By John Golden, Posted September 15, 2014 – 

Every day it seems like K–12 students have more powerful tech tools at their disposal. But what are they to do with these tools? And when is it appropriate for them to use these tools? If the answer is “not much more than to watch lectures and take quizzes,” then we are missing the boat. When I’m asked for tech recommendations for middle school and high school, there are two apps that I most want to get in students’ hands: Desmos and GeoGebra.

Desmos is what graphing calculators want to be (see fig. 1). This app is free to anyone with a browser and is built on HTML5. It is also on iOS, with an update coming soon. Desmos graphs in color, adds sliders seamlessly, and can be saved via a URL. It has free accounts and regular updates, often in response to teachers’ requests. (Check out https://teacher.desmos.com/.) One of the wonderful features of Desmos is some truly amazing support for assessing student understanding with real-time monitoring of the students’ work. Curious? Here’s a post on the Desmos blog by Lee Bissett about using it in a middle school, and there are several Fawn Nguyen posts on using Desmos in her classroom. There’s even a collaborative blog dedicated to challenging Desmos creations.

fig. 1 An actual graph on Desmos

As great as Desmos is, my heart belongs to another program.

Research on the impact of dynamic geometry programs goes back decades. Do a quick search of MTMS dynamic software and you can find a bevy of great articles, covering teaching ideas about geometry, algebra, and data. However, the cost of the software has often been a barrier for schools, and if schools had the software, students didn’t have access at home.

When I heard about GeoGebra (GGB) (version 2, I think), I was quick to give it a try. Instant mathcrush. Every object you make has a geometric and an algebraic identity; it’s Descartes’ fantasy world. Students took to it quickly and even started using it from home. GeoGebra is free and open source. Starting with version 4, the developers began a great support/feature/service: GeoGebraTube. Modeled on YouTube, teachers and students can upload their GGB work to save or share. Students can submit work this way or access activities selected by their teacher. The activities can be downloaded or opened directly in the browser. GeoGebraTube is searchable, so you can usually find an activity on your topic made by someone else as well as elementary school topics up through research mathematics. You can also make GeoGebra “books”: collections of activities that you can share with a single link. (See Jennifer Silverman’s book on straightedge and compass.)

As students are engaged in a GGB activity, they are exploring and making connections on their own. It’s one of the best tools I’ve ever used to get students to make conjectures, discover reasons for an argument or proof, or build experience to generate intuition. Examples from my blog include Percent Game Remixing and Flip Flop (about reflections). The ability to incorporate pictures into sketches makes for great modeling practice. (Here are several quadratic ideas.) Because the interface is often just click and drag, there’s no start-up time needed for students who are learning how to use the program. Read how Jed Butler uses the applets to promotearea formula understanding.

But the most exciting opportunity is for the students to start to learn the program. For example, Audrey McLaren and her students produced astounding projectile projects. The program is powerful and a tool that will grow with students throughout the rest of their career. I’m literally learning new things every day through GGB, often through trying to “dynamicize” a classic theorem (like the Euclidean algorithm for the GCD; see fig. 2) or some intriguing art with mathematical properties (like Truchet squares; see fig. 3). (See, also, fig. 4’s Matheo on GeoGebraTube.)

fig. 2 Euclid’s GCD Algorithm on GeoGebraTube 

fig. 3 Truchet squares on GeoGebraTube

 fig. 4 Mathio on GeoGebraTube

Because GeoGebra is an open-source community, you will find support. We’ve just started #ggbchat on Twitter (every other Wednesday at 8 p.m. EST), and the GeoGebra forums are helpful to novices and masters. Or ask me, I love collaborating on GeoGebra. 

 John Golden, @mathhombre, is a member of the department of mathematics at Grand Valley State University in Michigan. He teaches math and elementary and secondary teacher preparation courses. At mathhombre.blogspot.com, he blogs about math games, geometry and GeoGebra, lesson ideas, and teacher prep.