An investigation into grade 12 teachers' understanding of Euclidean Geometry.

Abstract

The main focus of the research was to investigate the understanding of Euclidean Geometry of a group of Grade 12 mathematics teachers, who have been teaching Grade 12 mathematics for ten years or more. This study was guided by the qualitative method within an interpretive paradigm. The theoretical framework of this research is based on Bloom’s Taxonomy of learning domains and the Van Hiele theory of understanding Euclidean Geometry. In national matriculation examination, Euclidean Geometry was compulsory prior to 2006; but from 2006 it became optional. However, with the implementation of the latest Curriculum and Assessment Policy Statement it will be compulsory again in 2012 from Grade 10 onwards. The data was collected in September 2011 through both test and task-based interview. Teachers completed a test followed by task-based interview especially probing the origin of incorrect responses, and test questions where no responses were provided. Task-based interviews of all participants were audio taped and transcribed. The data revealed that the majority of teachers did not posses SMK of Bloom’s Taxonomy categories 3 through 5 and the Van Hiele levels 3 through 4 to understand circle geometry, predominantly those that are not typical textbook exercises yet still within the parameters of the school curriculum. Two teachers could not even obtain the lowest Bloom or lowest Van Hiele, displaying some difficulty with visualisation and with visual representation, despite having ten years or more experience of teaching Grade 12. Only one teacher achieved Van Hiele level 4 understanding and he has been teaching the optional Mathematics Paper 3. Three out of ten teachers demonstrated a misconception that two corresponding sides and any (non-included) angle is a sufficient condition for congruency. Six out of ten teachers demonstrated poor or non-existing understanding of the meaning of perpendicular bisectors as paths of equidistance from the endpoints of vertices. These teachers seemed to be unaware of the basic result that the perpendicular bisectors of a polygon are concurrent (at the circumcentre of the polygon), if and only if, the polygon is cyclic. Five out of ten teachers demonstrated poor understanding of the meaning and classification of quadrilaterals that are always cyclic or inscribed circle; this exposed a gap in their knowledge, which they ought to know. Only one teacher achieved conclusive responses for non-routine problems, while seven teachers did not even attempt them. The poor response to these problems raised questions about the ability and competency of this sample of teachers if problems go little bit beyond the textbook and of their performance on non-routine examination questions. Teachers of mathematics, as key elements in the assuring of quality in mathematics education, should possess an adequate knowledge of subject matter beyond the scope of the secondary school curriculum. It is therefore recommended that mathematics teachers enhance their own professional development through academic study and networking with other teachers, for example enrolling for qualifications such as the ACE, Honours, etc. However, the Department of Basic Education should find specialists to develop the training materials in Euclidean Geometry for pre-service and in-service teachers.