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A presentation of the following article was given at the Joint Mathematics Meetings, January 10-13, 1996, Orlando, Florida.

Computer Applications in the Teaching of Mathematics

Hubert J. Ludwig
Department of Mathematical Sciences
Ball State University

During the 1987-88 academic year the Department of Mathematical Sciences at Ball State University developed a new secondary mathematics undergraduate teaching major. Looking ahead, we placed among the required course work a new course Mathematics 331 (Computer Applications in the Teaching of Mathematics).

The new major was not listed in the University Undergraduate Catalog until the 1990-92 edition and hence we would not have students for whom the course was required until the Fall Semester 1992. However the Catalog did indicate that the course could be used as an elective on a "Computer Endorsement" program. Registrations for the Fall Semester 1990 showed that Mathematics 331 had attained the "magic" level of 10 registrants and hence it was taught for the first time during that semester.

The Department Chair asked if I would be willing to teach the course. Why? What were my qualifications? The department computer laboratory was filled with Apple II's of several designations and in various states of repair. At that time I was the only Apple user in a Department in which those who were using computers were using IBM-compatibles, in a University which was (and is) quite IBM-compatible oriented. I was also the person that people came running to when they had problems while working with the Apples, so it was assumed that I knew something about all of this. What else could I do? I said yes.

What did I have available to work with? I had a laboratory filled with approximately 20 Apple II's (ranging from Apple II's to unenhanced Apple IIe's). I had Krell Logo, I had Green Globs and Graphing Equations, I had the Geometric Supposers, I had the NCTM's Short Programs for Teaching and Learning High School Mathematics from Preparing Teachers to Use Computers in Mathematics Classrooms (a 1986 project), and with my "hacker" mentality I had my own collection of Public Domain mathematics software. I don't remember how much time I applied to each of the topics we worked with but I made things up as I went along and we made it through the semester and the students indicated that they felt that they had had a useful experience.

By the beginning of the Fall Semester 1992 we had enough students who were required to take the course to insure that a section would be taught on a regular basis. This semester (Spring 1996) the course is being taught for the eighth time. With this semester's work I will have taught it six of the eight times that it has been taught.

The nearly six years which have passed since my first experience with the class have seen many changes. I acquired a Macintosh LC in the fall of 1992 and have gradually changed all my computer work from the Apple II to the Macintosh. The Department computer laboratory has been upgraded several times and is now equipped with an assortment of LC's and Performas all of which have IIe emulation cards installed. We have site licenses for Terrapin Logo and for the Geometer's Sketchpad and our collection of other software (primarily for the Macintosh) has grown.

The course is required of all secondary mathematics education majors. The class meets for two seventy-five minute sessions each week during each semester. As a beginning axiom I would assert that in a fifteen week semester it is not possible to become an expert in dealing with very many software packages or possible uses of a computer in a teaching situation. Thus my approach to the course is to make it a "tool-awareness" experience. My desire is that the students be exposed to as many different ways that computers can be used to teach mathematics at the secondary level as possible.

The students are required to learn something about telling a computer what to do by working with Logo. We spend nine class periods working with material excerpted from the Logo Discoveries series by Margaret L. Moore. They must read several required articles about Logo and discover and read several more on their own. On the basis of their readings and laboratory work they must submit a paper on the possible educational uses of Logo. A Logo project must also be submitted. I usually refer them to a journal article or to a textbook containing a Logo routine and ask them to get the routine up and running and then make some improvements to it. The students often find this assignment to be a bit frustrating but they generally agree that it is a useful experience. The most rewarding comment I have heard was that a student now appreciated the fact that creators of software do need to be compensated for their efforts and that there is certainly a lot of injustice involved in software piracy.

The next twelve class meetings are spent doing a unit on commercially produced software comparison during which they examine several software packages for the teaching of geometry and several for the teaching of algebra. There are some software packages which are examined every semester and some which vary. I do make certain that the Geometer's Sketchpad and Maple are always included. During the Spring Semester 1996 they looked at the Geometer's Sketchpad (2 class meetings), Maple (2 class meetings), Geometry Inventor (1 class meeting), Cabri Geometry (1 class meeting), Geometric Connectors: Coordinates and Transformations (1 class meeting), Math Connections: Algebra I and Algebra II (1 class meeting), Green Globs and Graphing Equations (1 class meeting), Algebra Expresser (1 class meeting), Algebraic Patterns together with the Function Supposer (1 class meeting). They also spend one class meeting examining the demonstration versions of Tesselmania! and Cabri II.

Students must then submit reports on their experiences with the software. There are three required reports, one on the geometry software, one on the algebra software, and a third report on the demonstration versions. My directions for the preparation of reports on the algebra and geometry software contains the following statement. "Your report should compare and contrast the software packages. Speculate on their usefulness in the classroom. Are there topics for which one package would be more useful than the others? Is one package much more useful than the others? Is any package basically useless? Show that you have done some reflection on your experiences with the software in the laboratory. If you were allowed to purchase only one of these packages for classroom use which one would it be? If you could purchase all but one, which one would you leave out? (Do not let comparative costs enter into your decision here.) Explain your choices." I also indicate to them that my opinion of their report will be enhanced if they can find and comment on reviews of the various software packages.

This is, in a way, the most difficult section of the course to prepare for. We have the site license for the Geometer's Sketchpad. For other software packages we have at least two copies, for Maple we have ten copies. The fun part here is scheduling activities so that each student manages to look at all the software packages and no copyright laws (or licenses) are violated. As we move into this part of the course I take the opportunity to emphasize to the students what their obligations in this regard will be when they begin to work in a classroom setting.

