Exploring pre-service teachers’ understanding of similarity and proofs in Euclidean geometry.

Abstract

This qualitative study explores pre-service teachers’ understanding of similarity and proofs in Euclidean geometry in one South African university from KwaZulu-Natal (KZN) province. Such insight is vital for addressing pre-service teachers’ geometric knowledge, which has been found lacking. The research participants were 34 pre-service teachers (PSTs) in their first year of study towards a Bachelor of Education degree specialising in mathematics at the FET phase. A pen and paper test and semi-structured interviews were employed in gathering the required data for this study. The Van Hiele levels of geometric thought were used as a theoretical framework, which formed the basis for the analysis and discussion of findings. The findings indicated that most pre-service teachers performed adequately on familiar items but struggled with those unfamiliar, which were not typical grade 12 examinable questions. A follow-up of semi-structured interviews was conducted with seven PSTs of mixed abilities to probe the originality of their written responses. Although all interviewed PSTs indicated an improvement when responding to research items verbally than in writing, they did not reach the expected acquisition necessary to teach geometry effectively. Overall, this study found that many PSTs displayed poor levels of understanding similarity and proofs, including (1) limited understanding of the definition of similarity to triangles; (2) poor understanding of how to prove two figures are similar; (3) the haphazard use of geometric theorems in devising proofs; (4) a display of higher Van Hiele levels of understanding for familiar items but lower levels of understanding for unfamiliar items. These findings raised concerns about this group of PSTs teaching geometry, especially if certain concepts require more complex skills that are slightly beyond the secondary school curriculum. It is recommended that professional teacher education training offered to pre-service teachers should include aspects such as (1) Improving PSTs’ geometry content knowledge, (2) Teaching geometry for understanding and (3) Improving PSTs’ written mathematical responses. These factors may be pivotal in improving pre-service teachers’ geometric knowledge beyond the scope of the secondary school curriculum.