Macromedia Flash as a Tool for Mathematics Teaching and Learning

School Science and Mathematics, Feb 2004 by Garofalo, Joe, Summers, Tim

Macromedia Flash is a powerful and robust development tool. Because of its graphical, sound, and animation capabilities (and ubiquitous browserplug-in), major companies employ it in their website development (see www.nike.com or www.espn.com). These same features also make Flash a valuable environment for building multi-representational "movies" (a Flash term analogous to "applet" in Java) to illustrate mathematical ideas or simulate mathematical situations.

The National Council of Teachers of Mathematics' (NCTM, 2000) Principles and Standards for School Mathematics includes the Technology Principle as one of the six principles characterizing high-quality mathematics education. The Technology Principle states, "Technology is essential in teaching and learning mathematics; it influences the mathematics that is taught and enhances students' learning."

In making a case for incorporating technology in mathematics teaching, the NCTM stated,

The graphic power of technological tools affords access to visual models that are powerful but that many students are unable or unwilling to generate independently... technology enriches the range and quality of investigations by providing a means of viewing mathematical ideas from multiple perspectives (p. 25)

In this column we present brief descriptions of Flash movies we have created to support mathematics learning and investigations through visualization, as envisioned in the Principles and Standards.

Flash as an Environment for Mathematics Development

Flash works where elementary graphics and elementary programming mesh. It has many capabilities that make it a useful environment for mathematics movie development. General Flash features include graphical flexibility for displaying discrete arrangements or representations in various ways, an accessible scripting language (ActionScriptj, and animation and sound capabilities.

Flash movies can incorporate displayed equations, graphs constructed from these equations, tables and spreadsheet-like tools, imported images, and sounds. These capabilities allow developers to include multiple representations of functions and simulations of realistic situations, with graphics and elements tailored to specific contextual, conceptual, and personal goals. Because of these capabilities, Flash has the potential - on its own, and not as "educational" software - both to utilize and demonstrate important mathematical concepts. It also has the potential to encourage virtually endless constructive playing.

Learning to Develop in Flash

Since it is used for major projects and for producing mildly magical animation effects, you might assume Flash to be difficult to learn. On the contrary, starting out with Flash is surprisingly easy. After you learn how to operate in the drawing stage, which is similar to that of most drawing and photo programs, you can quickly make things move.

You can save drawings onstage as discrete objects in a "library" of objects. From the library, you can remove as many copies of an object as needed. Each obj ect in the library has its own origin and defines its own "space." These spaces can be compressed, expanded, translated, rotated, and animated as the user likes.

The library, even without an ounce of code, contains a very important programming idea : Items built on a computer can (and should) be shared. If one student makes an animation in his library, another can easily borrow itto her library and use it in her own movies. This pattern, with portable "objects" in the library and "instances" on stage, traces the basic shape of Object-Oriented Programming (OOP).

Once you are comfortable drawing items, saving them to the library, and animating them, you will be able to begin making things happen with ActionScript, an obj ect-oriented language containing all the major operations of any computer language. Most commands can be double-clicked from a list of commands in the "Actions" window.

The following are examples of Flash movies we have created to help students visualize and understand difficult mathematical ideas. These movies can be freely downloaded from our website at http:// www.teacherlink.org/content/math/interactive/flash/ home.html

Example 1. Kids and Cookies

We developed Kids and Cookies to simulate a fairsharing situation (see Figure 1). A parent or teacher can select a number of kids and a number of cookies and ask children to share the cookies fairly. Cookies are dragged to the cutting board and cut, and then cookie pieces are distributed by dragging. Cookies can also be reconstituted. The movie includes on/off options for directions to be read aloud, music to accompany movements, fraction names to be read as children drag them, fraction symbols to be displayed, and totals to be tallied.

Example 2. Linear Programming

Linear programming is a three dimensional topic traditionally taught in two dimensions. Many students (and teachers) cannot explain the rationale behind the usual algorithm. Our Linear Programming movie allows users to input up to four constraint functions and a function to be optimized (see Figure 2). It then graphs constraint functions and displays the feasibility region in two dimensions, like you would see in a textbook. As the cursor is moved over the feasibilityregion, the x and y coordinates are displayed, along with the value of the objective function. Such two-dimensional representations, while helpful, do not necessarily help students understand the linear programming algorithm. However, once the feasibility region and objective functions are displayed in three dimensions, it is easier for students to understand why the algorithm only calls for testing corner points.