The next five class periods are spent learning about possible uses for a spreadsheet in a mathematics classroom. The students begin by working with some material which I have written that is designed to get them used to some spreadsheet preliminaries and especially to replication processes. This includes creating tables of values, investigating roots of polynomials, looking at limits, and examining properties of Fibonnaci-like sequences. They then work with How to Use the Spreadsheet as a Tool in the Secondary Mathematics Classroom by William J. Masalski. I provide them with references to several journal articles on the use of spreadsheets in the classroom and occasionally I will also give them copies of handouts from presentations which I have attended at NCTM conferences. I also tell them to find several more articles dealing with the uses of spreadsheets in mathematics classrooms. They must then submit a report which is based on their laboratory experiences and their readings. Along with the report they must submit some spreadsheets for teaching mathematical topics. These can be original or improvements on spreadsheets found in the Masalski text or in the journal articles.

The next two sessions are spent examining some shareware and freeware which might possibly be utilized in a classroom setting. For the current semester I have examined archives of software at the Macintosh archive at the University of Michigan the Geometry Forum, the mathematics software archive at the University of Tennessee - Knoxville, and the Geometry Center. I have selected approximately forty mathematics-related programs and placed them in an archive on hard-drives of computers in the laboratory. In determining which programs to place in the archives I am relying simply on the description supplied by the various archivers. Thus I do not guarantee any "gems" for the students to discover. During the two class I allow them to examine as many of these programs as they have time for. The students are required to write a report on their experiences with this material. The directions for the report include the following statement. "In your report you should indicate which programs you have examined. List your top five, list your bottom five. Give reasons for your opinions. Comment on the general quality of this material. If your school had Macintosh computers and the administration told you that there was no money for software purchases could you use some of these programs to enhance your teaching of mathematics? Be rather specific in your comments. Which programs would you use, where and how would you use them? For which aspects of secondary school mathematics would this material be most useful? In the case of shareware programs indicate whether or not you feel the material would be worth the requested shareware fee. Some of these materials are demonstration versions, not intended for classroom use, designed to entice people into purchasing the full-blown version. If you examined some demonstration versions, did they make you want to have access to the program itself?" As a result of this section I hope to make them aware of the concept of freeware and shareware and also to inculcate in them the concept that if you utilize a piece of shareware for the purpose which its author intended you must pay the shareware fee. I indicate to the students that they may take any of these materials for their own use that they wish, reminding them that if they have selected some shareware and intend to utilize it that they are responsible for seeing that the shareware fee is taken care of.

The last two class meetings are spent examining some BASIC routines from Short Programs for Teaching and Learning High School Mathematics which were a part of the NCTM's 1986 project Preparing Teachers to Use Computers in Mathematics Classrooms. I have this as a part of the course primarily to set a historical perspective. Just ten short years ago this is what many people thought the future of computers in the mathematics classroom involved. I require a written report on this experience. In the directions for the report I include the following statement. "In your report you should indicate which routines you have examined. Be willing to examine these routines in their historical context. List your top five, list your bottom five. Give reasons for your opinions. Comment on the general quality of this material. If your school had Apple II computers and the administration told you that there was no money for software purchases could you use these programs to enhance your teaching of mathematics? Be rather specific in your comments. Which programs would you use, where and how would you use them? For which aspects of secondary school mathematics would this material be most useful?" Since these materials are in the public domain I give all class members copies of the routines.

Early in the semester I require the students to do a library search for resources related to the use of computers in the teaching of mathematics and submit a report on that search.

The course grade is based on my evaluation of the reports and projects which the students submit. There are no examinations given.

I have given a description of what I did when I started to teach this course and what I am doing now. What do I envision for the future? What would I like to do? One thing which is missing in the course is active involvement with the Internet. This is not an intentional omission. However it is going to be a while before Ball State University will equip the laboratory in such a way that we will be able to access the Internet. My office was not so equipped until late 1995. Even then other problems will arise. The architecture of the Macintosh LC's and Performas indicates that an ethernet card must go in the PDS slot. When an ethernet card is placed in the PDS slot this means that the Apple IIe emulation card must go and hence that the IIe software which we are now working with must be replaced. This means that the budget must not only handle the cost of ethernet cards and their installation it must also provide funds for replacement software.

I would like to make students aware of HyperCard and some of its uses. Here I find two problems. One arises from the fact that HyperCard seems to be more difficult for students to gain some facility with in a short time. The other is that Apple no longer supplies HyperCard as a part of the Macintosh software and hence that once again the budget rears its head.

The course has evolved and will continue to evolve until I retire (no later than June 2001).

I welcome any comments that readers may have on what I am doing, or questions about it. I would especially like to hear from anyone who feels that it would be possible to do with HyperCard what I am managing to do with Logo in the same amount of time and can direct me to materials which might accomplish this. Then I would have only the budget to contend with.

Preparing pre-service mathematics teachers to use computers in their future teaching assignments is an important task. Those of us who have an opportunity to do so are really experimenters. I realize that in many instances courses like this are taught in Colleges (or Departments) of Education. At Ball State we are fortunate that the course is housed in the Department of Mathematical Sciences. I feel that all teacher training institutions should do it this way (not because we do it that way but because it is the best way). Perhaps this is a starting point for a discussion group. Is there a way to have such courses taught as content courses in Mathematics Departments and still avoid the "turf wars" which might be inevitable (at least in a large University)? I think that there might be.

Hubert Ludwig
Ball State University
email: 00hjludwig@bsuvc.bsu.edu
home page: http://www.cs.bsu.edu/~hjludwig/

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