One, two, three, four, nothing more: an investigation of the conceptual sources of the verbal counting principles

Since the publication of [Gelman, R., & Gallistel, C. R. (1978). The child's understanding of number. Cambridge, MA: Harvard University Press.] seminal work on the development of verbal counting as a representation of number, the nature of the ontogenetic sources of the verbal counting principles has been intensely debated. The present experiments explore proposals according to which the verbal counting principles are acquired by mapping numerals in the count list onto systems of numerical representation for which there is evidence in infancy, namely, analog magnitudes, parallel individuation, and set-based quantification. By asking 3- and 4-year-olds to estimate the number of elements in sets without counting, we investigate whether the numerals that are assigned cardinal meaning as part of the acquisition process display the signatures of what we call "enriched parallel individuation" (which combines properties of parallel individuation and of set-based quantification) or analog magnitudes. Two experiments demonstrate that while "one" to "four" are mapped onto core representations of small sets prior to the acquisition of the counting principles, numerals beyond "four" are only mapped onto analog magnitudes about six months after the acquisition of the counting principles. Moreover, we show that children's numerical estimates of sets from 1 to 4 elements fail to show the signature of numeral use based on analog magnitudes - namely, scalar variability. We conclude that, while representations of small sets provided by parallel individuation, enriched by the resources of set-based quantification are recruited in the acquisition process to provide the first numerical meanings for "one" to "four", analog magnitudes play no role in this process.     

Priming reveals differential coding of symbolic and non-symbolic quantities

Number processing is characterized by the distance and the size effect, but symbolic numbers exhibit smaller effects than non-symbolic numerosities. The difference between symbolic and non-symbolic processing can either be explained by a different kind of underlying representation or by parametric differences within the same type of underlying representation. We performed a primed naming study to investigate this issue. Prime and target format were manipulated (digits or collections of dots) as well as the numerical distance between prime and target value. Qualitatively different priming patterns were observed for the two formats, showing that the underlying representations differed in kind: Digits activated mental number representations of the place coding type, while collections of dots activated number representations of the summation coding type.     

Memory for sequences of tone bursts in the rat: Reliance on temporal cues and evidence for an instructional ambiguity explanation of the choose-few effect

Two groups of rats were trained in a symbolic delayed matching-to-sample task with a 0-s delay to discriminate sample stimuli that consisted of sequences of tone bursts. For one group, sequences varied in number with total sequence duration controlled. For the other group, total sequence duration, sum of the tone durations, and sum of the gap durations were all controlled. The former group of rats acquired the discrimination, but the latter group of rats were not able to learn the discrimination with the additional temporal cues controlled. Retention functions for the group that acquired the discrimination exhibited a choose-many bias at the 1-s delay and a choose-few bias at the 8-s delay regardless of the similarity in the intertrial interval and delay interval illumination condition. The asymmetrical retention functions appeared to be due to instructional ambiguity which resulted from the similarity of the delay interval to temporal features of the tone burst sequence, such as the gap between tone bursts and the total duration of tone. In Experiment 2, diagnostic tests provided evidence that rats discriminated the sequences of tone bursts by timing and summing the duration of individual tone bursts rather than using an event switch to count tones. Like pigeons, rats use multiple temporal features rather than number to discriminate sequences of successive events.     

Mathematical skill in individuals with Williams syndrome: evidence from a standardized mathematics battery

Williams syndrome (WS) is a developmental disorder associated with relatively spared verbal skills and severe visuospatial deficits. It has also been reported that individuals with WS are impaired at mathematics. We examined mathematical skills in persons with WS using the second edition of the Test of Early Mathematical Ability (TEMA-2), which measures a wide range of skills. We administered the TEMA-2 to 14 individuals with WS and 14 children matched individually for mental-age on the matrices subtest of the Kaufman Brief Intelligence Test. There were no differences between groups on the overall scores on the TEMA-2. However, an item-by-item analysis revealed group differences. Participants with WS performed more poorly than controls when reporting which of two numbers was closest to a target number, a task thought to utilize a mental number line subserved by the parietal lobe, consistent with previous evidence showing parietal abnormalities in people with WS. In contrast, people with WS performed better than the control group at reading numbers, suggesting that verbal math skills may be comparatively strong in WS. These findings add to evidence that components of mathematical knowledge may be differentially damaged in developmental disorders.     

Language-relative construal of individuation constrained by universal ontology: revisiting language universals and linguistic relativity

Objects and substances bear fundamentally different ontologies. In this article, we examine the relations between language, the ontological distinction with respect to individuation, and the world. Specifically, in cross‐linguistic developmental studies that follow Imai and Gentner (1997) , we examine the question of whether language influences our thought in different forms, like (1) whether the language‐specific construal of entities found in a word extension context ( Imai & Gentner, 1997 ) is also found in a nonlinguistic classification context; (2) whether the presence of labels per se, independent of the count‐mass syntax, fosters ontology‐based classification; (3) in what way, if at all, the count‐mass syntax that accompanies a label changes English speakers' default construal of a given entity? On the basis of the results, we argue that the ontological distinction concerning individuation is universally shared and functions as a constraint on early learning of words. At the same time, language influences one's construal of entities cross‐lingistically and developmentally, and causes a temporary change of construal within a single language. We provide a detailed discussion of how each of these three ways language may affect the construal of entities, and discuss how our universally possessed knowledge interacts with language both within a single language and in cross‐linguistic context.     

