Number and type of toys affect joint attention of mothers and infants

Establishing joint attention with a caregiver on a physical object provides an optimal environment for language learning for infants. In the present study, we investigated whether 12-month-olds and their mothers establish higher quality joint attention interactions in the presence of fewer compared to more toys. As a secondary goal, we investigated how different types of toys affect how mother-infant dyads establish joint attention. In a five-minute free play setting, mothers and infants participated in either Five Toy (n = 48) or Twelve Toy (n = 33) groups. They were given organizational (i.e., toys that require arrangement of parts), responsive (i.e., toys that emit sounds via manipulation), and symbolic toys (i.e., toys that elicit pretend play). Results showed that compared to the Twelve Toy group, joint attention interactions in the Five Toy group were less frequent, lasted longer, were more likely to be initiated by maternal following than by maternal directing of infants’ attention, and more likely to be coordinated in which infants demonstrated awareness of the mothers’ simultaneous attentional focus by looking at their mothers, vocalizing, or turn-taking. We further found longer joint attention durations on organizational compared to symbolic toys, which were preferred to a lesser extent by the dyads. With responsive toys, mothers were more likely to initiate joint attention by following their infants’ attention. Joint attention interactions lasted longer and were more likely to be coordinated in the second half compared to the first half of the play session, suggesting that over time it became easier for the mothers and infants to settle on certain toys for more elaborate play. In sum, mothers and infants establish higher-quality joint attention with fewer toys in general and with organizational toys in particular.     

Too much to count on: impaired very small numbers in corticobasal degeneration

Patients with corticobasal degeneration (CBD) have calculation impairments. This study examined whether impaired number knowledge depends on verbal mediation. We focused particularly on knowledge of very small numbers, where there is a precise relationship between a cardinality and its number concept, but little hypothesized role for verbal mediation. We evaluated accuracy and reaction time (RT) for matching dot arrays and Arabic numerals involving smaller (2-4) and larger (5-9) cardinalities in non-aphasic patients with CBD (n=16), frontotemporal dementia (FTD; n=23), and healthy controls (n=15). CBD were less accurate and slowed at judging smaller Arabic numeral-dot array stimuli compared to FTD patients and controls. Moreover, only CBD showed longer RTs judging successively larger number-dot array pairs among the smaller cardinalities. Difficulty judging very small numbers is impaired in CBD, suggesting degraded representation of precise number knowledge that does not depend on language functioning.     

Children’s representation of symbolic magnitude: The development of the priming distance effect

The comparison distance effect (CDE), whereby discriminating between two numbers that are far apart is easier than discriminating between two numbers that are close, has been considered as an important indicator of how people represent magnitudes internally. However, the underlying mechanism of this CDE is still unclear. We tried to shed further light on how people represent magnitudes by using priming. Adults have been shown to exhibit a priming distance effect (PDE), whereby numbers are processed faster when they are preceded by a close number than when they are preceded by a more distant number. Surprisingly, there are no studies available that have investigated this effect in children. The current study examined this effect in typically developing first, third, and fifth graders and in adults. Our findings revealed that the PDE already occurs in first graders and remains stable across development. This study also documents the usefulness of number priming in children, making it an interesting tool for future research.     

The development of arithmetic principle knowledge: How do we know what learners know?

This paper reviews research on learners’ knowledge of three arithmetic principles: Commutativity, Relation to Operands, and Inversion. Studies of arithmetic principle knowledge vary along several dimensions, including the age of the participants, the context in which the arithmetic is presented, and most importantly, the type of knowledge assessment (e.g., application of procedures, evaluation of examples). The vast majority of studies utilize single-faceted knowledge assessments, which can lead to incomplete or misleading views of learners’ knowledge. Both context and type of knowledge assessment can influence conclusions about learners’ arithmetic principle knowledge. However, relatively few studies directly address the possible effects of context or type of knowledge assessment on their results. To move the field forward, research that utilizes multifaceted knowledge assessments is needed.     

