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The phenomenon where small numbers produce faster left than right responses and large numbers produce faster right than left responses (The SNARC effect) has been used as evidence for obligatory activations of magnitude. In two experiments, I used the SNARC effect to examine crosstalk using a psychological refractory period (PRP) paradigm. In Experiment 1, subjects made a parity judgment to the second number, while ignoring the first number in session 1 or performing a magnitude judgment on the number in session 2. In Experiment 2, subjects performed a magnitude judgment on the second number. They ignored the first number in session 1 or performed a parity judgment on the number in session 2. The results supported two conclusions. First, presentation of a number produced obligatory representations of magnitude even if the number was to be ignored. Second, early representations of magnitude resulted in crosstalk on processing of the subsequent number.

Two event-related potential (ERP) experiments were conducted to investigate spatial-numeric associations of response codes (SNARC) and attentional cuing (SNAAC). In the SNARC effect, people respond faster when making a left-hand response to report that a number is small, and when making a right-hand response to report that a number is large. Experiment 1 examined effects of SNARC-compatibility and prior response-probability in a number comparison task. Lateralized readiness potentials (LRPs) showed that SNARC-compatibility influenced an intermediate stage of response-selection, and prior response-probability influenced both earlier and later stages. The P300 ERP component was also modulated by SNARC-compatibility and prior response-probability, suggesting parietal involvement in the SNARC effect. In the SNAAC effect, attention is directed to left-side regions of space upon viewing small-magnitude numbers, and to right-side regions of space upon viewing large-magnitude numbers. Experiment 2 investigated whether ERPs evoked by peripheral visual probes would be enhanced when probes appeared in the left hemifield after small-magnitude digits and when they appeared in the right hemifield after large-magnitude digits. ERPs to peripheral probes were not modulated by numerical magnitude of digit pre-cues.

Aachen, Techn. Hochsch., Diss., 2008.

The present thesis deals with the mental representation of numbers in space. Generally it is assumed that numbers are mentally represented on a mental number line along which they ordered in a continuous and analogical manner. Dehaene, Bossini and Giraux (1993) found that the mental number line is spatially oriented from left­-to­-right. Using a parity­-judgment task they observed faster left-hand responses for smaller numbers and faster right-hand responses for larger numbers. This effect has been labelled as Spatial Numerical Association of Response Codes (SNARC) effect. The first study of the present thesis deals with the question whether the spatial orientation of the mental number line derives from the writing system participants are adapted to. According to a strong ontogenetic interpretation the SNARC effect should only obtain for effectors closely related to the comprehension and production of written language (hands and eyes). We asked participants to indicate the parity status of digits by pressing a pedal with their left or right foot. In contrast to the strong ontogenetic view we observed a pedal SNARC effect which did not differ from the manual SNARC effect. In the second study we evaluated whether the SNARC effect reflects an association of numbers and extracorporal space or an association of numbers and hands. To do so we varied the spatial arrangement of the response buttons (vertical vs. horizontal) and the instruction (hand­related vs. button­-related). For vertically arranged buttons and a button­related instruction we found a button-­related SNARC effect. In contrast, for a hand-­related instruction we obtained a hand­-related SNARC effect. For horizontally arranged buttons and a hand­related instruction, however, we found a button­related SNARC effect. The results of the first to studies were interpreted in terms of weak ontogenetic view. In the third study we aimed to examine the functional locus of the SNARC effect. We used the psychological refractory period paradigm. In the first experiment participants first indicated the pitch of a tone and then the parity status of a digit (locus­-of-­slack paradigma). In a second experiment the order of stimulus presentation and thus tasks changed (effect­-propagation paradigm). The results led us conclude that the SNARC effect arises while the response is centrally selected. In our fourth study we test for an association of numbers and time. We asked participants to compare two serially presented digits. Participants were faster to compare ascending digit pairs (e.g., 2-­3) than descending pairs (e.g., 3-­2). The pattern of our results was interpreted in terms of forward­associations (“1­-2-­3”) as formed by our ubiquitous cognitive routines to count of objects or events. / Die vorliegende Arbeit beschäftigt sich mit der räumlichen Repräsentation von Zahlen. Generell wird angenommen, dass Zahlen in einer kontinuierlichen und analogen Art und Weise auf einem mentalen Zahlenstrahl repräsentiert werden. Dehaene, Bossini und Giraux (1993) zeigten, dass der mentale Zahlenstrahl eine räumliche Orientierung von links­-nach­-rechts aufweist. In einer Paritätsaufgabe fanden sie schnellere Links-hand­ Antworten auf kleine Zahlen und schnellere Rechts-hand Antworten auf große Zahlen. Dieser Effekt wurde Spatial Numerical Association of Response Codes (SNARC) Effekt genannt. In der ersten Studie der vorliegenden Arbeit ging es um den Einfluss der Schriftrichtung auf den SNARC Effekt. Eine strenge ontogenetische Sichtweise sagt vorher, dass der SNARC Effekt nur mit Effektoren, die unmittelbar in die Produktion und das Verstehen von Schriftsprache involviert sind, auftreten sollte (Hände und Augen). Um dies zu überprüfen, forderten wir Versuchspersonen auf, die Parität dargestellter Ziffern durch Tastendruck mit ihrem rechten oder linken Fuß anzuzeigen. Entgegen der strengen ontogenetischen Hypothese fanden wir den SNARC Effekt auch für Fußantworten, welcher sich in seiner Charakteristik nicht von dem manuellen SNARC Effekt unterschied. In der zweiten Studie gingen wir der Frage nach, ob dem SNARC Effekt eine Assoziation des nicht-­körperbezogenen Raumes und Zahlen oder der Hände und Zahlen zugrunde liegt. Um dies zu untersuchen, variierten wir die räumliche Orientierung der Tasten zueinander (vertikal vs. horizontal) als auch die Instruktionen (hand-­bezogen vs. knopf­-bezogen). Bei einer vertikalen Knopfanordnung und einer knopf-­bezogenen Instruktion fanden wir einen knopf­bezogenen SNARC Effekt. Bei einer hand-­bezogenen Instruktion fanden wir einen hand-­bezogenen SNARC Effekt. Mit horizontal angeordneten Knöpfen gab es unabhängig von der Instruktion einen knopf-­bezogenen SNARC Effekt. Die Ergebnisse dieser beiden ersten Studien wurden im Sinne einer schwachen ontogenetischen Sichtweise interpretiert. In der dritten Studie befassten wir uns mit dem funktionalen Ursprung des SNARC Effekts. Hierfür nutzten wir das Psychological Refractory Period (PRP) Paradigma. In einem ersten Experiment hörten Versuchspersonen zuerst einen Ton nach welchem eine Ziffer visuell präsentiert wurde (locus-­of-­slack Paradigma). In einem zweiten Experiment wurde die Reihenfolge der Stimuluspräsentation/Aufgaben umgedreht (effect­-propagation Paradigma). Unsere Ergebnisse lassen vermuten, dass der SNARC Effekt während der zentralen Antwortselektion generiert wird. In unserer vierten Studie überprüften wir, ob Zahlen auch mit Zeit assoziiert werden. Wir forderten Versuchspersonen auf zwei seriell dargebotene Zahlen miteinander zu vergleichen. Versuchspersonen waren schneller zeitlich aufsteigende Zahlen (z.B. erst 2 dann 3) als zeitlich abfolgenden Zahlen (z.B. erst 3 dann 2) miteinander zu vergleichen. Unsere Ergebnisse wurden im Sinne unseres vorwärtsgerichteten Mechanismus des Zählens („1-­2-­3“) interpretiert.

