As the power of desktop computers increases, it is becoming more feasible to provide interactive software to aid students in better understanding the various mathematical theories and principles. Visualization applications have been widely adopted in the academic community to help make technical education more authentic and tangible. More than 3,500 universities around the world use such products for teaching and research in a broad range of technical disciplines.

There are numerous math software available worldwide, with more being developed each day. Let us briefly take a look at three popular visualization software products and how lecturers in NUS are using them.

Maple is a powerful mathematical problem-solving and visualization system which is used in education, research, and industry. Its principal strength is its symbolic problem solving algorithms. Unlike conventional math software, which can only work with floating-point numbers, Maple can solve problems involving formal mathematical definitions and return answers as mathematical objects.

Its mathematical engine has an extensive library which covers:

• Calculus, precalculus
• Differential equations (ODEs, PDEs)
• Linear algebra and high performance matrix computation
• Engineering mathematics including transforms
  (Laplace, Fourier, Z, FFT)
• Pure mathematics including discrete mathematics, number theory and group theory
• Over 3,500 functions

Dr. Victor Tan, from the Department of Mathematics, uses Maple in Advanced Calculus modules where multivariable concepts are introduced. His students attend computer laboratory sessions where all the PCs are installed with the Maple software. Students learn how to use Maple commands to plot a variety of module-related graphs. Particularly, students construct three dimensional objects such as “surfaces” and “space curves” which are difficult to visualize and plot by hand.

Maple also enables students to rotate the 3D graphics and view the plotted objects from different angles. Two or more graphs can be plotted on the same axes for comparison, with different colours distinguishing the graphs. These functions assist the students in performing calculations and in interactively visualizing concepts. Maple usage helps students to understand and internalize mathematical theories.

MATLAB is a high-level technical computing language and interactive environment for algorithm development, data visualization, data analysis and numerical computation. Using MATLAB, users can solve technical computing problems faster than traditional programming languages, such as C, C++, and Fortran. It can be used in a wide range of applications, including signal and image processing, communications, control design, test and measurement, financial modeling and analysis, and computational biology.

Some of the key features of MATLAB are:

• High-level language for technical computing
• Development environment for managing code, files, and data
• Interactive tools for iterative exploration, design, and problem solving
• Mathematical functions for linear algebra, statistics, Fourier analysis, filtering, optimization, and numerical integration
• 2-D and 3-D graphics functions for visualizing data
• Tools for building custom graphical user interfaces
• Functions for integrating MATLAB-based algorithms with external applications and languages, such as C, C++, Fortran, Java, COM, and Microsoft Excel

In the Department of Chemical and Biomolecular Engineering, Dr. Lakshminarayanan Samavedham’'s students use MATLAB to solve nonlinear ordinary differential equations and partial differential equations. The software is useful for solving linear and nonlinear algebraic equations. These mathematical concepts are applied in process simulation, control and process data analysis.

With the 2-D and 3-D plotting capabilities of MATLAB, students are able to see the response surfaces, contour plots, etc. They use FEMLAB, an application built on MATLAB, to understand problems involving multiphysics such as chemical or biological reactions plus electrokinetic flow. Dr. Laksh and his students also simulate 3-D models of systems, visualizing 2-D or 3-D graphical results, including the systems’ evolution over time.

Another interesting product is LiveMath. This is a computer algebra system that allows interactive and graphical experimentation via the web browser by using a free plugin. The product suite comprises of LiveMath Maker, LiveMath Plugin, LiveMath Viewer and MathEQ Equation Editor. Using the LiveMath Maker, the lecturer can create “notebooks” which can then be embedded in any web page. Students can access these LiveMath notebooks through the LiveMath Plugin.

Beyond seeing static math symbols on a screen, the plugin enables the student to interact with algebra, investigate graphs, and explore live mathematics. They can modify the input expressions and watch the computations change without the need for any math engine running on the server side.

The key features of this product are:

• Looks like Math - WYSIWYG - What You See Is What You Get
• "Drag and Drop" intuitive actions to complete math operations
• "Show Steps" feature that allows you to see and work through intermediate steps
• Not programming language-based
• Convenient symbol entry using a mouse click palette or the keyboard
• Object-oriented, powerful 2-D and 3-D graphing with easy-to-use investigatory tools
• Publish to the web for free, interactive viewing with most web browsers

There are many other useful tools and utilities listed in the references below which are available either as Shareware or Freeware. We hope that the usage of mathematical visualization software will enhance your teaching.

References
1. List of Mathematical Software
2. Guide to Available Mathematical Software
3. Maple
4. MATLAB
5. LiveMath

For more information on Mathematics software and other courseware, contact Mr .