Mathematical Analysis in Engineering: Reform of Calculus in the United States

GREENBERG, William

Virginia Tech, Dept. of Mathematics, Blacksburg, U.S.A., greenberg@vt.edu

Abstract: The calculus reform in the United States has spawned two programs within the first year engineering calculus sequence at Virginia Tech. The Emerging Scholars Program was designed to deal with the lack of preparedness of a large segment of incoming engineering students. The Mathematics Emporium weds information technology to the teaching of mathematics. Both programs have significant implications for the future of engineering education.

Keywords: calculus reform, engineering calculus, emporium, emerging scholars

1  Introduction

Over the past decade considerable resources have been devoted nation-wide to assessing and reformulating the teaching of calculus in the first year university curriculum. In large measure this has been motivated by the poor performance of science and engineering students in these courses. Although there is to date still no consensus on how this reform of the calculus should be carried out, our experiments in calculus reform at Virginia Tech have led us to some definite conclusions on what is needed.

I will describe two types of experiments we have carried out at Virginia Tech. These represent, in some sense, two extremes in dealing with revision of the calculus. Both of these projects have achieved different but unexpected levels of success. Both have been implemented at the freshman calculus level (introductory differential and integral calculus), a course which at many engineering schools serves as a filter to eliminate unprepared or unqualified students. However, a number of educators have argued [1] that the importance of basic calculus in the science and engineering curricula militates against using it to weed out students.

The first of these experiments we refer to as the Emerging Scholars Program (ESP), which is specifically formulated to deal with the different preparedness of today's engineering students. The second is the Mathematics Emporium, which attempts to utilize the latest developments of information technology in university education.

2  Emerging Scholars Program

At Virginia Tech an index (mathematics readiness score or MRS) is computed for all students planning to take first year calculus. This score is derived from a regression model which includes high school grade point average in mathematics courses, mathematics S.A.T. score, and whether or not the student has taken any calculus in high school. We have found such a score to be a far better predictor of success in college calculus than mathematics pre-testing. We refer to students with low MRS as at-risk students.

Our university offers a variety of introductory calculus courses for different majors. In particular, we offer distinct courses for business majors; for architecture majors; for life, social and biological science majors; and for engineering, mathematics, and physical science majors. For brevity we refer to this last course, the highest level of introductory calculus, as the engineering calculus sequence. Approximately 1300 students each year enter the engineering calculus sequence, and it is in this sequence that we have developed the ESP program.

Prior to 1996, at-risk students were required to take a remedial pre-calculus course, which is the conventional approach to the problem of dealing with such students. The results were hardly an endorsement of this approach. In 1995, 72% of the students assigned to the remedial course and intending to enter the engineering calculus failed to achieve a grade of C, thereby failing to qualify for entrance into the first semester of that course. Only 12% proceeded into the engineering calculus and earned a grade of C or better.

Success within the calculus course itself was also a matter of concern. Prior to 1996 all calculus sequences were presented in traditional lecture format: 3 hours weekly of lecture, assigned homework, and in-class written examinations. Altogether, 38% of all students in first semester differential calculus in spring 1995 and the same percentage of students in second semester integral calculus of the engineering sequence received grades below C.

The ESP program in engineering calculus is a supplemental instructional program built around individualized instruction by faculty and peer interaction. The idea of supplemental instruction to improve classroom performance goes back more than 90 years to John Dewey [2]. Cooperative learning has become a benchmark of the educational reform in the engineering curriculum [3], [4], but has made very limited inroads into the teaching of mathematics. The ESP program at Virginia Tech was first introduced as a limited experiment in the 1997-98 academic year, and then fully implemented the following academic year. It is based on Uri Treisman's collaborative workshops with underachieving minorities introduced at the University of California - Berkeley in the 1980's. Similar experiments have been conducted at the Universities of California - Davis, Texas and Wisconsin.

At the beginning of the 1998-99 academic year, approximately 50% of the students enrolling in the engineering calculus were identified as at-risk, based on their MRS. Students with high MRS were enrolled in 17 traditional lecture sections. Students classified as at-risk were enrolled both in 17 separately identified lecture sections and in associated work sessions, our so-called ESP laboratories, supplementing the lectures. In both cases the lecture sections lasted 3 hours per week for 15 weeks each semester, and had enrollments of approximately 35 students each. The ESP laboratories lasted two and one half-hours each week. Each lecture section was taught by a mathematics professor or experienced graduate teaching assistant. Each ESP laboratory was taught by a different mathematics professor or graduate teaching assistant, assisted by 3 undergraduate teaching assistants. A strong dialog was maintained between the lecture instructor and the laboratory instructor. In the laboratory, students divided into groups and labored on problem sheets designed to review algebra skills, reinforce calculus notions presented in the lectures, and introduce engineering-based applications. Problem sheets were modified as needed to deal with particular difficulties observed in the lecture section. Different instructors were encouraged to experiment with different teaching strategies within the ESP laboratory. The model described below was typical, and arguably the most successful.

