Developing geometrical reasoning (original) (raw)

The Interplay of Teacher and Student Actions in the Teaching and Learning of Geometric Proof

Educational Studies in Mathematics, 2005

Proof and reasoning are fundamental aspects of mathematics. Yet, how to help students develop the skills they need to engage in this type of higher-order thinking remains elusive. In order to contribute to the dialogue on this subject, we share results from a classroom-based interpretive study of teaching and learning proof in geometry. The goal of this research was to identify factors that may be related to the development of proof understanding. In this paper, we identify and interpret students' actions, teacher's actions, and social aspects that are evident in a classroom in which students discuss mathematical conjectures, justification processes and student-generated proofs. We conclude that pedagogical choices made by the teacher, as manifested in the teacher's actions, are key to the type of classroom environment that is established and, hence, to students' opportunities to hone their proof and reasoning skills. More specifically, the teacher's choice to pose open-ended tasks (tasks which are not limited to one specific solution or solution strategy), engage in dialogue that places responsibility for reasoning on the students, analyze student arguments, and coach students as they reason, creates an environment in which participating students make conjectures, provide justifications, and build chains of reasoning. In this environment, students who actively participate in the classroom discourse are supported as they engage in proof development activities. By examining connections between teacher and student actions within a social context, we offer a first step in linking teachers' practice to students' understanding of proof.

Teaching geometry in schools: an investigative rather than instructive process

2007

Research has documented the prevalence of lessons characterised by homework check, followed by teacher lecture and demonstration, followed in turn, by learner practice sequence of classroom instructional activities in our classrooms. This sequence of classroom activities does not allow for the development of sound mathematics practices and mathematical proficiency. Meanwhile, curriculum reforms in South Africa as well as in other parts of the world recommend classroom activities where teachers create opportunities for, listen to and extend learners. This paper presents a sequence of activities to be used in the teaching of geometry and surface areas of solid shapes in a grade 8 classroom. The sequence portrays the teaching of these concepts as an investigative rather than instructive process.

Exploring pre-service teachers' knowledge of proof in geometry

2013

My gratitude is extended to the following people for their contributions and supports in making this study a success: My Supervisor and mentor Dr Jayaluxmi Naidoo, who tirelessly guided and monitored my p rogress t hroughout my s tudy. Her s upport, i nspiration, pa tience, e ncouragement, and unde rstanding w ith m y s hortcomings w ere ve ry ov erwhelming. Her w ise an d constructive remarks have been highly appreciated. My wife Nokukhanya, for l ove, m oral s upport, encouragement, words of i ntelligence, and everlasting patience. My da rling da ughters, Zenande a nd A benathi f or c onstant l ove, l aughs, i nsightful debates about the merits of the study and for allowing me to use the time I suppose to spend with them. My c olleague, M rs Lindiwe P. B uthelezi, w ho i s the E nglish e ducator a nd H ead o f Department (HOD) for languages, for her extended assistance she has provided.

The Teaching of Geometry in Schools

The Mathematical Gazette, 1929

Research has documented the prevalence of lessons characterised by homework check, followed by teacher lecture and demonstration, followed in turn, by learner practice sequence of classroom instructional activities in our classrooms. This sequence of classroom activities does not allow for the development of sound mathematics practices and mathematical proficiency. Meanwhile, curriculum reforms in South Africa as well as in other parts of the world recommend classroom activities where teachers create opportunities for, listen to and extend learners. This paper presents a sequence of activities to be used in the teaching of geometry and surface areas of solid shapes in a grade 8 classroom. The sequence portrays the teaching of these concepts as an investigative rather than instructive process.