Stages of Grouth of Geometric Understanding in Triangles Using Geogebra for the Eighth Grade Students

This study aimed at identifying the levels of geometric understanding of the eighth grade learners, studying the unit of "Triangle", depending on the theory of mathematical understanding developed by Pirie and Kieren (1991). The participants in the research were eighth grade students in a school in one of the Directorate of Education private schools in Nablus, where the students' achievement in mathematics was at least 80, and they worked as three groups consisting of 2-3 students each group. The researcher monitored and followed up the actions and discussions of the students during their work in groups, and videoed their learning actions and discussions. The researcher also interviewed the participants to get information about their perceptions of learning with geogebra.The researcher, based on the grounded theory (Strauss & Corbin, 1998), and the theory of Pirie and Kieren, analyzed the data of the experiment, where the participants understanding stages were identified, as well as the characterizes of every stage, when learning the topic of triangle with geogebra. The researcher also studied what affected the participants' transition from stage to stage of understanding. The researcher found that the participants passed some stages of understanding according to Pirie and Kieren during their learning the subject of triangles. These stages were different according to the topic that they learned. In all the topics they passed through the six stages of primitive knowing, image making, image having, property noticing, formalizing, and observing. At the same time, they were unable to reach the advanced stages of understanding: structuring and Inventising. This situation could be related to the nature of the indirect applications which were provided by their teacher, which were not constructive for them to get to the new concepts. The results showed that the stages of understanding acheived by the participants in each topic were distinguished from the stages experienced by the participants in other topics in terms of the occurrence of Folding back or not, and the presence of boundaries between the stages of understanding or lack of it. In addition they were distinguished by the participants actions and discussions Based on the results of the research, the researcher stresses on the importance of giving the students the opportunity to deal with new geometric theorems based on the theorems they have learned, and giving them the opportunity to explore them by themselves without much help from the teacher. The researcher also recommends to perform further qualitative studies on mathematical understanding, because of the lack of Arabic studies focusing on students' understanding and using the theory of Pirie – Kieren. The researcher also recommends the use of GeoGebra by school students to further their mathematical knowledge. GeoGebra would help them explore new mathematical situations and problems, and progress to higher stages of understanding, and thus improve their attainment in mathematics. The researcher suggests planning training courses for administrators and mathematics teachers on the integration of GeoGebra in teaching mathematics, as well as planning the upgrade of the mathematics curriculum in Palestine to include GeoGebra activities.