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Hands on History

Hands on History: A Resource for Teaching Mathematics

Edited by Amy Shell-Gellasch
Series: MAA Notes
Volume: 72
Copyright Date: 2007
Edition: 1
Pages: 191
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  • Book Info
    Hands on History
    Book Description:

    This volume is a compilation of articles from researchers and educators who use the history of mathematics to facilitate active learning in the classroom. The contributions range from simple devices such as the rectangular protractor that can be made in a geometry classroom, to elaborate models of descriptive geometry that can be used as a major project in a college mathematics course. Other chapters contain detailed descriptions on how to build and use historical models in the high school or collegiate mathematics classroom. Some of the items included in this volume are: sundials, planimeters, Napier's Bones, linkages, cycloid clock, a labyrinth, and an apparatus that demonstrates the brachistocrone in the classroom. Whether replicas of historical devices or models are used to represent a topic from the history of mathematics, using models of a historical nature allows students to combine three important areas of their education: mathematics and mathematical reasoning; mechanical and spatial reasoning and manipulation; and evaluation of historical versus contemporary mathematical techniques.

    eISBN: 978-0-88385-976-6
    Subjects: Mathematics

Table of Contents

  1. Front Matter
    (pp. i-vi)
  2. Preface
    (pp. vii-viii)
    Amy Shell-Gellasch
  3. Introduction
    (pp. ix-x)

    With the advent of computers and classroom projectors, educators have a world of resources at their fingertips. However, when we let our fingers do the walking in the virtual world, we and our students miss the physical and creative aspects of learning. Numerous studies have shown that doing (as opposed to simply listening or reading) is the best way to learn.

    Educators have always employed physical models to aid in instruction. During much of the nineteenth century and the twentieth up to the advent of classroom computers, makers of mathematical devices for the classroom abounded. SeeMulti-Sensory Aids in Teaching...

  4. Table of Contents
    (pp. xi-xii)
  5. Learning from the Medieval Master Masons: A Geometric Journey through the Labyrinth
    (pp. 1-16)
    Hugh McCague

    In recent years, there has been a resurgence in understanding, constructing, and walking mazes and labyrinths, many originating from the Middle Ages. For the student of mathematics, these novel geometric objects can be an experience in the power, beauty and utility of mathematics.

    These qualities of mathematics were well appreciated by medieval master masons, the designers and builders of the great pavement labyrinths in some of the medieval churches and cathedrals. Master masons worked their way up the ranks of stonemasonry starting with demanding years of apprenticeship. There were many tools for stone cutting and sculpting to master. The key...

  6. Dem Bones Ain’t Dead: Napier’s Bones in the Classroom
    (pp. 17-28)
    Joanne Peeples

    The year is about 1610; the place is Merchiston Castle near Edinburgh, Scotland; and the Baron of the castle, John Napier, is sitting at his desk working on his latest invention — logarithms. The computation of all of the tables of logarithms involved required many calculations, which had to be correct. Napier, ever the inventor, realized that his need to be able to calculate correctly, as well as the needs of many others at this time, made instruments that would mechanically compute of special interest. So … Napier picked up his bones (or rods as they were sometimes called) and started...

  7. The Towers of Hanoi
    (pp. 29-34)
    Amy Shell-Gellasch

    The classic Towers of Hanoi puzzle (Figure 1) is known to many of us, either from our school days or as educators. The puzzle works as follows: given some number of discs (washers) of varying diameters stacked in a pyramid on one of three posts, move the stack one disc at a time to one of the other posts in as few moves as possible. The catch is that no disc can rest on top of a disc of smaller diameter. This is a wonderful problem for school age students and many people first encounter it in elementary or middle...

