Malcolm A. Cutchins &Thomas H. Shumpert
Professors of AE &EE, and
Phillip L. Zenor, Professor of Mathematics
Auburn University, Auburn, AL 36849
In a referenced paper an innovative approach and an associated curriculum structure through the sophomore year is proffered. This new Pre-Engineering curriculum of two years is constructed around a thorough mathematical foundation. However, in this new approach, this mathematical foundation is addressed in a systematic format which simultaneously integrates fundamental engineering principles and concepts, problem solving skills, and technologically-advanced computation/visualization tools to provide a stimulating interaction which should better motivate the students of the ``Internet Surfing...'' generation. To accomplish this, faculty from several disciplines must interact in innovative ways. This paper addresses some of the background, barriers to, and solutions for faculty/faculty; faculty/topic; and faculty/students interaction if such an approach is to have any chance of working.
Faculty from the Departments of Mathematics, Physics, and Aerospace, Electrical, and Mechanical Engineering form the nucleus for developing and carrying out the instruction of this new curriculum. The curriculum (and concomitant experiences) are structured in such a way to provide a dynamic interdisciplinary approach to early engineering education.
Few issues encourage interaction between engineering faculty and the faculty in other disciplines, particularly mathematics. Could this be a major flaw in the discouraging statistics that of over 600,000 calculus grades during a recent year, there were approximately 220,000 failing grades [1]? It would be easy to blame math departments or poor high school backgrounds, but attrition rates at various engineering schools vary from 30 percent up as well. With the preponderance of students entering engineering being ``the cream of the crop,'' it is apparent that we must do a better job with the input group, no matter what influences may dictate their success.
Under the auspices of a variety of supporting agencies and interested foundations, a significant number of efforts are underway nationally to redefine the first year (or two) of science and engineering education. The approach of many of the collaborative efforts underway around the nation is to provide the beginning engineering student with hands-on, group-interactive, project-oriented classroom and laboratory experiences which will provide early stimulation, motivation, and ``real-world'' experiences to instill the desire to continue their pursuit of formal education in one of the demanding, challenging, and rewarding disciplines of engineering.
While admirable in their intentions and in their zeal to grab the attention and interest of the beginning student, these approaches could ``miss the mark'' in providing critically important mathematical concepts, principles, and tools which will be needed by the well-rounded engineer of the future, one that is more often now required to function in an interdisciplinary engineering work environment. Many of the programs being developed simply ignore the need to integrate mathematics and physics into the beginning engineering experiences and exclude mathematical instruction in the format in which it has existed during the past several decades. Other programs advertise their introduction of the mathematical principles and tools as a catchy, but dangerous, ``just in time'' necessity.
We believe that neither of these approaches is adequate for the radically different experiences being faced by the engineering student. The unique feature of the Pre-Engineering Curriculum being proposed by Auburn University is that it is built fundamentally around a sound and thorough mathematical base, vector calculus, with engineering projects and classroom experiences being chosen specifically to enhance, expand, and amplify the mathematical concepts and principles presented.
Successful engineers of the future cannot afford to have backgrounds which leave them deficient or even ignorant of a fundamental mathematical foundation. That underpinning will be absolutely essential to mankind in finding creative and innovative solutions to the complex problems (many not yet even defined) to be confronted within the first part of the next century.
Recognizing the need (and the national trend) of restructuring the beginning curriculum and associated activities of our engineering students, faculty from the College of Engineering, and the College of Mathematics and Sciences have jointly developed an entirely new and completely restructured two-year Pre-Engineering Curriculum [2], related to an in-publication calculus text [3]. Examples from the areas of fundamental electricity and mechanics offer motivation and ``real-world'' opportunities to learn and apply mathematical concepts and skills related to vector calculus, linear algebra, and differential equations and the fundamental natural laws and relationships introduced in physics. This paper discusses some of the details, merits, potential advantages and disadvantages, and related matters associated with implementing the program described in [2] from an engineering perspective.
Currently, engineering students generally do not begin to use their mathematical
concepts and skills immediately upon being exposed to them in the traditional
freshman (and sophomore) mathematics courses.
Applications (use) of these
important mathematical ideas and tools in the engineering curriculum are
just too far removed from the math learning experience. The ``shelf life''
of these important conceptual experiences and skills is just too short
to wait for their application to junior and senior years of the engineering
program. As is being painfully experienced daily by the general public,
common storage media such as home video magnetic recording tapes, older
computer floppy diskettes, outmoded or forwardly incompatible digital computer
format schemes, etc. render perfectly good data essentially non-retrievable
and therefore useless. It is the opinion of these authors that this digital
data analogy applies quite effectively to the experiences of the beginning
engineering student with respect to his (her) retention (and subsequent
retrieval) of essential mathematical data. Their storage media, ``disks''
or ``tapes'' have been either ``erased,'' or at least the file arrangement
and status on these media have become too ``fragmented'' to recover the information
in a timely or useable form.
