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Prado J, Knops A. Spatial attention in mental arithmetic: A literature review and meta-analysis. Psychon Bull Rev 2024:10.3758/s13423-024-02499-z. [PMID: 38565841 DOI: 10.3758/s13423-024-02499-z] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Accepted: 03/06/2024] [Indexed: 04/04/2024]
Abstract
We review the evidence for the conceptual association between arithmetic and space and quantify the effect size in meta-analyses. We focus on three effects: (a) the operational momentum effect (OME), which has been defined as participants' tendency to overestimate results of addition problems and underestimate results of subtraction problems; (b) the arithmetic cueing effect, in which arithmetic problems serve as spatial cues in target detection or temporal order judgment tasks; and (c) the associations between arithmetic and space observed with eye- and hand-tracking studies. The OME was consistently found in paradigms that provided the participants with numerical response alternatives. The OME shows a large effect size, driven by an underestimation during subtraction while addition was unbiased. In contrast, paradigms in which participants indicated their estimate by transcoding their final estimate to a spatial reference frame revealed no consistent OME. Arithmetic cueing studies show a reliable small to medium effect size, driven by a rightward bias for addition. Finally, eye- and hand-tracking studies point to replicable associations between arithmetic and eye or hand movements. To account for the complexity of the observed pattern, we introduce the Adaptive Pathways in Mental Arithmetic (APiMA) framework. The model accommodates central notions of numerical and arithmetic processing and helps identifying which pathway a given paradigm operates on. It proposes that the divergence between OME and arithmetic cueing studies comes from the predominant use of non-symbolic versus symbolic stimuli, respectively. Overall, our review and findings clearly support an association between arithmetic and spatial processing.
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Affiliation(s)
- Jérôme Prado
- Centre de Recherche en Neurosciences de Lyon (CRNL), INSERM U1028 - CNRS UMR5292, Université de Lyon, Lyon, France
| | - André Knops
- Université Paris Cité, LaPsyDÉ, CNRS, F-75005, Paris, France.
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Haman M, Lipowska K. Preschoolers prior formal mathematics education engage numerical magnitude representation rather than counting principles in symbolic +/-1 arithmetic: Evidence from the Operational Momentum effect. Dev Sci 2022; 26:e13322. [PMID: 36069221 DOI: 10.1111/desc.13322] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 09/01/2021] [Revised: 06/28/2022] [Accepted: 07/25/2022] [Indexed: 11/26/2022]
Abstract
In numerical cognition research, the operational momentum (OM) phenomenon (tendency to overestimate the results of addition and/or binding addition to the right side and underestimating subtraction and/or binding it to the left side) can help illuminate the most basic representations and processes of mental arithmetic and their development. This study is the first to demonstrate OM in symbolic arithmetic in preschoolers. It was modeled on Haman and Lipowska's (2021) non-symbolic arithmetic task, using Arabic numerals instead of visual sets. Seventy-seven children (4-7 years old) who know Arabic numerals and counting principles, but without prior school math education, solved addition and subtraction problems presented as videos with 1 as the second operand. In principle, such problems may be difficult when involving a non-symbolic approximate number processing system, whereas in symbolic format they can be solved based solely on the successor/predecessor functions and knowledge of numerical orders, without reference to representation of numerical magnitudes. Nevertheless, participants made systematic errors, in particular, overestimating results of addition in line with the typical OM tendency. Moreover, subtraction and addition induced longer response times when primed with left- and right-directed movement, respectively, which corresponds to the reversed spatial form of OM. These results largely replicate those of non-symbolic task and show that children at early stages of mastering symbolic arithmetic may rely on numerical magnitude processing and spatial-numerical associations rather than newly-mastered counting principles and the concept of an exact number. Adding and subtracting 1 in a symbolic format formally requires only knowledge of numerical orders and the predecessor/successor function, but not numerical magnitude processing Preschoolers knowing the counting principles and Arabic numerals, but without prior mathematics education, demonstrated operational momentum by overestimating results of symbolic addition of 1 In the same arithmetic task children showed faster reactions for addition primed with an object moving leftward and in subtraction primed with rightward motion These effects replicate findings from non-symbolic ±1 arithmetic, indicating that preschoolers use magnitude representation and spatial-numerical associations for symbolic calculation This article is protected by copyright. All rights reserved.
