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.

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.

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Over the past twenty years, the Approximate Number System (ANS), a cognitive system for representing non-symbolic quantity information, has been the focus of much research attention. Psychologists seeking to understand how individuals learn and perform mathematics have investigated how this system might underlie symbolic mathematical skills. Dot comparison tasks are commonly used as measures of ANS acuity, however very little is known about the cognitive skills that are involved in completing these tasks. The aim of this thesis was to explore the factors that influence performance on dot comparison tasks and discuss the implications of these findings for future research and educational interventions. The first study investigated how the accuracy and reliability of magnitude judgements is influenced by the visual cue controls used to create dot array stimuli. This study found that participants performances on dot comparison tasks created with different visual cue controls were unrelated, and that stimuli generation methods have a substantial influence on test-retest reliability. The studies reported in the second part of this thesis (Studies 2, 3, 4 and 5) explored the role of inhibition in dot comparison task performance. The results of these studies provide evidence that individual differences in inhibition may, at least partially, explain individual differences in dot comparison task performance. Finally, a large multi-study re-analysis of dot comparison data investigated whether individuals take account of numerosity information over and above the visual cues of the stimuli when comparing dot arrays. This analysis revealed that dot comparison task performance may not reflect numerosity processing independently from visual cue processing for all participants, particularly children. This novel evidence may provide some clarification for conflicting results in the literature regarding the relationship between ANS acuity and mathematics achievement. The present findings call into question whether dot comparison tasks should continue to be used as valid measures of ANS acuity.

Thesis advisor: Sara Cordes / The format of our quantity representations is a contentious topic of study in the field of numerical cognition with researchers debating whether we use discrete (i.e. number) or continuous (e.g. area, time, volume or density) cues to make quantity judgements. It has been proposed (through the Sense of Magnitude Theory) that continuous quantities are more perceptual in nature and thus do not require the higher order cognitive processes needed to represent abstract number, making it unlikely that number is tracked in the presence of perceptual quantities. In the current dissertation, I examined claims made by the Sense of Magnitude theory by 1) investigating the accuracy with which we represent continuous quantities and the mental processes we may engage in when representing these quantities and by, 2) comparing the relative salience of discrete and continuous quantities and how this may change across development. In Project 1, I investigated the accuracy with which infants make element size discriminations and whether this ability becomes more precise with age. Project 2 examined the precision with which adults track cumulative area and uncover the process by which they do so. Lastly, Project 3 explored the relative salience of number for preschoolers by assessing their “Spontaneous Focusing on Number.” Together, findings from these three projects undermine claims stating that humans at all stages of development are better at, and prefer to, attend to continuous quantities over discrete number. Instead I propose that this dissertation suggests that humans at all stages of development are strongly attuned to number in their environment. This work not only provides insight into the way we represent quantity in our day to day lives, but it can help us understand where individual difference in mathematical achievement in school may stem from. / Thesis (PhD) — Boston College, 2019. / Submitted to: Boston College. Graduate School of Arts and Sciences. / Discipline: Psychology.