This thesis concerns the formulation of novel developmental robotics models of embodied phenomena in number learning. Learning to count is believed to be of paramount importance for the acquisition of the remarkable fluency with which humans are able to manipulate numbers and other abstract concepts derived from them later in life. The ever-increasing amount of evidence for the embodied nature of human mathematical thinking suggests that the investigation of numerical cognition with the use of robotic cognitive models has a high potential of contributing toward the better understanding of the involved mechanisms. This thesis focuses on two particular groups of embodied effects tightly linked with learning to count. The first considered phenomenon is the contribution of the counting gestures to the counting accuracy of young children during the period of their acquisition of the skill. The second phenomenon, which arises over a longer time scale, is the human tendency to internally associate numbers with space that results, among others, in the widely-studied SNARC effect. The PhD research contributes to the knowledge in the subject by formulating novel neuro-robotic cognitive models of these phenomena, and by employing these in two series of simulation experiments. In the context of the counting gestures the simulations provide evidence for the importance of learning the number words prior to learning to count, for the usefulness of the proprioceptive information connected with gestures to improving counting accuracy, and for the significance of the spatial correspondence between the indicative acts and the objects being enumerated. In the context of the model of spatial-numerical associations the simulations demonstrate for the first time that these may arise as a consequence of the consistent spatial biases present when children are learning to count. Finally, based on the experience gathered throughout both modelling experiments, specific guidelines concerning future efforts in the application of robotic modelling in mathematical cognition are formulated.

This study is an investigation into the relationship between numeric cognition and aging. Specifically, older and younger adults engaged in an experimental protocol that allowed observation of number comparison accuracy and response time latencies associated with the SNARC effect, the distance effect, and number format. The experimental protocol featured a computerized magnitude comparison task wherein the participants were prompted to identify the larger of two numbers. Half of the trials featured whole numbers and half featured fractions. The number stimuli were consistently mapped such that half of all trials were at near distance (i.e., difference of 2) or far distance (i.e., difference of 4) and half of all trials had the larger numerosity on the left side of space and the other half with the larger numerosity on the right side of space. Older adults were significantly slower and less accurate than young adults. Both age groups were significantly slower and less accurate when comparing fractions as opposed to comparing whole numbers. The SNARC effect impaired accuracy in both age groups but did not significantly impact response times. The distance effect impacted both age cohorts in accuracy but differentially impacted older adult response times more than young adult response times. The results of this study support the model of numeric cognition as an automatic process when comparing whole numbers at a far distance and this process is not disrupted by the SNARC effect but is when comparing whole numbers at near distance. The results also indicate that fraction comparison is a controlled process even when the fraction stimuli are consistently mapped. Further investigation is necessary to understand the amount of cognitive resources necessitated by fraction processing and if training can improve fraction comparison.

