By the Editors: Doris Schattschneider and James KingDynamic is the opposite of "static." Dynamic also connotes action, energy, even hype. Dynamic geometry 1 is active, exploratory geometry carried out with interactive computer software. The papers in this volume will convince you, we think, that dynamic geometry is full of action, energy, and, yes, even hype--the hype of excited individuals (students, teachers, researchers) who can't help but communicate their enthusiasm as they discuss the many implications of the software.
Mathematicians all know the power that a figure can provide--often a quick sketch or a diagram can make everything clear. We say "I see" and mean "I see and understand." In geometry, figures seem to be essential for most descriptions and proofs. Yet mathematicians also know the danger in relying on figures--inevitably, extra assumptions are made (suggested by a sketch), special cases are missed (omitted from a sketch), or absurd results are derived (from an inaccurate sketch). A classic case of the last occurence is the following oft-cited "theorem" and proof 2.
Theorem. All triangles are isosceles.
Proof: Let ABC be a triangle with l the angle bisector of A, m the perpendicular bisector of BC cutting BC at midpoint E, and D the intersection of l and m. From D, draw perpendiculars to AB and AC, cutting them at F and G, respectively. Finally, draw DB and DC. Figure 1 shows a sketch of the situation.
Triangle ADF is congruent to Triangle ADG (aas), so AF is congruent to AG and DF is congruent to DG. Triangle BDE is congruent to Triangle CDE (sas), so BD is congruent to CD. This implies Triangle BDF is congruent to Triangle CDG (hyp-leg), so FB is congruent to GC. Thus AB is congruent to AF + FB = AG + GC is congruent to AC, and so Triangle ABC is isosceles. QED
What is wrong with this proof? Absolutely nothing is wrong with the chain of reasoning--the conclusion must follow if the figure is accurate. But even though the sketch is plausible, the configuration cannot occur as drawn. D, the intersection of l and m, is never inside Triangle ABC.
This fallacious proof has been offered by some as a reason to ban the drawing of any figures! But they miss the point.
Quite to the contrary, this example offers compelling reasons to emphasize the need for acurately constructed figures, rather than quick sketches, especially when they are to be used in deductive aguments. As E. P. Northrop put it, "It shows how easily a logical argument can be swayed by what the eye sees in the figure and so emphasizes the importance of drawing a figure correctly, noting with care the relative positions of points esential to the proof. Had we at the start actually constructed--by means of rule and compass--the angle bisector and the various perpendiculars, we should have saved ourselves a good deal of trouble."3 The "proof" also underscores the need for a variety of constructions, rather than just one. (Authors who discuss this particular "proof" display a variety of inaccurate sketches of various cases as well as accurate ones obtained by construction.)
Even careful constructions can be inaccurate, as anyone who has labored with a wobbly compass or a thick lead can testify. Complicated constructions are time-consuming and difficult to make precise. The time (and tedium) required to make a variety of figures by hand to illustrate various cases of a geometric configuration is beyond what almost anyone is willing to spend.
With dynamic geometry software, the proof above would be most unlikely. With the software, an investigator could make a single accurate sketch of the geometric configuration described in which the constraints on l (angle bisector of A), m (perpendicular bisector of BC), D (intersection of l and m), F and G (feet of perpendiculars from D to AB and AC, respectively) are specified. Then the shape of Triangle ABC could be deformed at will with all the constraints being preserved in the newly created configuration. Almost instantaneously, a (seemingly infinite) family of accurately drawn configurations would be produced by manipulating a vertex of the elastic figure. The investigator would see that the only time that l and m do not intersect outside Triangle ABC is when l and m coincide (and B = F, C = G), and that is exactly the case in which Triangle ABC is isosceles. Moreover, in all but that special case, the equation in the last line of the "proof" is false. Figure 2 captures only a few of the different configurations that occur.
Today there are new generations of personal computers in the schools (and on teachers' and researchers' desktops and laps). The dynamic geometry software that has been developed takes advantage of the new hardware (with vastly increased memory, speed, and mouse interface for graphics) and builds on the early ideas. The most commonly used dynamic geometry software is The Geometer's Sketchpad, Cabri Geometry II, Geometry Inventor, and the successor to the Supposers, the Geometric SuperSupposer. These all provide means for accurate construction (and measurement) of geometric configurations (of points, segments, rays, lines, arcs, circles) and have the ability to replay a construction. What is most exciting about the new software, however, is that unconstrained parts of the configurations (arbitrary segments or points, for example, that are not dependent on any other objects) are moveable--they can literally be grabbed with a cursor (using the mouse) and be dragged or stretched--and as they move, all other objects in the configuration automatically self-adjust, preserving all dependent relationships and constraints. In doing this, a smooth sequence of instances of the configuration can be viewed, morphing contnuously from the first drawing to the final one that remains when the dragging is stopped. If the configuration is a general one (beginning with an "arbitrary" line, circle, or polygon, for instance) there can be no question that there are many (even an infinite number) of cases, and the visual evidence of invariant properties (such as concurrence of lines, collinearity of points, or ratios of certain measurements) is compelling.
