 On Constructivism  Susan Hanley
"The constructivist approach opens new avenues for learning as well as challenges for the teacher trying to implement it." Hanley outlines problems in traditional teaching methods, describes constructivism as a possible solution, and reviews research
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 On Deciding Whether a Surface is Parabolic or Hyperbolic [PDF]  Peter G. Doyle
An application of Rayleigh's method to the classical type problem for Riemann surfaces, which is really just the same random walk problem, only instead of walking around on an infinite graph the walker is diffusing around on a surface. The aim will be
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 OnePage Proof of Fermat: Critique (Math Chat)  Frank Morgan, MAA Online
Critique the following short proof of Fermat's Last Theorem sent in by
reader Rob Connelly. (In perhaps the biggest mathematics news of the century, Andrew Wiles recently came up with a very long and complicated proof of this 350year old problem.) Answer:
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 One size fits all?  Mike Robison; Mathematics Dept., Michigan State University
Age based tracking versus ability grouping in elementary school mathematics. A research paper on ability and age grouping that reviews the literature and makes suggestions for math education.
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 One Trip to the Attic? (Math Chat)  Frank Morgan, MAA Online
In the lobby of a hotel there are three on/off switches. One is attached to a light bulb on the third floor; the other two are attached to nothing at all. You may work the three switches as often as you please. How many times do you need to visit the
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 On Four 4s and Speed Limits (Math Chat)  Frank Morgan; Christian Science Monitor
Can you get past 30 in forming numbers 0, 1, 2, 3, ... from four 4s and the standard mathematical operations +,  , x, /, decimal point, square root, powers, and factorial(!)?
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 Online Innovation Institute (OII)  Ferdi Serim
The OII provides educators with a learning environment to support integrating the Internet into their individual teaching styles, offering a combination of online and onsite collaborations in which participants develop projects to use in their classrooms.
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 Online journal publication: two views from the electronic trenches  Jonathan Borwein (CECM), Richard Smith (CPROST)
A report on the experiences of two sets of scholarly publications, the Canadian Mathematical Society and the Canadian Journal of Communication, during their adoption and deployment of new electronic media, drawing on the authors' firsthand experience
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 On Motivation and Understanding  Cut the Knot!, Alexander Bogomolny
Fourth in a series of columns that draw on Frank Morley's fundamental paper On the Metric Geometry of the Plane nLine  dealing with what amounts to his proof of the Trisector Theorem. Bogomolny outlines Morley's reasoning first and then supplies necessary
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 On the Bass Note of a Schottky Group [PDF]  Peter G. Doyle
Using a classical method from physics called Rayleigh's cutting method, Doyle proves the conjecture of Phillips and Sarnak that there is a universal lower bound L2 > 0 for the lowest eigenvalue of the quotient manifold of a classical Schottky group,
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 On the Evolution of Islands  Doyle, Mallows, Orlitsky, Shepp
On islands and seas: an island is a maximal set of adjacent marked cells in a ring or row; a sea is a maximal set of adjacent unmarked cells. An inductive proof and a countingargument proof. A paper that appeared in the Israel Journal of Mathematics.
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 On the Horizon (OTH)  School of Education, University of North Carolina (UNC)
OTH Online: the electronic version of the print publication, with the full text of prior volume years. Also the HORIZON mailing list, seminars and workshops, conferences, and presentations to explore and extend the educational community's thinking about
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 On the Minimax Sphere Eversion  Rob Kusner, Mathematics and GANG, Univ. of Massachussetts at Amherst
A brief paper describing the evolution of the animation of the minimax sphere eversion. Does not include the animation itself.
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 On the Puzzles with polyhedra and numbers  Jorge Rezende
Several articles in PDF format, with related links: one on a set of didactic puzzles published by the Portuguese Mathematical Society (SPM); "Jogos com poliedros e permutações", an article in Portuguese describing puzzles with polyhedra
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 On the Teeth of Wheels (Computing Science)  Brian Hayes, American Scientist
A little history on the importance of gears in the history of computing and the importance of computing in the history of gears. "Designers of gear trains have not merely borrowed ideas from mathematics but have also developed some of those ideas on their
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 Operations Research, Mathematical Programming  Dave Rusin; The Mathematical Atlas
A short article designed to provide an introduction to operations research, which may be figuratively described as the study of optimal resource allocation.
Depending on the options and constraints in the setting, this may involve linear programming,
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 Operator Algebra Resources  N. Christopher Phillips
Lists of some resources useful to mathematicians working in the subject of operator algebras. General operator algebra sites and publications, information on regular series of conferences, and some important general mathematical resources.