Students were divided into groups of three to work on the problem sheets, with minimal structured lecturing, but peer interplay providing the basis for much of the learning. The undergraduate assistants and the instructor circulated from group to group, contributing on-the-spot corrections, additional motivation, and individualized strategies for learning. Each group was required to solve problems successively, not proceeding to the next problem until all group members were satisfied with the solution of the previous problem.

The importance of the peer interaction can not be overemphasized. Groups solved problems sometimes by working as a team to develop a strategy for obtaining the solution, and sometimes by one group member explaining a solution to other members of the group. The instructor and undergraduate assistants served only as facilitators.

An equally important contribution of the ESP program to these first year students was in teaching them how to study mathematics. Indeed, the lack of study skills was one of the most notable differences between these students and those of earlier generations. On this matter the instructor had to provide the training, rather than the undergraduate assistants. For this reason, he had to circulate and interact with all groups.

Another essential feature of the program was the ability of the instructor and his assistants to catch and correct errors while they were being made, rather than as corrections in returned homework or quizzes. We have found that today's students do not respond to corrective measures after the fact, but do respond if corrective measures are interactive, that is to say, immediate.

Finally, the ESP instructor served as motivator. Students were urged to keep up their focus and their pace. They were reassured that they were, in fact, very capable students, with perhaps some deficiencies in their background. Indeed, these students were typically high performing students in their high schools. In fact, this may have been a contributing factor in their at-risk performance. Precisely because they had talent in math and science, they may have succeeded in high school without the need for serious study habits.

3  Performance

All engineering calculus sections, as with other calculus sequences at Virginia Tech, have a common final examination. The same common examination was given to both ESP and non-ESP sections. Such examinations provide a measure to compare performance between different sections and different faculty members. In 1998-99, the common final in the engineering calculus contained 17 problems.

At the outset the mean difference in mathematics S.A.T. score between the ESP sections and the non-ESP sections was a remarkable 60 points. Based on previous experience, it was estimated that the failure rate in the ESP sections would exceed 30%. The table below indicates the actual results in differential calculus in fall 1998. (Engineering differential calculus is labeled Math 1205)

Table 1. Comparison of Traditional and ESP Sections in Differential Calculus

Numerical grades are given the usual designation (A=4, B=3, C=2, D=1). A grade of C is generally required to continue in the various engineering programs. Exam grade is number correct of 17 problems on common final. Failure is a grade of F.

As one would expect, non-ESP students outperformed the at-risk ESP students. However, the remarkably small difference in the common examination results and the failure rates, and the overall low failure rate, give a clear indication of the benefit of the ESP strategy.

The results in integral calculus (Math 1206) the following semester were even more impressive. In the past scores have declined by an entire grade in this second semester. Here is what actually happened.

Table 2. Comparison of Traditional and ESP Sections in Integral Calculus

The fact that there is virtually no difference in exam performance and in the failure rate between non-ESP and ESP sections provides evidence that the ESP students continue to catch up with the non-ESP students in the course of the academic year.

Our conclusion is that today's engineering students are in fact strong and can be motivated. However, we must replace our traditional teaching methods with strategies keyed to the strengths of this TV / computer generation.

4  Mathematics Emporium

The Mathematics Emporium is designed to integrate computer technology into the teaching of mathematics courses. The facility is located in a former department store building with 7000 square meters of space, and consists of five hundred Pentium PC and Macintosh PowerPC workstations in pods of six, networked to the Internet. Most of the course specific software and record keeping resides on a SUN Enterprise 3000 workstation. The furnishings are arranged in an open space design with minimal use of partitions. Software includes Netscape Navigator, RealAudio, Macromedia Shockwave, Mathematica, Matlab, and Office 95. Regions on the periphery are divided into small units: one large lecture area, two classroom areas, two lounge areas, and partitioned spaces for small group sessions.

The Emporium opened in the fall semester of 1997. Virtually all first and second year mathematics courses have technology assignments which may be carried out at the Emporium, including engineering differential and integral calculus. However, the linear algebra course required of all engineers is taught entirely at the Emporium. This course covers topics included earlier in the first year calculus sequence for engineers, but presently taught as a separate course. As this represents the cutting edge of technology in teaching, it is this course which will be described.

Technology continues to play an increasing role in the teaching of mathematics. Technology enhances the ability of the teacher to present concepts not just visually, but also graphically, symbolically and numerically. Technology allows for individualized instruction targeted to specific students and correlated to their previous recorded performance. Technology promises to allow the development of creative ways to deal with the economic pressure of serving more students with fewer resources. And technology provides preparation for the contemporary and future workplace.

