Analysis of the Level of Geometric Thinking of Pupils in Slovakia

Abstract

This study is focused on the analysis of the level of geometric thinking of 15-year-old Slovak pupils in relation to the difficulty of geometric problems, their gender, and their assessment in mathematics. The main aim of this study was to determine the level of geometric thinking of 15-year-old Slovak pupils, to examine the relationship between their mathematics assessment and the level of geometric thinking, and to find out gender differences in relation to the different levels of geometric thinking. The van Hiele test was adapted and applied to a representative sample of 15-year-old Slovak pupils to determine the level of geometric thinking. We used reliability/item analysis. The reliability of the knowledge test (after adaptation) was assessed using Cronbach's alpha (0.64). The validity of the test was demonstrated by the correlation of the Usiskin test results with pupils' mathematics grades (Goodman-Kruskal's gamma, p < 0.05). Statistical analysis showed that 15-year-old Slovak pupils achieve different levels of geometric thinking depending on the difficulty of the tasks. Pupil achievement declined significantly as task difficulty increased. Pupils had the greatest difficulty with tasks classified as the fifth (rigorous) and partly the fourth (deductive) van Hiele level, which require a deep understanding of geometric systems and the ability to prove logically. The lower-level tasks (visualization, analysis, and abstraction) were able to differentiate students according to different levels of geometric thinking. The results showed a significant positive relationship (Goodman-Kruskal's gamma, p < 0.05) between the pupils' overall mathematics scores (expressed as a grade) and their level of geometric thinking as detected by the van Hiele test. The analysis of gender differences (Duncan's test, p < 0.05) showed that in the less challenging tasks, corresponding to the first three van Hiele levels (visualization, analysis, abstraction), girls performed statistically significantly better than boys. In the more challenging tasks, classified as the fourth (deductive) and fifth (rigorous) levels of geometric thinking, there were no statistically significant differences between boys and girls. In the more challenging tasks, the performances of both genders were comparable. The presented study identifies significant deficits in the development of higher levels of geometric thinking among 15-year-old Slovak pupils. These findings strongly imply the necessity for the transformation of the curriculum, textbooks, and didactic approaches with the aim of systematically developing deductive and rigorous reasoning, while it is essential to account for the demonstrated gender differences in performance.