Adding Apples and Oranges: Alignment of Semantic and Formal Knowledge

Bassok, Miriam; Chase, Valerie M.; Martin, Shirley A.

doi:10.1006/cogp.1998.0675

Cited by 97 publications

(106 citation statements)

References 53 publications

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“…The entities in a problem situation evoke semantic relations (e.g., tulips and vases evoke the functionally asymmetric ''contain" relation), which people align with analogous mathematical relations (e.g., the noncommutative division operation, tulips/vases) (Bassok, Chase, & Martin, 1998;Guthormsen et al, 2015). Rapp, Bassok, DeWolf, and Holyoak (2015) found that a form of semantic alignment guides the use of different formats for rational numbers, fractions and decimals.…”

Section: Semantic Alignment and The Ontology Of Quantity Typesmentioning

confidence: 99%

Conceptual and procedural distinctions between fractions and decimals: A cross-national comparison

et al. 2016

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a b s t r a c tPrevious work has shown that adults in the United States process fractions and decimals in distinctly different ways, both in tasks requiring magnitude judgments and in tasks requiring mathematical reasoning. In particular, fractions and decimals are preferentially used to model discrete and continuous entities, respectively. The current study tested whether similar alignments between the format of rational numbers and quantitative ontology hold for Korean college students, who differ from American students in educational background, overall mathematical proficiency, language, and measurement conventions. A textbook analysis and the results of five experiments revealed that the alignments found in the United States were replicated in South Korea. The present study provides strong evidence for the existence of a natural alignment between entity type and the format of rational numbers. This alignment, and other processing differences between fractions and decimals, cannot be attributed to the specifics of education, language, and measurement units, which differ greatly between the United States and South Korea.

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Section: Semantic Alignment and The Ontology Of Quantity Typesmentioning

confidence: 99%

Conceptual and procedural distinctions between fractions and decimals: A cross-national comparison

et al. 2016

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“…Instead of creating word problems with different cover stories or abstract schemata that can be instantiated with different insight problems, domains can be selected that naturally and intrinsically instantiate principles of complex adaptive systems. The advantages are that the systems are inherently, rather than arbitrarily, connected to their abstract principles (see Bassok, 1996;Bassok, Chase, & Martin, 1998), the principles have external validity in that they are of authentic scientific interest, and comprehending the principles is challenging even for college students. Complex adaptive systems offer a unique laboratory for exploring abstract and concrete thought because the domains of inquiry embody abstract formalisms.…”

Section: Complex Adaptive Systemsmentioning

confidence: 99%

The transfer of abstract principles governing complex adaptive systems

Goldstone

Sakamoto

2003

Cognitive Psychology

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217

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“…This paper focuses on one essential step: the creation of a notational expression-in this case, an equation-that models a situation. One possible account suggests that much of the time this equation-construction process involves selecting an appropriate template-say, a memorized equation pattern such as F=ma, y=mx+b, ax 2 +bx+c=0, or base+change (Carlson-Radvansky & Logan, 1997;Fisher Borchert, & Bassok, 2011;Sherin, 2001)-and applying structure-mapping (Gentner, 1983;Hummel & Holyoak, 1997) to align that structure with other representations of the scene, such as written descriptions, pictures, or mental simulations (Bassok, Chase, and Martin, 1998). For example, Abstract numeric relations 5 asked to find the total weight of a case with 6 apples, each apple weighing 10 ounces, one might retrieve an a times b frame, and fill in the mapping a=6, b=10 via alignment to yield 6*10.…”

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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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Adding Apples and Oranges: Alignment of Semantic and Formal Knowledge

Cited by 97 publications

References 53 publications

Conceptual and procedural distinctions between fractions and decimals: A cross-national comparison

Conceptual and procedural distinctions between fractions and decimals: A cross-national comparison

The transfer of abstract principles governing complex adaptive systems

Abstract numeric relations and the visual structure of algebra.

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