Mapping kindergartners’ quantitative competence

In this study we investigated the structure of quantitative competence of kindergartners by testing a hypothesized four-factor model of quantitative competence consisting of the components counting, subitizing, additive reasoning and multiplicative reasoning. Data were collected from kindergartners in the Netherlands (n = 334) and in Cyprus (n = 304). A confirmatory factor analysis showed that the four-factor structure fitted the empirical data from the Netherlands. For the Cyprus data a one-factor structure was found to have a more adequate fit. Regarding the effect of country on performance, a comparison at item level showed that the kindergartners in the Netherlands outperformed those in Cyprus in the majority of quantitative competence items. Analyses of variance revealed for each country a significant effect of kindergarten year on performance, with children in K2 (second kindergarten year) outperforming those in K1 (first kindergarten year). A statistical implicative analysis at item level revealed that in both countries the relevant implicative chain, showing what successful solving of an item implies for correct solving of another item, reflects by and large the sequential steps mostly followed in teaching kindergartners early number. This sequence starts with counting and subitizing, then continues with additive reasoning and finally multiplicative reasoning. These implicative chains also clearly show that the development of early quantitative competence is not linear. There are many parallel processes and cross-connections between the components of quantitative competence.publishedVersionPaid Open Acces

Long-short term memory networks for modelling embodied mathematical cognition in robots

Mathematical competence can endow robots with the necessary capability for abstract and symbolic processing, which is required for higher cognitive functions such as natural language understanding. But, so far, only few attempts have been made to model mathematical cognition in robots. This paper presents an experimental evaluation of the Long-Short Term Memory networks for modeling the simple mathematical operation of single-digits addition in a cognitive robot. To this end, the robotic model creates an association between the proprioceptive information from finger counting and the handwritten digits of the MNIST dataset. In practice, the model executes two tasks concurrently: it recognizes the handwritten digits in a sequence and sums them. The results show that the association with fingers can improve the robot precision, as observed in children. Also, the robot makes a disproportionate number of split-five errors similarly to what observed in studies with children and adults, hence giving evidence to support the hypothesis that these errors are due the use of a five-fingers counting system

Using Rasch modeling to investigate the construct of motor competence in early childhood

Purpose: The present study investigated the dimensionality and homogeneity of motor competence, which is defined as the ability that underlies the performance of a wide variety of motor skills, in early childhood using a large set of items. Method: A total of 1467 children (aged 3-6 years) were measured with the Motor Proficiency Test for 4- to 6-Year-old Children (Motoriktest für vier-bis sechsjährige Kinder [MOT 4-6]), which consists of 17 items. Results: Analyses using the Partial Credit Model and mixed Rasch model revealed a one-dimensional structure (CR = 1.964, pCR = .06; P-χ2 = -.227, pp-χ2=.24). Due to unordered threshold parameters, five items were excluded. These items have a scoring system that counts the amount of successful trials (0-2). Conclusion: The study shows item and person homogeneity within a validated motor score, using 12 items of the MOT 4-6. Thus, it provides evidence of a single latent construct (i.e., motor competence), which underlies the performance of motor skills in early childhood. Furthermore, it shows that counting the number of successful trails may be less suitable as a scoring system in motor competence assessment. Present findings also support the use of validated composite scores in motor assessment

