Calculus, Technology, and Coordination

Calculus, Technology, and Coordination

L. E. Levine, V. Mazmanian, P. Miller, and R. Pinkham

Department of Mathematical Sciences

Stevens Institute of Technology

Hoboken, NJ 07030

Computing at Stevens Institute of Technology

In 1983 Stevens was the first school in the country to require a group of undergraduates to purchase a PC and use these machines throughout their academic careers. This naturally led to a requirement that all undergraduates have a PC. However, beginning with the fall of 1999 the PC requirement was replaced with a requirement that all students have a laptop. The goal is that students will use their laptops as part of the teaching/learning experience both within laptop classrooms that contain network and power connections and outside of the classroom while studying and completing assignments. Beginning in the summer of 1999 Stevens supported several projects designed to achieve this goal. This article deals with the calculus project.

The Calculus Project

Approximately 300 incoming first year students at Stevens take calculus. Hence this is a natural course for the use of laptops and technology as teaching and learning tools. With this in mind the first year calculus course was redesigned. Four lecturers, four graduate teaching assistants, and two undergraduates were assigned to this course.

It was decided to use Scientific Notebook (SNB) throughout calculus I and II. SNB is a word processing package that incorporates a Maple kernel. This Maple kernel allows one to perform a wide variety of mathematical operations. A particularly nice feature of SNB is that one can use the Maple kernel without typing any code. One simply types the mathematical expression and then clicks on the appropriate operation to be performed. SNB then performs the operation. Prior experience of one of the lecturers had shown that students could learn to use SNB in a relatively short time. Furthermore, extensive Help files preclude the need to spend very much time teaching students how to use the program.

(More information about SNB can be found at In particular, at one can download a free Scientific Notebook 30-day Trial Version as well as a free Scientific Viewer. These can then be used to view the various files referred to in the rest of this paper.)

Work on this project to redesign first year calculus began during the summer of 1999 and continued throughout the 1999-2000 academic year. The goals of the calculus project were:

To use computer technology to enhance the learning/teaching experience in a calculus course that has a relatively traditional course content.

To develop computer homework assignments and projects that extend the material taught in the lectures;

To develop a web site containing extensive resources for use by students as they learn the subject matter;

To provide the students with a complete set of SNB tex file “notes” that they could download and use;

To have students access the Internet and use SNB during drill sessions in Stevens newly renovated laptop classrooms;

To emphasize understanding of concepts over the ability to do routine manipulations;

To have four different lecturers teach a unified, coordinated course with one agreed upon syllabus, uniform exams, homework assignments, and projects;

To grade as much of the students’ work as uniformly as possible so that all students are graded by the same standards.

We shall now discuss these items in some detail.

Computer technology as an enhancement to learning/teaching

The use of computer technology to enhance the learning/teaching in these courses was done in several ways.

Six computer projects were written, three for each course. The first semester projects were Functions and Curves, Insulated Pipe, and Riemann Sums. The second semester projects were Cubics, Series, and Determinants & Curvature. Each project required the students to use SNB. The project entitled Insulated Pipe was developed in conjunction with a colleague in the Department of Chemical Engineering. This project represents a “real” application of mathematics to engineering.

A number of homework assignments that required the use of SNB were developed. These generally involved modifications and extensions of problems in the text.

In some lectures a laptop connected to the Internet was used. At times the laptop was used to present demonstrations that visually illustrated the topics being taught. Sometimes the laptop was used to present notes dealing with a topic. These notes were then made available on the web. At other times web sites were accessed that contained material related to what was being taught. One such web site that was found to be particularly useful was

Extensive web sites for each course were prepared. These are at and These sites contain the syllabi, course policies, calendars, lecture notes, homework assignments, old exams, and projects. In short they served as an extensive resource for students taking these courses.

A complete set of notes was prepared using SNB and made available on the Web. This resource was in presented in two formats. The first was the WebCalC format first developed at Texas A&M University by Don Allen and his colleagues. ( Since this resource deals essentially with first semester calculus, material dealing with the topics in Stevens second semester calculus was prepared. Permission was granted to put the Texas A&M material on our server, so that students could access everything at Stevens. To see this material one uses File, Open Location in SNB and puts in the URL

The WebCalC material was also used by one of the lecturers to prepare a complete set of notes for each lecture in the course. There is a link to these notes on the main Web pages for each course that are given above.

Use of networked laptop classrooms

During the first part of 1999 Stevens renovated three classrooms that were designed for use by students with laptops. Each classroom has power and network connections for 24 students so that each student can plug in his/her laptop and connect to the Internet. The classrooms are also equipped with projection systems, video players and a variety of other multimedia devices. In short these are state of the art classrooms with a multitude of technological potential.

Our calculus courses (each carrying 3 credits) met for two 50-minute lectures per week and for one 75-minute recitation (drill) session per week. Professors give the lectures whereas graduate teaching assistants handle the recitations. The original plan was to have the students bring their laptops to recitation weekly and use them for a part of each class. The rest of the recitation was to be used for discussing homework. Given that the recitations had at most 24 students in them, it was felt that this class size was ideal for this approach.

Unfortunately, we were not able to utilize laptops in the majority of the recitation meetings. At the beginning of the first semester we encountered a host of technical problems. For example, on the first day of recitation almost all students in classes that met in the morning were able to access the Internet whereas in the afternoon no one could connect. While most of the technical problems were eventually solved, these initial difficulties led to a scaling back of the use of laptops in recitation. (Please note that lecture rooms did not have laptop facilities for students.)

