Volume 18, Number 1 1999
Learning: The Web
Edited by David A. Smith
Duke University, Durham, NC, USAAssisted by David J. DeVries,
Georgia College & State University, Midgeville, GA, USAJohn S. Robertson
The wise through excess of wisdom is made a fool.
—Ralph Waldo Emerson
It seems that the Internet, and more specifically, the set of linked sites knowncollectively as the Web, is changing...well, just about everything. And that includes theteaching and doing of mathematics.
The volume of material, both technical and pedagogical, is enormous. It is easilymeasured in terabytes (240 bytes), and in our lifetimes may well call for the introductionof the petabyte (250 bytes) as the standard reference unit for measuring the volume ofinformation. A large fraction of all this is relevant to our community, and frankly it isquite exciting to have it all at our fingertips. So what’s the problem? It turns outthat there are three.
The problem most often faced by mathematics educators is the difficulty in findingmaterial relevant to our specific needs. Mathematicians are not the only ones with thisproblem, and a variety of solutions have been developed to help with this challenge. Theseare the so-called search engines: Yahoo, Excite, and AltaVista just to name a few. Arecent check of Yahoo showed 32 sites devoted to chaos theory, 206 to numerical analysis,165 to statistics, and 9 to ethnomathematics. One can easily find the algorithm fortrisecting an angle (with a marked ruler, of course) or download the FORTRAN source codefor computing Fresnel integrals. What a cornucopia of information!
But try looking for information about Airy functions. If you search for“airy,” you may get lots of hits on towns in North Carolina, Maryland, andGeorgia, a resort in the Pocono Mountains of Pennsylvania, as well as many products andservices before you find the first reference to Airy functions. The results are better ifyou search for “airy functions”, but many of the hits would be on pages forcomputer algebra systems, not the place to find out about analytical properties of thosefunctions. The secret is to be able to narrow the search down quickly by using just theright keywords. And that can take practice as well as skill. Depending on the subject andyour ability to specify it to a search engine, you might be better off in the library. Andsuppose you want to find something really obscure, such as, the number of finite groups oforder 1024. You might try key words like “group theory,” “finitegroups,” or “group tables”—and you’d be led through what seemslike an unending morass of hits, none of which will have the answer to this question.
It has been my experience in many cases that a combination of search engines and“surfing” will locate what I want. The search engine can point to any number ofrelevant sites that almost always contain links or connections to still other sites thatmay not have ever entered the scope of the search engine’s database. This techniquetakes some patient practice, but is well worth the effort, whether you are teaching numbertheory or mathematics for middle school teachers.
On the other hand, it can sometimes be very frustrating, particularly if you suspectthat the information you are seeking ought to be commonly known. It is frequently the casethat too many hits (and not too few) overwhelm the person searching the web. The solutionis to learn the features of your favorite search engine, features that allow you to lookfor multiple keywords at the same time, exact phrase matches, and more complex types ofrelational searches. They all have convenient help features, often a button or iconlabeled “advanced search” or something similar. The search engine will usuallysuggest methods or strategies for refining the search in ways that make sense.
A second problem that comes with the Web is a bit more subtle: There is no real qualitycontrol. Material obtained from commercial sites, especially if one must pay a fee, can becounted on for quality, but material from free sites can run the gamut from the superb tothe horrific. Imagine the problem faced by our students, who often lack the discerningexperience we bring to the process. For them, finding a hit on a certain topic is greatnews. And yet the hit may be poorly-written and just plain wrong. It happens.
[Editor’s note: For example, while looking for information on the Lotka-Volterrapredator-prey equations, you might find a very interesting site on predation (whose URLshall be nameless) that is actually a set of lecture notes for a course in ecology. Butthe biologist-author makes the assertion that if either population deviates even a littlefrom equilibrium, then both populations go into wild oscillations. He doesn’t say howhe produced the oscillatory graphs, but he must have blindly followed Euler’s methodwithout realizing that it cannot give a satisfactory solution to these differentialequations. DAS]
Misinformation is even more likely to happen when the material that one seeks containsa philosophical or political component. Such is the case when sifting through the enormousamount of mathematics education material available on the web. One must keep in mind thatmuch of it has never undergone peer review. Our students need to approach web sources likethis with a healthy dose of skepticism.
A final problem comes from the darker side of such an abundance of information in ahighly portable format. Academic integrity is undermined when it is trivial to plagiarize.In the old days, blatant plagiarism meant at least having to type purloined material byhand. Nowadays, it’s just a simple cut and paste, and someone else’s work can byquickly and effortlessly integrated into your own. It is so easy to do that many studentshave difficulty seeing the danger of academic dishonesty. But the danger is therenonetheless. Already, one can find entire term papers perfectly written, spell-checked andbeautifully formatted on the Web. Before long, one will be able to do the same forsolutions to homework problems in popular calculus books. The Web makes it so easy tocheat. Faculty must remain vigilant if our charges are not to acquire habits that willhave awful consequences later in their lives.