Associations between space, time, numbers and quantities have suggested that there must be a shared representational format which underlies them. One of the more prominent theories of an underlying system of representation is that of ATOM, which suggests that the there exists a generalized system of magnitude which processes these dimensions. However, there has been no study that examines whether speed is also processed within such a system. This study investigates this by seeing if a SNARC or Problem-Size effect arise in two speed estimation tests. The fundamental task for the participants in both tests was to judge which of two stimuli was the fastest, with the stimuli moving towards a wall located in the center of a computer screen. Their correct discrimination and response time was recorded. The results found that there were no significant results for either effect. This might be due to low sample size, methodological difficulties, and low robustness. Future research might amend these difficulties by having a more consistent setup with more trials reflecting the effect being tested and fix some potential problems with the test environment.

Four experiments, using both humans and monkeys as participants, were conducted to investigate the similarities and differences in human and nonhuman primate numerical cognition. In Experiment 1 it was determined that both humans and monkeys display a SNARC effect, with similar symbolic distance effects for both species. In addition, both species were found to respond faster to congruent stimulus pairs. In Experiment 2 both species were found accurately to recognize quantitative stimuli when presented for durations of 150 msec in a divided visual field paradigm. Performance for humans and monkeys for numerals and dot-patterns was almost identical in terms of accuracy and response times. In Experiment 3 participants were required to make relative numerousness judgments in a divided visual field paradigm. Both species responded faster and more accurately to stimuli presented to the right visual field. Species differences appeared, with monkeys performing equally well on both trial types whereas the humans performed better on numeral trials than on dot trials. In Experiment 4 repetitive transcranial magnetic stimulation (rTMS) was combined with the divided visual field paradigm. Accuracy was significantly disrupted for both species when compared to a no stimulation condition. A facilitation effect was also evident with both species exhibiting significant decreases in response time for all trials. Right-handed participants took longer to respond to stimuli presented to the left visual field. These findings add to the body of knowledge regarding both the similarities and differences of how quantitative stimuli are processed by humans and monkeys.

Abstract concepts seem to be related to space dimension. Evidence of this relation refers to the domain of numerical cognition. An example is the SNARC effect (Spatial Numerical Association of Response Codes, Dehaene, Bossini, and Giraux 1993), which consists in the observation that people react faster to small number with the left hand and to large number with the right hand. This number-space interaction has been explained according to the mental number line hypothesis (e.g. Restle 1970; Dehaene, Bossini, and Giraux 1993), which claims that the representation of numbers has the form of a horizontal line upon which numbers are represented from left to right. Recently, an alternative account suggests that the association between numbers and space results from a decision process to categorize numbers as “small” and “large” before being associated with space dimension (e.g. Gevers et al. 2006b, 2010; Van Opstal and Verguts 2013). The first goal of this thesis is investigating the spatial coding of numbers. In a first study, magnitude concepts such as “small” and “large” were observed to be spatially organized like numbers. In a second study, these magnitude concepts were intermixed with numbers in a reversal design (e.g. Notebaert et al. 2006). In this study, responding as incompatible to magnitude concepts with hand or foot was observed to reverse the spatial mapping of numbers, supporting the idea that the congruency between numbers and space results from conceptual coding of magnitude (e.g. Gevers et al. 2006b, 2010; see also Van Opstal and Verguts 2013). Further evidence of association between abstract concepts and space has been provided also in the domain of emotion. On one hand, Casasanto (2009a) demonstrated that people spontaneously associate positive valence with the side of space congruent to the dominant hand. On the other hand, Holmes and Lourenco (2011) observed that emotional expressions are left-to-right spatially organized with increasing in happiness/angriness rather than positive/negative valence. A second aim of this thesis is focused on investigating the spatial coding of emotion. This was meant to understand how general are the spatial mechanisms. In a third study, the reversal paradigm (e.g. Notebaert et al. 2006) was adopted to investigate the processing mechanism underlying spatial coding of numbers and emotional valence concepts. Manipulation of the mapping between valence concepts and lateralized responses did not influence the spatial coding of numbers, suggesting a separate underlying architecture. Finally, in a fourth study, spatial coding of emotion was observed according to both valence and arousal dimensions (Casasanto 2009a; Holmes and Lourenco 2011). / Doctorat en Sciences psychologiques et de l'éducation / info:eu-repo/semantics/nonPublished