Department of
Mathematical Sciences
to be presented at
the Frontiers in Engineering
Education Conference, November 5,2002, Boston, MA
A course taken to meet
distribution requirements with questionable intrinsic value to the science and
engineering students enrolled in it is studied and modified. By augmenting the
core material with team projects focusing on the particular majors of the team
members, the education of the students is extended in a number of ways important
to the curricula they are in. The added work does not infringe on the required
core material; data analysis indicates that understanding of it, as measured by
final exam performance, may have even improved.
WPI is a private engineering university in central Massachusetts of about 2500 undergraduates. The more highly populated majors are Biology, Computer Science, Civil Engineering, Electrical and Computer Engineering, and Mechanical Engineering.
Students
at WPI are as a rule required to take 6 mathematics courses to satisfy most
undergraduate degree requirements.
Certain degrees require additional, specific courses. Most students satisfy the requirement of 6
courses by taking Calculus I through IV, and two other courses from Discrete Math, Statistics Differential
Equations, and Linear Algebra I. The latter two courses are taught each of the
4 terms of the year and presented in a “large lecture” format. The latter means
lectures 4 times per week to a class of about 150 and one conference.
Engineering students usually take Differential Equations as one course,
Computer Science majors take Discrete Math, and Bio majors favor
Statistics. Almost all of these
students take Linear Algebra as their other course to complete the total of 6.
Referring
to the Linear Algebra course, the presentation has been quite traditional:
lectures on standard topics, homework, 2 exams and a final exam. The standard topics were mandated by the
Department of Mathematical Sciences and included systems of equations, matrix
algebra, determinants, basis and coordinates, and eigenvalues/eigenvectors.
The
course did not have much of a reputation for being interesting. A nickname for
it among students was “Mattress Algebra” presumably referring to the number of
students either staying in bed when class met or falling asleep during class
due to, what for them, was not stimulating material. Class attendance was sometimes as low as 50%. A survey showed students putting in about 8
hours per week of work on the course, including class time. On the other hand,
WPI assumes its students will put in about 14 hours a week per course for 7
weeks. Yet students continued to take the course each year; 150 students per
term for 4 terms. One had to conclude
that many were taking it simply because they needed credit for a math course.
In some accreditation circles, this is part of what is called “bean counting”.
The issue of student questioning the value of lower level courses had shown up before (WPI,1991). In that study of both students and alumni, between 50 and 75% questioned whether their initial studies in science and mathematics were important to their future academic work at WPI.
Another
reason for concern was the fact that approximately 55% of the linear algebra
class were freshmen. That fact alone
merits attention; the freshmen year is a critical period for a student.
Furthermore, large classes are not their best friend – research shows a strong
correlation between the number of small classes taken and their personal
satisfaction , while other research shows a correlation between the number of
small classes taken and their grades (Light, 45)
I
classified the overall conditions of this course as a problem for the
following reasons: given the intellectual expectations of many students, the
demands of future employers, and the cost of a private education, a student
should get more from the course than a check mark on a list.
While teaching in the 90s in various courses to the same population described earlier, I often grappled with the issue of motivating students and making their education more meaningful. In 1995, while teaching multivariable calculus (Calculus IV) also to a large class of 145 in a similar format, I tried one approach which gave me a glimmer of hope.
WPI has a project oriented curriculum, being a pioneer in such education, so it is common to have students work in teams on projects of various sizes. In that course, (Goulet,1997), I had students work on a project during the course in the area of fractals. Specifically, I had them work in teams of 3 arranged according to their major. In such teams, they had to investigate the fundamentals of fractals, and then research applications of fractals to whatever their field of study was. It was the latter that really got them interested. Additionally, their view of the credibility of the calculus course increased substantially.
Also
in the 90s, the university took part in an NSF sponsored Institute Wide Reform
(IWR) project. One component of this, which I had been an active participant
in, was that of seeking and developing connections, or “bridges” between
courses. The more general philosophical axiom which came with that was that
courses should not be viewed as isolated intellectual activities, but that they
have relations to other entities: courses, majors, attitudes and so on. A great deal of my enthusiasm for this
project came from participation in the IWR program in the area of course
bridging. This course, however, did not lend it self to bridging for logistical
reasons. The 150 students, from the freshmen and sophomore years were simply
taking far too many courses as a group to realistically considering bridging to.
When
planning a large class of Linear Algebra in 1999, I viewed the course in a
different light than merely delivering one of the 6 math courses they were
required to take. My self defined goal
became to decide how to deliver this course, given its parameters, in a way
which would make the greatest contribution
to the education of the students in the course.
(By “parameters” is meant the given constraints of departmental requirements of material covered, large class size, and student backgrounds).
Such
a question has many answers, many of which depend on how one defines
“education”. Given my prior
experiences, I arrived at the following answer: it should do as much as
possible to relate linear algebra to each students chosen major.
