Computer Environments

Students and teachers can access virtual manipulatives on many websites, including the NCTM Illuminations and National Library of Virtual Manipulatives sites. Virtual manipulatives are simulations of actual concrete manipulatives or representations such as graphs, created using JAVA, a robust, but neutral, dynamic computer language (Heath, 2002). JAVA “applets” are immediately accessible to the student; these tools don’t require special keystrokes or syntax like other software. They have applications from kindergarten through graduate-level mathematics. Applets can also be created by using the directions found on many websites, such as the English/Japanese site “Manipula Math with JAVA.”

Another powerful learning tool for mathematics are learning objects, modular digital resources that include various forms of software such as simulations, calculators, animations, tutorials, video clips, graphs, and assessments (Wiley, 2001). Learning objects (sometimes called “widgets”) have the potential to provide individualized learning experiences with teacher-selected instructional objectives and can be used with any content area or level. The Wisconsin Online Resource Center includes the following criteria for quality learning objects: small (2 to 15 minutes), independent, stored in a searchable data base, based on a clear instructional strategy, interactive, reusable, and groupable. An example of a learning object on the Washington State University website is the Dollars and Cents Widget for practicing making change up to $5.00, estimating cash back, and identifying exact amounts for purchases (Miller, Brown, & Robinson, 2002).

Simulation software creates microworlds that can be manipulated by the user, who is able to view the consequences of manipulations immediately. Simulations can range from real-world applications (e.g., flight simulator, electric motor, whitewater rafting) to gamelike fantasy worlds. The earliest mathematics simulation tool was LOGO, built into microcomputers in the mid 1980s (Papert, 1980). LOGO is a programming language for moving objects (in early versions, an abstract “turtle”; later, “robots”) around space and analyzing spatial relationships and properties. Available today through commercial sources or online freeware in many versions, LOGO activities offer countless mathematics applications (Logo Foundation, 1991). Another microworld for mathematics is Blocks Microworld (Thompson, 1992), based on Dienes’ blocks. Children can create and explore their own algorithms or follow formal routines for addition and subtraction. Other microworlds include Conservation of Area and its Measurement (Kordaki & Potari, 2002) and Mathwright 32 Author (Bluejay Lispware).

A final note is warranted about the use of technology to support mathematics instruction. There is no doubt technology is powerful and can assist student learning. However, teachers must make decisions about how it is used. Goldenberg offered six principles “for thinking about technology use in math classrooms” (2000, p. 2):

• choose technologies that further existing learning goals (rather than create artificial goals so technology can be used)
• allow calculator use when computation gets in the way of instruction’s purpose
• consider when the analysis (process) of the problem is as important as the answer
• be cautious when technology might be replacing the student’s development of important thought processes
• be aware of the effects of removing content from the curriculum just because the technology can substitute (e.g., square roots, trigometric functions)
• encourage students to learn a few tools well rather than attempting to expose students only superficially to many tools

Like their students, teachers must have time to learn new technologies and their applications for mathematics. Calculator and computer technologies require significant resources and support services from the school and district level.