Approximate number sense,symbolic number processing,or number–space mappings: What underlies mathematics achievement?

In this study, the performance of typically developing 6- to 8-year-old children on an approximate number discrimination task, a symbolic comparison task, and a symbolic and nonsymbolic number line estimation task was examined. For the first time, children’s performances on these basic cognitive number processing tasks were explicitly contrasted to investigate which of them is the best predictor of their future mathematical abilities. Math achievement was measured with a timed arithmetic test and with a general curriculum-based math test to address the additional question of whether the predictive association between the basic numerical abilities and mathematics achievement is dependent on which math test is used. Results revealed that performance on both mathematics achievement tests was best predicted by how well children compared digits. In addition, an association between performance on the symbolic number line estimation task and math achievement scores for the general curriculum-based math test measuring a broader spectrum of skills was found. Together, these results emphasize the importance of learning experiences with symbols for later math abilities.     

Is Dunbar's number up?

This article is a commentary on ‘Relationships and the social brain: Integrating psychological and evolutionary perspectives’ ( Sutcliffe, Dunbar, Binder, & Arrow, 2012 ).     

The number–weight illusion

Psychonomic Bulletin & Review - When objects are manually lifted to compare their weight, then smaller objects are judged to be heavier than larger objects of the same physical weights: the...     

Beyond the mental number line: A neural network model of number–space interactions

It is commonly assumed that there is an interaction between the representations of number and space (e.g., Dehaene et al., 1993, Walsh, 2003), typically ascribed to a mental number line. The exact nature of this interaction has remained elusive, however. Here we propose that spatial aspects are not inherent to number representations, but that instead spatial and numerical representations are separate. However, cultural factors establish ties between them. By extending earlier models (Gevers et al., 2006, Verguts et al., 2005) based on this hypothesis, the authors present computer simulations showing that a model incorporating this idea can account for data from a series of studies. These results suggest that number–space interactions are emergent properties resulting from the interaction between different brain areas.     

Incongruence in number–luminance congruency effects

Congruency tasks have provided support for an amodal magnitude system for magnitudes that have a "spatial" character, but conflicting results have been obtained for magnitudes that do not (e.g., luminance). In this study, we extricated the factors that underlie these number-luminance congruency effects and tested alternative explanations: (unsigned) luminance contrast and saliency. When luminance had to be compared under specific task conditions, we revealed, for the first time, a true influence of number on luminance judgments: Darker stimuli were consistently associated with numerically larger stimuli. However, when number had to be compared, luminance contrast, not luminance, influenced number judgments. Apparently, associations exist between number and luminance, as well as luminance contrast, of which the latter is probably stronger. Therefore, similar tasks, comprising exactly the same stimuli, can lead to distinct interference effects.     

The auditory redundant signals effect: An influence of number of stimuli or number of percepts?

In two experiments, we examined simple reaction times (RTs) for detection of the onsets and offsets of auditory stimuli. Both experiments assessed the redundant signals effect (RSE), which is traditionally defined as the reduction in RT associated with the presentation of two redundant stimuli, rather than a single stimulus. In Experiment 1, with two identical tones presented via headphones to the left ear, right ear, or both, no RSE was found in responding to tone onsets, but a large RSE was found in responding to their offsets. In Experiment 2, with a pure tone and white noise as the two stimulus alternatives, RSEs were found for responding to both onsets and offsets. The results support the notion that the occurrence of an RSE depends on the number of percepts, rather than the number of stimuli, and on the requirement to respond to stimulus onsets versus offsets. The parallel grains model (Miller & Ulrich, 2003) provides one possible account of this pattern of results.     

Do children's number words begin noisy?

