My Mathematical Journey - Project CALC
By: David Bressoud @dbressoud
David Bressoud is DeWitt Wallace Professor Emeritus at Macalester College and former Director of the Conference Board of the Mathematical Sciences. davidbressoud.org
Of all my experiences at Penn State between deciding to stop chasing research grants and the move to Macalester College, none was more important or formative than my implementation of Project CALC.
Project CALC (Calculus As a Laboratory Course) was one of the many calculus reform projects that came out of the NSF Calculus Initiative in the early 1990s. Created by Lang Moore and David Smith at Duke, its distinguishing features included
A course that was built around real-world projects, starting with problems in dynamical systems, and that was heavily reliant on computer labs,
A textbook that required students to read, think about what they had read, and fill in their own conclusions and explanations,
A heavy dependence on student writing,
And an expectation for group work.
David and Lang had visited Penn State to talk about their materials. I was excited by the prospect of trying them out. As it happened, David had a sabbatical coming up in 1993–94. Penn State was attractive because he had relatives who lived nearby. When I learned that Wayne Roberts would be running workshops on the different materials developed under the NSF initiative, I signed up to host one at Penn State based on Project CALC.
That year I offered an honors section of Calculus I and II using these materials. David also taught calculus using his materials, though not to an honors section. We met once a week over lunch to compare notes. The Project CALC materials had originally been developed to be used with Mathcad, but a version was also developed to be used with Mathematica. I have since used some of these projects with Mathematica. It has too many black boxes for my taste. The Mathcad that we used in 1993–94 was very simple and intuitive.
We met Monday, Wednesday, and Friday in a regular classroom, Tuesday and Thursday in a computer lab. There were only half as many computers as students, which forced students to double up. This wound up being one of the most important strengths of the program because it forced interaction between students. There was always a lot of talking in these sessions. I would introduce a new idea in class, have students play and experiment with it in lab, then the next class day summarize what we had learned and prepare for the next idea.
Progress was slow. We did not cover everything in the same way it would have been covered in a traditional class. But between the emphasis on reading and the encouragement to experiment and play with the mathematical ideas, I found that there was a lot that I could leave with the students to learn on their own. We were somewhat behind the traditional course at the end of the first semester, but once we got into the second semester, my students were able to take responsibility for much of their own learning, and we moved much faster than the traditional class.
Project CALC would eventually be published by MAA as Calculus: Modeling and Application, with a second edition issued in 2010.
And the students seemed to understand it better. One of the greatest signs of success was when one of the engineering majors told me that she had had an engineering exam where she had forgotten the appropriate formula needed to answer one of the questions. But then she remembered what she had learned in calculus and used it to correctly solve the problem. Transfer. It is what we hope to accomplish but so seldom achieve.
I have never since taught a math class without an emphasis on interesting and unfamiliar problems that force students to work together and creatively build new understandings. A huge part of this is the requirement for them to write out an explanation of their solution. It was David who gave me the courage to give such writing assignments by providing guidance for grading them.
David Smith and Lawrence (Lang) Moore
One goes into a first experience in teaching with a heavy emphasis on writing with a fear—at least back in the 1990s—that one will need to pay attention to every spelling and grammatical mistake. David and Lang encouraged those using their materials to take a more wholistic approach: How easy is it for someone to read and understand a student’s report? This includes not just spelling and grammar, but the cumulative effect of things that make it more difficult for the reader: sentences that are not in active voice and that have not placed both subject and verb near the beginning, paragraphs that lack a clear connection to the previous paragraph or lack a single dominant theme that is highlighted by a key sentence in a stress position, reports that lack an introduction that sets the stage for the reader or a conclusion that summarizes what the report claims to accomplish.
Most important for me would be correct use of mathematical terminology. In number theory classes, I would always cringe when students used “modulo” as anything other than a preposition. I finally started spending time pointing out how seldom one gets to learn a new preposition, but here one was. Use it as a preposition and only as a preposition.
I also came to use the analogy of making duck soup: you need a duck, you need to cook it, and you need to flavor the soup to make it tasty. A math paper must begin with correct mathematics, that is the duck. But the author must also thoroughly understand both the problem and its solution and present them correctly. That is the cooking. All the other aspects of effective writing are what makes the soup tasty. Time spent on the last is pointless without the first two.
Over the years I developed several different rubrics for grading writing. I finally borrowed one from a colleague in Sociology at Macalester that perfectly fit how I wanted to deal with my students’ writing assignments. You can download it at https://docs.google.com/document/d/1eN9lSO-Hz064SHk95EKfyOozL_Uj6PNQ/edit?usp=sharing&ouid=105471153373385575166&rtpof=true&sd=true
Especially early in the semester, most student first drafts are terrible. But I found it was worthwhile to go over each these with extensive comments. The next version, the one submitted for the grade, was almost always much easier to read and grade. To keep the workload manageable I would have pairs, or occasionally a team of three, submit a common report, at least on the first draft. I usually had each student write up their own final version of the report. Even though I encouraged my students to share their further drafts with the other team members and critique each other’s work, I was able to identify who on the team really understood and who was still struggling.
It is now almost thirty years since my first immersion into teaching for active learning. I am forever indebted to David Smith for helping me begin to learn how to do it effectively.