Abstract – The WPI approach to engineering education has as its outcomes team projects which work on problems of contemporary significance in both engineering and society. This approach has been highly successful since its inception in the 1970s. In terms of curriculum, the implementation of this approach has been done in a top-down manner. This paper is concerned with how a mathematics course – linear algebra – taken at the freshmen level might be modified to be consistent with the WPI educational philosophy while still maintaining traditional mathematical goals. After several years of doing this, it seems possible to achieve these goals. Final exam data is provided to substantiate these claims.

BACKGROUND

WPI is a private engineering college located in Worcester, Massachusetts, founded in 1865. In the early 1970s, it instituted a curriculum, the WPI Plan, which had not the usual 120 credits as its outcome, but 3 projects: one in the students major, one attacking a problem at the intersection of technology and society, and one in the Humanities. This project based approach to education has been highly successful; indeed it has recently been copied by other colleges. Prior to these projects, however, students still took many conventional courses which should provide them with the background they would ultimately need in their projects and their eventual jobs.

In the late 1990’s, WPI put considerable effort into developing links between courses, called “bridges”, thanks to efforts of Professors Judy Miller and Art Heinricher, as well as grants from the National Science Foundation and the Davis Foundation. Thanks to the intellectual atmosphere generated by these grants, instructors began to look at courses as more than isolated activities, and the author was motivated to examine a mathematics course, Linear Algebra.

Linear Algebra at WPI

At the first and second year level, still lie many traditional courses, yet to be reformed to meet new demands of the WPI Plan. Since the college now has distribution requirements, students in many majors find they must take as many, if not more, than 6 mathematics courses. A common situation is for someone to take Calculus 1 through 4, Differential Equations and Linear Algebra, by the end of the sophomore year. Linear algebra is taken by between 80 and 85% of the students at the college. Only a tiny fraction are mathematics majors. A survey showed that some 80% of the students did indeed take the course to satisfy the distribution requirement, as opposed to any specific interest in the content of it.

The latter data is cause for concern. When students are asked to work for an external goal, rather than due to intrinsic interest in the course, it is known ([1] and [2]) that their effort and participation may suffer.

The course is offered four times per year in a “large lecture” format, meaning that the class has between 130 and 160 students, all meeting 4 times per week for lectures, and once for smaller conferences.

The required syllabus mandates topics including systems of linear equations, matrix algebra, vector spaces, linear transformations and diagonalization. Anyone familiar with an introductory linear algebra course offered in the last 100 years will recognize these topics.

The Goal

Merely taking a course to increase the total amount of math courses taken clearly seems insufficient. One can desire a greater contribution to a student’s overall education. How to do this? The large number of students at any time was a problem, as was their academic diversity. In a given class of 150 students, one might find as many as 10 different majors. Yet any one of those students deserved a course which made a meaningful contribution to their education, beyond mere credit.

Thanks to grants from both the NSF and Davis Foundation, the university had, in the late 1990s, explored the possibilities of linking courses together so as to more clearly demonstrate applicability to the students enrolled (this is hereafter referred to as course “bridging”). Motivation increased in mathematics courses when the students were shown that the material was directly applicable to another subject they were studying, apparently giving it credibility and removing arbitrariness. This worked well for small classes but was logistically impossible in the case described here.

Finally, based upon smaller scaled prior efforts, I made the following decision: a goal of the course would be to relate the material to the majors of each student in the course.

IMPLEMENTATION([12])

Consistent with the rest of the college’s curriculum, the students would work on applications to their majors by way of projects. These projects would be done in groups, outside of class. What is a project? My working definition was that of a collection of activities with a well defined theme, taking a nontrivial amount of time to complete (unlike homework assignments in most textbooks where many problems take 2-5 minutes to complete). It would be interdisciplinary in nature, and might require the students to find resources not readily provided. The theme should be something of contemporary interest, perhaps having some social component. Having it done outside of class would allow me to follow the departmentally mandated course outline in class, essentially covering core material needed by everyone, yet still add on the important component of applicability. Giving the students experience in a “learn-to-learn” situation earlier than they ordinarily would have, this was a plus, especially given that their junior and senior projects would demand such.

This also meant I was asking the students to do more work. Would they do it? Research had shown that in the past students in this course had put in far less effort than WPI assumes people invest in a course. So all I was doing was regaining what should have been put in anyway (or at least I hoped to).

Was this a reasonable way to do things? Since their later projects would involve working in groups, outside of classes, with people they might not know, this seemed to be good preparation.