How do children acquire exact meanings for number words like three or forty‐seven? In recent years, a lively debate has probed the cognitive systems that support learning, with some arguing that an evolutionarily ancient “approximate number system” drives early number word meanings, and others arguing that learning is supported chiefly by representations of small sets of discrete individuals. This debate has centered around the findings generated by Wynn's ( 1990 , 1992 ) Give‐a‐Number task, which she used to categorize children into discrete “knower level” stages. Early reports confirmed Wynn's analysis, and took these stages to support the “small sets” hypothesis. However, more recent studies have disputed this analysis, and have argued that Give‐a‐Number data reveal a strong role for approximate number representations. In the present study, we use previously collected Give‐a‐Number data to replicate the analyses of these past studies, and to show that differences between past studies are due to assumptions made in analyses, rather than to differences in data themselves. We also show how Give‐a‐Number data violate the assumptions of parametric tests used in past studies. Based on simple non‐parametric tests and model simulations, we conclude that (a) before children learn exact meanings for words like one, two, three, and four, they first acquire noisy preliminary meanings for these words, (b) there is no reliable evidence of preliminary meanings for larger meanings, and (c) Give‐a‐Number cannot be used to readily identify signatures of the approximate number system.     

Who can escape the natural number bias in rational number tasks? A study involving students and experts

Many learners have difficulties with rational number tasks because they persistently rely on their natural number knowledge, which is not always applicable. Studies show that such a natural number bias can mislead not only children but also educated adults. It is still unclear whether and under what conditions mathematical expertise enables people to be completely unaffected by such a bias on tasks in which people with less expertise are clearly biased. We compared the performance of eighth‐grade students and expert mathematicians on the same set of algebraic expression problems that addressed the effect of arithmetic operations (multiplication and division). Using accuracy and response time measures, we found clear evidence for a natural number bias in students but no traces of a bias in experts. The data suggested that whereas students based their answers on their intuitions about natural numbers, expert mathematicians relied on their skilled intuitions about algebraic expressions. We conclude that it is possible for experts to be unaffected by the natural number bias on rational number tasks when they use strategies that do not involve natural numbers.     

Does action make the link between number and space representation? Visuo-manual adaptation improves number bisection in unilateral neglect

If the visual world is artificially shifted by only 10 degrees, people initially experience difficulty in directing their actions toward visual goals, but then rapidly compensate the visual distortion. The consequence of such adaptation can be measured as visual and proprioceptive aftereffects, as well as by performance on pointing tasks without visual feedback. Recent work has shown that more cognitive deficits can be improved following prism adaptation in patients with unilateral neglect. Here we show that a short visuo-manual adaptation to prisms improves performance on a mental number-bisection task recently shown to be impaired in unilateral neglect. The association previously found between space and number representation (the mental number line) may thus be grounded in common action principles. Our results suggest that visuo-motor plasticity functionally links parietal areas involved in space and number representation.     

How are number words mapped to approximate magnitudes?

How do we map number words to the magnitudes they represent? While much is known about the developmental trajectory of number word learning, the acquisition of the counting routine, and the academic correlates of estimation ability, previous studies have yet to describe the mechanisms that link number words to nonverbal representations of number. We investigated two mechanisms: associative mapping and structure mapping. Four dot array estimation tasks found that adults' ability to match a number word to one of two discriminably different sets declined as a function of set size and that participants' estimates of relatively large, but not small, set sizes were influenced by misleading feedback during an estimation task. We propose that subjects employ structure mappings for linking relatively large number words to set sizes, but rely chiefly on item-by-item associative mappings for smaller sets. These results indicate that both inference and association play important roles in mapping number words to approximate magnitudes.     

Choosing the number of clusters in Κ-means clustering

Steinley (2007) provided a lower bound for the sum-of-squares error criterion function used in K-means clustering. In this article, on the basis of the lower bound, the authors propose a method to distinguish between 1 cluster (i.e., a single distribution) versus more than 1 cluster. Additionally, conditional on indicating there are multiple clusters, the procedure is extended to determine the number of clusters. Through a series of simulations, the proposed methodology is shown to outperform several other commonly used procedures for determining both the presence of clusters and their number.     

What is magic about the magical number four?

