Abstract: Our primary goal for the Core Curriculum Conference was to restructure our calculus sequence so that the Calculus II course would be more attractive, effective, and useful for students than it had been, but so that placing high school and transfer students with calculus experience in courses other than Calculus I would remain possible. Prior to the conference, we covered---using "calculus reform" texts and pedagogy--- differential equations and differentiation in Calculus I; differential equations, integration, and series in Calculus II; and traditional multivariable calculus topics in Calculus III. During the conference, we devised a plan to interchange and integrate some of the Calculus II and III topics. Specifically, Calculus II would include both single and multiple integration, as well as an introduction to vectors, whereas Calculus III would include the vector calculus from gradients to Green's Theorem, along with sequences and series. We believe that this restructuring of the calculus sequence would make the Calculus II and III curricula more commensurate with students' interests and abilities, resulting in greater student success in these courses and greater student retention in the mathematics program.

Description of School and Student Audience

The University of Redlands is an independent, coeducational, comprehensive university located in the city of Redlands, California. It enrolls 1500 undergraduates in the arts and sciences and in small professional programs in communicative disorders, business administration, and music. The student/faculty ratio is approximately 13:1. The University operates under a 4-1-4 academic calendar, with 13-week fall and spring semesters and a 3 1/2-week January term.

The Department of Mathematics has eight full-time faculty members, including two professors, three associate professors, two assistant professors, and one lecturer. An additional mathematics professor serves as the University's Director of Academic Computing. The Department offers two degrees, a B.S. degree in mathematics and a B.S. degree in mathematics leading to a California secondary teaching credential. Requirements for these majors as well as for the minor in mathematics are listed in Appendix I. The Department graduates approximately nine majors per year, with approximately one third of these pursuing the secondary teaching credential. The Department also graduates approximately 13 mathematics minors each year.

In addition to offering courses supporting its major and minor programs, the Mathematics Department offers service courses for biology, chemistry, physics, environmental studies, computer science, economics, business administration, and liberal studies (elementary education) majors, as well as courses satisfying the University's general education requirement in Quantitative Reasoning. In order to fulfill the Quantitative Reasoning requirement, most students take a multi-section Finite Mathematics course which has a minimal high school algebra prerequisite. Approximately 200 students per year, however, take at least one calculus course. We offer six sections of Calculus I with approximately 24 students per section, five sections of Calculus II with approximately 22 students per section, and three sections of Calculus III, with approximately 12 students per section, each year. We also offer a Pre-Calculus course for students who wish to take Calculus I but are not adequately prepared for it. Our calculus sequence serves as our introductory mathematics sequence, and we attempt to convince as many students as possible to study calculus and to entice as many calculus students as possible to major or minor in mathematics.

There is just one calculus track, and, as a result, the students in the first year calculus courses (Calculus I and II) have varied interests and abilities. Although many intend to major in mathematics, science, or computer science, a fair number of these students plan to major in business or in one of the social sciences. A large number are preparing for graduate study in medicine, dentistry, or other health professions. Approximately one-third of the students in Calculus I have studied calculus previously. With the exception of new students who place into Calculus III, we find our students' high school preparation to be generally weak; students often need extensive review of the pre-calculus and calculus topics they studied in high school. The students who continue from Calculus II to Calculus III are primarily mathematics, physics, or chemistry majors or minors.

Description of Current Status and Why Reform Is Needed

Our core curriculum for the first two years currently consists of Calculus I, II, and III, Problem Solving Seminar (or Discrete Mathematical Structures), and Linear Algebra. (See Diagram 1.) We discuss the latter three courses first.

Problem-Solving Seminar. The January term Problem-Solving Seminar is our "bridge to abstraction" course, emphasizing proof techniques as well as problem-solving strategies. Students learn these skills "in context"; an appropriate topic, such as graph theory or knot theory, is selected each year based on student and faculty interest. The course enrolls approximately 26 students per year with nearly all going on to major or minor in mathematics. As shown in Diagram 1, the prerequisite for the Problem-Solving Seminar is Calculus II. In most years, this prerequisite is meant only to ensure a certain level of mathematical maturity; although the majority of Problem-Solving Seminar students have just completed Calculus III, we sometimes encourage talented Calculus I students to take the course during the January term immediately following their Calculus I course.

