PROJECT CALC: CALCULUS WITH MATHEMATICA Bowdoin College, Brunswick, Maine A Test Site for Duke University's Project CALC. Program Director at Bowdoin: William H. Barker Supported in part by NSF Grants USE-9053397 and USE-9052224, and by a grant from the George I. Alden Trust, Worcester, Massachusetts. WHAT IS PROJECT CALC? Project CALC ("Calculus As a Laboratory Course"), developed at Duke University with support from the National Science Foundation, is part of a nationwide effort to revitalize the teaching of calculus. The project represents a major revision of the calculus curriculum, built around (1) selected problem areas in the natural and social sciences that motivate the introduction of calculus techniques, (2) a discovery-based computer laboratory, conducted in much the same way as a laboratory for a science course, and (3) writing assignments which place the emphasis of the course on conceptual understanding, not solely on computational skills. At Bowdoin the Project CALC materials have been reformulated around the computer mathematics package Mathematica, a sophisticated integration of text-processing, numeric and symbolic computation, graphics, and programming. To support this conversion, Bowdoin has been awarded grants from both the National Science Foundation and the Alden Trust. In particular, these grants have allowed the College to construct two Macintosh computer laboratories to support the program. GOALS OF THE PROJECT. The goals of Bowdoin's revised calculus sequence are to (1) instill an understanding of calculus as the primary language of science and applied mathematics, and (2) develop the skills necessary to solve those real world problems which require the techniques of calculus. These skills consist of the abilities to (a) model a scientific or mathematical problem in the language of calculus, (b) select tools most appropriate to the problem, (c) compute and analyze answers for the problem, and (d) communicate the meaning and derivation of the computational results in clear prose. MEANS FOR ACHIEVING THE GOALS. (1) Calculus as the language of science. The Project CALC curriculum studies real, non-contrived, non-trivial problems in mathematics, science, and economics. Calculus techniques are developed as needed to solve such problems.(2) Skills. (a) Modelling. Certain problem areas from science and economics are considered from a variety of viewpoints, resulting in a variety of competing models. Some problem areas are returned to multiple times, resulting in a sequence of increasingly sophisticated models. The construction of the models is given as much emphasis as the computational results which flow from them. (b) Selection of Tools. The collection of tools available to our students is larger than in a traditional course. In addition to exact symbolic computations, strong emphasis is given to approximation procedures, graphical analysis, and data analysis-- often in the context of empirical and experimental problem solving as opposed to deductive and theoretical approaches. Computing is also emphasized as a tool, and its use permeates the program. (c) Computation and Analysis. In addition to developing the usual "paper and pencil" skills, heavy emphasis is placed on achieving an understanding of the basic concepts of calculus. Without such understanding, meaningful analysis of results is impossible. (d) Communication of results. A large number of computer laboratory experiments and class projects result in written reports. These reports are first submitted in draft form and then revised on the basis of the comments received. Developing problem solving skills is not easily done in traditional lecture classes. Problem solving requires the active involvement, interest, and creative energy of the students, attitudes which are not readily developed in a lecture settings. For this reason Project CALC uses a variety of teaching methods, all designed to promote self- discovery of mathematical fact, cooperation with other students, and personal involvement in the learning process. These methods include (1) weekly computer laboratories, (2) in-class group projects, (3) in-class data collection exercises, (4) interactive classroom computer demonstrations, (5) extended discussions built from student questions, and (6) reading assignments intended to convey the basic factual information of the course. MATHEMATICA AND PROJECT CALC. The Project CALC curriculum, the text materials, and the basic content of most of the computer laboratory experiments have been developed at Duke University. The primary innovation which Bowdoin brings to the program is the reformulation of the laboratory materials and classroom demonstrations into Mathematica notebooks. A "notebook" is an interactive document produced by Mathematica's sophisticated integration of text-processing, numeric and symbolic computation, graphics, and programming. For example, a Bowdoin laboratory assignment is given in the form of a notebook; the preliminary discussion and all the specific instructions are supplied, in addition to executable Mathematica commands that allow the investigations to be performed "in" the notebook itself. The results of the laboratory sessions are recorded automatically into the notebook, utilizing the document as the lab record book as well as the lab instruction manual. If required, a final laboratory narrative is written and attached to the end of the notebook, and the notebook is then submitted electronically as the laboratory report. Mathematica has also proven to be a nearly perfect tool for developing interactive computer demonstrations for classroom use, especially demonstrations making significant use of graphics. Interactive in this context means the package is easy to use yet retains