Andre Knops

Recruitment of an Area Involved in Eye Movements During Mental Arithmetic

Science, 2009

The arithmetic problems were created from a set of standardized problems with 50 as first operand... more The arithmetic problems were created from a set of standardized problems with 50 as first operand and either a small (6) or a large numerosity (26) as the second operand. The operands were identical for addition and subtraction. Apart from the correct result, 8 deviant results were created for each arithmetic problem. These deviants were arranged as a geometric series (i.e. were linearly spaced on a logarithmic scale) and ranged from double the correct result to half of the correct result (technically, they were generated as round(c x r i/4 ), where c is the correct result, r the maximal ratio between correct outcome and the lowest/highest deviant, and i an index ranging from -4 to +4). Because prior research indicated that subjects could achieve a greater precision in symbolic than in non-symbolic notation (1), in order to maintain an approximately similar level of difficulty the ratio r varied between notations. Deviants for Arabic problems were more closely centered on the correct outcome (r = 1.41) than were the deviants for non-symbolic problems (r = 2.5). To avoid a strategy of always selecting the response falling in the middle of the proposed range, only seven out of those nine possible results were presented. In 50% of the trials we presented the upper seven (high range), and thus the correct result was the third largest numerosity. In the other 50% of the trials the lower seven choices were shown (low range), and the correct result was therefore the fifth largest numerosity.

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Asymmetric activation spreading in the multiplication associative network due to asymmetric overlap between numerosities semantic representations?

Cognition, 2015

Simple multiplication facts are thought to be organised in a network structure in which problems ... more Simple multiplication facts are thought to be organised in a network structure in which problems and solutions are associated. Converging evidence suggests that the ability for solving symbolic arithmetic problems is based on an approximate number system (ANS). Most theoretical stances concerning the metric underlying the ANS converge on the assumption that the representational overlap between two adjacent numbers increases as the numerical magnitude of the numbers increases. Given a number N, the overlap between N and N + 1 is larger than the overlap between N and N À 1. Here, we test whether this asymmetric overlap influences the activation spreading within the multiplication associative network (MAN). When verifying simple multiplication problems such as 8 Â 4 participants were slower in rejecting false but related outcomes that were larger than the actual outcome (e.g. 8 Â 4 = 36) than rejecting smaller related outcomes (e.g. 8 Â 4 = 28), despite comparable numerical distance from the correct result (here: 4). This effect was absent for outcomes which are not part of either operands table (e.g., 8 Â 4 = 35). These results suggest that the metric of the ANS influences the activation spreading within the MAN, further substantiating the notion that symbolic arithmetic is grounded in the ANS.

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Mental subtraction and multiplication recruit both phonological and visuospatial resources: evidence from a symmetric dual-task design

Psychological Research, 2015

Previous studies pointed out a selective interaction between different working memory subsystems ... more Previous studies pointed out a selective interaction between different working memory subsystems (i.e., phonological and visuospatial) and arithmetic operations (i.e., multiplication and subtraction). This was interpreted to support the idea that multiplication and subtraction predominantly rely on a phonologically or spatially organized number code, respectively. Here, we investigated this idea in two groups (multiplication and subtraction group) using a dual-task paradigm. Going beyond previous studies, we carefully controlled and balanced the difficulty of both working memory and calculation tasks within and across participants. This allowed us to test the reciprocal impact of calculations on working memory. We observed no selective interaction between different working memory subsystems and arithmetic operations. Instead, both types of arithmetic operations were impaired by both types of concurrent working memory tasks. Likewise, both types of working memory tasks were impaired by both types of concurrent arithmetic. Our findings suggest that multiplication and subtraction depend on both phonological and visuospatial codes and highlight the importance of balancing task demands within and between participants in the context of dual-task studies.

