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Bert De Smedt - One of the best experts on this subject based on the ideXlab platform.
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Fact retrieval or compacted procedures in Arithmetic a neurophysiological investigation of two hypotheses
bioRxiv, 2020Co-Authors: Roland H. Grabner, Clemens Brunner, Valerie Lorenz, Stephan E Vogel, Bert De SmedtAbstract:There is broad consensus that adults solve single-digit multiplication problems almost exclusively by Fact retrieval (i.e., retrieval of the solution from an Arithmetic Fact network). In contrast, there has been a long-standing debate on the cognitive processes involved in solving single-digit addition problems. This debate has evolved around two theoretical accounts. The Fact-retrieval account postulates that these are solved through Fact retrieval, just like multiplications, whereas the compacted-procedure account proposes that solving very small additions (i.e., problems with operands between 1 and 4) involves highly automatized and unconscious compacted procedures. In the present electroencephalography (EEG) study, we put these two accounts to the test by comparing neurophysiological correlates of solving very small additions and multiplications. A sample of 40 adults worked on an Arithmetic production task involving all (non-tie) single-digit additions and multiplications. Afterwards, participants completed trial-by-trial strategy self-reports. In our EEG analyses, we focused on induced activity (event-related synchronization/desynchronization, ERS/ERD) in three frequency bands (theta, lower alpha, upper alpha). Across all frequency bands, we found higher evidential strength for similar rather than different neurophysiological processes accompanying the solution of very small addition and multiplication problems. This was also true when n + 1 and n x 1 problems were excluded from the analyses. In two additional analyses, we showed that ERS/ERD can differentiate between self-reported problem-solving strategies (retrieval vs. procedure) and even between n + 1 and n + m problems in very small additions, demonstrating its high sensitivity to cognitive processes in Arithmetic. The present findings clearly support the Fact-retrieval account, suggesting that both very small additions and multiplications are solved through Fact retrieval.
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Language and Arithmetic: The Potential Role of Phonological Processing
Heterogeneity of Function in Numerical Cognition, 2018Co-Authors: Bert De SmedtAbstract:Abstract The association between language and numerical cognition has been subject to a long-standing debate. This chapter focuses on one area of language, namely phonological processing and its association with one domain of numerical cognition, i.e., Arithmetic. The existing brain and behavioral studies suggest, in line with the original assumption of the triple-code model, that there is an association between phonological processing and Arithmetic, which is most prominent in the context of Arithmetic Fact retrieval, particularly in multiplication. The association is most often found for phonological awareness and is particularly observed in the context of atypical numerical development. This suggests that poor phonological skills might be one risk Factor for Arithmetic Fact retrieval difficulties. As the existing behavioral and brain imaging data are mainly cross-sectional, longitudinal studies are critical to make headway in our understanding of the association between phonological processing and Arithmetic Fact retrieval.
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The role of physical digit representation and numerical magnitude representation in children’s multiplication Fact retrieval
Journal of experimental child psychology, 2016Co-Authors: Alice De Visscher, Marie-pascale Noël, Bert De SmedtAbstract:Abstract Arithmetic Facts, in particular multiplication tables, are thought to be stored in long-term memory and to be interference prone. At least two representations underpinning these Arithmetic Facts have been suggested: a physical representation of the digits and a numerical magnitude representation. We hypothesized that both representations are possible sources of interference that could explain individual differences in multiplication Fact performance and/or in strategy use. We investigated the specificity of these interferences on Arithmetic Fact retrieval and explored the relation between interference and performance on the different Arithmetic operations and on general mathematics achievement. Participants were 79 fourth-grade children (Mage = 9.6 years) who completed a products comparison and a multiplication production task with verbal strategy reports. Performances on a speeded calculation test including the four operations and on a general mathematics achievement test were also collected. Only the interference coming from physical representations was a significant predictor of the performance across multiplications. However, both the magnitude and physical representations were unique predictors of individual differences in multiplication. The frequency of the retrieval strategy across multiplication problems and across individuals was determined only by the physical representation, which therefore is suggested as being responsible for memory storage issues. Interestingly, this impact of physical representation was not observed when predicting performance on subtraction or on general mathematical achievement. In contrast, the impact of the numerical magnitude representation was more general in that it was observed across all Arithmetic operations and in general mathematics achievement.
