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!d!7dDynamic Manipulation of Mathematical Objects
DATE: June 25, 1998
Authors: William Finzer and Nicholas JackiwKey Curriculum Press1150 65th StreetEmeryville, CA 94608
Note: An electronic version of this paper, with interactive illustrations, is available for Java-compatible web browsers at .
What is Dynamic Manipulation?
On the computer screen sits an equilateral triangle with three constructed medians. The medians concur in a single point. Do medians always concur in a point, or is this only true for equilateral triangles? You click and drag a vertex with the computer mouse. As you drag, the triangle changes shape and orientationbut the medians continue to concur, no matter how you deform the triangle.
We believe that technology can best foster mathematical inquiry and learning through dynamic manipulation experiments such as this, in which students explore, experiment with, and build mathematical knowledge interactively. Dynamic manipulation environments are characterized by three attributes:
Manipulation is direct. You point at the triangles vertex and you drag it. The cognitive distance between what is on the screen and the mathematics behind it is minimal. You do not feel inclined to say, Im moving the mouse, which drags this small circle on the screen, which changes the coordinates of the triangles vertex. You say, Im dragging the triangles vertex.
Motion is continuous. Change takes place during the drag. The mathematical objects represented on the screen stay coherent and whole at all times. As the triangles vertex moves from point A to point B, you can see all the intermediate states.
The environment is immersive. Your experience is that you are involved with the objects you are manipulatingsurrounded by them, exploring them, playing with them. The interface is minimally intrusive so that your focus is on how to accomplish your mathematical goals, not on how to drive the technology.
Computer technology inevitably changes what happens in the classroom. The ease with which learners can now dynamically manipulate mathematical objects changes the path along which students progress toward mathematical power. In this paper, working from examples, we propose experiences that students should have, and insights that should come from these experiences.
Dynamic Manipulation in the Technology Landscape
To varying degrees, dynamic manipulation ideas, concepts, and ways of learning may find form in many technological mediaas well as in off-line activities involving physical manipulatives and thought experiments. However, at the end of the 1990s, the fully continuous, direct, and immersive manipulation environments described here remain available to students and educators only as software packages for desktop computers. We wish to be absolutely clear that we do not regard the current generation of graphing calculators, CD-ROM, or the Internet as capable of providing fully-realized dynamic manipulation environments. For the most part, calculators user interfaces evolve from the tradition of teletype (line printing) devices and plotters, rather than from the graphical user interfaces, multimodal input devices, and direct manipulation interface paradigms from which dynamic manipulation potential emerges. Calculators provide almost no opportunities for dynamic interaction: the motion, if any, is almost never continuous, and their small, monochrome screens do not immerse the user in mathematics. Nor is the dynamic manipulation approach frequently found in multimedia CD-ROM-type applications. While able to present compelling animations and other pre-formed content, the CD-ROM-type medium is poorly suited to the high level of direct user interaction stipulated by a dynamic manipulation pedagogy. Finally, though the Internet and World-Wide-Web often afford a much greater degree of interactivity (through following hypertextual links), this interactivity is rarely structured according to a model of coherent and continuous conceptual development. (Only recently have Java applets, embedded in web pages, begun to make close and structured interactivity possible within net-based media.) There are many compelling reasons to use graphing calculators, CD-ROMs, and the Internet in the teaching of mathematics when one considers issues of portability and price on the one hand, and information density, diversity, and accessibility on the other. But these media cannot yet compete with conventional desktop computers in realizing the dynamic manipulation potential of technology.
The Authors Bias
We acknowledge that the two of us, having worked with dynamic manipulation software for the past seven and ten years respectively, as creators, designers, users, and advocates, have a considerable investment in the continued and growing success of such software in mathematics classrooms. In this paper, we draw on our experience to propose standards based on our firm belief that dynamic manipulation of mathematical objects using software has both unexplored potential and already-demonstrated utility in the learning and teaching of mathematics. Examples in this paper are based on The Geometers Sketchpad (Jackiw, 1991) for geometric visualization; Fathom (Finzer et al., 1998) for statistical exploration and data analysis; and NuCalc (Avitzur, 1994) for graphing and symbol manipulation. JavaSketchpad (Jackiw, 1997) was used to create the interactive illustrations for the electronic version of this paper.
Another admission: We never had as much fun learning and doing mathematics before we could play with mathematical objects, and build working mathematical models, on the computer screen. Maybe we cant force kids to enjoy math, but we can try!
