Investigating grade 10 learners’ understanding of quadrilaterals in Euclidean geometry.

Abstract

With the growing focus on geometry education and the challenges South African learners face in understanding Euclidean geometry with reference to quadrilaterals, it is imperative, therefore, to examine the understanding and reasoning of learners, specifically in quadrilaterals. This study focused on investigating Grade 10 learners’ understanding and reasoning of quadrilaterals in Euclidean geometry, within the South African context. Being the entry grade into the Further Education and Training (FET) phase, a sound foundation in geometric reasoning at this FET stage is important for academic progression in mathematics. The objectives of the study seek to identify key factors that influence the understanding of learners in quadrilateral concepts, unearth common misconceptions they hold, and gain insights into learners’ reasoning in terms of the van Hiele model of geometric thought. A qualitative research approach was adopted within an interpretivist paradigm, which aids in understanding the phenomenon being studied. Data was collected from 21 participants from Grade 10 learners, by means of questionnaires, (multiple-choice and open-ended tasks) and semi-structured interviews. The data were analysed using a thematic approach aimed at interpreting the responses of learners to the given tasks. The analysis focused on learners’ reasoning abilities, conceptual understanding, and challenges with properties of geometry, proofs, and classification of quadrilateral shapes. The research findings reveal that learners experience significant challenges, such as misidentification of parallelograms and trapeziums, difficulty in classification of quadrilaterals, and mixing up ‘necessary’ and ‘sufficient’ conditions in geometric-related proofs. Further findings reveal that learners depend excessively on visual cues over analytical reasoning. Procedural errors in constructing proof and misappropriation of geometric terminology were also evident. Most learners’ reasoning in terms of the van Hiele level of geometry is still at the lower levels of Visualization and Informal Deduction, which impedes progression towards the higher level of deductive reasoning. The emerging critical factors that hinder the development of learners’ geometric reasoning were cognitive challenges, language barriers, and inadequate instructional support. The study highlights the importance of considering the use of structured, level-appropriate instructional approaches, in line with the van Hiele model. These include the use of reinforced vocabulary, visual aids, and guided informal proof tasks, which improve the geometric reasoning skills of learners. This will ensure that the identified challenges are addressed, and recommendations for curriculum development, classroom practices, and teacher development are provided as well.