flexibility to c over a wide range of input variables and options. Using such a package--as opposed to a prearranged computer presentation--allows a true discussion to be organized around the computer. The instructor can respond to student questions and suggestions, explore topics collaboratively with the students in an experimental setting, and uncover results that are surprising even to seasoned calculus veterans. The Bowdoin program focuses on mathematics, not Mathematica. Because of this, the students are not required to learn more about Mathematica then they absolutely require. In the first semester they learn how to generate graphs and scatter plots, define functions, and perform elementary manipulations with data lists. In the second semester they are introduced to the basics of symbolic computation: differentiation, integration, and methods for simplifying expressions and solving equations. More complicated procedures (e.g., generating a sophisticated animation) are handled by special "packages", generally defining new Mathematica commands that produce the desired results. The code for such a package is always available to the students. In the third semester the students learn to use the Mathematica tools that are relevant to multi-variable calculus. Utilizing Mathematica in this fashion has minimized student complaints concerning the difficulty of using the computer--in fact, almost all our students cite the use of computing as one of the most positive aspects of the new program. OTHER SOFTWARE. Mathematica alone handles all the major computing functions of the program. However, some enhancements are possible by use of auxiliary software. Expressionist provides typeset quality formulas in Mathematica notebooks. SuperPaint produces more freeform graphics than can be gotten from Mathematica, and can be used to modify Mathematica produced graphics. Tempo II Plus, a macro program for the Macintosh, is a significant aid to instructors when correcting electronic files. EQUIPMENT. Currently Bowdoin's Project CALC has two Macintosh computer laboratories. Each lab consists of 10 Macintosh IIcx computers with 8 megs of RAM, 40 meg internal hard disks, and 13 inch color monitors. In addition, each lab has one IIci with a 100 meg internal hard disk connected to a video display panel for computer demonstrations. All the machines are linked together via an AppleShare network--a Macintosh SE/30, equipped with a total of 340 megs of storage, is used as a dedicated file-server. One lab also has an ImageWriter II dot-matrix printer and a LaserWriter IINT printer, both networked to the lab computers. Two more Macintosh IIci's (configured in a similar fashion to the lab demonstration machines) are located on movable carts for use in classroom demonstrations. Each is connected to a video display panel, one of which is a color panel (well worth the expense). Additional IIci computers are located in instructors' offices, and are connected to the laboratories via the AppleShare network. This allows for electronic submission, correction, and return of student reports. All the machines are further connected to a UNIX DECstation 5000/200, a workstation that contains its own copy of Mathematica. This allows a user to run the Mathematica notebook front end on a Macintosh while performing all computations on the remote Mathematica kernel; computational speed will sometimes increase by more than tenfold when compared against working entirely on a Macintosh. (Not surprisingly, the increase in speed is inversely proportional to the number of users.) This equipment configuration has been entirely satisfactory. The IIcx is an excellent machine for a student laboratory, the n-View video display panels produce large, well-focused video images in both the lab and the classroom, and the AppleShare system has performed all the necessary file-server tasks in a reliable and convenient manner. In order to handle the demands of a growing program, we anticipate the need for at least one further Macintosh laboratory and additional classroom display equipment in the not- too-distant future. LABORATORY AND DEMONSTRATION NOTEBOOKS. All thirty-seven of the Project CALC laboratories for the three semesters of Project CALC have been recast into Mathematica Version 2.0 notebooks. Many of the original Duke laboratories have been modified considerably to make the best use of the strengths of Mathematica. Although a significant number of classroom demonstration notebooks have also been constructed, we have increasingly turned to the Mathematica packages written for the laboratories to drive the classroom demonstrations. Student reaction to the labs has been positive--they are frequently cited on evaluations as comprising one of the best aspects of the course and as significant aids in developing an understanding of the material. The students were even more positive about the value of in-class computer demonstrations--there was a near unanimous desire to have more demonstrations in the future. TEACHING METHODS. It is a basic tenet in Project CALC that lecturing to first year calculus students is generally an ineffective teaching method. Because of this, many alternate teaching techniques are used. The use of weekly computer laboratories has already been discussed. In addition, many of the regular class sessions are devoted to group work on in-class projects. In such a class students are given a problem and told to solve it by working cooperatively in groups of approximately three or four. The instructor acts as a facilitator, traveling around the room to provide help and advice to the students as necessary. Often the students are required to write up their work as essay reports on the computer. A few classes are devoted to data collection, e.g., empirically