Teaching the Perceptual Structure of Algebraic Expressions: Preliminary Findings from the Pushing Symbols Intervention

Teaching the Perceptual Structure of Algebraic Expressions: Preliminary Findings from the Pushing Symbols Intervention Erin Ottmar (erin.ottmar@richmond.edu) David Landy (dlandy@richmond.edu) Robert L. Goldstone (rgoldsto@indiana.edu) 1101 E. 10 th St., Indiana University Bloomington, IN 47405 USA Department of Psychology, 28 Westhampton Way University of Richmond, VA 23173 USA Abstract We describe an intervention being developed by our research team, Pushing Symbols (PS). This intervention is designed to encourage learners to treat symbol systems as physical objects that move and change over time according to dynamic principles. We provide students with the opportunities to explore algebraic structure by physically manipulating and interacting with concrete and virtual symbolic systems that enforce rules through constraints on physical transformations. Here we present an instantiation of this approach aimed at helping students learn the structure of algebraic notation in general, and in particular learn to simplify like terms. This instantiation combines colored symbol tiles with a new touchscreen software technology adapted from the commercial Algebra Touch software. We present preliminary findings from a study with 70 middle-school students who participated in the PS intervention over a three-hour period. Keywords: Algebra education; learning; perception; mathematical cognition Introduction The core conceptual content of algebra is extraordinarily simple: it is largely exhausted by the properties of addition and multiplication over the real numbers, such as commutativity, associativity, and distributivity, together with basic properties of functions and equivalence relations over the same structure. This formal simplicity belies the great difficulty students have in mastering basic algebra content (NAEP, 2011) — and especially the notation universally used to express algebraic claims (McNeil, 2008; Koedinger & Alibali, 2008). One way to explain the difficulty of algebra is that unlike number cognition, algebraic reasoning does not seem to fit neatly into a core conceptual domain (Dehaene, 1997; Carey, 2009). Children may then face the challenge of assembling new cognitive tools appropriate to algebraic interactions. This task is made more challenging because typical instruction in basic algebraic notation is often brief and involves an emphasis on memorization of abstract rules. Algebraic literacy—the fluent construction, interpretation, and manipulation of algebraic notations—involves not just memorizing rules, but also learning appropriate perceptual processes (Goldstone, Landy, & Son, 2010; Kirshner, 1989; Landy & Goldstone, 2007, 2008, 2010; Kellman, Massey, & Son, 2010). Like other formal diagrammatic systems (such as, for example, Venn diagrams) algebraic notation aligns the structure of the content domain with automatic perceptual properties and necessary physical laws (Cheng, 1999; Landy, Allen, and Anderson, 2011; Landy, 2010). In this way reasoning that is properly cognitive can be accomplished by perceptual-motor systems such as attention (Patsenko & Altmann, 2010) or perceptual organization (Landy & Goldstone, 2007; Novick & Catley, 2008). Although such transformation of cognitive work into perceptual processing may the carry distinctive risk of mistaking perceptual properties of representations for content principles (Novick & Catley, 2007; Kirshner & Awtry, 2004), it may also be critical to reducing cognitive load in complex operations (Sweller, 1994). Successful students often use perceptual and visual patterns available in notations to solve mathematical problems. Like many skills learned from long practices learning algebra involves perceptual training- learning to see equations as structured objects (Landy and Goldstone, 2007; Kellman et al., 2008; Kirshner & Awtry, 2004). For instance, people seem to group symbols into perceptual chunks and use these groups, rather than just calculation rules, to perform mathematics. Although in some cases the appropriate perceptual patterns are fairly easy to see (Kirshner & Awtry, 2004), in other cases understanding the visual forms requires that a learner internalize an appropriate way of seeing a piece of notation. Real-world motion, changes, and transformations are naturally memorable and easy to acquire, making these processes natural tools for helping students grapple with algebra (Landy, Some successful object-centered transformations, however, may not be as immediately obvious as others in traditional instruction. Therefore, training students to see the structure of algebra may be a promising approach to teaching algebraic ideas. While this perceptual-motor understanding of algebraic forms is a potentially rich and powerful source of student understanding, it also stands as a barrier to learning if visual patterning is not taught in a controlled manner. While some students learn easily, others latch on to incorrect perceptions and, consequently, generalizations (Marquis, 1988; Kirshner, 1989; Nogueira de Lima & Tall, 2007). Our goal is to find instructional and pedagogical paths through which students can make use of the strength of perceptual patterns in algebraic notation without falling prey to misleading visual structures or overly procedural, low-level understandings. Pushing Symbols: Teaching the Structure of Algebraic Expressions The purpose of the PS intervention is to explore an alternative method of algebra instruction that focuses