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The Thinking Characteristics of the Shifts from Visualization to Geometrical Reasoning by Students in Senior One
Reasoning and argumentation are essential for mathematical studies and learning. Deductive reasoning is the main learning objective of geometry in junior secondary school; however, it is one of the hardest skills to master for students. In the process of geometrical reasoning, figures and concepts interact simultaneously. The process from visualization to geometrical reasoning is mainly supported by the connection between the information from the figures and the ideas from individuals’ mind. An important reference can be provided by revealing students’ characteristics during their reasoning process. With the rapid development of science and technology, technological tools can provide effective support for construction, visualization, and reasoning. However, geometrical reasoning is still difficult for students and the tools are also not yet widely accepted in the learning of geometrical reasoning. This study aims at discussing the reasoning process and characteristics for the shifts from visualization to geometrical reasoning by students in senior one who have already learnt geometry. Using a semi-structured interview, three reasoning tasks are used. Students are asked to orally present their ideas about the tasks with a paper-and-pencil or dynamic geometry environment. Their reasoning process is analyzed by their transcripts and captions using the combination of Toulmin’s argument model and the reasoning features of dynamic geometry environment. Results showed that selecting argumentative strategies, finding the invariances, and using intuitive judgment are the main elements that can influence the reasoning process for students with different levels of knowledge. The geometrical imagination mainly supports an effective conjecture and then the dynamic geometry environment encourages the generalization of reasoning results. Hence, the shifts from visualization to geometrical reasoning can be described as the connection between geometrical knowledge and geometric objects by four levels: appearance-oriented, element-oriented, knowledge-oriented, and logic-oriented. By the structure of the figure and the relation of the sub-figures, the knowledge-oriented and logic-oriented levels are also described by two sub-levels respectively. Future studies may consider designing geometrical tasks that include various stages of reasoning process and different visualization levels for geometrical reasoning instruction. Students’ performance and difficulties with these geometrical tasks are required to analyze and discuss for developing suitable visualizations with different levels of geometrical reasoning.
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