The Emporium is open 24 hours a day, 7 days a week, during the school semesters. Throughout the university campus terminals are available which indicate in real time the number of free computers in the Emporium, so waiting time for a free computer can be minimized or eliminated. This same information is accessible on a Web page, and thus available to students at any location with a modem or Ethernet connection.

Engineering linear algebra is totally computer based, with on-line instruction and testing. Instruction is based on the availability of multiple approaches and on open scheduling. At any time of the day or night, students may log onto the weekly assignment, which is explained by an interactive computer tutorial. During the tutorial itself the student participates in answering questions on the material. After the tutorial, the student can take practice examinations as frequently as he wishes. Mistakes on the practice examinations are corrected as they are made at each problem. At all times (except between midnight and early morning) there is a corps of circulating floor helpers to assist students in understanding the mathematics. There is also support staff available to deal with hardware and software questions. At scheduled times during the week, there are topical lectures and demonstrations in the peripheral classrooms. In addition, some of the lectures have been recorded on CD's which may be checked out by students and played on the computer. These last alternatives accommodate students who prefer the more traditional approach, although in fact the bulk of the students master the material exclusively from the computer tutorial. Finally, there is a designated area in the Emporium where extended individualized help may be obtained from teaching assistants.

An examination on the material must be taken before a weekly deadline. The examinations are on-line and are similar to the practice examinations. Longer cumulative midterm and final examinations are also administered at scheduled times. These two tests are the only ones which cannot be taken at any time convenient to the student. A record of all scores obtained is available to the student. The final grade is determined by the results of the weekly, midterm and final examinations.

Initially computer utilization was concentrated in afternoon hours, with very low utilization in early morning hours. Over the 18 months the Emporium has been operational, this distribution has changed, with significantly more utilization of early morning hours. For example, during Saturdays in September of 1998, utilization from 3.00 am to 5.00 am averaged 65% of the utilization from 3.00 pm to 5.00 pm. This evidently resulted from the gradual adjustment of the student body to the freedom offered by open scheduling.

The Mathematics Department recognizes the importance of successful evaluations of the Emporium project. In particular, it is necessary to evaluate educational effectiveness of the project, as well as student reaction. Because of the short time the Emporium has been in operation, only limited information is available at the present time. A few conclusions can be drawn.

To the extent that students interact with others while on-line, the students choose to communicate with other students far more often than with floor helpers. Surprisingly, a slight majority of students surveyed in early 1998 did not feel that their experience in the Emporium helped increase their independence in learning. There are indications that this opinion has been decreasing as students adjust to their freedom and responsibilities with technology based education, and as bugs and breakdowns in the software decrease.

It is difficult to interpret comparisons of common time examinations between different years. Nevertheless, it is to be noted that the average score on the common time final examination in the Emporium engineering linear algebra was 69% (fall 1997) compared to an average of 55% in pre-Emporium final examinations in fall 1995 and 1996.

5  Conclusions

Experience with the Emerging Scholars Program indicates surprising academic strength and receptivity even in students predicted to fail to make the grade in their freshman courses. There is no doubt that changing cultural values, the failure of public schools to teach adequate study skills, and the broadening of the base of entering students are responsible for the failure of students in the traditional curriculum. Our new approach builds on the strengths of these students by finding new ways to present material. In this manner we have attained some remarkable results without the need to weaken the content of our mathematics courses.

On the down side, the ESP program is more labor intensive than the traditional lecture course. This additional cost is somewhat vitiated by the much higher pass rate: the need to repeat courses, or to drop out of disciplines entirely, is a direct cost to the university.

The Emporium program will, in the long run, result in a dramatic decrease of labor costs, with only a very modest equipment cost. This program is still in its formative period. It is clear, however, that the pressure to develop technology-based education will increase throughout university settings in the United States, simply because of immense economic pressures. The goal of the educator is to see that the outcome of this pressure is a technology based system which equals or exceeds traditional methods. Given the inherent ability of computers to be interactive, their intrinsic visual capabilities, and the computational nature of mathematics, there is every reason to believe that, properly developed, technology courses can be developed to exceed the best of conventional presentations.

References

[1]  MALCOM, S.M., & TREISMAN, U. Calculus Success NOT a Filter, Edited by Steen, L.A. for the Board on for All Students, Calculus for a New Century: A Pump. Mathematical Sciences and the Mathematical Sciences Education Board of the National Research Council, Washington, DC: National Colloquium, 1987.

[2]  ARCHAMBAULT, R.D. Dewey on Education, New York: Random House, 1966.

[3]  WANKAT, P.C., & OREOVICZ, F.S. Teaching Engineering. New York: McGraw Hill, 1993.

[4]  FELDER, R.M., and BRENT, R., Effective Teaching: A Workshop, Virginia Tech, Blacksburg, VA, 1995.