**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.

*, but should be a tool we use to verify concepts that we have developed through investigation.*

**Proof should not be just an act of regurgitating nonsensical nitpicking**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.

*.”(Stojanovskay, 2009)*

**These are crucial life skills**

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*?

**Geometer’s Sketchpad **

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.

## Geogebra

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.

* *

## References:

**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/**

Hey Greg,

This is a great site with a lot of comprehensive material for math teachers, students, and people who just want to know more about math! I also like that you bold-faced the most important notes in your post. It really helps the reader find the focus of your post and learn more. My biggest question is, especially having strugged with math in high school-geometry in particular, will these programs help students making math more meaningful and applicable to their lives, creating more “buy-in” and understanding? I feel as if they could, but I wanted to know your take on it.

-Katie

Thanks for the response, Katie.

I think math is more meaningful when you discover it for yourself. Sketchpad and other dynamic programs allow students to discover geometric concepts by manipulating drawings, seeing what changes and what doesn’t. The feature of Geogebra that allows you to include a background picture could be used to make the excercise relevant. I’ve used a graphing calculator to try to match the shape of parabolas in photos of the St Louis Arch, McDonalds arches, etc. If you could give students Geogebra files with digital photos, they could modify function equations to match the given photo. Similarly, photos of famous or local building could be used and students could try to match the slopes of their roofs. The concept of slope becomes more meaningful when applied to a physical entity (or in this case the photographic representation of it.

Greg

I agree totally. I have incorporated more Internet based learning for the students every year, because let’s face it, teaching and relating a language that is not spoken anymore is not the easiest sell. I really like the idea of using architecture to help understand geometry. I know, though again it was not my strong point, that it would have helped my geometry if I was applying it to real world items!

-Katie

Greg, you are doing a great job with this site. I feel like I am so far behind. Keep putting out great ideas, it has been a big help for me.

Chris

Greg, students are going to love this site. Students will be able to concentrate better without all the disruptions accosiated with school. This will make your job in the classroom a lot easier.

Good job,

George Hughes

Wow! You really went above and beyond in your first posting! I am blown away by your professional approach to your material and your ability to break down concepts into terms that novices can understand.

The media elements you chose complement the content well. I also appreciate how you made a point of highlighting major points in your post. I honestly wish that my school learning materials would have been as concise as this. Keep it up!

I’m curious what you think of these iPad/iPhone apps. Any experience with these?

http://appfinder.lisisoft.com/ipad-iphone-apps/coordinate-geometry-software.html

I don’t have an Ipad or Iphone, but seeing some of the great stuff available for them, I may need to get one! Thanks for the great link!

Greg-

Wow – this IS impressive. What a great first post, adding to an already great site. I think that you are spotlighting a very important point here. For students to really be able to learn geometry, then the visual represenations are key. This software sounds like it is a great way to make geometry dynamic and help students truly understand the concepts.

I was never very good at math, but if the process of learning it was fun and relevant, then I would have gotten much more out of it. The free online tool “GeoGebra” sounds fascinating…I think if students can use this software at home, then have the concepts reinforced in the classroom, then it will really solidify their understanding.

I like how you set up your blog, along with the sentences in bold to catch your reader’s attention. A very visually appealing post!

Reading you post reminded me of something a bit more old school…last November I went to a day long seminar with Edward Tufte, an artist and an expert in visual representation. His examples of the best ways to represent information visually are based on models from centuries ago. One example that stuck with me was a book that he showed the participants, Euclid’s “Geometry,” published in 1570. Within the book, Tufte shows us how Euclid provided 3-D model of a triangle to illustrate the concept; it was essentially a “pop-up” with three flaps of triangular paper that create a 3-D model of a triange. What a nifty concept! Here is his site if interested: http://www.edwardtufte.com/tufte/

My first thought about your first blog post was how comprehensive and informative a job you did on this topic. I can really tell that you know your subject and did some great research to present to your colleagues. I must admit that you make great points for using both software programs, but with the financial constraints of school districts, it certainly seems as though Geogebra is the way to go.

Your videos were really helpful. The first video, although short, made me see very easily that the idea behind the software is to make Math meaningful in a real life way. I always did well in Math, but I never really understood the point of it beyond the basics for all students. I realize the importance for certain career fields, but I always thought that the specific Math could be taught at the University level. Why do you feel that it is important for all students to be able to do algebra, geometry, etc…? What percentage of the population uses math beyond say the 7th or 8th grade level? I think that this site and these programs will certainly help to answer those questions. I was just curious as to your thoughts.

I was reading this review of Geogebra and thought it might be interesting to you…

http://www.scribd.com/doc/16975395/GeoGebra-in-the-Secondary-Mathematics-Classroom-A-Literature-Review-

I don’t know how much you could actually use from this site, but I think since it deals with architecture, you could have a lot of opportunites. I thought it seemed better cool anyway!

http://archkidecture.org/

-Katie