<p> This research investigates human perception of brief auditory events presented sequentially. Following an analysis of current, relevant theories, two experiments arc described. The results of the first experiment show that two trains consisting of n and n+l pulses become more difficult to discriminate from each other as (i) the time between the pulses decreases (ii) the number of pulses in the trains increase and (iii) the size of the set of stimulus trains increases. The results of Experiment 2 indicate that discrimination between a pair of "empty" intervals differing only by a constant duration depends on the time between the pulses marking the beginning and end of the intervals and not on the duration of the markers. Discrimination decreases as the durations of the pair of intervals increases. This research investigates human perception of brief auditory events presented sequentially. Following an analysis of current, relevant theories, two experiments arc described. The results of the first experiment show that two trains consisting of n and n+l pulses become more difficult to discriminate from each other as (i) the time between the pulses decreases (ii) the number of pulses in the trains increase and (iii) the size of the set of stimulus trains increases. </p> / Thesis / Doctor of Philosophy (PhD)

The capacity to estimate the number of objects (numerosity) in the environment is ontogenetically precocious and phylogenetically ancient. In animals, this ability holds significant adaptive advantages, directly influencing survival and reproductive success. In humans, it may serve an additional purpose by providing a start-up kit for the acquisition of symbolic numbers, thus making it a potential focus for mathematics education and intervention strategies. Behavioral, neurophysiological, and neuroimaging findings suggest that numerosity information is directly extracted from the environment. However, numerosity is inherently linked with other visual characteristics of sets (such as larger sets often occupy more space or are more densely spaced), making it challenging to determine the extent to which the observed response to numerosity is distinct from the response to other visual attributes. In my PhD research I provide experimental evidence through neuroimaging and computational modeling techniques elucidating where, when, and how numerical information is encoded in the human brain. This work therefore provides a threefold contribution. First, I show that numerosity is represented over and above nonnumeric visual features in a widespread network of areas starting from early visual areas and further amplified in associative areas along the dorsal but also notably the ventral stream, and that the neural representational geometries of regions across the two steams are substantially identical. Second, I showed that numerosity is represented at an early stage and seemingly in parallel across of a set of regions including early visual, parietal, and temporal, preceding the emergence of non-numeric features that could indirectly contribute to numerosity computation. Finally, by comparing the fMRI data with a convolutional neural network (CNN) to explore similarities and differences between the model and human brain data, I discovered that although the CNN can perform approximate numerosity comparisons and the structure of their representation in their hidden layers captures well numerosity representation in early visual areas of humans, it falls short of fully simulating the way in which associative brain regions represent numerosity. Taken together, the findings of this thesis provide experimental evidence supporting the notion that number is a primary visual feature, encoded independent from other visual features quickly and widely across the human brain. Furthermore, they emphasize the need for additional investigation to unravel the computational mechanisms underlying numerosity in the human brain.

Children show large individual differences in numerical skills, even before they begin formal education. These early differences have significant and long-lasting effects, with numerical knowledge before school predicting mathematical achievement throughout the primary and secondary school years. Currently, little is known about the dispositional factors influencing children's numerical development. Why do some children engage with and succeed in mathematics from an early age, whilst others avoid mathematics and struggle to acquire even basic symbolic number skills? This thesis examines the role of two dispositional factors: First, spontaneous focusing on numerosity (SFON), a recently developed construct which refers to an individual's tendency to focus on the numerical aspects of their environment; and second, mathematics anxiety (MA), a phenomenon long recognised by educators and researchers but one which is relatively unexplored in young children. These factors are found to have independent effects on children's numerical skills, thus the empirical work is presented in two separate parts. The SFON studies start by addressing methodological issues. It is shown that the current measures used to assess children's SFON vary in their psychometric properties and subsequently a new and reliable picture-based task is introduced. Next, the studies turn to theoretical questions, investigating the causes, consequences and mechanisms of SFON. The findings give rise to three main conclusions. First, children's SFON shows little influence from parental SFON and home numeracy factors. Second, high SFON children show a symbolic number advantage. Third, the relationship between SFON and arithmetic can be explained, in part, by individual differences in children's ability to map between nonsymbolic and symbolic representations of number. The MA studies focus primarily on gender issues. The results reveal no significant differences between boys' and girls' overall levels of MA; however, there are gender differences in the correlates of MA. Specifically, boys' (but not girls') MA is related to parents' MA. Moreover, the relationship between MA and mathematical outcomes is stronger for boys than it is for girls. Possible causal explanations for these gender differences are explored in two ways: First, by examining the reliability of the scales used to assess MA in boys and girls. Second, by investigating the relationship between girls' (and boys') mathematics anxiety and their societal math-gender stereotypes. The findings from both sets of studies draw a link between children's emerging dispositions towards mathematics and their early numerical skills. Future research needs to examine how these dispositional factors interact with other (cognitive and non-cognitive) predictors of mathematics achievement.