Cultural Considerations

Aside from seeing applications of mathematics to projects in their areas, a number of other, less technical, considerations were taken into account. First, students would work in teams. This meant putting people together who had the same major but had never met one another. This potentially fostered relationships between people which might last for four years or more. It also meant having people work in teams who might never have done such before. This does not always come easily, but good or bad, the experience is important. Students gain exposure to their chosen major. In some cases they have yet to even take a course in their major. Both the project and their partners provide them with some insight into what the major will be like. This may even result in a decision to change to another major, but that is part of what being a freshman is about. Just as significant, perhaps, is that you have many freshmen who have just left home, friends and family and may be in need of new ties to make the transition to college possible. Working in project teams does more to allow that to happen than sitting in a lecture does.

THE PROJECTS

What follows is a description of my travels through the various majors of the students in the course.

Biology and Biotechnology

This major at WPI (best described at [3]) has five possible concentrations within it: bioprocess, cell & molecular biology and genetics, computational biology, ecology, and organismal biology. Beyond traditional uses of statistics, none of them have a quantitative demand to them. Telling students that linear algebra would at some point be a part of their arsenal would be quite a stretch. However, two projects were developed allowing them to relate the two fields. Each was based upon the Leslie population model (see [4]). In short, the Leslie model uses matrices to dynamically describe a population in terms of age groups within it

The first project focused on controversy surrounding possible over fishing of the George’s Bank fishing ground off New England. The goal of the project was first to implement the model in computer software (Maple) and then to see if one could adjust the rate of harvesting so as to allow a stable population along with adequate fishing activity. The final product was a modified model, along with a paper explaining how the model had been changed , interpreting the results from simulation, and how the fishing would be effected both in the short and long term.

The project is always a popular one and is easily updated since newspaper articles regularly appear discussing the plight of the fishing grounds. From an instructors point of view, it reinforces many fundamental linear algebra topics in the process of being developed, especially eigenvalue analysis.

The second project applies the Leslie model to the global, human population. In the first stage, students gather data so as to build a Leslie matrix based on 10 age cohorts, with a 10 year time increment. Once this has been implemented into software, they see if it produces results relatively consistent with existing projections (which they must also gather). In the second stage, they adapt the model to reflect the global spread of HIV. In the final stage, they consider possible adaptations of a hypothetical HIV vaccine and what impact it might or might not have on spread shown in the second stage of the project. They are encouraged to take into account cultural and economic obstacles. The final product includes the model itself, its implementation in software, and a paper discussing all assumptions and simplifications made.

A project currently being developed for the next group of students to consider is one which considers the acoustics of sounds generated by various species of whales. As will be described later in the paper, software exists allowing students to study sound files (such as those in the WAV format) with Fourier methods. Thanks to the Internet, there are a great many sites with files available containing substantial background material as well as actual whale recordings (see [5] and [6], for example). The actual project will have students gather background on whale acoustics as well as skills at using the software for acoustical analysis. Further, they will try to identify what some of the current issues and concerns are in the area. They then will make a determination as to whether effective research might be done using this tool.

Civil Engineering

Civil engineers often take on positions as managers. Two things they manage are resources (building materials, human resources, environmental resources) and projects. A fantastic application of linear algebra, since the 1950s, has been Linear Programming (a poor name; it is really an optimization technique so powerful it resulted in a Nobel Prize award). Linear Programming is ideal for providing the best management decision for a given set of resources; exposing civil engineers to it seemed imperative.

Residents of Massachusetts know that civil engineers are always involved in project management because of a huge construction project in Boston known as the Big Dig ([11]). Mistakes in management have extended the duration of the project and added billions of dollars to the tab. A by product of the defense industry in the late 1950s has been the Critical Path Method (CPM) for managing complex projects. Interestingly, this is a refinement of Linear Programming. Almost every construction company has software for implementing it. Students at WPI use the software in project management courses ([see [7]). This course thus had the chance to show them the mathematics and assumptions behind the software. One unexpected by-product in the first use of this material was when computations done by hand in the linear algebra course revealed a bug in the software then being used by the civil engineering course. This was a valuable lesson!

The actual work of the civil engineering students in this project involves: 1) studying the fundamentals of linear programming and applying them to textbook problems, 2) studying the fundamentals of network graphs of projects, especially the concept of Critical Paths, 3) using commercial software and comparing results obtained by hand, and 4) writing a paper on how management of the Big Dig might have been improved.

