Journal of Effective Teaching

Fostering Multimedia Instruction in Mathematics

Gabriel G. Lugo and Russell L. Herman

Abstract

We introduce a coordinated effort to teach Mathematics, Physics and Chemistry through the use of Multimedia presentations, electronic Data Acquisition experiments and computer assisted instruction.

Keywords: Hypermedia, Toolbook, Electronic Data Acquisition.

The MCP Project.

The institutional Mission Statement of the University of North Carolina at Wilmington cites Undergraduate Teaching Excellence as its primary goal. The MCP project represents a major effort by several faculty in mathematics, physics and chemistry at UNCW, to revitalize our instructional methods in undergraduate education. The project was created as a response to alleviate three major problems in our mathematics and science programs:

The MCP project was started in the Fall of 1992 with the support of an NSF grant. The idea of the project was to redefine teaching methods through the use of multimedia technology, and to restructure the first year sequences in Mathematics, Physics and Chemistry with a number of goals and pedagogical methods common to the three disciplines.

Our instructional methods have been well received by our colleagues. The MCP faculty, in conjunction with the Center for Teaching Excellence, has sponsored a large number of two and a half day workshops on Hypermedia Instruction and more than a third of our campus faculty have attended the workshops. The University administration also supports our efforts fully. Partially due to the success of our teaching innovations, we have seen a recent influx of a large quantity of instructional technology equipment in our campus. Presently, there is at least one full multimedia technology classroom in every academic building on campus, and all faculty who have requested an office computer have been provided one by the University. Construction of new wing for the Mathematics building is nearly completed. The addition to the building includes two new, state-of-the-art technology classrooms with forty, 133 Mhz pentium workstations each. This brings the number of PC's in the Mathematical Sciences classrooms alone to 120. A large Science building is also under construction with a completion date projected in the Summer of 1996. The new Science building will also house several instructional technology master classrooms. In addition, the Center for Teaching Excellence maintains cutting-edge hypermedia production equipment for faculty to use in curriculum development.

Project Goals

The MCP project emerged out of our common concern for both locally and nationally decreasing enrollment and retention rates of students in mathematics and the physical sciences [1]. In addition, while we are not aware of many documented studies on the ability of students to transfer and utilize their knowledge from one subject to another, we are well aware that a problem exists.

It is not uncommon to hear anecdotes of students who have just learned to integrate f(x) =1/x in their calculus course and yet have trouble recognizing the integral of f(T) = 1/T in the context of entropy in their physics course. We have also heard chemistry professors complain that their students do not know any physics, and physics students claim that they were never taught the mathematical concepts that their professors expect them to know. This story is all too familiar.

In view of these problems, the MCP project was envisioned with the following goals in mind:

Transferability: Students should enhance their ability to transfer knowledge from one discipline to another. The importance of this goal was based on our continued observations that often students were not able to recognize concepts in one course which they had previously encountered in another.

Motivation: Traditional chalkboard lectures seem to have lost their appeal to all but a few of the best students. We wanted a new and creative idea to renew in our students the excitement and curiosity for learning, which we all possessed when we were children.

Retention: We are concerned with both the ability of the students to retain knowledge as they progress to higher level courses in their fields, and our ability to retain our majors. A common complaint we have often heard from our students was "Oh, we never learned that" when we knew quite well that the concept under discussion had just been introduced in a previous course, or sometimes even in the same course.

Synthesis. We want our students to crystallize the concepts and tools we teach them so that they can recognize when these concepts can be used to solve and model real problems. This requires that they gain a clear global understanding of the subject matter without sacrificing rigor and attention to detail.

There have been numerous national studies suggesting that we need to make changes in the way we teach. The National Research Council's 1991 report Moving Beyond Myths: Revitalizing Undergraduate Mathematics, states [2]

"The way mathematics is taught at most colleges - by lectures - has changed little over the past 300 years, despite mounting evidence that the lecture-recitation method works well only for a relatively small portion of students. Moreover, the syllabi of many undergraduate mathematics courses and the template-style textbooks are detached from the life experiences of students and are seen by many students as irrelevant."

Mathematics is not the only discipline to which this statement can be applied. Many of our faculty have very similar concerns in their disciplines. This has been supported through discussions with the participants of our numerous workshops on hypermedia instruction, which we have given recently. Typical comments we hear are: The students are not learning the material; they do not find it relevant; they are not excited enough to work; and, they do not retain what they have learned. In summary, we are all faced with the same problems and are looking for new instructional techniques.

The Classroom

Our Calculus classroom serves a dual role as both a mathematics laboratory and a master classroom. The classroom tables are arranged in a double horseshoe with all of the student computers aligned along the walls and a multimedia instructor station facing the opening of the horseshoe. This arrangement makes it possible for the instructor to see all the computer screens from the front of the classroom. This also proves to be the most convenient arrangement when students are taking tests, since the only computer directly in front a student is his/her own.