What kinds of alternative possibilities are required of the folk concept(s) of choice?

Our concept of choice is integral to the way we understand others and ourselves, especially when considering ourselves as free and responsible agents. Despite the importance of this concept, there has been little empirical work on it. In this paper we report four experiments that provide evidence for two concepts of choice—namely, a concept of choice that is operative in the phrase having a choice and another that is operative in the phrase making a choice. The experiments indicate that the two concepts of choice can be differentiated from each other on the basis of the kind of alternatives to which each is sensitive. The results indicate that the folk concept of choice is more nuanced than has been assumed. This new, empirically informed understanding of the folk concept of choice has important implications for debates concerning free will, responsibility, and other debates spanning psychology and philosophy.     

Conceptual Hierarchies in a Flat Attractor Network: Dynamics of Learning and Computations

The structure of people's conceptual knowledge of concrete nouns has traditionally been viewed as hierarchical ( Collins & Quillian, 1969 ). For example, superordinate concepts ( vegetable ) are assumed to reside at a higher level than basic-level concepts ( carrot ). A feature-based attractor network with a single layer of semantic features developed representations of both basic-level and superordinate concepts. No hierarchical structure was built into the network. In Experiment and Simulation 1, the graded structure of categories (typicality ratings) is accounted for by the flat attractor network. Experiment and Simulation 2 show that, as with basic-level concepts, such a network predicts feature verification latencies for superordinate concepts ( vegetable ). In Experiment and Simulation 3, counterintuitive results regarding the temporal dynamics of similarity in semantic priming are explained by the model. By treating both types of concepts the same in terms of representation, learning, and computations, the model provides new insights into semantic memory.     

The association between children's numerical magnitude processing and mental multi-digit subtraction

Children apply various strategies to mentally solve multi-digit subtraction problems and the efficient use of some of them may depend more or less on numerical magnitude processing. For example, the indirect addition strategy (solving 72–67 as “how much do I have to add up to 67 to get 72?”), which is particularly efficient when the two given numbers are close to each other, requires to determine the proximity of these two numbers, a process that may depend on numerical magnitude processing. In the present study, children completed a numerical magnitude comparison task and a number line estimation task, both in a symbolic and nonsymbolic format, to measure their numerical magnitude processing. We administered a multi-digit subtraction task, in which half of the items were specifically designed to elicit indirect addition. Partial correlational analyses, controlling for intellectual ability and motor speed, revealed significant associations between numerical magnitude processing and mental multi-digit subtraction. Additional analyses indicated that numerical magnitude processing was particularly important for those items for which the use of indirect addition is expected to be most efficient. Although this association was observed for both symbolic and nonsymbolic tasks, the strongest associations were found for the symbolic format, and they seemed to be more prominent on numerical magnitude comparison than on number line estimation.     

Processes of religious individualisation: stocktaking and issues for the future

Abstract

The paper spells out implications of a perspective on religious individualisation that is both comparative and analytical and discusses the various issues that this line of research has to confront. Arguing against notions of modern Western exceptionalism, it points to the selectivity of Western forms of (religious) individualisation and makes a case for the inclusion of additional forms of articulating individual agency and ‘selfhood'. The paper takes individualisation as an inter-subjective, social phenomenon and connects it to the concept of social imaginaries. Opposing standard concepts of modernisation the paper nevertheless brings processuality centre stage, emphasising contingency. The paper discusses the possibility of the convergence of contextual developmental trends and the question of transculturality of notions of self and agency. With reference to the dimension of critique included in visions of religious individualisation, the paper finally points to the effects processes of religious individualisation can have on human actors within, and beyond, the realm of spirituality, including the recognition of subjectivities and the strengthening of actors’ resilience.     

Patterns of Response Times and Response Choices to Science Questions: The Influence of Relative Processing Time

We report on five experiments investigating response choices and response times to simple science questions that evoke student “misconceptions,” and we construct a simple model to explain the patterns of response choices. Physics students were asked to compare a physical quantity represented by the slope, such as speed, on simple physics graphs. We found that response times of incorrect answers, resulting from comparing heights, were faster than response times of correct answers comparing slopes. This result alone might be explained by the fact that height was typically processed faster than slope for this kind of task, which we confirmed in a separate experiment. However, we hypothesize that the difference in response time is an indicator of the cause (rather than the result) of the response choice. To support this, we found that imposing a 3‐s delay in responding increased the number of students comparing slopes (answering correctly) on the task. Additionally a significant proportion of students recognized the correct written rule (compare slope), but on the graph task they incorrectly compared heights. Finally, training either with repetitive examples or providing a general rule both improved scores, but only repetitive examples had a large effect on response times, thus providing evidence of dual paths or processes to a solution. Considering models of heuristics, information accumulation models, and models relevant to the Stroop effect, we construct a simple relative processing time model that could be viewed as a kind of fluency heuristic. The results suggest that misconception‐like patterns of answers to some science questions commonly found on tests may be explained in part by automatic processes that involve the relative processing time of considered dimensions and a priority to answer quickly.