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Exploring the boundary conditions of unconscious numerical priming effects with continuous flash suppression

Consciousness and cognition, 2015

The scope and limits of unconscious processing are a controversial topic of research in experimen... more The scope and limits of unconscious processing are a controversial topic of research in experimental psychology. Particularly within the visual domain, a wide range of paradigms have been used to experimentally manipulate perceptual awareness. A recent study reported unconscious numerical processing during continuous flash suppression (CFS), which is a powerful variant of interocular suppression and disrupts the conscious perception of visual stimuli for up to seconds. Since this finding of a distance-dependent priming effect contradicts earlier results showing that interocular suppression abolishes semantic processing, we sought to investigate the boundary conditions of this effect in two experiments. Using statistical analyses and experimental designs that precluded an effect of target numerosity, we found evidence for identity priming, but no conclusive evidence for distance-dependent numerical priming under CFS. Our results suggest that previous conclusions on high-level numeric...

No conclusive evidence for numerical priming under interocular suppression

Psychological science, 2014

Micro and macro pattern analyses of FMRI data support both early and late interaction of numerical and spatial information

by Andre Knops and Klaus Hinckeldey

Frontiers in human neuroscience, 2011

Numbers and space are two semantic primitives that interact with each other. Both recruit brain r... more Numbers and space are two semantic primitives that interact with each other. Both recruit brain regions along the dorsal pathway, notably parietal cortex. This makes parietal cortex a candidate for the origin of numerical-spatial interaction. The underlying cognitive architecture of the interaction is still under scrutiny. Two classes of explanations can be distinguished. The early interaction approach assumes that numerical and spatial information are integrated into a single representation at a semantic level. A second approach postulates independent semantic representations. Only at the stage of response selection and preparation these two streams interact. In this study we used a numerical landmark task to identify the locus of the interaction between numbers and space. While lying in an MR scanner participants decided on the smaller of two numerical intervals in a visually presented number triplet. The spatial position of the middle number was varied; hence spatial intervals we...

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Operational Momentum in Multiplication and Division?

PLoS ONE, 2014

Biases are commonly seen in numerical cognition. The operational momentum (OM) effect shows that ... more Biases are commonly seen in numerical cognition. The operational momentum (OM) effect shows that responses to addition and subtraction problems are biased in the whole-number direction of the operation. It is not known if this bias exists for other arithmetic operations. To determine whether OM exists in scalar operations, we measured response bias in adults performing symbolic (Arabic digits) and non-symbolic (dots) multiplication and division problems. After seeing two operands, with either a multiplication (6) or division (4) sign, participants chose among five response choices. Both nonrandom performance profiles and the significant contribution of both operands in a multiple regression analysis predicting the chosen values, suggest that adults were able to use numerical information to approximate the outcomes in both notations, though they were more accurate on symbolic problems. Performance on non-symbolic problems was influenced by the size of the correct choice relative to alternatives. Reminiscent of the bias in addition and subtraction, we found a significant response bias for non-symbolic problems. Non-symbolic multiplication problems were overestimated and division problems were underestimated. These results indicate that operational momentum is present in non-symbolic multiplication and division. Given the influence of the size of the correct choice relative to alternatives, an interaction between heuristic bias and approximate calculation is possible.

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What does the absence of operational momentum in 6-to 7-year-old children tell us about the mental number representation?

Transkranielle Magnetstimulation

Attentional cueing in numerical cognition-A Commentary on Zanolie & Pecher (2014)

Lateralisiertes priming in Kopfrechenaufgaben

Die semantische Größenrepräsentation: Existiert mehr als nur ein numerischer Zahlenstrahl

by Andre Knops and Klaus Hinckeldey

On the structure and neural correlates of the numerical magnitude representation and its influence in the assessment of verbal working memory

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The scope and limits of numerical priming under continuous flash suppression

In How Many Ways is the Approximate Number System Associated with Exact Calculation?

by Andre Knops and Klaus Hinckeldey

PLoS ONE, 2014

The approximate number system (ANS) has been consistently found to be associated with math achiev... more The approximate number system (ANS) has been consistently found to be associated with math achievement. However, little is known about the interactions between the different instantiations of the ANS and in how many ways they are related to exact calculation. In a cross-sectional design, we investigated the relationship between three measures of ANS acuity (non-symbolic comparison, non-symbolic estimation and non-symbolic addition), their cross-sectional trajectories and specific contributions to exact calculation. Children with mathematical difficulties (MD) and typically achieving (TA) controls attending the first six years of formal schooling participated in the study. The MD group exhibited impairments in multiple instantiations of the ANS compared to their TA peers. The ANS acuity measured by all three tasks positively correlated with age in TA children, while no correlation was found between non-symbolic comparison and age in the MD group. The measures of ANS acuity significantly correlated with each other, reflecting at least in part a common numerosity code. Crucially, we found that non-symbolic estimation partially and non-symbolic addition fully mediated the effects of non-symbolic comparison in exact calculation.