The net result was that in-class laptop usage was primarily limited to introducing the students to the six projects. During these classes the students downloaded the projects and were led through some of the steps that had to be taken to do the project. This proved quite effective. We did give one quiz in recitation in which the students had to use SNB to take the quiz.

We had envisioned more extensive in-class laptop usage, and it may well be that this will be achieved the second time the course is taught in this manner.

One unified course taught by four different people

The lecturers had agreed at the outset that they would all follow the same syllabus, that they would give the same homework assignments, that the same exams would be taken by all students, etc. However, each of us was to prepare and deliver his own lectures. Thus we envisioned one unified course taught by four different people.

The coordination of this effort turned out to be a time consuming and not always easy effort. For example, a draft of each exam was generated by one or two of the lecturers and then circulated to others. Not infrequently it took three or four versions of a given exam before all agreed to a version that we were all (sort of) comfortable with. A similar approach was taken with the projects.

We met together with the teaching assistants several times each semester. At times the meetings were heated. However, we were fortunate in that each person involved in the course always showed the greatest respect for the others and always approached each topic of discussion with willingness to compromise. However, it was not easy.

We had four student graders to grade homework. The teaching assistants graded exams and projects. For uniformity one teaching assistant graded all of a particular problem on a given exam. This was also done with the final exam, but here the professors and the teaching assistants did the grading with one person responsible for a given problem on all examination papers.

Once the results of an exam were tabulated the course coordinator would propose a scale for this exam and ask the other lecturers for their opinions. Finally a scale for all students was agreed upon. Wherever possible uniformity was the rule. Thus in the end each student in the course received the same grade for a given score regardless of who his/her lecturer was.

Observations

No extensive assessment of these courses was done. To do this would have required resources and expertise that were not available. Thus the conclusions given below are essentially anecdotal.

The integration of computer technology into the learning/teaching experience is certainly not easy. The first difficult question to be dealt with is what sort of technology will be incorporated into which topics. Then one must decide how this will be done. Finally, one must develop the software that is to accomplish this. All of this takes a great deal of time and effort. Indeed, we had to cut back on our original plans, since everything took at least twice as long as we estimated.

The use of technology can serve to enhance understanding and facilitate learning. For example, certain animations were shown that definitely helped the students grasp the concepts being illustrated. However, these demonstrations do take away from class time, and if there are any technical problems with the equipment, considerable time can be wasted. Faculty need appropriate support so that use of computer technology in the classroom becomes not much more difficult than using the blackboard.

Related to the comments just made is the fact that it is crucial that faculty have support staff that can deal with and solve hardware and other technical problems in a timely fashion. If this is not provided, then most instructors will not even attempt to use the equipment that the school has provided. Things must work, and they must work easily.

Computer projects and homework assignments are a good way to integrate technology into the teaching/learning experience. They not only serve as a vehicle for encouraging the students to learn how to use the software related to the projects and homework, but also can be leveraged so that students are exposed to extensions of basic concepts introduced during lecture.

There is no question that the approach taken with this course is very different from the scenario in which a group of people all teach the same course but essentially go their own ways by giving individual exams, homework assignments, etc. The coordination effort was considerable. Since all professors were experienced faculty, each of us had our own way of doing things. Having to change one’s approach to accommodate others was a challenge for each of us.

Students in such a large course like the fact that things are done uniformly despite the fact that they have different lecturers. They feel that this approach is very fair. However, to achieve uniformity is again time consuming and requires much coordination.

Students appreciate and utilize a good web site. It serves as a valuable resource that students visit often for all sorts of information about the course. In particular, students like being able to download sample exams and exams given in previous years. It is very important to provide solutions to these exams. As a result of student requests we even provided the solutions to the optional homework problems during the second semester.

We wanted to have as much control over our web site as possible, and, as a result, put our web site on a server that resides in the office of one of the lecturers. Stevens does make WebCT available to faculty, and a number of faculty members at Stevens are using it. However, for our purposes we felt it would be best if we went our own way. On the whole this turned out to be true. However, during the second semester we did use WebCT to post student’s grades.

The web site developed should serve as a resource for the first year calculus course in future years. It provides a record of what was done that can be modified and expanded by those who teach the course in the future. This is important, because all too often something new is tried in a given year, and then it disappears when the people involved move on to other things.

Conclusions

The teaching of mathematics, and for that matter all subject matters, must undergo constant change if students are to be prepared to enter a technological world which is evolving rapidly. It is no longer a questions of whether to use technology in the teaching/learning experience. It is now a question of what technology to use and how and when to use it. There are no readily available answers to this question, and there is probably no one answer to it. The challenge today is for instructors to experiment with a variety of approaches and to then select and build on the ones that work. As our experience has shown, this is certainly far from a simple task.

Acknowledgements

The authors wish to express their thanks to the following individuals who assisted us in teaching these courses:

Dr. George DeLancey, Professor of Chemical Engineering, who played a major role in writing the Insulated Pipe Project.

Maureen McLaughlin-Brearley, Kerry Smith-McNeil, James Orthmann, Paul Rossi, and Natalie Vanatta, who devotedly served as our teaching assistants;

Melissa Garofala, Eve Naim, Danijela Sain, and Nicole Saigal, who graded homework for us;

Jonathan Kay and Natalie Vanatta, who provided us with invaluable technical assistance.