  8. Rectangular Protractors and the Mathematics Classroom
    (pp. 35-40)
    Amy Ackerberg-Hastings

    It may not occur to contemporary students that architects, surveyors, and drafters once did their work by hand, without the aid of computers. The nineteenth-century drawing instruments used to prepare engineer ing drawings may seem unusual in their form and materials to a twenty-first century audience. Meanwhile, although they likely used protractors to measure and draw angles in middle school and learned a general proof for bisecting the angle in high school, even mathematics majors may never realize that someone had to divide the circle precisely in order to manufacture instruments that measure angles. The rectangular pro tractor was a...

  9. Was Pythagoras Chinese?
    (pp. 41-48)
    David E. Zitarelli

    This article presents two self-contained proofs of the Pythagorean Theorem that are strictly geometric, involving neither measurements nor numbers. The first might have been discovered by Pythagoras in the sixth century BC. The second is due to Liu Hui from about 300 AD. The two proofs show how mathematicians in two ancient civilizations—one in the West (ancient Greece) and the other in the East (ancient China)—deduced a result about right triangles from strictly geometric arguments. We also briefly contrast the geometric approaches with an arithmetic method employed by mathematicians from a third great ancient civilization—the Babylonians. The...

  10. Geometric String Models of Descriptive Geometry
    (pp. 49-62)
    Amy Shell-Gellasch and Bill Acheson

    Many art galleries exhibit sculptures constructed of taut strings or wires strung on wood or metal frames. The genesis of much of this form of art is the static string models originally devised and constructed by Gaspard Monge in the late eighteenth century, and the subsequent articulated models of his student Theodore Olivier in the nineteenth century. These models were constructed as three-dimensional aids in the teaching of descriptive geometry in the nineteenth century.

    Simple models that exhibit surfaces such as hyperboloids and warped planes can be constructed for classroom use by the instructor or teams of students. These models...

  11. The French Curve
    (pp. 63-70)
    Brian J. Lunday

    Humankind designs and constructs tools, furniture, vehicles, buildings and other structures with deliberate precision. And while it is relatively easy to draw objects using only lines and circles, these shapes are insufficient to represent the constructs of our world. Although it is possible to represent some simple curvatures in two dimensions with a linear projection of an arc or a circle to an ellipse, a parabola, or a hyperbola, these too are insufficient in their lack of complexity. Within these limitations, gross representation of curved lines is not sufficient for engineering purposes. In the words of Professor Thomas French, the...

  12. Area Without Integration: Make Your Own Planimeter
    (pp. 71-88)
    Robert L. Foote and Ed Sandifer

    Clay tablets from Mesopotamia and papyri from Egypt provide evidence that work with area has been part of mathematics since its early history. These Ancients knew how to find areas of squares, circles, triangles, trapezoids, and a number of other shapes for which we no longer have names.

    Like many other physical quantities, we usually measure area indirectly. That is, we measure something else, such as lengths, a radius, or angles. Then we do some calculations to find area based on appropriate formulas. The object determines the formula we use and the measurements we make.

    There are a number of...

  13. Historical Mechanisms for Drawing Curves
    (pp. 89-104)
    Daina Taimina

    If you have a collection of straight sticks that are pinned (hinged) to one another, then you can say you have a linkage like in the windshield wipers in your car or in some desk lamps. Linkages can also be robot arms. It is possible that our own arms caused people to start to think about the use of linkages.

    In this paper I will discuss how linkages and other historical mechanisms (that involve sliding in groove or rolling circles) can be used for drawing different curves and in engineering to design machine motion. This knowledge was very popular at...

  14. Learning from the Roman Land Surveyors: A Mathematical Field Exercise
    (pp. 105-114)
    Hugh McCague

    In the development and rise of civilizations and empires, land surveying has played a major role because it is crucial to the imposition and maintenance of system, order, and control of the landscape through the demarcation of properties, boundaries and roads. The key behind this system and order is always mathematics. We will focus on the Romans who had a highly developed system of land surveying as attested by their surveying manuals, and land divisions, town plans, architecture and engineering works still to be seen throughout the wide expanse of the earlier Roman Empire and Republic [1]. Indeed, some of...

