We believe mathematics education should be a participatory, interactive experience for our students. A successful model is one in which mathematics students think of their class as a "community of learners" nurtured by an experienced professor guiding them to an understanding and appreciation of mathematical concepts. We want our students not simply to be proficient at mathematical manipulations but to understand and appreciate mathematics and to be able to relate mathematical concepts to the world around them. Our teaching is rooted in a value system focusing on aesthetics and usefulness. For students in introductory mathematics courses, we begin with the realization that students come not only with a diversity of preparation but with many fears and misconceptions about mathematics. While mathematics is a demanding subject which operates within the model of the scientific method, students' conceptions are more often motivated by the psychology of anxiety and the belief that mathematics is rigid, unyielding and "cold." We want them to relax and open their minds to see that within the structure of mathematics are creativity, beauty and "warmth." Only then will students be open to actually learning mathematics as a tool for problem solving.
We also begin with the realization that applications make mathematical concepts "real." Most students learn best by relating new knowledge to previous knowledge. Applications using familiar ideas from related disciplines help the students understand new mathematical concepts in a comfortable setting. Moreover, new computer technologies present a context in which students can discover theoretical concepts through visualization as well as exploration of numerical examples.
Our methodology is based on the framework of both interactive and Socratic discourse. We attempt to help mathematics students discover an individual balance between "working things out for themselves" and collaborative learning. Often the model for doing mathematics homework is to disappear into an isolated niche in the library with a pad and pencil to "do mathematics." While it is true that "doing mathematics" is often a private and inner struggle with oneself and that eventually one must be able to stand alone, it is also true that productive thinking and learning occur in the group process. Interacting with other students through talking, writing, and using the computer helps students reach the point where individual study is practical and useful. Although the traditional model for mathematics classes is to lecture and then send students home to study alone, students who obtain mathematics related jobs in industry usually find their work efforts organized around a project team or group.
In advanced mathematics courses, we also recognize the effectiveness of the Socratic method. One mathematical version of this method is called the "Moore method." In most instances of this method the students present work at the blackboard while the instructor and other students listen and critique. The result is that the student learns not only the mathematics but also how to communicate the mathematics effectively. We use the Moore method in varying degrees beginning in the last term of the sophomore year.
We integrate writing as a tool in both introductory and advanced courses to help students understand mathematics. When students articulate solutions to problems in writing, they learn to use the language of the discipline and develop a deeper understanding of the logical reasoning required to solve problems and prove theorems. Writing has an even more dramatic impact when coupled with collaborative learning in a discovery environment.
The following example illustrates this point. Ten years ago members of the department began to make innovative yet incremental changes in the freshman calculus sequence. This initiative originated in part from the participation of members of the department in Hendrix's Writing Across the Curriculum program as well as participation in the Washington colloquium "Calculus for a New Century." These experiences reinforced our assessment that greater emphasis should be placed on conceptual understanding by adopting the following pedagogical techniques: (1) presenting calculus concepts in a meaningful context; (2) using writing as an effective learning tool; (3) nurturing collaborative learning among the students; and (4) developing an environment for discovery learning.
By evaluating and adapting to our needs the best ideas and materials generated in the national writing across the curriculum and calculus reform movements, we gradually modified the calculus courses to meet our objectives. The cumulative result of these incremental changes is a very dynamic, interactive, writing intensive environment in Calculus I and II. These courses contain most of the traditional single-variable calculus topics, but the topics are organized in a more intuitive and evocative progression. Writing assignments and computer laboratories using a computer algebra system emphasize discovery learning in a collaborative setting. These experiences integrate writing with numeric, graphic, and symbolic computations.
The mathematics program is supported by three full-time professors, as well as a faculty member who divides time between teaching and administrative duties; they work closely with the department's three computer science faculty, including one professor with a mathematics PhD who teaches both. The department offers a traditional undergraduate curriculum in mathematics including sequences in algebra and analysis. This curriculum supports a
major in mathematics and a
minor in mathematics.
Over half of our graduates pursue post baccalaureate degrees in mathematics or computer science. In fact, Hendrix ranks 36th among all four-year colleges in the number of graduates (30) earning doctoral degrees in mathematics during the period 1920-95.
Through the interactive nature of its curriculum and the accessibility of its faculty, the department intentionally fosters a community of learners. Our introductory students benefit from the close mentorship of faculty and upper-class students, the latter as assistants who staff the
Mathematics Help Center & Computer Laboratory on afternoons and evenings. The curriculum of the majors is designed as a springboard for their required senior projects under the direction of the faculty.
A distinctive feature of the curriculum is its undergraduate research program which began in 1969 and is now partially funded by an endowment from faculty and mathematics alumni. During the academic year, the faculty direct students in independent projects which often lead to papers published in undergraduate journals and talks presented at regional and national conferences sponsored by NCUR (National Conference on Undergraduate Research), Pi Mu Epsilon, and the Mathematical Association of America. A problem-solving seminar is also available to students in the department. For the past four years, students in the department have also received external funding to support student research in the summer. For most of the past eight years, students in the department have been accepted to summer Research Experiences for Undergraduates programs sponsored by the National Science Foundation.
Over the past twenty years there has been only one comprehensive revision of the curriculum. Our history of making incremental rather than revolutionary changes is based on a recognition of local circumstances such as institutional resources, academic calendar, and faculty teaching styles and loads. We also believe that changes which are responsive to student and faculty interests are more likely to be sustained. The calculus reform initiative described above is one example of this approach; our introduction of a computer science emphasis in 1986 and our introduction of a computer science and mathematics major are others. We anticipate maintaining this process of incremental improvement in the future.