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.
The project Visualization Tools for 3D NSF-DUE-995256 1999-2003 was a proof of concept grant that addressed the concerns about visualization in multivariable calculus. This project created a primitive set of tools to help students of science and engineering visualize concepts relating to points, surfaces, planes, curves, contours, and vectors in three dimensions. The results of this grant were very promising and correspondingly the NSF approved a follow-up grant entitled Full Realization ofVisualization Tools for 3D. This follow-up project produced a marketable version of a 3D kit and a draft of some applications that use this kit in multivariable calculus. Reviewing numerous textbooks, it was found that in both the differential and integral calculus, textbook authors commonly assume that: a) students can generalize associations between representations in 2D to associations between representations of the same mathematical concept in 3D on their own and b) explicit discussions of these representations are not necessary. For example, in the presentation of 3D derivatives, it is assumed that students will understand a 3D slope without it ever being explicitly presented. Preliminary interviews prior to this project indicated that such assumptions may be erroneous. This project continued this work of the previous two kit projects by creating materials that allow this kit to be effectively implemented in the classroom and conducting research to verify their effectiveness. In the process of creating these materials, considerable attention was given to explicitly include the aforementioned missing representations. And the evaluation paid considerable attention to the effect of the inclusion of these representations. The following table shows the results of interviews for students using the 3D kit and its accompanying manual (labelled the experimental group) and a control group. The results are divided into 4 categories: Differential topics only seen in the manual (These are slopes as the manual presents 3D slopes while traditional multivariable calculus courses do not.); Standard differential topics seen by both groups; Integral topics seen only by the experimental group; And integral topics seen by both groups. Nature of Interview Tasks Control Group Experimental Group 3D Slopesa (Seen only by the experimental group) 4% 59% Standard 3D Derivativesa (Seen by both groups 23% 58% 3D Integral Representationsb (Seen only by the experimental group) 15% 78% Standard 3D Integral Representationsb (Seen by both groups) 29% 80% a Interview conducted during fall semester, 2012, b Interview conducted during spring semester 2012 The following table shows the results on common exam questions associated with derivatives for these two groups where the common exam questions were considered appropriate for any multivariable calculus class. Common Questions Administered During the Fall 2012 Semester Control Group (n=32) Experimental Group (n=36) 1.If f is represented by the above surface (standard 3D surface not presented) Draw the cross sections x = 0 and y = 0 Identify the signs of the following derivatives where u is in the direction <-1,1> a. fx(2,2) b. fy(2,-1) c. Duf(2,2) 53% 76% 2.If f(x,y) = sin(x2y), find formulas for the following: fx(x,y) b. fy(x,y) 88% 83% x=1 x=3 y=1 3 5 y=3 7 4 3. If the function f is represented by the above table* A. Find the best approximations for fx(1,1) and fy(1,1) B. Find the formula for the tangent plane to f at the point (1,1,3) and use it to approximate f(1.1, 1.2) 37% 61% *p<0.01, **p >0.5 The following table presents the results on common exam questions for these two groups for topics associated with integration that would be appropriate for just about any multivariable calculus class. Common Questions Administered During the Spring 2012 Semester Control Group (n=68) Experimental Group (n=36) 1. Find the volume over the xy-plane and between the surfaces y =0 and z = 10 – x2- y.* 26% 53% 2. Find the volume over the plane z = 2, below the surface z = 9 – x2 and bounded by the planes y =-1 and y =4.* 47% 75% *p<0.01 Both the interviews and common exam questions would indicate that visualization of missing representations may be important in understanding mathematical concepts associated with integral and differential topics in Multivariable Calculus. Since the data provide evidence that the inclusion of the missing representations does indeed impact student understanding but requires no more time than traditional instruction, there are strong implications that instructional modifications in Multivariable Calculus should perhaps be considered. .
