Approximate Numerical Magnitudes Research Articles

The approximate number system and mathematics achievement: it's complicated. A thorough investigation of different ANS measures and executive functions in mathematics achievement in children

ABSTRACT The ability to represent approximate numerical magnitudes is often referred to as the approximate number system (ANS) and has regularly been proposed as foundational to mathematics achievement. However, some argue that the relation between ANS acuity and mathematics achievement ceases to exist when controlling for domain-general cognitive abilities. The current debate in the literature on whether ANS acuity remains a predictor to mathematics after strict control is applied leads to the need to simultaneously investigate domain-specific and domain-general foundational abilities in different ages. 174 Children took part in two computerised ANS tasks, two executive function tasks, a verbal skills task, two intelligence subscales, and a mathematics achievement task (i.e. global, formal, and informal mathematics achievement). Results demonstrated that, when controlling for intelligence and visuospatial memory, the relation between ANS acuity and mathematics achievement ceased to exist, and that ANS acuity might only play a predictive role in early informal mathematics.

Differences in Counting Skills Between Chinese and German Children Are Accompanied by Differences in Processing of Approximate Numerical Magnitude Information.

Human beings are supposed to possess an approximate number system (ANS) dedicated to extracting and representing approximate numerical magnitude information as well as an object tracking system (OTS) for the rapid and accurate enumeration of small sets. It is assumed that the OTS and the ANS independently contribute to the acquisition of more elaborate numerical concepts. Chinese children have been shown to exhibit more elaborate numerical concepts than their non-Chinese peers, but it is still an open question whether similar cross-national differences exist with regard to the underlying systems, namely the ANS and the OTS. In the present study, we investigated this question by comparing Chinese and German preschool children with regard to their performance in a non-symbolic numerical magnitude comparison task (assessing the ANS) and in an enumeration task (assessing the OTS). In addition, we compared children’s counting skills. To ensure that possible between-group differences could not be explained by differences in more general performance factors, we also assessed children’s reasoning ability and processing speed. Chinese children showed a better counting performance and a more accurate performance in the non-symbolic numerical magnitude comparison task. These differences in performance could not be ascribed to differences in reasoning abilities and processing speed. In contrast, Chinese and German children did not differ significantly in the enumeration of small sets. The superior counting performance of Chinese children was thus found to be reflected in the ANS but not in the OTS.

Looking for more food or more people? Task context influences basic numerosity perception

Approximate numerical magnitude (or numerosity) is thought to represent one of the fundamental sensory properties driving perceptual choices. Recent studies indicate that numerosity judgment on a dot array is primarily driven by its numerical magnitude, largely independent from its other non-numerical visual dimensions. Nevertheless, these findings do not preclude the possibility that non-numerical cues such as size or spacing of a dot array influence numerosity judgment. Here, we test the hypothesis that numerosity judgment is influenced by non-numerical dimensions of a dot array depending on the context to which those non-numerical cues could be useful. Participants were asked to choose the more numerous of two dot arrays in two different contexts that differed only in one aspect. In one condition, the task was framed as choosing a set with more fruits to consume. In the other condition, the task was framed as choosing a group with more people to join. The results demonstrate that the influence of non-numerical cues – and particularly of the dimension of size – was significantly smaller when participants made quantitative choices about people than when they made choices about food, illustrating that the representation of discrete magnitude is more pronounced in the former case. These findings suggest that the information pooled to reach a decision about numerosity is flexibly determined according to the context and the goals of such judgment.

Differences in arithmetic performance between Chinese and German adults are accompanied by differences in processing of non-symbolic numerical magnitude.

Human beings are assumed to possess an approximate number system (ANS) dedicated to extracting and representing approximate numerical magnitude information. The ANS is assumed to be fundamental to arithmetic learning and has been shown to be associated with arithmetic performance. It is, however, still a matter of debate whether better arithmetic skills are reflected in the ANS. To address this issue, Chinese and German adults were compared regarding their performance in simple arithmetic tasks and in a non-symbolic numerical magnitude comparison task. Chinese participants showed a better performance in solving simple arithmetic tasks and faster reaction times in the non-symbolic numerical magnitude comparison task without making more errors than their German peers. These differences in performance could not be ascribed to differences in general cognitive abilities. Better arithmetic skills were thus found to be accompanied by a higher speed of retrieving non-symbolic numerical magnitude knowledge but not by a higher precision of non-symbolic numerical magnitude representations. The group difference in the speed of retrieving non-symbolic numerical magnitude knowledge was fully mediated by the performance in arithmetic tasks, suggesting that arithmetic skills shape non-symbolic numerical magnitude processing skills.

The Faculty of Language Integrates the Two Core Systems of Number.