<p>There is a long-standing claim that humans and nonhuman primates share an evolutionarily ancient system of nonverbal number representation. By and large, the focus in the field has been on providing existence proofs of numerical competence in wide-ranging taxa or using individual species as models for comparisons with humans. Recent findings in numerical cognition have suggested that evidence for approximate numerical abilities in nonhuman species may indicate that humans and animals share a cognitive system for representing numerosities nonverbally. To date, little is known about the contextual and quantitative limits of that system, or how those limits differ between species. The studies presented here take a comparative, behavioral approach to characterizing species differences and similarities in the approximate number system, and the contexts that affect that system. Collectively, this set of studies provides evidence that the approximate number system evolved in primates as a malleable system in which numerical representations are accessed spontaneously and improved through training. Despite the sensitivity of the system to experience and context individual differences in sensitivity are greater than species differences suggesting that the selective pressures that constrained its evolution were early and general and that species variation in social group size and diet have less influence on the ANS. Finally my studies indicate that the ANS supports approximate arithmetic and is consistent with the idea that ANS representations evolved to allow animals to calculate the world around them.</p> / Dissertation

The ability to discriminate relative and absolute number has been researched widely in both human and nonhuman species. However, the full extent of numerical ability in nonhuman animals, and the nature of the underlying numerical representation, on which discriminations are based, is still unclear. The aim of the current research was to examine the performance of pigeons and humans in tasks that require the discrimination of relative number (a bisection procedure), and absolute number (in a reproduction procedure). One of the main research questions was whether numerical control over responding could be obtained, above and beyond control by temporal cues in nonhuman animals, and if so, whether it was possible to quantify the relative influences of number and time on responding. Experiment 1 examines nonhuman performance in a numerical bisection task; subjects were presented with either 2 and 6, 4 and 12, or 8 and 24 keylight flashes across three different conditions, and were required to classify these flash sequences as either a “large” or “small” number, by pecking the blue or white key, respectively. Subjects were then tested with novel values within and 2 values higher and lower than the training values. Experiments 2-4 investigate responding in a novel numerical reproduction procedure, in which pigeons were trained to match the number of responses made during a production phase to the number of keylight flashes (2, 4, or 6) in a recently completed sample phase. Experiments 2 and 2A examined discrimination performance when the temporal variables, flash rate and sample phase duration, were perfectly correlated (Experiment 2) or only weakly correlated (Experiment 2a) with flash number. Acquisition of performance in the numerical reproduction procedure was investigated in Experiment 3. For Experiments 1-3, hierarchical regression analyses showed significant control by number over responding, after controlling for temporal cues. Additionally, positive transfer to novel values both within and outside the training range was obtained when the temporal organization of test sequences was similar to baseline training. Experiment 4 investigated the effects of increasing or decreasing the retention interval (RI) on performance in the reproduction procedure, and found this produced a response bias towards larger numbers, contrary to predictions based on previous RI research, and suggested responding was not affected by memorial decay processes. The structure of the representation of number developed by subjects in the bisection and reproduction procedures was investigated using analyses of responding and response variability in Chapters 2 and 6, respectively. Bisection points obtained in Experiment 1 were located at the arithmetic, not geometric mean of all three scales, and coefficients of variation (CVs) obtained in both the bisection and reproduction experiments tended to decrease as flash number increased. Additionally, analyses of the acquisition data found differences in average response number was better fit by a linear than logarithmic scale. These results show that responding did not conform to scalar variability and is largely inconsistent with previous nonhuman research. Together these results suggest responding appeared to be based on a linear scale of number with constant generalisation between values, similar to that associated with human verbal counting, rather than a logarithmic scale with constant generalisation or a linear scale with scalar generalisation between values. Experiment 5 compared pigeons’ and humans’ verbal and nonverbal discrimination performance with numbers 1-20 in analogous bisection, reproduction and report tasks. Human verbal and nonverbal performance in the three tasks was similar and resembled nonhuman performance, although verbal discriminations were more accurate and less variable. The main findings from Experiments 1 and 2A were replicated with humans; bisection points were located at the arithmetic mean, average response number increased linearly as sample number increased, though there was a tendency to underestimate sample number, and decreasing CVs were also obtained for values less than 8. An additional, interesting finding was that CVs showed scalar variability for values greater than 8, suggesting a less exact representation and discrimination process was being used for these values. Collectively, these five experiments provide new evidence for a nonverbal ability to discriminate relative and absolute number with increasing relative accuracy resembling human verbal counting in both human and nonhumans.

