Calculus, Technology and Coordination
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Stevens was the first schoolin the country to require a group of undergraduates to purchase PCs and use these machines throughout theiracademic careers. This naturally led to a requirement that each undergraduate havea PC. Then, beginning in the fall of 1999, it became a requirement that allstudents have a laptop. The goal is that students will use their laptops as part of the teaching and learning experience, findinguses for the devices within network-connected laptop classrooms, as well asoutside of the classroom. Beginning in the summer of 1999, Stevenssupported several projects designed to achieve this goal. Thisarticle deals with the calculus project.
The Calculus Project
Approximately 300 incomingfirst-year students at Stevens take calculus. This is a natural course for theuse of laptops and technology as teaching and learning tools. With this inmind, the first-year calculus course was redesigned. Four lecturers, fourgraduate teaching assistants, and two undergraduates were assigned to thiscourse.
Scientific Notebook (SNB)was used throughout calculus I and II. SNB is a word processing package thatincorporates a Maple kernel. This Maple kernel allows one to perform a widevariety of mathematical operations. A particularly nice feature of SNB is thatthe Maple kernel can be used without typing any code. One simply types themathematical expression and clicks on the appropriate operation to beperformed. SNB then performs the operation. Prior experience of one of thelecturers had shown that students could learn to use SNB in a relatively shorttime. Furthermore, extensive help files preclude the need to spend very muchtime teaching students how to use the program itself. 1
Work on this project toredesign first-year calculus began during the summer of 1999, and continuedthroughout the 1999-2000 academic year. The goals of the calculus project were:
1. To usecomputer technology to enhance the learning and teaching experience in acalculus course that has a relatively traditional course content.
2. Todevelop computer homework assignments and projects that extend the materialtaught in the lectures.
3. To developa Web site that contains extensive resources that can be used by students asthey learn the subject matter.
4. To provide the students with a complete set ofSNB text file “notes” that they can download and use.
5. To havestudents access the Internet and use SNB during drill sessions in Stevens’newly renovated laptop classrooms.
6. Toemphasize understanding of concepts over the ability to do routinemanipulations.
7. Tohave four different lecturers teach a unified, coordinated course with one agreed-uponsyllabus, uniform exams, homework assignments and projects.
8. Tograde the students’ work as uniformly as possible so that all students aregraded by the same standards.
Computer Technology as an Enhancement to Learning and Teaching
The use of computer technology to enhance learning andteaching in these courses was done in several ways.
· Six computerprojects were written, three for each course. The first semester projects wereFunctions and Curves, Insulated Pipe, and Riemann Sums. The second semesterprojects were Cubics, Series, and Determinants & Curvature. Each projectrequired the students to use SNB. The project that was titled “Insulated Pipe”was developed in conjunction with a colleague in the Department of ChemicalEngineering. This project represents a real application of mathematics toengineering.
· A number ofhomework assignments that required the use of SNB were developed. Thesegenerally involved modifications and extensions of problems in the text.
· Insome lectures, a laptop connected to the Internet was used. At times the laptopwas used to present demonstrations that visually illustrated the instructionaltopics. Sometimes the laptop was used to present notes dealing with aparticular topic. These notes were then made available on the Web. At othertimes, sites that contained related materials were accessed. One site we foundto be particularly useful was http://archives.math.utk.edu/visual.calculus/.· Extensive Websites for each course were prepared. These are at http://golem.math.stevens-tech.edu/ma115/
and http://golem.math.stevens-tech.edu/ma116/.These sites contain the syllabi, course policies, calendars, lecture notes,homework assignments, old exams and projects. In short, they serve as anextensive resource for students taking these courses.· A completeset of notes was prepared using SNB, and made available on the Web. Thisresource was presented in two formats. The first was the WebCalC format firstdeveloped at Texas A&M University by Don Allen and his colleagues. Sincethis resource deals essentially with first semester calculus, materialspecifically dealing with the topics in Stevens’ second semester calculus wasprepared. Permission was granted to put the Texas A&M material on ourserver, so that students could access everything they needed at Stevens.
· TheWebCalC material was also used by one of the lecturers to prepare a completeset of notes for each lecture in the course. There is a link to these notes onthe main page for each course given above.
