> # DRAFT

> #

> # Session 2620

> Project Links:

> Interactive Web-Based Modules For Teaching Engineering

> Kenneth S. Manning, Ph. D.

> Rensselaer Polytechnic Institute

> Abstract

> Project Links, an NSF supported project, is a cooperative effort at Rensselaer between post-secondary educators to develop materials to link the mathematical topics with their applications in engineering and science. The primary product of this effort is a set of interactive, web-based learning modules that rely heavily on hypertext, animations, and interactive Java applets. Most questions and examples are intentionally open-ended.

> Integrating the concepts of higher math with their applications in science and engineering lies at the heart of the mission of Project Links. Our method relies on interactive web-based modules used in the studio classroom environment, pioneered at Rensselaer, to engage students in guided learning. The intent is to provide students with a unique experience unavailable in traditional lecture or textbook lessons. These modules are designed for use in more than one course, with a topic-qualified instructor and assistant available in the classroom during use. They are not intended as self-paced learning modules, nor as text replacements, but are to supplement existing courses with a degree of interactivity and universality not available before the advent of the World Wide Web.

> There are currently 45 modules in development. Four of particular interest to engineering educators are highlighted in this paper. These include Drag Forces, The Bicycle, Mechanical Oscillations, and Mass Transport.

> I. Introduction

> Project Links is a five-year, NSF supported undertaking to develop web-based interactive modules that integrate mathematical concepts with contemporary topics in science and engineering. The project is based at Rensselaer Polytechnic Institute, with collaboration from the University of Delaware, Virginia Polytechnic Institute, Hudson Valley Community College, and Siena College.1

> These modules are to be used in a studio setting, with an instructor present, and with student-access to the Internet. The modules are topic-specific, intended for use over one to three days in the normal course of the term. They rely very heavily on hypertext construction, animations, interactive Java applets, and students in small group interaction. Most questions and examples are purposely left open-ended to encourage teamwork and self-discovery. The intent is to supplement existing courses, and is not for independent study away from the classroom, though parts of it may be assigned that way as follow-up.

> A module is a self-contained conceptual unit intimately linking a fundamental mathematics topic with its application in a science and/or engineering course. Most modules have a mathematics and an applied subject professor as developers, and are targeted for use in at least one course in each area. The examples used in the modules are based on real-world situations. To achieve this, the module developers must incorporate actual experimental results, demonstrations, or design problems. Our modules use videos, real-time experiments run over the web, animations of experimental results, and data-reduction.

> II. Background

> Many forces are coming together to alter the methods used to teach today’s students in the technical fields. Many years ago new multimedia tools became available that allowed those with the time, talent, and the hardware to produce innovative teaching methods. Use of many of these new ideas was limited because of the difficulty inherent in purchasing and transporting equipment that was relatively uncommon.2 At that time, however, multimedia essentially meant the use of videos, simple computer animations, and possibly some hypertext documents resident on a local computer.

> Now, with the advent and unprecedented popularity of the World Wide Web and laptop computers, multimedia-based educational methods have come to mean something new entirely. Several courses are based wholly on presentation via CD-ROM in a laptop environment.3

> As educational learning theories progress more effort is applied to the introduction of collaborative learning4 and interactive learning5 in the engineering classroom. This fosters teamwork and personalizes feedback for the students.

> Project Links, known formally as Mathematics and its Applications in Engineering and Science: Building the Links, was conceived to intimately tie (or link) crucial topics in mathematics with one or more corresponding areas of contemporary application in engineering and science fields. It is funded under the National Science Foundation initiative “Mathematical Sciences and Their Applications Throughout the Curriculum”6.