Summary Previous studies have shown that the apprehension of number can be represented by three models according to the experimental procedure and the data analysis. The present experiment was designed to test the effect of figural characteristics of pattern on the response time. The subjects were asked to perform a same/different judgment, i.e., they were requested to decide whether a dot pattern, shown on a monitor, equalled a previously defined target number (n=2–6) or not. Different types of pattern were used and learning effects were studied. As was expected, the slopes for random and linear patterns were not so steep when the target number was low. With patterns in the dice mode, however, the slope was zero. Repeated presentations led to a slight reduction in slope for random and linear patterns only. In the case of the patterns in the dice mode, the repeated presentations caused only a change in the absolute reaction times (RTs) but had no effect on the slope. When the target numbers were larger (n=5–6), the repeated presentations led to remarkable reductions in slope for random and linear patterns. The slope discontinuity at n=4 occurred with all types of pattern but it became less pronounced in the course of training at least in the case of random and linear patterns. This result is explained by clustering effects, use of figural cues, and a more efficient scanning process.     

Does number data entry rely on the phonological loop?

Two experiments investigated effects of articulatory processing on number data entry. Participants entered four‐digit numbers presented as either words or numerals on a keyboard, either under an articulatory condition or in silence. In Experiment 1, the articulatory condition was articulatory suppression; in Experiment 2, it was vocalisation. In Experiment 1, the articulatory suppression group typed initial digits faster than the silent group, but for subsequent digits, the opposite pattern occurred at least with word stimuli. In Experiment 2, the silent group typed initial digits faster but typed subsequent digits somewhat slower than the vocalisation group. Thus, articulation of numbers, which promotes entry into the phonological loop of working memory, retards processing of initial digits but enhances processing of subsequent digits.     

Short-term memory capacity: magic number or magic spell?

Previous experiments have found that memory span is greater for items that can be pronounced more quickly. For a variety of materials the span equals the number of items that can be pronounced in about 1.5 s, presumably the duration of the verbal trace. This suggests a model for immediate recall: The probability of correctly recalling a list equals the probability that the time to recite the list is less than the variable duration of the trace. Recall probability for lists of various lengths was determined for six materials. Later, subjects read the lists aloud. The standard normal deviates corresponding to probability of correct recall were linear in pronunciation time. Evidently, over subjects, a normal distribution is a reasonable approximation of the distribution of the trace duration. The mean and variance of the trace duration were estimated. The mean (1.88 s) agrees well with previous estimates, and the model accounts for 95% of the variance in immediate recall.     

Does number data entry rely on the phonological loop?

Two experiments investigated effects of articulatory processing on number data entry. Participants entered four-digit numbers presented as either words or numerals on a keyboard, either under an articulatory condition or in silence. In Experiment 1, the articulatory condition was articulatory suppression; in Experiment 2, it was vocalisation. In Experiment 1, the articulatory suppression group typed initial digits faster than the silent group, but for subsequent digits, the opposite pattern occurred at least with word stimuli. In Experiment 2, the silent group typed initial digits faster but typed subsequent digits somewhat slower than the vocalisation group. Thus, articulation of numbers, which promotes entry into the phonological loop of working memory, retards processing of initial digits but enhances processing of subsequent digits.     

Why is number word learning hard? Evidence from bilingual learners

Young children typically take between 18 months and 2 years to learn the meanings of number words. In the present study, we investigated this developmental trajectory in bilingual preschoolers to examine the relative contributions of two factors in number word learning: (1) the construction of numerical concepts, and (2) the mapping of language specific words onto these concepts. We found that children learn the meanings of small number words (i.e., one, two, and three) independently in each language, indicating that observed delays in learning these words are attributable to difficulties in mapping words to concepts. In contrast, children generally learned to accurately count larger sets (i.e., five or greater) simultaneously in their two languages, suggesting that the difficulty in learning to count is not tied to a specific language. We also replicated previous studies that found that children learn the counting procedure before they learn its logic – i.e., that for any natural number, n, the successor of n in the count list denotes the cardinality n + 1. Consistent with past studies, we found that children’s knowledge of successors is first acquired incrementally. In bilinguals, we found that this knowledge exhibits item-specific transfer between languages, suggesting that the logic of the positive integers may not be stored in a language-specific format. We conclude that delays in learning the meanings of small number words are mainly due to language-specific processes of mapping words to concepts, whereas the logic and procedures of counting appear to be learned in a format that is independent of a particular language and thus transfers rapidly from one language to the other in development.     