The instructor's station is a complete multimedia, 75 Mhz Pentium computer with a 16-bit sound card, a video overlay capture board and dual spin CD-ROM. Connected to the system, we have a full color, full video projection panel, a pioneer laser disk player and an Cannon Visual Presenter unit which serves as our electronic "blackboard". The instructor's file server is connected to the campus communication fiber optic backbone and to a second multimedia file server used to support the classroom stations. The 15 classroom multimedia stations are 486, 33 MHz computers with 8Mb of RAM and dual spin CD-ROMS. Each station runs locally Mathcad, Maple, Toolbook and Netscape. Additional software packages such as Matlab, Amipro and MSOffice run from the file server. The system is served by two HP laser heavy duty laser printers and a black and white HP scanner.

Our calculus classes meet for fifty minutes, five days per week. On the average, two of these days are allocated to hypermedia presentations. These presentations are designed to cover the particular topic at hand, the fundamental theory and a few selected examples. The concepts are further enhanced by a few simple simulations, sound events, or a short video clip. In preparing these presentations, one needs to be careful not to pack more information than is typically put into a class lecture. The idea is to explore a variety of ways to get across that one point, which is important to the theme of the day. Students respond and learn in a variety of ways, so that having the availability of several media to present a concept will increase the chances that a student will relate to the material.

The MCP project consists of a coordinated (as opposed to "integrated") method of instruction targeted to the first year sequence of courses in Calculus, Physics and Chemistry. The courses are separately taught by faculty in their corresponding departments, but we share the same technology, software and pedagogical techniques.

In our Calculus course the emphasis is modeling. For example, rather that just talking about projectile motion, we get students to collect and analyze their own data. This can be done either with our Data Acquisition systems, or by analyzing digitized videos. We have developed an image analysis Toolbook which with allows the user to step through the frames of a video file and click on points to extract calibrated coordinates to a spreadsheet. All the major ideas in our course are introduced analytically, graphically and numerically. These are carried out through multimedia presentations; labs using data acquisition, numeric and symbolic computations; class projects, and writing lab reports.

Multimedia Materials

While multimedia instruction has received a much attention by publishers and other commercial enterprises, very few commercial materials have been produced for mathematics instruction. Consequently, we have for the most part developed our own materials.

Our centerpiece for multimedia instruction is a Toolbook 3.0 template developed by the MCP Project Principal Investigator, Dr. Dick Ward. The template incorporates menu items to launch other applications, and has built-in buttons to navigate throughout the instructional toolbooks. It also contains a variety of features which make it extremely easy to create audio and video clips and to "clean up" pages when you leave them.

The MCP template has become very popular in our campus, and most faculty begin all their Toolbook applications using this template as a starting point. Many faculty from other institutions have also participated in our workshops and seem to be pleased with the versatility of the template.

Our toolbox are arranged by chapters, which follow closely the standard "fat" calculus text typically selected by departmental committees. However, since many of the interesting situations that we want our students to model involve exponential functions and simple first order differential equations, we strive to introduce derivatives of transcendental functions and Euler's method early in the curriculum.

The presentations, which we produce for the project, use Toolbook as a control center to navigate through video and audio events, other Windows and DOS applications, and simple simulations within the module itself. For example, imagine the favorite example of the cycloid. Typically, the professor will draw a wheel on the board and after waving his/her hands in the air, will hope to convince the class that a point on the rim of the wheel will trace out a particular path. In our presentation, we show an actual wheel rolling across the screen. This motion can be repeated over and over giving the class the opportunity to describe the path traced by the moving point. After a few guesses, the actual path is displayed plotted on the background and the wheel is again allowed to roll across the screen. However, this time the class can see that the point on the rim does indeed follow the prescribed path (Toolbook users may also view an active version of the cycloid).

One of the benefits of such hypermedia presentations is this ability to go back and easily repeat the events. This is particularly useful when reviewing materials from a previous class. The students can actually see the exact images and hear the same sounds as they had observed the day before. Typically, it only takes a couple of minutes at the beginning of each class to review the material covered in the last lecture, as well as emphasizing what is important.

Another of the activities facilitated by hypermedia is the possibility of showing events that cannot be produced on the board or are impossible demonstrate in the classroom. This means that we can exemplify the concepts with applications that we never would have considered before. For example, in one of the projects in the first semester of calculus students used digitized data to calculate the amount of paint that would be required to cover the campus water tower. We can do more to show that our subject is exciting and relevant to the real world. By using this technology appropriately, we have the opportunity to widen the experiences of our students.

Finally, using a presentation device, such as Toolbook, we can access other software programs at the click of a mouse. There are numerous mathematical packages, which have appeared on the market in the past decade. Many of us have developed programs in languages such as Basic, Pascal and FORTRAN, which bring out interesting features, or problems, encountered in our courses. These can often be brought up and projected on the screen. In our classes we rely on software such as Mathcad, Visual Basic, and Excel. The Mathcad program is very useful for doing quick and/or tedious computations. It opens up the possibility of exploring the limitations of functions, strategies and techniques. What if questions can be posed and answered on the spot, instead of postponing the question until the next day (when the students have forgotten why they asked it!).