Dynamic Manipulation and Mathematical Learning
Reasoning about Continuity
Across the curriculum, students learning mathematics confront dichotomies of discrete and continuous phenomena, of constancy and change. Yet the tools we give them for thinking about these opposing ideas rarely bridge the span between them. We show them a drawing on the blackboard; but this offers only a single examplea case studyof a mathematical idea. In it, one might see that some condition is true, but rarely how or why it came to be so, or whenperhapsit might no longer obtain. We then deliver a symbolic expression that generalizes all possible related examples. But where in this fixed symbolism can one find the rich mathematical diversity it encodes?
Dynamic manipulation software bridges this gap. As students vary a parameter directly, they seeand more, they generatea near-infinite number of continuously-related case examples. Their figure is no longer merely illustrative; through dynamic manipulation, it approaches the general case.
ExampleThe Orthocenter of a Triangle
Given the figure at left, a student might observe that the altitudes of a triangle concur in a point, and that this point, the orthocenter, is located inside the triangle. Other examples will demolish this conjecture, and show that sometimes the orthocenter must fall outside the triangle. But will these examples reveal when the orthocenter falls inside its triangle, and when outside? Or why?
Experimenting in a dynamic manipulation environment, a student observes (by dragging) that each of the three vertices contribute equally to the location of the orthocenter. Dragging one vertex, she finds it possible to push the orthocenter outside the triangle, and that when it leaves the triangle, the orthocenter always exits through a vertex. Investigating each vertex at the moment the orthocenter passes through it, she realizes that each exit or re-entry of the orthocenter occurs as the vertex angle passes through 90 degrees: once the angle is greater, the orthocenter must fall outside the triangle. Thus, only acute triangles have interior orthocenters; and obtuse triangles must have exterior orthocenters. Moreover, the case in which the orthocenter coincides with a vertexthe right triangleemerges not as some third and separate mythical entity of the geometry curriculum, but instead as the natural border between obtuse and acute, where opposing tensions are held in equilibrium.
Proposed Standard
Starting in about grade 6, students should experience problems and situations in which continuity between one state and another allows them to reason about intermediate states. Since dynamic manipulation software helps students to create and work with such problems, students should have some of these experiences using such software at each grade level.
Linkages -> Dependencies -> Causality ->Implication
As you drag one object on the screen, the objects that are linked to it change as well. Sometimes you think of these linkages as dependencies: The size of this residual depends on the location of this point. Sometimes you see causality: Increasing the exponent causes the curve to go up more sharply. Or you describe an implication: As this vertex angle becomes 90 degrees, the side opposite has to get closer to being a diameter of the circumcircle. These insights characterize the heart of mathematics as the study of relationships; and dynamic manipulation provides learners with tools for experiencing and investigating such relationships.
ExampleLeast Squares Regression
An early prototype of Fathom, a computer learning environment for data analysis and statistics, illustrates how dynamic manipulation can reveal the workings of the algorithm for computing a least squares regression line.
Each small square is constructed from a residual, the difference between the value predicted by the fitted line and the actual data value. The large squares area is the sum of the area of the small squares. As you drag the line you see the squares change size and you can adjust the line for a minimum sum of squares. We are convinced, even without a controlled experiment, that playing with this model demystifies how this algorithm works, suggests other algorithms for fitting a line, and provides insight into how an outlier can have a great deal of influence over the slope of the fitted line. These discoveries contextualize mathematical knowledge, helping us understand how an analysis works, and when and why we might wish to apply it.
Proposed Standard
Given a dynamic mathematical model, students should be able to discover and describe in mathematical language the relationships that exist between the models parts.
Building Something that Works
Mathematics in the classroom is too often an end in itself. There must be times when students see math as a tool to be used for some desired goal. Applications-type problems frequently postulate some external context in which the mathematics currently under study is useful. Constructivist dynamic manipulation environments offer another, more immediate, context. In them, students can solve problems and address challenges by building interactive, manipulable, mathematical models. Mathematics drives the visual display of these environments, and learners, ourselves included, will go to extraordinary lengths to understand and use math to cause a (simulated) frogs tongue to wiggle just right, a bicycle wheel to turn smoothly, or a Luxo lamp to extend and swivel with properly-constrained mechanical motion. These interactive model-building challenges are often more satisfying than traditional applications problems, because we are not simply told that our solution is physically correct (as we are told that the motion of a projectile follows a parabola), but we can see that it is so. (In such environments, we also quickly visualize the limitations of our models; and frequently go on to improve them in an iterative fashion.)
