\documentstyle{article}
\title{Assessment}
\author{David M.\ Bressoud}
\date{}
% version of 12 May, 1995
\begin{document}
\maketitle
\begin{quote}
The biggest problem with being a student is that you're always too busy getting an
education to learn anything. ---Richard Feynman\cite{schwarz}
\end{quote}
\section{Introduction}
Richard Feynman has put his finger on the reason why it is so difficult to get
mathematicians to take assessment seriously. Most of us recognize the
educational ideal to which he is alluding: the teacher should share hard won
insights, signal traps, coach, challenge, and encourage while the student
struggles, stumbles and rises, and---by dint of hard work---comes to a personal
understanding of the subject. This is what learning should be, and within this
scenario testing seems at best a distraction.
We do use tests of knowledge, skill, and understanding. We know
that we must, not only because our institutions insist upon it, but for
our own peace of mind. Whatever our ideal may be, we live in an imperfect world in
which such certification is the motivating force behind much of the learning that
takes place. We bemoan the perennial question, ``Will this be on the test?'', but
we know that, in fact, this question is inevitable and even rational given our
educational system and the forces that send students to us to learn mathematics.
I begin with the assumption that I will always have students who will not study
without such external motivation, and that I have an obligation to teach them.
I have found that most of these students want to learn, that they do want to
understand, but that there are limits to their single-mindedness of
purpose, to their willingness to face repeated failures in the quest to
understand. I have also found that almost all of them carry a spark which, if
nurtured and encouraged, can burst into the flame of a passion to learn.
Assessment constitutes one of the bags of tools at my disposal. It is not a recent
discovery that it contains more than tests and can accomplish far more than
certification. There are assessment techniques that can help me keep my finger on
the pulse of the class so that I can react when confusion is spreading. There are
tools to help a student recognize misconceptions in time to correct them, and
confirm when personal progress has been made. Assessment can enable the
student to integrate seemingly disparate topics and can provide challenging
and summative learning experiences. These tools are not new. Good teachers have
used them for generations. But not all of them are widely known or widely used.
As we proceed with the task of reinvigorating our teaching of calculus, we need
all the help we can get. The purpose of this article is to display some of the
tools of assessment that others have found useful, and to challenge each of us to
think about what assessment means and how we want to use it. My own paradoxical
experience is that when it is used well and wisely it can actually move me and my
students closer to that ideal in which testing has no place.
\bigskip
This is necessarily a very personal and anecdotal treatment of assessment. I
hope that what I have learned may be useful to others. I will also dare to make
general statements about assessment. In a few instances they will be supported by
research or statistics, but in the spirit of John Ewing's comments on assessment
in the {\it Monthly} \cite{ewing}, I am relying primarily on presenting a case
that will make sense.
\section{Basic principles}
Much of the current work on assessment involves the entire undergraduate program
in mathematics and asks what the students are learning and how well they are able
to use what they learn. This is the focus of the assessment report of the {\sc
maa}'s Committee on the Undergraduate Program in Mathematics \cite{cupm}. We
shall look at these questions within a much narrower context: the individual
calculus course. Within a specific class, assessment is intimately tied to
evaluation and the process of assigning grades. This is appropriate, but we
need to be aware of the wider uses of assessment even here.
As we look for more effective ways of assessing what is happening in our
classes, there are three important principles:
\begin{enumerate}
\item What is assessed and how it is assessed must be seen by both faculty and
students as emerging from clearly articulated goals and objectives.
\item The assessment must be an integral part of the course.
\item The information gathered from assessment must be used to improve both our
own teaching and student performance.
\end{enumerate}
\subsection{Goals and objectives}
Calculus tests are usually constructed by looking over the syllabus for the past
few weeks, determining what knowledge the students should have picked up, and
finding problems that use this knowledge. The first calculus test invariably
includes the obligatory derivative of a polynomial (to build confidence)
followed by problems that use an assortment of techniques and applications of
differentiation. We are assessing the students'
ability to use this knowledge, and the message that we send is that the mastery of
the techniques of differentiation---with some sense of how to apply them to
standard problems---has been the objective of the first weeks of class.
If we actually ask students and faculty what students should be learning,
the answer goes beyond manipulative skill and well practiced applications.
Both students and faculty use the word ``understanding'' when describing the
objectives of a calculus course. It is useful to take this concept apart so that
we can get a better handle on what it would mean to evaluate understanding.
