The Alignment of Ordering and Space in Arithmetic Computation

The Alignment of Ordering and Space in Arithmetic Computation David Landy (dlandy@indiana.edu) Department of Computer Science Indiana University, Bloomington, IN 47408 USA Robert L. Goldstone (rgoldsto@indiana.edu) Department of Psychological and Brain Sciences Indiana University, Bloomington, IN 47408 USA Abstract may be larger or more salient than another, or bear other non-formal but readily accessible features and relations. Although these properties are not formally meaningful, it is quite likely that reasoners are sensitive to such regularities, and use them to build successful interpretations. The major goal of this paper is to explore the role that one such property—physical spacing—plays in simple arithmetic computations. Arithmetic equations are often non-uniformly laid out on a page. For instance, in mathematical typesets and typesetting programs such as LaTeX, multiplications are automatically spaced more closely than additions. Handwritten expressions are more variable in their appearance, but our data suggests that this spacing regularity also exists in handwritten expressions (Landy & Goldstone, In Press A). In this paper we explore three hypotheses about how reasoners might use regular addition and multiplication spacing in building arithmetic interpretations. According to the operator feature hypothesis, participants might be sensitive to the relational spacing regularities of various operators when they identify operator type. If an operator is widely spaced in a particular expression, therefore, it is assumed to be more likely an addition than a multiplication. This assumption may lead to people interpreting 2+4 * 7 to equal 56, if they interpret the “+” as a “*” because the close spacing leads them to perform the calculation involving the “2” and “4” before the calculation involving “4” and “7. ” The proximity-precedence alignment hypothesis proposes that people will be sensitive to the visual hierarchy of perceptual groups present in an equation, and that this sensitivity will cause closer objects to be combined first (Kirshner, 1989; Kirshner & Awtry, 2004). Thus, the expression 2+4 * 7 is likely to be interpreted as 42, rather than (the formally correct) 30, because the spatial proximity of the threes will incline perceivers to group them formally. In this paper, we will refer to expressions as consistent if higher-precedence operations are more closely spaced, inconsistent if higher-order operations are more widely spaced, and neutral if evenly spaced. Finally, because equations are typically read from left to right, we expect that expressions with products placed to the left of sums will be easier to solve (the expression reading hypothesis). These hypotheses will be measured against a null hypothesis of no effect of physical layout, which although not essential to any particular theory, has generally served as a default view in discussions of symbolic reasoning (e.g., Anderson, 2005; Stenning, 2002). Two experiments explore the role that spatial format has on arithmetic computations. In printed arithmetic expressions containing both multiplications and additions, terms that are multiplied are often placed closer together than terms that are added. These experiments test whether that external tendency plays a role in how reasoners construct interpretations of simple compound arithmetic expressions (such as “2 + 3*4”). Evidence is found to support two such influences: First, reasoners use spacing information when choosing an operation to apply (e.g., whether to add or to multiply two numbers). Second, reasoners also use spacing patterns to select an order in which to apply operations in compound expressions. Terms placed closer together tend to be computed sooner. Although spatial relationships besides order are entirely formally irrelevant to expression semantics, reasoners use these regular spatial relationships to support their success with various formal properties. Keywords: symbolic processing, mathematics, embodied cognition, relational reasoning Introduction One of the central challenges facing the cognitive study of mathematical reasoning is symbolic interpretation: how do people use symbol strings as carriers of meanings? In the domain of mathematics, formal rules specify how terms are supposed to be interpreted. Despite the simplicity and explicitness of these rules, numerous studies have noted that difficulties generating solutions from mathematical expressions often result from failures to correctly read and understand symbolic notation (Koedinger & MacLaren, 1997; Koedinger & Nathan, 2004; Sfard & Linchevski, 1994). Understanding and predicting such difficulties requires that cognitive scientists learn more about how reasoners process formal notational systems. Cognitive conceptions of abstract formal interpretation generally follow formal logics by assuming that reasoners explicitly represent rules of combination, and apply those rules to symbolic expressions (Fodor, 1975; Marcus, 2001). In this view, the role of perception is to identify and represent for internal consumption individual symbols written in the external notation. Problems that involve more, or more difficult, rules are predicted to be harder to solve, but perceptual factors should only affect the transcription of the external notation into internal symbols. In addition to their formal properties, commonly used symbol systems have many informal properties as well: pairs of symbols may be similar or dissimilar, or one symbol