Connections Built and Lost within and Across Geometries with Transformations in School Geometry
The main theme of this talk is connections built and lost within and across geometries with a look into historical reforms in school geometry. The presentation has three parts. First, I exemplify my classroom experiments to advance teachers’ knowledge of connections between different geometries by making familiar objects unfamiliar in alternative geometries. Definitions and the extensibility of the geometric objects such as rhombus or squares across Euclidean and non-Euclidean geometries are examined to build an advanced knowledge of geometry for preservice secondary mathematics teachers. Teachers’ interpretations of intensional content of the geometric objects are analyzed and discussed with their connections within and across geometries. Second part presents classroom experiments with preservice elementary teachers in designing and implementing mathematical modeling task sequences to foster disciplinary connections of geometry and other mathematical disciplines. Dual Modeling Framework by Matsuzaki and Saeki (2013) is here extended to study the theory and practice of geometry with mathematical modeling perspective. The crosscutting idea between the modeling tasks is the Golden Ratio building on proportional reasoning. Participants, for the geometric modeling task, use physical manipulatives and digital manipulatives such as GeoGebra in geometric modeling of Stars as in Texas Lone Stars, extending the idea to the constructible numbers and Golden ratios. For the modeling tasks bridging geometry and statistics, students analyze their data to describe goodness of a star building on the proportional relationships observed in their sample of stars including their hand-drawn stars. Following Blum and Ferri (2009) and Anhalt and Cortez (2015), analysis of students’ responses are given along the phases of mathematical modeling with a focus on mathematizing and validation phases in the dual mathematical modeling tasks with geometrical and statistical approach. The last part of the presentation is about the connections lost by providing a critical look at SMSG axioms with its consequences on the geometry instruction towards removing the practice of constructability from school geometry. The SMSG revised the axiomatic foundations for school geometry building on the earlier reform by David Hilbert’s Foundations of Geometry where the algebra of segments was introduced as a continuation of Descartes’ Geometric Algebra perspective. The emerging student practices during the classroom experiments are examined on the algebra of segments. The unintended consequences of curricular reforms will be discussed with implications on disrupting the continuity of historical practices on school geometry.
Duration: 60 minutes
Format: Online seminar via Zoom web meeting software with questions and discussion. Detailed instructions for joining the seminar will be emailed to registered participants.