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Lee Smith
© 1997 Copyright on this material is held by the authors.
Most dynamic software models provide sliders that control one parameter of a plotted algebraic expression [1] or the behaviors of an endless number of physical objects [4]. A knob is positioned directly over parameters of a polynomial formula in video [2]. As the knob turns the polynomial plot smoothly updates. Yet none of these formula-based graphic models allow direct manipulation of the plotted line itself!
Existing learning models have overlooked this interaction style not due to a lack of computation power or to irreversible models, but, we believe, due to a failure to appreciate published educators' observations [5]. Learners need to move back and forth from one representation to another and to view the same model in different conceptual representations. Learning happens through the experience of directly manipulating change in a model and observing coupled behaviors in other representations of that same model [3].
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| Figure 1. An illustration of a Responsive Graph interactor showing the direct manipulation of graph elements. |
Continuous interactivity mandates the use of an underlying mathematical model that is computed in real-time. The Java(TM) programming language was used to construct custom interactions within an object-oriented framework. Chosen over other WWW-based interactive plug-ins, Java(TM) supports a robust computational environment that delivers interactivity on multiple platforms via the World Wide Web.
Figure 2. An example of using Responsive Graphs to explain the concept of Amplitude Modulation (AM radio). The curve in the left graph, arrows in the middle graph, and sliders in the right panel can all be directly manipulated in a correlated manner. In this example, the user is manipulating one of the sidebands in the frequency domain. A verbal interpretation appears below the graph during the interaction. This & other models are available on the World Wide Web. http://www.hp.com/go/tminteractive
An example of an engineering concept explained using Responsive Graphs is shown in Figure 2. This mathematical model explains Amplitude Modulation, the technique by which AM radio works. This applet is part of a WWW page describing AM techniques along with an in-depth reference.
Even when graph interactions can be reverse-computed into parameter values, the mapping functions are often non-linear requiring the real-time inversion of a non-linear equation for each mouse event. Non-linear mappings modify slider behavior. While most of a slider's range may result in minuscule graph changes the last few slider-pixels may result in jumpy graph updates. Producing corresponding smooth graph and slider movements requires the design and implementation of appropriate non-linear scaling functions.
2. Blinn J., Project MATHEMATICS! Polynomials, video & workbook, California Institute of Technology, 1991.
3. Jackson S.L., Stratford S.J., Krajcik J., & Soloway E., A Learner-Centered Tool for Students Building Models, ACM Communications, April 1996, Vol. 39, No. 4.
4. Smith R., Experiences with the Alternate Reality Kit, ACM CHI+GI'87 Proceedings, 61-67, 1987, & video.
5. Soloway E., Interactive Learning Environments, ACM SIGCHI'95 Tutorial & workbook, May 1995.
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