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Numerical cognition and its underlying mechanisms have been extensively investigated because of their crucial role in the survival and reproduction of biological organisms. Evidence shows that animals possess a “number sense” to estimate quantities without counting, which assists behaviors such as foraging and social interaction. Thus, contrary to traditional beliefs that link numerical concepts only to language and culture, a growing body of research demonstrates that non-symbolic numerical cognition is shared across species, ages, and cultures. This number sense relies on the Approximate Number System (ANS), which is a fundamental cognitive mechanism that enables the estimation and comparison of quantities without the need for language or formal numerical symbols, such as digits or counting. The ANS supports the ability to make approximate judgments about the number of objects or events in a set and to perform basic arithmetic operations, such as addition and subtraction, although with inherent imprecision. Research has shown that many species, from primates to invertebrates, possess these abilities. For example, primates can perform numerical tasks and display ratio-dependent accuracy similar to humans, while domestic chicks show proto-arithmetic abilities comparable to those of human infants’. These findings suggest that the ANS might represent an evolutionarily ancient capability for processing numerical information, supporting the ability to navigate numerical tasks in natural environments. However, ongoing debates continue about the balance between innate predispositions and cultural influences on the ANS. In this thesis, we aimed to investigate key features of the ANS across different populations and species to deepen our understanding of the possible innate components of this system. In the first study, we investigated the Spatial-Numerical Association (SNA) phenomenon. Since its discovery, SNA has been largely attributed to cultural factors, such as reading and writing habits, while more recent research indicates that SNA is also manifested in individuals lacking cultural exposure, including non-human animals and pre-verbal children, suggesting that SNA may be influenced by both evolutionary and cultural factors. Thus we explored whether different SNA behaviors could emerge as a function of culture, age, and stimuli presentation. Results from Himba adults, Italian preschoolers, and Western adults revealed different SNA patterns in explicit and implicit tasks. Specifically, non-symbolic numerosities elicited a left-to-right SNA across all the populations in an implicit task, suggesting that SNAs are not solely the product of cultural influence. Additionally, we further investigated how symbolic and non-symbolic numerosities elicit asymmetrical SNA effects in literate adults. Our findings suggest that symbolic and non-symbolic numerosities might trigger distinct hemispheric activations thus affecting the directionality of SNAs. In the second study, we aimed to expand our understanding of numerical cognition in non-human animals (i.e., domestic chicks, Gallus gallus) by extending research from the visual domain to the auditory domain. Our findings demonstrated that the predispositions observed with visual stimuli are also present with acoustic stimuli, highlighting the consistency of the ANS across different sensory modalities. The role of potential confounding variables and the need for further research are discussed. In the third study, we examined another key aspect of the ANS, namely cross-modal numerical transfer. For both humans and non-human animals, the ability to assess and compare numerical quantities across different sensory modalities has been reported. Moreover, it has been demonstrated that multisensory information can improve numerical accuracy in children. Therefore, cross-modal numerical ability may offer evolutionary advantages, providing additional support for the idea that the ANS is shared and maintained across different species. We investigated this ability specifically in domestic chicks by presenting them with auditory and visual stimuli, both in spontaneous choice and imprinting paradigms. While our preliminary results may be not yet conclusive, they suggest the potential for cross-modal numerical abilities in chicks. Limitations of the current studies and proposals for further research are discussed. In summary, this thesis advances our understanding of numerical cognition by exploring SNA and numerosity perception across populations not subjected to cultural inputs. The findings point to a complex interaction between innate mechanisms and cultural influences, with implications for both human and animal cognition. Cross-modal mapping emerges as a particularly promising area for future research, especially in non-human species.

The Triple Code Model (TCM) of numerical cognition has become one of the most predominantly theories for how humans perceive, manipulate, and communicate numerical information. It builds on the notion that there exist three functionally distinct but neurologically connected codes that handle manipulations of different numerical input (non-symbolic magnitudes, symbolic representations, and verbal number words). In this study, we add a developmental perspective by collecting child data and comparing it to existing adult data. The main question is whether or not children elicit the same neural correlates as adults while performing three different number comparison tasks in line with TCM. Neuroimaging data using fMRI were collected for a total of 20 participants (ten children and ten adults). The results suggest that children rely on more right-lateralized regions and that a developmental shift towards the left hemisphere and associated language areas occur during acquisition of mathematical proficiency. / <p>VG</p>

Children who struggle to learn math are often identified by their poor performance on common math learning activities, such as number line estimations. While such behavioral assessments are useful in the classroom, naturalistic neuroimaging of children engaged in real-world math learning activities has the potential to identify concurrent behavioral and neurological correlates to poor math performance. Such correlates may help pinpoint effective teaching strategies for atypical learners, and may highlight instructional methods that elicit typical neurological response patterns to such activities. For example, multisensory stimulation that contains information about number enhances infants' and preschool children's behavioral performance on many numerical tasks and has been shown to elicit neural activation in areas related to number processing and decision-making. Thus, when applied to math teaching tools, multisensory stimulation may provide a platform through which both behavioral and neural math-related processes may be enhanced. Common approaches to neuroimaging of math processing lack ecological validity and are often not analogous to real-world learning activities. However, because of its liberal tolerance of movement, near-infrared spectroscopy (NIRS) provides an ideal platform for such studies. Here, NIRS is used to provide the first concurrent examination of neurological and behavioral data from number line estimation performance within children and adults. Moreover, in an effort to observe the behavioral and neurological benefits to number line estimations that may arise from multisensory stimulation, differential feedback (i.e., visual, auditory, or audiovisual) about estimation performance is provided throughout a portion of the task. Results suggest behavioral and neural performance is enhanced by feedback. Moreover, significant effects of age suggest young children show greater neurological response to feedback, and increase in task difficulty resulted in decreased behavioral performance and increased neurological activation associated with mathematical processing. Thus, typical math learners effectively recruit areas of the brain known to process number when math activities become increasingly difficult. Data inform understanding typical behavioral and neural responses to real-world math learning tasks, and may prove useful in triangulating signatures of atypical math learning. Moreover, results demonstrate the utility of NIRS as a platform to provide simultaneous neurological and behavioral data during naturalistic math learning activities.