Discrete Mathematical Structures. Beginning in Fall, 1995, we will offer a lower division Discrete Mathematical Structures course for students interested in mathematics and computer science. Students with some calculus experience may take this course before beginning the calculus sequence, simultaneously with a calculus course, or as a "break" from the calculus sequence. Mathematics majors and minors may take Discrete Mathematical Structures instead of the Problem Solving Seminar.

Linear Algebra. Most students proceed from Calculus III to Linear Algebra and/or Differential Equations (spring semester courses) or to Probability Theory (fall semester course). Since Linear Algebra is a prerequisite for many upper division mathematics courses and since it appeals to so many potential mathematics majors, we strongly encourage students to take it as early as possible. In fact, the curriculum reform plan we introduce here allows students to study linear algebra topics and to take the Linear Algebra course even earlier than they previously could.

Our sophomore-level Linear Algebra course enrolls approximately 26 students per year, and, in combination with the Problem-Solving Seminar (or the Discrete Mathematical Structures course), is intended to prepare students for abstract thinking and theorem-proving in their upper division mathematics courses. The course covers the standard elementary linear algebra topics, but also emphasizes applications and uses computing as an integral tool. The course meets in a classroom in which each two students share a computer, enabling students to use the MATLAB package virtually every class day for computations, for formulating and testing conjectures, and for exploring applications.

In fact, all sections of our linear algebra, differential equations, calculus, and pre-calculus courses meet in classrooms equipped with one computer for each two students. We currently have two such classrooms, set up with University and NSF funding (NSF-ILI Grant No. DUE-9351491). These classrooms also are used for afternoon and evening tutorial sessions. Appendix II contains a diagram showing the floor plan for one of these two classrooms.

Calculus sequence. The three-semester calculus sequence serves as the backbone of our core mathematics curriculum. This is due primarily to the high demand for calculus instruction from partner disciplines, and our reluctance to track entering students by discipline or ability in different mathematics sequences or even in different calculus sequences. The most practical reason for not offering various mathematics or calculus tracks is that, with only 150 or so first year students (and 200 students in all) in calculus courses each year, offering only a few sections each of various types and levels of mathematics courses would create severe scheduling problems for students. Our main reason for not tracking students, however, is that we want students to have as much flexibility as possible in choosing a major---that is, we do not want entering students to have to decide on the first day of college if they are mathematics or physics or economics majors and we want students to be able to switch majors as easily as possible during their first two years of college. Since there is a large demand for calculus instruction from client disciplines and since calculus is an interesting, vital, and useful area of mathematics, we intend to retain the calculus sequence as our introductory mathematics sequence. The main disadvantages of having a single introductory mathematics sequence emphasizing calculus are that students may not get an accurate view of the nature and scope of mathematics and some students may not have the opportunity to study the type of mathematics that appeals most to them. The Problem-Solving Seminar and Discrete Mathematical Structures courses were created in part to address these concerns. The present curriculum plan attempts to address these issues not only by making our calculus courses more appealing, but also by allowing students to study linear algebra topics earlier than they previously could.

During the academic years 1992-93 and 1993-94, we used the innovative, computer-based Calculus in Context curriculum [6] in all of our Calculus I and II courses. Created by members of the mathematics departments of the Five College Consortium (Amherst, Hampshire, Mount Holyoke, and Smith Colleges, and the University of Massachusetts, Amherst) with funding from the National Science Foundation (NSF), this curriculum develops calculus concepts in the context of scientific applications, and uses computer programs and graphing packages as exploratory tools. For example, on the first day of Calculus I class, students begin to construct a model for a measles epidemic. After setting up rate (differential) equations for the susceptible, infected, and recovered populations, the students use these equations to predict future sizes of the populations, first by hand and then using simple True BASIC computer programs. As they use the computer to calculate and plot future population sizes, they soon notice that their approximations seem to approach limiting values when they recompute them using smaller and smaller step sizes. In this way, students discover, and gain an intuitive understanding of, the limit process. Later in the first course, the derivative is introduced as the slope of the straight line the students see when they look at a small section of the graph of a (locally linear) function under a "computer microscope." It became apparent early on that our students were gaining a better understanding of the limit process and of the derivative as slope than they ever had in the past. Indeed, the Calculus in Context curriculum forces students to focus on concepts and to communicate clearly---both orally and in writing---about them. As an added bonus, the percentage of students in fall semester Calculus I courses who continue to spring semester Calculus II courses has increased from 45% to 85% since we instituted the Calculus in Context curriculum.

