Promoting Development of Geometry Concepts: Interfacing Multiple Embodied Representations with a Computer Game

Promoting Development of Geometry Concepts: Interfacing Multiple Embodied Representations with a Computer Game Jonathan Michael Vitale (JMV2125@Columbia.Edu) Department of Human Development, Teachers College, 525 W. 120 th Street New York, NY 10027 USA John B. Black (Black@Exchange.TC.Columbia.Edu) Department of Human Development, Teachers College, 525 W. 120 th Street New York, NY 10027 USA Michael Swart (MIS2125@Columbia.Edu) Department of Human Development, Teachers College, 525 W. 120 th Street New York, NY 10027 USA Abstract In this study we examined the effectiveness of a computer- based geometry learning tool with 4 th grade students. In both an experimental and control condition children constructed a series of common four-sided figures to fit a set of visual constraints. Additionally, children in the experimental condition were required to validate the presence of parallel segments, congruent segments, or right angles embedded in their figures with the assistance of a visual depiction of the property’s spatial significance. Following training we conducted six identification tasks in which participants attempted to discriminate two valid members of a given polygon class from four displayed polygons. In all six instances, varying polygon, instruction, and feedback type, children in the experimental condition were more likely to correctly identify both class members than children in the control condition. Keywords: embodied cognition, mathematical development, education, cognitive development Introduction Core mathematical abilities may emerge from either innate processes or general human experiences (Lakoff & Nunez, 2000). Yet, the development of mature concepts and skills depends on one’s cultural experiences – such as schooling. For example, while number sense may emerge from general spatial processes (Dehaene, 1997), the development of mature numerical representation is moderated by exposure to formal and informal activities (Siegler & Ramani, 2008). Furthermore, Siegler and Ramani (2008) successfully demonstrated that persistent SES-based disadvantages could be overcome by effective use of appropriate tools (i.e., linear board games). Cognitive research in geometry may follow a similar trajectory. The core systems approach – which has been applied successfully in numeracy – has recently been invoked to explain the cognitive basis of Euclidean geometry. Specifically, Spelke, Lee, and Izard, (2010) claim that geometric concepts are grounded in two core cognitive systems, typically associated with spatial navigation and object perception. The latter system is hypothesized to support the concepts of lengths and angles, which are critical to school-based geometric activities. Yet, while lengths and angles are certainly perceivable attributes of objects, other qualities, such as bilateral symmetry (Quinlan & Humphreys, 1993), may be more salient. In some cases these perceptually salient features may interfere with formal classifications of geometric objects. For example, Mach (1886/1959) demonstrated that a square may be perceived as a diamond when its diagonal is oriented vertically. Similarly, in a study that applied nonmetric multidimensional scaling to a set of four-sided figures – a technique in which perceived dissimilarities between objects are spatially represented for analysis – perceived attributes confounding lengths and angles emerged, including “dispersion” (irregularity), “elongation”, and “jaggedness” (Behrman & Brown, 1968). Likewise, Shepard and Chipman (1970) found similar dimensions in participants’ categorizations of U.S. state shapes. While a variety of factors affect natural object perception, perhaps the more narrow set of educationally-relevant geometric figures (e.g. square, equilateral triangle) better afford conceptualization by normative (Euclidean) class- based properties (e.g. parallel lines). However, interview data demonstrates that young children categorize polygons based on informal perceptual attributes, such as “slanty”, “pointy”, or “skinny” (Clements, Swaminathan, Hannibal, & Sarama, 1999). As the Mach (1886/1959) square- diamond demonstration suggests, informal concepts play a role in geometric representation into adulthood. Yet, over two thousand years of geometry as a formal discipline suggests that humans are not bound by superficial percepts. Abstractly, geometric concepts may be conceived in terms of strict hierarchy, facilitating deductive reasoning. Such organization suggests an internal representation similar to the discrete, hierarchical, propositional network of Quillian (1966). Yet, rather than viewing the development of geometry concepts as a shift from perception to abstraction, one may conceive of development in terms of perceptual