Formal notations are diagrams: Evidence from a production task

DAVID LANDY 0 ROBERT L. GOLDSTONE 0 0 Indiana University , Bloomington, Indiana Although a general sense of the magnitude, quantity, or numerosity of objects is common in both untrained people and animals, the abilities to deal exactly with large quantities and to reason precisely in complex but well-specified situationsto behave formally, that isare skills unique to people trained in symbolic notations. These symbolic notations typically employ complex, hierarchically embedded structures, which all extant analyses assume are constructed by concatenative, rule-based processes. The primary goal of this article is to establish, using behavioral measures on naturalistic tasks, that some of the same cognitive resources involved in representing spatial relations and proximities are also involved in representing symbolic notationsin short, that formal notations are a kind of diagram. We examined self-generated productions in the domains of handwritten arithmetic expressions and typewritten statements in a formal logic. In both tasks, we found substantial evidence for spatial representational schemes even in these highly symbolic domains. - It is clear that mathematical equations written in modern notation are, in general, visual forms and that they share some properties with diagrammatic or imagistic displays. Equations and mathematical expressions are often set off from the main text, use nonstandard characters and shapes, and deviate substantially from linear symbol placement. Furthermore, evidence indicates that at least some mathematical processing is sensitive to the particular visual form of its presentation notation (Cambell, 1999; McNeil & Alibali, 2004, 2005). Despite these facts, notational mathematical representation is typically considered sentential and is placed in opposition to diagrammatic representations in fields as diverse as education (Stylianou, 2002; Zazkis, 1996), philosophy of science (Galison, 1997; Perini, 2006), computer science (Iverson, 1980), and cognitive modeling and problem solving (Anderson, 2005; Stenning, 2002). The standard conception of mathematical notation is best understood via Palmers (1978) classic distinction between intrinsic and extrinsic representational schemes. A representation is intrinsic whenever a representing relation has the same inherent constraints as its represented relation (p. 271). For example, Line As being shorter than Line B can be intrinsically represented by the representational element that corresponds to As being shorter, taller, brighter, or larger than the element representing Bin other words, by any relation that is inherently asymmetric and transitive. Representations are extrinsic when their inherent structure is arbitrary. They model the represented world by explicitly building the structure that is needed to conform to the world. Palmer argued that analog representations are intrinsic, in that correspondences and inferences between represented and representing worlds come for free because of their shared intrinsic structure. Propositional representations, including language, logic, and mathematics, are extrinsic and hence come to represent objects by explicitly establishing relations with whatever structure is needed. The only intrinsic relation necessary to propositions is the leftright concatenation of basic symbols. Although representations in mathematics and logic are traditionally understood as extrinsic, it is possible that they nonetheless possess intrinsic and analog properties, and it is this possibility that we empirically pursue here. In a separate study (Landy, Havas, Glenberg, & Goldstone, 2007), we consider the case for language. Stenning (2002) tried to characterize the apparent distinction between diagrams on the one hand and formal equations and language on the other while also recognizing that both are frequently visual and schematic/abstract representational formats. Stenning proposed that diagrams represent relational structures directly, whereas notationsformal or otherwisehave structural information mediated via rules governing individual elements. That is, assuming that the represented domain consists of relation r governing objects in a set {a} in a directly represented (diagrammatic) representation schema, there is a metric property of the representation R (such as spatial proximity) such that R and r act in direct proportion. In an indirectly represented language, r does not correspond to any metric feature of the representation. Instead, r(a1, a2) is expressed via a set of rules governing concatenative strings, in which the only relevant property of a display is the order in which terms appear. Specifically, we propose that formal notations are diagrammatic as well as sentential and that the property conventionally described as syntactic structure is cognitively mediated, in part, by spatial information. Elements of expressions are bound together through perceptual grouping, often induced by simple spatial proximity. Thus, our claim is that mathematical formalizations of syntax are not themselves the direct cognitive mechanisms typically employed in processing that syntax. The former really are concatenative, but we propose that people use space and spatial relationships in representational schemas to facilitate the processing of syntax. We are not claiming here that the execution of each individual step in a proof or computation is inherently spatial or processed exclusively using sensorimotor mechanisms. We do suggest that spatial reasoning with regard to the physical layout of notational forms is common in reasoning with formal languages and that spacing practices play a significant role in human reasoning using notations. We have argued previously that a broadly similar interference of metric (non-order-related) spatial properties on syntactic judgments provides evidence that spatial processes and representations implement syntax in typical human judgments (Landy & Goldstone, 2007; see also Kirshner, 1989; Kirshner & Awtry, 2004). To study the influence of perceptual grouping on mathematical reasoning, we tasked undergraduate participants with judging whether an algebraic equality was necessarily true. The equalities were designed to test the students abilities to apply the order of precedence of operations rules (e.g., the rule that multiplication precedes addition). Although our participants knew these rules, we were interested in whether perceptual and form-based groupings would be able to override their general knowledge of the order of precedence rules. We tested this by having grouping factors either consistent or inconsistent with order of precedence. For example, a participant might be asked whether n*w y*b was necessarily equal to y*b n*w. In this example, the physical spacing around the operators was consistent with the order of operators, with more space around the plus symbols than the multiplication symbols. On other trials, the spacings were incons (...truncated)