Computer Science

While linear algebra is hardly a key component of a computer scientists background, it turned out that this population of students was the easiest to satisfy of all. Linear algebra does have any number of algorithms in it, depending on how it is taught. The Gaussian Elimination method is one. The Computer Science majors were given the task of writing interactive software to carry out the algorithms that the rest of the class did by hand. Many said they enjoyed it because it was the first time they had to write software that performed a clearly useful function and where the answers simply had to be correct. One student, several years later, told me that he could not imagine learning the linear algebra any other way except by writing code to implement it.

Electrical Engineering

Perhaps the most elegant and powerful applications of Linear Algebra exist in Electrical Engineering, through Fourier Analysis of periodic functions. The Electrical Engineering department uses a great deal of Fourier Analysis in its Signal Processing course [8] . Once a dialogue was established with the instructor, it became clear that department was teaching a great deal of linear algebra within that course, as illustrated by materials used within that department ([10]). Taking advantage of both the applicability and duplication was a sheer delight.

The specific application we settled on was in the area of voice print analysis. The students were asked to make a decision on the following: can a person be uniquely identified based upon their voice?

Students began by gathering data by recording team members pronouncing vowels into WAV files. The Matlab software package was then used to graph the files and to generate a Fourier Transform of the vowels. This allowed comparisons between individuals, vowel by vowel, at a detailed level. They also did background reading out of a most unusual and stimulating book [9].

The final portion of the work they turned in was to make a recommendation, one way or the other, as to whether voiceprint identification might work, based upon the data they had collected and generated and their analysis of it. There were no “wrong” answers except those not supported.

In the process of doing the work, they became more skilled at applying Matlab to periodic functions, skills needed in the Signal Processing course, among other places. This work has been helped considerably by a grant from WPI’s Center for Educational Technology, Development and Assessment (CEDTA) which is currently developing on-line tutorials in Matlab and Signal Processing. These materials will greatly facilitate the efforts of students working in such project teams as described here.

Mechanical Engineering

At a computational level, linear systems of algebraic equations and linear differential equations, such as those arising in mechanics, look very different to most students. Looked at correctly, they are really the same; one needs both homogeneous and particular solutions to generate the most general solution. This idea alone does not cause much enthusiasm in students; a good application is needed. This was achieved in two steps. The first was to establish the notion of resonance in a vibrating system. This, simply put, is when the frequency of the external force acting on a system is the same as the natural frequency of the system. While this certainly moves more in the direction of the areas that mechanical engineers study, it still suffers for want of demonstrative examples. The vast collection of WPI Major Qualifying Projects provided several such examples. One was a video tape of an actual mass-spring system with a motor driving it. The tape shows both visually and numerically the occurrence of resonance, and the energy considerations involved with getting the driving frequency to be greater than the natural frequency. An additional benefit of this was that the instructor for the corresponding mechanical engineering course covering resonance gave a guest lecture, giving students a chance to meet a future instructor, as well as to meet an expert in the area.

The products of the actual project done by the students in this area were 1) derivation of the solution to the mass-spring system near resonance using calculus and trig, 2) derivation of the resonant solution 3) a summary of the mechanical engineering professor’s talk on resonance, 4) analysis of the two videos on occurrences of resonance, relating the theory and the phenomenon.

A second, very similar project focused on a video-taped phenomenon where the goalposts on the football field were seen oscillating mysteriously, only in the presence of a mild, constant breeze. The guest lecturer finally revealed that oscillations in eddys were acting as the driving force, and that the classical resonance model was still applicable.

This work also reflects my attitude that their education should have some vertical integration in that it is desirable to expose freshman to senior level work.

ASSESSMENT

The object of performing assessment was to see if the students comprehension of core material was as good as before the project component was added on. To that end, the same final exam was given in 1997,1999 and 2000. 1997 was the control year as the project component was not used then. There were five identical questions on the three finals. The class averages on each question, as well as the overall exam are shown below. The final exams were not returned at any point, so cheating by way of reading older exams was not an issue.

The conclusion I draw from this is that I was able to add the project component to the course with no adverse impact on the students performance on core material, and in some cases an improvement.

	1998 average n=143	1998 std dev	1999 average n=100	1999 std dev	2001 average n=150	2001 std dev	1999 vs 1998	2001 vs 1998
question 1	67.0	23.1	79.2	30.3	68	24.0	p=.0003	p=.61
question 2	58.9	21	74.9	31.3	73	24.9	p=.0000	p=.01
question 3	81.0	19.0	81.7	28.1	86	22.4	p=.4081	p=.0197
question 4	57.1	39	60.1	38.1	55	30.1	p=.2679	p=.72
question 5	81.1	19.2	83.43	25.5	78	20.2	p=.2096	p=.90
Exam*	71.4	22.0	79.23	16.3	75.1	17.2	p=.0007	p=.054