Abstract:

A circle is one of the basic geometrical concepts that students are usually introduced to at primary school. This contribution deals with the understanding of this concept among Czech students at various levels of education. In contrast to the English language, Czech mathematical terminology makes a distinction between the term circle (closed curve) and disk (an area enclosed by a circle). In this paper, we use the term circle in the sense of a curve.

As part of our broader research into geometrical terms, students were assigned the task to determine the number of intersections of a circle and a straight line passing through its centre. The tests were given to approximately 1,500 Czech students from primary schools, lower secondary schools, and upper secondary schools including universities. The students’ responses were coded and subjected to quantitative analysis. Gender response dependencies and connections with other tasks in the test were also investigated. The possible reasons for the students´ responses were subsequently examined in semi-structured interviews with several other students. To gain a comprehensive view of the possible reasons, we also analysed the Czech textbooks used, specifically with regards to the introduction of the concept of a circle and mutual positions of a straight line and circle.

This paper presents the results of quantitative analysis of students' solutions and discusses the students' thought processes. The results revealed, surprisingly, that many students consider the centre of a circle to be a point of the circle. At the end of primary education, more than 36% of students answered that the given straight line has 3 points in common with the given circle. This misconception persists at the end of lower secondary education, where 29% of students gave the same answer. It is not until the transition to university education that the percentage of students who answered “3 points” drops to 13%. Statistically significant differences between the answers of males and females were confirmed only in the oldest students, i.e. at university level, whereby males answered more frequently correctly. The paper concludes with recommendations for pre-service mathematics teachers’ education.

Keywords:

Circle, straight line, intersection, geometrical concept.