A working draft of resources and reports from an NSF-sponsored project intended to strengthen the role of mathematics in Advanced Technological Education (ATE) programs. Intended as a resource for ATE faculty and members of the mathematical community. Comments are welcome by e-mail to the project directors: Susan L. Forman or Lynn A. Steen.
However, the way many ATE mathematics activities are developed makes such use problematic. Typically faculty observe (or work with) technicians or engineers to see how what they do can be used to motivate the study of mathematics. This process generally produces innovative contextual problems that illustrate mathematical concepts. The authentic origin of the problem allows curriculum developers to embed particular mathematical topics in interesting and rich applications. Nonetheless, most of the mathematics lessons thus produced, whether used in labs, lectures, or structured projects, do not reflect the way people actually use mathematics in their jobs.
For example, one ATE project uses building codes for stairway construction to teach linear inequalities and feasible regions. But interior designers don't use this kind of mathematical analysis to do their work; instead, they make several design drawings, trying to insure that codes are satisfied, and persist until they find one they like that also satisfies the codes.
Another common practice that is amenable to nice mathematical analysis is the challenge of efficiently arranging storerooms for retail stores (e.g., a shoe store with dozens of styles, colors, sizes, and manufacturers). This problem can provide wonderful motivation for many algorithmic and combinatorial problems, yet store clerks rely more on intuition than analysis in actually making their decisions.
Although basic skills remain important as a foundation for all applications of mathematics, changes in the workplace have created new priorities for other parts of mathematics. For example, while statistical and modeling skills are more important than ever, traditional secondary school algorithms such as the quadratic formula and trigonometric identities are rarely used in the average workplace. Neither, of course, are the fake applications found in so many mathematics textbooks. So if by "real" mathematics one means the mathematics that people really use, much of what is taught in school is not real.
Mathematics teachers often say, quite correctly, that even though advanced mathematics may not actually be used, it could be used to good advantage if workers only possessed the necessary skill and habits of mind. Yet one reason many technicians (and even engineers) don't use calculus or other mathematical tools on the job is that they learned their mathematics as a collection of rules divorced from application. If everyone learned mathematics in context, such as ATE-like environments, more people would approach problems mathematically instead of using heuristic tools. More people would use mathematics if they knew how to.
Many of those who resist change believe deeply that traditional skills are as important as ever and don't want to introduce anything that may dilute the established syllabus of skills that dominates developmental and intermediate algebra. Others offer a more general criticism of ATE: If the system is not broken, why change?
On one campus the curriculum council blocked a proposed change to permit inclusion of new ATE-generated material by changing their own procedures to require a full meeting and formal vote if any single person objected. This obstruction left the ATE developers with only one viable strategy: to use individual modules here and there in regular courses. This strategy minimizes disruption of traditional syllabi while developing faculty and student confidence in the value of these new materials.
Publishers often adopt a different strategy to get faculty on board: they sponsor weekend and summer workshops to introduce new materials and new approaches to teaching, especially those that use technology. Yet these workshops tend to attract the same people over and over again. It sometimes seems as if there is no middle left: many faculty resist proposals for change, while others--a minority--eagerly jump at every opportunity to learn about new applications or new methods of teaching.
However, curriculum specifications for mathematics courses are often expressed as lists of topics interpreted as being of equal weight. Because these lists pack a great deal into each course, there is virtually no flexibility to introduce different emphases (much less different topics) that may be suggested by the ATE program. One option has worked in some cases: to seek high-level agreement that syllabi should specify only two-thirds of each course, leaving the other third to the instructor's option.
Arguments about how a new ATE course fits into the extant curriculum go to the heart of the question "What is mathematics?" Traditional mathematics moves from axioms to theorems to applications. In contrast, applied (ATE) mathematics moves from assumptions to data to analysis. For example, the canonical treatment of population growth is first based on unrealistic assumptions of limitless (idealized) growth, and only later modified to consider limited growth based on economic assumptions or petri dish conditions. In ATE mathematics, these comparative assumptions would be introduced, scrutinized, and highlighted right from the beginning.
The most common resolution of these issues, especially in higher education, is to offer a variety of courses that reflect different perspectives and priorities. But for this to work without disadvantaging students who move from one program to another, there needs to be widespread agreement on a reasonable mixture of common (prescribed) and individualized topics in course syllabi.
Copyright © 1999.
Last Updated: October 12, 1999.
Comments to:
Susan L. Forman or
Lynn A. Steen.