**Rationale**

*Integration of dynamic geometry software enhances the learning of mathematics.*

The importance of technology is recognized and incorporated in many Canadian curricula (e.g. Ontario) and it is one of the principals that National Council of Teacher Mathematics determines for school mathematics. Technology is essential in mathematics learning and instruction while improving students’ learning experience (Ross, Bruce, & Sibbald, 2011).

*A powerful technological tool that can be utilized in learning mathematics is a dynamic geometry software (DGS) allowing students to see that math can become “alive”.*

There is limited research done when it comes to the effective integration of GeoGebra into teaching and learning mathematics (Hohenwarter, Hohenwarter, & Lavicza, 2009). Research suggest Mathematical frameworks need to be changed to reflect new possibilities created by online tools such as GeoGebra, an online resource for both teachers and students. Integration of DGS into teaching would benefit if it is done with a more systematic development on a larger scale and this process is requiring of explicit recognition when it comes to the development of instrumental and mathematical knowledge (Ruthven, 2008).

Piaget believed that learning goes through developmental stages: sensorimotor (birth - 2 year), preoperational (2 - 7), concrete operational (6 - 12) and formal operational (12 and up).

Piaget’s basic premise is “Intelligence is seen as a.process of active adaptation to the physical and social environment and all intelligent acts presume underlying "structures" that develop in an organized, interrelated manner.” (Good, 1978, p. 688). According to Piaget, active engagement is essential to learning, therefore as educators we need to consider how to actively engage students within the context of technology. Stols suggested that Piaget’s theory can be applied to learning and it could be extended with Van Hiele’s (VH) theory in the context of technology. VH’s theory suggests that there is progress through thinking on sequential levels and it is a result of experience which deals specifically with the geometric thought and how it develops. Stols found in his research that technology does help with improving the conceptual geometric growth of students on VH’s levels that deal with geometric visualization, recognition of properties of geometric figures (analysis), and the construction of proofs( deduction) (Stols, 2012).

*Inquiry based learning and student-centered construction of knowledge could be used when incorporating GeoGebra in instruction*.

GeoGebra offers grounds for active engagement of students and learning. Students need to collaboratively investigate mathematics while a teacher is guiding them to ensure that learning is effective. This happens when students work with others that are more capable: teachers and classmates who are then able to push their zones of proximal development while using timely prompts. Therefore, it would be possible to use GeoGebra in providing opportunities where students learn mathematics by doing it while engaging in mathematical discussion (Hahkioniemi, & Leppaaho, 2012). GeoGebra designed inquiry tasks would be ideal for student exploration where appropriate guidance practices should be in place.

The students should follow a series of steps such as the launch, the exploration, and then the summary/discussion phase in inquiry-based lessons. These were the steps followed by the research team in one of the studies (Hahkioniemi & Leppaaho, 2012). The students are introduced to the problem in the first stage, in the next stage students explore and work together while the teacher guides them, and in the last one there are class discussions where solutions, methods, insights and conjectures are shared. The teacher provided open problem activities with more paths of solving them. In this last phase, the teacher ensures that all is sound and that the concepts are clear and any misconceptions are addressed. The central theme for constructing knowledge is that students work together while discussing and negotiating their steps and are able to justify them. Therefore, this particular research is also in line with the other research study where it’s claimed that in this way students are able of developing and demonstrating deeper understanding of mathematical concepts and therefore, can investigate more advanced mathematical contents than in when in traditional teaching environments (Hohenwarter et al., 2009). However, it is important to consider the student-centered approaches and principles when it comes to using computers as well (Ozgen & Bindak, 2012).