Only humans possess the faculty of language that allows an infinite array of hierarchically structured expressions (Hauser et al., 2002; Berwick and Chomsky, 2015). Similarly, humans have a capacity for infinite natural numbers, while all other species seem to lack such a capacity (Gelman and Gallistel, 1978; Dehaene, 1997). Thus, the origin of this numerical capacity and its relation to language have been of much interdisciplinary interest in developmental and behavioral psychology, cognitive neuroscience, and linguistics (Dehaene, 1997; Hauser et al., 2002; Pica et al., 2004). Hauser et al. (2002) and Chomsky (2008) hypothesize that a recursive generative operation that is central to the computational system of language (called Merge) can give rise to the successor function in a set-theoretic fashion, from which capacities for discretely infinite natural numbers may be derived. However, a careful look at two domains in language, grammatical number and numerals, reveals no trace of the successor function. Following behavioral and neuropsychological evidence that there are two core systems of number cognition innately available, a core system of representation of large, approximate numerical magnitudes and a core system of precise representation of distinct small numbers (Feigenson et al., 2004), I argue that grammatical number reflects the core system of precise representation of distinct small numbers alone. In contrast, numeral systems arise from integrating the pre-existing two core systems of number and the human language faculty. To the extent that my arguments are correct, linguistic representations of number, grammatical number, and numerals do not incorporate anything like the successor function.

Effects of Non-Symbolic Approximate Number Practice on Symbolic Numerical Abilities in Pakistani Children.

Current theories of numerical cognition posit that uniquely human symbolic number abilities connect to an early developing cognitive system for representing approximate numerical magnitudes, the approximate number system (ANS). In support of this proposal, recent laboratory-based training experiments with U.S. children show enhanced performance on symbolic addition after brief practice comparing or adding arrays of dots without counting: tasks that engage the ANS. Here we explore the nature and generality of this effect through two brief training experiments. In Experiment 1, elementary school children in Pakistan practiced either a non-symbolic numerical addition task or a line-length addition task with no numerical content, and then were tested on symbolic addition. After training, children in the numerical training group completed the symbolic addition test faster than children in the line length training group, suggesting a causal role of brief, non-symbolic numerical training on exact, symbolic addition. These findings replicate and extend the core findings of a recent U.S. laboratory-based study to non-Western children tested in a school setting, attesting to the robustness and generalizability of the observed training effects. Experiment 2 tested whether ANS training would also enhance the consistency of performance on a symbolic number line task. Over several analyses of the data there was some evidence that approximate number training enhanced symbolic number line placements relative to control conditions. Together, the findings suggest that engagement of the ANS through brief training procedures enhances children's immediate attention to number and engagement with symbolic number tasks.

Electrophysiological markers of newly acquired symbolic numerical representations: the role of magnitude and ordinal information

It is not yet understood how children acquire the meaning of numerical symbols and most existing research has focused on the role of approximate non-symbolic representations of number in this process (see Piazza, (Trends in Cognitive 14(12):542–551, 2010). However, numerical symbols necessitate an understanding of both order and magnitude, therefore order likely also plays a role in learning about number. Here, we used an artificial symbol-learning paradigm to contrast learning approximate numerical magnitude with learning numerical order. Thirty-two adult participants were randomly assigned to either the magnitude training group, in which they were trained to associate novel abstract symbols with non-symbolic numerical magnitudes, or the order training group, in which they were taught the ordinal sequence of the symbols, in analogy to the count sequence. Subsequently, electroencephalographic (EEG) data were recorded while participants completed a magnitude comparison task with the newly learned symbols. Comparing these newly acquired symbols affected event related potentials (ERPs) in a way that resembled comparisons of real numerical symbols [e.g. Temple & Posner, (Proc Natl Acad Sci USA 95(13):7836–7841, 1998)]. Furthermore, these ERP effects did not differ across learning groups, suggesting that adults formed similar representations regardless of method of instruction. In turn, the current findings highlight the potential role of ordinal information in symbolic acquisition.

Spatial Attention Determines the Nature of Nonverbal Number Representation

Coordinated studies of adults, infants, and nonhuman animals provide evidence for two systems of nonverbal number representation: a "parallel individuation" system that represents individual items and a "numerical magnitude" system that represents the approximate cardinal value of a group. However, there is considerable debate about the nature and functions of these systems, due largely to the fact that some studies show a dissociation between small (1-3) and large (>3) number representation, whereas others do not. Using event-related potentials, we show that it is possible to determine which system will represent the numerical value of a small number set (1-3 items) by manipulating spatial attention. Specifically, when attention can select individual objects, an early brain response (N1) scales with the cardinal value of the display, the signature of parallel individuation. In contrast, when attention cannot select individual objects or is occupied by another task, a later brain response (P2p) scales with ratio, the signature of the approximate numerical magnitude system. These results provide neural evidence that small numbers can be represented as approximate numerical magnitudes. Further, they empirically demonstrate the importance of early attentional processes to number representation by showing that the way in which attention disperses across a scene determines which numerical system will deploy in a given context.

Spatiotemporal dynamics of processing nonsymbolic number: an event-related potential source localization study.

