Assessing Readiness for Geometry in Mathematically Talented Middle School Students

Abstract

Mathematically talcnted m i d & grade students are traditionally served by beginning Algebra I (and the traditionalpre-caliussequence) in seventh, or even sixth, grade. They thus enroll in geometry early, with lit& attention paid to the fact that readiness for and sucyss in algebra requires diferent skillr and types of understanding than readiness for and success in geomeq. Research about geometric understanding in regular andgifed students, as well as research predicting success in high school geometry chses, sug;gests that the decision to place mathematically talented students in geometry classes at a younger than typicai age should be made on the basis of more infirmation than the successfii completion ofAlgebra L This paper describes a procedure for assessing geometry readiness in mathematically ablc mi&k school studnts.

The difficulties in geometry: A quantitative analysis based on results of mathematics competitions in Italy

“…These include the theory of the development of geometric thinking by Dutch teachers Pierre van Hiele and Dina van Hiele-Geldof (Fuys et al, 1984;van Hiele, 1986). Noting the difficulties their students had in geometry, they developed a theory that introduces different levels of thinking that students go through as they move from simply recognizing a figure to being able to describe a formal proof (Clements & Battista, 1992;Crowley, 1987;Mason, 2009, Mason & Moore, 1997Sbaragli & Mammarella, 2010;Usiskin, 1982). Five levels are indicated by van Hiele (1986), which are sequential and hierarchical: level 1 (visualization), in which one recognizes figures only by their appearance without perceiving their properties; level 2 (analysis), in which one recognizes the properties of figures without, however, perceiving the relationship between them; level 3 (abstraction), in which one perceives the relationships between properties and figures without, however, yet understanding the role and significance of formal deduction; level 4 (deduction), in which one is able to construct demonstrations, understand the role of definitions and axioms and know the meaning of necessary and sufficient conditions; and, finally, level 5 (rigor), in which one understands the formal aspects of a demonstration.…”

Investigating Pre-Service Secondary Mathematics Teachers’ Reasoning When Learning Trigonometry Using The Line Segment Approach

“…Mathematical reasoning being dynamic and unique and individual-based, we adopted Lithner's (2008) characterization of the different types of reasoning to investigate the types of reasoning pre-service teachers engage in while learning trigonometry. Several researchers have contended that mathematical reasoning is best evaluated by how a learner's performs on geometric tasks (Mason & Moore, 1997;Wu, 1996). In other words, geometric reasoning follows from successfully establishing mathematical reasoning (Clements &Battista, 1992).…”