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Today's problems have become more complex than ever. In order to solve these problems, society is required to find ways of obtaining information about subjects that they have little knowledge or time to acquire that knowledge. The result is the use of tools to provide relevant information making solving the problems possible. These software tools maybe available to users but at a cost that is too high to warrant their purchase. One alternative to purchase is using open source software.
Technology continues to invade all aspects of U.S society. Most problems require electronic tools in order to be resolved with as much efficiency as possible. Therefore, students need training in technology used for numerical or graphical data manipulation. This training can also be time consuming and expensive. Thus, schools try to find ways to minimize these costs and time requirements. Again, open source software tools assist in these cost savings.
This action research centers on students employing GeoGebra to solve algebraic and geometric problems. GeoGebra is free open source software that users also receive customer support for free. This means that schools can implement GeoGebra as a viable math and science tool at no cost. I will emphasize the engaging quality of GeoGebra in the math classroom and thus it's increase in students' motivation.
Technology in Education
Technology is not just used in every aspect of society, but is part of society. With the development of the modern computer and its applications, problems that required many hours of labor can be solved in minutes. According to National Council of Teachers of Mathematics (NCTM 2011), technology is integrated into society and the workplace; Math classrooms need to do the same. As we move forward in our society, it has become increasingly important for students to be trained in Technology. Students are motivated to learn technology tools because the see the relevance that this training has on their career and personal life. This training has not always been available to students in high school. NCTM (2011) notes combinations of math tools are utilized for society's problems. Indeed, from data collection to statistical manipulation, computers are taking over. Thus the training in software tools is paramount. Accordingly, I believe that technology in various forms is important in any classroom but especially the math classroom. The benefits that technology can provide to a classroom are numerous. According to NCTM (2011), students can use technology to find patterns, relationships and connections, to think strategically and form conjectures.
Open Source Software
Open source software is software that is developed and maintained by a consortium. This consortium is made up of volunteers that assist in creating and extending the software product. According to von Hippel (2002), "open source" software products are being developed, distributed, and supported on a voluntary basis by users themselves-no supplier required. Any person can request a change or extension to open source software. If members of the consortium feel that the change or extension is a great idea, it will be added to the program. Any person may join the consortium or download the source code and make the changes for their use. With the world using computers for almost everything, the cost of software can become quite expensive. In schools and family homes, open source software is a valid alternative to spending money on software and licensing fees. Open source software runs on existing operating systems (Waters, J. K. 2010). Waters (2010) also states that open source software is meant to co-exist with existing technology. This interoperability means that schools do not need to upgrade existing systems to make use of open source software. This results in a cost savings to the school. According to Waters (2010), schools can be sure the same computer programs will be available to the students outside of school by implementing open source software into their computer labs. This availability of the software to the students in non-school hours can be a valuable reason for schools to use open source software.
Open source software has another benefit over commercial software. According to Lakhani and von Hippel (2000), "open source software does not generally involve a charge for field support. Instead, some product users voluntarily provide answers to the questions of other users - for free." It is supported by the users and many provide training and customer support because they believe in the product and want the recognition for their efforts. Commercial products often charge for support on a per user basis and this can become expensive for a school. One of the only drawbacks to using open source software according to Waters (2010), is people are suspicious about something that is free. In reality though, open source software may be free but the developers receive recognition which helps their career. Thus they are highly motivated to provide robust software to the users.
What is GeoGebra?
GeoGebra is dynamic mathematics software joining geometry, algebra, tables, graphing, calculus and statistics (Geogebra, 2011). They also state all level of education can implement GeoGebra. In my experience, GeoGebra's unique in its ability to assist in solving algebra, geometry, calculus and statistics problems. Other programs could solve one or the other but not all. According to Lavicza, & Papp-Varga (2010), "GeoGebra is a free, multi-platform, open-source dynamic mathematics software." They also state students and teachers are free to use it both in class and at home. They go on to add that because of GeoGebra's open-source nature, there is no licensing issue. One of the reasons that GeoGebra is usable by all levels is that it is easy to use and learn. Bulut, M., & Bulut, N. (2011) added that GeoGebra demonstrates a link between algebra and geometry. The ability for GeoGebra to show changes in the equations as changes are modeled geometrically, allows students to make connections. According to Edwards, J., & Jones, K. (2006), it is easy to navigate GeoGebra (p. 2). Garber, K., & Picking, D. (2010) reaffirm this, noting GeoGebra is straightforward and little instructional or set-up time is needed. Aydin, H., & Monaghan, J. (2011), praise GeoGebra as quickly mastered and used. The fact that GeoGebra is easy to learn and use it to solve mathematical problems is one of its greatest selling points.