In addition to construction capabilities, the programs allow certain affine transformations (translations, reflections, rotations, dilations) to act on figures or their parts. These transformations can be determined by "fixed" data or by "dynamic" data, such as a rotation by a fixed angle (e.g. 45 degrees) or by a moveable angle that is in some sketched triangle, for instance. The software allows loci to be traced as parts of figures are dragged freely or guided along a specified circle or segment. The programs also have the special feature of being able to "record" the sequence of steps that leads to a particular configuration and then be able to use that construction as a "macro" tool in producing new configurations. For example, a script or macro that produced a tangent to a circle from a point outside the circle might be employed several times in producing a complicated figure. The software has components that link synthetic geometric constructions to analytic equations, coordinate representations, and graphs. Conics are also implemented in at least one program. Information on all of the software discussed by our authors is at the back of this volume.
Although this volume is printed in a conventional manner, with illustrations in every article, most of the illustrations beg to be played with. You will read descriptions of how certain configurations behave when manipulated, but not be able to tweak the diagram on the printed page. We want you to be able to experience some of the explorations described by our authors. To make that possible, dynamic sketches that use Geometer's Sketchpad or Cabri II have been made available by several of the authors and are posted on this Web site maintained by the Mathematics Forum.
This page can also be accessed by visiting the MAA web site.
For those who do not have access to the software, demonstration copies of the software can be downloaded and used to open the sketches.
Overview of Contents
Dynamic geometry software has had a profound effect on classroom teaching wherever it has been introduced. Although not originally intended by its developers, it has also become an indispensible research tool for mathematicians and scientists. The papers in this volume give a good idea of the ways in which the software can be used, and some of the effects it can have. Authors also make clear that the software raises various questions for teaching and research, and its continuing evolution raises questions on the design of the software itself. Many of the papers here expand on presentations that were given at the annual meeting of the MAA in San Francisco in January 1995 in a special session devoted to dynamic geometry.
In an attempt to give an overview of the contents of the papers, we address the basic question "What is dynamic geometry software good for?"
Accuracy of construction. Dynamic geometry software provides an accurate constructor for any ruler-and-straightedge construction in Euclidean geometry, any configuration that can arise by applying affine transformations (isometries and dilations) to a Euclidean construction, or any locus of an object (or set of objects) that arises when part of a construction is moved along some path. Reliance on the accuracy of the software's geometric sketches and measurements is so basic that it is not discussed explicitly in any article, but it is implicit in every discussion. (Accuracy is, of course, limited to the tolerences of internal computation, screen display, allowed numerical display, and printer fidelity. And accuracy is occasionally diminished by the choice of heuristic for an algorithm or by pesky bugs that are bound to occur in such complex software.) Arnold and Bernadette Perham offer several suggestions for investigations of constructions that exhibit the collinearity of 3 points or exhibit invariance of the ratio of two segments. Tony Hampson describes how the software easily implements the techniques for perspective drawing and dramatically displays the principle of parallel lines converging at vanishing points, even as the viewpoint is changed. In every article, the sketches are produced by software; these attest to the author's reliance on it as a faithful constructor.
Visualization. As a demonstration tool in the classroom, dynamic geometry software can help students see what is meant by a general fact. Douglas Brumbaugh tells how an animated triangle got the attention of middle school students and demonstrated that the area of a triangle is determined by its base and height. Jim Morrow observes, "Dynamic geometry software can be used to aid the process of visualization in all mathematics classes, not just in the study of geometry. Students can construct, revise, and continuously vary geometric sketches. While visualization in itself is a powerful problem-solving tool, the capacity for students to make instantaneous and precise variations to their own visual representations adds a new dynamic dimension whose implications are only beginning to be understood." His several examples illustrate how the abstract concepts of variable, function, and invariance take on meaning through visualization. Al Cuoco and Paul Goldenberg observe "By allowing students to investigate continuous variation directly (without intermediary algebraic calculation), dynamic geometry environments can be used to help students build mental constructs that are useful (even prerequisite) skills for analytic thinking." Cathy Gorini shows how the software can help students in calculus see (and vary continuously) relationships between quantities in min-max applications and related-rates problems.