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 Operator Theory  Dave Rusin; The Mathematical Atlas
A short article designed to provide an introduction to operator theory, which studies transformations between the vector spaces studied in Functional Analysis,
such as differential operators or selfadjoint operators. The analysis might study the spectrum
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 Optics2001
Online optical library with links to papers, theses, and articles on optics for beginner level to researchers. Contents include: General Optics; Optical Detectors; Optical Fibers; Image Processing; Design Techniques; Optical Elements; Optical Instruments;
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 Optics, Electromagnetic Theory  Dave Rusin; The Mathematical Atlas
A short article designed to provide an introduction to optics, electromagnetic theory: the study of the propagation and evolution of electromagnetic waves,
including topics of interference and diffraction. Besides the usual branches of analysis, this
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 Orbits and Gravitation  MacTutor Math History Archives
A hypertext essay covering Copernicus through Poincaré, LeviCivita, and Birkhoff, with References.
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 Order, Lattices, Ordered Algebraic Structures  Dave Rusin; The Mathematical Atlas
A short article designed to provide an introduction to ordered sets. Ordered sets, or lattices, give a uniform structure to, for example, the set of subfields of a field. Various special types of lattices have particularly nice structure and have applications
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 Ordinary Differential Equations  Dave Rusin; The Mathematical Atlas
A short article designed to provide an introduction to ordinary differential equations, equations to be solved in which the unknown element is a function rather than a number, and in which the known information relates that function to its derivatives.
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 Ordinary Differential Equations Project  Blanchard, Devaney, Hall; Boston University
This project included the development of a textbook (publisher Brooks/Cole Publishing Co.) as well as various workshops on teaching differential equations. Instructors manuals and selected articles are available from this site: "Teaching Differential
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 The Organic Mathematics Project  Centre for Experimental and Constructive Mathematics (CECM)
A collaborative research project involving a number of researchers at the Centre for Experimental and Constructive Mathematics (CECM). Involves in part the development of WWWbased and Mapleassisted tools for document annotation and math activation.
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 Organic Mathematics Workshop: The Proceedings  Borwein, Borwein, Corless, Jörgenson, Sinclair, eds.
Proceedings of a 1996 workshop held at Simon Fraser University, Vancouver, Canada. Speakers (top experimental and computational mathematicians); their articles, with equations, code, photos, videos, and annotations; and related resources and links. Invited
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 Orthogonal Polynomials, Integral Transforms and Solutions for Dynamics of Multilevel Systems  Vadim Zelenkov; International Sakharov Environmental University
Papers and preprints by a small group (Savva, Mazurenko, Zelenkov) that studies multiphoton excitation of molecules by infrared laser radiation in an attempt to find analytic solutions of the dynamic equations using miscellaneous orthogonal polynomials
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 Ostrich Llama Count  Examining Solution Methods  Suzanne Alejandre
Activity, alignments, directions, extensions, and activity sheets that detail five ways to solve the Problem of the Week (PoW) "Ostrich Llama Count." Originally published in the March, 2013, issue of the CMC ComMuniCator, journal of the California Mathematics
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 The Other Crisis in American Education  Singal (Atlantic Monthly)
A look at potentially high achievers whose SAT scores have fallen, and who "read less, understand less of what they read, and know less han the top students of a generation ago." Includes statements like "schools that had 'severely declining test scores'
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 Overcoming Greatness: The Decline and Recovery of British Mathematics in the PostNewton Era  Robert Bruen; Harvard University
An historical essay on Newton and Leibniz, British Progress, European Progress, and the story of the overthrow of Newton's notation at Cambridge. Includes a bibliography.
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 An overview of Babylonian mathematics  MacTutor Math History Archives
Linked essay exploring the arithmetic and geometry of the Babylonians, who lived in Mesopotamia, a fertile plain between the Tigris and Euphrates rivers. The Babylonian civilization replaced the Sumerian civilisation and the Akkadian civilisation; this
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 An overview of Egyptian mathematics  MacTutor Math History Archives
Linked essay on the development of the Egyptian number systems, and the arithmetical methods they devised to work with their numerals. With references and other related web sites.
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 An Overview of IMP Years 1 and 2  Kim Mackey
An indepth review of the Interactive Mathematics Program (IMP) by a teacher. Contents include: Introduction: Why be concerned about IMP? Brief Synopsis of Year 1 and Year 2 of the IMP curriculum; A more indepth look at Year 2 IMP unit "Solve it!"; A
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 An overview of Indian mathematics  MacTutor Math History Archives
Linked essay on the contributions of Indian mathematics, in particular the number system invented by the Indians, on which much of mathematical development has rested. With references and other related web sites.
