## About Calculus in Context at the University of Redlands:

CONTENTS OF THIS REPORT:

Course Information for Calculus in Context I and II

NSF ILI Grant for Classroom Computer Laboratories

Plan to Integrate Calculus II and III Curricula

Calculus II Course Information

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), the Calculus in Context 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.

Text: Calculus in Context, The Five College Calculus Project, James Callahan and Kenneth Hoffman, W.H. Freeman, 1995

****************

2. Calculus Reform and Calculus in Context at Redlands: We have used Calculus in Context for all of our Calculus I courses since Fall, 1992. We used Calculus in Context for our Calculus II courses for three semesters. The University of Redlands mathematics faculty began planning for and implementing calculus reform during the summer prior to the 1990-91 academic year, when four faculty members planned computer demonstrations and assignments for an experimental section of our Calculus I course. In academic year 1991-92, we added sections of Calculus I and II with biweekly discovery-based laboratory sessions, and in AY 1992-93, we adopted the Five College Calculus in Context curriculum for all sections of Calculus I and II. Although we have used the Calculus in Context curriculum in our Calculus I course since Fall, 1992, we have used the Harvard Calculus Consortium texts for all or part of our Calculus II and all of our Calculus III course since Fall, 1994. The reform of our calculus curriculum continues: during the 1995-96 academic year, we are implementing a plan to integrate our Calculus II and III curricula. (More information about our current Calculus II and III curricula appears in Section 7, at the very bottom of this document.)

Reasons for abandoning Calculus in Context for Calculus II include:

A) We found the Calculus in Context II materials too unpolished for part-timers (and full-timers who weren't willing to make a huge time commitment) to use. Text was unrealistic about students' prerequisite knowledge of, e.g., inverse trigonometric functions.

HOWEVER: Many of these problems have been alleviated in the newest edition.

B) Calculus in Context never did come out with Calculus III materials.

HOWEVER: The text does contain some multivariable material. The text certainly contains enough material for three semesters of work by average students!

C) Adopting a more traditional Calculus II curriculum helped us resolve some high school to Calculus II and Calculus II to Calculus III articulation problems we had when we used Calculus in Context in Calculus II.

HOWEVER: We are using it again for part of our Calculus II course; specifically, for single variable integration.

3. Facilities: Each of our calculus courses meets in a classroom equipped with 12-14 computers arranged in a U-shape around the outside of the classroom, tables arranged in a U-shape inside of these, and chairs on wheels between the two U's. Meeting in a computer classroom with students working two to a computer allows students to work cooperatively to explore and discover calculus concepts and to solve messy real-world problems during every class session. Having the computers arranged around the outside of the classroom, rather than in rows or clusters, allows the instructor (and students) to see every computer screen from virtually anywhere in the classroom. The wheeled chairs allow the students to move easily between computers and tables; having the tables arranged in a U-shape encourages class discussion. In summary, the entire configuration encourages the instructor to coach rather than orate, and the students to actively discover rather than passively receive knowledge.

NSF ILI Grant for Classroom Computer Laboratories

4. General Impressions: We've had what I think are the usual successes and failures with such projects. I think that students really are understanding the calculus better and are more excited about it. I'm especially excited about the opportunities the curriculum affords students to discover and understand the concepts of calculus. I believe that introducing my students to the limit process by having them make successive approximations in the context of a measles epidemic has given them a better understanding of the limit process than my former students have had. Introducing the derivative as the slope of the straight line the students see when they look at a graph under a "computer microscope" has had a similar effect. I've introduced the derivative in this way before, but have always switched quickly to the secant line characterization. I think that my past students believe that the derivative gives the slope of the graph, but I think more recent students "feel it in their bones," as Ernst Snapper would say.

On the other hand, students also are more frustrated than they were in the past because of their lack of a sense of mastery of the material. Students constantly are pushed to explore and discover new and challenging concepts on their own and have few techniques and formulas to latch onto, so of course they're always a bit uncertain and sometimes frustrated.

I do try to take full advantage of the opportunities for discovery- and activity-based learning that the curriculum affords. This is time-consuming (both class time and preparation time), and perhaps students' adjustment to context- and computer-based learning is hard enough without this additional adjustment to investigative and interactive learning. I probably should do a better job of warning students about this, of discussing frequently how this course differs from math courses they've had in the past (and why I think this is a good thing), but I just hate spending precious class time on anything but calculus! I also should do a better job of reassuring students that they are doing well. Again, I think the biggest problem in a course like this is the students' invariable sense of uncertainty; it is much more difficult for students to gain a sense of mastery when they constantly are discovering and experimenting with new ideas rather than memorizing formulas and techniques introduced by the instructor.

Despite all this, I really have fun teaching this course and believe it is an effective and successful course.

