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Mathematics departments are increasingly being asked, both internally by other departments and externally by accreditors, to reflect on their role on campus. At one time, a mathematics department was acting within the expectations of a college by providing research and teaching in a way that weeded out the poorer performers and challenged and pushed forward the elite. With the diversity of today's student clientele, changing expectations on the nature of education, the jobs crisis, and the downsizing of mathematics departments, we are having to examine the broader issue of how we can provide a positive, well-rounded learning environment for all students.
As more and more disciplines require mathematics, outside expectations of mathematics departments have risen; other departments are now competing for ownership of some of our courses. The issue of effective teaching in mathematics, for example, is a frequently discussed problem for students and faculty on campus. So is the content we are teaching, say, in the calculus sequence. And we shake our heads at what to do about general education courses in physics and chemistry in which students can't use mathematics because they can't do the simple, prerequisite algebra. However, assessment does not function at its best when aspects are pulled apart and examined under a microscope. The evaluation of mathematics teaching or the curriculum or the quantitative capabilities of our students cannot be separated from the evaluation of broader issues for which the department as a whole is responsible. Some of these issues involve finding ways to properly place and advise students, networking with client disciplines, examining new curricula and methods of teaching, and demanding and maintaining basic quantitative literacy requirements that will assure our students graduate even minimally competent.
All of these major departmental initiatives must operate within an assessment framework. Until recently, however, mathematics departments have not generally been at ease with the idea of going public. Even if we move beyond the idea that the word "assessment" is mere jargon, perhaps we feel we are being expected to compromise our high standards, which are naturally opaque to others, or that our freedom to teach in ways that intuitively feel best is being stripped from us. Many of us, overworked as we are, may be content to leave assessment issues to administrators. However it has become clear through a variety of publicized situations that if we neglect to deal with new expectations, then decisions about us will be made by others, with the result that we will be marginalized.
Some mathematics departments do not change; others may adopt changes (such as a change in a text or curriculum, a change in the use of technology, or a change in a policy of placement or admittance of transfer students) without ever paying explicit attention to the results of these changes. This unexamined approach to teaching will probably not be acceptable in the future.
In this section we have searched for efforts by departments which have created changes that feature assessment as an integral component. The method of assessment-as-process helps create an educational environment that is open and flexible in ways which benefit faculty, students and the institution as a whole. For example, everyone benefits when placement is running efficiently. We are not as concerned that the final results of the projects should be awesomely successful (although some are) as we are by the processes by which departments have examined and made public the results of their efforts. Good assessment is not a finished product but the input for more changes. In fact, most of the reform efforts of departments presented here are not at a stage of completion. Rather, the purpose of this section is to acknowledge that on campuses now there is some ferment on the issue of assessment, and that questions are being raised and brooded about for which we hope to have provided some plausible starting solutions.
Our contributions are highly diverse; we will see the efforts of small and large schools, research institutions and small inner city schools. Retention of majors is the issue in some cases, while elsewhere the focus is on developmental mathematics or general education mathematics. Some programs are personally focused on listening to students; others emphasize dealing collaboratively with faculty from other departments, and still others (not necessarily exclusively) have conducted massive analytical studies of student performance over the years for purposes of placement or for charting the success of a new course. We will learn about the efforts of one school with a bias against assessment to lift itself up by the bootstraps, and about some other schools' large-scale, pro-active assessment programs that operate with an assessment director/guide and encouraging, assessment-conscious administrators. We will also hear from some experts on quantitative literacy across campus, learning theory, and calculus reform so we may view concrete assessment practices against this theoretical backdrop.
In the diversity of methods exhibited here, it is interesting to observe how the rhetorical dimensions of assessment function; how the use of language in the interpretation of results has the power to influence opinion and persuade. Rhetorical interpretations of assessment procedures may indeed be intended to sell a program, but if this strikes us as an unnatural use of assessment, we would be equally unaccepting of the language of politics, law, business, and journalism. We would never recommend blanket adoption of these programs — this would not even be feasible. Rather we hope that the spectrum of efforts presented here, achieved through the obvious commitment and energy of the mathematics departments represented, will suggest ideas about what is possible and will serve to inspire the examined way.
Here is an overview of this section.
Placement and Advising
Judith Cederberg discusses a small college, nurturing a large calculus clientele in a flexible, varied calculus program, which recognizes the need for a careful placement procedure and studies the effectiveness of this effort with statistics that help predict success. Placement is also the issue at a large comprehensive school, in an essay by Donna Krawczyk and Elias Toubassi explaining, among other initiatives the department took to improve its accessibility, their Mathematics Readiness Testing program. Statistics about the reliability of the testing are studied, and the authors observe that their results have been found useful by other departments on their campus. At many institutions, advising operates at a minimal level — it may be the freshmen, or possibly even the seniors who are forgotten. Steve Doblin and Wallace Pye, from a large budding university, have concentrated on the importance of advising students at all levels; from providing mentors for freshmen to surveying graduating students and alumni, this department operates with continuing feedback. Materials which carefully describe the program are distributed to students.
