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This study investigates the impact of a dynamic geometry environment, GeoGebra, in enabling grade 9 students to explore a geometrical task related to the Euler's line. The research subjects were grade 9 students in India and Sweden who had no prior experience of using dynamic geometry. Their approach to geometrical reasoning, argumentation and conjecture making has been analysed using Marton's theory of variation, Pea's theory of technology as amplifier and reorganizer and with Marrades and Gutiérrez categorization of proofs. Results indicate that when students of both countries worked on geometrical tasks in GeoGebra, they arrived at similar conclusions. For both groups of students, GeoGebra played the role of amplifier and reorganiser in enabling their explorations. Further, in some of the construction and dragging episodes it was observed that the four functions of variation, namely, contrast, generalisation, separation and fusion played a significant role in developing students' geometrical thinking. Results further indicate that the dynamic geometry environment (DGE) facilitated the making of conjectures and developing empirical proof. However, when it came to developing a deductive proof, students required adequate scaffolding. In general, it concluded that students perform adequately when given investigatory problems in such an environment. They learn new mathematics and learn to handle the DGE at the same time. [ABSTRACT FROM AUTHOR]

In recent years, several extensions of the Hadoop system have been proposed for dealing with spatial data. SpatialHadoop belongs to this group of projects and includes some MapReduce implementations of spatial operators, like range queries and spatial join. the MapReduce paradigm is based on the fundamental principle that a task can be parallelized by partitioning data into chunks and performing the same operation on them, (map phase), eventually combining the partial results at the end (reduce phase). Thus, the applied partitioning technique can tremendously affect the performance of a parallel execution, since it is the key point for obtaining balanced map tasks and exploiting the parallelism as much as possible. When uniformly distributed datasets are considered, this goal can be easily obtained by using a regular grid covering the whole reference space for partitioning the geometries of the input dataset; conversely, with skewed distributed datasets, this might not be the right choice and other techniques have to be applied. for instance, SpatialHadoop can produce a global index also by means of a Quadtree-based grid or an Rtree-based grid, which in turn are more expensive index structures to build. This paper proposes a technique based on both a box counting function and a heuristic, rooted on theoretical properties and experimental observations, for detecting the degree of skewness of an input spatial dataset and then deciding which partitioning technique to apply in order to improve as much as possible the performance of subsequent operations. Experiments on both synthetic and real datasets are presented to confirm the effectiveness of the proposed approach. [ABSTRACT FROM AUTHOR]