USING TECHNOLOGY IN AN INTEGRATED CURRICULUM — PROJECT IMPULSE1 Robert E. Kowalczyk and Adam O. Hausknecht University of Massachusetts Dartmouth Mathematics Department, 285 Old Westport Road, N. Dartmouth, MA 02747-2300
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[email protected] Increasing problems with the motivation and performance of first year engineering and science students prompted a group of faculty at UMass Dartmouth to team up to look for solutions. The central objective was to find and implement changes which would improve learning in those courses that form the foundation of students' academic and professional careers. As a result of this effort, a pilot first year integrated engineering curriculum called IMPULSE ( Integrated Math, Physics and Undergraduate Laboratory Science, English and Engineering) was developed. The primary goals of this pilot program are for students to: • • • •
become motivated life long learners, work effectively in teams, develop oral, written, and visual communication skills, understand and apply the fundamentals of math and science to a wide range of problems, • synthesize diverse knowledge to create solutions to complex problems, • and use computers effectively for analysis, design, and communication. The IMPULSE program is modeled after the research-tested engineering curriculums at the NSF-funded Foundation Coalition Universities: Arizona State University, Maricopa Community Colleges, Rose-Hulman Institute of Technology, Texas A&M University, Texas A&M University-Kingsville, Texas Woman's University, and the University of Alabama. The implementation of IMPULSE stresses: • the integration of knowledge across the disciplines of calculus, physics, chemistry, engineering, and English; • cooperative and active learning; • and computer assisted learning. The IMPULSE classroom is a state-of-the-art computer studio where students use the latest technological tools to solve complex problems. IMPULSE has 48 students divided into twelve teams of four members each. Each team of four students sit at a custom designed table rather than individually at their own folding-top desk chair (see Figures 1 and 2). The tables have a computer at either end with an interface box for attaching sensors (force, motion, temperature, ...) when doing experiments in physics, engineering, and mathematics. One computer can be used for data collection while the other can be used to write the lab report or to do some mathematical analysis. All students are facing each other which naturally leads to effective discussions, brainstorming, and group problem solving. In almost every class, individual students or teams present the results of team projects, solutions to homework problems, or solutions to impromptu questions that come up in class. Thus, they are constantly practicing and developing their oral, written, and visual communication skills. By working in teams, students learn many different types of problem-solving strategies, share their diverse knowledge backgrounds, and become active, cooperative team members. 1This article appeared in "Proceedings of the Eleventh Annual International Conference on Technology in Collegiate Mathematics", Addison-Wesley Publishing Co., 2000, p. 235-239.
Figure 1 IMPULSE Classroom
Figure 2 IMPULSE Team
How many times have we as instructors of mathematics heard our students ask, “When will I ever use this?” In our efforts to make calculus more relevant to engineering students, we designed an integrated curriculum that provides an immediate answer to the above question: “You'll use it in your physics (or engineering) class tomorrow to solve real-world applied problems.” To make this happen, we had to modify the traditional calculus sequence so that it would integrate well with physics and engineering. Vectors, line integrals, 3D graphing, double and triple integrals and other topics from the third semester of traditional calculus were moved into the first two semesters of integrated calculus. Additionally, the concept of integration is introduced earlier in the semester, and less emphasis is given to techniques of integration and convergence of series. Using the methods of the reform calculus movement, the integrated calculus courses place more emphasis on the understanding of the mathematical concepts, and technology is used to do the traditional drill work. The CAS software package Maple ® is used by the students as a tool to help them solve homework and practice problems, to perform mathematical explorations of calculus concepts both in and out of class, and to do general problem solving in physics and engineering. Specially designed computer explorations are used to help students learn and understand the many important concepts presented in the first year engineering curriculum. Students use computers to simulate physical experiments, gather data, model data, perform symbolic manipulations, visualize complex phenomenon, and write reports. The mathematical software package TEMATH is often used as a demonstration and visualization tool for presenting the foundations and applications of calculus. A variety of CAS software packages were reviewed for use in IMPULSE and Maple was chosen based on its universal use by all NSF-funded Foundation Coalition Universities, its popularity in academia, the abundance of mathematical materials available, and the large number of platforms that it runs on. During the first two months of the course, student reaction to using Maple has been mixed, ranging from “Maple is a powerful program and can be useful” to “Maple is very hard to use and complicated for simple functions”. Many students have used graphing calculators in high school and are very comfortable using them to solve problems. During many of our classes, we have observed our students using their graphing calculators instead of Maple to solve problems requiring graphing, function evaluation, and numerical computations. They struggle with Maple's syntax and the difference between exact and numerical arithmetic. They find it frustrating when they keep getting answers like sin(1) and ln(e) instead of the numerical approximations. 1This article appeared in "Proceedings of the Eleventh Annual International Conference on Technology in Collegiate Mathematics", Addison-Wesley Publishing Co., 2000, p. 235-239.
