Mathematics In Engineering: Identifying, Enhancing, And Linking The Implicit Mathematics Curriculum

A study is undertaken to lay out in a structured manner the mathematics skills required of undergraduate students in the Department of Aeronautics and Astronautics at the Massachusetts Institute of Technology. The key objective of the research is to identify barriers to deep mathematical understanding among engineering undergraduates. Data from engineering course syllabi and interviews with engineering and mathematics faculty are combined to form an implicit mathematics curriculum, which lists the mathematical skills relevant to core engineering classes along with the flow of learning and utilization. Several problematic areas are identified, including the concept of a function, linearization, and vector calculus. Interview results show that many engineering faculty have an inadequate knowledge of mathematics class syllabi, and often do not know where or how the skills they require are taught, while mathematics instructors often have a limited understanding of how mathematical concepts are applied in downstream engineering classes. A number of recommendations are made, including increased communication between mathematics and engineering faculty, development of joint resources for problematic areas, and dissemination of a formal catalogue of mathematical skills and resources to engineering students and faculty. Background Inadequate mathematical skills present a widespread problem throughout engineering undergraduate programs; however, specific, well-documented examples of student difficulties are often lacking, and the exact nature of the difficulty is frequently uncertain. Moreover, there is often little communication between engineering and mathematics faculty dedicated to or addressing mathematics skills related issues. Engineering faculty assume that certain concepts are taught in the mathematics courses, but they are often not familiar with the specifics of the mathematics curriculum, or the methods utilized (for example: terminology and context of use). The level of mathematics skills of sophomores and juniors at MIT has been identified as a problem by a number of the faculty that teach core subjects in the Department of Aeronautics and Astronautics. This issue manifests itself in a number of ways and, in particular, has a negative impact on students’ ability to grasp engineering subject material. Specific problems are observed during lectures, where questions often arise regarding basic mathematic manipulations. These questions are also posed in the form of “muddy cards” – cards on which students anonymously write down the muddiest part of the lecture. Some examples of such muddy cards taken from a junior-level controls class are shown in Table 1. In all cases shown, the question relates to material that a typical junior is expected to know when entering the class. The questions on these cards strongly suggest that lack of mathematical understanding presents a Proceedings of the 2004 American Society for Engineering Education Annual Conference & Exposition Copyright © 2004, American Society for Engineering Education barrier to deep understanding of the control systems concepts, which are the focus of the lectures. Other evidence of mathematics problems has been observed on class quiz results and homework problems. Table 1: Example muddy card comments from Principles of Automatic Control (juniorlevel class), fall 2002 and fall 2003. Lecture subject Muddy card comment Control system analysis “Laplace is muddy” Steady-state errors “How did you go from v K C E s = to v dc K e dt = ?” State-space analysis “What is a non-singular transformation” “What does singular mean” Diagnostics have been performed by several faculty members to document this problem. Figure 1 shows the results from a diagnostic quiz given to students entering the junior class Principles of Automatic Control in 2001. Although the questions were graded very leniently, the results show that many of the students are unable to perform an integration by parts or calculate the eigenmodes of a second-order system. This issue is of great concern, since these mathematical skills are fundamental to much of the material covered in the course. If the students are stumbling on the mechanics of the problem, it is unlikely that they are grasping the true underlying physical principles and core material of the course. A similar diagnostic was performed in another class in the Department of Aeronautics and Astronautics, Computational Methods in Aerospace Engineering, which is taken primarily by seniors and second-semester juniors. The mathematical concepts tested were Taylor series, firstorder ordinary differential equations (ODEs), eigenvalues, integration by parts, minimum finding, mean/standard deviation, root finding, and numerical ODE integration. The results showed that, with the exception of eigenvalues, many students lacked the ability to correctly approach these basic problems. For example, only 20% of students were able to calculate the mean and standard deviation of a linear function. Of particular interest is the result for the eigenvalue question. This was the highest scoring question over 80% of the students were able to correctly calculate the eigenmodes of a second-order system. This result is in direct contrast to that shown in Figure 1; however, it is interesting to note that all students in the computational methods class had previously completed Principles of Automatic Control, which not only revisits the concept of eigenvalues, but also ties this mathematical concept to application for aerospace systems. Proceedings of the 2004 American Society for Engineering Education Annual Conference & Exposition Copyright © 2004, American Society for Engineering Education Figure 1: Results from mathematical diagnostic quiz taken by 65 juniors. Questions were each worth two points, and are as follows. 1a: plotting complex numbers; 1b: conversion from Cartesian to polar coordinates; 1c: multiplication and addition of complex numbers; 2a: integration of a function; 2b: integration by parts; 3a: matrix-vector multiplication; 3b: calculate eigenvalues and eigenvectors of a second-order system. This problem is not unique to students at MIT. The question of how to best teach mathematics in an engineering program has been considered by a number of researchers (for example, [1], [4], [5]). Recently, at the University of Hartford, faculty teaching the freshman engineering design, physics, and calculus courses worked closely together and developed shared outcomes for the three courses. The evaluation showed that this unified approach enabled students to gain better understanding of the linkages between engineering, physics and calculus. In a study to assess mathematics proficiency of students at Grand Valley State University, it was determined that student problems in this area are widespread and originate from many sources. Some resources exist that attempt to address these problems. Examples include the dAimp project, 2 which is currently developing online resources for engineering mathematics. The goal is to put together a series of manipulatives that lend greater understanding of mathematical concepts to engineering undergraduates. Project Links aims to link the concepts of higher mathematics to real-world applications through interactive web-based modules. 8 One of the major challenges associated with developing such resources is the creation of an effective bridge between mathematics and engineering. The first step to bridging the gap between mathematics and engineering is to comprehend the barriers to deep mathematical understanding among engineering undergraduates. In order to gain such understanding, it is critical to identify specifically what mathematical skills are expected and where in the engineering curriculum these skills are gained. While there were many suppositions regarding this issue in the Department of Aeronautics and Astronautics at MIT, such identification had not been formally carried out or documented. This paper describes an effort to formally identify and document the implicit mathematics curriculum in the undergraduate degree program. Proceedings of the 2004 American Society for Engineering Education Annual Conference & Exposition Copyright © 2004, American Society for Engineering Education Approach The implicit mathematics curriculum is a comprehensive list of topics in mathematics relevant to the core undergraduate engineering curriculum in the Department of Aeronautics and Astronautics. The core engineering classes are Thermodynamics, Fluid Dynamics, Structures, Signals and Systems, Computation, and Dynamics for sophomores, and Thermodynamics and Controls for juniors. At MIT, all sophomore courses except Computation are taught together as one subject called Unified Engineering. Many of the mathematics skills are taught in required freshman and sophomore mathematics courses; a few skills are taught explicitly in engineering courses. An initial list of mathematics topics was collected from the syllabi and measurable outcomes documents of the core engineering classes and then organized by subject. For example, eigenvectors and eigenvalues, extracted from the Unified curriculum, were listed under the heading of “Linear Algebra”, together with matrix algebra, and linear systems of equations. As found later, disagreement exists about where certain topics belong between the engineering and mathematics community. Our list of topics was modified continuously to approximate a consensus among faculty, but also to serve our original purpose of focusing on key mathematics topics in the context of engineering education in the department. It should be noted that while forming this list, we often found overlap in different disciplines and decided that our classification / organization is not unique. The disagreement among faculty on terms and their organization was also the first pointer towards problematic areas in the students’ understan