It has been estimated that 700,000 students are enrolled in college-level calculus courses in the U. S. in any given year. Of these, 100,000 are in Advanced Placement courses in high schools, 125,000 in two-year colleges, and the rest in four-year colleges or universities [17]. A very small percentage of these students intend to take any mathematics beyond calculus, let alone major in mathematics or do graduate study or become a mathematician. Most of this enrollment is generated either by general education requirements or by prerequisites for subsequent course work. To cite just one example, Duke University has flexible distribution requirements that allow students to avoid calculus, but it also has 24 major programs that require one or more semesters of calculus. Even though many students enter with Advanced Placement credits, some 80% of our first-year students take a calculus course. About 2% of each class graduates with a major in mathematics. Thus, most students are not motivated to study calculus except as it serves some other goal -- e.g., keeping open options for a major.
American colleges provide liberal, vocational, and/or pre-professional education to students who overwhelmingly see themselves as participants in pre-professional or vocational programs. A small percentage contemplate academic graduate study, but only the tiniest fraction have any concept of liberal education and its potential importance in their lives. Parents, who pay the bills through tuition and/or taxes, usually see things the same way: The objective is for their child to become productive and self-supporting. And potential employers of graduates at all levels have definite expectations for the skills and abilities of their employees. Since these employers collectively influence support for and accountability from institutions of higher education, public or private, it behooves us to pay attention to what they want.
Here is what employers want, expressed in seven ''skill groups'' [3]:
With the possible exception of item 2, this is very different from what educators usually think of as ''basic skills.'' If students enter college lacking most of these skills, college must be where they learn them. Indeed, this list defines the goals of higher education in the broad sense: liberal, vocational, and pre-professional. The job of teaching these skills belongs to the entire faculty, including the Mathematics Department -- and not just for ''computation'' and ''problem-solving.'' To get a consistent message from the faculty and to have a good chance of graduating with these skills in place, students must encounter most of them in almost every course.
At least one American college, Alverno, has clear expectations that every course will address a similar list of learning objectives [1], so we know this is possible. Professional schools are more directly accountable for these outcomes than are liberal arts schools, so it is no accident that many of them are ahead of the undergraduate institutions in transforming themselves to achieve these objectives. The story of such a transformation at the management school of Case Western Reserve University is told in [2]. We will see in later sections why addressing these objectives enables more students to learn more mathematics.
| Title page |
| Reform or Renewal?
| Problems |
| Cognitive Psychology
| Brain Research |
| Technology and Learning
| Technology and Curriculum |
| Renewal in Calculus Courses
| References |