Using Maple In the IMPULSE classroom, our students use Maple as a tool to help them model data, develop general algorithms for solving various classes of problems, and for extending the scope of a mathematical concept. At the beginning of the semester, we study the many different types of functions that will be used in calculus, physics, and engineering. This is the ideal time to introduce the methods of modeling data. As an example, the following problem is placed in an engineering context and is part of a Maple exploration assigned to our students. This problem is based on an example from the text: Finney, Thomas, Demana, Waits, Calculus A Graphing Approach, Addison-Wesley Publishing Co., 1993. The Trans-Alaska Pipeline starts in Prudhoe Bay in the High Arctic, crosses the Alaskan Arctic tundra, and ends in Valdez on Prince Williams Sound. Oil must be heated so that it can easily flow through the pipeline. However, the heat from the oil is transferred to the pipe and its supports. The engineers responsible for designing and building the pipeline had to use insulated pads at the bottom of the pipe supports to keep the heat from the hot oil from melting the permafrost (permanently frozen soil) beneath. If not, the pipe would eventually sink into the ground. In designing the pads, the engineers needed to take into account the variation in air temperature throughout the year, that is, they needed a mathematical model for air temperature. As an example, since the pipeline passes through Fairbanks , Alaska, the engineers obtained the historical normal mean air temperature for Fairbanks for each day of the year. Using this data, develop a mathematical model for the temperature in Fairbanks, Alaska. Using a Maple worksheet, students enter and plot the data, estimate the amplitude, period, horizontal shift, and vertical shift of the sine function that is necessary to model the temperature data, and plot their function f (x) = asin(b(x + c)) + d fit along with the data to see how well the sine function models the temperature data. Figure 3 shows the sine function model and the temperature data.
Figure 3 Using the Sine Function to Model Temperature Data
1This article appeared in "Proceedings of the Eleventh Annual International Conference on Technology in Collegiate Mathematics", Addison-Wesley Publishing Co., 2000, p. 235-239.
Using TEMATH to Model a Hanging Cable Pictures taken by a digital camera can be imported into the new version of TEMATH which we are currently developing. This feature can be used to demonstrate to students that a cable hanging between to posts has the shape of a catenary y = Acosh(x A) where A = T w is the ratio of the tension T of the cable at its lowest point to the weight w of the cable (see problem 16, page 280 of [4]). To simplify the fitting process, we center the image of a hanging cable so that the cable's lowest point is at the origin. This changes the model to y = A(cosh(x A) −1) . Once this is done, it's a simple matter of trying different values of the parameter A until one is found that gives a good visual fit (see figures 4- 6).
Figure 4 Fit of Chain Cable by y(x) = 1.26(cosh(x/1.26)-1)
Figure 5 Fit of Rope Cable by y(x) = 1.1(cosh(x/1.1)-1)
Figure 6 Fit of a Chain Cable by y(x) = .713(cosh(x/.713)-1) For a differential equations course, data points could be sampled from the imported image and used to estimate y and y . An estimate for the parameter A of the catenary 1This article appeared in "Proceedings of the Eleventh Annual International Conference on Technology in Collegiate Mathematics", Addison-Wesley Publishing Co., 2000, p. 235-239.
differential equation y
1 (y )2 A could then be obtained quantitatively by fitting a
least-squares-line through the origin to a plot of the estimated data pairs ( 1 ( y )2 , y ). Using TEMATH and Parametric Equations To Write Your Name A fun way to introduce parametric equations is to have students use parametric equations to write their names. We show students that the vector equation s(t) = [x1 + (x 2 − x1 )t, y1 + (y2 − y1 )t] where 0 ≤ t ≤ 1 parameterizes the line segment between the two points (x1,y1) and (x2 , y2 ) . Also, we show them that the vector equation e(t) = [h + a cos( + t), k + bsin( + t)] where 0 ≤ t ≤1 parameterizes an arc of an ellipse centered at (h,k) with horizontal radius a and vertical radius b. The arc starts at angle α and sweeps through an angle of β. For example, the equations I1 (t) = [1+ t, 1], I2 (t) = [1.5, 1+2 t], I3(t) = [1+ t, 3] C1(t) = [3.5+ 0.5cos(2π 6 +9 π 6t), 2+sin(2 π 6 +9 π 6t)] T1 (t) = [5.5,1+2t], T2 (t) = [5+ t,3] C2( t) = [7.5+ 0.5cos(2π 6 +9π 6t), 2+sin(2 π 6 +9 π 6t)] M1 (t) = [9,1+2t], M2(t) = [9+1 2 t,3 −1 2t], M3 (t) = [9.5+1/ 2t, 2.5+ 0.5t], M4 (t) = [10, 3 − 2t] generate ICTCM as shown in figure 7.
Figure 7 ICTCM Written Using Parametric Equations Bibliography [1] TEMATH - Tools for Exploring Mathematics Version 2.0a by Robert Kowalczyk and Adam Hausknecht, 1997. [2] Documentation for TEMATH 2.0 - Tools for Exploring Mathematics — Draft by Robert Kowalczyk and Adam Hausknecht, 1997. [3] Labs for TEMATH 2.0, by Robert Kowalczyk and Adam Hausknecht, in preparation. [4] Calculus Single and Multivariable, Hughes-Hallet, Gleason, McCallum, et al., John Wiley & Sons, Inc., 1998. 1This article appeared in "Proceedings of the Eleventh Annual International Conference on Technology in Collegiate Mathematics", Addison-Wesley Publishing Co., 2000, p. 235-239.
