INTERACTIVE INTERNET-BASED MULTIVARIABLE CALCULUS

June 26-29, 2005
Brown University
Providence, RI


This workshop is designed for college and university teachers who are interested in using software for interactive communication and computer-generated demonstrations in their undergraduate mathematics classes. Here is a presentation highlighting several aspects of the course. Although we will concentrate on multivariable calculus as our main topic, the concepts and methods from this project can be transferred to other courses such as linear algebra, introductory differential geometry of curves and surfaces, and geometry for liberal arts students and potential elementary and secondary school teachers. No prior knowledge of computer programming is necessary for students to use our software, and none will be assumed for the participants in the workshop.

Participants completing the workshop will:

The workshop will meet for two sessions each morning and two each afternoon on Monday and Tuesday, with two final sessions on Wednesday morning. Each session will feature a presentation by the workshop organizer, followed by opportunities to work online with the concepts, aided by student assistants. In each case, this will be followed by discussion of the pedagogical aspects of the topic at hand.

CASE STUDY

At St. Louis University, Professor Michael May has created localized implementations of Professor Banchoff's visualization applet.

Professor May's Applets for Multivariable Calculus, which include Applet Worksheets written by Dr. Steve Harris, are fine examples of how this software can be customized for specialized lessons:

  • The Cross Section applet looks at how the graph of a 2 variable function is built up from the cross graphs of the cross sections, which are each graphs of single variable functions.

  • The Level Curve applet builds up the graph from its level sets.  It connects the contours of a three dimensional surface with the contour graph obtained by looking down the z-axis.  (Currently, this one has bugs with Safari.)

  • The DirectionalSlice applet lets you take a cross section of a surface that includes a given line segment on the x-y plane.

  • The Continuity applet looks at a visualization of the definition of continuity as it applies to functions of two variables.  This is a generalization of an applet that looks at single variable continuity.

  • The applet Tangent Planes, connects the tangent lines of the x and y cross sections at a point to the tangent plane to the surface at that point.

  • The Differentiability applet explores the definition of differentiability as an extension of the definition of continuity.  This is a generalization of an applet that looks at single variable differentiability.

Prior to attending the workshop, the participants will have the opportunity to complete a short tutorial (provided by the presenters) on the use of the Java applets for the course. During the workshop, participants will complete "exercises", submit them online, and receive comments, following the procedure used in an undergraduate class. This experience will form the basis of discussion on the mechanical and pedagogical aspects of this particular Internet-based approach to teaching and learning.

The webpage for the workshop will continue to be available to the participants after the workshop, for further discussion and for further sharing of materials for multivariable calculus.

 
Click the image above to try interactive Java applets

 

 
About Brown University
Directions to Brown University
Directions to Brown Math Department