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Are Individual Differences in Arithmetic Fact Retrieval in Children Related to Inhibition
Frontiers in psychology, 2016Co-Authors: Elien Bellon, Wim Fias, Bert De SmedtAbstract:Although it has been proposed that inhibition is related to individual differences in mathematical achievement, it is not clear how it is related to specific aspects of mathematical skills, such as Arithmetic Fact retrieval. The present study therefore investigated the association between inhibition and Arithmetic Fact retrieval and further examined the unique role of inhibition in individual differences in Arithmetic Fact retrieval, in addition to numerical magnitude processing. We administered measures of cognitive inhibition (i.e., numerical and non-numerical stroop tasks) and a complementary, more ecologically valid measure of children's inhibition in the classroom (i.e., teacher questionnaire), as well as numerical magnitude processing (i.e., symbolic and non-symbolic numerical magnitude comparison) and Arithmetic Fact retrieval (i.e., two verification tasks) in 86 typically developing third graders. We used a correlation, a regression and a Bayesian analysis. This study failed to observe a significant association between inhibition and Arithmetic Fact retrieval. Consequently, our results did not reveal a unique contribution of inhibition to Arithmetic Fact retrieval in addition to numerical magnitude processing. On the other hand, symbolic numerical magnitude processing turned out to be a very powerful predictor of Arithmetic Fact retrieval, as indicated by both frequentist and Bayesian approaches.
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The association between symbolic and nonsymbolic numerical magnitude processing and mental versus algorithmic subtraction in adults.
Acta psychologica, 2016Co-Authors: Sarah Linsen, Joke Torbeyns, Lieven Verschaffel, Bert Reynvoet, Bert De SmedtAbstract:There are two well-known computation methods for solving multi-digit subtraction items, namely mental and algorithmic computation. It has been contended that mental and algorithmic computation differentially rely on numerical magnitude processing, an assumption that has already been examined in children, but not yet in adults. Therefore, in this study, we examined how numerical magnitude processing was associated with mental and algorithmic computation, and whether this association with numerical magnitude processing was different for mental versus algorithmic computation. We also investigated whether the association between numerical magnitude processing and mental and algorithmic computation differed for measures of symbolic versus nonsymbolic numerical magnitude processing. Results showed that symbolic, and not nonsymbolic, numerical magnitude processing was associated with mental computation, but not with algorithmic computation. Additional analyses showed, however, that the size of this association with symbolic numerical magnitude processing was not significantly different for mental and algorithmic computation. We also tried to further clarify the association between numerical magnitude processing and complex calculation by also including relevant Arithmetical subskills, i.e. Arithmetic Facts, needed for complex calculation that are also known to be dependent on numerical magnitude processing. Results showed that the associations between symbolic numerical magnitude processing and mental and algorithmic computation were fully explained by individual differences in elementary Arithmetic Fact knowledge.
K. Orbinian Moeller - One of the best experts on this subject based on the ideXlab platform.
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Time Reading in Middle and Secondary School Students: The Influence of Basic-Numerical Abilities
The Journal of genetic psychology, 2020Co-Authors: Katharina Lambert, Silke M. Wortha, K. Orbinian MoellerAbstract:Time reading skills are central for the management of personal and professional life. However, little is known about the differential influence of basic numerical abilities on analog and digital time reading in general and in middle and secondary school students in particular. The present study investigated the influence of basic numerical skills separately for analog and digital time reading in N = 709 students from 5th to 8th grade. The present findings suggest that the development of time reading skills is not completed by the end of primary school. Results indicated that aspects of magnitude manipulation and Arithmetic Fact knowledge predicted analog time reading significantly over and above the influence of age. Furthermore, results showed that spatial representations of number magnitude, magnitude manipulation, Arithmetic Fact knowledge, and conceptual knowledge were significant predictors of digital time reading beyond general cognitive ability and sex. To the best of our knowledge, the present study is the first to show differential effects of basic numerical abilities on analog and digital time reading skills in middle and secondary school students. As time readings skills are crucial for everyday life, these results are highly relevant to better understand basic numerical processes underlying time reading.