Use of Networked Laptop Classrooms
During the first part of1999, Stevens renovated three classrooms that were designed for laptop use.Each classroom has power and network connections for 24 students, so that eachstudent can plug in his or her laptop and connect to the Internet. The classroomsare also equipped with projection systems, video players and a variety of othermultimedia devices. In short, these are state-of-the-art classrooms with a highdegree of technological potential.
Our calculus courses (eachcarrying three credits) met for two 50-minute lectures per week and for one75-minute recitation (drill) session per week. Professors gave the lectures andgraduate teaching assistants handled the recitations. The original plan was tohave the students bring their laptops to recitation weekly and use them for apart of each class. The rest of the recitation was to be used for discussinghomework. It was felt that this class size was ideal for this approach sincethe recitations had, at most, 24 students in them.
Unfortunately, we were notable to utilize laptops in the majority of the recitation meetings. At thebeginning of the first semester we encountered a host of technical problems.For example, on the first day of recitation, almost all students in classesthat met in the morning were able to access the Internet, but in the afternoonno one could connect. While most of the technical problems were eventuallysolved, these initial difficulties led to a scaling back of laptop use. (Itshould be noted that lecture rooms did not have laptop facilities forstudents.)
The net result was thatin-class laptop usage was primarily limited to introducing the students to thesix projects. During these classes, the students downloaded the projects andwere led through some of the required steps. This proved quite effective. Wedid give one quiz in which the students had to use SNB. We had envisioned moreextensive in-class laptop usage, and it may well be that this will be achievedthe second time the course is taught in this manner.
One Unified Course Taught by Four People
The lecturers had agreed at the outset that they would allfollow 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 uswas to prepare and deliver his own lectures. We envisioned one unified coursetaught by four different people.
The coordination of thiseffort turned out to be a time-consuming and not always easy effort. Forexample, a draft of each exam was generated by one or two of thelecturers and then circulated to others. Frequently, ittook three or four versions of a given exam before all agreed to a version thatwe were (sort of) comfortable with. A similar approach was taken with theprojects.
We met with the teaching assistants several times eachsemester. At times the meetings were heated. We were fortunate in that eachperson involved in the course always showed the greatest respect for the othersand always approached each topic of discussion with a willingness tocompromise. However, it was not easy.
We had four student aides to grade homework. The teachingassistants graded exams and projects. For uniformity, one teaching assistantgraded all of a particular problem on a given exam. This was also done with thefinal exam, but here the professors and the teaching assistants did thegrading, with one person responsible for a given problem on all examinationpapers.
Once the results of an examwere tabulated, the course coordinator would propose a scale for the exam andask the other lecturers for their opinions. Finally a scale for all studentswas 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 regardlessof who his or her lecturer was.
Observations
No extensive assessment ofthese courses was done. To do this would have required resources and expertisethat were not available. Therefore, the conclusions given below are essentiallyanecdotal.
· Theintegration of computer technology into the learning and teaching experience iscertainly not easy. The first difficult question to be dealt with is what sortof technology will be incorporated into which topics. Then one must decide howthis will be done. Finally, one must develop the software that is needed toaccomplish these things. All of this takes a great deal of time and effort.Indeed, we had to cut back on our original plans, since everything took atleast twice as long as we estimated.
· Theuse of technology can enhance understanding and facilitate learning. Forexample, certain animations were shown that definitely helped the studentsgrasp the concepts being illustrated. However, these demonstrations do takeaway from class time, and if there are any technical problems with theequipment, considerable time can be wasted. Faculty need appropriate support sothat use of computer technology in the classroom becomes only slightly moredifficult than using the blackboard.
· It iscrucial that faculty have support staff that can deal with and solve hardwareand other technical problems in a timely fashion. If this is not provided, thenmost instructors will not even attempt to use the equipment that the school hasprovided. Things must work, and they must work easily.
· Computerprojects and homework assignments are a good way to integrate technology intothe teaching and learning experience. They not only serve as a vehicle forencouraging the students to learn how to use the necessary software, but alsocan be leveraged so that students are exposed to extensions of basic conceptsintroduced during lecture.
