The Visual Syntax of Algebra

Kirshner, David

doi:10.2307/749516

Cited by 58 publications

(63 citation statements)

References 3 publications

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“…Although deliberate application of learned procedures is clearly important (even for expressions presented with inconsistent spacing, errors were relatively rare), the rules of operator precedence do appear to be enforced, in part, through a fundamentally perceptual mechanism. When one considers that inexperienced participants benefit from physical spacing of symbols that is consistent with the rules of operator precedence (Kirshner, 1989), and that both handwritten expressions and typeset algebraic notations employ consistent spacing (Landy & Goldstone, 2007b), it makes sense that experienced participants might have learned to implicitly utilize this cue for deciding which part of a compound expression should be evaluated first.…”

Section: Resultsmentioning

confidence: 99%

Persistent perceptual grouping effects in the evaluation of simple arithmetic expressions

Rivera

Garrigan

2016

Mem Cogn

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Landy and Goldstone (2007a demonstrated that an explicit rule, operator precedence for simple arithmetic expressions, is enforced in part by perceptual processes like unit formation and attention. When perceptual grouping competes with operator precedence, errors increase. We replicated this result (Exp. 1) and investigated whether perceptual grouping effects persist when the visual stimulus is presented briefly and then masked (Exp. 2) and when verbal recoding is encouraged through vocal expression (Exp. 3). We found that perceptual-grouping effects persisted in the masking condition, suggesting that the mental representations of arithmetic expressions retain visuospatial characteristics. Similarly, verbalization of the expressions did not eliminate perceptualgrouping effects, suggesting that participants were not verbally recoding. In sum, the persistent effects of unit formation and spatial attention emphasize the importance of perceptual processing in the development of human expertise in this domain.

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Section: Resultsmentioning

confidence: 99%

Persistent perceptual grouping effects in the evaluation of simple arithmetic expressions

Rivera

Garrigan

2016

Mem Cogn

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“…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 ; 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.…”

Section: Methodsmentioning

confidence: 99%

Formal notations are diagrams: Evidence from a production task

Landy

Goldstone

2007

Mem Cogn

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“…Recent work has repeatedly demonstrated that detailed physical properties and relations-and the perceptual systems that process them-play an important role in reasoning using visual displays in a variety of abstract contexts including puzzles (Patsenko & Altmann, 2010), biological taxa (Novick & Catley, 2007), weather diagrams (Hegarty Canham, and Fabrikant 2010), and algebra (Kirshner, 1989). …”

Section: Introductionmentioning

confidence: 99%

Abstract numeric relations and the visual structure of algebra.

Landy¹,

Brookes²,

Smout³

2014

Journal of Experimental Psychology: Learning, Memory, and Cogni

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Formal algebras are among the most powerful and general mechanisms for expressing quantitative relational statements, and yet even university engineering students, who are relatively proficient with algebraic manipulation, struggle with and often fail to correctly deploy basic aspects of algebraic notation (Clement, 1982). In the cognitive tradition, it has often been assumed that skilled users of these formalisms treat situations in terms of semantic properties encoded in an abstract syntax that governs the use of notation without particular regard to the details of the physical structure of the equation itself (Anderson, 2005;Hegarty, Mayer, & Monk, 1995). We explore how the notational structure of verbal descriptions or algebraic equations (e.g., the spatial proximity of certain words, or the visual alignment of numbers and symbols in an equation) plays a role in the process of interpreting or constructing symbolic equations. More specifically, we propose that construction processes involve an alignment of notational structures across representation systems, biasing reasoners toward the selection of formal notations that maintain the visuo-spatial structure of source representations. For example, in the statement "There are five elephants for every three rhinoceroses," the spatial proximity of "five" and "elephants; and "three" and "rhinoceroses" will bias reasoners to write the incorrect expression 5E = 3R because that expression maintains the spatial relationships encoded in the source representation. In three experiments, participants construct equations with given structure, based on story problems with a variety of phrasings. We demonstrate how the notational alignment approach accounts naturally for a variety of previously reported phenomena in equation In the case of symbolic algebra, a tension between formal syntactic rules and visual patterns can arise. The formal syntax of algebra encodes relations among quantities by abstract rules specified in terms of ordinal relations among symbol tokens. The same mathematical expression can be rearranged in a number of different ways without altering the formal content of the equation. On Abstract numeric relations 4 the other hand, a particular symbol system such as mathematics uses spatial relations in unique ways to convey additional meaning (Sherin, 2001). In the usual algebraic syntax, this tension creates visual structures that are sometimes aligned with (Landy & Goldstone, 2007a), and sometimes opposed to (Kirshner, 1989;Kirshner & Awtry, 2004;Marquis, 1988), underlying mathematical content. A rich notational calculus such as modern algebra replaces some conceptual relations with concrete organizational relations specified by principles of visualization and perceptual organization (Dantzig, 1930;Malle, 1993;Goldstone, Landy, & Son, 2010). For example, the relatively large space surrounding the equals sign in an arithmetic equation allows spatial grouping to serve as a proxy for the conceptual separation of an equation into left-hand and right-hand sides ...

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The Visual Syntax of Algebra

Cited by 58 publications

References 3 publications

Persistent perceptual grouping effects in the evaluation of simple arithmetic expressions

Persistent perceptual grouping effects in the evaluation of simple arithmetic expressions

Formal notations are diagrams: Evidence from a production task

Abstract numeric relations and the visual structure of algebra.

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