Another selling point is suggestions for software changes from users are quickly integrated into new versions of state GeoGebra Lavicza, & Papp-Varga (2010). GeoGebra has a large online community that is devoted to training and promoting its development. Developers and users volunteer their time and effort to promote GeoGebra. "Online help is very concise, easy to access, and professionally done" states Grandgenett, N. (2007). Users can ask a question at any time and fellow users are available to answer them quickly. This community also has many lesson plans and models available for public use. Grandgenett, N. (2007) indicates that the contributed resources on many math topics are available. Teachers can use these resources to advance the knowledge that students attain through the use of GeoGebra.
GeoGebra is widely used throughout the world. If you do a search in Google for "GeoGebra", it will turn up almost 2.8 million hits. If you include pages translated to English, the number goes over 12 million. Bulut, M., & Bulut, N. (2011), indicate GeoGebra has multilingual menus and commands. This means that any person who uses GeoGebra can solve problems in their own languages. Thus it is available around the world and also in multi-cultural communities. The only drawback is that some families or societies may not have any computers available. Since GeoGebra was created using Java, it is available for all platforms minimizing this drawback. Inexpensive tablets and internet appliances can run GeoGebra and assist low income families in being able to use GeoGebra.
Figure 1. GeoGebra ScreenshotThe best way to understand GeoGebra is with a screenshot. See Figure 1.
Figure 1 shows a creation using GeoGebra 4.2. Data has been entered into the spreadsheet view on the left of the figure. All of the statistics and plots on the right part of the figure were created in 1 step by GeoGebra. This figure was created in less than 20 seconds and required very little instruction. Being able to enter data and create and view plots and graphs directly is only one benefit of GeoGebra. As we can see a large amount of information can be created and viewed in GeoGebra that would take a person a long time by other means. Below is another screenshot created in GeoGebra in about the same time.
Figure 2. GeoGebra Screenshot
Figure 2 shows a point A, a circle with center A, a tangent line to the circle at point B. Notice the information available in the Algebra window. We can see the coordinates of the points, the equation of the circle, and the standard form of the tangent line. As point A or B is dragged on the screen using the computer's mouse the circle or the tangent line dynamically changes to demonstrate the new coordinates. According to Garber, K., & Picking, D. (2010), GeoGebra allows students to graph linear and nonlinear equations, as we can see by figure 1. Notice also that in GeoGebra, equations can be entered directly into the input bar on the bottom of the window (Bulut, M., & Bulut, N. 2011). Also we can see that Geogebra contains graphic and text windows which blend algebra, geometry, and calculus (Grandgenett, N. 2007). These elements show why GeoGebra is easy to learn and use.
Students learning and GeoGebra
We have witnessed the easy at which GeoGebra is able model math. In the classroom is where GeoGebra shines. According to Hohenwarter, M., & Fuchs, K. (2006, January 26) GeoGebra is designed for the use in schools. Because of it straightforward approach to modeling mathematics, students are able to grasp the topics easily. According to GeoGebra [the organization] (2011, October 15), GeoGebra is for learning and teaching at all levels of education. Since GeoGebra requires little instruction, children of all ages can learn and use it effectively. "Students can discover relationships between mathematical concepts through different types of representations such as geometric, algebraic, and graphical," states (Bulut, M., & Bulut, N. 2011). These relationships build connections between topics that the students have previously learned to new ones. Edwards, J., & Jones, K. (2006) also noted that GeoGebra demonstrates a link between algebra and geometry.
The ability of GeoGebra to display geometric ideas dynamically provides opportunities for students to learn. It can model relationships between everyday objects and the math that each includes. According to Aydin, H., & Monaghan, J. (2011), student realize math is not just about school. They also noticed that students learned not only Geogebra but recognized math in real world applications. They believe that "GeoGebra enables using real-life examples for teaching and learning mathematical concepts." Many others have touted the effectiveness of GeoGebra in student learning. According to GeoGebra [the organization] (2011, October 15), GeoGebra facilitates the creation of mathematical constructions and models by students.
