The use of semiotic representations in reasoning about similar triangles in Euclidean geometry

Abstract

Many pre-service mathematics teachers in South Africa are apprehensive about the content of Euclidean geometry, because they did not study Euclidean geometry in high school but will be expected to teach the content when they start their teaching career. This article reports on a study that explored the role of semiotic representations in pre-service teachers’ reasoning about the similarity relationship between triangles. Data were generated from the written responses of 65 pre-service mathematics teachers as well as three semi-structured interviews. Duval’s notions of conversions and treatments were used as a framework to understand the pre-service teachers’ struggles with negotiating movements between the visual and symbolic registers of representation. The findings revealed that many pre-service teachers struggled with identifying the similarity relationship between triangles appearing in various configurations of geometric objects. While some participants were easily able to draw upon the two registers to express the relationships, one student who initially made many errors was only able to discern the necessary relationships with the help of a concrete representation that could be physically manipulated. The study therefore provided an example of how a student’s errors could be used as a learning resource to lead to meaningful learning.