation calculus students was consistently slightly higher on this common final exam than students from previous years.
We faculty have developed a greater array of instructional techniques. More than that, we have increased our sensitivity to the issues of learning mathematics, not just the problems of teaching mathematics. We have increased our respect for each other's teaching; we have learned from each other; we have achieved 100% participation. And, as Judy Lambert relates, "we all had a spirit of learning, even the instructors!"
Reference
Steen, Lynn Arthur (Ed.). "Calculus for A New Century: A Pump, Not a Filter." MAA Notes 8 (1988): n. pag.