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Guez A, Piazza M, Pinheiro-Chagas P, Peyre H, Heude B, Ramus F. Preschool language and visuospatial skills respectively predict multiplication and addition/subtraction skills in middle school children. Dev Sci 2022; 26:e13316. [PMID: 36028996 DOI: 10.1111/desc.13316] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 10/19/2021] [Revised: 05/13/2022] [Accepted: 07/25/2022] [Indexed: 11/03/2022]
Abstract
A converging body of evidence from neuroimaging, behavioral and neuropsychology studies suggests that different arithmetic operations rely on distinct neuro-cognitive processes: while addition and subtraction may rely more on visuospatial reasoning, multiplication would depend more on verbal abilities. In this paper, we tested this hypothesis in a longitudinal study measuring language and visuospatial skills in 358 preschoolers, and testing their mental calculation skills at the beginning of middle school. Language skills at 5.5 years significantly predicted multiplication, but not addition nor subtraction scores at 11.5 years. Conversely, early visuospatial skills predicted addition and subtraction, but not multiplication scores. These results provide strong support for the existence of a double dissociation in mental arithmetic operations, and demonstrate the existence of long-lasting links between language/visuospatial skills and specific calculation abilities. Using structural equation modelling, we analyzed longitudinal data from 358 children. Language skills in preschool significantly predicted multiplication, but not addition nor subtraction scores in middle school. The reverse was true for preschool visuo-spatial skills. Importantly, this pattern remained unchanged when we controlled addition and multiplication operations for overall difficulty and for the magnitude of the operands. These results provide strong support for the existence of a double dissociation in mental arithmetic operations. This article is protected by copyright. All rights reserved.
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Affiliation(s)
- Ava Guez
- LSCP, Département d'études cognitives, ENS, EHESS, PSL University, CNRS, Paris, France
| | - Manuela Piazza
- Center for Mind/Brain Sciences - CIMeC, University of Trento, Rovereto, TN, Italy
| | - Pedro Pinheiro-Chagas
- Laboratory of Behavioral and Cognitive Neuroscience, Stanford Human Intracranial Cognitive Electrophysiology Program, Stanford University, Stanford, CA, 94305, USA
| | - Hugo Peyre
- LSCP, Département d'études cognitives, ENS, EHESS, PSL University, CNRS, Paris, France.,Department of Child and Adolescent Psychiatry, Robert Debré Hospital, APHP, Paris, France.,INSERM U1141, Université Paris Diderot, Sorbonne Paris Cité, DHU Protect, Paris, France
| | - Barbara Heude
- Université de Paris, CRESS, INSERM, INRAE, F-75004, Paris, France
| | - Franck Ramus
- LSCP, Département d'études cognitives, ENS, EHESS, PSL University, CNRS, Paris, France
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Alcaraz-Carrión D, Alibali MW, Valenzuela J. Adding and subtracting by hand: Metaphorical representations of arithmetic in spontaneous co-speech gestures. Acta Psychol (Amst) 2022; 228:103624. [PMID: 35667244 DOI: 10.1016/j.actpsy.2022.103624] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 11/22/2021] [Revised: 04/05/2022] [Accepted: 05/20/2022] [Indexed: 11/28/2022] Open
Abstract
This study investigated the spontaneous co-speech gestures produced by speakers who were talking about the concepts of addition and subtraction in a television news setting. We performed a linguistic and co-speech gesture analysis of expressions related to the concepts of addition (N plus N, addition, add) and subtraction (N minus N, subtraction, subtract). First, we compared the linguistic frequency of these structures across several corpora. Second, we performed a multimodal gesture analysis, drawing data from a television news repository. We analyzed 423 co-speech gestures (169 for subtraction and 254 for addition) in terms of their axis (e.g., lateral, sagittal) and their direction (e.g., leftwards, away from their body). Third, we examined the semantic properties of the direct object that was added or subtracted. There were two main findings. First, low-frequency linguistic expressions were more likely to be accompanied by co-speech gestures. Second, most gestures about addition and subtraction were produced along the lateral or sagittal axes. When people spoke about addition, they tended to produce lateral, rightwards movements or movements away from the body. When people spoke about subtraction, they tended to produce lateral, leftwards movements or movements towards the body. This co-speech gesture data provides evidence that people activate two different metaphors for arithmetic in spontaneous behavior: arithmetic is motion along a path and arithmetic is collecting objects.