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More than simple Facts: cross-linguistic differences in place-value processing in Arithmetic Fact retrieval
Psychological Research-psychologische Forschung, 2018Co-Authors: Julia Bahnmueller, Silvia Pixner, Verena Dresen, Silke M Gobel, K. Orbinian MoellerAbstract:Linguistic specificities such as the inversion property of number words (e.g., in German 43 is spoken dreiundvierzig, literally three and forty) moderate Arabic number processing. So far, cross-linguistic studies have mostly focused on inversion-related effects on simple (e.g., number comparison) and calculation-based (e.g., multi-digit addition) magnitude processing of numerical information. Despite the assumption that multiplication Facts are represented in verbal format, not much attention has been paid to inversion-related influences on multiplication Fact retrieval. Accordingly, the current study evaluated inversion-related effects on the processing of place-value information in multiplication. In a verification paradigm, the decade consistency effect (i.e., more errors when the decade of a solution probe shares the decade digit with the correct solution) was larger for English- than German-speaking participants for table-related probes. Processing of decade digits might be prioritised in English-speaking participants because the decade digit is named first in English number words, whereas in German number words the unit digit is named first. Our results indicate that (1) the influence of specificities of a verbal number word formation on place-value processing generalise to Arithmetic Fact retrieval and (2) inversion of number words might even be advantageous in specific cases.
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White matter neuro-plasticity in mental Arithmetic: Changes in hippocampal connectivity following Arithmetic drill training.
Cortex; a journal devoted to the study of the nervous system and behavior, 2018Co-Authors: Elise Klein, Klaus Willmes, Silke M. Bieck, Johannes Bloechle, K. Orbinian MoellerAbstract:Verbally-mediated Arithmetic Fact retrieval has been suggested to be subserved by a left-lateralized network including angular gyrus and hippocampus. However, the contribution of these areas to retrieval of Arithmetic Facts has been under debate lately, challenging the prominent role of the angular gyrus in Arithmetic Fact retrieval. In the present study, we evaluated changes in structural connectivity of left hippocampus and left angular gyrus in 32 participants following a short extensive drill training of complex multiplication. We observed a significant increase of structural connectivity in fibers encompassing the left hippocampus but not the left angular gyrus. As such, our findings substantiate that the left hippocampus plays a central role in Arithmetic Fact retrieval. While both structures, left angular gyrus and left hippocampus seem to be parts of the network processing Arithmetic Facts, hippocampus actually seems to subserve encoding and retrieval of Arithmetic Facts. In turn, the role of the left angular gyrus might rather be to mediate the Fact retrieval network as to whether or not processes of Fact retrieval are referred to.
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Behavioural evidence for sex differences in the overlap between subtraction and multiplication
Cognitive Processing, 2016Co-Authors: Belinda Pletzer, K. Orbinian Moeller, Frank Domahs, Andrea Scheuringer, Hubert H. Kerschbaum, Hans-christoph NuerkAbstract:The present study aims to identify Factors that may influence the dissociability of number magnitude processing and Arithmetic Fact retrieval at the behavioural level. To that end, we assessed both subtraction and multiplication performance in a within-subject approach and evaluated the interdependence of unit-decade integration measures on the one hand as well as sex differences in the interdependence of performance measures on the other hand. We found that subtraction items requiring borrowing (e.g. 53–29 = 24, 3 3), thereby demonstrating a borrowing effect, which has been suggested as a measure of unit-decade integration in subtraction. Furthermore, we observed that multiplication items with decade-consistent distractors (e.g. 6 × 4 = 28 instead of 24) are more error prone that multiplication items with decade-inconsistent distractors (e.g. 6 × 4 = 30 instead of 24), thereby demonstrating a decade-consistency effect, which has been suggested as a measure of unit-decade integration in simple multiplication. However, the borrowing effect in subtraction was not correlated with the effect of decade consistency in simple multiplication in either men or women. This indicates that unit-decade integration arises from different systems in subtraction and multiplication. Nevertheless, men outperformed women not only in subtraction, but also in multiplication. Furthermore, subtraction and multiplication performance on correct solution probes were correlated in women, but unrelated in men. Thus, the view of differential systems for number magnitude processing and Arithmetic Fact retrieval may not be universal across sexes.
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Fact learning in complex Arithmetic-the role of the angular gyrus revisited.
