The AMATYC Review
A refereed publication of the American Mathematical Association of TwoYear Colleges

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From the Editor 
Feature Articles

Regular Articles
 Book ReviewEdited by Sandra DeLozier Coleman
 Software Review Edited by Brian E. Smith
 The Problems Section Edited by Stephen Plett and Robert Stong

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The Mathematics of Supply and Demand
Philip Cheifetz and Ted Pyne


Phil Cheifetz has taught mathematics at Nassau Community College for 34 years. He was a founder of AMATYC and served as its fourth president. For the past six summers, he has taught graduate courses in quantitative methods at the JFK School of Government at Harvard. His hobbies include solving crossword puzzles and performing closeup magic. cheifep@ncc.edu


Ted Pyne teaches physics, astronomy, and mathematics at Harvard College and in the Harvard Extension School. For the past two years has taught graduate courses in quantitative methods at the JFK School of Government. Trained as an astrophysicist, his research interests are in general relativity and cosmology. He is also the guitarist in the rock band euphonic. pmconant@aol.com

In this paper, we present an example from economics, and show how the ideas of domains, parameters, functions, compositions, and translations are all necessary for a full understanding of the concept of market equilibrium. A particularly rich extension of the mathematics of equilibrium analysis is necessitated by the introduction of taxation. A surprising result is that the amount of the tax paid by consumers and the amount of the tax paid by suppliers is independent of whether the tax is imposed on consumers or suppliers.

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The Archimedean Trisection Method Revisited
Richard Francis

Richard Francis received a BS degree from Southeast Missouri State University and master’s and doctoral degrees from the University of Missouri (Columbia). His major interests include number theory and the history of mathematics. Previous articles in these areas have appeared in the publications of the National Council of Teachers of Mathematics, the American Mathematical Society, Consortium for Mathematics and Its Applications, and the Mathematical Association of America (from which he received the George Polya Award). He currently resides in Cape Girardeau, Missouri. C714scm@semovm.semo.edu

The three famous problems of antiquity provided a challenge to geometers of the middle Greek period. Such an early quest foreshadowed centuries of diverse attempts to discover, among other things, a valid Euclidean method of trisecting the general angle. Today, due to the works of Wantzel (1837) and others, the construction is known to be impossible.
Among the many contributors to the overall story of trisection attempts is Archimedes (287212 B.C.). His famous method, though fallacious, permits an interesting generalized look at the trisection matter and, by revisiting, suggests extended questions of a multisection kind.

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Students with Disabilities in Mathematics: Barriers and Recommendations
D. Patrick Kinney

Pat Kinney is a mathematics instructor at Wisconsin Indianhead Technical College in Ashland, Wisconsin. His interests include teaching developmental mathematics with interactive multimedia software. Previously, he was an assistant professor in the General College of the University of Minnesota. pkinney@witc.tec.wi.us

This article examines barriers encountered by students in mathematics courses at the postsecondary level. Interviews with students with disabilities are used to provide suggestions for more effectively meeting their needs. Attempting to understand the challenges facing students with disabilities is an important step towards more effectively meeting their needs, carrying out our responsibilities as representatives of the institution in which they have enrolled, and working towards increasing the number of people with disabilities in the workforce in the fields of science, mathematics, and engineering.
The concept of Universal Instructional Design is examined as one approach that may be useful in better meeting the needs of all students, not just those with disabilities. The basic premise of Universal Instructional Design is that curriculum should include alternatives to make it accessible and applicable to students with different backgrounds, learning styles, abilities, and disabilities. The "universal" in universal design does not imply that one size fits all; instead, it stresses the need for flexible, customizable content, assignments, and activities (Curriculum Transformation and Disability, 2000).

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The Gambler’s RuinHow Long Does It Take?
Thomas C. McMillan

Tom McMillan is Professor of Mathematics at the University of Arkansas at Little Rock, where he serves as department chair. He previously taught for 21 years at Radford University, as professor of both mathematics and computer science. He has degrees in mathematics from the University of Idaho and the University of Utah and a degree in computer science from Virginia Polytechnic Institute. tcmcmillan@ualr.edu

A problem known as "the gambler's ruin" (two players flip a coin until both players have won all the coins) is reviewed, as is the probability that each player has of winning all the coins. A formula that gives the expected number of coin flips required until one of the players wins all the coins is derived. The derivation, which uses summation to solve a difference equation, can be used as an example in an elementary discrete mathematics course. A computer simulation that validates the mathematical analysis is presented. It is noted that similar mathematical techniques can be applied to such problems at the analysis of computer network performance.

