ABSTRACT

The second year mathematics courses at Stevens Institute of Technology deal with ordinary and partial differential equations, linear algebra, multiple integration, and surface integrals. During the 1996-97 academic year the author used Scientific Workplace (SWP) and Scientific Notebook (SNB) as tools in teaching these courses to 260 students. SWP and SNB are technical word processors which produce .tex files.
They each contain a Maple kernel which allows the performance of a large number of mathematical procedures such as algebraic manipulation and simplification, graphing, differentiation and integration, solving algebraic and differential equations both exactly and numerically, matrix manipulation, etc. The World Wide Web was used as a vehicle for transmitting software and files as well as a learning tool. Several projects were prepared in SWP and SNB which required the student to use Maple. One project dealing with a mass-spring-damping system is interactive and represents a rather striking balance between analytic solution, use of Maple, and simulation. A second project combines Web searching, software downloading and installation, and SWP or SNB to study some first order differential equations. Others deal with matrices, Fourier series, numerical solutions for first order differential equations, and multiple integration.

This paper discusses the benefits gained by integrating computer technology into these courses as well as the problems encountered.

The Course in Differential Equations

Differential equations is a first semester sophomore course at Stevens Institute of Technology. It covers such standard topics as first and second order ordinary differential equations, undetermined coefficients, variation of parameters, LaPlace transforms, Euler's method, and infinite series solutions. The course meets four hours per week with two hours of lecture and two hours of recitation (drill). The author lectured twice a week to 150 students divided into two groups. Two teaching assistants handled the twelve hours of recitation per week. For recitation the students were divided into six groups of 25 students each. The text used was the second edition of Fundamentals of Differential Equations and Boundary Value Problems by R. Nagle and E. Saff published by Addison -Wesley. The software used was Scientific Workplace which is produced by TCI Software Research under the auspices of the Brooks/Cole Publishing Company.

Software Distribution: Every Stevens student is required to have a computer. In addition, all dormitory rooms are connected to the campus network. Thus the first issue to be dealt with was the distribution of SWP to the students in the course. This was done by compressing the program files which fill 17 diskettes and putting the resulting .exe file on the network. A Web page was then created which allows a student to simply click on a link which then downloads SWP. Those students still running Win 3.11 instead of Win 95, had to download a win32.exe file also. Once downloaded the files are unzipped by running them, and then the student is lead through an installation procedure. Thus students living in the dormitories were able to get SWP by going to the Web site, clicking, and then following the installation procedure.

About 30% of the students live either in fraternity or sorority houses which are not networked or off-campus. These students could not use the download procedure just described given the size of the program (about 60 MB), the time required to download the program, and the automatic timed shut-off built into the Stevens dial-up procedure. To solve this problem CDs with SWP were made and given to those who did not live in the dormitories.

Web Pages: The author prepared several Web pages to go with the course. These pages consist of frames with appropriate buttons. The starting page consists of a buttonbar on the left, a title bar on the top, and the home page of the author in the middle. The student then clicks on the button related to the course s/he is taking. From there one goes to another set of buttons dealing with various aspects of the course such as a course overview, the grading policy, course notes and exams given in previous years, information about the text, the syllabus, projects, homework assignments, meeting times, etc.

One button leads to a display of the students' grades. Grades were tracked using Excel. An html file was created periodically and posted from the Excel files using the Internet Assistant add-on for this program. Names were removed from the original Excel file and social security numbers were sorted from lowest to highest to insure privacy. Thus students could monitor their grades as the course progressed and contact the instructor if they perceived any errors. Students liked the idea that as soon as an exam was graded the results were posted allowing them to know their grades before the exams were returned.

It is worth noting that the main course Web page was accessed almost 2000 times by the students during the Fall 1996 semester.

Classroom Demonstrations: During many of the lectures the instructor had an IBM Thinkpad CDV available for his use. This Thinkpad is so constructed that the back of the screen can be removed and placed on a high intensity overhead projector so that whatever is on the screen of the laptop can be projected for the entire class to see. Students were shown how to access the Web pages, download files, and use SWP. At appropriate times SWP was used in class to solve differential equations, graph the solution to an equation, evaluate an integral, find a derivative, find the first few terms in the series solution of an equation, etc. Students were also shown how to set up the Projects that were assigned.

Projects: Three projects were assigned: one on the Web and differential equations, one on the mechanical vibrations of a mass-spring-damping system, and one on Euler's method for solving first order equations. These projects were written in SWP and required the student to use SWP in specific ways in conjunction with employing the standard analytic tools taught in class. Each project is discussed in some detail below.