Subitizing, a quick apprehension of the numerosity of a small set of items, is inconsistently utilized by preschool educators to support early number understandings (Sarama & Clements, 2009). The purpose of this qualitative study is to investigate the relationship between children’s number understanding and subitizing activity. Sarama and Clements (2009) consider students’ subitizing activity as shifting from reliance upon perceptual processes to conceptual processes. Hypothesized mental actions carried into subitizing activity by children have not yet been empirically investigated (Sarama & Clements, 2009). Drawing upon Piaget’s (1968/1970) three mother structures of mathematical thinking, the theoretical implications of this study consider expanding the scope of Piaget’s (1968/1970) definition of topological thinking structures to include patterned orientations. Increasing the scope of this definition would allow for the investigation of the development of topological thinking structures and subitizing activity. An 11-week teaching experiment was conducted with six preschool aged children in order to analyze student engagement with subitizing tasks (Steffe & Ulrich, in press). To infer what perceptual and conceptual processes students relied upon when subitizing, tasks were designed to either assess or provoke cognitive changes. Analysis of interactions between students and the teacher-researcher informed this teacher-researcher of cognitive changes relative to each student’s thinking structure. Results indicated that students rely upon the space between items, symmetrical aspects of items, and color of items when perceptually subitizing. Seven different types of subitizing activity were documented and used to more explicitly describe student reliance upon perceptual or conceptual processes. Conceptual subitizing activity was redefined in this study, as depending upon mental reversibility and sophisticated number schemes. Students capable of conceptual subitizing were also able to conserve number. Students capable of conserving number were not always capable of conceptual subitizing. The symmetrical aspects of an item’s arrangement elicited students’ attention towards subgroups and transitioning students’ perceptual subitizing to conceptual subitizing. Combinations of counting and subitizing activity explained students’ reliance upon serial and classification thinking structures when transitioning from perceptual subitizing to conceptual subitizing. Implications of this study suggest effectively designed subitizing activity can both assess students’ number understandings, and appropriately differentiate preschool curriculum. / Ph. D.

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.

This work is focused on numerical competence in primates specifically focusing on relative numerosity, one of the many aspects of these cognitive abilities. Relative numerosity is an ability to discriminate a larger quantity from a smaller amount or smaller quantity from a larger amount and could be classified as one of the easiest numerical competence. In this work I have summarized the present knowledge of numerical competences in primates, which have been so far studied only in a few species of primates, most of them was rhesus macaque (Macaca mulatta) and chimpanzee (Pan troglodytes). In the experimental part I focused on the research of relative numerosity in rhesus macaque (Macaca mulatta). The goal of my work was to find out whether macaques are able to solve the problem of relative abundance with different types of stimuli. Another goal was to find out whether they are able to generalize information about relative abundance and whether they can apply it for new design of the task. The results confirm that macaques possess the ability of relative numerosity and are able to abstract stimuli that are differing in their character. This proves that they are not learning to recognize a particular stimul, but are able to use this numerical skill on any type of stimul. I also found out that...

Cognition allows animals to acquire, process, and store sensory information from the environment and use it to adapt to their surroundings. A battery of behavioral tests was used to assess the cognitive abilities of black-handed spider monkeys (Ateles geoffroyi). Black and white cups were used to assess (1) object permanence by showing the animals under which cup the reward was placed, (2) associative learning by concealing where the reward was placed, and (3) long-term memory by repeating the second task after a 4-month break; petri dishes with varying amounts of food were used to assess (4) relative quantity discrimination; and boxes fitted with dotted cards were used to assess discrete number discrimination with (5) equallysized dots and (6) various-sized dots. For each task, one session comprised 10 trials (i.e. responses). All nine animals succeeded in all tests and, as a group, reached the learning criterion of 70% correct responses on session two in the object permanence and associative learning tasks; on session eleven in the quantity discrimination task; on session sixteen in the numerosity task with equally-sized dots; on session three in the numerosity task with various-sized dots; and averaged 84.4% correct responses in the long-term memory task. Their prompt high score in the numerosity task with various-sized dots suggests that the animals acknowledged the task for its numerical properties as opposed to the size or pattern of the dots. These cognitive abilities are thought to shape the necessary behaviors for the ecological and social needs of the species.

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.