*When implementing GeoGebra it is important to pay attention to the teacher dimension.*

There has been some research done in this area and some suggest that the process of integrating DGS is not a straightforward one and there are many barriers in place. Lavitza also adds the standards driven curricula, time constraints, lack of support for teachers coming from their peers and administrators are some additional barriers (Hohenwarter et. al, 2009). Additionally, teachers need to be able to see the value of technology in terms of perceived usefulness, ease of use, and the value of replacing the old. However, it is noted that teachers need to have experience with these technologies in order to see and then implement these changes. It makes sense then to note that for some teachers there may be still a long way from being able to effectively integrate it into their teaching (Stols & Kriek, 2011).

However, in order to have GeoGebra successfully integrated, the teacher needs to be able to promote higher level thinking, discussions, and be able to motive students in their inquiry learning if the goal is to foster a deeper understanding of mathematics. Therefore, the teacher/researcher would need to be mindful of Bloom’s taxonomy and ask questions that would promote higher level math thinking,

As mentioned by Hahkioniemi & Leppaaho (2012) the teacher would need to motivate students by promoting further discussion and lead them correctly to justify their answers and for them to realize any misconceptions they might have made. Therefore, it seems that the teacher needs to be fully aware of the role of GeoGebra and also able to react and adapt accordingly to students’ way of thinking and use those opportunities to further student inquiry. It is important to note that this last role might reach expert or near expert levels with teacher experience along with the awareness.

**GeoGebra Tutorial Overview**

The Math department of a distance education school is concerned about the level of engagement and practice the Math students are getting in their online Math courses. Because students only log into their online learning platform, Elluminate, once a week for live instruction, they are not participating in enough learning opportunities outside of live “class” time that support active engagement in learning and collaboration with classmates. The students particularly struggle in connecting their learning to real-world application in Geometry and Algebra. The students prefer instead to read examples of math questions provided on their class Moodle site and follow the same process to solve questions that appear on their assignment sheets.

The Math instructors would like students to take more ownership for learning and constructively develop “big picture” understanding of math concepts. They would also like to see students solve math questions using principles of Math instead of always following examples provided on their Moodle site. They believe that accessing online resources such as GeoGebra will support their ability to foster active student engagement within a collaborative learning environment.

**What is needed from teachers for a successful integration?**

The teacher needs to possess expert knowledge in Math pedagogy as well as a solid command of the software being used. Expertise in these areas are substantiated by the Koehler-Mishra’s Technological Pedagogical Content Knowledge (TPCK) framework, In other words, teachers require pedagogical techniques that use technologies effectively and constructively when it comes to teaching content (Hahkioniemi & Leppaaho, 2012).

To be able to use the software, Hohenwarter et. al (2009) suggest a series of high quality workshops where teachers get familiar with technological tools directly related to the benefits of using GeoGebra in learning environments. Their research demonstrated that teachers were satisfied with GeoGebra rated the software as user-friendly, intuitive, and potentially beneficial for teaching Mathematics in secondary schools.

If we are to apply Wegner’s communities of practice (COPs) in this context, it would follow that teachers could benefit from joining GeoGebra communities of practice in their respective schools, school boards, or a virtual and global GeoGebra’s COP (on GeoGebra’s website) where teachers share common interest of teaching with GeoGebra and are able to move from a novice to an expert in the field (Wegner-Trayner, 2014).

**How sequencing in teaching could be done when it comes to implementing GeoGebra?**

After reading the literature, one becomes more aware that what matters the most is the time students spent on the task, even when the teacher is incorporating online learning opportunities into a lesson sequence. Learning by doing is, hence, very important. However, it is important to note that both students and teachers preferred an approach in which technology-based activities were interspersed with teacher explanations and demonstrations (Ross, Bruce, & Sibbald, 2011).

What also would allow students to spend more time on trying different methods with GeoGebra is if teachers prepare GeoGebra files beforehand and are ready to be used The parameters which students can alter are ready for use. This helps the students to work more easily with the software because they do not have to know how to use all the functions of GeoGebra (Hahkioniemi, & Leppaaho, 2012).This would also allow students to learn the software with a gradual release of responsibility in a manner that is more stress-free.