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Three Proofs of the Countability of the Rationals
Students: Jonathan Dandy, David Duke, and Gabriel Foust
Dean B. Priest

Dr. Dean Priest is Dean of the College of Sciences and Professor of Mathematics at Harding University, Searcy, Arkansas. He served as a member of the publications committee of NCTM and as a member of the project task force for AMATYC’s Crossroads in Mathematics: Standards for Introductory College Mathematics Before Calculus. dpriest@harding.edu

The set M is said to be countable if there exists an "onto" function from the set of positive integers I+ to M. Three students produced three separate functions from I+ "onto" the set of positive rationals thereby producing three additional proofs of the countability of the rationals.

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Square Roots: Greek Ladders, Farey Fractions, and Pascal’s Triangle
Robert J. Wisner

Robert J. Wisner has been at New Mexico State University since 1963, retiring as Professor of Mathematics in 1999. He received a PhD in 1953 from the University of Washington, taught at Haverford College, and was Executive Director of CUPM. He was founding editor of The SIAM Review and for many years was the Consulting Editor of Mathematics for Brooks/Cole. He has authored or coauthored numerous books as well as expository and research papers. rwlht@juno.com

Most undergraduate textbooks omit two potent and fertile topics, Greek ladders and Farey fractions, though they feature Pascal’s triangle in many of its guises. Such is also true of most undergraduate courses. This expository and historical work shows that not only do the three topics of its title appear in arrays that vaguely resemble one another, but also the three subjects are intertwined in the quest for fractional square root approximations.
The three topics are presented in separate parts of the paper. The reader does not need to have ever heard of Greek ladders or Farey fractions to see in explicit detail how they operate to provide fractional approximations to square roots of positive whole numbers. Both can also provide similar approximations to irrational numbers in general, and such an example is provided. Then the paper culminates in demonstrating how Pascal’s triangle also provides a setting for approximating square roots of positive integers, and in the process reveals how the three subjects are intertwined.
The references are accessible to the interested undergraduate student.

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Using the TI89/92 Text Editor for Exploration and Discovery
Thomas W. Shilgalis

Thomas W. Shilgalis is Professor of Mathematics at Illinois State University, Normal, Illinois. He earned his PhD in mathematics education at the University of Illinois at UrbanaChampaign. toms@math.ilstu.edu 
The TI89/92 text editor provides a way to save the work involved in a moderately complicated solution to a problem and to use the saved script to solve similar problems by editing just a few lines. Beyond that helpful but somewhat uninspiring use, the text editor provides opportunities for discovering formulas by successively generalizing the parameters in a particular problem initially involving constants. An example illustrating the effect of a linear transformation on distances is included, showing that distances in a given direction are affected the same, with the multiplier depending only on the direction and the entries of the twobytwo matrix representing the transformation.

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The Implications of HandHeld Computer Algebra System Calculators Throughout the Mathematics Curriculum
Sheldon P. Gordon and Arlene H. Kleinstein

Dr. Sheldon Gordon is Professor of Mathematics at SUNY Farmingdale. He is a member of a number of national committees involved in undergraduate mathematics education. He is the principal author of a precalculus text and a coauthor of the texts developed under the Harvard Calculus Consortium. gordonsp@farmingdale.edu

Dr. Arlene Kleinstein is Associate Professor of Mathematics at SUNY Farmingdale. Dr. Kleinstein has been a recipient of several NSF grants and has given regional and national presentations in the areas of developmental mathematics and interdisciplinary studies in physics and engineering technology. She is a coauthor of the developmental/college algebra textbook series. kleinsah@farmingdale.edu

The authors review the changes in curriculum and pedagogy that have transformed undergraduate mathematics education in the last decade. Aspects of the reform movement in calculus have significantly influenced the way in which calculus is taught as well as the way in which its precursor and successor courses have evolved. The first generation of graphing calculators has allowed students to view topics through a triple lens, the Rule of Three, with its algebraic, graphical, and numerical perspectives. The latest generation of handheld calculators incorporates formal symbolic manipulation into the mix. These calculators will factor, simplify, solve, integrate, and differentiate, in closed form, almost any expression or function that we would ever consider assigning our students; and the results are displayed almost instantaneously.
The paper discusses the implication of this new technology on mathematics education in the coming decade. The authors have rejected the option of ignoring this technology, and, therefore, grapple with the potential benefits and disadvantages of implementing CAS throughout the entire mathematics curriculum. This may involve actually utilizing the technology in more advanced courses and simultaneously rethinking the content and emphasis in courses below that level to reflect changing prerequisite needs because of the subsequent use of CAS technology. The authors believe that implementation should not occur in an individual course or by an individual instructor. Rather, the impact of CAS technology on the way in which quantitative courses are taught and learned requires departmentwide, if not collegewide, consensus. The authors offer some ground rules and recommendations for the deliberations that will need to occur at each institution that contemplates rethinking its offerings in light of this extraordinary technology.

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