• WEBDE: This project begins by directing the student to a Web site from which s/he has to download a .pdf file which requires Adobe Acrobat Reader 2.1 to be read. The student is then told to search the WEB, find Adobe Reader, download it, install it and then use SWP to solve two of the equations in the downloaded .pdf file. It turns out that close examination of the solution which SWP gives to one of these equations shows that SWP misinterprets its own solution and reveals a "bug" in the Maple software. (TCI Software, Inc. has acknowledged that this "bug" is indeed present and hopes to fix it in future releases of SWP.) Thus students learn in a striking way that one cannot simply take what a computer returns at face value.
• Vibrations: A rather lengthy project dealing with a mass-spring-damping system was prepared as an application of the techniques taught to solve second order constant coefficient differential equations. This project deals with the standard topics of mechanical vibrations and simple harmonic motion, damped free vibrations, and forced vibrations. Students are required to use analytical techniques and the Maple kernel in SWP. Each is used to shed light on the other.
• A unique feature of this project is that the .tex file describing it is linked to a simulation program which vividly illustrates the motion. Thus a student can select a certain combination of mass, spring constant, and damping factor with certain initial conditions, analyze the motion analytically by solving the resulting initial value problem, check his/her solution using the Maple in SWP, use SWP to graph the solution, and then view a simulation of the motion for this combination by clicking on a link in the SWP file which takes one to the simulation. Closing the simulation returns one to the project file. Thus the project has the novel feature of being interactive.

The simulation program to which the vibrations project is linked was developed by Virginia Polytechnic Institute and State University as part of their SUCCESS Project. It is public domain software.

• Euler's Method: The third project is one which deals with Euler's Method for numerically solving first order differential equations. The student is first introduced to the simple Euler Method and shown by comparing the numerical results to the exact solution of a simple equation that the numerical procedure in this method leads to error rather quickly. A more precise Improved Euler's Method based on a trapezoid scheme is then presented. This approach leads to a predictor-corrector method.

The Maple kernel in SWP does not allow for the option of writing routines. As a result, the simple and improved Euler Method routines had to be written in Maple V R3 and then imported into SWP.

Problems: Several problems were encountered. The first was that a number of students had trouble getting SWP running on their machines. Three students majoring in computer science who were enrolled in the course volunteered to serve as trouble shooters and were able to solve almost all of the problems. A second problem was that SWP has a number of bugs in it, and this frustrated some students. A third was that the instructor had to be careful not to use the computer "too much" in lecture. After all, the instructor did not want to give the impression that the computer was "taking over the course." Those students who were not particularly adept with computers were especially concerned about this. In order to get feedback on how things were going students were asked to complete a questionnaire about the course on the Web about half-way through the course.

The Second Semester Math Course

The second semester sophomore mathematics course at Stevens deals with eigenvalue problems, Fourier series and separation of variables for partial differential equations, matrices and determinants, multiple integration, surface integrals, and the theorems of Green, Stokes, and Gauss. In addition to the Nagle and Saff text which is used for the first part of the course, the second edition of Intermediate Calculus by M. Protter and C. Morrey, Jr. published by Springer-Verlag is used for the non-differential equations topics.

In early January the author became a beta tester for Scientific Notebook (SNB), so students were encouraged to use SNB in place of SWP. However, this was not required. About half of the 110 students enrolled in this course did opt to use SNB. Since not all of the students in this second semester course had taken differential equations with the author in the fall, these "new" students all used SNB.

SNB is similar to SWP, but it contains a number enhancements and simplifications. It uses the latest version of Maple in its kernel, and allows for connection to the Web. Thus using SNB one can configure Netscape so that a tex file can be downloaded and opened in SNB directly. This avoids the necessity of having to wrap tex files and then zip them in exe format for downloading and extraction for use in SWP. Any tex file prepared in SWP can be opened in SNB and vice versa.

Web pages: Web pages similar to those described above for the differential equations course were prepared for this course. Thus students were able to get all relevant information regarding the course via these pages. A midi file which automatically plays music was embedded in the titlebar as an added "attraction". During the Spring 1997 semester the author's WEB page was accessed more than 4200 times. Clearly students are using the WEB pages developed for the course as an integral part of their learning.

Projects: Three projects were assigned: one dealing with Fourier series, one dealing with matrices, and one dealing with multiple integration. These projects were written in SNB and required the student to use SNB (or SWP) in specific ways. We discuss each in some detail.