Human brain mapping, 2016Co-Authors: Johannes Bloechle, K. Orbinian Moeller, Julia Bahnmueller, Klaus Willmes, Stefan Huber, Johannes Rennig, Seda Cavdaroglu, Elise KleinAbstract:In recent theoretical considerations as well as in neuroimaging findings the left angular gyrus (AG) has been associated with the retrieval of Arithmetic Facts. This interpretation was corroborated by higher AG activity when processing trained as compared with untrained multiplication problems. However, so far neural correlates of processing trained versus untrained problems were only compared after training. We employed an established learning paradigm (i.e., extensive training of multiplication problems) but measured brain activation before and afte training to evaluate neural correlates of Arithmetic Fact acquisition more specifically. When comparing activation patterns for trained and untrained problems of the post-training session, higher AG activation for trained problems was replicated. However, when activation for trained problems was compared to activation for the same problems in the pre-training session, no signal change in the AG was observed. Instead, our results point toward a central role of hippocampal, para-hippocampal, and retrosplenial structures in Arithmetic Fact retrieval. We suggest that the AG might not be associated with the actual retrieval of Arithmetic Facts, and outline an attentional account of the role of the AG in Arithmetic Fact retrieval that is compatible with recent attention to memory hypotheses. Hum Brain Mapp 37:3061-3079, 2016. © 2016 Wiley Periodicals, Inc.
Hans-christoph Nuerk - One of the best experts on this subject based on the ideXlab platform.
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Behavioural evidence for sex differences in the overlap between subtraction and multiplication
Cognitive Processing, 2016Co-Authors: Belinda Pletzer, K. Orbinian Moeller, Frank Domahs, Andrea Scheuringer, Hubert H. Kerschbaum, Hans-christoph NuerkAbstract:The present study aims to identify Factors that may influence the dissociability of number magnitude processing and Arithmetic Fact retrieval at the behavioural level. To that end, we assessed both subtraction and multiplication performance in a within-subject approach and evaluated the interdependence of unit-decade integration measures on the one hand as well as sex differences in the interdependence of performance measures on the other hand. We found that subtraction items requiring borrowing (e.g. 53–29 = 24, 3 3), thereby demonstrating a borrowing effect, which has been suggested as a measure of unit-decade integration in subtraction. Furthermore, we observed that multiplication items with decade-consistent distractors (e.g. 6 × 4 = 28 instead of 24) are more error prone that multiplication items with decade-inconsistent distractors (e.g. 6 × 4 = 30 instead of 24), thereby demonstrating a decade-consistency effect, which has been suggested as a measure of unit-decade integration in simple multiplication. However, the borrowing effect in subtraction was not correlated with the effect of decade consistency in simple multiplication in either men or women. This indicates that unit-decade integration arises from different systems in subtraction and multiplication. Nevertheless, men outperformed women not only in subtraction, but also in multiplication. Furthermore, subtraction and multiplication performance on correct solution probes were correlated in women, but unrelated in men. Thus, the view of differential systems for number magnitude processing and Arithmetic Fact retrieval may not be universal across sexes.
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Considering structural connectivity in the triple code model of numerical cognition: differential connectivity for magnitude processing and Arithmetic Facts.
Brain structure & function, 2014Co-Authors: Elise Klein, K. Orbinian Moeller, Julia Suchan, Hans-otto Karnath, André Knops, Guilherme Wood, Hans-christoph Nuerk, Klaus WillmesAbstract: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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Menstrual cycle variations in the BOLD-response to a number bisection task: implications for research on sex differences.
Brain research, 2011Co-Authors: Belinda Pletzer, Hans-christoph Nuerk, Martin Kronbichler, Gunther Ladurner, Hubert KerschbaumAbstract:Numerical processing involves either number magnitude processing, which has been related to spatial abilities and relies on superior parietal regions, or Arithmetic Fact retrieval, which has been related to verbal abilities and involves the inferior parietal lobule. Since men score better in spatial and women in verbal tasks, we assume that women have advantages in Fact retrieval, while men have benefits in number magnitude processing. According to findings on menstrual cycle variations in spatial and verbal abilities, Fact retrieval should improve during the luteal phase and magnitude processing during the follicular phase. To dissociate sex- and menstrual cycle-dependent effects on Fact retrieval and number magnitude processing, we applied a number bisection task in 15 men and 15 naturally cycling women. Multiplicative items (e.g. 12_15_18) are part of a multiplication series and can be solved by Fact retrieval, while non-multiplicative items (e.g. 11_14_17) are not part of a multiplication series and require number magnitude processing. In men and women in their luteal phase, error rates were higher and deactivation of the medial prefrontal cortex and the bilateral inferior parietal lobules was stronger for non-multiplicative compared to multiplicative items (positive multiplicativity effect), while in the follicular phase women showed higher error rates and stronger deactivation in multiplicative compared to non-multiplicative items (negative multiplicativity effect). Thus, number magnitude processing improves, while Arithmetic Fact retrieval impairs during the follicular phase. While a female superiority in Arithmetic Fact retrieval could not be confirmed, we observed that sex differences are significantly modulated by menstrual cycle phase.