Bloom's taxonomy \cite{bloom} of the major categories of cognition is helpful:
\begin{enumerate}
\item {\bf Knowledge:} the ability to remember previously learned material.
\item {\bf Comprehension:} the ability to grasp the meaning of material. This
can be demonstrated by translating (for example between graphical and
numerical or symbolic representations) or by explaining or summarizing.
\item {\bf Application:} the ability to use learned material in new and concrete
situations.
\item {\bf Analysis:} the ability to break the material down into its constituent
parts so that its structure may be understood.
\item {\bf Synthesis:} the ability to recombine constituent parts into something
that is new.
\item {\bf Evaluation:} the ability to judge the appropriateness of using what has
been learned for a particular purpose.
\end{enumerate}
True understanding involves all of these. If we value more than knowledge, then
we must test more than what can be memorized.
Our first task is to establish clear goals for the students who will be
taking our course. Bloom's taxonomy is generic. We must
translate these categories into specific goals that are appropriate for
calculus. An example of such a set of specific objectives for first-semester
calculus is the is one developed by Smith and Moore for the
students in {\it Project CALC\/} \cite{smith}:
\bigskip
\noindent ``Specific goals for this semester are for you to
\begin{enumerate}
\item understand the concept of a function in a variety of representations,
\item understand the concept of a derivative and its relationship to rates of
change including linear approximations, Newton's method, Euler's method, tangent
lines, and instantaneous velocity,
\item formulate problems involving rates of change as initial value problems,
\item solve initial value problems both numerically and formally, and be able to
explain and use your solutions,
\item develop familiarity and facility with a computer as a tool for
understanding mathematics and for solving mathematical problems,
\item write reports using a word processor,
\item differentiate functions using the standard techniques of differentiation,
\item explain the relationship between data and theoretical models as a means of
examining real world
phenomena.''
\end{enumerate}
Whatever others may think of this particular set of goals, the point is that
they go well beyond what we usually test and they give definition to what it will
mean to understand calculus. There is no universal set of goals for any course.
Each department must determine objectives that are consistent with the nature and
expectations of the institution and the preparation of its students. The
individual instructor then has the responsibility to flesh these out, keeping in
mind personal strengths and preferences and any special circumstances of the
actual students that he or she will be facing. What is important is that both we
and our students know where this course is going and what is expected.
\subsection{Making assessment part of the course}
A wonderful set of goals is useless unless students are held accountable
for achieving them. Most students come into a calculus class with a desire to
learn and understand, but few have a passion for learning. Many things
compete for their time and attention, and most students will do the minimum of
work that is required to attain what they see as a satisfactory grade.
If your course objectives include the ability to summarize and explain a theorem,
or to analyze its proof, or to clearly and cogently lay out the reasoning behind
the solution to a problem, or to choose an appropriate technique to explore and
solve a new and unfamiliar problem, then you must test these abilities. Our
standard, timed examinations are not sufficient for this purpose. Few students can
demonstrate higher level thinking skills when they are under pressure. Such
examinations must be supplemented by other assessment tools such as take-home
exams, projects, reports, and oral presentations.
If you want the student to do the assigned homework problems, then collect and
grade them. If you want the student to learn to read a textbook and digest the
main points before you give your presentation, then give periodic and unannounced
quizzes on the material they were to read. If you want the student to master a
basic collection of techniques of differentiation, then test these techniques and
insist on perfection (see section~3.1). If you want the student to be able
to write clearly, then collect, critique, and grade written exposition.
\subsection{Using assessment to improve teaching and learning}
The third principle is critical. How often have we constructed a final
examination that we feel really gets to the core of what we want our students to
be able to do, only to find that the entire class performs abysmally? We have
put ourselves in the untenable position of failing everyone or of giving A's
and B's that we know do not mean what we want them to mean. I myself plead
guilty to having revised the goals for a course in the light of the results of
the final exam.
The point of the third principle is that there should be no such surprises at
the end of the semester. This should not be the first time that we evaluate
comprehension or the ability to apply, analyze, or synthesize, that we ask
students to be creative and insightful. Our expectations of what they are to
achieve must be made clear from the very beginning of the course, and we must
use the results of early assessment to help the students move toward the goals.