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Jang S, Cho S. Operational momentum during children's approximate arithmetic relates to symbolic math skills and space-magnitude association. J Exp Child Psychol 2021; 213:105253. [PMID: 34419664 DOI: 10.1016/j.jecp.2021.105253] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 11/17/2020] [Revised: 05/21/2021] [Accepted: 07/03/2021] [Indexed: 01/29/2023]
Abstract
Operational momentum (OM) refers to the behavioral tendency to overestimate or underestimate the results of addition or subtraction, respectively. The cognitive mechanism of the OM effect and how it is related to the development of symbolic math abilities are not well understood. The current study examined whether individual differences in the OM effect are related to symbolic arithmetic abilities, number line estimation performance, and the space-magnitude association effect in young children. In this study, first-grade elementary school children manifested the OM effect during approximate addition and subtraction. Individual differences in the OM effect were not correlated with number line estimation error. Interestingly, children who showed a greater degree of the OM effect performed not worse, but better on the symbolic arithmetic task. In addition, the OM effect was correlated with the space-magnitude association (size congruity) effect measured with the Numerical Stroop task. More specifically, the OM bias was correlated with the ability to inhibit interference from competing information on the incongruent trials of the Numerical Stroop task. Our results suggest that the inaccuracy of numerical magnitude representations is not the source of the OM effect. Given that children with better math ability showed a greater OM bias, a stronger OM effect may reflect better intuition in arithmetic operations. Altogether, we carefully interpret these findings as suggesting that a greater OM effect reflects superior intuition or fundamental knowledge of arithmetic operations and a more adult-like maturation of the reorienting component of the attentional system.
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Affiliation(s)
- Selim Jang
- Department of Psychology, Chung-Ang University, Seoul, South Korea; Department of Psychology, University of Illinois at Urbana-Champaign, Champaign, IL 61820, USA
| | - Soohyun Cho
- Department of Psychology, Chung-Ang University, Seoul, South Korea.
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Díaz‐Barriga Yáñez A, Couderc A, Longo L, Merchie A, Chesnokova H, Langlois E, Thevenot C, Prado J. Learning to run the number line: the development of attentional shifts during single‐digit arithmetic. Ann N Y Acad Sci 2020; 1477:79-90. [DOI: 10.1111/nyas.14464] [Citation(s) in RCA: 6] [Impact Index Per Article: 1.5] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 02/13/2020] [Revised: 07/10/2020] [Accepted: 07/20/2020] [Indexed: 11/28/2022]
Affiliation(s)
- Andrea Díaz‐Barriga Yáñez
- Lyon Neuroscience Research Center (CRNL), INSERM U1028 ‐ CNRS UMR5292 University of Lyon Lyon France
| | - Auriane Couderc
- Lyon Neuroscience Research Center (CRNL), INSERM U1028 ‐ CNRS UMR5292 University of Lyon Lyon France
| | - Léa Longo
- Lyon Neuroscience Research Center (CRNL), INSERM U1028 ‐ CNRS UMR5292 University of Lyon Lyon France
| | - Annabelle Merchie
- Lyon Neuroscience Research Center (CRNL), INSERM U1028 ‐ CNRS UMR5292 University of Lyon Lyon France
| | - Hanna Chesnokova
- Lyon Neuroscience Research Center (CRNL), INSERM U1028 ‐ CNRS UMR5292 University of Lyon Lyon France
| | - Emma Langlois
- Lyon Neuroscience Research Center (CRNL), INSERM U1028 ‐ CNRS UMR5292 University of Lyon Lyon France
| | | | - Jérôme Prado
- Lyon Neuroscience Research Center (CRNL), INSERM U1028 ‐ CNRS UMR5292 University of Lyon Lyon France
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Haman M, Lipowska K. Moving attention along the mental number line in preschool age: Study of the operational momentum in 3- to 5-year-old children's non-symbolic arithmetic. Dev Sci 2020; 24:e13007. [PMID: 32567767 DOI: 10.1111/desc.13007] [Citation(s) in RCA: 4] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 10/16/2018] [Revised: 05/03/2020] [Accepted: 06/02/2020] [Indexed: 01/29/2023]
Abstract