Lest readers of this column mistake me for a Luddite, let me state quite clearly myenthusiasm for the power brought to mathematics and teaching by this technology. But thereis a giddiness about in the land, and one that masks the pitfalls, frustrations, anddangers of this technology for us and our students. There is a great responsibility forall of us to help each other and our pupils to tame this beast, and to ensure, as best wecan, that it remains our servant and not our master.
We’ve all heard that a million monkeys banging on a million typewriters willeventually reproduce the works of Shakespeare. Now, thanks to the Internet, we know thisis not true.
The Effect of Instruction with Graphing Calculators on How General MathematicsStudents Naturalistically Solve Algebraic Problems
Michelle Merriweather and Marcia L. Tharp
Old Dominion University Hampton Boulevard Norfolk, VA 23529, USAMmerriweather@wcupa.edu
This study investigated the effect of instruction with the TI-82 graphing calculator oneighth grade general mathematics students. Specifically, this study focused on changes inattitude toward mathematics and calculator use and changes in how general mathematicsstudents naturalistically solve algebraic problems.
The students were grouped into experimental (calculator) and control (non-calculator)classes. A pre and postsurvey was given to both the control and experimental groups toassess the students’ attitudes toward calculators and mathematics. The survey alsodetermined whether a student is rule-based (i.e. a student who is only concerned where andwhen to use a rule or formula and not why the rule works). In addition, in depthinterviews were conducted to determine students’ preference in solving algebraic wordproblems.
Results found that the rule-based students used an equation to solve the algebraic wordproblem whereas the non-rule based students used a numeric method. Although the studentsdid not use the graphing calculator to solve the word problem, many students became moreexcited and involved in mathematics.
Dana L. Roberts
Benson High School Omaha Public Schools Omaha, NE 68104, USALarry J. Stephens
Dept. Of Mathematics University of Nebraska At Omaha Omaha, NE 68182, USAstephens@cwis.unomaha.edu
This year long study compared students of average ability in three high school Geometryclasses that utilized computer software in varying amounts. One class utilized thesoftware Geometry Inventor twice a week, one class utilized the software once a week andthe other class did not use the software. The means for the three groups were compared ontwelve chapter tests as well as first and second semester tests. The only significantdifferences found were for the introductory chapter and the chapter on transformationalGeometry. The group not utilizing the software scored higher in both instances. Theseresults indicate that using computer software may not be beneficial when teaching certaintopics in Geometry. It was observed that using computer software did improve studentinterest and participation in Geometry.
Azita Manoucherhri
Mathematics Education Maryville University-St. Louis, 13550 Conway Road St. Louis, Mo 63141, USAAzita@maryville.edu
Current mathematics reform has encouraged the use of computers in learning and teachingmathematics. Recent research also provides strong evidence of the usefulness of computersin mathematics learning. National surveys, however, indicate that computers are not beingused by teachers in mathematics classrooms.
A survey study was conducted to investigate the extent in which computers were beingused by middle and high school mathematics teachers in the state of Missouri. Moreover, itwas intended to develop an understanding of the basis for teachers’ decisionsregarding the use of computers, or lack of, in instruction. The data indicated that theteachers did not use computers for purposes other than drill and practice. This was due tolack of adequate knowledge about when and how computers could be used in mathematicsinstruction, and lack of sufficient training. Mathematics teacher educators, then, need tohelp increase teachers’ knowledge about the usefulness of computers by engaging themin extensive experiences with computers and a variety of educational software. Teachersalso need to be assisted in building curricular models that integrate the use ofcomputers.
Paul G. Shotsberger
Mathematical Sciences Department University of North Carolina at Wilmington, 601 S. College Road Wilmington, NC 28403-3297, USAshots@cms.uncwil.edu
This paper reports on the implementation of a World Wide Web site for teacherprofessional development which can address the shortcomings of typical face-to-facetraining. Results of a semester-long pilot project involving four high school and middleschool mathematics teachers highlight three specific ways in which teachers can benefitfrom Web professional development: consistent opportunities for reflection and sharing; ashortened cycle for training, implementation and evaluation; and teacher empowermentthrough direct access to information. A tentative model of just-in-time professionaldevelopment using the World Wide Web is introduced as a means of representing the complexdynamics of this learning environment.
An Educational MONTE CARLO Simulation/Animation Program for the Cosmic RaysMuons and a Prototype Computer-Driven Hardware Display
G. Kalkanis and M.M. Sarris
University of Athens, Pedagogical Department P.E. Physics, Technology, and Environment Section Solonos 104, GR-106 80, Greecegkalkan@atlas.cc.uoa.gr
An educational software for the study and the detection methods of the cosmic raysmuons, passing through several light transparent materials (water, air, etc.), isdescribed. Muons are created and Cherenkov photons are produced. Their paths andinteractions are simulated and visualised/animated on the computer screen using MONTECARLO methods/techniques (which employ random numbers following certain distributions).Finally the photons are detected. The simulated detector used is a Cherenkov counter,which is an array of photodetectors. The photodetectors are presented on the computerscreen or/and by a prototype computer-driven hardware display, which has been/can bedeveloped by students using simple electronics.