The first time I used the CCP linear algebra materials, I was teaching at Duke University, and I was fortunate to have the mentorship of the primary authors. Both Lang Moore and David Smith had taught linear algebra with the CCP materials they had largely created, and I was able to modify their syllabi for my course. This sharing of syllabi worked well in part because the student audience for our courses was essentially the same, though I readily admit that making their syllabus work at another institution, with a different student audience, would take more effort. Aside from the advantage of a tried and true syllabus, the fact that Moore and Smith were colleagues was perhaps more of a psychological crutch than a real one. I would have been more tentative about diving in and using the materials for such a large portion of the course had I not known that I could always consult with them if I encountered difficulties. But we rarely discussed either the contents of the modules or the logistics of using them, while the course was in progress. (On the other hand, we discussed the modules at great length the summer after the course.) The modules are so well written, the accompanying worksheets so clear, and the questions so perfectly tailored for student exploration, that I rarely had any question about the intent of the modules or their authors. We used David Lay’s Linear Algebra and its Applications (Lay, 1997) for the course, and the modules fit well with the text, in the sense that when we were ready to use a particular module, the students already had the necessary linear algebra prerequisites, whether or not they had completed “earlier” CCP modules.
My linear algebra class met three times each week, twice in a traditional classroom and once in a computer laboratory. Over the course of the semester, we spent twelve class sessions in the computer lab and completed eleven CCP modules. Most of the students, having taken a lab calculus course that used Maple, were already familiar with the computer algebra system, but even those who were not caught on quickly with the Maple Tutor for Linear Algebra, the first CCP module we completed, which provides a great introduction to the software.
On lab day each week, the students would come to lab, launch a web browser that opened with the Materials for Linear Algebra page as the default homepage, and set up the day's assigned module by downloading the Maple file and arranging the browser and Maple windows. After the first two days in the lab, students never waited for me to tell them what to do – they just got started on the day's module, the title of which was on the chalkboard when they arrived. Their self-motivation was wonderful, and I was elated to see that they were taking responsibility for their own learning. Part of their motivation stemmed from wanting to make good use of their time. Our lab sessions were only 50 minutes long, but the modules generally took between 90 and 120 minutes to complete. Most students strongly preferred to be able to consult with one another, and with me, as they worked through the modules, so they worked hard during class time. Students who did not have another scheduled after ours often stayed and finished their modules immediately, rather than going back to them later.
The careful integration of a computer algebra system with the web-based materials has great advantages to students. In many cases, using Maple for computations allowed students to focus on the bigger picture, rather than becoming lost and frustrated during the course of a long computation, such as finding the inverse of a 4 by 4 matrix. Because they could do many computations quickly, they were also able to look for patterns in ways that would have been horribly inefficient were they doing the computations by hand.
Students made strong gains in their conceptual understanding of linear algebra as they used Maple to complete the modules. One of my favorite examples is in the Inverses and Elementary Matrices module, where students are prompted, “In general, for any two row equivalent matrices A and B, describe how to find a matrix P such that PA = B.” This requires considerable thought, effort, scratch work and verification (usually with Maple), on the part of students, but working together, they are almost all able to give a general construction of P and an explanation of how and why it works.
Using the CCP modules influenced other aspects of the course as well. I was fortunate to have Maple demonstration capabilities in the regular classroom, so I often used Maple to illustrate points or do computations even when we were not in the lab. The materials also affected how I assessed student learning. The Maple worksheets were submitted by pairs of students after the completion of each module and graded as homework. Moreover, because students tend to equate the importance of a classroom activity with its effect on their final grades, and I wanted to communicate my belief in the importance of the laboratory activities and the utility of Maple as a tool, I followed Moore’s lead, and gave exams in two parts: a traditional in-class paper and pencil exam, and a take-home portion that required (or was made easier by) the use of Maple. This arrangement worked well and allowed me to evaluate hand computation and proof writing, while still allowing for longer, application-oriented or computationally intensive questions on the take-home portion of the test.