PCMI, Summer 2008
Secondary School Teacher Program

Modular Origami in the Secondary Geometry Classroom
by Peg Cagle

Cube Instructions

 By the time students are in a secondary high school course, they are expected to be operating at Van Hiele level three (informal deduction) and moving to level four (formal deduction). Yet most of the curricula they have encountered in their k-8 mathematics preparation only address geometry at level one (visualization; recognizing figures by their physical appearance), focusing on naming shapes and relationships such as parallel or perpendicular, with at best, infrequent opportunities to develop geometric reasoning. Today's children do not play in the same ways as children of past generations and this is having a significant negative impact on the development of their spatial reasoning and visualization abilities. People of earlier generations grew up routinely engaged in activities that built spatial reasoning. They learned to sew, following diagrams and using patterns to transform two-dimensional fabric into three-dimensional garments. They played with Lincoln Logs, Tinkertoys and other blocks well past the age of four and may have even graduated to Erector sets complete with motors and gears. They assembled jigsaw puzzles and built scale models. They climbed trees, played on jungle gyms and built impromptu forts from card tables and sofa cushions rather than playing with pre-fabricated plastic environments. Wood, metal and plastics shop classes were a staple in junior high schools across the country, but are now virtually gone. While many of today's students play with simple puzzles and build with Lego's as young children, most quickly move to video games, where they spend their time manipulating two-dimensional images of three-dimensional worlds. As a result, they arrive in secondary geometry classrooms with weak spatial reasoning and poor visualization skills. Given this seemingly daunting confluence of issues, it is reasonable to ask why origami should be added to the already over-crowded curriculum. The answer is that origami can provide multiple opportunities to remediate these areas of weakness in an inherently engaging manner. It is accessible, affordable, non-routine and allows students to produce objects of their own. Building a model for one's self provides a profoundly deeper understanding of the geometry of an object, than engaging in a study of either a pre-made model or a two-dimensional representation. Educators need to intentionally move students from concrete to abstract and in order to do that, students need to be provided with support along the way with intermediate steps. Before it is possible to accurately visualize in three-dimensions, students need to be taught to look at and analyze three-dimensional objects. Developing this ability does not happen overnight, but can be built over time through repeated exposure to meaningful exercises.

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