Mathematical Visualization Toolkit is a set of plotting and solving applets that would be useful to students of calculus and differential equations and researchers in applied mathematics. Please see the author's MERLOT-Carnegie snapshot at http://www.cfkeep.org/html/snapshot.php?id=83229305765805 for a brief description of the history and development of MVT.
The MVT applets are designed to help the user visualize the functions and fields present in calculus-related problems.
Target Student Population:
Educators and students at and above the calculus level.
Prerequisite Knowledge or Skills:
This varies from one applet to the next. Some require knowledge of functions, other of single-variable calculus, multi-variable calculus, or ODE's.
Type of Material:
At least a browser equipped with the Java 1.2 (not 1.3) plugin. The reviewers could only rarely get the applets to run in Netscape browsers. However, they almost always worked well in Internet Explorer browsers.
Evaluation and Observation
MVT contains several modules: Function Plotter, ODE Integrator, Parametric Plotter, Contour Plotter, Implicit Plotter, Vector Field Plotter, Gradient Field Plotter, ODE Vector Field Plotter, Numerical Integrator, and Numerical Rootfinder. All are excellent, fast implementations that handle almost all continuous situations (and some discontinuous ones) very well. The user has good control over the appearance of the generated graphs (domain, range, color, variable names, etc.). Reasonably informative error messages appear at appropriate times. In spite of the concerns mentioned below, this is an excellent and very useful package.
The Function Plotter module handles discontinuities well, but other modules (like Parametric Plotter) are less successful and produce incorrect results that could confuse a student. (Try, for example, x(t)=abs(t)/t, y(t)=cos(t), t in the range [-2*pi,2*pi]). Implicit Plotter misses branches of some graphs (such as x^2-y^2=0). Again, an experienced mathematician would likely spot this, but a student could become confused. An instructor could turn these faults into learning experiences by asking students to find the errors and try to explain why they occur. Given all the other fine graphical modules, it would have been nice to see some graphical representations of the processes involved in the Numerical Integrator and Rootfinder modules.
Potential Effectiveness as a Teaching Tool
MVT is a truly excellent teaching tool. It would be valuable throughout the standard calculus/ODE sequence, and in a wide range of science and engineering classes. The modules are easy to use and have a standard interface, so the software does not get in the way of the mathematics. Students could quickly and efficiently explore a wide range of situations,
aiding in the development of important graphical interpretation and reasoning skills. Features such as the ability to overlay consecutive plots or examine numerical output open up learning opportunities in a wide variety of situations. Even the flaws can be turned into learning opportunities by asking students to find errors and try to explain why they occur.
Because the plots produced in some situations are inaccurate, an instructor should warn students of potential difficulties before assigning exercises using these tools.
Ease of Use for Both Students and Faculty
The modules feature a unified appearance and user interface, serving to reduce confusion. The controls are easy to use, clearly labeled, and self-explanatory in function. For those who need explanation, there is an excellent set of online documentation pages. Available options give the user good control over the appearance of the graphical output of the modules. For entirely new users, a list of standard syntax and available functions is included in the Help Pages.
Controls on the plot range are placed in the Options Menu instead of on the main module screen. If students aren't aware of this, the default range choice on some graphs will render them either confusing or useless. The comprehensive MVT online documentation is well done, but would be easier to use if broken into sections and integrated into the currently non-functioning Help Menu indicators in the individual modules.