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What makes multiplication Facts difficult. Problem size or neighborhood consistency
Experimental psychology, 2006Co-Authors: Frank Domahs, Margarete Delazer, Hans-christoph NuerkAbstract:Two current models of Arithmetic Fact retrieval, the network interference theory (NIT; Campbell, 1995) and the interacting neighbors (IN) model (Verguts & Fias, 2005a), predict that errors in simple multiplication should be more probable, if they include the same digit as the correct result (i.e., if they are "consistent," compared with "inconsistent" errors). In a reanalysis of error data originally reported by Campbell (1997), we provide first empirical evidence for this prediction. Furthermore, these results support the notion of different quantity representations for decades and units as proposed by Nuerk, Weger, and Willmes (2001). However, the NIT and IN-model differ in their explanations of the problem-size effect, a hallmark finding robustly observed in Arithmetic Fact retrieval. Only the IN-model predicts that a correct answer's neighborhood consistency can fully account for the problem-size effect, which was confirmed in our analysis.
Frank Domahs - One of the best experts on this subject based on the ideXlab platform.
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Behavioural evidence for sex differences in the overlap between subtraction and multiplication
Cognitive Processing, 2016Co-Authors: Belinda Pletzer, K. Orbinian Moeller, Frank Domahs, Andrea Scheuringer, Hubert H. Kerschbaum, Hans-christoph NuerkAbstract:The present study aims to identify Factors that may influence the dissociability of number magnitude processing and Arithmetic Fact retrieval at the behavioural level. To that end, we assessed both subtraction and multiplication performance in a within-subject approach and evaluated the interdependence of unit-decade integration measures on the one hand as well as sex differences in the interdependence of performance measures on the other hand. We found that subtraction items requiring borrowing (e.g. 53–29 = 24, 3 3), thereby demonstrating a borrowing effect, which has been suggested as a measure of unit-decade integration in subtraction. Furthermore, we observed that multiplication items with decade-consistent distractors (e.g. 6 × 4 = 28 instead of 24) are more error prone that multiplication items with decade-inconsistent distractors (e.g. 6 × 4 = 30 instead of 24), thereby demonstrating a decade-consistency effect, which has been suggested as a measure of unit-decade integration in simple multiplication. However, the borrowing effect in subtraction was not correlated with the effect of decade consistency in simple multiplication in either men or women. This indicates that unit-decade integration arises from different systems in subtraction and multiplication. Nevertheless, men outperformed women not only in subtraction, but also in multiplication. Furthermore, subtraction and multiplication performance on correct solution probes were correlated in women, but unrelated in men. Thus, the view of differential systems for number magnitude processing and Arithmetic Fact retrieval may not be universal across sexes.
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Modulating Arithmetic Fact retrieval: a single-blind, sham-controlled tDCS study with repeated fMRI measurements.