The need for early assessment as part of the teaching process is especially
critical in calculus or any first-year course. Many students are not accustomed to
analyzing, explaining, or applying what they have learned to new and unfamiliar
settings. If all we do is say that they are now responsible for these higher
levels of cognition, test them and find them wanting, and then throw up our hands
and walk away, then we are abnegating our responsibilities as teachers. It is our
duty to help them achieve the course objectives.
This can be done by front loading the course with many projects and challenging
problems at increasing levels of difficulty. We must set aside the time to
critique student efforts, to use their work as a springboard for
discussing expectations, to show examples of what their classmates
have done that is praiseworthy. It is especially important that
students have the opportunity to revise and rewrite their early work. Assessment
is not just for our benefit. We must use it to help our students recognize where
their level of understanding is falling short of what we expect of them and what
they need to do to correct this.
One example of how this can work comes from my own experience in first-year
calculus. The problem was to find population, $P$, as function of time, $t$,
given actual population data and the differential equation that $P$ is assumed to
satisfy. One pair of students found a correct solution, but their explanation of
how they found it revealed a lack of understanding. They described the problem as:
``to get $dP/dt = k\,P^{1+r}$ in the form of $P^r = 1/rk(T-t)$ [where]
$T=1/krP_0^r$.'' This suggested that they were not solving a differential
equation, they were performing a series of manipulations. This impression was
confirmed when they used the data to approximate the value of $r$. They correctly
recognized that $\ln P$ is a linear function of $\ln (T-t)$, but their explanation
revealed that they had no understanding of why this was true: ``For any function
$P(t)$, the graph of $\ln P$ versus $\ln(T-t)$ must be a straight line because
population growth is dependent on time.'' I now knew that I needed to back up and
hit certain key ideas again.
Ferrini-Mundy and Graham \cite{fmg} are among a number of researchers
in mathematics education who have been examining student conceptions of
such basic ideas as limits, derivatives, and integrals. Much of what they
find can upset our notions of what we should be testing:
\begin{itemize}
\item Confusion between approximation as it is used in the definition of a limit
and as it pertains to the actual limit: ``It [$.999\overline{9}$\,] wouldn't be
quite 1 [but it is] close enough to solve the problems that we needed to solve.''
\item The ability to correctly locate local maxima, minima, and inflection
points in order to sketch the graph of a function, while the same person has a
complete inability to explain the relationship between the derivative and the
tangent line.
\item A recognition of the definite integral both as a process (a signal to
anti-differentiate and substitute values) and as a representation of the area
under a curve without having formed any connection between these
two representations.
\end{itemize}
Their work can alert us to serious misconceptions that are common and which our
current tests may not uncover. We have the responsibility to determine whether
these misconceptions are present and, if so, to modify our own teaching
accordingly.
\section{Common tools of assessment}
\subsection{Traditional tests, quizzes, and homework}
If speed and accuracy in executing well established algorithms
are what we value most, then multiple choice tests with less than five minutes per
problem are appropriate. If we want to discover whether our
students are capable of applying the ideas of calculus to
unfamiliar problems, then we must give few problems and sufficient time to
actually think about them. There is, of course, middle ground between these two
extremes of what we want and what we feel we are forced to use by constraints of
time or class size. The point is that we must be aware of how far what we test
diverges from our true objectives so that we can continually improve the tests
and so that we are aware of what other evaluative tools we will need if students
are to perceive that we really do expect them to achieve the stated goals.
Student perception is very important. It is not enough that we can look
over the questions we have written and see that they are consistent with our
objectives. The students must also see this. This is the greatest flaw of the
multiple choice test. No matter how well the questions and possible answers have
been constructed, most students believe that the test favors speed and accuracy
above thoughtfulness and understanding. They also see the
incorrect solutions as traps that we have set for those who make minor
errors.\footnote{These points were articulated in the survey of student
attitudes \cite{bressoud}.} With this perception of our tests, it should not
surprise us if students believe that the purpose of the calculus course is to
reduce the number of science and engineering majors.