• Fourier Series: This projects deals with a 5^th degree polynomial. The student must first graph this polynomial, and is then required to find the first term in the Fourier sine series for the polynomial and insert its graph in the same axes in which the polynomial is graphed. The student next finds the first two terms in the Fourier sine series and graphs that. The procedure is continued until the student has found and graphed the first 5 terms in the Fourier sine series. With each succeeding graph the students sees how the partial Fourier sum better approximates the given polynomial. The student is asked to perform a similar procedure for the partial sums in the Fourier cosine series for this polynomial. SNB (or SWP) easily evaluates the integrals which yield the coefficients in the Fourier series. It is important to keep in mind that the time involved in doing these calculations by hand is prohibitive.
  
• Matrices: This project deals with routine matrix manipulations of matrices such as evaluation of determinants, powers of a matrix, finding inverses, finding eigenvalues, etc. The point here is that for the matrices chosen the performance of these operations by hand would be extremely tedious and time consuming.
  
• Multiple Integration: Students often have trouble sketching the region over which a double or triple integral is to be evaluated. This project uses SNB (or SWP) to assist them in sketching regions and then evaluating double integrals. Surfaces in three dimensions are also dealt with.

Homework: Wherever appropriate students were encouraged to check the answers they obtained to homework problems using pencil and paper by solving the problems in SNB or SWP. In the past the instructor had assigned the odd numbered problems almost exclusively, since the answers to these are to be found in the texts. However, with SNB or SWP the student can easily find the answers to many of the even numbered problems, so these were also assigned.

Use of the Internet: Besides posting course information on the Web, the instructor tried to encourage the students to explore the vast array of information available on the Web. This was done by periodically sending out E-mail indicating interesting Web sites. For example, attention was drawn to PointCast, a program which gives daily updates on news, stock quotes, weather, etc.; to Travelocity, a Web page which allows one to find inexpensive airfares; to Mapquest, where one can get a detailed map of any city in the country as well as directions on how to drive from point A to point B. All in all the goal was to make the student aware that the Web is a tremendous resource which can assist one in an unbelievable variety of ways.

Performance: Student performance on in class hourly examinations has been the highest that the author has seen in the more than 10 years that he has been teaching the sophomore mathematics sequence at Stevens. While it is difficult to analyze precisely the reasons for this given that different (but nonetheless similar examinations) are given each year, the fact still remains that the students appear to have mastered the material better this year when SNB/SWP were incorporated into the teaching/learning experience than in earlier years when this software was not available to them.

A contributing factor may be that students can now concentrate on understanding the mathematics and leave the "drudge work" to the software. For example, students traditionally find the topics of multiple integration and surface integrals difficult. While they may set up a problem correctly, they often make mistakes in evaluating the integrals they have set up and do not get the answer in the back of the book. On the other hand, they may come up with the wrong integral, not get the answer in the back of the book, and attribute it to a calculation error rather than an incorrect set-up. Also, the time involved in evaluating many of the double, triple and surface integrals precludes the student doing more than a couple of problems.

With Scientific Notebook available the student sets up the problem and then uses the program to evaluate the integral in seconds. If s/he gets the answer in the back of the book, then the set-up is correct. If not, then the student goes back and looks at her/his set-up to see where it is wrong. Thus homework assignments are focused on understanding the material rather than on doing tedious calculations. Furthermore, the student can work on more problems than before in the time allocated for the homework. The author believes that this process is a contributing factor to improved performance.

Conclusions: There is no question that the use of a program such as Scientific Notebook or Scientific Workplace in traditional mathematics courses at Stevens added new and important dimensions to these courses. Students were given the experience of using a sophisticated piece of software as a powerful tool in their study of classical subjects in mathematics. In addition, they gained experience in using the Web as a resource in their studies. This use of computer technology tends to add a more "participatory" dimension to learning the mathematics which is lacking when one uses the traditional mode of instruction.

Student evaluations and discussions indicate that many students felt that the experience was interesting and valuable. There are some, however, who are not comfortable with computers and thus found some of the projects somewhat intimidating. However, even these students felt that they gained much from doing the projects.

The question of the balance between the use of software such as SNB and the teaching of standard techniques is difficult to deal with. This author is opposed to the elimination of the teaching of all pencil and paper activities that can be done with software. On the other hand, having something like SNB available encourages one to think about how the presentation of material should be changed, what material should be eliminated and what should be added. The right mix of computer activities and pencil and paper activities is something which will certainly evolve over time as software develops. Our challenge is to incorporate the new without doing away with the key benefits of the old.

Aknowledgement: The author wishes to express his thanks to James Orthmann, who was his teaching assistant during the 1996-97 academic year.