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What Is Dynamic Manipulation?

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On your 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 orientation--but 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:

  1. Manipulation is direct. You point at the triangle's 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, "I'm moving the mouse, which drags this small circle on the screen, which changes the coordinates of the triangle's vertex." You say, "I'm dragging the triangle's vertex."
  2. 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 triangle's vertex moves from point A to point B, you can see all the intermediate states.
  3. The environment is immersive. Your experience is that you are involved with the objects you are manipulating--surrounded 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 media--as well as in off-line activities involving physical manipulatives and thought experiments. However, at the end of the 1990's, 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 Geometer's 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 as when we could play with mathematical objects, and build working mathematical models, on the computer screen. Maybe we can't force kids to enjoy math, but we can try!

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