A Multivariable Calculus Manual to Accompany 3D Manipulatives

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Mathematical Sciences (21): This project, in two phases, is producing and testing an inexpensive kit, along with accompanying materials for its use, which allows students to effectively visualize points, vectors, lines, planes, and surfaces in 3-D. Many students currently visualize only 2D topics in their natural dimension and then must generalize to n dimensions. Mathematical software packages such as Mathematica, Maple, MathCAD, and Matlab have provided enormous aid to students and professors wishing to visualize concepts in three dimensions. However, there are many concepts where the two dimensional nature of a computer screen can limit the effectiveness of these packages, particularly if students have a weak geometrical background. For example, directional derivatives require the tangent line to a surface in a given direction associated with the xy plane. In three dimensions, a surface can be placed over the xy plane, the direction on the xy plane can be indicated, and the concept can be visualized quite easily. However, visualizing a precise direction and its associated tangent line on a 2D computer screen is often difficult for students. Correspondingly, a more effective pedagogical approach for this and many other concepts is to use physical 3D manipulatives. These allow visualization and motivation of concepts in a real three dimensional space. Manipulatives often prove more effective than a projection of three dimensions onto a two dimensional computer screen, particularly when students are first being introduced to multivariable functions. With this kit, students can visualize both 2D and 3D topics in their natural dimension before generalizing to n dimensions. The initial phase of this project has produced the basic set of manipulatives and they have been successfully tested. This new phase of the project is producing the materials which allows both students and teachers to use the manipulatives effectively.