Our construction of each tool required some additional work beyond the development via a Sketchpad 4.01 document. Due to *JavaSketchpad's* limited command structure, particular constructions available in Sketchpad were not replicable in *JavaSketchpad*. For instance, when attempting to use the coordinates of a point as part of a computation, *JavaSketchpad* would permit one to obtain the coordinates but would not separate the abscissa and ordinate. Consequently, we used trigonometric relationships and relating lengths to the unit length to compute the needed values.

Other structures unsupported by *JavaSketchpad* are arcs, angle bisectors, dashed lines, points on polygonal interiors, to name a few. In addition, the *Geometer's Sketchpad* to HTML converter does not provide seamless conversion but attempts to convert the elements of a sketch that are permissible. Generally, the converter identifies sequences of geometric constructions that are not supported either because the construction utilizes an illegal object construction or a parameter not understood by the converter. This in turn required us to reconsider the underlying construction and develop it in a way to avoid the conflict.

Perhaps one of the most difficult limitations of JavaSketchpad is its inherent 2D nature. Although it would be beneficial to have students interact with the geometry of linear algebra concepts in higher dimensions, creating appropriate tools to aid those investigations using *Geometer's Sketchpad* is problematic. We have attempted to overcome these limitations, and this continues to be a work in progress.

To understand the visual limitation of rendering 3D on a 2D presentation device, we developed a **Transformer3D** tool to try to address some of the conclusions of Sierpinska et al. (1999) and Sierpinska (2000), who indicated that students who interact with linear algebra concepts from solely a two-dimensional perspective via Cabri geometry might misgeneralize concepts. Consequently, we thought appropriate and effective pedagogical activities are needed that integrate a variety of technological tools, such as web-based sketches with multiple dimensions, MATLAB activities, Maple or Mathematica investigations, and by-hand computations, thereby allowing students to investigate relationships in many dimensions and to develop their personal and collective understandings.

Next or or page: **14.** Transformer3D Tool and Sample Activity