Volume 19, Number 1 2000
Edited by David A. Smith, Assisted by David J. DeVries 3
David J. DeVries
Georgia College & State UniversityMilledgeville, GA 31061 USAddevies@mail.gac.peachnut.edu
In his first column in this space, David Smith noted one of theremarkable features of the Web: It opens possibilities for all of usto explore mathematics and science through original sources in newand exciting ways. However, as noted in that column, many of theoriginal sources available on the Web are “little more than thesame text you could read—less conveniently, but morecomfortably—in any good library.” We have continued tosearch for examples of web sites that use technology to enrich ourunderstanding of original sources. Unfortunately, we have not beenable to locate other sites like those on Euclid and Mendel that Smithbrought to our attention in his 1998 column. Perhaps some of ourreaders can share with us their knowledge of additional“original text” sites.
In surfing the Web, I did come across a related type of sitewhich, although not an original source, is based on such a source. Ithas been made available to us by students at the Museum of theHistory of Science, Oxford, and can be found in the museum’s website (www.mhs.ox.ac.uk) in the “Student Space.” This onlineexhibition is entitled “Cosmographia: A Close Encounter.”It is not an easy site to find—I tried several search enginesusing the word “cosmography” and got no hits on thisparticular site. 16th century to own a copy of Cosmographia in orderto associate themselves with the increasing power of mathematics insolving real world problems.
One of the more interesting features of the site is the inclusionof mathematical instruments whose illustrations in the book arereproduced and linked to photographs of instruments in themuseum’s collection. Apparently the book provided a convenientway for Gemma, an instrument maker, to advertise his products.However it might also provide an interesting project for students toinvestigate how some of these instruments worked and try to uncoverthe mathematics involved in the construction and use of these earlycosmographical instruments.
The ruling classes of the time believed in the usefulness ofcosmography, as demonstrated by records of large navigationalinstrument orders and the establishment of astronomicalobservatories. As it turns out, this belief was not misplaced. Whenthe problem of determining longitude at sea was finally solved in the18th century, the solution was based on techniques first suggested byApianus (measuring lunar distances) in Cosmographia and Gemma (usingportable clocks to determine differences in local time) in a laterpublication.
The site also includes pages devoted to the cartography inCosmographia, which in turn are linked to other texts available inthe museum’s collection. Cartography was intimately related toastronomy, navigation, and instrument making in the 16th century. (Weknow Gerard Mercator primarily as a mapmaker—less well known isthe fact that he worked in Gemma’s shop as an instrument maker.)Apianus and Gemma highlight the relationships between cartography andinstrument making, as well as their respective roles incosmography.
For insight into the vital role that Renaissance mathematics wasabout to play in the development of the modern world, Cosmographia: AClose Encounter is clearly a site worth visiting.
Editorial notes: “Learning: The Web” is a deliberatedouble entendre. On the one hand, it means the reader is learningabout resources available on the World Wide Web. On the other hand,it means the column is about resources for learning. That is, weconcentrate on web sites that provide innovative environments inwhich students can be expected to learn.
Your learning experience cannot end with reading this column. Infact, there is no way to convey in a static print medium any realsense of how learning from a Web site can take place. At best, thiscolumn will stimulate you to find and experience the subject website(s) yourself. (The technical term is “check it out.”)
We (the editors) do not intend to write all the columns ourselves.Contributions that are consistent with the first paragraph areeagerly solicited, as are comments, suggestions, corrections,criticisms, or any other kind of feedback. Send communications todas@math.duke.edu.
DAS
Zhonghong Jiang and Edwin McClintock
Department of Subject Specialization. EB344BFlorida International UniversityMiami, FL 33199 USAjiangz@fiu.edu
In mathematics, the same concept or the same mathematicalstructure is often encompassed by different representations such asgraphical/geometric, numerical/arithmetic, and symbolic/algebraicrepresentations. Research studies have provided evidence that usingmultiple representations of a mathematical concept can help studentsbetter construct the concept (Dufour-Janvier, Bednarz, & Belanger1987; Kaput 1992). Different representations usually suggestdifferent approaches to mathematical problem solving; and with thesame representation, there may be different solution methods as well.Working with both secondary school students and preservicemathematics teachers for many years, our experience has indicatedthat encouraging multiple approaches to problem solving plays animportant role in facilitating students’ understanding ofmathematical concepts and their grasp of methods of mathematicalthinking. Innovative technology tools in mathematics education,featuring multiple, linked representations, have greatly enhanced thestudents’ potential to develop multiple solutions to variousproblem situations. This article will describe the explorationprocesses of our students in solving a mathematical problem, andillustrate how they took full advantage of the power of technology inpursuing multiple solution methods, verifying the validity of thesemethods, and thereby enhancing their conceptual understandings.
Sergei Abramovich
State University of New York at Potsdam44 Pierrepont AvenuePotsdam, NY 13676-2294 USAabramovs@potsdam.edu
This paper deals with computational applications of number theoryconcepts developed from arithmetic properties of spreadsheet-basedlattices. It reflects on activities explored with inservice andpreservice teachers in an egalitarian, student-centered universityclassroom. The proposed pedagogy emphasizes the emergence of theconcepts as a response to a particular problematic situation thatarises via design of a computational environment. A discussion of thecurrent mathematics education reform incorporates comments by theteachers.
The old, the near, the accustomed, is not that to which butthat with which we attend; it does not furnish the material of aproblem, but of its solution. John DeweyConstance P. Hargrave and Jeffrey M. Kenton
N128 Lagomarcino HallIowa State UniversityAmes, Iowa 50014cph@iastate.edu
With the increased availability of powerful computers in theclassroom, simulations have become viable instructional tools. Yetthese technological advances are not coupled with advances inpedagogy but are stifled by conventional instruction. Typically inthe classroom, a simulation is used to deliver content to students,reinforce content already delivered, or test students’knowledge. In this article, we argue that designing simulations forstudent use prior to formal instruction changes their learningexperiences. In addition, instructional time can be used to addressstudents’ alternative conceptions and further develop theirconceptual understanding.
Eric C. R. Hehner
Department of Computer ScienceUniversity of TorontoToronto, ON M5S 3G4, CanadaHehner@cs.utoronto.ca
Boolean algebra is simpler than number algebra, with applicationsin programming, circuit design, law, specifications, mathematicalproof, and reasoning in any domain. So why is number algebra taughtin primary school and used routinely by scientists, engineers,economists, and the general public, while boolean algebra is nottaught until the university level, and not routinely used by anyone?A large part of the answer may be in the terminology and symbolsused, and in the explanations of boolean algebra found in textbooks.The subject has not yet freed itself from its history and philosophy.This paper points out some of the problems delaying the acceptanceand use of boolean algebra, and suggests some solutions.