Longitudinal study of low and high achievers in early mathematics

Background. Longitudinal studies allow us to identify, which specific maths skills are weak in young children, and whether there is a continuing weakness in these areas throughout their school years. Aims. This 2-year study investigated whether certain socio-demographic variables affect early mathematical competency in children aged 5–7 years. Sample. A randomly selected sample of 127 students (64 female; 63 male) participated. At the start of the study, the students were approximately 5 years old (M = 5.2; SD = 0.28; range = 4.5–5.8). Method. The students were assessed using the Early Numeracy Test and then allocated to a high (n = 26), middle (n = 76), or low (n = 25) achievers group. The same children were assessed again with the Early Numeracy Test at 6 and 7 years old, respectively. Eight socio-demographic characteristics were also evaluated: family model, education of the parent(s), job of the parent(s), number of family members, birth order, number of computers at home, frequency of teacher visits, and hours watching television. Results. Early Numeracy Test scores were more consistent for the high-achievers group than for the low-achievers group. Approximately 5.5% of low achievers obtained low scores throughout the study. A link between specific socio-demographic characteristics and early achievement in mathematics was only found for number of computers at home. Conclusions. The level of mathematical ability among students aged 5–7 years remains relatively stable regardless of the initial level of achievement. However, early screening for mathematics learning disabilities could be useful in helping low-achieving students overcome learning obstacles.This material is based on work supported by the Spanish Ministry of Science & Technology grant no. SEJ2007-62420/EDUC and Junta de Andalucia grant no. P09-HUM-4918

The Parliament of the Experts

In the administrative state, how should expert opinions be aggregated and used? If a panel of experts is unanimous on a question of fact, causation, or prediction, can an administrative agency rationally disagree, and on what grounds? If experts are split into a majority view and a minority view, must the agency follow the majority? Should reviewing courts limit agency discretion to select among the conflicting views of experts, or to depart from expert consensus? I argue that voting by expert panels is likely, on average, to be epistemically superior to the substantive judgment of agency heads, in determining questions of fact, causation, or prediction. Nose counting of expert panels should generally be an acceptable basis for decision under the arbitrary and capricious or substantial evidence tests. Moreover, agencies should be obliged to follow the (super)majority view of an expert panel, even if the agency\u27s own judgment is to the contrary, unless the agency can give an epistemically valid second-order reason for rejecting the panel majority\u27s view

Preschool predictors of mathematics in first grade children with autism spectrum disorder

AbstractUp till now, research evidence on the mathematical abilities of children with autism spectrum disorder (ASD) has been scarce and provided mixed results. The current study examined the predictive value of five early numerical competencies for four domains of mathematics in first grade. Thirty-three high-functioning children with ASD were followed up from preschool to first grade and compared with 54 typically developing children, as well as with normed samples in first grade. Five early numerical competencies were tested in preschool (5–6 years): verbal subitizing, counting, magnitude comparison, estimation, and arithmetic operations. Four domains of mathematics were used as outcome variables in first grade (6–7 years): procedural calculation, number fact retrieval, word/language problems, and time-related competences. Children with ASD showed similar early numerical competencies at preschool age as typically developing children. Moreover, they scored average on number fact retrieval and time-related competences and higher on procedural calculation and word/language problems compared to the normed population in first grade. When predicting first grade mathematics performance in children with ASD, both verbal subitizing and counting seemed to be important to evaluate at preschool age. Verbal subitizing had a higher predictive value in children with ASD than in typically developing children. Whereas verbal subitizing was predictive for procedural calculation, number fact retrieval, and word/language problems, counting was predictive for procedural calculation and, to a lesser extent, number fact retrieval. Implications and directions for future research are discussed

Developmental neuroscience of time and number: implications for autism and other neurodevelopmental disabilities

Estimations of time and number share many similarities in both non-humans and man. The primary focus of this review is on the development of time and number sense across infancy and childhood, and neuropsychological findings as they relate to time and number discrimination in infants and adults. Discussion of these findings is couched within a mode-control model of timing and counting which assumes time and number share a common magnitude representation system. A basic sense of time and number likely serves as the foundation for advanced numerical and temporal competence, and aspects of higher cognition—this will be discussed as it relates to typical childhood, and certain developmental disorders, including autism spectrum disorder. Directions for future research in the developmental neuroscience of time and number (NEUTIN) will also be highlighted

Testimony and Children’s Acquisition of Number Concepts

An enduring puzzle in philosophy and developmental psychology is how young children acquire number concepts, in particular the concept of natural number. Most solutions to this problem conceptualize young learners as lone mathematicians who individually reconstruct the successor function and other sophisticated mathematical ideas. In this chapter, I argue for a crucial role of testimony in children’s acquisition of number concepts, both in the transfer of propositional knowledge (e.g., the cardinality concept), and in knowledge-how (e.g., the counting routine)