A Multivariable Calculus Manual to Accompany 3D Manipulatives

数学科学(21)：这个项目分两个阶段，正在生产和测试一种廉价的工具包，以及它的使用附带材料，它允许学生有效地将点、矢量、直线、平面和表面可视化为3-D。许多学生目前只在自然维度中可视化2D主题，然后必须将其概括到n维。数学软件包，如Maple、Maple、MathCAD和MatLab，为希望将三维概念可视化的学生和教授提供了巨大的帮助。然而，在许多概念中，计算机屏幕的二维特性可能会限制这些程序包的有效性，特别是当学生的几何背景较弱时。例如，方向导数需要与XY平面关联的给定方向上的曲面的切线。在三维中，曲面可以放置在XY平面上，可以指示XY平面上的方向，并且概念可以很容易地可视化。然而，对于学生来说，在2D计算机屏幕上可视化精确的方向及其关联的切线通常是困难的。相应地，对于这一概念和许多其他概念，更有效的教学方法是使用物理3D操作。这使得概念可以在真实的三维空间中可视化和激发动力。事实证明，操控往往比将三维投影到二维计算机屏幕上更有效，特别是在学生第一次接触多变量函数的时候。有了这个工具包，学生可以将2D和3D主题在他们的自然维度上可视化，然后再概括到n维。这个项目的初始阶段已经产生了一套基本的操纵器，它们已经成功地进行了测试。这个项目的新阶段是制作材料，使学生和教师都能有效地使用操纵器。

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.