Students can use their knowledge of the van Hiele levels, proof, and the NCTM Standards to evaluate geometry curriculum for appropriate sequencing. Throughout the instruction at different van Hiele levels, students can continually evaluate the role that proof is playing for the learners and understand how to adjust their instruction to ensure that proof activities are meaningful. TeachingStudents can assess the van Hiele level at which their students are operating and design tasks and lessons to move learners to a higher van Hiele level, eventually to the level of deduction and proof. They can use these understandings to prove general cases of geometric theorems and apply these theorems to solve problems in Euclidean geometry. Euclidean GeometryStudents understand the central concepts, conditions, definitions, theorems, assumptions, structure and extensions of high school (Euclidean) geometry. Critical StanceStudents can apply principles of quality research to analyze and critique research on the teaching and learning of geometry, and understand the affordances and constraints of research paradigms and methodologies in this body of research. They can use this knowledge to evaluate current instructional, curricular, and policy recommendations of leaders in mathematics education regrading the teaching of geometry in the public schools. ScholarshipStudents understand and can evaluate the role of proof in geometry and in mathematics in general, the theories of how students progress through the van Hiele levels in learning geometry and how these two topics are related.
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