One of the major purposes of the proposed Pre-Engineering curriculum at Auburn University is to address this problem directly with the goal of ``solidly encoding'' and/or ``embedding'' the essential mathematical concepts and tools into a solid physical and engineering ``matrix'' which will preserve the organized structure (format) and permit its retrieval, use, and development in subsequent years (both during the remaining years of matriculation as well as many years thereafter in professional practice).
The President of one university [5] has observed:
``I believe all educated people in America today should be fluent in at least two essential languages-English &calculus ... I have found that those who speak calculus, and are also fluent in their native tongue, can generally learn relatively rapidly the rudiments of practically any science or technology,''
If he is correct (and the authors believe his statement is probably even more appropriate now, eight years later), we must find ways to improve student mastery of both ``essential languages.'' Recently postulated and implemented solutions concerning the former (English) include more core courses and ``writing through the curriculum'' requirements. Solutions concerning the latter (calculus and associated subjects), are going to require the same approach of more than just-a-one-course fix. Such solutions are also going to require more interdisciplinary faculty cooperation if they are going to have any chance of succeeding. Just a few solutions of a widely-varying nature which are already proposed appear in references [6-8].
Some of the matters which tend to fragment the faculty and which prevent or make difficult cooperation between faculty in different disciplines/departments are often ignored. For example, in the current ``priorities and planning'' activities (TQM, strategic planning, etc.) taking place at Auburn University and also at many other institutions around the country, cooperation between academic units (departments) is really discouraged because as one professor aptly stated, ``my efforts in conjunction with your efforts might make your department look better than my department.''
And on a similar note, as old as academe itself, ``my'' evaluation for tenure and promotion may be frowned upon if I don't show enough ``independent, non-group related'' productivity.
Most faculty are overachievers and highly individualistic. Many of them have perfectionist traits, and most have a history of ``doing it all one's self.'' Exceptions to this are more rare than we might like to admit. If there are no influences to counter these traits, faculty cooperation will in general not happen.
As another example; the students in ``our'' department might be tempted to change over to ``your'' department if we were to work together in some joint course activities and they saw the fascinating applications on which ``you guys'' work.
Students are also fragmented in many ways of their own choosing. For example, one department's students are fearful of taking a chance on any interdisciplinary types of courses or projects because they suspect that faculty in other departments will not treat them with the same degree of appreciation that their ``home'' department will. Something needs to be done to remove these fears of students (which are totally unfounded, but very real).
Could faculty/student/software interaction during the Pre-Engineering program in team-taught engineering /math sequences be a mechanism which could impact these barriers? With current software and the computer-oriented students of today, the math/engineering interface should provide an opportunity for approaching an 85 percent success rate. To reach this level would necessarily involve more than just lectures, more that just a rearrangement of traditional material, more than just the current Pre-Engineering structure of isolated courses and isolated faculty.
As we in higher education move into the 21st century, it is imperative, especially in engineering and science education, that we introduce and employ the most modern technology to communicate the fundamental concepts, principles, and practices which will prepare our students for exciting and challenging careers in the increasingly high-tech society of the future. We can ill afford engineers that will be seriously math-deficient, especially in the competitive, new-product, high-tech world of our future. Some examples of such ``modern technology'' are discussed in references [9-11]. As two illustrations, reference [10] has resulted in a CD-ROM which includes both some remarkable visualizations (using Mathematica) and authoring software which eases course modernization. Reference [11] describes the development and use of a C++ program running in a Visual Basic ``front end.'' The software involves the student in an interactive mode, with ease of input of a wide range of shapes and the manipulation of their 3-D views, both which enable the interactive user to ``get a feel'' for the dynamic-related properties of the object and the quick resolution of its mathematically complex 3-D eigenanalysis.
It is possible that no lecture/classroom improvement in some students' programs will be sufficient to drastically change the alarming early math failure statistics. However, use of summer programs may have a chance. Our approach is that they would be required for high-risk students. A ``students teaching students'' feature is an important element, too, much like the use of recent players as assistant coaches in college football, a formula that has been quite successful. (Older coaches are sometimes too far removed from long ago learned fundamentals. Could older professors sometimes be too far removed from the fundamentals as well?)
Several mechanisms have worked for interdepartmental cooperation; two of which relate to AU's Honors Program and its Minority Introduction To Engineering (MITE) Program. In the latter, engineering math concepts organized by an EE and an AE professor [12], are presented to exceptional minority students in an abbreviated summer session format. In the former, an architect, an EE and an AE [13] combine to teach creativity and problem-solving from an engineering perspective. A portion of this three-faculty effort (steps a through f below with more of a math application) is planned for implementation in the Pre-Engineering effort described herein:
Other illustrations will be given at the conference.
In [14], the President of a major university has recently noted ``we are facing a watershed time for higher education...'' The problems and opportunities he then describes are all very complex ones which will require major changes in the way faculty interact, and certainly an increased understanding of engineering &mathematics by those entering the engineering field. This interaction will have to be among not only those in different engineering and mathematics disciplines, but also Colleges of Business and other fields. In this paper, arguments for a different way of approaching the traditional Pre-Engineering curriculum are proffered. Successful implementation of this curriculum will require an increase in the ability of engineering and mathematics faculty to interact with one another, and promote ways which should advance better understanding of engineering mathematics.
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