ExampleSymmetry Animations
The model of a frog at left was developed by Alexis McClean (while a student at Mountain Brook Junior High) during an exploration of bilateral symmetry. The frog and flies around it were assembled from compass-and-straightedge constructions which parameterize the orientation of parts of the animalslimbs, wings, eyebrowsthrough the location of key points (not shown) in the construction. When software is used to animate these points along circular or linear trajectories, the frog fidgets in anticipation as the flies flit about its head.
Proposed Standard
Students should have experiences in which they use mathematics to build working, manipulable models with both physical materials and computer tools.
The Behavior of Mathematical Objects
Consider the following two descriptions of an isosceles triangle.
Two sides are equal in length and the two angles opposite them have equal angle measure.
No matter how you transform any part of it, the two sides adjust to remain equal in length and the two angles equal in measure.
The first lists two properties of the triangle while the second describes its behavior. The first is declarative and static; the second, imperative and dynamic. The difference is one of emphasisa particular isosceles triangle representing all isosceles triangles in the first, versus a triangle that can become all isosceles triangles in the second. In learning and problem-solving contexts, the imperative representation of knowledge is richer in generative potential: where a static list of properties lends itself only to recapitulation, a model of dynamic behavior leads to predication and extrapolation. We believe that if you could see inside the minds of mathematics practitioners, you would find imperative, active knowledge that models behaviors of objects moving and changing in response to stimuli.
ExampleDrag Algebra
Lets take an unlikely example from symbolic algebra. NuCalc is a remarkable algebra and graphing program that has been bundled on over six million Power Macintoshes. In the illustration at right, the user is dragging x from one side of an equation to the other. As x moves inside the squared term, it acquires a square root, which disappears as it moves out again on the right. As x moves across the equals sign, it drops into the denominator. The x has a behavioror, the equation as a whole has a behaviorin response to the constraint that the two sides remain equal.
ExampleTaylor Series
Consider another example of mathematical behavior, again using NuCalc. Each of the series of graphs at right shows the function EMBED Unknownplotted against the sum of the first n terms of the Taylor series, EMBED Unknown, where n varies from 0 to 4.
NuCalc provides a slider with which you can vary the value of n. As you drag the slider, you see the Taylor series plot flip its tails and smooth itself along the sine curve. The opaque algebraic expression becomes a transparent and, eventually, friendly object through graphing and dynamic manipulation.
Proposed Standard
In addition to listing properties of mathematical entities, students should be encouraged to think about and describe their behaviors. Throughout their careers, students should have experience with computer technology that, through dynamic manipulation, encourages this view of mathematics.
Problem Posing and Generalization
An important part of mathematical thinking involves considering limiting situations, going beyond the initially imposed constraints, and generalizing to broader domains. Dynamic manipulation lends itself, in very seductive ways, to fostering these attitudes towards mathematics. As you drag things around, you often stumble onto unexpected treasures. Another example will help.
ExampleConic Conundrum
The top illustration at right shows a Geometers Sketchpad construction of the set of perpendicular bisectors to a given segment as one end of that segment moves around a circle. The envelope of these lines is a hyperbola.
As you drag the right end of the line segment, the envelope changes. Inevitably you drag the point inside the circle, getting something similar to the middle illustration, in which the envelope appears to be an ellipse. When your dragged point reaches the center, the ellipse becomes a circle. Hyperbola, ellipse, circlewhats missing? Aha, you think, dragging the point onto the circle will give me a parabola! Wrong. Now you have to explain how these three envelopes are related and figure out how to get the missing conic.
Proposed Standard
Students should be given ample opportunity to make mathematical discoveries, to propose generalizations, to ask what if questions, and to engage in open-ended investigations. Dynamic manipulation software, as it provides an environment in which serendipity and the unexpected abound, should be used throughout a students mathematical career to encourage such behavior.
Summary
Dynamic manipulation of mathematical objects provides a way of learning and understanding mathematics that has already proven itself in the classroom. In thinking about the next round of mathematics standards, we recommend that there be explicit mention of use of dynamic manipulation technologies, and we have proposed example standards that do so.
This material is based in part upon work supported by the National Science Foundation under awards numbered III-9400091 and DMI-9623018. Any opinions, findings, and conclusions or recommendations expressed in this paper are those of the authors and do not necessarily reflect the views of the National Science Foundation.
Dynamic Manipulation of Mathematical Objects June 25,1998
Finzer and Jackiw PAGE 2
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