Students become motivated to learn math through the use of GeoGebra. The ease of use of GeoGebra and the ease of learning helps students fell that they have mastered the program quickly. Garber, K., & Picking, D. (2010) state that GeoGebra serves as a means to motivate and help students explore algebra and geometry. Being able to see the math in action is important to students. Students are also able to recognize the relevancy of the math topics that they are working on. According to Edwards, J., & Jones, K. (2006), GeoGebra supports students discovering rules themselves leading to a more powerful learning experience. They indicate the power that GeoGebra demonstrates forces actions with mathematical relationships. Attaining the knowledge themselves, instead of by lecture is a powerful motivator for students. Grandgenett, N. (2007) asserts GeoGebra can interrelate with other disciplines such as social studies and contribute to engaging instruction. This allows for connections to be made by the students.
From the teacher's point of view, GeoGebra is a great tool for classroom discussions and presentations. Hohenwarter, M., & Fuchs, K. (2006, January 26) states that GeoGebra is a cooperation, communication and representation tool. Using GeoGebra as a teaching and presentation tool allows students the opportunity to learn more quickly. Its power in the classroom demonstrates flexibility, teachers and students become co-learners, enables student-centered learning, and develops thinking skills (Mainali, B. R., & Key, M. B. 2012). By students and teachers sharing in the learning, students feel like their input is more important and therefore learn more effectively. Mainali, B. R., & Key, M. B. (2012) also support that GeoGebra allows students to learning from feedback, seeing patterns, making connections and dynamic images. Seeing the results of their creations and how changes to the model affect the outcomes allow students to feel motivation and ownership of their education.
Many studies have been done on motivation in students. These studies have lead educators and society to demand that students must learn technology. According to Arnone, M., Reynolds, R., & Marshall, T. (2008), the new standards make show information skills alone are not sufficient for student success. They also state that the value a student associates with these skills are as important. As well they state that students need to use these skills in a productive and responsible manner, and have the motivational disposition to support successful research and independent lifelong learning. AASL describe dispositions as the learning behaviors, attitudes, and habits of mind, which transform a learner from being able to learn to actually learning (AASL, 2008). These dispositions include students' perceived competence in their skills, their confidence in these skills and the intrinsic motivation to use them.
According to Skinner EA & Belmont MJ (1993) a child's need for competence is fostered when their classrooms are optimal in structure. Structure refers to the amount of information in the context about how to effectively achieve desired outcomes. They go on to state that teachers can provide structure by communicating their expectations clearly, issuing consistent responses, and by offering help and support. Next they state that a children's need for learning autonomy is increased as they experience autonomy support. This refers to the amount of freedom a child is given to determine his or her own behavior. Educators can increase a child's autonomy by allowing students choices with the learning activities. Also connections with the student's interest need to be pronounced to the students. For this autonomy to occur teachers need to minimize external rewards, controls, and pressures.
GeoGebra is one of the best modeling programs available. Using GeoGebra, Students are able to create mathematical models quickly and with little instruction. It is a valid replacement for a graphing calculator and other mathematical software. It is free to the users of all languages and available on all platforms. For this reason alone, GeoGebra is more readily available to students than other technology. Modeling is what GeoGebra does best and the process is very straightforward. Students feel closeness to math that is not possible without modeling. Thus the use of GeoGebra should motivate and engage students in Mathematics.
Annotated Bib indicating Refences.
American Association of School Librarians [AASL]. (2008). Standards for the 21st century learner. Retrieved on November 9, 2008, from
Arnone, M., Reynolds, R., & Marshall, T. (2008). Contribution of affect to information
literacy knowledge: The role of perceived competence in information skills and
disposition for reading.
Aydin, H., & Monaghan, J. (2011). Bridging the divide-seeing mathematics in the world
through dynamic geometry. Teaching Mathematics & Its Applications, 30(1), 1-9.
Students see more math in pictures with dynamic geometry software.P2
Each student receives 15 minutes of individual instruction in GeoGebra. Afterwards all students used it at home.
Realize math is not just about school. P7
GeoGebra is quickly mastered and used appropriately.P8
Learned not only Geogebra but recognized math in real world applications.P8
Geogebra forces actions with mathematical relationships.P9
Bulut, M., & Bulut, N. (2011). Pre-service teachers' usage of dynamic mathematics software. Turkish Online Journal Of Educational Technology - TOJET, 10(4), 294-299.
