Dynamic geometry software is a wonderful tool for teaching Geometry in a discovery mode. Students create and manipulate geometric figures and are encouraged to make conjectures based on what they see. Constructed figures can be modified by dragging points around the screen, but underlying relationships are unchanged. By recognizing the aspects that remain constant, students identify geometric concepts.
Manizade claims that “the process of making and testing a conjecture is made easier by … dynamic geometry software” and that “one way to booster proof writing skills may lie in the ability to make conjectures or hypotheses based on empirical results.”(Manizade, 2009) Further, she claims that conjecture building is “an activity found to help students understand and improve proof writing skills.(Manizade, 2009) The Von Hiele geometric model also asserts that students are more ready for formal proof when they first understand the underlying geometric concepts. In teaching proof, I try to emphasize that developing conjectures based on observations (inductive reasoning) and then proving those conjectures through deductive logic are two stages of a process. Putting proof in this prospective, I believe, provides a justification for formal proof. Proof should not be just an act of regurgitating nonsensical nitpicking, but should be a tool we use to verify concepts that we have developed through investigation.
While many students see formal proof as irrelevant, it’s inclusion in the curriculum has traditionally been defended based on the need to develop logical methods of thinking. “Given any problem in life,” Stojanovskay asserts, “one needs to be able to think about it logically. This means, understand what the problem is, organize data into knowns and unknowns, explore possibilities and assess solutions. These are crucial life skills.”(Stojanovskay, 2009)
Assuming that a dynamic geometry program will help students identify geometric concepts, create hypothesis and make conjectures that will, in turn, help them develop more proficiency in proof writing and logical constructs, what software should we use?
is probably the “gold standard” of geometry programs and I have had very good results using it. The construction tools are fairly intuitive and students can quickly learn to construct circles, segments, parallel and perpendicular lines and many other geometric figures. The measuring tools are fairly easy to use, especially since Sketchpad makes available only those measurement tools that can be applied to highlighted figures. You can’t choose to measure an angle, if you have a segment highlighted. You can also, with experience, create fairly elaborate geometric figures. So why consider anything else?
Wikipedia lists over 40 dynamic geometry software products, many of them free. One of these has garnered a fair amount of attention, even from key curriculum press, Sketchpad’s distributor.
is a free open-source program, available online as a Java applet or downloadable using either a Web start or an installed front end for users without internet access. Many of the construction tools have a very similar look to those in Sketchpad. The menu system is more visual, which may appeal to some students. The system has been translated into dozens of different languages for international use.
“The unique feature of GeoGebra”, according to Manizade, “is the integration of dynamic geometry software and a computer algebra system into a single tool for mathematics education“.(Manizade, 2009) “The GeoGebra software itself is a blend of basic geometry, algebra, and calculus tools”.(Geogebra, 2007) While Sketchpad also includes tools for exploring Algebraic and Calculus topics, Geogebra makes the connections between visual/graphical representations and their algebraic definitions more explicit through the use of a split or dual screen. While the graphic/Geometric concepts are presented on the right side, their algebraic definitions or equations are shown on the left allowing students to “observe how changes in a formula in the algebra window are affected by manipulation of the construction”.(Manizade, 2009) Stojanovskay believes that “student[s] will explore visually/geometrically until the idea of how to solve the problem algebraically/symbolically comes to [them].”(Stojanovskay, 2009) Displaying both representations helps students make this connection. One demonstration I saw also made use of Geogebra’s ability to import background images (i.e. digital photos) and to overlay geometric of graphical elements, visually tying the mathematical concepts to real world examples. Being able to manipulate the algebraic form to adjust a graphical representation to conform to aspects of the underlying photo provides a relevant experience of the exercise, answering the ever-present question: when am I going to use this?”
“The biggest argument in favor of continued use of Geometer’s Sketchpad”, according to Wysocki, “is the sheer wealth of material out there for many different subjects and topics that specifically reference GS.” (Wysocki, 2008) This is being addressed by the community of Geogebra users who contribute lessons, files and support through the Geogebra wiki, the ever-growing set of tutorials and the international professional development support organization. “The biggest argument in favor of GeoGebra is its cost. It is free.”(Wysocki, 2008)
Geometer’s Sketchpad has a variety of purchasing options for educators, district and for individual students. A student license is about $29.00. For students who can’t (or won’t) even provide their own calculator, requiring them to purchase this package might be unreasonable. Unless provisions were made for students who could not afford the program, making the home software available for purchase might create an issue of equity, since some students may benefit from its use while others cannot. With the free download of Geogebra, just about every student should be able to access the program. The non-internet version even makes the software available to students with computers but without internet access. That said, I did find a site offering a free download of a Geometer’s Sketchpad demo version. It is an earlier version, without all the bells and whistles and restricted so that images cannot be exported, cut, copied or printed, but it seems to have most of the functionality and could provide a reasonable alternative for home use and exploration. Still, the Geogebra option is more equitable, since all students would have access to the same version of the software, which is the same version available for classroom use.
Another way to make content available and accessible is through the creation of “interactive web pages or dynamic worksheets.”(Manizade, 2009) While Geometer’s Sketchpad can output sketches as interactive web pages, these are somewhat limited in terms of how they can be manipulated and they require installation of a Javaserver to play them. While I haven’t used the Geogebra version, it is purported to be both easier to create and more powerful to use.
Wysocki states: “I expect my students to explore, make discoveries, make conjectures, and write proofs. Three of those four can be aided by a piece of dynamic geometry software”. (Wysocki, 2008) and Stojanovskay claims that “the creation of dynamic GeoGebra files by pupils and students to explore, discover and then understand mathematics is both viable and useful.” (Stojanovskay, 2009) If we want to make dynamic geometry software available to all our students and want to promote the connection between algebraic and graphical/geometrical representations, it seems that Geogebra may be the better choice. In any case, it may be a wise alternative to make available to students for home use, even though it may require some additional in-school training.
Wikipedia comparison of Interactive Geometry Software: http://en.wikipedia.org/wiki/Interactive_geometry_software
Geogebra (2007) Mathematics & Computer Education retrieved from http://web.ebscohost.com/ehost/pdfviewer/pdfviewer?vid=7&hid=113&sid=2da6ec92-8814-4a3d-9013-efab13105892%40sessionmgr114
Manizade, Agida Gabil & Beth Lundquist, Beth (2009) Learning About Proof by Building Conjectures Conference Papers — Psychology of Mathematics & Education of North America retrieved from http://web.ebscohost.com/ehost/pdfviewer/pdfviewer?vid=7&hid=113&sid=2da6ec92-8814-4a3d-9013-efab13105892%40sessionmgr114
Fahlberg-Stojanovskay, Linda & Stojanovski, Vitomir (2009) GeoGebra freedom to explore and learn Teaching Mathematics and Its Applications retrieved from http://web.ebscohost.com/ehost/pdfviewer/pdfviewer?vid=7&hid=113&sid=2da6ec92-8814-4a3d-9013-efab13105892%40sessionmgr114
Wysocki, Jim (2008) Geometer’s Sketchpad vs. GeoGebra retrieved from http://jimmy13.wordpress.com/2008/09/07/geometers-sketchpad-vs-geogebra/