# The Potential and Challenges of the Use of Dynamic Software in Upper Secondary Mathematics : Students’ and Teachers’ Work with Integrals in GeoGebra Based Environments

Abstract: An introduction of computer software into mathematics classrooms makes the didactical situation more complex compared with previous learning environments (Blomhøj, 2005). A technological tool becoming a mathematic work tool in the hands of the students is a process that has turned up unexpectedly complex (Artigue, 2002). In addition to this problem, the teachers as the users of the tool go through the same process, while, at the same time, trying to integrate the tool into their teaching activities in a meaningful way. For these reasons it seems important to contribute to the research focused on the learning and teaching conditions in environments, where computer software is newly introduced, in order to better understand impacts of the introduction of different software in mathematics classrooms.In this study the dynamic mathematical software GeoGebra was used. GeoGebra is freely available for a number of platforms and has drawn much attention during the last years with growing user communities (www.GeoGebra.org). However, being generally available just recently, there are, comparatively, few studies on the use of GeoGebra in classroom settings.In this thesis the introduction and integration of GeoGebra was investigated in two studies with different perspectives. In the first study students’ work with GeoGebra in their mathematical activities related to the integral concept has been researched. In the second study teachers’ utilization of the didactical potential has been investigated. The results of the two studies show that GeoGebra as a mathematical tool in the hands of the students and the teachers can have a significant role in supporting their mathematical work if exploited in a, from a didactical perspective, adequate way. A learning and teaching environment based on GeoGebra bring with it a possibility to work with mathematical concepts in a broader way compared with blackboard based classrooms. GeoGebra’s facilities makes it possible to communicate mathematics in different ways and expressing mathematical concepts in different representations in a more direct way than in non dynamical environments. Communicating mathematics in different ways and expressing mathematics knowledge through different representations is of significant importance for students, not least in relation to the new curriculum for mathematics in Sweden (The Swedish National Agency for Education, 2011), where these aspects are explicitly named as aims for students to work towards.On the other hand, the investigations also showed that the introduction and the integration of GeoGebrawas a complex process for both the students and the teachers in this research. The introduction and integration of the software in the students’ mathematical activities made the didactical situation more complex and a differentiation of students’ work with the software was observed. For some students the use of the software seemingly supported their mathematical work, and at the same time for some students the result was the opposite; the use of the software was seen as a disturbing factor in their mathematical activities. When it comes to the study of teachers’ work with GeoGebra the investigations revealed that they encountered different types of obstacles that prevented them from utilizing the full didactical potential of the software in their teaching of mathematics. Three different types of obstacles were identified:technical - a teacher is not able to operate the software in the intended way;epistemological - a teacher is not aware of the didactical potential of GeoGebra and howto exploit it in in a way that supports students’ learning of integrals;didactical - a teacher is not aware of the complexity of technology based environments or he/she is aware of this aspect, but not comfortable with his/her competence in carrying out the process of integration of the software into his/her teaching without external help and support.Even if it is difficult to see the software detached from the context in this research, it seems that many of the obstacles perceived by the teachers in the experimental group, as well as difficulties students perceived in their work with the software, were related to the fact that they were inexperienced with the software and, consequently, lacked in knowledge in how to exploit its features in their mathematical activities. As it seems, the teachers would encounter the same obstacles every time they try to integrate a new, to them unfamiliar, software into their teaching practice. Also many of the students would experience same difficulties if they are not adequately supported in this process. Based on this, there are reasons to believe that problems with integration of GeoGebra into mathematics classrooms identified in this research would be similar in relation to integration of other dynamic mathematic software into mathematics classrooms, or even broader, other types of software as e.g. Computer Algebra Systems (CAS), as long as the integration considers the use of an unfamiliar software.

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