Interference of prototypical geometry representations in students’ construction of concepts and development of proofs

ABSTRACT

This study investigates the impact of prototypical geometric representations on students’ ability to construct accurate geometric concepts and their subsequent performance in proving tasks related to those concepts. Ninety-one 11th-grade students participated in the study. We intentionally designed and utilized a questionnaire as the primary data-collection tool. We used the questionnaire items and findings as a basis for semi-structured interviews that we conducted with approximately 10% of the study participants. This allowed the integration of the quantitative and qualitative analyses, to provide a comprehensive understanding of the learning processes. Our analysis reveals that many Grade 11 students struggle with geometric concept formation due to an overreliance on prototypical examples, which in turn hinders their ability to construct proofs concerning the attributes of these concepts, highlighting the need for more diverse representations in geometry instruction.

KEYWORDS:

• Students’ difficulties in geometric concept construction
• perceptual difficulties
• prototypical phenomenon
• impact of difficulties on students’ proving ability

Disclosure statement

No potential conflict of interest was reported by the author(s).

Ethical approval

The authors fulfilled all ethical responsibilities of authors applicable for this journal.

Data availability statement

The data that support the findings of this study are available on request from the corresponding author.

Notes

1. Teachers frequently use squares as diagrams to represent general quadrilaterals, which can lead to confusion. However, this issue is less prevalent in diagrams used as illustrations for textbook proofs. This observation introduces another potential explanation for students’ difficulties with Task 1. It’s possible that some students accepted Salim’s proof because they resolved the conflict between the stated hypothesis (general quadrilateral) and the diagram (square) by assuming the hypothesis was misstated, and that a square was intended.

Additional information

Notes on contributors

Aehsan Haj Yahya

Aehsan Haj-Yahya holds a bachelor’s degree in mathematics and computer science, complemented by two additional degrees in Teaching Mathematics. With a wealth of experience, Aehsan has dedicated years to teaching mathematics in high schools and leading as a coordinator for the subject. Transitioning to academia, he has spent over a decade instructing prospective teachers in teacher training colleges, specializing in mathematical education.

Rina Hershkowitz

Rina Hershkowitz – Was a founding member of the Mathematics Group in the Science Teaching Department at the Weizmann Institute of Science Israel. Presently she is an Academic Consultor in the department. Her activities focus on curriculum development, teachers’ education and research. Her main research interests lie in the domains of: “Cognitive thinking and learning in Geometry of students and teachers”; “Learning Mathematics with computers” and the “Abstract in Context theory”.