Dissertation committee:
Θεοδόσιος Ζαχαριάδης, Ομότιμος Καθηγητής, Ε.Κ.Π.Α. (επιβλέπων)
Δέσποινα Πόταρη, Καθηγήτρια, ΕΚΠΑ (μέλος τριμελούς συμβουλευτικής επιτροπής)
Γεώργιος Ψυχάρης, Αναπληρωτής Καθηγητής, ΕΚΠΑ (μέλος τριμελούς συμβουλευτικής επιτροπής)
Χαράλαμπος Σακονίδης, Καθηγητής, Δ.Π.Θ.
Χρυσαυγή Τριανταφύλλου, Επίκουρη Καθηγήτρια, ΕΚΠΑ
Αγγελική Μάλη, Αναπληρώτρια Καθηγήτρια, Πανεπιστήμιο Κρήτης
Λεωνίδας Κυριακίδης, Καθηγητής, Πανεπιστήμιο Κύπρου
Summary:
The concept of infinity permeates various scientific fields, including Mathematics, Physics, and Philosophy, and has been a subject of systematic study for centuries. The real world as we know it is finite and therefore there is no reason to discuss the infinity. The concept of infinity is contradictory, as it confronts our cognitive schemas that have been adapted to a finite world.
Despite the frequent reference of the concept of infinity in the mathematical textbooks, it is rarely studied extensively as it is often avoided or obscured while teaching. As a result, a right intuitive approach to the concept could help not only in a better understanding but also in achieving the learning objectives set for the teaching of mathematics at school.
This thesis examines the intuitive perceptions of elementary and secondary school students regarding the concept of infinity. Specifically, it attempts to identify the prevailing perceptions that students form about the concept of infinity, to explore the factors that shape them and to identify the differentiations that arise at different stages of school education.
The present study is combining quantitative and qualitative research. The quantitative data were collected from the written responses of 377 students (124 6th graders, 154 9th graders, 99 12th graders) to a questionnaire. In addition, 39 individual interviews were conducted (17 6th graders, 16 9th graders, 6 12th graders). Data analysis was conducted using both quantitative and qualitative methods. Initially, the quantitative data were categorized and analyzed, a procedure which highlighted the basic perceptions of infinity in the three educational levels studied. Afterwards, through the analysis of the interviews, their origin was investigated.
The results revealed that the perceptions about infinity can be categorized into four main categories. The first two are connected to the potential and active infinity. These are the procedural perception, in which infinity is viewed as an endless procedure and the actual perception, in which infinity is considered to be something completed such as the set of all numbers. The third perception is the finite one, according to which, students consider infinity to be something very large but finite. The fourth perception considers infinity to be something indefinite.
Furthermore, from the study emerged that the origin of perceptions of infinity is complex and highly influenced by education, everyday experiences, and the socio-cultural environment. The use of the word "infinity" in everyday language differs from its mathematical concept, a fact that creates difficulties in understanding it.
Based on the analysis of students' perceptions and performances on infinity-related topics in the three grades studied, no statistically significant differences were found in the above perceptions. This observation suggests that the development of school mathematical knowledge does not seem to significantly affect perceptions about infinity.
The study concludes that teaching the concept of infinity requires attention and flexibility and the diverse perceptions of students must be carefully considered. Utilizing different learning strategies, engaging with infinity-related topics and educators' awareness of the basic perceptions and common misconceptions that students exhibit, can contribute to the creation of correct intuitions about infinity in school mathematics education.
Keywords:
infinity, intutitive perception, teaching of calculus, primary education, secondary education
