Multi-digit Number Processing

2014 ◽  
Author(s):  
Hans-Christoph Nuerk ◽  
Korbinian Moeller ◽  
Klaus Willmes
Keyword(s):  
Numerical Cognition ◽  
Distance Effect ◽  
Number Processing ◽  
Place Value ◽  
Digit Number ◽  
Single Digit ◽  
Numerical Development ◽  
Conceptual Distinction ◽  
Place Identification

Only recently the focus in numerical cognition research has considered multi-digit number processing as a relatively new and yet understudied domain in mathematical cognition. In this chapter: (i) we argue that single-digit number processing is not sufficient to understand multi-digit number processing; (ii) provide an overview on which representations and effects have been investigated for multi-digit numbers; (iii) suggest a conceptual distinction between place-identification, place-value activation, and place-value computation; (iv) identify language influences on multi-digit number processing along that conceptual distinction; and (v) argue that for numerical development indices of multi-digit number processing may be more suitable predictors of later arithmetical performance than classical single-digit measure such as the distance effect or non-numerical variables (e.g., working memory). In the final section, we summarize the important issues in multi-digit number processing, outline future directions and try to encourage readers to contribute to a new, exciting, yet understudied domain of numerical cognition.

scholarly journals Tracking the continuous dynamics of numerical processing: A brief review and editorial

2018 ◽  
Author(s):  
Thomas J. Faulkenberry ◽  
Matthias Witte ◽  
Matthias Hartmann
Keyword(s):  
Numerical Cognition ◽  
Mental Arithmetic ◽  
Special Section ◽  
Past Research ◽  
Number Processing ◽  
Hand Tracking ◽  
Single Digit ◽  
Future Directions ◽  
Numerical Processing ◽  
Computer Mouse

Many recent studies in numerical cognition have moved beyond the use of purely chronometric techniques in favor of methods which track the continuous dynamics of numerical processing. Two examples of such techniques include eye tracking and hand tracking (or computer mouse tracking). To reflect this increased concentration on continuous methods, we have collected a group of 5 articles that utilize these techniques to answer some contemporary questions in numerical cognition. In this editorial, we discuss the two paradigms and provide a brief review of some of the work in numerical cognition that has profited from the use of these techniques. For both methods, we discuss the past research through the frameworks of single digit number processing, multidigit number processing, and mental arithmetic processing. We conclude with a discussion of the papers that have been contributed to this special section and point to some possible future directions for researchers interested in tracking the continuous dynamics of numerical processing.

scholarly journals Language influences on numerical development—Inversion effects on multi-digit number processing

2013 ◽  
Vol 4 ◽  
Author(s):  
E. Klein ◽  
J. Bahnmueller ◽  
A. Mann ◽  
S. Pixner ◽  
L. Kaufmann ◽  
...  
Keyword(s):  
Number Processing ◽  
Digit Number ◽  
Numerical Development ◽  
Inversion Effects

scholarly journals No Evidence for an Influence of Grammatical Number in Two-Digit Magnitude and Place-Value Processing in Adults

2020 ◽  
Author(s):  
Julia Huber ◽  
Mojtaba Soltanlou ◽  
Krzysztof Cipora ◽  
Katarzyna Lipowska ◽  
Frank Domahs ◽  
...  
Keyword(s):  
Distance Effect ◽  
Snarc Effect ◽  
Place Value ◽  
Comparison Task ◽  
Digit Number ◽  
Response Level ◽  
Response Side ◽  
Plural Form ◽  
Number Comparison ◽  
Grammatical Number

Numerous studies revealed effects of some linguistic properties like inversion or reading/writing direction on number processing. However, it remains more controversial, whether influences at a syntactic level, such as singular vs. plural form associated with certain numbers, can also influence magnitude and place-value processing and vice versa. In this study, we investigated for the first time in a classical two-digit number comparison task whether grammatical number also affects magnitude and place-value processing (and vice versa). To do so, we used a peculiarity of the Polish language, where the inflection of a verb depends on the unit digit of a number (singular for 25-29, 35-39, etc. and plural for 22-24, 32-34, etc.). This systematic pattern allows the manipulation of congruency between grammatical number and magnitude information, both on an item and a response level (i.e., the grammatical number is either compatible or incompatible to the magnitude information or the response side). We observed no significant interference effects, neither between grammatical number (i.e., associated singular/ plural inflection of the number) and magnitude information, nor between grammatical number and the response side. Model comparisons revealed that models without grammatical number, could explain our data best. Hence, grammatical number did not contribute to the explanation of the data beyond unit-decade compatibility, distance effect and SNARC effect and, thus, seems to be negligible in two-digit number comparison. Task characteristics, which might contribute to this finding are discussed.

scholarly journals Effects of attentional shifts along the vertical axis on number processing: an eye-tracking study with optokinetic stimulation

2021 ◽  
Author(s):  
Arianna Felisatti ◽  
Mariagrazia Ranzini ◽  
Elvio Blini ◽  
Matteo Lisi ◽  
Marco Zorzi
Keyword(s):  
Eye Tracking ◽  
Numerical Cognition ◽  
Response Times ◽  
Eye Gaze ◽  
Distance Effect ◽  
Vertical Axis ◽  
Number Processing ◽  
Numerical Magnitude ◽  
Optokinetic Stimulation ◽  
Comparison Task