Exploration and Discovery. In a traditional geometry course, students are told definitions and theorems and assigned problems and proofs; they do not experience the discovery of geometric relationships, nor invent any mathematics. As Schwartz and Yerushalmy observed, "This constitutes a kind of satire on the nature of mathematical thinking and the way new mathematics is made."6 Dynamic geometry software is perfectly suited for exploration and discovery--either guided, or completely open-ended (a.k.a. research). Tim Garry observes that it allows students to "test their own mathematical ideas and conjectures in a visual, efficient and dynamic manner and--in the process--be more fully engaged in their own learning." The range of investigation and the amount of guidance provided varies greatly with the level and experience of the students (or researchers).
Our authors provide a variety of of examples of investigations for students and in many cases, discuss the process and the outcomes of the tasks. For example, Tim Garry had his students investigate the ratio of perimeter to diameter for regular polygons and also explore questions concerning circles tangent to a given circle. Zhonghong Jiang and Edwin McClintock had preservice teachers find the shortest path (with restrictions) that connects towns A and B separated by one or more rivers and also find various constant ratios that arise in a construction of a triangle within a triangle. Students using the software often discover surprising things that are not in any text, and not known to the teacher. As Kathy Boehm points out, "Any teacher using...dynamic geometry software needs to be prepared for students to ask unexpected questions. I have had to answer, 'I don't know; let me get back to you,' many times." Michael Keyton describes the process of his students moving from guided discovery of well-known properties of families of quadrilaterals to open-ended investigation of new families of quadrilaterals (with newly-invented names) proposed by the students themselves. Fadia Harik gives her students (often teachers) tasks with deliberately ambigious directions and many possible solutions, designed to prompt questions and guide the process of formulating hypotheses.
Although it may seem a cliché, dynamic geometry software empowers. Students can get hooked pursuing open-ended problems. Michael Keyton reports that some of his students have come back after graduating to tell him some new result on an unfinished investigation begun in his class. Although ninth-grade student Ryan Morgan's discovery using dynamic geometry software received national attention7, there are many more students who have experienced a similar thrill of discovery. And not only students. Douglas Hofstadter gives a riveting account here of his own pursuit of a geometric problem that tantalized him. Mike de Villiers describes how, with dynamic geometry software, he could follow the irresistible lure of "what if?" to find some new results related to known theorems.
Proof. While dynamic geometry software cannot actually produce proofs, the experimental evidence it provides produces strong conviction which can motivate the desire for proof. Dan Bennett was tantalized by a 40-year-old unsolved problem from the American Mathematical Monthly for which there was compelling computer evidence of its truth; he was determined to find a proof (and succeeded). Mike de Villiers discusses the synergy between using dynamic geometry software and proving the conjectures that spring from its use. Conviction is necessary for undertaking the (often difficult) search for a proof, he contends, and in addition, the software may even give insight into geometric behavior that can help with a proof. Subtle geometric relationships may be not at all obvious, but be revealed in experimenting with dynamic figures. Some teachers have been reluctant to use the software because they fear that visually convincing evidence will replace proofs of theorems. Our authors describe the opposite situation: when students make their own conjectures based on their explorations with the software, they know it is not enough to stop with the evidence; they need to devise a proof. Zhonghong Jiang and Edwin McClintock record for us the process by which their students arrived at proofs of their conjectures.
Transformations. Dynamic geometry software can transform figures in front of your eyes. Isometries and similarities are important examples of functions. In witnessing the action of these transformations moving and scaling figures, students see that functions are not synonymous with symbolic formulas. Doris Schattschneider points out that students can visually test fundamental properties of composition such as commutativity and inverses, as well as other abstract concepts encountered in the theory of groups. Jim Parks describes how students can identify a transformation by producing a sequence of points in its orbit.
Ross Finney's 1970 article "Dynamic Proofs of Euclidean Theorems"8 (written long before the current dynamic geometry software was even an idea) begins, "Simple observations about [affine] transformations of the plane lead to elegant proofs of unusual Euclidean theorems." Proofs such as Finney's can be brought to life with the software--to show that one figure is the transformed image of another, one only needs to perform the transformation to see if the two figures coincide. Jim King's article is devoted to proofs that rely on similarities. Mike de Villiers also gives proofs that depend on the action of transformations.
