Academic journal article Memory & Cognition

Formal Notations Are Diagrams: Evidence from a Production Task

Academic journal article Memory & Cognition

Formal Notations Are Diagrams: Evidence from a Production Task

Article excerpt

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 situations-to behave formally, that is-are 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 notations-in 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 Palmer's (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 A's being shorter than Line B can be intrinsically represented by the representational element that corresponds to A's being shorter, taller, brighter, or larger than the element representing B-in 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. Prepositional 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 left-right 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 notations-formal or otherwise-have structural information mediated via rules governing individual elements. …

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