Many of the new elementary sCHOOL and middle school mathematics curricula provide students with opportunities to encounter geometric transformations. Some of the curricula (e.g., MATHThematics, Math Trailblazers, and Investigations in Number, Data, and Space) introduce transformations (such as translations, reflections, and rotations) as motions (slides, flips, and turns) undergone by familiar shapes. How can we make the most of these experiences to ensure that our students are prepared for high school geometry courses? One way is to find out what students understand about transformations and to address any misconceptions they have regarding transformations. In working with eight eighth-grade prealgebra students, I identified one possible misconception as they completed a set of activities using The Geometer's Sketchpad (Jackiw 2001). The students focused on the path followed by a shape, rather than on the relationship between the preimage and image shapes, thereby leading them to incorrect conclusions regarding the equivalency of transformations. The technology-based activities helped some of the students begin to rectify this misconception.
A Historical Process Analysis and Extension of Division into Equal Parts in Middle School Geometry