Loci. It is virtually impossible for most people to imagine a point moving in a configuration (in which other several parts may also may be moving) and be able to describe the locus of the point's path as it travels. Dynamic geometry software, with its built-in feature to trace the locus of any specified object is ideally suited to show how a locus is generated and to reveal the shape of its traced path. Except for the most simple loci (circles, as the locus of points equidistant from a fixed center, and perhaps the conics), this rich subject has been avoided in most geometry texts. In fact, most of the classic curves arose as loci, and one must go to long out-of-print books to find these nonanalytic descriptions of them. Classic locus problems and intriguing generalizations of these, as well as surprising new loci, are discussed by our authors. Hampson's students investigate conics. Cuoco and Goldenberg describe various locus experiments--they create generalized ellipses, explore a locomotion problem, and solve some optimization problems by means of loci. Heinz Schumann and David Green describe some classic and non-classic curves that are constructed as loci and also provides some useful applications of loci. John Olive describes how the Joukowski transformation (which uses inversion in a circle) produces airfoil shapes as loci.
Simulation. Dynamic geometry software's special features of dragging, animation (of points on line segments or on circles), tracing loci, and random point generation provide many opportunities to simulate a surprising variety of situations. Our authors present imaginative examples: moving robot arms (Jim Morrow), a sine tracer (Tim Garry), a 2-dimensional random walk machine (Tim Garry), and an airfoil tracer (John Olive). David Dennis and Jere Confrey simulate a mechanical linkage devised by Rene Descartes (1637) that produces points in a geometric series and then pair those points with points in an arithmetic series to produce logarithmic curves. In this simulation, the curves can "flex and bend as the arithmetic and geometric sequences are manipulated," since the manipulation effectively changes the base of the logarithm. Ben Backus describes some of the special simulations devised to help in the teaching of optics and in vision research. Susan Addington and Stuart Levy use a 3D interactive viewer, Geomview, to produce a simulation of what is seen in an Ames room, a distorted room-size box that tricks our perception. Fadia Harik uses The Physics Explorer: One Body to simulate the action of a billiard ball bouncing off the sides of a square table.
Microworlds. Dynamic geometry software produces an environment in which Euclidean geometry can be explored. Three papers in our collection, all by designers of this type of software, discuss new environments that can be created with software. Jean-Marie Laborde (Cabri-géomètre) and Nick Jackiw (The Geometer's Sketchpad) each discuss ways in which their differing software can create a "Poincaré world" of hyperbolic geometry. Jackiw also discusses how other microworlds can be created through the use of scripts which produce new "tools" that replace the Euclidean tools and allow exploration fully within a new geometry. Allen and Trilling describe a program (GéoSpécif) under development that adds a new dimension to the dynamic geometry programs currently in use. Their program takes as input the specifications for a construction and then automatically produces the construction (if it is possible); using this, students can explore to discover various dependent relationships among the parts of the construction. All these software developers acknowledge the challenges that dynamic geometry software presents--there is a delicate balance between what might be desired and what is feasible, and there is an ongoing dialogue as to what should be the design of software that can best enhance learning for all.
Notes and References
1. Here, and throughout this volume, the term "dynamic geometry" is used in its broadest sense. Although Key Curriculum Press has trademarked the phrase, our authors' use of the phrase does not refer to any particular computer software or published software support materials.
2. This fallacious proof has been in the literature for over 100 years; it is probably even older. It appears, with many "cases" discussed, in the following sources: Mathematical Recreations and Essays, by W.W. Rouse Ball and H.S.M. Coxeter, 13th ed., Dover 1987 (1st edition 1892); Riddles in Mathematics, by E.P. Northrop, Van Nostrand, 1944; Fallacies in Mathematics, by E.A. Maxwell, Cambridge [Eng.] University Press, 1959; and Mistakes in Geometric Proofs, by IA.S. Dubnov, Heath, 1963.
3. Northrop (see 1. above), page 101.
4. J. Schwartz and M. Yerushalmy, designers, The Geometric Supposers. Pleasantville, NY: Sunburst Communications, 1983-1991.
5. D. Chazen and R. Houde, How to Use Conjecturing and Microcomputers to Teach Geometry, National Council of Teachers of Mathematics, 1989.
6. J.L. Schwartz and M. Yerushalmy, "Using Microcomputers to Restore Invention to the Learning of Mathematics," Contributors to Thinking, D. Perkins and R. Nickerson, eds. Lawrence Erlbaum Associates, Hillsdale, NJ, 1986, pp. 293-298.
7. T. Watanabe, R. Hanson, and F. D. Nowosielski, "Morgan's Theorem," Mathematics Teacher, 89 (May 1996) 420-423.
8. R. Finney, "Dynamic Proofs of Euclidean Theorems," Mathematics Magazine 43 (1970) 177-185.
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