determining the relationship between the frequency of a pendulum and its length. In all cases the intent is to later compare the data to a theoretical model. The subsequent comparison of measurement against theory is beneficial--it reinforces the notion that calculus has a value in real, honest problems. Students are expected to read the appropriate sections of the text, generally before the material is encountered in the class. Not all of this material is repeated in class sessions--instead, the sessions focus on the most central and/or most difficult concepts. In addition, many classes are organized solely around student questions. WRITING INSTRUCTION. The program places strong emphasis on written work. Each semester a total of five labs and/or projects require significant written reports. Given this substantial but not overwhelming number of reports, the writing component of the course has been well-received by our students. The use of draft and revision procedures is particularly important: they guarantee that instructor comments on the drafts are carefully read and contemplated, and they allow the students to rethink and deepen their understanding of the issues and concepts under consideration. At Duke the writing component of Project CALC is supported by a strong, University-wide writing program. At Bowdoin no such program exists. For this reason, the approach to judging student writing in Bowdoin's Project CALC is simply to require understandable prose. If a sentence or paragraph is confused--no matter what the reason--then it is bad. On the other hand, if a passage conveys its message in a clear, unambiguous manner, then it is good. The instructors pass back into mathematics when judging the accuracy of the message. STUDENT REACTION TO PROJECT CALC. We have averaged approximately 20 students per semester in Project CALC courses over the last two years--the course evaluations submitted by these students at the end of each semester have been, on the whole, very positive. In general, the students approved of the emphasis placed on (1) concepts over computational skills, (2) real world applications, (3) collaborative learning (although some abuses were cited), and (4) written work (especially when it involved a draft and a revision process). The students also generally liked the labs, and cited the computer in positive comments far more than in complaints. The first year of the program produced a howl of complaints about the workload--these complains essentially disappeared during the second year after we reduced the number of significant written reports to five per semester. Our biggest problem in attracting large numbers into the program remains the apprehension of entering students about a mathematics course that emphasizes writing, computing, and collaborative learning. Given the more familiar and comfortable alternative, it is not surprising that most entering students choose the traditional calculus sections. THE QUALITY OF STUDENT WORK. The overall quality of the students' work on the laboratory and project reports has been good. Roughly a quarter of the first drafts of these reports would have major problems or errors; few fell into this category upon resubmission. In many cases the students went further than requested or expected, using the power of Mathematica to make and then test additional conjectures about the problems under consideration. The student performance on examinations has improved from mediocre during the first year to reasonably good (though still not ideal) during the second. The Project CALC examinations are more conceptual than their traditional counterparts--reflecting the values of the program--so the students cannot prepare themselves simply by drilling on template problems. In order for our students to understand and work with more difficult and sophisticated ideas, class work and assignments must clearly highlight the desired concepts and return to them constantly during the term. During the second year of our program we were far more attuned to this need; we believe this to be the primary reason for the improvement in student performance on examinations. The students' classroom performance has been gratifying, not only when working on group projects, but also during student generated discussion sessions. There has been a level of enthusiasm and genuine interest in the material that is rarely seen in traditional calculus. Moreover--especially during computer demonstrations-- many good questions are asked ( and often answered by other students), and many insightful comments are offered. Thinking is going on in the classroom. DISSEMINATION. The Bowdoin materials should be of value to any institution interested in using Mathematica in calculus instruction, especially in conjunction with the Project CALC textbook and curriculum. Our materials are made available to anyone wishing to have them. In addition, visits to Bowdoin by people interested in viewing the program are encouraged and welcomed. In the fall of 1992 D. C. Heath will publish a preliminary version of the Project CALC textbook as well as a preliminary version of the Mathematica laboratory manual for the first two semesters of Project CALC. This manual will consist of printed copies of the first twenty-five of our thirty-seven labs. For information on ordering this manual, contact the College Mathematics Department, D. C. Health and Company, 125 Spring Street, Lexington, MA 02173. For further information concerning Bowdoin's implementation of Project CALC, and for information on how to obtain copies of our Mathematica notebooks, please contact: Prof. William H. Barker (207)725-3571 Department of Mathematics (207)725-3123 (FAX) Bowdoin College wbarker@.bowdoin.edu Brunswick, Maine 04011