People tend to underestimate subtraction and overestimate addition outcomes and to associate subtraction with the left side and addition with the right side. These two phenomena are collectively labeled 'operational momentum' (OM) and thought to have their origins in the same mechanism of 'moving attention along the mental number line'. OM in arithmetic has never been tested in children at the preschool age, which is critical for numerical development. In this study, 3-5 years old were tested with non-symbolic addition and subtraction tasks. Their level of understanding of counting principles (CP) was assessed using the give-a-number task. When the second operand's cardinality was 5 or 6 (Experiment 1), the child's reaction time was shorter in addition/subtraction tasks after cuing attention appropriately to the right/left. Adding/subtracting one element (Experiment 2) revealed a more complex developmental pattern. Before acquiring CP, the children showed generalized overestimation bias. Underestimation in addition and overestimation in subtraction emerged only after mastering CP. No clear spatial-directional OM pattern was found, however, the response time to rightward/leftward cues in addition/subtraction again depended on stage of mastering CP. Although the results support the hypothesis about engagement of spatial attention in early numerical processing, they point to at least partial independence of the spatial-directional and magnitude OM. This undermines the canonical version of the number line-based hypothesis. Mapping numerical magnitudes to space may be a complex process that undergoes reorganization during the period of acquisition of symbolic representations of numbers. Some hypotheses concerning the role of spatial-numerical associations in numerical development are proposed.
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Affiliation(s)
- Maciej Haman
- Faculty of Psychology, University of Warsaw, Warszawa, Poland
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Dorneles B. Numerical cognition in Brazil: a narrative review of a growing research field ( Cognición numérica en Brasil: una revisión de un campo de investigación en desarrollo). STUDIES IN PSYCHOLOGY 2020. [DOI: 10.1080/02109395.2020.1748998] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 10/24/2022] Open
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Dunn H, Bernstein N, de Hevia MD, Cassia VM, Bulf H, McCrink K. Operational momentum for magnitude ordering in preschool children and adults. J Exp Child Psychol 2019; 179:260-275. [PMID: 30562633 PMCID: PMC6311425 DOI: 10.1016/j.jecp.2018.11.017] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.6] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Grants] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 04/02/2018] [Revised: 11/15/2018] [Accepted: 11/21/2018] [Indexed: 11/18/2022]
Abstract
When adding or subtracting quantities, adults tend to overestimate addition outcomes and underestimate subtraction outcomes. They also shift visuospatial attention to the right when adding and to the left when subtracting. These operational momentum phenomena are thought to reflect an underlying representation in which small magnitudes are associated with the left side of space and large magnitudes with the right side of space. Currently, there is limited research on operational momentum in early childhood or for operations other than addition and subtraction. The current study tested whether English-speaking 3- and 4-year-old children and college-aged adults exhibit operational momentum when ordering quantities. Participants were presented with two experimental blocks. In one block of trials, they were tasked with choosing the same quantity they had previously seen three times; in the other block, they were asked to generate the next quantity in a doubling sequence composed of three ascending quantities. A bias to shift attention to the right after an ascending operation was found in both age groups, and a bias to overestimate the next sequential quantity during an ascending ordering operation was found in adults under conditions of uncertainty. These data suggest that, for children, the spatial biases during operating are more pronounced than the mis-estimation biases. These findings highlight the spatial underpinnings of operational momentum and suggest that both very young children and adults conceptualize quantity along a horizontal continuum during ordering operations, even before formal schooling.
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Affiliation(s)
- Hannah Dunn
- Barnard College of Columbia University, New York, NY 10027, USA
| | - Nicky Bernstein
- Barnard College of Columbia University, New York, NY 10027, USA
| | - Maria Dolores de Hevia
- Université Paris Descartes, 75006 Paris, France; Laboratoire Psychologie de la Perception, CNRS UMR 8242, Centre Biomédical des Saints-Pères, Université Paris Descartes, 75270 Paris, France
| | - Viola Macchi Cassia
- Università di Milano-Bicocca, 20126 Milano, Italy; Milan Center for Neuroscience (NeuroMI), Università di Milano-Bicocca, 20052 Monza, Italy
| | - Hermann Bulf
- Università di Milano-Bicocca, 20126 Milano, Italy; Milan Center for Neuroscience (NeuroMI), Università di Milano-Bicocca, 20052 Monza, Italy
| | - Koleen McCrink
- Barnard College of Columbia University, New York, NY 10027, USA.