My examinations have very few but very challenging problems, and I allow
students to improve their grade by correcting the answers that were wrong. I
have also found it useful to supplement such examinations with a {\bf gateway
test}. This is a straightforward test of the techniques of differentiation or
integration that students are allowed to retake as many times as necessary. The
grade is not averaged into the course grade, but a pass at the level of 80\% or
90\% or 100\% is required for passing the course. I require a pass at 100\%, but
I do test exactly the same set of techniques---with different problems---on
each retest. Starting with the second retest, I let the student know which
answers are incorrect and give the opportunity to change them before I grade the
test. In two years of using gateway tests, every student has passed. I have also
found that my weakest students appreciate this test the most; it forces
them to a minimal level of competency. The experience of others is that those who
never pass the gateway test are extremely rare. Usually, they are already failing
the course. Occasionally, it may be expedient to drop the course grade by
one letter rather than to fail them.
\medskip
We should collect and grade {\bf homework problems} if we expect them to be done
faithfully and completely. If grading is not possible, then {\bf quizzes} can be
used as a check that the problems have been done. I use unannounced quizzes in
addition to graded homework to verify that students have done the required reading
and kept up with assignments. Such quizzes are short and do not address more
than a single, simple idea.
\subsection{Writing assignments, expositions, and proofs}
One of the best ways of evaluating student understanding is through the use of
writing assignments. These can take a variety of forms from a
careful explanation of how a particular technique works and when it is appropriate
to employ it to an extensive write-up of the solution to a demanding problem. It
is not clear to me when we should begin to require the ability to read and
analyse proofs, but whenever that time comes, students will understand the
structure of a proof much faster if they write and rewrite their own proofs.
Nothing forces the student to clarify a personal understanding and reveals where
there are gaps in that understanding as effectively as the requirement to explain
it in writing.
Very few students turn in mathematics assignments that are in any sense of
the word readable. This does not necessarily imply an inability to write. I have
known many students who turn in beautiful exposition when it is required, and a
page of scrawl with circled answers when they feel that that is sufficient. We
must be clear about our expectations, and we must be prepared to give the support
that may be needed to achieve them. Most students have to learn how to
write mathematics.
There are several things that aid in the development of the ability to write
mathematics. One is to insist that students use the first person most of the time
and active voice all of the time. Another is to critique early versions of
the reports before they submit the draft that is to be graded, especially early
in the year. It may be necessary to set aside time to talk about what makes for
good writing in mathematics. And it helps if students are collaborating, at least
on the first few assignments. They do learn from each other.
\subsection{Major projects and laboratory work}
{\bf Projects} give students the chance to exercise higher level thinking skills:
to apply their knowledge, to integrate concepts, to synthesize new solutions, to
evaluate the validity of the solutions they find. They can also make class
exciting. Through them, I have seen students discover the creative and engaging
side of mathematics. I find nothing in teaching so exhilarating as a classroom
abuzz with small groups of students tackling a difficult problem.
Group work is the right way to handle most major projects. It builds
confidence, it forces students to think through their insights as they explain
them to their collaborators, it corrects many errors before they become embedded
in the solution, and it facilitates discoveries that might not have been made by
individuals. There are many ways of organizing such group work and of maintaining
individual accountability. I leave the organization
of each group up to the participants, shuffle the students after each project,
and require them to sign the report, signifying that each has contributed to it.
Once students are accustomed to writing such reports, it is useful to have
some projects that are written up individually. This is most appreciated by the
best students---who are given the chance to show what they are capable of doing
alone---and the weakest students---who often feel that they have only been able
to make minimal contributions to group projects.
The literature is now rich with such projects. The five volumes of {\it
Resources for Calculus\/} \cite{roberts} constitute one starting point.
\medskip
{\bf Laboratory work} is similar but has its own set of opportunities and
hazards. At its best, it is strongly integrated into the classroom experience:
a new concept is introduced in class, the next day students use computers or
graphing calculators to explore it in various contexts and to
begin to build an understanding of how it is used and how it relates to what they
already know. They then return to the classroom where questions can be answered
and the concept further developed. Student evaluations of the labs in {\it
Project CALC\/} \cite{barker} indicated that this is what was happening:
\begin{description}
\item{\quad} ``helpful in clearing up confusing concepts in text''
\item{\quad} ``Several times I found myself thinking well, I'll just wait till
Tuesday, maybe that will help, and it did.''
\end{description}
But there are dangers in laboratory work. It usually has a fixed time limit. It is
difficult to make it challenging and engaging without putting it beyond what
can be completed in the time allotted. To err in the other direction is worse.
Labs lose their value if they are made mechanical, if they can be executed without
forcing students to think about what is happening. And if they are not directly
tied to the classroom---used and built upon by the classroom instructor---then
they are meaningless.