Quantitative Literacy
When our students graduate from four or more years of college, are they prepared for the quantitative understanding the world will require of them? This issue, which could be one of the most important facing mathematics departments today, has suffered some neglect from the mathematics community, in part because of the varied ways in which liberal arts courses are perceived and the general confusion about how to establish uniform standards and make them stick. Guidelines from the MAA for quantitative literacy have been spelled out in a document authored by Linda Sons who offers an assessment of how this document was actually used in a quantitative literacy program at her large university. For a variety of liberal arts mathematics courses and calculus, this school tested students and graded the results by a uniform standard, then compared results with grades in the courses. Results led to making curricular changes in some of the courses. Some of us may have tried student-sensitive teaching approaches in higher level courses, but when it comes to teaching basic skills, there seems to be a general taboo about deviating from a fixed syllabus. Cheryl Lubinski and Albert Otto ask, what is the point of teaching students something if they're not learning? They describe a general education course that operates in a learning-theoretic mold. For them, the content is less important than the process in which the students construct their learning. Elsewhere, Mark Michael describes a pre- and post-testing system in a liberal arts mathematics course, that among other things, raises some interesting questions about testing in general, such as how students may be reluctant to take such testing seriously, and why some students may actually appear to be going backwards in their learning. With an even broader perspective, how does one assure that all students graduate quantitatively literate at a large collective bargaining school with little coordination in its general education program and a tradition of diversity and disagreement among faculty? Is the mathematics requirement continually being watered down in other general education courses, in effect bowing to students' inability? What do students perceive and how can one interpret the broad findings? A non-mathematician, Philip Keith, discusses the ramifications of accommodating various departments' views of quantitative literacy and reports on using a simple survey to get an institution on track. How realistically can such broad data be interpreted? The value of this survey lay in its effect to prod the faculty to begin thinking and talking about the subject.
Developmental Mathematics
Developmental (remedial) mathematics is the subject of Eileen Poiani's report from an inner city school with a diverse, multi-cultural clientele. Deeply committed to raising students' ability in mathematics, this school has a richly expansive supportive program. They ask: does developmental mathematics help or is it a hopeless cause? On the receiving end of grants, an assessment plan was in order, and three major studies were completed over the course of ten years, with observations about retention and graduation data and performance in developmental and subsequent mathematics courses. As this study continues, it looks more in depth at student characteristics.
Mathematics in a Service Role
How is a mathematics department supposed to please its client disciplines? From a comprehensive university, Curt Chipman discusses a variety of ways in which an enterprising mathematics department created a web of responsible persons. Goals to students and instructors are up-front and across the board, course leaders are created, and committees and faculty liaisons are used to the fullest for discussions on placement and course content. In another paper, William Martin and Steven Bauman discuss how a university which services thousands of students begins by asking teachers in other disciplines — not for a "wish list" — but for a practical set of specific quantitative knowledge that will be essential in science courses. As determined by the mathematics department and client disciplines, pre-tests are designed, the questions on the tests asking about students' conceptual understanding relating to the determined skills. Results are used for discussions with students, the mathematics faculty, and faculty in client disciplines, creating a clear view for everyone of what needs to be learned and what has been missed. From a small university in Florida, Marilyn Repsher and J. Rody Borg, professors of mathematics and economics respectively, discuss their experiences in team teaching a course in business mathematics in which mathematical topics are encountered in the context of economic reasoning. Opening up a course to both mathematics and business faculty stimulates and makes public dialogue about the issues of a course that straddles two departments. It is hard not to notice that seniors graduating in mathematics and our client disciplines often take longer to graduate, and minority students, even longer. Martin Bonsangue explores this issue in his "Senior Bulge" study, commissioned by the Chancellor's Office of the California State University. The results directly point to the problems of transfer students, and Bonsangue suggest key areas for reform. At West Point, when changes were felt to be in order, Richard West conducted an in-depth assessment study of the mathematics program. With careful attention to the needs of client disciplines (creating a program more helpful to students), the department created a brand new, successful curriculum—cohorts are studied and analyzed using a model by Fullan. Robert Olin and Lin Scruggs describe how a large technical institute has taken a "pro-active" stance on assessment: hiring Scruggs as an assessment coordinator, this institute has been able to create an assessment program that looks at the full spectrum of responsibilities of the department. Among other things, it includes statistical studies to improve curriculum, teaching, and relations with the rest of the university. These authors point out the benefits of using assessment to respond carefully to questions that are being asked and that will be asked more and more in the future. An unexpected benefit of this program proved to be increased student morale.
New Curriculum Approaches
In a discussion piece, Ed Dubinsky explains an approach to teaching based on learning theory. He asks how one can really measure learning in a calculus course. This question stimulates an approach in which continual, on-going assessment feeds back into the learning environment. The efforts of Dubinsky and RUMEC have often focused on the assessment of learning; one can observe in this volume how the methods he suggests here have influenced many of our authors.
Reform efforts in teaching (particularly in calculus)
have been the subject of much discussion, and we could not
begin to accommodate all the research currently conducted as to
the success of such programs. Here we touch on a few that
represent different schools and different approaches. Sue Ganter
offers a preliminary report of a study at the National
Science Foundation discussing what NSF has looked for and what
it has found, with directions for future study. While it's true
that the emphasis has shifted from teaching to student learning,
she reports, we are not simply looking at what students learn,
but how they learn, and what sort of environment
promotes learning. At some departments, even where change is
desirable, inertia takes over. Darien Lauten, Karen Graham, and
Joan Ferrini-Mundy have created a practical survey that might
be useful for departments to initiate discussion about goals
and expectations about what a calculus course should
achieve. Nancy Baxter Hastings discusses a program of assessment
that stresses breadth, from pre- and post-testing, to
journals, comparative test questions, interviews of students, end
of semester attitudinal questionnaires, and more. Good
assessment is an interaction between formative and summative.
Larger programs demand more quantified responses. In their
separate papers, Joel Silverberg and Keith Schwingendorf (both
using C4L) and Jack Bookman (Project Calc) describe their
efforts with calculus programs that have looked seriously at the
data over time. These longitudinal studies are massive and
broad, and not necessarily replicable. However, they indicate what
is important to study, and suggest the ways in which
gathering data can refine the questions we ask.
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