Evolution of Calculus Courses at NMSU
Joe Lakey and David Pengelley
The calculus program for science and engineering students at New Mexico State University has gone through several major evolutionary changes, each prompted by involvement from new groups of faculty. Today essentially all faculty teaching calculus are voluntarily involved in this ongoing process. Our experiences can serve as an example of how widespread change can occur in a mathematical sciences department of over 30 research faculty at a state university.
The introduction of two-week student project assignments in 1987 led to the National Science Foundation funded development of a "student research projects" program at NMSU and collaborating institutions, initiated by five NMSU faculty. These projects get students to think for themselves on major multistep take home problems, working individually or in groups. They alter fundamentally students' views of what mathematics is all about and build their self confidence in what they can achieve through imaginative thinking. The projects resemble mini research problems, and most of them require creative thought. All of them engage students' analytic and intuitive faculties, often weaving together ideas from many parts of calculus. While many of the projects are couched in a real world setting, often with an engaging story line, they are all in a sense theoretical. One cannot do them without an appreciation of the ideas behind the method. Students must decide what the problem is about, what tools from calculus they will use to solve it, find a strategy for its solution, and present their findings in a written report. This approach yields an amazing level of sincere questioning, energetic research, dogged persistence, and conscientious communication from students. Moreover, our own opinion of our students' capabilities skyrocketed as they rose to the challenges presented by these projects. After a couple of years we had settled on a program in which we assigned 2-3 major projects per semester. Over 100 projects were developed during this period for single and multivariable calculus, available in the book Student Research Projects in Calculus authored by Marcus Cohen, Ed Gaughan, Art Knoebel, Doug Kurtz, and David Pengelley, published by the Mathematical Association of America in 1992. In addition to the projects, the book contains several chapters detailing the logistics of assigning projects and other advice for instructors.
In 1990 the NMSU program expanded and branched in various directions, some with further support from NSF. Other faculty in the department volunteered to use calculus projects in their classes; national and regional workshops were presented to disseminate the program; we developed a discovery-project based vector calculus and differential equations curriculum, in which a continuous sequence of discovery projects forms the context for learning all the material of the course (email@example.com); and we started a collaborative program with local high school teachers to bring projects into high school mathematics courses (firstname.lastname@example.org).
As new faculty became involved in teaching with projects, they introduced fresh ideas into the program and the projects approach itself evolved. Beginning in 1991 a group of faculty pioneered a major new emphasis on cooperative self-learning both in and out of the calculus classroom. They developed structured in-class assignments called "themes" for differential and integral calculus. A distinct change is that themes are used to introduce the core material of the course and much class time is spent working on them, with less time on lecture, whereas the projects were completed outside of class and contained material over and above day-to-day course work.
In a theme assignment, students learn and write about core course material while working in groups with the instructor serving as a resource. When themes were first assigned, students completed a written theme report every week. Experience tempered this pace somewhat, and for some time we assigned three to six themes a semester, along with midterm and final exams. Typically the first theme of the term was completed by individual students and the others were solved by groups of students. One theme may be a two week assignment, possibly in the spirit of the original projects; the rest were 7 day assignments. During theme weeks, students spend considerable class time working on their theme, with help from the instructor as needed. For example, concerning techniques of integration the teacher may introduce the class to substitution and integration by parts and assign a theme on developing a strategy of integration using a table of integrals; regarding vector algebra and applications, the class might "cover" the algebra of vectors, after which students would work on a theme concerning lines and planes.
Through time the project and theme approaches blended with each other, and with the discovery-project methods developed in vector calculus and differential equations courses. Today individual instructors are using a variety of combinations of these teaching tools. In the three semester single and multivariable calculus sequence, our current typical class section now has two or three major projects, which are worked on both in and out of the classroom, and many instructors involve students in active work in the classroom at other times as well, with traditional lecture playing a smaller role than in the past. The introduction of projects and themes also helped us begin other desirable changes, such as a greater emphasis on student reading and writing, and a shift on midterm and final exams away from rote calculation towards more conceptual and essay questions. Another effect of theme assignments was to prompt us to incorporate structured means of improving student skills at good mathematical and prose report writing.