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Didino D, Pinheiro-Chagas P, Wood G, Knops A. Response: Commentary: The Developmental Trajectory of the Operational Momentum Effect. Front Psychol 2019; 10:160. [PMID: 30787896 PMCID: PMC6372510 DOI: 10.3389/fpsyg.2019.00160] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.4] [Reference Citation Analysis] [Key Words] [Track Full Text] [Download PDF] [Journal Information] [Subscribe] [Scholar Register] [Received: 12/07/2018] [Accepted: 01/16/2019] [Indexed: 11/13/2022] Open
Affiliation(s)
- Daniele Didino
- Department of Psychology, Faculty of Life Sciences, Humboldt-Universität zu Berlin, Berlin, Germany
| | - Pedro Pinheiro-Chagas
- Laboratory of Behavioral and Cognitive Neuroscience, Stanford Human Intracranial Cognitive Electrophysiology Program, Department of Neurology and Neurological Sciences, Stanford University, Stanford, CA, United States
| | - Guilherme Wood
- Department of Psychology, University of Graz, Graz, Austria.,BioTechMed-Graz, University of Graz, Graz, Austria
| | - André Knops
- Department of Psychology, Faculty of Life Sciences, Humboldt-Universität zu Berlin, Berlin, Germany.,CNRS UMR 8240, Laboratory for the Psychology of Child Development and Education, Paris, France.,University Paris Descartes, Sorbonne Paris Cité, Paris, France
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Fischer MH, Shaki S. Repeating Numbers Reduces Results: Violations of the Identity Axiom in Mental Arithmetic. Front Psychol 2018; 9:2453. [PMID: 30568623 PMCID: PMC6290039 DOI: 10.3389/fpsyg.2018.02453] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 09/05/2018] [Accepted: 11/20/2018] [Indexed: 11/13/2022] Open
Abstract
Even simple mental arithmetic is fraught with cognitive biases. For example, adding repeated numbers (so-called tie problems, e.g., 2 + 2) not only has a speed and accuracy advantage over adding different numbers (e.g., 1 + 3) but may also lead to under-representation of the result relative to a standard value (Charras et al., 2012, 2014). Does the tie advantage merely reflect easier encoding or retrieval compared to non-ties, or also a distorted result representation? To answer this question, 47 healthy adults performed two tasks, both of which indicated under-representation of tie results: In a result-to-position pointing task (Experiment 1) we measured the spatial mapping of numbers and found a left-bias for tie compared to non-tie problems. In a result-to-line-length production task (Experiment 2) we measured the underlying magnitude representation directly and obtained shorter lines for tie- compared to non-tie problems. These observations suggest that the processing benefit of tie problems comes at the cost of representational reduction of result meaning. This conclusion is discussed in the context of a recent model of arithmetic heuristics and biases.
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Affiliation(s)
- Martin H. Fischer
- Division of Cognitive Science, University of Potsdam, Potsdam, Germany,*Correspondence: Martin H. Fischer,
| | - Samuel Shaki
- Department of Behavioral Sciences, Ariel University, Ariel, Israel
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Fischer MH, Miklashevsky A, Shaki S. Commentary : The Developmental Trajectory of the Operational Momentum Effect. Front Psychol 2018; 9:2259. [PMID: 30524343 PMCID: PMC6262352 DOI: 10.3389/fpsyg.2018.02259] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Track Full Text] [Download PDF] [Journal Information] [Subscribe] [Scholar Register] [Received: 08/27/2018] [Accepted: 10/30/2018] [Indexed: 11/23/2022] Open
Affiliation(s)
- Martin H. Fischer
- Psychology Department, University of Potsdam, Potsdam, Germany
- *Correspondence: Martin H. Fischer
| | | | - Samuel Shaki
- The Department of Behavioral Sciences, Ariel University, Ariel, Israel
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