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The neural basis of number processing

Considering structural connectivity in the triple code model of numerical cognition: differential connectivity for magnitude processing and arithmetic facts

by Klaus Hinckeldey and Andre Knops

Brain Structure and Function, 2014

The current study provides a generalizable account of the anatomo-functional associations as well... more The current study provides a generalizable account of the anatomo-functional associations as well as the connectivity of representational codes underlying numerical processing as suggested by the triple code model (TCM) of numerical cognition. By evaluating the neural networks subserving numerical cognition in two specific and substantially different numerical tasks with regard to both grey matter localizations as well as white matter tracts we (1) considered the possibility of additional memory-related cortex areas crucial for arithmetic fact retrieval (e.g., the hippocampus); (2) specified the functional involvement of prefrontal areas in number magnitude processing, and, finally; (3) identified the connections between these anatomo-functional instantiations of the representations involved in number magnitude processing and arithmetic fact retrieval employing probabilistic fiber tracking. The resulting amendments to the TCM are summarized in a schematic update, and ideas concerning the possible functional interplay between number magnitude processing and arithmetic fact retrieval are discussed.

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A shared, flexible neural map architecture reflects capacity limits in both visual short-term memory and enumeration

by Andre Knops and Rakesh Sengupta

The Journal of neuroscience : the official journal of the Society for Neuroscience, Jan 23, 2014

Human cognition is characterized by severe capacity limits: we can accurately track, enumerate, o... more Human cognition is characterized by severe capacity limits: we can accurately track, enumerate, or hold in mind only a small number of items at a time. It remains debated whether capacity limitations across tasks are determined by a common system. Here we measure brain activation of adult subjects performing either a visual short-term memory (vSTM) task consisting of holding in mind precise information about the orientation and position of a variable number of items, or an enumeration task consisting of assessing the number of items in those sets. We show that task-specific capacity limits (three to four items in enumeration and two to three in vSTM) are neurally reflected in the activity of the posterior parietal cortex (PPC): an identical set of voxels in this region, commonly activated during the two tasks, changed its overall response profile reflecting task-specific capacity limitations. These results, replicated in a second experiment, were further supported by multivariate pa...

Attentional cueing in numerical cognition

Frontiers in psychology, 2014

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Recruitment of an Area Involved in Eye Movements During Mental Arithmetic

Science, 2009

The arithmetic problems were created from a set of standardized problems with 50 as first operand... more The arithmetic problems were created from a set of standardized problems with 50 as first operand and either a small (6) or a large numerosity (26) as the second operand. The operands were identical for addition and subtraction. Apart from the correct result, 8 deviant results were created for each arithmetic problem. These deviants were arranged as a geometric series (i.e. were linearly spaced on a logarithmic scale) and ranged from double the correct result to half of the correct result (technically, they were generated as round(c x r i/4 ), where c is the correct result, r the maximal ratio between correct outcome and the lowest/highest deviant, and i an index ranging from -4 to +4). Because prior research indicated that subjects could achieve a greater precision in symbolic than in non-symbolic notation (1), in order to maintain an approximately similar level of difficulty the ratio r varied between notations. Deviants for Arabic problems were more closely centered on the correct outcome (r = 1.41) than were the deviants for non-symbolic problems (r = 2.5). To avoid a strategy of always selecting the response falling in the middle of the proposed range, only seven out of those nine possible results were presented. In 50% of the trials we presented the upper seven (high range), and thus the correct result was the third largest numerosity. In the other 50% of the trials the lower seven choices were shown (low range), and the correct result was therefore the fifth largest numerosity.

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