\subsection{Oral presentations}
It is not enough to be able to write well and to interact constructively in
small groups. We would like our students to reach the level of familiarity and
confidence at which they are able to present and defend solutions before an
audience. This can be as simple as a homework problem whose solution is
explained by one student, or as complex as the digestion of a special topic to
be presented to the entire class. One way of guaranteeing everyone's participation
in a group project is to hold each liable for presenting the group's
solution to the rest of the class.
\subsection{Class journals and individual portfolios}
Students take justifiable pride in their best work, but interest in what they
have done quickly wanes once a grade has been assigned. One way of maintaining
interest and pride is to publish a {\bf class journal}, taking one or perhaps two
examples of the very best reports from each major assignment. In addition to
serving as recognition for students who have put in extra effort, it
demonstrates to the entire class what you consider to be exemplary work.
Another means of preserving a student's best work is to require that each keep a
{\bf portfolio} of those reports or examinations that represent the student's
best work. This enables subsequent instructors to see what the student has done,
is capable of doing, and still needs to do. It helps the student track the
progress that has been made.
\subsection{Ungraded assessment}
One of the great myths of education is that the experienced college teacher can
tell how well the class is following from the level of perceived attentiveness.
That is simply not true. I have been teaching long enough to know how easily I can
deceive myself by concentrating on the best students. We need other measures of
what is happening in the classroom, not only for our benefit but to help our
students recognize when they do not understand a concept that is important. Some
of the techniques that I have found helpful are
\begin{itemize}
\item Ask a question and force the students to come up with an answer that they
are willing to defend. Take a few minutes for them to discuss their
solution with neighbors. This works particularly well when you give a
problem with two plausible answers and ask how many people favor one or the other
of the alternatives. When first presented, usually very few will be sufficiently
confident to raise their hands. After a few minutes of talking it over with
neighbors, you will find that most of the class is now willing to state a
preference (and that most have now correctly identified the right answer).
\item In the last minute of the period, have everyone write down brief
answers to two questions that ask for some sort of summary of the period: ``What
did we do this period?'' and ``What unanswered question do you still have?''
\item Have a colleague sit in on your class and observe your students. If you
really want information about student understanding, have your colleague
interview selected students and probe what they believe they have learned in the
course.
\end{itemize}
A good resource for assessment techniques is {\it Classroom Assessment
Techniques\/} by Angelo and Cross \cite{angelo}. It has a general overview of the
topic of ungraded assessment together with fifty examples that each include
a sample implementation, pros, and cons.
Ungraded assessment is almost always for our benefit. However modest or
ambitious it may be, it is a waste of time unless we use the information
obtained from it to improve our teaching.
\section{Student responsibilities}
Having described all of the things that we should be doing, I want to make the
point that it is {\it not\/} our responsibility to see that everyone achieves the
goals we have set. This is where I have the greatest difficulty with the {\sc
nctm} {\it Assessment Standards for School Mathematics\/} \cite{nctm} which cites
``equity'' as one of standards. Assessment in the classroom does have a role to
play as filter, identifying those students who have fallen short of the goals.
Our goals must be constructed with care: they must be honest, they must be
realistic, they must be challenging. And they must be upheld. A large part of the
purpose of assessment is to hold each student accountable for making satisfactory
progress toward these goals.
If we insist on this accountability, we can go a long way toward correcting one of
the great problems in our calculus classes: the meager amount of time that most
students spend studying calculus. From my own survey of students at The
Pennsylvania State University \cite{bressoud}, I have seen that the problem is
more one of knowing what to do when studying than a lack of willingness to study.
Our students come to college with the belief that studying mathematics means
doing the exercises at the end of the section. Furthermore, experience has taught
them that the really hard problems seldom appear on tests. I was surprised to
learn how very conscientiously almost all students will do the relevant problems
the same day they have been discussed in class, even when these exercises will
not be collected or graded. But that is all they know to do. The surveyed students
spend about an hour working these assigned problems. When they start to go beyond
the examples worked out in class---and thus beyond what they believe will be on
the test---most students will stop working. They have other demands on their
time. If the only assessment tool we use is the traditional test, then the only
studying most students will do is what has worked for them on tests in the past.