We have been quite pleased with the Calculus in Context curriculum in our Calculus I course and intend to keep it in place. However, although the percentage of students in Calculus I who continue to Calculus II has increased dramatically since we instituted the Calculus in Context curriculum, the number of students who continue from Calculus II to Calculus III has not increased as substantially. Currently, nearly 80% of our Calculus I students proceed to Calculus II, whereas only about one-third of our Calculus II students go on to Calculus III. We believe that this drop off in retention is due in large part to our continuing inability to make Calculus II an interesting and coherent course in which students can be successful. Both the traditional Calculus II syllabus and the more innovative Calculus in Context syllabus for the course (which includes dynamical systems in addition to single variable integration and Taylor series approximations) seem to students to be collections of unrelated topics, which become successively more difficult and which culminate in a topic most Calculus II students find nearly impossible to understand: Taylor series approximations.

In order to address students' concerns about the Calculus II course, we have tried changes in both pedagogy and content. Prior to adoption of the Calculus in Context curriculum, we used computer demonstrations as well as a series of laboratory sessions, some computer-based, in our Calculus II courses. (See [1], [2], [4], and [8] for details.) While this seemed to improve students' attitudes, understanding, and communication skills, the retention rate did not increase significantly. The Calculus in Context Calculus II curriculum, which we began using in Spring, 1993, includes exciting topics which build on students' work with differential equations in Calculus I. Unfortunately, during the three semesters in which we used it, we found that it greatly overestimated our students' prerequisite knowledge, making the Calculus II course as disjointed and difficult for them as our previous versions of the class. Furthermore, students who had studied Calculus I in high school or at another college or university were at an even greater disadvantage in this course, because of their lack of experience with differential equations. These problems precipitated what we hoped would be a temporary retreat to a more traditional Calculus II course.

During the academic year 1994-95, we used the Calculus in Context curriculum in the Calculus I course, but, in order to make it possible to place entering students with a traditional Calculus I background in Calculus II and to extend "reformed" calculus instruction to our Calculus III course, we used the more conventional NSF-funded Calculus Consortium at Harvard curriculum for the Calculus II and III courses [3]. In addition, in our Pre-Calculus course, we used preliminary materials written by members of the Bridge Calculus Consortium at Harvard, along with the laboratory manual, Precalculus in Context: Functioning in the Real World [5]. Although our experience with the Calculus in Context curriculum and, more generally, with technology in the classroom led us to adapt the Harvard materials in a manner which emphasizes discovery-based learning and in which computing is used as an integral tool, we have not been satisfied with the Harvard curriculum's traditional approach to single-variable calculus. Our only misgiving about our Calculus III course, which has traditional multivariable calculus content and in which we've used the Maple computer algebra system for two years and the Harvard Consortium's preliminary Multivariable Calculus text for one year, is that we try to cover too much material in too little time, resulting in inadequate student understanding. The present curriculum reform plan attempts to remedy this situation by distributing multivariable calculus topics throughout the Calculus II and III curricula.

Our immediate goal then is to make our Calculus II course more intrinsically interesting, coherent, and accessible than it has been, with the ultimate goal of enticing more Calculus II students to continue their study of mathematics. This means, of course, that we must ensure that the core curriculum courses that follow Calculus II, especially Calculus III, also are attractive courses for students. Hence, while our main objective for our core curriculum is to provide students with a solid foundation in mathematical ideas, methods, and thought processes within a reasonable time frame, we also want to help ensure that students continue their mathematical study by making the core mathematics curriculum, and especially our Calculus II course, as appealing and as flexible as possible for them.

Model

Goals for student growth. Members of the University of Redlands mathematics faculty are concerned primarily that our students understand mathematical concepts and that they learn how to think mathematically---that is, analytically, creatively, logically, abstractly and, when appropriate, concretely. Our goal is for students to develop mathematics skills and, more generally, thinking skills they actually can use both in and beyond our classrooms. We want our students to become independent learners and thinkers who take responsibility for their own learning, who explore and discover mathematics on their own, and who take risks in problem solving. Such students must be able to learn in a variety of contexts, including abstract, concrete, and applied settings, and by various methods, including exploring examples, solving problems, reading, discussing and listening. Furthermore, we believe that expecting students to communicate mathematics clearly---both orally and in writing---not only deepens their understanding of mathematical concepts, but builds their confidence and gives them practical skills which they will use both in and after college. Most importantly, we want our mathematics students to develop enough ability, confidence, and enthusiasm not only to continue to the next mathematics course, whatever it may be, but to continue to learn and use mathematics throughout their lives.

