Transformative Applications in Education/GeoGebra

Introduction


GeoGebra is an outstanding technology tool that improves mathematics education. It is a quality 21st century transformative application that enhances and supplements the traditionally memorization-based study of geometry. Through simulations, students create concrete representations of geometric theorems instead of abstractly configuring images in the mind.

Relevance to 7-12 Education: The Traditional Geometry Class
The conventional classroom study of geometry is based on students' innate trust in their teachers and textbooks which, they assume provide mathematically sound information. Traditionally, teachers introduce theorems to their students who apply these memorized theorems in proving or disproving given geometric statements.

Relevance to 7-12 Education: The Transformative Geometry Class
No longer should students accept the traditional classroom geometry experience as complete! GeoGebra is an exciting, web-based application that eliminates the unknowns and ideas students previously inherently accepted as truth, and instead allows users to see the big picture by learning geometry through discovery rather than memorization. With the addition of GeoGebra in the classroom, students transform from passive classroom observers to active, excited participants! Meaningful learning occurs as the students manipulate and test data within the GeoGebra

Description of Application
GeoGebra located at http://www.geogebra.org is a dynamic software package that encourages students to visualize geometric transformations and observe geometric theorems in real-time. It is a free, open-source tool that broadens the study of geometry to incorporate algebra and calculus. GeoGebra was specially designed by Markus Hohenwarter for middle school and high school students and has won several prestigious awards such as the European Academic Software Award (2002), the Austrian Educational Software Award (2003), and the German Educational Software Award (2004). GeoGebra is multi-faceted in that it can be used as both a teaching and a learning tool. Teachers can instruct students by providing step-by-step instructions in order that students may construct certain, preconceived figures and/or shapes. In contrast, learners may independently build their own geometric creations using the principles of constructivism to generate knowledge.

Transformative Potential
GeoGebra is user-friendly as evidenced by the ease of use by twelve through fourteen-year-old eighth grade students. What sets this geometry tool apart from other programs is its dynamic link to the algebraic representation of the data, and the manner in which this interconnectivity is presented. The manner is simple, straightforward, and by no means overwhelming. GeoGebra provides a seamless link between the various mathematical sciences: Geometry, algebra, and calculus. GeoGebra is a free program making it completely accessible to students with a computer with internet access. It is engaging, responsive, and empowering and will enhance the study of geometry with appropriate integration into the traditional curriculum. GeoGebra makes learning geometry fun and promotes the study of mathematics and technology amongst secondary students. It uses a constructivist approach to learning and encourages exploration, creativity, and learning by doing.

How It Works
Users have three options in creating GeoGebra files: Menu options, toolbar icons, or manual entry in the input text box displayed at the lower left of the screen. The menu options include commands such as changing the settings and properties such as point style, language, axes, right angle style, and coordinates; toolbar customization meaning the user can change or remove icons; and additional features such as creating custom tools, saving files, and print preview. The toolbar icons differ from the menu options in that it focuses solely on the construction of objects such as points, lines, vectors, and polygons. Toolbar icons contain drop-down boxes with more specific options (see figure 2).

For instance, if a user selects the line option, the drop-down menu includes perpendicular line, parallel line, line bisector, angular bisector, tangents, polar or diameter line, and locus. The toolbar icons are not available from the menu options. The manual entry box offers users the opportunity to input algebraic versions of geometric figures. For instance, a user may type in a line in the format y=mx+b, replacing m and b with slope and y-intercept values. Upon completion of the formula, the user slects the "Input" button, and the figure (in this case, a line) appears on the screen.

In figure 3 below, note that each blue point displayed on the screen has its corresponding coordinates displayed numerically in blue font in the algebra window on the left side of the screen. Likewise, the light blue line has its algebraic representation (also in light blue font) in the algebra window. Each green line segment in figure 3 has its corresponding length located in the algebra window in green font. The coordination of the geometric and algebraic representations is easily recognizable due to the color-coding.



The purpose of GeoGebra is to explore and manipulate relationships between the algebraic and geometric representations of figures. This allows students to make connections between various courses of study in mathematics and provides a richer, deeper understanding of the mathematical sciences as a result. In figure 4 below, the table describes how to create simple geometric figures using both the toolbar and the algebra window in order to explore these relationships.



Sample Activities
•Coordinate Geometry - Students may determine the distance between two points

•Coordinate Geometry - Students may locate the midpoint of a line segment

•Coordinate Geometry - Students may determine if lines are parallel or perpendicular and derive differences and similarities in algebraic representation of these lines.

•Coordinate Geometry - Students may find the slope of a line and manipulate data to change the slope of the line.

•Coordinate Geometry - Students may determining the slope-intercept form of an equation given a line.

•Basic Geometry - Students may determine perimeter and area of given polygons and circles.

•Basic Geometry - Students may discover properties of polygons and circles.

•RECOMMENDATION: Allow students to use GeoGebra to explore relationship between algebra and geometry through a constructivist approach (rather than providing structured, step-by-step instructions or activities). The more time students spend using the program in this manner, the greater their understanding of the interconnectivity of mathematics will be.