The Vanx Hiele Model of Geometric Understanding and Mathematically Talented Students

Abstract

Mathematically talented students typically begin the traditional precalculus sequence by completing Algebra I in seventh grade or earlier. Consequently, they enroll in geometry early based on their successful completion of Algebra I. Little or no attention is paid to their readiness for geometry as indicated by such measures as their van Hiele level of geometric understanding. Logical reasoning ability is a characteristic often used to identify mathematically talented students, but how it applies to reasoning about geometry is unknown. This study investigated the geometric understanding and reasoning about geometry of mathematically talented students in the sixth through eighth grades prior to a formal course in geometry. This paper describes and analyzes the responses from 120 students who completed the van Hiele Geometry Test, developed by the Cognitive Development and Achievement in Secondary School Geometry Project (Usiskin, 1982), and 64 students who participated in 30–45 minute individual interviews, using an abbreviated version of Mayberry's van Hiele protocols.

Reform-Based Curriculum and Motivation in Geometry

“…Usiskin (1982), Mayberry (1983), and Burger & Shaughnessy (1986)) confirmed the validity of the levels and investigated students' behavior on tasks. Others (Usiskin (1982), Senk (1989), Gutierrez, Jaime, & Fortuny (1991), Mason (1997), and Gutierrez & Jaime (1998) evaluated and assessed the geometric ability of students as a function of van Hiele levels. Moreover, there have been some studies with pre-service elementary and secondary mathematics teachers regarding their reasoning stages in geometry (e.g., Mayberry, 1983).…”

“…Over the past few decades, researchers have documented that many students encounter difficulties and show poor performance in geometry classrooms in both middle and high schools (e.g., Burger & Shaughnessy, 1986;Crowley, 1987;Fuys, Geddes, & Tischler, 1988;Gutierrez, Jaime, & Fortuny, 1991;Mason, 1997;Halat, 2007). Moreover, research shows a decline in students' motivation toward mathematics courses (c.f., Gottfried, Fleming & Gottfried, 2001).…”

Pre-Service Classroom Teachers' Proof Schemes in Geometry: A Case Study of Three Pre-service Teachers

“…Despite the importance of geometry in our school and daily lives, there have been numerous studies concerned with the failure of students in geometry (Burger & Shaughnessy, 1986;Mason, 1997;Gutierrez, Jaime & Fortuny, 1991;Yavuz Mumcu & Cansiz Aktas, 2015). Since proofs are the heart of mathematics, and proving is complex, teachers should help their students develop these processes in the early grades.…”

Understanding kindergarten teachers’ perspectives of teaching basic geometric shapes: a phenomenographic research

“…However, many relevant researches have shown that teachers cannot adapt their geometry teaching to their students' levels (e.g. Mason 1997;Gutierrez and Jaime 1998).…”