Neuropsychologia, 2013Co-Authors: Benjamin Clemens, Stefanie Jung, Mikhail Zvyagintsev, Frank Domahs, Klaus WillmesAbstract:Transcranial direct current stimulation (tDCS) is a non-invasive technique which has been used to modulate various cognitive functions in healthy participants as well as stroke patients. Despite the increasing number of tDCS studies, it still remains questionable whether tDCS is suitable for modulating performance in Arithmetic tasks and whether a single tDCS session may cause brain activity changes that can be detected with functional magnetic resonance imaging (fMRI). We asked healthy participants to repeatedly solve simple multiplication tasks in three conditions: STIMULATION (anodal tDCS over the right angular gyrus, AG), SHAM (identical electrode set-up without stimulation), and CONTROL (no electrodes attached). Before and after tDCS, we used fMRI to examine changes in brain activity. Behavioural results indicate that a single session of tDCS did not modulate task performance significantly. However, fMRI measurements revealed that the neural correlates of multiplication were modified following a single session of anodal tDCS. In the bilateral AG, activity was significantly higher for multiplication problems rehearsed during active tDCS, as compared to multiplication problems rehearsed without tDCS or during sham tDCS. In sum, we present first neuro-functional evidence that tDCS modulates Arithmetic processing. Implications of these findings for future tDCS studies and for the rehabilitation of acalculic patients with deficits in Arithmetic Fact retrieval are discussed.
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Rehabilitation of Arithmetic Fact retrieval via extensive practice: A combined fMRI and behavioural case-study
Neuropsychological rehabilitation, 2008Co-Authors: Luisa Zaunmüller, Elise Klein, Frank Domahs, Katharina Dressel, Jan Lonnemann, Anja Ischebeck, Klaus WillmesAbstract:The present study investigates the effects of a training of Arithmetic Fact retrieval in a patient suffering from particular difficulties with multiplication Facts. Over a period of four weeks simple multiplication Facts were trained extensively. The outcome of the training was assessed behaviourally and changes in cerebral activation patterns were investigated using fMRI. The training led to a change in calculation strategies: Prior to training, the patient used predominantly time-consuming back-up strategies, after training he relied increasingly on the direct retrieval of Arithmetic Facts from long-term memory. Regarding the fMRI results, prefrontal activations were observed for untrained problems, which can be attributed to the application of back-up strategies strongly relying on fronto-executive functions. Interestingly, significant foci of activation for both trained and untrained items were found in the angular gyrus of the right hemisphere, which, however, differed in their exact localisation. For the trained condition, activations were observed in anterior parts of the angular gyrus which may be related to the training-based automatisation in Fact retrieval. Activations in the untrained condition were found in a more posterior portion of the angular gyrus, that might be attributable to one of the patient's back-up strategies, namely to recite a whole multiplication row to get to the correct answer.
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What makes multiplication Facts difficult. Problem size or neighborhood consistency
Experimental psychology, 2006Co-Authors: Frank Domahs, Margarete Delazer, Hans-christoph NuerkAbstract:Two current models of Arithmetic Fact retrieval, the network interference theory (NIT; Campbell, 1995) and the interacting neighbors (IN) model (Verguts & Fias, 2005a), predict that errors in simple multiplication should be more probable, if they include the same digit as the correct result (i.e., if they are "consistent," compared with "inconsistent" errors). In a reanalysis of error data originally reported by Campbell (1997), we provide first empirical evidence for this prediction. Furthermore, these results support the notion of different quantity representations for decades and units as proposed by Nuerk, Weger, and Willmes (2001). However, the NIT and IN-model differ in their explanations of the problem-size effect, a hallmark finding robustly observed in Arithmetic Fact retrieval. Only the IN-model predicts that a correct answer's neighborhood consistency can fully account for the problem-size effect, which was confirmed in our analysis.