GeoGebra has multilingual menus and commands.P295
GeoGebra , equations can be entered directly.P295
"Students can discover relationships between mathematical concepts through different types of representations such as geometric, algebraic, and graphical." P296
"GeoGebra enables using real-life examples for teaching and learning mathematical concepts." P297
GeoGebra can interrelate with other disciplines such as social studies. P297
Edwards, J., & Jones, K. (2006). Linking geometry and algebra with GeoGebra. Mathematics Teaching, (194), 28-30.
GeoGebra supports students discovering rules themselves leading to a more powerful learning experience. P2
It is easy to navigate around GeoGebra. P2
GeoGebra will modify the change in function automatically. P3
Demonstrates a link between algebra and geometry. P4
Garber, K., & Picking, D. (2010). Exploring algebra and geometry concepts with GeoGebra.
Mathematics Teacher, 104(3), 226-228.
A means to motivate and help students explore algebra and geometry.p226
GeoGebra allows students to graph linear and nonlinear equations.P227
Learning to use GeoGebra is straightforward; little instructional and set-up time needed.P228
Grandgenett, N. (2007). GeoGebra. Mathematics And Computer Education, 41(3), 276-278.
Geogebra contains graphic and text windows which blends algebra, geometry, and calculus.P276
Online help is very concise, easy to access, and professional done.P277
Contributed resources on many math topics.P277
Can contribute to engaging instruction.P278
GeoGebra. (2011, October 15). GeoGebra quickstart [PDF]. Retrieved from
For learning and teaching at all levels of education
GeoGebra facilitates the creation of mathematical constructions and models
by students that allow interactive explorations by dragging objects and
Hohenwarter, M., & Fuchs, K. (2006, January 26). Combination of dynamic geometry, algebra and calculus in the software system GeoGebra [Microsoft Word]. Retrieved from http://www.geogebra.org/publications/pecs_2004
The construction protocol of GeoGebra makes it possible to redo constructions at any time, insert new elements and even change its order with hindsight.
A cooperation, communication and representation tool.
Teachers have more time to concentrate on fundamental ideas and mathematical reasoning.
designed for the use in schools.
Lakhani, K. and von Hippel, E., 2000, 'How Open Source Software Works: 'Free' User-to-User Assistance', Working Paper 4117, Sloan School of Management, Massachusetts Institute of Technology.
Lavicza, Z., & Papp-Varga, Z. (2010). Integrating GeoGebra into IWB-equipped teaching
environments: preliminary results. Technology, Pedagogy and Education, 19(2), 245-252.
Mainali, B. R., & Key, M. B. (2012). Using dynamic geometry software GeoGebra in developing countries: A case study of impressions of mathematics teachers in Nepal. International Journal for Mathematics Teaching and Learning.
Reasons to use dynamic geometry software
Flexibility, teachers and students become co-learners, enables student-centered learning, and students develop thinking skills p.3
Ways students are engaged
Learning from feedback, seeing patterns, making connections and dynamic images p.3
National Council of Teachers of Mathematics (NCTM). Focus in High School Mathematics: Reasoning and Sense Making. Reston, Va.: NCTM, 2011.
Connections between math and real-world problems developed in modeling provide value, incentive and context of topics. P13
Real-world problems require a combination of mathematical tools. P13
Technology is an integral part of society and the workplace; Math classrooms need to reflect this.P14
Technology can be used to advance the goals of reasoning and sense making.P14
Tools are useful in finding patterns and relationships and in forming conjectures. P14
Technology can relieve students of computations giving them the freedom to think strategically.P14
Skinner EA & Belmont MJ. 1993. Motivation in the classroom: reciprocal effects of teacher behavior and student engagement across the school year. J. Educ. Psychol. 85:571-81
von Hippel, E., 2002. Open source projects as user innovation networks. MIT Sloan School of Management Working Paper 4366-02, June.
Waters, J. K. (2010). Prepare for impact. T H E Journal, 37(5), 20-25.
People are suspicious about open-source software. (p. 3)
Run on existing operating systems and save money. (p. 5)
Students can run open source software at home.(p. 5)
Open source software can reach a larger audience.(p. 6)
Open source is meant to be interoperable with existing technology. (p. 7)