We, along with many other mathematics educators across the nation, have come to believe that one of the best ways to ensure that individual students grapple with, understand, and learn to use mathematical ideas is to facilitate for them what our colleagues in education call constructive learning, or what we in mathematics tend to call discovery-based learning. This view, which is based on extensive study of existing programs as well as on our own experiences in incorporating computer use into our calculus, differential equations, linear algebra, and other courses, is consistent with the recommendations made in such publications as Moving Beyond Myths: Revitalizing Undergraduate Mathematics [7]. In recent years, we have attempted to facilitate discovery-based learning in all of our courses, but especially in our core curriculum courses, primarily through computer activities and other cooperative learning strategies.

Curriculum Reform Plan. While we believe that our recent improvements in pedagogy are helping us achieve our general goals for student growth as well as our more specific goal of making our core curriculum more effective and attractive for students, we feel that changes in content also are necessary in order to achieve this latter goal. As described above, we believe that the weak link in our core curriculum is the Calculus II course. In order to make this course more attractive, effective, and rewarding for students than it currently is, we propose the following integration of our Calculus II and III curricula.

Calculus II:

Single and multiple integration
Integration as accumulation, introduced in the context of
applications
Fundamental Theorem of Calculus
Techniques of integration: substitution, parts
Brief introduction to multivariable functions and partial
differentiation
Multiple integration
Nonrectangular coordinate systems

Introduction to vectors
Dot product and projections, cross product
Equations of lines and planes in three-dimensional space

For more information, please see Calculus II Course Information.

Calculus III:

Vector calculus
Multivariable functions (again) and their limits
Partial differentiation (again), chain rule
Gradient, directional derivative
Parametric equations
Line and surface integrals
Stokes', Divergence, and Green's theorems

Sequences and series
Sequences, series, power series, Taylor series
Transcendental functions

Both courses would continue to be computer-based, employing True BASIC computer programs and the Maple computer algebra system. While the actual topics proposed for the new Calculus II and III courses are quite traditional, we would continue to emphasize discovery-based learning in which computing is used as an integral tool, and, whenever possible, we would continue to introduce concepts in the context of scientific applications. Themes carried over from Calculus I to Calculus II and III would include successive approximation and multivariable functions, but less attention would be paid to differential equations in the new Calculus II and III courses. This should make it possible for students to succeed in the new Calculus II course whether their previous course was our Calculus in Context Calculus I course or a traditional high school or college Calculus I course.

Advantages of curriculum reform plan: We believe that the topics we've included in the Calculus II course form a more coherent whole than do the topics in our current Calculus II curriculum. We also believe that we have selected interesting, accessible concepts for the Calculus II course. The Calculus III topics also are interesting, of course, but we have found that they are more difficult for students than those we've included in the Calculus II course. We anticipate, however, that by the time students reach the Calculus III course, they will be able to understand these concepts. In particular, we predict that students will have a much better chance of understanding Taylor series approximations if their introduction is delayed to Calculus III. We also anticipate that students will understand the traditional multivariable calculus topics better if they are distributed throughout the Calculus II and III curricula. In summary, we believe that the proposed Calculus II and III curricula are more commensurate with students' interests and abilities than are our current curricula for these courses, and we expect students to be more successful in our new versions of these courses.

Our curriculum plan also has the advantage of allowing students to progress straight from Calculus II to Linear Algebra, as shown in Diagram 2. (We note also that introducing vector topics at the end of Calculus II will give us an opportunity to introduce matrices and determinants in that class.) Of course, students also may proceed to Calculus III or to both Linear Algebra and Calculus III. This makes their core curriculum schedule slightly more flexible. We hope that reaching Linear Algebra sooner, together with the option of enrolling in our new Discrete Mathematical Structures course early on, ensures that more students who prefer discrete mathematics and algebra over calculus continue to study mathematics.

Note also that this curriculum plan ensures that physics students get some of the calculus concepts they need---most notably vectors and multiple integration---earlier than they previously did.

Disadvantages of curriculum reform plan: The primary disadvantages of the plan are that 1) new students who formerly placed into Calculus III now must begin with Calculus II, 2) increased flexibility in student schedules may make scheduling courses more difficult for us, and 3) we have been unable to find textbooks that support this sequence of topics.