· There is noquestion that the approach taken with this course is very different from thescenario in which a group of people all teach the same course but essentiallygo their own ways by giving individual exams, homework assignments, etc. Thecoordination effort was considerable. Since all professors were experiencedfaculty, each of us had his or her own way of doing things. Having to changeone’s approach to accommodate others was a challenge for each of us.
· Studentsin such a large course like the fact that things are done uniformly, despitethe fact that they have different lecturers. They feel that this approach isvery fair. However, to achieve uniformity is time-consuming and requires muchcoordination.
· Studentsappreciate and utilize a good Web site. It serves as a valuable resource thatstudents visit often for all sorts of information about the course. Inparticular, students like being able to download sample exams and exams givenin previous years. It is very important to provide solutions to these exams. Inresponse to student requests, we even provided the solutions to the optionalhomework problems during the second semester.
· We wanted tohave as much control over our Web site as possible, and, as a result, put oursite on a server that resides in the office of one of the lecturers. Stevensd'es make WebCT available to faculty, and a number of faculty members at Stevensare using it. However, for our purposes, we felt it would be best if we wentour own way. On the whole this turned out to be true. However, during thesecond semester we did use WebCT to post students’ grades.
· The Web sitethat we developed should serve as a resource for the first year calculus coursein future years. It provides a record of what was done that can be modified andexpanded 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 itdisappears when the people involved move on to other things.
Conclusions
The teaching of mathematics, and all subjects, for thatmatter, must undergo constant change if students are to be prepared to enter arapidly-evolving technological world. It is no longer a question of whether touse technology in the teaching and learning experience. It is now a question ofwhat technology to use, and how and when to use it. There are no readilyavailable answers to this question, and there is probably no single answer toit. The challenge today is for instructors to experiment with a variety ofapproaches and then to select and build on the ones that work. As ourexperience has shown, this is certainly not a simple task.
Lawrence E. Levine has beena professor of Mathematical Sciences at Stevens Institute of Technology formore than 30 years. Presently he is a visiting professor at the United StatesMilitary Academy at West Point. His research interests lie in the areas ofordinary and partial differential equations and perturbation methods. He hasbeen interested in the utilization of technology in the teaching of mathematicsfor many years. This interest has led him to state-of-the-art applications ofcontinuously changing technologies.
E-mail: [email protected]
Veroujan Mazmanian was bornin Lebanon and is of Armenian extraction. He has studied at the AmericanUniversity of Beirut, and at Stevens Institute of Technology. He has taught atStevens for the past 30 years and is known for his interesting and thoroughlecturing style. He was very involved with the Stevens Technical EnrichmentProgram, a program to help students from disadvantaged backgrounds succeed in acompetitive, technical environment.
E-mail: [email protected]
Dr. Patrick Miller has beenan assistant professor in the Department of Mathematical Sciences at StevensInstitute of Technology since September 1997. Dr. Miller’s main research focusis in the application of dynamical systems to transport and mix geophysicalflows. Prior to his appointment at Stevens, Dr. Miller was a postdoctoralfellow in the Division of Applied Mathematics at Brown University. He receivedhis Ph.D. from the University of Massachusetts in 1994 with a dissertation onthe stability of traveling waves for reaction-diffusion systems.
E-mail: [email protected]
Roger Pinkham was born onthe coast of Maine to seafaring parents and has been in love with mathematicssince doing synthetic geometry problems on the floor of the family boat as ayoung boy. In 1982 he taught the first class at Stevens Institute of Technologythat was required to buy their own personal computers. First, students learnedto program, then they employed their newfound skills in Linear Algebra.
E-mail: [email protected]
Acknowledgements
The authors wish to expresstheir thanks to the following individuals who assisted them in teaching thesecourses:
Dr. George DeLancey,professor of Chemical Engineering, who played a major role in writing the InsulatedPipe Project.
Maureen McLaughlin-Brearley,Kerry Smith-McNeil, James Orthmann, Paul Rossi, and Natalie Vanatta, whodevotedly served as our teaching assistants.
Melissa Garofala, Eve Naim,Danijela Sain, and Nicole Saigal, who graded homework for us.
Jonathan Kay and NatalieVanatta, who provided us with invaluable technical assistance.
This article originally appeared in the 12/01/2000 issue of THE Journal.