Descriptions of methodologies that can be used to create items for assessing teachers' professionally situated knowledge are lacking in mathematics education research literature. In this study, researchers described and used the Delphi method to design an instrument to measure teachers' pedagogical content knowledge. The instrument focused on a specific area of middle school geometry and measurement, decomposing and recomposing figures in 1-D and 2-D. The Delphi methodology used included three rounds of surveys of a diverse panel of 20 experts. The data analysis and data collection were done reflectively. Initial items were developed based on the research literature in the field of geometry education and teacher knowledge. New categories for the analysis emerged from the data and were used to complete the analysis. Consensus among the experts was reached after the third round of surveys, producing the assessment instrument. The researchers described strengths and limitations of this methodology for designing assessment tools of teachers' knowledge. The researchers suggested that Delphi methodology could also be used to design and validate broader instruments related to other types of professional knowledge of teachers in different fields.

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.

When calculating the area of a trapezoid, students use a range of problem-solving strategies and measurement concepts.