Math as a Way of Reasoning: CRESMET Piloting Reform of College Algebra
“Our course is focusing more on the key ideas of algebra, which students initially explore through hands-on exercises, than on having students perform long worksheets of repetitive calculations,” said Dr. Carlson. “Students do need practice to learn procedures, but that learning sticks with them better and longer when they can connect the mathematical procedures to mathematical ideas.” She said these key ideas in algebra include variation and co-variation, rate of change, function, and additive and multiplicative reasoning.
Class sessions in the CAR course typically involve the instructor introducing a novel problem that engages students in exploring the meaning of these key ideas. Problems are designated to promote student interest and sense making. Students work in groups to develop solutions that they share with the class. Guided by the instructor, who prompts students to explain their reasoning and support it with evidence, the students come to articulate for themselves the ideas and mathematical principles at play in the problem.
“This way of learning mathematics is far more engaging for students than meaningless drills,” said April Strom, co-chair of the mathematics department at Scottsdale Community College and a senior member of the CAR team. “When students are engaged by doing math in real world contexts, instead of disembodied equations on worksheets, they come to understand mathematics as a powerful way to model events in the physical world.”
One innovation of the CAR project is this devotion of major portions of class time to group work on challenging problems. The course designers carefully construct these problems and activities to include logical channels that, as students follow them, lead the learners to discover mathematical ideas and patterns. According to the pedagogical theory known as constructivism that lies behind the CAR project, students learn most effectively when they engage in this kind of reasoning-through of the logic of an idea for themselves.
Another innovation in the new course is that instructors assign most procedural practice homework online, in an intelligent tutoring program that helps students with hints and reminders when they get stuck.
Attendance is mandatory in CAR classes. Students must also spend three hours per week utilizing the tutoring services of the university’s math tutoring center.
In addition to developing enhanced curriculum and introducing computer-based homework support, the CAR project has developed workshops to train tutors and instructors and is employing research-developed assessments.
In particular, the project employs the Precalculus Assessment Instrument (PCA), a tool developed over 15 years by Dr. Carlson in collaboration with Michael Oehrtman, assistant professor in the mathematics department, to assess students’ understanding of the concepts research points to as crucial for succeeding in calculus.
The CAR team, which includes Scottsdale Community College Mathematics Instructor Phil Clark, leads the training of the tutors of the CAR sections. Dr. Clark, a recent graduate of ASU’s Ph.D. concentration in mathematics education, assists the tutors in acquiring questioning strategies that are based in the research on learning key concepts of College Algebra.
These tutors are also being recruited for ASU's new teaching certification program for math majors in the Department of Mathematics and Statistics whose development is being led by Associate Professor Michelle Zandieh and Professor Pat Thompson.
The CAR team, which includes senior member Lance Ward, an instructor in the Department of Mathematics & Statistics, is also serving as a critical interface with the traditional sections of College Algebra.
Early results from CAR are “encouraging,” said Dr. Carlson, “and we definitely think we’re on the road to generating an effective model that can be scaled up.”
In a pre- and post-course administration of the PCA, CAR students gained 15 percentage points, which is significantly greater than previous administrations of PCA to students taught in a traditional format.
In addition, students in CAR sections overwhelmingly reported that the instruction had been more meaningful than their previous instruction in mathematics. Dr. Carlson is forming a national advisory board for the College Algebra Reform project that will have its inaugural meeting this summer.
The CAR project is funded by ASU and a $50,000 grant that Dr. Carlson and Dieter Armbruster, interim chair of the mathematics department, won from the Arizona Board of Regents (ABOR). The Regents have contracted with the National Center for Academic Transformation to provide support for faculty who undertake reforms of large-enrollment undergraduate courses.