When we used Calculus in Context for both semesters, I covered the following topics in my classes.

Calculus I

Successive Approximations - measles epidemic model, Euler's method approximations by hand and by computer, successive approximations and limits

The Derivative - rates of change, "computer microscope" and local linearity, definition of derivative, local linearity ("microscope equation"), derivative as function, chain rule, product and quotient rules

Differential Equations - population and predator-prey models, closed form solutions, exponential and logarithm functions, review of Euler's method

Differentiation, continued - review of techniques, application to shapes of graphs, application to optimization

Calculus II

The Integral - Riemann sums and integral introduced via applications, FTC

Techniques of Integration - antiderivatives, substitution, parts, separation of variables, partial fractions, trigonometric integrals, Simpson's Rule, improper integrals

Periodicity - periodic behavior, periodic functions, differential equations with periodic solutions

Series and Approximations - Taylor polynomials and series, power series, convergence tests, l'Hopital's Rule

Course information for Calculus in Context I and II

6. Specific advice on teaching the first semester course:

We cover Chapters 1-5 only during the first semester, putting off integration until the second semester. Of course, since differential equations is one of the main topics, students compute many antiderivatives.

I probably assign too many homework problems for students to complete outside of class as thoroughly and thoughtfully as I like. Since I'm not willing to delete any of the problems I currently assign, I have students work on some homework problems during class. This may sound rather high school-ish, but it is what the CiC authors intended and it seems to be effective. I, of course, listen in and push and prod the students as they work. I usually assign specific problems for class work and follow up with discussion to unify or extend topics. Our MWF, 1 hour, 20 minute class sessions facilitate this in-class work; if you aren't able to work with students in class as much as I do, you may find that you need to cut back on the amount of homework.

As advised by the CiC authors, I spend a lot of time on Chapters 1 and 2, and especially on the SIR model. In Chapter 2, I begin using world population growth as my primary class example. We do exponential growth and then logistic growth. In fact, I omit supergrowth in order to spend more time on logistic growth. Eventually, I give them (and we check together) the formula/solution for logistic growth (which doesn't appear in the text until later). I also use as a recurring class example boatloads of bunnies landing on desert isles with 1) an unlimited supply of food and water, 2) a limited supply of food and water, and 3) a population of foxes (with carrying capacity for the rabbits---see Section 4.1). Yes, it's silly, but it does seem to get the ideas across.

Students have much difficulty understanding what it really means for a function to be a solution to a differential equation. I don't know what to tell you here because I think the authors' discussion of this in Section 4.2 is excellent. You can arrange this discussion more effectively on the blackboard than the authors have done in the text, and this helps students. I have students do a very mechanical lefthand side vs. righthand side check, and eventually most of them get it (I hope).

Students' other major difficulty is with the computer. It's hard to convince them that they do not have to know how to program in order to succeed in the course. I think you'll find that the more you teach them about programming, the happier they'll be. In fact, what they seem to want is for you to teach them programming as a separate topic; I resist this, because I keep thinking I can convince them that programming is an easy, but integral tool.

I also try to introduce all of the algebra the students are supposed to know but don't in context, i.e. as needed. This means that I introduce/review topics from Section 1.2 as they are needed. This has the advantage of not disrupting our study of the SIR model in Chapter 1. As much as I believe in my approach, I must warn you: students seem to prefer chunks of information over information as needed. Also, I've found in recent years that students know/remember very little about exponential and logarithm functions; I find that I pretty much have to start from scratch with logarithm functions.

I teach the Product Rule and the Quotient Rule at the end of Chapter 3, rather than waiting until the start of Chapter 5. I don't usually cover partial derivatives.

7. New plan:

And now let me ask you for some advice about what we're trying to implement this year:

After finding that the Calculus II course, whether traditional or reformed, seemed to most students to be disjointed, uninteresting, and, in places, frustratingly difficult, we revised the Calculus II and III curricula in order to make them more commensurate with students' interests and abilities. Briefly, we've included both single and multiple integration, as well as an introduction to vectors, in Calculus II; and the vector calculus from gradients to Green's Theorem, along with sequences and series, in Calculus III. In these courses, we continue to emphasize computer use, applications, and cooperative learning. Now for my questions:

2) Keeping in mind that we're still teaching differential equations and differentiation (using the Calculus in Context text) in Calculus I, can you suggest a text for our Calculus II and III courses? Thanks in advance for any advice you are able to offer.

Plan to Integrate Calculus II and III Curricula

Calculus II Course Information

****************

or to contact me at:

Dr. Janet L. Beery
Department of Mathematics
University of Redlands
Redlands, California 92373
(909) 793-2121, ext. 3118
beery@uor.edu

updated 5/96

Calculus Index Page