Abstract numeric relations and the visual structure of algebra.

Abstract

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 ...

Relational Priming Based on a Multiplicative Schema for Whole Numbers and Fractions

“…Although the division format (non‐relational order) did not yield facilitation between inverse equations (which depends on recognizing relational similarities across equations), it may have facilitated activation of common factors within each individual equation. This possibility is consistent with previous work showing that different visual representations of arithmetic equations can differentially highlight the elements that bind across operations (e.g., Kirshner, ; Landy, Brookes & Smout, 2014; Landy & Goldstone, 2007b, ). In Experiment 4 case, the division format made it easier to simplify either one of the numerator numbers with the denominator.…”

Mathematics students demonstrate superior visuo-spatial working memory to humanities students under conditions of low central executive processing load

“…Moreover, general visuo-spatial skill was a significant independent predictor of mathematics performance. This is in line with previous findings that general visuo-spatial processing has a role in complex mathematics such as algebra (Landy et al, 2014) and interpreting graphs (Hegarty & Waller, 2005), arithmetical reasoning (Geary et al, 2000) and generally in mathematics (Casey et al, 1995;Casey et al, 1997;Friedman, 1995). The finding that visuo-spatial working memory storage capacity also significantly and uniquely predicted calculation even after general visuo-spatial skills were accounted for suggests that the mathematics students' superior capacity cannot simply be explained by a better general ability to deal Hubber, Gilmore, & Cragg with visuo-spatial information but that the two skills accounts for separate variance in mathematics performance.…”

“…However, they did not examine whether the relationship between visuo-spatial storage ability and mathematics could be explained by general visuo-spatial skills. Previous research has implicated the use of general visuo-spatial resources in the solving of mathematical problems (Delgado & Prieto, 2004;Friedman, 1995;Jiang, Cooper, & Alibali, 2014;Landy, Brookes, & Smout, 2014;Marghetis, Núñez, & Bergen, 2014;Pinhas, Shaki, & Fischer, 2014;Wiemers, Bekkering, & Lindemann, 2014). Casey, Nuttall, Pezaris, and Benbow (1995) and Casey, Nuttall, and Pezaris (1997) found links between mental rotation ability and scores on the SAT-M math test, whilst Geary, Saults, Liu, and Hoard (2000) found mental rotation was related to arithmetical reasoning ability.…”

Analogy Lays the Foundation for Two Crucial Aspects of Symbolic Development: Intention and Correspondence

“…For example, by comparing the phonological characteristics of sound symbolic words to their referents, children may notice the similarity between them, which helps them to establish correspondences, leading to the understanding that people use words to represent meanings and ultimately facilitating early vocabulary development. Not only has analogical comparison been applied to the learning of spoken (Gentner & Namy, ; Namy & Gentner, ) and written language (White, ), it has also been applied to the learning of many other conventionalized symbolic systems, such as mathematical notations (Landy, Brookes, & Smout, ), geosciences visualizations (Jee et al., ; Resnick, Shipley, Newcombe, Massey, & Wills, ), gestures (Cooperrider, Gentner, & Goldin‐Meadow, ), and sketches and diagrams (Forbus, Usher, Lovett, Lockwood, & Wetzel, ). All of this evidence points to the important role that analogical comparison plays in facilitating the learning of symbol systems.…”