> The four main objectives of Project Links are:

> 1. To stimulate greater cooperation in educational development among faculty in mathematics and other disciplines.
> 2. To encourage interactive teaching and learning strategies and to produce instructional materials for use in workshop or studio-type courses.
> 3. To create a library of interactive learning materials that link topics in mathematics with applications in engineering and science.
> 4. To continue Rensselaer's pioneering efforts in the application of contemporary technology for educational purposes and to encourage the widespread distribution of the results of these efforts

> Our approach has been to produce instructional modules that exploit the Internet and its attendant technologies of the World Wide Web, and the Java programming language. These modules are designed to be used in the studio classroom7, with an instructor present, with significant student-to-student interaction, and with many open-ended challenges included. Project Links modules are not intended to replace textbooks, professors, or entire courses. It is meant to allow flexibility to the instructor in her efforts to emphasize certain well-contained topics that are a one- to three-day part of a regular course.

> Ideally (though not always in practice) each module is developed by a team of at least on mathematics professor and one science/engineering professor, for use in a mathematics course and a science/engineering course. The Project Links Technical office supplies these interdisciplinary teams with the Internet expertise, programmers, and guidance.

> Project Links has engaged the aid of the Evaluation Consortium of the State University of New York at Albany for the purposes of judging the educational efficiency of the modules. This group has standardized the process for module development and documentation, and has developed and is implementing the evaluation plan. The consortium conducts three levels of testing: classroom observation, educational technology, and student usability. Each of the three is described in detail at the Project Links Web site.

> The modules currently under development are shown in the appendix. Four of them of most interest to the engineering educator are highlighted in the remainder of this paper.

> III. The Project Links Web Site And Modules

> Four mature Project Links modules will now be discussed. They are Drag Forces on Solid Objects, Constrained Optimization, the modules of the Mechanical Oscillations Group, and Mass Transport.

> Every module in Project Links has the same basic format. Shown in Figure 1 is a representative front page; this one is from the Drag forces module. There are three normal modes of navigation through a module.

> Shown marked 1 are the PRIOR/NEXT arrows. These follow a sequence preset by the module developers. This is the path through the module the developing professors recommends, and intends to follow in normal use. This is akin to the chapter sequence chosen by a textbook author. This sequence can be entered at any point, but is not visible to the user. The sequence will be a part of the preparatory materials made available to the using professor.

> Marked 2 in Figure 1 is the side navigation bar (known as the “side navbar”). This is a clickable list of the main module topics. Each of these are reached when using the PRIOR/NEXT buttons, but this side list allows the user to jump around, as one would skip through parts of a textbook. A triangular icon appears next to the current topic shown in the main frame. None is shown in Figure 1 because it is of the introductory page. This page can always be reached by clicking “Module Home” at the bottom of the side navbar..

> In the figure, area 3 is the top navbar, used to branch off of the main topic areas listed in area 2. There are five chooses here, available to the developing professor during design of the module. They are “Concepts”, “Discover”, “Applications”, “Collaboration”, and “Practice”. Any or all may be used from any one page. Again, a small triangular icon appears next to the current branch, available choices are in bold blue, and those unavailable from the current page are grayed-out (as all are in this view). “Concepts” are main topics, usually those that appear on the side navbar. “Discover” pages lead to questions or exercises that allow the student to push into a new area with information from the “Concepts” pages. “Applications” are current uses of the topic in real-world situations. “Collaboration” supplies challenges that must be solved with a partner or by discussion between groups, perhaps with instructor prodding and guidance. “Practice” contains problems that the student must answer to allow the professor to assess the learning that has taken place. These can have the style of pencil-and-paper worksheet problems, applets, or form submissions.

> IV. The Drag Forces on Solid Objects Module8

> In this module, simple differential equations are used to study the effects of the drag force, which slows the motion of a body moving through a fluid. In this case the fluid is air, and this air resistance is responsible for the safe landing of the skydiver. This module is a part of our Calculus set of modules. This would be an appropriate module for use in a course in Differential Equations and Fluid Mechanics. Use could also be found in Physics and Dynamics courses.

> This module leads the student through an exploration of the physics involved in objects falling through both a vacuum and a fluid by an application of a Newton’s Second Law force balance. There are several QuickTime9 video presentations to illustrate the nature of falling objects. An innovative Java applet allows students to predict the velocity-time and acceleration-time curves for a skydiver in free fall subject to the effects of air resistance. The students first sketch in their predictions, then run the applet to compare those to the answer. Probing questions allow the student to reinvestigate their ideas and make another attempt. This applet is used again later, with the parachute deploying in mid fall.