Previous studies suggest that associations between numbers and space are mediated by shifts of visuospatial attention along the horizontal axis. In this study, we investigated the effect of vertical shifts of overt attention, induced by optokinetic stimulation (OKS) and monitored through eye-tracking, in two tasks requiring explicit (number comparison) or implicit (parity judgment) processing of number magnitude. Participants were exposed to black-and-white stripes (OKS) that moved vertically (upward or downward) or remained static (control condition). During the OKS, participants were asked to verbally classify auditory one-digit numbers as larger/smaller than 5 (comparison task; Exp. 1) or as odd/even (parity task; Exp. 2). OKS modulated response times in both experiments. In Exp.1, downward attentional displacement increased the Magnitude effect (slower responses for large numbers) and reduced the Distance effect (slower responses for numbers close to the reference). In Exp.2, we observed a parity by magnitude interaction that was amplified by downward OKS. Moreover, eye tracking analyses revealed an influence of number processing on eye movements both in Exp. 1, with eye gaze shifting downwards during the processing of numbers 1-2 as compared to 8-9; and in Exp. 2, with leftward shifts after large even numbers (6,8) and rightward shifts after large odd numbers (7,9). These results provide evidence of bidirectional links between number and space and extend them to the vertical dimension. Moreover, they document the influence of visuo-spatial attention on processing of numerical magnitude, numerical distance and parity. Together, our findings are in line with grounded and embodied accounts of numerical cognition.

scholarly journals Tracking the continuous dynamics of numerical processing: A brief review and editorial

2018 ◽  
Vol 4 (2) ◽  
pp. 271-285 ◽  
Author(s):  
Thomas J. Faulkenberry ◽  
Matthias Witte ◽  
Matthias Hartmann
Keyword(s):  
Numerical Cognition ◽  
Mental Arithmetic ◽  
Special Section ◽  
Past Research ◽  
Number Processing ◽  
Hand Tracking ◽  
Single Digit ◽  
Future Directions ◽  
Numerical Processing ◽  
Computer Mouse

Many recent studies in numerical cognition have moved beyond the use of purely chronometric techniques in favor of methods which track the continuous dynamics of numerical processing. Two examples of such techniques include eye tracking and hand tracking (or computer mouse tracking). To reflect this increased concentration on continuous methods, we have collected a group of 5 articles that utilize these techniques to answer some contemporary questions in numerical cognition. In this editorial, we discuss the two paradigms and provide a brief review of some of the work in numerical cognition that has profited from the use of these techniques. For both methods, we discuss the past research through the frameworks of single digit number processing, multidigit number processing, and mental arithmetic processing. We conclude with a discussion of the papers that have been contributed to this special section and point to some possible future directions for researchers interested in tracking the continuous dynamics of numerical processing.

Attentional Strategies in Place-Value Integration

2011 ◽  
Vol 219 (1) ◽  
pp. 42-49 ◽  
Author(s):  
Anne Mann ◽  
Korbinian Moeller ◽  
Silvia Pixner ◽  
Liane Kaufmann ◽  
Hans-Christoph Nuerk
Keyword(s):  
Compatibility Effect ◽  
Developmental Trajectories ◽  
Number System ◽  
Number Processing ◽  
Place Value ◽  
Digit Number ◽  
Magnitude Comparison ◽  
Attentional Processes ◽  
Arabic Number ◽  
Value Integration

To process a multi-digit number its constituting digits need to be integrated into the place-value structure of the Arabic number system. For two-digit numbers, processes of unit-decade integration are reflected by the compatibility effect in magnitude comparison. Recent research in adults indicated that the size of the compatibility effect increases when stimuli prevent to focus on the decade digits as achieved by the inclusion of within-decade items (43_47). In the present study within- and between-decade items (47_62) were used to assess the compatibility effect in children. We observed reliable compatibility effects that increased with grade level and that were larger than in a previous study without within-decade stimuli. Furthermore, evaluation of the developmental trajectories showed that two-digit number processing develops to more automatic parallel processing of the constituent digits of tens and units. From these results we conclude that (i) even for children attentional processes can strongly influence multi-digit number processing and (ii) with increasing age and experience more parallel and automated understanding of two-digit numbers develops which seems to remain relatively stable once achieved.

scholarly journals Automatic place-value activation in magnitude-irrelevant parity judgement

2018 ◽  
Author(s):  
Krzysztof Cipora ◽  
Mojtaba Soltanlou ◽  
Stefan Smaczny ◽  
Silke Melanie Goebel ◽  
Hans-Christoph Nuerk
Keyword(s):  
Congruency Effect ◽  
Judgment Task ◽  
Number Processing ◽  
Place Value ◽  
Digit Number ◽  
Target Task ◽  
Unit Digit ◽  
Relevant Task ◽  
Cross Lingual ◽  
Parity Judgment