A common complaint from students in nontraditional calculus classes is that
too much work is required. Limited data suggests that
the C4L program of Dubinsky, Schwingendorf, and Mathews comes close to doubling
the amount of time students spend studying calculus \cite{dub}. As we broaden
student perception of what they are required to know and be able to do in
mathematics, we can expect that they will study more and get more out of the time
that they spend studying.
Student responsibilities are real and cannot be ignored, but we must first show
them how to study effectively and then hold them accountable for so doing.
\section{Conclusion}
What I have described in this paper is at the opposite pole from the ideal with
which I began. My classes will usually involve tests, graded homework
assignments, unannounced quizzes, gateway tests, major projects, lab reports,
written exposition and proofs, as well as various forms of ungraded assessment.
It begins to look like a class that is so busy getting educated that no one has
time to learn. But my own experience has been that something amazing emerges when
evaluation is this constant and this varied: {\it the evaluation ceases to be
the focus of the course}.
Students no longer study just for the test. Because they are held accountable
for all aspects of the course, they work at all aspects of the course.
Because no single evaluation will determine their grade, no single evaluation
dominates their horizon. They learn what it means to understand
mathematics, and they begin to deepen their own understanding.
Teaching in this way is unsettling, and it means more work for the professor.
It is uncomfortable to learn how ineffective our beautiful lectures can be.
It takes time to determine where and how we can improve what we are doing and to
settle into new modes of teaching. The increased assessment is time-consuming.
There are ways to do the assessment efficiently: collaborative projects rather
than individual ones reduce the number of papers, some assignments can be
self-graded or critiqued by other students, hired graders can be used
intelligently, but it still requires more effort on our part than the traditional
lecture--exam format. The question is not whether we can afford this extra
effort, it is whether we can afford not to take assessment this seriously.
\begin{thebibliography}{99}
\bibitem{angelo} Thomas A.\ Angelo \& K.\ Patricia Cross, {\it Classroom
Assessment Techniques: a handbook for college teachers}, 2nd edition,
Jossey-Bass, San Francisco, 1993.
\bibitem{barker} William H.\ Barker, {\it Mathematica Laboratory Manual to
accompany The Calculus Reader}, revised preliminary edition, D.~C.~Heath, 1994.
\bibitem{bloom} B.~S.~Bloom {\it et al.}, {\it Taxonomy of Educational
Objectives}, vol.\ 1: {\it Cognitive Domain}, McKay, New York, 1956.
\bibitem{bressoud} David Bressoud, Student Attitudes in Calculus, {\it Focus},
{\bf 14}, June 1994, 6--7.
\bibitem{cupm} Committee on the Undergraduate Program in Mathematics, {\it
Assessment of student learning for improving the undergraduate major in
mathematics}, The Mathematical Association of America, Washington, DC 1995.
\bibitem{ewing} John Ewing, Comment, {\it The American Mathematical Monthly\/},
{\bf 102}, Feb.\ 1995, 98.
\bibitem{fmg} Joan Ferrini-Mundy \& Karen Graham, Research in calculus learning:
understanding of limits, derivatives, and integrals, {\it Research Issues in
Undergraduate Mathematics Learning}, Kaput \& Dubinsky, editors, {\sc maa}
Notes \#33, {\it The Mathematical Association of America}, Washington, DC, 1994.
\bibitem{dub} David Mathews, Time to study: the C4L experience, to appear in
{\it UME Trends}, 1995.
\bibitem{smith} Lawrence C.\ Moore \& David A.\ Smith, {\it Project CALC
Instructor's Guide}, Preliminary Edition, D.C.\ Heath, 1992.
\bibitem{nctm} National Council of Teachers of Mathematics, {\it Assessment
Standards for School Mathematics}, Working Draft, Reston, VA, 1993.
\bibitem{roberts} {\it Resources for Calculus}, vols.\ 1--5, A.\ Wayne Roberts,
Project Director, {\sc maa} Notes numbers 27--31, The Mathematical Association
of America, Washington, DC, 1993.
\bibitem{schwarz} Patricia M.\ Schwarz, Feynman's Theorem on Being a
Student, INTERNET newsgroup SCI.MATH, 26 Dec 1993.
\end{thebibliography}
\noindent{\sc Department of Mathematics and Computer Science, Macalester
College, St.\ Paul, MN 55105, USA; bressoud@macalstr.edu}
\end{document}