> One of the most popular features is the applet that allows students to choose two from among several types of objects, whether they fall in a vacuum or a fluid, and what the drag coefficient is. The objects are then released to fall, and their respective v-t and a-t graphs are plotted during the fall. The student can then directly compare the affects their choices have made on the rate at which an object falls.

> The main part of the module concludes with an experiment of a bob falling in a tube, with the students doing the analysis first without, and then with drag included. They are lead through the analysis, breaking at several points to discuss things with classmates before continuing. A brief treatise is included on dynamic similarity.

> The module ends with some answers and some hints to the questions posed throughout in the “Collaboration” and “Practice” sections.

> V. The Constrained Optimization Module10

> This module, part of the Advanced Math Methods group, helps the students grasp the principles of optimization of multivariable functions with equality constraints. This is done with the use of geometric experimentation.

> Students examine the geometry of navigating in a plane, and translate those geometric ideas into the language of calculus. They then reformulate the navigation problem as a problem in calculus and attempt its solution. Extensive use is made of graphical solution techniques through Java applets.

> Later in the module, students construct a mathematical model of the output of a business, and examine geometrically the associated optimization problem. Students learn to transform geometric observations into analytical statements by extracting the mathematical principles underlying single-constraint optimization. In the later parts of the module the instructor has the option of letting students extend the optimality conditions to include multiple constraints. The module ends with practice problems and more applications.

> VI. The Mechanical Oscillations Modules

> These are a collection of eight closely related modules. These are Dynamic Systems Investigation, Spring Mass, Forced Spring Mass, Linear Pendulum, Non-linear Pendulum, Spring Pendulum, Multiple Spring Mass System11, and Vibrating Strings12. Together these represent the combined efforts of over 40 people. They are part of the Differential Equations Group.

> The Mechanical Oscillations modules are all very closely tied to actual experiments conducted by the module authors and their students. The Spring Mass module, one of the more mature of the collection, begins by presenting the real life physical system and its actual dynamic behavior, using different spring mass combinations. The next step is to model the system, physically and mathematically. The behavior of the model is then analyzed using both exact formulas and numerical solutions. The final step is to compare the predicted dynamic behavior with the experimental data of the actual system and to evaluate the model's accuracy. If the predicted behavior matches the experimental data, then the model is adequate. If they don't match, then either the data collection or the modeling and analysis, or both, must be refined. This same process is repeated until the comparison is successful. The module also has a parallel track discussing damped oscillations of a spring mass system.

> The Vibrating Strings module uses an interactive simulation of the vibrating string, and also includes comparison with experimental data. The module then takes the student through an interactive derivation of the solution of the wave equation for the vibrating string. Many useful links are supplied to other animations and discussions outside of the Project Links site.

> VII. The Mass Transport Module13

> There are many instances in biology and in the environment where mass transport is an important phenomenon. Everything from cellular osmosis to glucose uptake to pollution in lakes and ponds can be related to mass transport. The Mass Transport module introduces the ideas of concentration and concentration difference and formalizes how these concepts are responsible for many common phenomena. The module contains a series of related demonstrations and exercises that build on one another, allowing students an incremental understanding of the subject through increasing levels of complexity.

> The module takes through the developmental ideas of mixing and transport in both single and multi-chamber configurations. The module addresses diffusion, transport across single membranes and multi-membrane systems, concentration profiles, and gas transfer. The module has interactive applets to illustrate the concepts and to guide the students toward understanding.

> VIII. Summary

> Project Links closely relates the instruction of topics in mathematics to the application of those topics in fields of science and engineering. It does this through the innovative use of the hypertext nature of the World Wide Web, animations, and interactive Java applets. These modules are meant to be used in an instructor-lead, studio setting. There are currently forty-seven modules in various stages of development. A consortium of experts in the field of education oversees the project with the intent of establishing educational effectiveness.

> Figure 1. This is a typical module introductory page. Shown is that of the Drag Forces module.

