By David Bigio and Susan L. Ganter
The MAA Committee on Curriculum Renewal Across the First Two Years (CRAFTY) is conducting the Curriculum Foundations (CF) Project, a major analysis of the first two years of the undergraduate mathematics curriculum. The goal of the project is to develop a curriculum document that will assist college mathematics departments as they plan their programs for the next decade. Much of the information for this curriculum document was gathered between Fall 1999 and Spring 2001 through a series of invitational disciplinary workshops, funded and hosted by a wide variety of institutions. This article, focusing on engineering, is part of a series of reports from these disciplinary workshops.
Format of the Curriculum Foundations Workshops
Each CF workshop focused on a particular partner discipline or on a group of related disciplines, the objective being a clear, concise statement of what students in that area need to learn in their first two years of college mathematics. The workshops were not intended to be dialogues between mathematics and the partner disciplines, but rather a dialogue among representatives of the discipline under consideration, with mathematicians present only to listen to the discussions and to provide clarification on questions about the mathematics curriculum. For this reason, almost all of the individuals invited to participate in each workshop were from the partner disciplines.
The major product of each workshop is a report or group of reports summarizing the recommendations and conclusions of the workshop. These were written by the representatives from the partner disciplines and address a series of questions formulated by CRAFTY. The reports from each workshop have been widely circulated within the specific disciplines as well as the mathematics community in order to solicit a broad range of comments. A curriculum conference that included invitees from all disciplines was convened in November 2001 to synthesize the workshop findings.
The MAA Engineering Workshop at Clemson University
One of the MAA Curriculum Foundations workshops was sponsored and hosted by Clemson University on May 47, 2000. This workshop focused on the needs of engineering from the first two years of college mathematics instruction. The workshop had thirtyeight invited participants, with roughly equal representation from each of four areas in engineering (chemical, civil, electrical, mechanical) and mathematics. The workshop resulted in four documents, one for each of the four engineering areas, addressing the MAA questions specified at the outset of the workshop.
This report focuses on the aggregate recommendations of the four groups at the engineering workshop. It is not intended to be a definitive document, but rather a working paper that generates discussion among mathematicians and engineers in order to provide additional feedback for the mathematics community. Therefore, the authors welcome comments and additional ideas.
Desired Student Outcomes
Before identifying any guiding principles for the mathematics curriculum, the workshop participants developed a list of general learning outcomes that should be achieved by all engineering students. Specifically, the workshop participants want students to be able to:

Statics, including large scale stresses and integration principle directions;

Dynamics, or a phenomenon in time; and,

Nondeterministic problems; i.e., those utilizing statistics and probability.
Engineering students need to understand the physics of a wide variety of problems and how mathematical equations can be used to describe the physics. Then, they need to solve these problems using the appropriate tools, understand the solution, and interpret the results.
Content and Timing
The issue of content is a complex one across the engineering disciplines. In general, the study of engineering can be divided into three areas:
1. Statics, including large scale stresses and integration principle directions;
2. Dynamics, or a phenomenon in time; and,
3. Nondeterministic problems; i.e., those utilizing statistics and probability.
Discussions at the engineering workshop made it clear that each of these areas requires different mathematical concepts at different times in the learning curve. And since mathematics is the language of engineering, then as with any ’foreignâ? language, students that learn mathematics years before their current engineering class may no longer be facile with the language. So, the workshop participants believe it is critical to integrate mathematics with applications in specific engineering disciplines. This ’justintimeâ? approach requires coordination between mathematics and engineering departments, perhaps resulting in a reconfiguration of the current collegiate mathematics curriculum for the first two years.
In engineering disciplines, problem solving requires the ability to understand a physical problem, place it in a mathematical context, solve the necessary equations, and interpret the results. Particularly in the first two years, students are more comfortable and adept at using sample problems to understand concepts. However, a full understanding of the problem solving process requires students to move through Bloom’s Taxonomy from mechanics to conceptualization to integration. This learning process can be solidified by extending the required mathematics courses for engineers into the third year, so that the material can be coordinated with major engineering courses.
Instructional Techniques
Instruction should be done in the context of physical concepts, not as isolated theoretical exercises. Students with mathematical training that focuses on symbolic techniques have great difficulty moving to more complex engineering problems. These students generally are too dependent on methodology and are unable to conceptualize based on broad principles.
Active learning is an important method for helping students to learn about openended problems. It also prepares them for real problems they will encounter in their future engineering careers.
Team Work and Interdisciplinary Collaboration
Educational reform in engineering?being driven in large part by ABET2000, the new standards of the Accreditation Board for Engineering and Technology?supports the use of active learning, problembased experiences, and team work. This has led to the use of problems that are open ended in engineering courses. The paradigms for teaching in engineering are evolving to include more varied methods, and to address a wide variety of student learning styles. If mathematics is taught in a way that does not support these new methodologies, it will not be effective in preparing students for engineering courses. Interdisciplinary team teaching can be used to promote the integration of mathematics and engineering content. Student teams also can help instructional effectiveness.
To accomplish these objectives, the workshop participants propose that mathematics departments consider developing small projects within mathematics courses that require team work and active learning so that the students have more opportunities to learn fundamental mathematics principles, beyond simple examples. One such project could utilize ordinary differential equations, moving from simple to complex mathematical situations. Projects also could cut across multiple engineering areas to further develop connections between disciplines.
Summary
The engineering workshop of the Curriculum Foundations Project certainly raised more questions than the answers it produced. For example, is it better pedagogically to teach mathematics to engineers as a homogeneous group, or together with nonengineering students? There was general consensus among the workshop participants that the mathematics community should look closely at the heterogeneous approach to the first two years?as well as other established traditions of the current course organization.
But one thing is clear: regardless of what new teaching methods are utilized, the needs of students and their learning processes are different than even just ten years ago. Students will work in a complex technological world and interface with problems from many disciplines. They need to understand how to use fundamental concepts in a variety of settings?and to appropriately integrate calculators, computers, and other technologies when solving physical problems.
Although this report is very general, the engineering workshop participants have provided more specificity and detail about specific mathematics topics in the full workshop reports (see http://academic.bowdoin.edu/faculty/B/barker/dissemination/Curriculum_Fo... there are four reports from this workshop, covering Civil, Chemical, Mechanical, and Electrical Engineering). The basic mathematical foundations presented in the reports are mostly the same as a decade ago, but the methods for helping students learn the ideas promoted by the reports implies the need for a pedagogical transformation that is critical to the training of engineers for the future.
David Bigio is a member of the Mechanical Engineering Department at the University of Maryland. Susan L. Ganter is in the Department of Mathematical Sciences at Clemson University.
This issue includes two articles on Curriculum Foundations, a project of CRAFTY, the MAA Committee on Curriculum Renewal Across 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), computer science students (May/June 2002), chemistry students (September 2002), students engaged in interdisciplinary programs (September 2002), students in the Life and Health Sciences (November 2002), and students in technical programs in twoyear colleges (November 2002). All of these reports can also be found on MAA Online, at /features/currfound.html. 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.