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Some Assumptions and Facts about Arithmetic Facts
Psychology Science, 2005Co-Authors: Frank Domahs, Margarete DelazerAbstract:Abstract In the present review we will deal with the following questions: What are Arithmetic Facts? How are they related to other cognitive functions? What are characteristic features of Fact retrieval performance in healthy adult subjects? Which typical patterns of breakdown can be observed in acalculia? How do current models account for the representation of and access to Arithmetic Facts? Key words: simple calculation, mental Arithmetic, Fact retrieval Simple Arithmetic problems, such as 3+6 or 5x4, are frequently encountered and routinely solved in every-day life. Those problems whose solution does not require further computational processes or strategies but can directly be retrieved from long-term memory are commonly referred to as Arithmetic Facts. Usually, problems with one-digit operands (2+4; 3x5; 6-3) are subsumed under this label, though a precise definition is rarely given. In the following, we will describe dissociations and associations of Arithmetic Fact retrieval and related cognitive functions, reviewing neuropsychological evidence as well as findings from experimental and developmental psychology. In this way we will try to characterize in more detail what Arithmetic Facts actually are. Finally, some approaches will be presented to model the acquisition, representation, and retrieval of Arithmetic Facts. Concepts, procedures, and Facts Three main types of Arithmetic knowledge can be distinguished: concepts, procedures, and Facts (Delazer, 2003). Conceptual knowledge provides understanding of Arithmetic operations and principles. This type of knowledge is the prerequisite to make inferences and to relate different information involved in Arithmetic. Conceptual knowledge is flexible, can be adapted and applied to new tasks and thus provides adaptive expertise (Hatano, 1988). Procedural knowledge guides the execution of algorithms. It can only be applied in familiar contexts. Accordingly, it can be characterised as routine expertise (Hatano, 1988). Arithmetic Facts, finally, can be conceptualised as stored in and directly retrieved from (declarative) long-term memory (e.g., Ashcraft, 1987; Campbell, 1995; Dehaene & Cohen, 1995; Rickard & Bourne, 1996; Siegler, 1988; for a different view see Baroody, 1994). During development, Arithmetic Facts evolve from conceptual and procedural knowledge. For instance, it has been demonstrated that the pattern of response latencies and errors in the retrieval of simple multiplication Facts of older children and adults reflects the number and type of errors made in the backup strategies used by children at an earlier developmental stage solving the same problems (Lemaire & Siegler, 1995; Siegler, 1988). Neuropsychological studies support the assumption that Arithmetic Facts are stored separately from other numerical skills. In a seminal paper Warrington (1982) reported a patient who was no longer able to retrieve simple Arithmetic problems from memory, but was able to give the approximate result of Arithmetic problems, both in simple and more complex calculation, to estimate visually presented quantities of dots, to give adequate numerical cognitive estimates, to judge the relative size of a number and to give accurate definitions of Arithmetic operations. Several later case reports confirmed the separate storage and selective vulnerability of Arithmetic Facts knowledge. However, other studies questioned the assumption that healthy, educated adults retrieve the solutions for all simple Arithmetic problems from memory (i.e., as Facts) (Lefevre, Bisanz, Daley, Buffone, Greenham, & Sadesky, 1996; Lefevre, Sadesky, & Bisanz, 1996; for methodological considerations see Kirk & Ashcraft, 2001). Lefevre and colleagues found that in single digit addition, non-retrieval strategies (e.g., counting) accounted for 29% of all trials and for about half the problems with sums above 10. In simple multiplication, up to 19% of all trials were solved by non-retrieval strategies (e. …
Klaus Willmes - One of the best experts on this subject based on the ideXlab platform.
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White matter neuro-plasticity in mental Arithmetic: Changes in hippocampal connectivity following Arithmetic drill training.
Cortex; a journal devoted to the study of the nervous system and behavior, 2018Co-Authors: Elise Klein, Klaus Willmes, Silke M. Bieck, Johannes Bloechle, K. Orbinian MoellerAbstract:Verbally-mediated Arithmetic Fact retrieval has been suggested to be subserved by a left-lateralized network including angular gyrus and hippocampus. However, the contribution of these areas to retrieval of Arithmetic Facts has been under debate lately, challenging the prominent role of the angular gyrus in Arithmetic Fact retrieval. In the present study, we evaluated changes in structural connectivity of left hippocampus and left angular gyrus in 32 participants following a short extensive drill training of complex multiplication. We observed a significant increase of structural connectivity in fibers encompassing the left hippocampus but not the left angular gyrus. As such, our findings substantiate that the left hippocampus plays a central role in Arithmetic Fact retrieval. While both structures, left angular gyrus and left hippocampus seem to be parts of the network processing Arithmetic Facts, hippocampus actually seems to subserve encoding and retrieval of Arithmetic Facts. In turn, the role of the left angular gyrus might rather be to mediate the Fact retrieval network as to whether or not processes of Fact retrieval are referred to.
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Fact learning in complex Arithmetic-the role of the angular gyrus revisited.