  15. Equating the Sun: Geometry, Models, and Practical Computing in Greek Astronomy
    (pp. 115-124)
    James Evans

    Ancient Greek planetary theory was geometrical in spirit. Each planet was deemed to ride on a circle, or combination of circles. Greek planetary theory, conceived in this way, originated late in the third century B.C. with the work of Apollonios of Perga. At first, such theories offered only a broad explanation of planetary phenomena: each planet generally travels eastward around the zodiac, but occasionally reverses direction and travels in retrograde motion toward the west for a few weeks or months (depending on the planet), before reverting to its normal eastward motion. Apollonius’s geometrical models were under-girded by Aristotle’s philosophy of...

  16. Sundials: An Introduction to Their History, Design, and Construction
    (pp. 125-138)
    J. L. Berggren

    The sundial is one of mankind’s oldest instruments for telling the time during daylight hours. The earliest surviving dials come from Egypt. There we find dials from as early as the 15thcentury B.C. with a short vertical block (called a gnomon)² of finished stone at the end of a horizontal stone ruler marked with a scale of hours [6, p. 59]. When the device is turned so the gnomon faces the sun and casts its shadow on the ruler, the end of the shadow shows the hour of the day according to an approximate arithmetic scheme.

    The first people...

  17. Why is a Square Square and a Cube Cubical?
    (pp. 139-144)
    Amy Shell-Gellasch

    Why is the algebraic process of forming a perfect quadratic expression referred to as completing the square? With simple cutouts for an overhead projector, the geometric underpinnings as well as many aspects of quadratic equations can be exhibited quickly and effectively. This can be expanded to the cubic with the use of five wooden blocks. This chapter will show how to use these items in the classroom to give students a geometrically intuitive as well as an historical understanding of quadratic and cubic expressions.

    At several points in the undergraduate curriculum, as well as in some secondary courses, we find...

  18. The Cycloid Pendulum Clock of Christiaan Huygens
    (pp. 145-152)
    Katherine Inouye Lau and Kim Plofker

    The cycloid was an important “new curve” attracting mathematicians’ attention in the seventeenth and eighteenth centuries. It turned out to be particularly significant in the study of the behavior of objects falling under the force of gravity: the cycloid is not only the brachistochrone (path of descent in shortest time) but also the tautochrone (path of descent in equal time from any point on the path). New mathematical tools such as the calculus made it possible to apply the study of such curves, and of concepts such as their “evolutes” and “involutes”, to mechanical problems.

    The significance of these developments...

  19. Build a Brachistochrone and Captivate Your Class
    (pp. 153-162)
    V. Frederick Rickey

    Cliff Long (1931–2002) [5] was a master teacher whose office was a wonderful place to visit, for it was crammed with a wealth of teaching devices. From his earlyBug on a Band[6], to his slides and flexible model of quadratic surfaces [3,4], to his head of Abraham Lincoln made with a computer-controlled milling machine [1], and his fascination with knots [7], Cliff was always on the lookout for new ways to illustrate mathematical concepts.

    As a young faculty member I went to his office whenever I wondered how best to present some topic in class. He had...

  20. Exhibiting Mathematical Objects: Making Sense of your Department’s Material Culture
    (pp. 163-174)
    Peggy Aldrich Kidwell and Amy Ackerberg-Hastings

    Most of the chapters in this volume suggest ways to use historical mathematical instruments or replicas of those instruments to highlight mathematical or pedagogical principles within the classroom. Yet, some teachers and professors may wish to bring these objects to a wider audience. An on-line or physical exhibit is one venue for increasing public awareness of mathematics and of one’s own mathematics department. This chapter outlines fundamental principles of exhibit planning to help professors, teachers, and students identify, understand, and arrange historical objects and books that might be available to them. It suggests methods appropriate to a range of projects,...

  21. About the Authors
    (pp. 175-178)