I realized
that in some ways over the years, my courses had perhaps been a one way street
in the sense that a certain course would be delivered and that would be
independent of who was in it. This did not mean I did not read the class list
carefully and consider the individuals and what I knew of them. But the course
content would not change. Sort of a “one size fits all” approach. Yet the class
consists of individuals, each with his or her own academic needs.
I
started seriously considering what those needs might be by looking at each
person’s major. Having done that, I then asked how can this course relate to
this particular major? This meant I had to do some learning. I have a lot
of experience with linear algebra and know many applications of it, yet to ask,
for example, what use is this course to a Civil Engineering major, meant I had
to do some work. It also meant the street was now two way. I was helping the
students learn mathematics and I was in turn learning about the students. Even
though the class consisted of 155 students, they were 155 individuals, with
their own needs.
Who
were they? For the class I instructed in the Spring of 2001, their majors broke
down as:
Bio
9 (6%)
Civil Engineering
4 (3%)
Computer Science
55 (35%)
Electrical and Computer Engineering 47 (30%)
Mechanical Engineering 29 (19%)
Others (Math, Physics, non declared) 11
(7%)
The course would have two components: a core component covering the traditional mathematics (see earlier list) required by the Department of Mathematical Sciences, and a project component organized according to major. The core material would be covered in class, still in a traditional manner (textbook, homework, exams and so on). The projects, on the other hand, would be done outside of class, in teams, and turned in weekly or biweekly. This represented a fairly gentle shift from an instructor oriented course to a more student oriented one, since the project work would be done entirely outside of class. Such approaches are supported elsewhere in engineering education (Catalano).
This would hardly the first time a course in Mathematical Sciences had weekly activity outside of the normal classroom based material, calculus labs being a regular thing. These had met with varying degrees of success and failure. One lesson learned by two of us at WPI was that the collection of activities should have a common theme throughout them, to the maximum possible extent. I had learned this while providing a collection of projects on statistical quality control (Goulet,1995), while Nick Kildahl of the Chemistry Department had found the same in a revised set of chemistry labs (WPI,2001). Since mathematics is often a cumulative, sequential subject, essentially one week would build upon the week before, with fundamentals being established in the first couple of weeks leading to more applied work in the last week. Viewed from an educational outcomes perspective, I was attempting to increase by one level, perhaps two. Specifically aimed at were application and synthesis (Bloom et al ,231-295).
Could
this course, modified as it was, still provide an adequate basic preparation in
linear algebra while also linking the material to the students majors in a substantial
way?
Biologists
would focus on populations and related issues. The foundation would be the
Leslie Population Model, the final goal being to mathematically model the
spread of aids in the US population. While the major has many facets to it,
almost all biologists ultimately consider issues relating to population whether
they are at the microorganism level, generating pharmaceuticals for human
populations, genetics, or the spread of disease.
Civil
Engineers manage resources and projects. The initial goal would be to study
Linear Programming which is concerned with optimizing results while maintaining
physical constraints. Eventually they
would apply this to understanding the quantitative side of project management,
through the Critical Path Method (CPM) model. Finally they would consider
problems of Massachusetts’ infamous Big Dig construction project.
Computer
Science students, by the very nature of the subject, study algorithms and write
code to implement algorithms. Since linear algebra is full of algorithms, they
are an easy group to serve. Their remarks always indicate they like writing
code to perform real tasks such as solving systems of equations and matrix
algebra. Most of the programs are interactive, which is good experience for
them.
Electrical and Computer Engineering majors study a variety of areas including signal analysis and circuits. Signal analysis rests, to a significant extent, on application of Fourier Series (Lathi), for example. Fourier Series, in turn, is a product of the concept of orthogonal basis in linear algebra. Closely related is the Fourier Transform, an important example of a linear transformation, a common topic in linear algebra courses. Such work also gave me the opportunity to introduce the students to an absolutely marvelous book on voice, Fourier Analysis and trigonometry (Gleason,1995).
One
pleasant, coincidental event took place: many of the ECE students in the course
were simultaneously taking a course in their own department in Signal
Processing which indeed did use Fourier Methods. For these students, the two
courses developed a natural, complementary relationship.
Mechanical
Engineers also have diverse concerns, but a common and important one is that of
vibrations. Previous links between
courses had built a strong relationship with one mechanical engineering
professor who enthusiastically offered advice and specialized lectures. The ultimate goal of studying vibrations, it
was decided, was to examine the phenomenon of resonance. Mechanical engineers
come to such a course with an intuitive notion of linearity, having seen
it in physics classes under the name of superposition. It was important
to take advantage of this.