Research on multi-digit number processing suggests that, in Arabic numerals, their place-value magnitude is automatically activated, whenever a magnitude-relevant task was employed: However, so far, it is unknown, whether place-value is also activated when the target task is magnitude-irrelevant. The current study examines this question by using the parity congruency effect in two-digit numbers: It describes that responding to decade-digit parity congruent numbers (e.g., 35, 46; same parity of decades and units) is faster than to decade-digit parity incongruent numbers (e.g., 25; 36; different parities of decades and units). Here we investigate the (a-)symmetry of the parity congruency effect; i.e. whether it makes a difference whether participants are assessing the parity of the unit digit or the decade digit. We elaborate, how and why such an asymmetry is related to place-value processing, because the parity of the unit digit only interferes with the parity of the decade digit, while the parity of the decade digit interferes with both the parity of the unit digit and the integrated parity of the whole two-digit number. We observed a significantly larger parity congruency effect in the decade parity decision than in the unit parity decision. This suggests that automatic place-value processing also takes place in a typical parity judgment task, in which magnitude is irrelevant. Finally, because of the cross-lingual design of the study, we can show that these results and their implications were language-independent.

Extending the Mental Number Line

2011 ◽  
Vol 219 (1) ◽  
pp. 3-22 ◽  
Author(s):  
Hans-Christoph Nuerk ◽  
Korbinian Moeller ◽  
Elise Klein ◽  
Klaus Willmes ◽  
Martin H. Fischer
Keyword(s):  
Everyday Life ◽  
Number Line ◽  
Mental Number Line ◽  
Number Processing ◽  
First Year ◽  
Digit Number ◽  
Single Digit ◽  
Comprehensive Review ◽  
Cognitive Research ◽  
Current Review

Multi-digit number processing is ubiquitous in our everyday life – even in school, multi-digit numbers are computed from the first year onward. Yet, many problems children and adults have are about the relation of different digits (for instance with fractions, decimals, or carry effects in multi-digit addition). Cognitive research has mainly focused on single-digit processing, and there is no comprehensive review of the different multi-digit number processing types and effects. The current review aims to fill this gap. First, we argue that effects observed in single-digit tasks cannot simply be transferred to multi-digit processing. Next, we list 16 effect types and processes which are specific for multi-digit number processing. We then discuss the development of multi-digit number processing, its neurocognitive correlates, its cultural or language-related modulation, and finally some models for multi-digit number processing. We finish with conclusions and perspectives about where multi-digit number processing research may or should be heading in following years.

scholarly journals Influences of hand action on the processing of symbolic numbers: a special role of pointing?

2021 ◽  
Author(s):  
Mariagrazia Ranzini ◽  
Carlo Semenza ◽  
Marco Zorzi ◽  
Simone Cutini
Keyword(s):  
Numerical Cognition ◽  
Distance Effect ◽  
Number Processing ◽  
Grounded Cognition ◽  
Special Role ◽  
Comparison Task ◽  
Functional Link ◽  
Number Comparison ◽  
Hand Action

Embodied and grounded cognition theories suggest that cognitive processes are built upon sensorimotor systems. In the context of studies on numerical cognition, interactions between number processing and the hand actions of reaching and grasping have been documented in skilled adults, thereby supporting embodied and grounded cognition accounts. The present study made use of the neurophysiological principle of neural adaptation applied to repetitive hand actions to test the hypothesis of a functional overlap between neurocognitive mechanisms of hand action and number processing. Participants performed repetitive grasping of an object, repetitive pointing, repetitive tapping, or passive viewing. Subsequently, they performed a symbolic number comparison task. Importantly, hand action and number comparison were functionally and temporally dissociated, thereby minimizing context-based effects. Results showed that executing the action of pointing slowed down the responses in number comparison. Moreover, the typical distance effect (faster responses for numbers far from the reference as compared to close ones) was not observed for small numbers after pointing, while it was enhanced by grasping. These findings confirm the functional link between hand action and number processing, and suggest new hypotheses on the role of pointing as a meaningful gesture in the development and embodiment of numerical skills.

The Influence of Structural Aspects of the English Counting Word System on the Teaching and Learning of Place Value

1998 ◽  
Vol 59 (1) ◽  
pp. 1-8 ◽  
Author(s):  
Ian Thompson
Keyword(s):  
Young Children ◽  
Teaching And Learning ◽  
Place Value ◽  
Single Digit ◽  
Value System ◽  
Number Words ◽  
Large Numbers ◽  
Structural Aspects

The influence of structural aspects of the English counting word system on the teaching and learning of place value In their discussion of the teaching of place value to young children Fuson and Briars (1990) describe the extent to which the English spoken system of number words constitutes a ‘named value’ system for large numbers. They argue that, because two-digit numbers are not ‘named value’, teachers should move from teaching single-digit calculations to teaching calculations with large numbers, only returning to two-digit numbers when children are familiar with the standard written algorithms. This article uses transcriptions of children calculating mentally to suggest that they appear to take advantage of the ‘partitionable’ aspect of the language associated with two-digit numbers - an aspect that Fuson and Briars (1990) appear to have ignored. These examples appear to raise questions about their recommendation that teachers should progress from single-digit to large number calculations.

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