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 Overview of the History of Mathematics  MacTutor Math History Archives
Linked overview from the Babylonians through topics from the twentieth century, with additional sections on notation and communication, brilliant discoveries, how we view history, and a challenge, as well as a 700item general bibliography.
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 Pacific Institute for the Mathematical Sciences (PIMS)
A networked institute based in Alberta and British Columbia, Canada. Primarily dedicated to the support and advancement of mathematical science in Western Canada and the Pacific Rim, it maintains physical offices at the Universities of Alberta, British
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 Pairing Like Students for Perseverance in Problem Solving  Marie Hogan
Article about how one particular duo of eighth grade PreAlgebra students, paired by ability level, overcame a contentious start to work together and "reach new heights of problemsolving proficiency." Originally published in the March, 2013, issue of
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 Palestinian Central Bureau of Statistics (PCBS)  The Palestinian National Authority
The agency that collects official statistics for Palestine. Tables of key social, economic and demographic statistics are online; some are available for download, or are linked to methodological information. Site also includes census information, list
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 Palindromic Primes  Ivars Peterson (MathTrek)
Palindromic primes are whole numbers, evenly divisible only by themselves and 1, that also have the same sequence of digits read forward or backward... The current record for the largest known palindromic prime is held by Harvey Dubner, a retired electrical
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 Panitz cooperative learning and WAC web page and much more  Ted Panitz, Cape Cod Community College
Workshops, articles, and other resources on cooperative learning and pedagogy.
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 Papers  A. R. D. Mathias
Downloadable papers in DVI PostScript format: a piece on Bourbaki; a piece on Stalin; a long piece on Mac Lane set theory; a short piece on Zermelo set theory; a recent piece on Strong Statements of Analysis; and three other papers, appdyn97, delays97
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 Papers by Dr. Bert K. Waits  Dr. Bert K. Waits
In PDF format, essays include "The Role of HandHeld Computer Symbolic Algebra in Mathematics Education in the TwentyFirst Century: A Call for Action!"; "The Role of Graphing Calculators in Mathematics Reform"; "The Evolution of Instructional Use of
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 Papers  Chaim GoodmanStrauss, Dept. of Mathematics, Univ. of Arkansas
On Composite Twisted Unknots; Matching Rules and Substitution Tilings; Aperiodic Hierarchical Tilings; Bounding the Results of Arithmetic Operations on Random Variables with Unknown Dependency; A Small Set of Planar Aperiodic Tiles; A Pair of Aperiodic
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 Papers Online: Advanced Chaos  The Chaos Group, Univ. of Maryland at College Park
An online collection of published articles and preprints on advanced chaos theory, covering the topics of: Control and Targeting, Dimensions, Fractal Basin Boundaries, General Chaos, Quantum Chaos, Riddled Basins, and Shadowing.
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 Papers on Riemannian Geometry  Alan Weinstein
Papers written by graduate students in the course Math 240, Riemannian Geometry, in the Spring Semester of 1995 at the Univ. of California at Berkeley. Each paper is either a survey of an area or a tutorial essay in a topic related to riemannian geometry.
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 Paradox (Encarta Encyclopedia 2000)  Microsoft Encarta Online
Paradox, in logic and mathematics, an apparently contradictory conclusion that is derived from what seem to be valid premises.
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 Parallel Coordinates  Alfred Inselberg
Download articles, read a course syllabus, and see images about a method for visualizing multidimensional spaces with industry applications that have included visual and automatic (classification) data mining, optimization, GIS, process control, decision
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 Partial Differential Equations  Dave Rusin; The Mathematical Atlas
A short article designed to provide an introduction to partial differential equations. Like ordinary differential equations, partial differential equations are equations to be solved in which the unknown element is a function, but in PDEs the function
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 Party Games  Ivars Peterson (MathTrek)
You're one of six people at a dinner party. You would undoubtedly find that the dinner party includes either a group of at least three people who all know one another or a group of at least three people who don't know one another. The reason for this
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 Pascal's Fractals  Ivars Peterson (MathLand)
It's possible to convert Pascal's triangle into eyecatching geometric forms. For example, one can replace the odd coefficients with 1 and even coefficients with 0. Continuing the pattern for many rows reveals an everenlarging host of triangles, of varying
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