Arabic digit naming speed: task context and redundancy gain

There is evidence for both semantic and asemantic routes for naming Arabic digits, but neuropsychological dissociations suggest that number-fact retrieval (2x3=6) can inhibit the semantic route for digit naming. Here, we tested the hypothesis that such inhibition should slow digit naming, based on the principle that reduced access to multiple routes would counteract redundancy gain (the response time advantage expected from parallel retrieval pathways). Participants named two single digit numbers and then performed simple addition or magnitude comparison (Experiment 1), multiplication or magnitude comparison (Experiment 2), and multiplication or subtraction (Experiment 3) on the same or on a different pair of digits. Addition and multiplication were expected to inhibit the semantic route, whereas comparison and subtraction should enable the semantic route. Digit naming time was approximately 15ms slower when participants subsequently performed addition or multiplication relative to comparison or subtraction, regardless of whether or not the same digit pair was involved. A letter naming control condition in Experiment 3 demonstrated that the effect was specific to digit naming. Number fact retrieval apparently can inhibit Arabic digit naming processes.     

Representational change and magnitude estimation: why young children can make more accurate salary comparisons than adults

Development of estimation has been ascribed to two sources: (1) a change from logarithmic to linear representations of number and (2) development of general mathematical skills. To test the representational change hypothesis, we gave children and adults a task in which an automatic, linear representation is less adaptive than the logarithmic representation: estimating the value of salaries given in fractional notation. The representational change hypothesis generated the surprising (and accurate) prediction that when estimating the magnitude of salaries given in fractional notation, young children would outperform adults, whereas when estimating the magnitude of the same salaries given in decimal notation, adults would outperform children.     

Strategies in subtraction problem solving in children

The aim of this study was to investigate the strategies used by third graders in solving the 81 elementary subtractions that are the inverses of the one-digit additions with addends from 1 to 9 recently studied by Barrouillet and Lépine. Although the pattern of relationship between individual differences in working memory, on the one hand, and strategy choices and response times, on the other, was the same in both operations, subtraction and addition differed in two important ways. First, the strategy of direct retrieval was less frequent in subtraction than in addition and was even less frequent in subtraction solving than the recourse to the corresponding additive fact. Second, contrary to addition, the retrieval of subtractive answers is confined to some peculiar problems involving 1 as the subtrahend or the remainder. The implications of these findings for developmental theories of mental arithmetic are discussed.     

Costs and benefits of representational change: effects of context on age and sex differences in symbolic magnitude estimation

Studies have reported high correlations in accuracy across estimation contexts, robust transfer of estimation training to novel numerical contexts, and adults drawing mistaken analogies between numerical and fractional values. We hypothesized that these disparate findings may reflect the benefits and costs of learning linear representations of numerical magnitude. Specifically, children learn that their default logarithmic representations are inappropriate for many numerical tasks, leading them to adopt more appropriate linear representations despite linear representations being inappropriate for estimating fractional magnitude. In Experiment 1, this hypothesis accurately predicted a developmental shift from logarithmic to linear estimates of numerical magnitude and a negative correlation between accuracy of numerical and fractional magnitude estimates (r = −.80). In Experiment 2, training that improved numerical estimates also led to poorer fractional magnitude estimates. Finally, both before and after training that eliminated age differences in estimation accuracy, complementary sex differences were observed across the two estimation contexts.     

Low working memory capacity impedes both efficiency and learning of number transcoding in children

This study aimed to evaluate the impact of individual differences in working memory capacity on number transcoding. A recently proposed model, ADAPT (a developmental asemantic procedural transcoding model), accounts for the development of number transcoding from verbal form to Arabic form by two mechanisms: the learning of new production rules that enlarge the range of numbers a child can transcode and the increase of the mental lexicon. The working memory capacity of 7-year-olds was evaluated along with their ability to transcode one- to four-digit numbers. As ADAPT predicts, the rate of transcoding errors increased when more production rules were required and when children had low working memory capacity, with these two factors interacting. Moreover, qualitative analysis of the errors produced by high- and low-span children showed that the latter have a developmental delay in the acquisition of the production rules.     

A validation of eye movements as a measure of elementary school children's developing number sense

The number line estimation task captures central aspects of children's developing number sense, that is, their intuitions for numbers and their interrelations. Previous research used children's answer patterns and verbal reports as evidence of how they solve this task. In the present study we investigated to what extent eye movements recorded during task solution reflect children's use of the number line. By means of a cross-sectional design with 66 children from Grades 1, 2, and 3, we show that eye-tracking data (a) reflect grade-related increase in estimation competence, (b) are correlated with the accuracy of manual answers, (c) relate, in Grade 2, to children's addition competence, (d) are systematically distributed over the number line, and (e) replicate previous findings concerning children's use of counting strategies and orientation-point strategies. These findings demonstrate the validity and utility of eye-tracking data for investigating children's developing number sense and estimation competence.