Coordinated studies with adults, infants, and nonhuman animals provide evidence for two distinct systems of nonverbal number representation. The "parallel individuation" (PI) system selects and retains information about one to three individual entities and the "numerical magnitude" system establishes representations of the approximate cardinal value of a group. Recent event-related potential (ERP) work has demonstrated that these systems reliably evoke functionally and temporally distinct patterns of brain response that correspond to established behavioral signatures. However, relatively little is known about the neural generators of these ERP signatures. To address this question, we targeted known ERP signatures of these systems, by contrasting processing of small versus large nonsymbolic numbers, and used a source localization algorithm (LORETA) to identify their cortical origins. Early processing of small numbers, showing the signature effects of PI on the N1 (∼150 ms), was localized primarily to extrastriate visual regions. In contrast, qualitatively and temporally distinct processing of large numbers, showing the signatures of approximate number representation on the mid-latency P2p (∼200-250 ms), was localized primarily to right intraparietal regions. In comparison, mid-latency small number processing was localized to the right temporal-parietal junction and left-lateralized intraparietal regions. These results add spatial information to the emerging ERP literature documenting the process by which we represent number. Furthermore, these results substantiate recent claims that early attentional processes determine whether a collection of objects will be represented through PI or as an approximate numerical magnitude by providing evidence that downstream processing diverges to distinct cortical regions.

Neural signatures of number processing in human infants: evidence for two core systems underlying numerical cognition.

Behavioral research suggests two cognitive systems are at the foundations of numerical thinking: one for representing 1-3 objects in parallel and one for representing and comparing large, approximate numerical magnitudes. We tested for dissociable neural signatures of these systems in preverbal infants, by recording event-related potentials (ERPs) as 6-7.5 month-old infants (n = 32) viewed dot arrays containing either small (1-3) or large (8-32) sets of objects in a number alternation paradigm. If small and large numbers are represented by the same neural system, then the brain response to the arrays should scale with ratio for both number ranges, a behavioral and brain signature of the approximate numerical magnitude system obtained in animals and in human adults. Contrary to this prediction, a mid-latency positivity (P500) over parietal scalp sites was modulated by the ratio between successive large, but not small, numbers. Conversely, an earlier peaking positivity (P400) over occipital-temporal sites was modulated by the absolute cardinal value of small, but not large, numbers. These results provide evidence for two early developing systems of non-verbal numerical cognition: one that responds to small quantities as individual objects and a second that responds to large quantities as approximate numerical values. These brain signatures are functionally similar to those observed in previous studies of non-symbolic number with adults, suggesting that this dissociation may persist over vast differences in experience and formal training in mathematics.

Two Systems of Non-Symbolic Numerical Cognition

Studies of human adults, infants, and non-human animals demonstrate that non-symbolic numerical cognition is supported by at least two distinct cognitive systems: a “parallel individuation system” that encodes the numerical identity of individual items and an “approximate number system” that encodes the approximate numerical magnitude, or numerosity, of a set. The exact nature and role of these systems, however, have been debated for over a 100-years. Some argue that the non-symbolic representation of small numbers (<4) is carried out solely by the parallel individuation system and the non-symbolic representation of large numbers (>4) is carried out solely by the approximate number system. Others argue that all numbers are represented by the approximate number system. This debate has been fueled largely by some studies showing dissociations between small and large number processing and other studies showing similar processing of small and large numbers. Recent work has addressed this debate by showing that the two systems are present and distinct from early infancy, persist despite the acquisition of a symbolic number system, activate distinct cortical networks, and engage differentially based attentional constraints. Based on the recent discoveries, I provide a hypothesis that may explain the puzzling findings and makes testable predictions as to when each system will be engaged. In particular, when items are presented under conditions that allow selection of individuals, they will be represented as distinct mental items through parallel individuation and not as a numerical magnitude. In contrast, when items are presented outside attentional limits (e.g., too many, too close together, under high attentional load), they will be represented as a single mental numerical magnitude and not as distinct mental items. These predictions provide a basis on which researchers can further investigate the role of each system in the development of uniquely human numerical thought.

Children's mappings of large number words to numerosities

Previous studies have suggested that children's learning of the relation between number words and approximate numerosities depends on their verbal counting ability, and that children exhibit no knowledge of mappings between number words and approximate numerical magnitudes for number words outside their productive verbal counting range. In the present study we used a numerical estimation task to explore children's knowledge of these mappings. We classified children as Level 1 counters (those unable to produce a verbal count list up to 35), Level 2 counters (those who were able to count to 35 but not 60) and Level 3 counters (those who counted to 60 or above) and asked children to estimate the number of items on a card. Although the accuracy of children's estimates depended on counting ability, children at all counting skill levels produced estimates that increased linearly in proportion to the target number, for numerosities both within and beyond their counting range. This result was obtained at the group level (Experiment 1) and at the level of individual children (Experiment 2). These findings provide evidence that even the least skilled counters do exhibit some knowledge of the form of the mapping between large number words and approximate numerosities.