Virtually all students who place into Calculus III have Advanced Placement (AP) Examination scores of 4 or 5, and are awarded four to eight units of credit at the University of Redlands. We hope that these credits and perhaps the waiver of an elective in the mathematics major or minor would be enough to entice them to major or minor in mathematics. Nevertheless, we view 1) as a serious disadvantage of our plan.

We are certain we can deal with 2), however, as of this writing, we have been unsuccessful in overcoming 3).

Reception of Model by Department, Partner Disciplines, Administration

The other members of the Department of Mathematics have approved our plan to revise the content of Calculus II and III, but share our concerns about choosing textbooks for these courses, placing new students in these courses, and scheduling mathematics courses for optimal student flexibility.

Faculty in partner disciplines are largely responsible for the continuing centrality of the calculus in our introductory mathematics sequence. The majority of these faculty have supported our move toward an applications- and computer-based calculus curriculum. In fact, many of them participated in a two-day workshop designed specifically to introduce faculty in client disciplines to the Calculus in Context curriculum and facilitated by the mathematics faculty during Summer, 1993. Physics faculty should appreciate students' earlier introduction to multiple integration and to vectors, which would occur now in Calculus II rather than in Calculus III.

The University's administration has been supportive of the mathematics faculty's curricular plans, and especially of our involvement in the calculus reform movement.

Time Line for Implementation

We plan to offer our new version of Calculus II beginning in Fall, 1995, and our new version of Calculus III in Spring, 1996. As of this writing, textbooks have not been chosen for these courses. Since Calculus in Context is the text for the Calculus I course, it probably would be used in the Calculus II course, along with a multivariable calculus text, such as the Harvard Consortium's text. The same multivariable calculus text would then be used in the Calculus III course. Despite the fact that we would use just two textbooks for three courses, we are concerned that students who take Calculus II only will suffer undue financial hardship as they would have to buy both textbooks for just one course. Ideally, we would locate or create a textbook for Calculus II and III that covers the topics we wish to cover in the order in which we wish to cover them, and that employs the applications-, computer-, and discovery-based learning techniques we wish to employ.

We currently offer one or two sections of Linear Algebra each spring semester; we plan to offer one section of Linear Algebra in Fall, 1996, to accommodate students just finishing Calculus II or Calculus III.

Acknowledgments: We wish to thank Professor Donald Small of the United States Military Academy for organizing the Seven-into-Four Curriculum Conference as well as the ensuing West Point Core Curriculum in Mathematics Conference, and Major Donald Engen, of the U.S. Military Academy, for helping to coordinate the latter conference. We also wish to thank Lieutenant Colonels Gary Krahn and Kelley Mohrmann for serving as our U.S. Military Academy faculty mentors during the latter conference.

Contact Person: Dr. Janet L. Beery
Department of Mathematics
University of Redlands
1200 E. Colton Ave.
Redlands, CA 92373
E-mail: beery@uor.edu
FAX: (909)793-2029
Office phone: (909)793-2121, extension 3118

Bibliography

[1] Beery, Janet L. Calculus with Weekly Exploratory Laboratories. Problems, Resources, and Issues in Mathematics Undergraduate Studies (PRIMUS). June, 1993.

[2] Beery, Janet L. My Second Favorite Lab---To This Point: Discovering Integral Formulas. Computer Algebra Systems in Education (CASE) Newsletter. July, 1993.

[3] Calculus Consortium based at Harvard. Calculus. 1994. Multivariable Calculus (Preliminary Version). 1995. New York: John Wiley & Sons.

[4] Cornez, Richard, Janet Beery, and Mary Scherer. A Computer-Based Calculus Curriculum. College Teaching. Spring, 1993.

[5] Davis, Marsha J., Judy Flagg Moran, and Mary E. Murphy. Precalculus in Context: Functioning in the Real World. 1993. Boston: PWS Publishing.

[6] Five College Calculus Project. Calculus in Context. 1995. New York: W.H. Freeman.

[7] Moving Beyond Myths: Revitalizing Undergraduate Mathematics. 1991. Washington, DC: National Academy Press.

[8] Scherer, Mary, Janet Beery, and Richard Cornez. Starting Small: Gradual Introduction of Computers into Calculus Courses. PRIMUS. June, 1993.

Major and Minor Requirements

Calculus II Course Information