Human brain mapping, 2016Co-Authors: Johannes Bloechle, K. Orbinian Moeller, Julia Bahnmueller, Klaus Willmes, Stefan Huber, Johannes Rennig, Seda Cavdaroglu, Elise KleinAbstract:In recent theoretical considerations as well as in neuroimaging findings the left angular gyrus (AG) has been associated with the retrieval of Arithmetic Facts. This interpretation was corroborated by higher AG activity when processing trained as compared with untrained multiplication problems. However, so far neural correlates of processing trained versus untrained problems were only compared after training. We employed an established learning paradigm (i.e., extensive training of multiplication problems) but measured brain activation before and afte training to evaluate neural correlates of Arithmetic Fact acquisition more specifically. When comparing activation patterns for trained and untrained problems of the post-training session, higher AG activation for trained problems was replicated. However, when activation for trained problems was compared to activation for the same problems in the pre-training session, no signal change in the AG was observed. Instead, our results point toward a central role of hippocampal, para-hippocampal, and retrosplenial structures in Arithmetic Fact retrieval. We suggest that the AG might not be associated with the actual retrieval of Arithmetic Facts, and outline an attentional account of the role of the AG in Arithmetic Fact retrieval that is compatible with recent attention to memory hypotheses. Hum Brain Mapp 37:3061-3079, 2016. © 2016 Wiley Periodicals, Inc.
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Considering structural connectivity in the triple code model of numerical cognition: differential connectivity for magnitude processing and Arithmetic Facts.
Brain structure & function, 2014Co-Authors: Elise Klein, K. Orbinian Moeller, Julia Suchan, Hans-otto Karnath, André Knops, Guilherme Wood, Hans-christoph Nuerk, Klaus WillmesAbstract: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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Modulating Arithmetic Fact retrieval: a single-blind, sham-controlled tDCS study with repeated fMRI measurements.
Neuropsychologia, 2013Co-Authors: Benjamin Clemens, Stefanie Jung, Mikhail Zvyagintsev, Frank Domahs, Klaus WillmesAbstract:Transcranial direct current stimulation (tDCS) is a non-invasive technique which has been used to modulate various cognitive functions in healthy participants as well as stroke patients. Despite the increasing number of tDCS studies, it still remains questionable whether tDCS is suitable for modulating performance in Arithmetic tasks and whether a single tDCS session may cause brain activity changes that can be detected with functional magnetic resonance imaging (fMRI). We asked healthy participants to repeatedly solve simple multiplication tasks in three conditions: STIMULATION (anodal tDCS over the right angular gyrus, AG), SHAM (identical electrode set-up without stimulation), and CONTROL (no electrodes attached). Before and after tDCS, we used fMRI to examine changes in brain activity. Behavioural results indicate that a single session of tDCS did not modulate task performance significantly. However, fMRI measurements revealed that the neural correlates of multiplication were modified following a single session of anodal tDCS. In the bilateral AG, activity was significantly higher for multiplication problems rehearsed during active tDCS, as compared to multiplication problems rehearsed without tDCS or during sham tDCS. In sum, we present first neuro-functional evidence that tDCS modulates Arithmetic processing. Implications of these findings for future tDCS studies and for the rehabilitation of acalculic patients with deficits in Arithmetic Fact retrieval are discussed.
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Rehabilitation of Arithmetic Fact retrieval via extensive practice: A combined fMRI and behavioural case-study
Neuropsychological rehabilitation, 2008Co-Authors: Luisa Zaunmüller, Elise Klein, Frank Domahs, Katharina Dressel, Jan Lonnemann, Anja Ischebeck, Klaus WillmesAbstract:The present study investigates the effects of a training of Arithmetic Fact retrieval in a patient suffering from particular difficulties with multiplication Facts. Over a period of four weeks simple multiplication Facts were trained extensively. The outcome of the training was assessed behaviourally and changes in cerebral activation patterns were investigated using fMRI. The training led to a change in calculation strategies: Prior to training, the patient used predominantly time-consuming back-up strategies, after training he relied increasingly on the direct retrieval of Arithmetic Facts from long-term memory. Regarding the fMRI results, prefrontal activations were observed for untrained problems, which can be attributed to the application of back-up strategies strongly relying on fronto-executive functions. Interestingly, significant foci of activation for both trained and untrained items were found in the angular gyrus of the right hemisphere, which, however, differed in their exact localisation. For the trained condition, activations were observed in anterior parts of the angular gyrus which may be related to the training-based automatisation in Fact retrieval. Activations in the untrained condition were found in a more posterior portion of the angular gyrus, that might be attributable to one of the patient's back-up strategies, namely to recite a whole multiplication row to get to the correct answer.