By Don Small and Kathi Snook
Engineers, physical scientists, and mathematicians came together for a Curriculum Foundations Interdisciplinary Workshop, held at the U.S. Military Academy at West Point. Dr. William Wulf, President of the National Academy of Engineers, gave the keynote address, entitled ’The Urgency of Engineering Education Reform.â? He noted that academia has not kept pace with changes in the professions and is failing to educate students to be technologically literate. With respect to mathematics, he encouraged changing the curriculum in order to spend less time on continuous, deterministic mathematics and more time on discrete and probabilistic mathematics.
Participants were divided into four groups to examine the following areas: (a) Interdisciplinary Culture; (b) Anticipated Advances in Technology; (c) Goals and Content of the Courses; (d) Instructional Techniques. Strong consensus was developed on the major issues listed below, although questions remain about how to implement the desired changes.
Incorporate more modeling into the curriculum. Emphasize real-world problem solving in the sense of modeling as illustrated in the diagram to the right, rather than in the sense of exercises.
Each group viewed modeling as an effective means of addressing their areas of interest. The Interdisciplinary group saw modeling as the best approach to break down current barriers to interdisciplinary cooperation. Real-world problem solving is inherently interdisciplinary. Technology is moving curriculums toward the modeling process and away from the solution process as it replaces hand computation and simplifies the ’what-iffingâ? process. Members of the Goals and Content group agreed with increasing the emphasis on modeling as a means to prepare students to become competent, confident, and creative problem solvers. The members expressed concern, however, over the conflict between the ’math wayâ? (e.g., emphasizing limits as the major ’primitiveâ?) and the ’science wayâ? (e.g., emphasizing rates of change as the major ’primitiveâ?). The Instruction group viewed modeling as an effective way to address multiple learning issues such as communication, interpretation, and sensitivity analysis.
Emphasize ’learning how to learn.â? Although not identical in meaning, the frequently heard phrases ’lifelong learner,â? ’learning to think,â? ’mental discipline,â? and ’learning the mathematical thought processâ? all offered perspectives on learning how to learn. Incorporating an emphasis on inquiry and rewarding intellectual curiosity were viewed as an effective approach to learning how to learn.
Effectively and appropriately use technology. Participants were particularly interested in using technology for visualization, discovery, and computation. Additionally, undergraduate technology experience should prepare students for the world into which they will graduate. Two important concerns were raised: How is technology effectively and appropriately integrated into the curriculum? How will the increased use of technology and loss of some hand calculation skills impact the curriculum?
Shift from teacher-centered to learner-centered instruction. This pedagogical shift necessitates less coverage and greater depth, less lecturing and more small group activities. Participants also recognized that this shift implies adjustments in resourcing time and in designing appropriate assessments. Reducing coverage provides the time required for student-learning activities, but may also pose a serious conflict to many instructors. Additionally, preparing for a class where students will be actively participating may require more planning time than preparing for a lecture. Traditional assessments are teacher-centered in the sense of minimizing grading time, maximizing coverage and focusing on well-constructed, well-defined skill type questions. In a curriculum that is changing to emphasize modeling ?solving real-world problems that are often neither well constructed or well defined, participants recognized that assessment practices must also change. There was not consensus on how to make these shifts in time allocation or assessment.
Value multiple learning activities. Throughout the workshop participants focused on process as being more important than content. As stated previously, there was consensus about modeling and problem solving and agreement that students must be actively engaged ?constructive experiences are more important than spectator experiences. Participants identified projects, discovery work, writing, presentations, calculator or computer laboratory sessions as examples of possible learning activities.
Integrate data analysis, statistics and probability into first and second year courses. Although process was seen as more important than content, these important concepts are many times missing from the early parts of the curriculum. As modeling is incorporated into the curriculum, the seeds of data analysis, statistics and probability concepts can be integrated with these real-world models.
Value interdisciplinary cooperation and interaction. Participants raised serious concerns with regard to the present state of interdisciplinary cooperation and interaction. Although there is (theoretical) agreement on the benefits of interdisciplinary cooperation, several barriers exist such as system inertia, fiefs and turfs, publish or perish syndromes focused on narrow results, entrenched attitudes, lack of a reward system, and time. The low level of interdisciplinary cooperation restricts student development as well as constraining reform efforts in mathematics, physics, and engineering.
The following major curriculum initiative was presented as a five-year plan. (This initiative is presently being developed at the U.S. Military Academy with planned implementation starting in fall 2003.)
Major Curriculum Initiative: Create a core sequence of courses focused on developing competent, confident, and creative problem solvers. The instruction, based on modeling and inquiry, would interweave continuous and discrete mathematics. Calculus topics of rates of change, accumulation, transformations, approximations, and others would arise through modeling realistic situations rather than studying specified subjects. Similarly, data analysis, statistics, probability, graph theory, matrix algebra and other discrete topics would also arise through modeling realistic situations. The program would be inherently interdisciplinary, as real-world situations are interdisciplinary.
In closing his keynote address, Bill Wulf provided an on-going challenge to the participants of the workshop as he told the story of Wayne Gretzky’s response to the question of what has made him such an effective hockey player. Wayne Gretzky said, ’he doesn’t skate to where the puck is, he skates to where the puck will be.â? Our challenge is to identify both the ’puckâ? and where it is going.
Volume 61 in the MAA Notes series, Changing Core Mathematics, (edited by Chris Arney and Don Small) is based on the Interdisciplinary Workshop.
This issue includes two articles on Curriculum Foundations, a project of CRAFTY, the MAA Committee on Calculus Reform and the First Two Years. Earlier articles have described the project as a whole (November 2000), the workshop on the mathematics courses needed by physics students (March 2001), and by computer science students (May/June 2001). Future articles will focus on other client disciplines. CRAFTY is a subcommittee of CUPM, the Committee on the Undergraduate Program in Mathematics, which is undertaking a review of the whole undergraduate curriculum.