How Space Guides Interpretation of a Novel Mathematical System

How Space Guides Interpretation of a Novel Mathematical System 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 for instance, the c is closer to the exponent than to the b, and b is closer to the c than to the a. This physical relationship maps directly onto the corresponding order of precedence: c should first be squared, then the result multiplied by b, and a should be added to the result. The correspondence is far from perfect, especially when expressions are hand-written (Landy & Goldstone, in press A), but nevertheless, there is a general relationship between physical and syntactic proximity in mathematics, exacerbated by the frequent omission of the multiplication sign in algebra (of four prominent algebra textbooks (McGraw-Hill 1998; Cord Communications 2004; Holt, Rinehart and Winston, 2004; McDougal Littell, 2004), each used at least one times sign convention which was more closely spaced than plus signs, besides omission. None ever did the reverse). People can learn and understand formal rule systems that lack the kind of perceptual-syntactic regularities algebra contains. The question addressed here is whether and in what ways salient perceptual regularities, when present, are used by people learning novel formal structures. This paper investigates how people build interpretations of compound mathematical expressions in a novel formal system. In traditional arithmetic, interpretations are guided by an order of precedence convention (times and division precede addition and subtraction). This order is supported by a spatial convention that supports the order of precedence. In the experiment described here, participants learned computation tables of two simple novel operators, and then were asked to discover a precedence rule. The operators were presented with a physical spacing convention that either aligned with the precedence order, opposed it, or randomly opposed or aligned with the precedence order. Participants were more likely to reach a criterion of successful performance when the order of operations aligned with the precedence order, and did so more quickly than either other group. The results indicate that reasoners integrate salient perceptual cues with formal knowledge following familiar conventions, even on novel systems. Keywords: Mathematical cognition, embodied cognition, formal reasoning, symbolic processing Introduction Background Kirshner (1989) explored the correspondence between spatial proximity and arithmetic syntax by creating a novel but natural set of symbols for the basic operations: M for multiplication, A for addition, E for exponentiation, and so on. Problems expressed in this language were presented with the natural spacing relationships to one group of high-school students, uniformly spaced to another. The participants who solved problems presented in the spaced language made many fewer errors compared to participants solving unspaced problems. Kirshner’s study demonstrates that arithmetic learners represent spacing regularities inherent in standard symbolic notation, and that they rely on the presence of those cues when developing interpretations of symbolic mathematical expressions. This is compatible with other studies that have shown that rule-based behavior uses irrelevant features of exemplars (McNeil & Alibali, 2004). One limitation of this work, however, is its use of standard arithmetic operations in the novel language. Because the stimuli are standard mathematical operations, it is difficult to determine the generality of the visual processes that govern order of operations behaviors. Participants have extensive experience with spacing in standard arithmetic symbology; this experience may drive their behavior when learning novel symbols for familiar operations without implying any general connection between syntax and spacing. The ability to understand abstract formal structures is one of humanity’s most distinctive and powerful cognitive traits. Arithmetical and algebraic notations, formal logic, and natural language syntax all contain underlying structure that at some level is entirely arbitrary and abstract. However, every actual notation has some particular physical presentation, and that notation always contains formally irrelevant physical relations. Often, especially when formal understanding is poor or partial, these relations may be more salient to a reasoner than the formally sanctioned abstract relations. The goal of this work is to explore whether and how people use irrelevant but salient visual information in exploring a novel formal system. This issue has special importance for understanding mathematical reasoning and learning. Although arithmetic notation may be the best-known example of a purely formal symbol system, arithmetic itself contains a variety of non- formal conventions that relate visual aspects of expressions to their formal structure. One of the most striking of these is a correlation between physical proximity and the order of precedence in three common operators: addition, multiplication, and exponentiation. In the typeset expression a + b c 2 ,