Abstract
This paper investigates and considers factors that affect success in solving a stand-alone geometrical problem by 182 students of the 7th and 8th grades of elementary school. The starting point for consideration is a geometrical task from the National Secondary School Leaving Exam in Croatia (State Matura), utilising elementary-level geometry concepts. The task was presented as a textual problem with an appropriate drawing and a task within a given context. After data processing, the key factors affecting the process of problem solving were singled out: visualisation skills, detection and connection of concepts, symbolic notations, and problem-solving culture. The obtained results are the basis of suggestions for changes in the geometry teaching-learning process.
Highlights
Throughout all stages of mathematical education, students are involved in solving various types of mathematical tasks
According to Duval, a true understanding and a success of problemsolving in geometry are achievable when a student is capable of establishing connections among the spoken language, visual representation, and the symbolic notation (Duval, 1999, p. 25), where the visualisation skills are essential
The observed difficulties indicate the existence of important factors that affect the success in determining the problem solution: the visualisation skill while reading complex drawings, the knowledge and experience when working with certain concepts, the skill to apply symbolic notations to the observed elements in the drawing and the conduction of computations based upon applicable rules
Summary
Throughout all stages of mathematical education, students are involved in solving various types of mathematical tasks. To achieve a deeper understanding of geometrical concepts, a flexible transition between the spoken language, visual representation, and symbolic notations within the problem is required This problem-solving process that utilises multiple representations is neither linear nor simple but can be mastered by learning and teaching (Duval, 1999). The visual representation of complex conceptual structures requires high cognitive effort to observe and establish connections between adequate elements of these structures As such, this process is not a routine, nor is there a procedure for students to rely upon as there is when working with formal symbolic notations (for instance, linear equations solving), which is one of the reasons why students give up visualisation (Dreyfus, 1991), meaning that they have difficulties in geometry (too). They do not see what the teacher sees or believes that they will see. (p. 160)
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