As more faculty became involved in changing our calculus program, we added mastery-based skills exams in each of the first two semesters, to ensure command of essential differentiation and integration skills separately from other types of assignments. These tests were modeled after examinations used at Duke University, though the implementation was quite different. The integration skills examination was based on a table of integrals that students were encouraged to use. Students could take the computer generated tests repeatedly, and mastery was generally considered to be perfect work on 9 of 10 questions. Currently, skills are tested on uniformly scheduled and coordinated evening exams during the semester. This format was implemented, in part, to provide course instructors some flexibility from one semester to the next to examine students on skills developed in class and through projects and themes, and partly in response to feedback from colleagues in engineering and sciences who need their students to carry out calculations requiring specific skills from calculus.
Even more recently we have made substantial changes in syllabus and incorporated the use of technology in meaningful ways. Unlike many other calculus reform programs, we concentrated for our first five years on developing our view and implementation of the types of assignments students should be working on, what intellectual activities they should engage in while working on them, what should happen in the classroom, and what we expect and foster in the way of student reading and writing. Once we had made some progress on these fundamental issues of how we expect students to learn calculus, we felt more ready to address how our syllabus should change, and how to incorporate technology in a way which enhanced learning without introducing distractions. In Fall 1997 we chose James Stewart's Calculus: Concepts and Contexts as our text for the three semester single and multivariable calculus sequence. It includes a number of elements our own changes had prepared us for, such as many more conceptually oriented exercises and challenges for students, some projects, deemphasis of certain specialized integration techniques (focusing now primarily on substitution and integration by parts as central), a chapter introducing differential equations and their applications, and incorporation of technology. We emphasize the use of calculators as exploratory tools using simple, readily linked programs for computing things like Riemann sums, Newton's method, and simple algorithms for numerical approximation of solutions of ODEs as specific uses; use of calculators is minimized when emphasizing essential skills like substitution and integration by parts. Instructors and students in the first two semesters currently use mostly graphing calculators to enhance what can be studied, and in the third semester, for multivariable aspects, we add either Maple or Scientific Notebook/Workplace (which integrates a sophisticated mathematical writing tool with Maple's computational capability and 3-dimensional graphing). We are proceeding in the continuing spirit of expecting initiative, independent thought, reading, and writing from our students, and we aim to focus ourselves and our students on the important ideas. The "content" in our syllabus is now more global in nature, oriented towards broad topics rather than merely regimented individual sections of a text. It is intended to encourage individual instructors to select directions for their students to delve into with greater depth, through the types of assignments described above. We are also discovering something very interesting and encouraging: when students are actively involved in collaborative self-learning in and out of the classroom, we are less pressed to "cover" material in class, so the syllabus feels less rushed; the difference may be substantial.
Evaluation of our program is ongoing, but there is statistically significant evidence of a substantial positive trend linking student participation in sections taught with projects and their success in later classes. Perhaps more important to many instructors, though, is the sense that we are engaging our students in exciting and challenging mathematics, and our students are rising to the occasion.
Where do we see our program going? We see a continuing evolution for both faculty and students. For faculty, collaboration in teaching innovation has made possible almost everything we have accomplished, in a way that has fostered individual faculty ownership of both the process and the results. We believe that without this faculty collaboration and individual ownership, we would accomplish little, and thus we see ourselves continuing to nurture these features for faculty as we walk further together down the path of substantially improving our teaching and our students' learning. Already we see the effects of our efforts spreading further locally; the types of teaching changes we have begun are subversive in nature, inducing faculty to make further changes, individually and collectively. Faculty are inexorably carrying these changes into the other courses they teach, and so the changes in our calculus program are having substantial effect on our department's teaching of everything ranging from general education Mathematics Appreciation to graduate courses in our Ph.D. program. For instance, we are developing discovery based learning in several earlier courses, including implementing a precalculus course using the book Functions Modeling Change. Although projects used in these courses are not quite as substantial as those used in the calculus sequence, they are being developed with the same emphasis on identifying variables, formulating descriptions of the real world in mathematical terms, and developing mathematical skills that allow one to draw conclusions or make predictions from mathematical derivations. Our hope is that when students pass from one level to the next, they will find the mathematical challenges posed by projects in the higher level courses to be welcome opportunities for them to hone their skills in solving real and meaningful problems. We anticipate this effect will grow, and we expect to see an ongoing evaluation of what our individual and collective resources are accomplishing toward the changes we desire in all our teaching.
For our students we expect to see further growth of discovery learning, of reading and writing to learn mathematics, the increased development of student-centered classrooms, less lecturing, and a host of innovative changes in what we ask of our students that we have not yet discovered. Alan Schoenfeld, in the preface to the book Mathematical Thinking and Problem Solving, referred to the changes we have begun as a Trojan Mouse.
Revised September, 2006Last modified: 2014-03-09 14:44:29 leisher