Persistent perceptual grouping effects in the evaluation of simple arithmetic expressions

Rivera, Jessie; Garrigan, Patrick

doi:10.3758/s13421-016-0593-z

Cited by 9 publications

(10 citation statements)

References 45 publications

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“…They found that operands are more likely to be summed when they are widely spaced, and more likely to be multiplied when the operands are narrowly spaced ( Landy and Goldstone, 2010 ). Consistently, Rivera and Garrigan (2016) replicated the study of Landy and Goldstone (2010) recently and confirmed that the physical spacing of formal equations has a large impact on successful evaluations of the expressions ( Rivera and Garrigan, 2016 ). Katz and Knops (2014) were the first to explore the operational momentum effect in multiplication and division.…”

Section: Introductionsupporting

confidence: 70%

Addition and Subtraction but Not Multiplication and Division Cause Shifts of Spatial Attention

Liu

et al. 2018

Front. Hum. Neurosci.

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Many studies have shown that solving addition and subtraction problems can induce overt shifts of spatial attention. In particular, right-side targets are detected faster than left-side targets when preceded by an addition operation, while left-side targets are detected faster than right-side targets when preceded by a subtraction operation. However, the interaction between space and arithmetic in multiplication or division is hardly studied and remains controversial. In order to make a strong case for the interaction between space and mental arithmetic, we attempted to replicate the spatial-arithmetic association in addition and subtraction (Experiment 1), and at the same time investigated whether shift of spatial attention would also be induced by multiplication or division operations (Experiment 2). We found that solving addition problems facilitated the detection of right-side targets, whereas left-side targets were detected faster after solving subtraction problems. However, no interaction between space and arithmetic operation was observed in multiplication or division. The implication of these findings is discussed.

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Section: Introductionsupporting

confidence: 70%

Addition and Subtraction but Not Multiplication and Division Cause Shifts of Spatial Attention

Liu

et al. 2018

Front. Hum. Neurosci.

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“…Other research indicates that people are heavily influenced by groupings based on perceptual properties when performing not only algebra but simple arithmetic as well. The influence of spacing persists even when the mathematical notation is only presented briefly and then masked and when verbal encoding is encouraged (Rivera & Garrigan, 2016), and these perceptual effects combine with traditional order-of-operations rules in compound expressions ( Jiang, Cooper, & Alibali, 2014). Effective training procedures for teaching math and science often deemphasize explicit, verbalizeable knowledge, instead focusing on perceptual skill learning through adaptive training (Kellman, Massey, & Son, 2010;Mettler & Kellman, 2014).…”

Section: Employing Visual Grouping Routines For Algebramentioning

confidence: 99%

Adapting Perception, Action, and Technology for Mathematical Reasoning

Goldstone

Marghetis

Weitnauer

et al. 2017

Curr Dir Psychol Sci

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Formal mathematical reasoning provides an illuminating test case for understanding how humans can think about things that they did not evolve to comprehend. People engage in algebraic reasoning by (1) creating new assemblies of perception and action routines that evolved originally for other purposes (reuse), (2) adapting those routines to better fit the formal requirements of mathematics (adaptation), and (3) designing cultural tools that mesh well with our perception-action routines to create cognitive systems capable of mathematical reasoning (invention). We describe evidence that a major component of proficiency at algebraic reasoning is Rigged Up Perception-Action Systems (RUPAS), via which originally demanding, strategically controlled cognitive tasks are converted into learned, automatically executed perception and action routines. Informed by RUPAS, we have designed, implemented, and partially assessed a computer-based algebra tutoring system called Graspable Math with an aim toward training learners to develop perception-action routines that are intuitive, efficient, and mathematically valid.

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“…And, indeed, people are sensitive to these visuospatial norms. When they judge the validity of an algebraic equation, performance is systematically worse if visual grouping or proximity conflicts with operator precedence (e.g., less space around addition than around multiplication), and systematically improved if visuospatial features align with operator precedence (Landy & Goldstone, 2007a; Rivera & Garrigan, 2016). Conversely, when adults write out algebraic expressions, they place terms connected by a higher-precedence operation (e.g., multiplication) closer together than those connected by a lower-precedence operation (e.g., addition; Landy & Goldstone, 2007b).…”

Section: Introductionmentioning

confidence: 99%

Mastering algebra retrains the visual system to perceive hierarchical structure in equations

2016

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Formal mathematics is a paragon of abstractness. It thus seems natural to assume that the mathematical expert should rely more on symbolic or conceptual processes, and less on perception and action. We argue instead that mathematical proficiency relies on perceptual systems that have been retrained to implement mathematical skills. Specifically, we investigated whether the visual system—in particular, object-based attention—is retrained so that parsing algebraic expressions and evaluating algebraic validity are accomplished by visual processing. Object-based attention occurs when the visual system organizes the world into discrete objects, which then guide the deployment of attention. One classic signature of object-based attention is better perceptual discrimination within, rather than between, visual objects. The current study reports that object-based attention occurs not only for simple shapes but also for symbolic mathematical elements within algebraic expressions—but only among individuals who have mastered the hierarchical syntax of algebra. Moreover, among these individuals, increased object-based attention within algebraic expressions is associated with a better ability to evaluate algebraic validity. These results suggest that, in mastering the rules of algebra, people retrain their visual system to represent and evaluate abstract mathematical structure. We thus argue that algebraic expertise involves the regimentation and reuse of evolutionarily ancient perceptual processes. Our findings implicate the visual system as central to learning and reasoning in mathematics, leading us to favor educational approaches to mathematics and related STEM fields that encourage students to adapt, not abandon, their use of perception.

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Persistent perceptual grouping effects in the evaluation of simple arithmetic expressions

Cited by 9 publications

References 45 publications

Addition and Subtraction but Not Multiplication and Division Cause Shifts of Spatial Attention

Addition and Subtraction but Not Multiplication and Division Cause Shifts of Spatial Attention

Adapting Perception, Action, and Technology for Mathematical Reasoning

Mastering algebra retrains the visual system to perceive hierarchical structure in equations

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