> IX. Acknowledgments

> Project Links gratefully acknowledges the support of the National Science Foundation under grant # DUE-9552465.

> X. Appendix

> Project Links Modules Grouped by General Mathematics Topic:

> Calculus

> Curvature and Curve Design

> Chemical Kinetics and Equilibria

> Drag Forces on Solid Objects

> Electrostatic Field and Potential

> The Gradient

> Moment of Inertia

> Advanced Math Methods

> Constrained Optimization

> Fourier Series

> Ampere's Law

> Electric Field

> Faraday's Law and Induction

> Gauss's Law

> Magnetic Field

> Maxwell's Equations

> Differential Equations

> Boundary Value Problems for ODEs

> Continuously Stirred Reactor

> Lake Pollution

> Mass Transport

> Sequential Batch Reactions

> Dynamic Systems Investigation Process

> Forced Spring Mass

> Linear Pendulum

> Non-Linear Pendulum

> Spring Mass

> Spring Pendulum

> Vibrating Strings

> Capacitance

> Current and Resistance

> Inductance and Inductive Circuits

> Geometrical Optics

> Electromagnetic Oscillations

> Discrete Mathematics

> Graph Theory: Industrial Drilling

> Graph Theory: Networking

> Graph Theory: Sperner's Lemma

> Linear Systems

> Bicycle

> Matrix Kit

> Probability and Statistics

> Conditional Probability

> Continuous Random Variables

> Inventory Control

> Means and Variances

> Poisson and Exponential Distributions

> Random Variable Relations

> Bibliography

> 1. The Project Links Web site, http://links.math.rpi.edu/

> 2. Neu, E. C. “Computer and Overheads vs. Multimedia in the Classroom”, ASEE Annual Conference Proceedings, 1996, Session 2220.

> 3. Gramoll, K. “Teaching Statics Online with only Electronic Media on Laptop Computers”, ASEE Annual Conference Proceedings, 1999, Session 1668.

> 4. Costanzo, F. and Gray, G. L. “Collaborative Learning in Undergraduate Dynamics Courses: Some Examples”, ASEE Annual Conference Proceedings, 1999, Session 3268.

> 5. Gray, G. L. and Costanzo, F. “Interactive Dynamics:A Collaborative Approach to Learning Undergraduate Dynamics”, ASEE Annual Conference Proceedings, 1999, Session 3268.

> 6. National Science Foundation, “Mathematical Sciences and Their Applications Throughout the Curriculum”, Program Announcements, 93-164 and 94-15.

> 7. Wilson, J. M., “The CUPLE Physics Studio”, Physics Teacher, 1994, 32, 518-523.

> 8. Littman, H. and Fleishman B., “Drag Forces on Solid Objects”, Project Links, http://links.math.rpi.edu/webhtml/CAindex.html, 1999.

> 9. The QuickTime video player, http://www.QuickTime.com/, Apple Computer, Inc., 1998.

> 10. Kapila, A. and Buhler, B., “Constrained Optimization”, Project Links, http://links.math.rpi.edu/webhtml/AMindex.html, 1999.

> 11. All are Seigmann. W., Boyce, W., et. al., Project Links, http://links.math.rpi.edu/webhtml/DEindex.html, 1999.

> 12. Kovacic, G., Project Links, http://links.math.rpi.edu/webhtml/DEindex.html, 1999.

> 13. Newall, J., Manson, R., and Drew, D., Project Links, http://links.math.rpi.edu/webhtml/DEindex.html, 1999.

> Biography

> KENNETH S. MANNING

> Ken Manning is the Technical Manager for Project Links, and an Adjunct Associate Professor for the Core Engineering Program at Rensselaer Polytechnic Institute in Troy, New York. He has also worked as a thermal-hydraulic design engineer for General Electric, first at the Knolls Atomic Power Laboratory, and then at the Corporate Research & Development Center. His B.S. is in Physics from the University of Oregon, received in 1976, his M.S. is in Mechanical Engineering from the University of Illinois at Chicago in 1984, and his Ph.D., also in Mechanical Engineering, is from Rensselaer in 1992.