Visual Basic is relatively easy to use. It is windows-based and object-oriented. The finished product can be professional looking and even looks like an extension of the Toolbook presentation. The use of such software allows one to write simple programs for producing simulations, based on mathematics. It is more useful in developing an interactive module for student explorations in the laboratory setting. Here, we show a simple module for demonstrating the addition of sine functions. In this case the student can control the amplitudes, wavelengths and phase difference for the two waves.

Some of the analysis of data obtained from computer based laboratory experiments is done in Excel. This package can be used to easily produce professional looking graphs. Such graphs can then be pasted into reports, or used to make points in class presentations.

AmiPro is a Windows-based word processor, which allows easy equation insertion. Our students use AmiPro to write their scientific reports, including proper mathematical grammar and graphs cut from other packages, such as Mathcad and Excel.

In addition to our toolbox, which are used mostly to introduce new material in a formal manner, we have also developed a large number of Mathcad worksheets and Visual Basic routines. These are provided to the students, or when appropriate, constructed by the students. With these interactive worksheets and routines, the students can explore and solve problems which are much more interesting, but otherwise intractable without the use of a computer. Many of our students use our course as the final leverage to convince their parents to purchase that computer they always wanted.

By the end of one semester of our calculus course, our students feel quite comfortable creating their own Mathcad worksheets, and finding their own way around Windows. These ideas are reinforced when they use the same skills and software programs in their physics and chemistry courses. In fact, it is possible for our calculus students to perform an experiment modeling the kinetics of bleaching a dye and later redo exactly the same experiment in their chemistry course, stressing the chemistry of the oxidation reaction.

The Labs

While our course is constrained by a traditional syllabus, it is our contention that mathematics is best learned when accompanied by exploration and applications. Therefore, we have written a laboratory manual to accompany the course. The labs range in scope from simple exercises to acquaint students with the sophisticated software packages used, to more challenging projects in calculus applied to physics, chemistry and other sciences.

There are basically three types of activities covered in the lab manual. To enjoy mathematics students must be given an opportunity to "try out" their own ideas and make their own discoveries, thus some of our lab activities are of an exploratory nature. The explorations may involve just looking at one parameter families of functions, or numerical simulations to understand the nature of derivatives and integrals. There are, of course, several highly structured labs focusing on specific skills that we want the students to acquire, such as the method of central differences, numerical integration and Newton's method. However, the concepts are introduced in an applied setting.

The emphasis of our laboratory instruction is mathematical modeling. The mathematical models may be continuous or discrete models with "canned" data, or they may involve modeling physical phenomena through electronic data acquisition. We often perform live experiments in the classroom, display the graph of the data "on the fly" with the projection system. Correlating the graph of the data as the event is taking place, greatly enhances understanding and makes a durable impression. The experience becomes indelible when the instructor engages the students in performing the experiment with different initial conditions.

Through the network, the students can then load the data into Mathcad or Excel for analysis. We use a data acquisition system by LOGAL. It is Windows based, so that all tables and graphs can be cut and pasted directly into a word processor, resulting in professional looking lab reports. Students are expected to convey their knowledge orally and in writing. The latter can be at times a painful activity even for the students who excel in their composition classes, but at the end most students recognize that scientific writing is an important and necessary skill.

As mentioned previously, we have the students make use of Mathcad. Currently we are using version 5.0; the instructor's station is equipped with Mathcad 5.0 Plus for presentations. The labs have been compiled in our lab manual [3]. Most of the calculus lab work is done using Mathcad for Windows, which is relatively free of the programming expertise needed in other mathematics packages, such as Mathematica or Maple. In fact, Mathcad has a menu for symbolic manipulation which utilizes the Maple kernel.

The labs typically involve graphing, analysis of functions, fitting data, and explorations of fundamental concepts in calculus. Data analysis begins with a linear regression lab, followed by power, exponential, and quadratic models. Examples of models in this portion of the course are predicting the age of the universe from Hubble's original data for moving galaxies, verifying Kepler's third law, and studying the exponential and logistic laws of population growth. We show a sample of a worksheet for the application of Newton's Method to determine the minimum surface area of a soup can , which has additional material for the more realistic problem of a container with seals. (Try to solve this problem in a traditional classroom!)

Assessment

The most challenging part of the MCP project is the assessment of the effectiveness of the program. Our exams are usually designed so that 50 to 60 percent of the problems are of the traditional type, and the rest necessitate the use of the computer. Our analysis show that our students perform about the same in the traditional problems as those who are not involved in the project. In this regard, we can at least say that no damage has been done. On the other hand there are so many other unmeasured skills that our students acquire that we now find it increasingly impossible to go back to the traditional classrooms. For example, by the end of the semester, we turn students who had never touched a mouse, into competent PC users. Their writing skills also show great improvement, and their final reports, after numerous corrections, exhibit features of good scientific writing. In addition, we have completely wiped out any absenteeism problems we may have had in the past. Our students enjoy coming to class, and they learn to enjoy sharing their mathematics experiences with other students.


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