All
groups worked on projects which were either 1 or 2 weeks in duration, for a
total of 5 weeks. They turned in one
document per team. The projects were cumulative in the sense that each one
built on the prior one. The earlier projects established basics while the final
project was much more applied, required a decision or recommendation to be
made, and tended to be more open ended. The earlier projects established links
to linear algebra; the latter projects moved towards science and engineering in
ways consistent with the philosophy of education they would see in upper level
work at WPI. The particular projects were:
Bio
week 1: the Leslie population model
week 2: the Leslie model applied to
the US population
week 3: a quantitative look at the
spread of HIV
weeks 4 and 5: develop a model showing
the potential impact of an
HIV vaccine
Civil
Engineering
week 1: Linear Programming
(optimization) formulation
week 2: the Simplex Algorithm
week 3&4: Project Management – the
Critical Path Method
week 5: the Big Dig – lessons learned
Computer
Science
weeks 1 and 2: software for
Gauss-Jordan Elimination
week 3: software for matrix algebra
week 4: choice of dominant eigenvalue
computation or graph theory
week 5: the Sierpinski Triangle
Electrical
and Computer Engineering
week 1: electrical circuits and
systems of equations
week 2&3: Fourier Basics
week 4: Sound and Sound Files
week 5: Voice Print engineering
Mechanical
Engineering
weeks 1 and 2: ordinary differential
equations and mass spring
systems (unforced)
week 3: forced mass spring systems –
superposition
week 4&5: Fourier Series and
Resonance
(A detailed course description, including projects, may be found at Goulet,2001 ).
I
taught the course without the major project tracks in 1998, and again with the
project tracks as described above in 1999 and 2001. In 1998 ,1999 and 2001 identical final exams were given, testing
only the core material. 1998 is treated as the control. Results were:
|
|
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 |
*weights by question of
.2,.2,.3,.1,.2 were used to compute the overall exam grade
For sake of completeness, the topics on the 5 questions were: behavior of dynamical systems, a proof of the coordinate formula for orthogonal coordinates, matrix diagonalization, a choice of proofs, and a problem on matrix algebra and limits.
Discussion of Data
My original hope was that the students might carry the additional load of working on projects and still learn as much core material as before. The data above suggests they may have done better than that. The overall final grade average for 1999 is significantly better than 1998, and the 2001 average is within a whisker of also being significantly better. In each case, it is achieved on the strength of two questions – 1 and 2 in 1999 and 2 and 3 in 2001. I am pleased that question 2 was consistently better than the control. The problem it is based on provides the basic algebraic result at the heart of Fourier Series. Since both the electrical and mechanical groups used Fourier Series, it is pleasing that they were able to also grasp the foundations as well as the applications.
We hope for many things to emerge out of the freshman and sophomore years beyond simple academic credit. The WPI education, as it applies to juniors and seniors, emphasizes project work and team work. This course gave students a chance to meet other students in their major and work with them, giving them an earlier start in both areas, as well as informal but valuable opportunities to learn what the major was really like. I did not fully appreciate the opportunity until I asked them during the first few classes to form teams according to their majors, and any number of them came to me and said “I don’t know anyone here in my major”. In the Harvard study, the following quote is very relevant here: “substantive work in the sciences should be structured to involve more interaction with other students and with faculty members” (Light 2001, 75 ).
As
the course evolved, I would get two kinds of visitors to my office: individual
students wanting help with core material (“can you show me how to do this problem?”),
and project teams wanting to discuss a project. The latter were most enjoyable.
Since the team members all had the same major, they took on an identity
because of that, and I treated them that way. I tried to ask at least as many
questions as I answered (“You guys are electrical engineers – how do YOU think
it ought to be done? Does this math make any sense?”). My attitude was that
engineering is much more than computations – with issues like interpretation
and decision making. Anything that fostered such events was desirable.
The
standard core material to the linear algebra course was retained while
augmenting it with tracks of projects by major. Students were able to perform as well or better than in the
control while carrying the additional workload. Additionally, contributions to other areas of their education
such as project work, team work, and problem solving may also have been made.
Finally, the course and its material had more relevance and meaning with regard
to their overall education.
Bloom B., Madaus,G., and Hastings, J.1981. Evaluation to Improve Learning. New York, McGraw-Hill.
Catalano, G. and Catalano, K. 1999. “Transformation: From Teacher Centered to Student Centered Engineering Education”. Journal of Engineering Education, 88(January):59-64.
Gleason, Alan (translator). 1995. Who is Fourier?. Boston, Language Research Foundation.
Goulet, John. 1995. “Calculus, Bell Shaped Curves and Global Competition.” Primus, 5(March): 37-42.
_____.
1997. ”Making Students Independent Learners – Fractal Projects” Creative
Math Teaching , 4(1): 1-3.
____.
2001. Course syllabus. http://www.wpi.edu/~goulet/ma2071/syll.htm.
Lathi,
B.P. 1998. Signal Processing and
Linear Systems. Carmichael CA, Berkeley-Cambridge Press: 171-234.
Light,
Richard J. 2001. Making the Most of College. Cambridge: Harvard
University Press, 2001:45.
Worcester Polytechnic Institute. “The First Year Learning Experience – A Report to the New England Association of Schools and Colleges”, WPI, June 1991.