What is the relationship between synaesthesia and visuo-spatial number forms?

This study compares the tendency for numerals to elicit spontaneous perceptions of colour or taste (synaesthesia) with the tendency to visualise numbers as occupying particular visuo-spatial configurations (number forms). The prevalence of number forms was found to be significantly higher in synaesthetes experiencing colour compared both to synaesthetes experiencing taste and to control participants lacking any synaesthetic experience. This suggests that the presence of synaesthetic colour sensations enhances the tendency to explicitly represent numbers in a visuo-spatial format although the two symptoms may nevertheless be logically independent (i.e. it is possible to have number forms without colour, and coloured numbers without forms). Number forms are equally common in men and women, unlike previous reports of synaesthesia that have suggested a strong female bias. Individuals who possess a number form are also likely to possess visuo-spatial forms for other ordinal sequences (e.g. days, months, letters) which suggests that it is the ordinal nature of numbers rather than numerical quantity that gives rise to this particular mode of representation. Finally, we also describe some consequences of number forms for performance in a number comparison task.     

How do we convert a number into a finger trajectory?

How do we understand two-digit numbers such as 42? Models of multi-digit number comprehension differ widely. Some postulate that the decades and units digits are processed separately and possibly serially. Others hypothesize a holistic process which maps the entire 2-digit string onto a magnitude, represented as a position on a number line. In educated adults, the number line is thought to be linear, but the “number sense” hypothesis proposes that a logarithmic scale underlies our intuitions of number size, and that this compressive representation may still be dormant in the adult brain. We investigated these issues by asking adults to point to the location of two-digit numbers on a number line while their finger location was continuously monitored. Finger trajectories revealed a linear scale, yet with a transient logarithmic effect suggesting the activation of a compressive and holistic quantity representation. Units and decades digits were processed in parallel, without any difference in left-to-right vs. right-to-left readers. The late part of the trajectory was influenced by spatial reference points placed at the left end, middle, and right end of the line. Altogether, finger trajectory analysis provides a precise cognitive decomposition of the sequence of stages used in converting a number to a quantity and then a position.     

Reconceptualizing the origins of number knowledge: A “non-numerical” account

This paper presents a new conceptualization of the origins of numerical competence in humans. I first examine the existing claim that infants are innately provided with a system of specifically numerical knowledge, consisting of both cardinal and ordinal concepts. I suggest instead that the observed behaviors require only simple perceptual discriminations based on domain-independent competencies. At most, these involve the formal equivalent of cardinal information. Finally, I present a “non-numerical” account that characterizes infants competencies with regard to numerosity as emerging primarily from some general characteristics of the human perception and attention system.     

Is the number of trials a primary determinant of conditioned responding?

Acquisition of conditioned responding is thought to be determined by the number of pairings of a conditioned stimulus (CS) and an unconditioned stimulus (US). However, it is possible that acquisition is primarily determined not by the number of trials but rather by quantities that often correlate with the number of trials, such as cumulative intertrial interval (ITI) and the number of sessions. Four experiments examined whether the number of trials has an effect on acquisition of conditioned responding, once cumulative ITI and number of sessions are equated. Results of the experiments with rats and mice favor the hypothesis that over an eightfold range, variation in number of CS-US pairings has little effect. It is suggested that learning curves might more accurately be plotted across cumulative ITI or number of sessions, and not across number of trials. Results pose a challenge to trial-centered accounts of conditioning, as demonstrated by simulations of the Rescorla-Wagner model, a simplified version of Wagner's standard operating procedure model (SOP), and Stout & Miller's sometimes competing retrieval model (SOCR). A time-centered account, rate estimation theory (RET), predicts the main finding but has trouble with other aspects of the learning process more easily accommodated by trial-centered models.