Multiple Representations in Number Line Estimation: A Developmental Shift or Classes of Representations?

Children's estimation patterns on a number line estimation task may provide information about the mental representation of the magnitude of numbers. Siegler and his colleagues concluded that children's mental representations shift from a logarithmic-ruler representation to a linear-ruler representation. However, there are important methodological issues with respect to their number-line studies that threaten the validity of the conclusions. We discuss these methodological issues and propose an alternative method to analyze estimation data. One hundred nineteen children from kindergarten, first, and second grade performed a number-line estimation task in which they had to estimate the position of 30 numbers on a 0-to-100 number line. The results supported the hypothesis that children show various kinds of estimation patterns. Five classes of children were distinguished, which were characterized by different estimation patterns. A remarkable result was that the logarithmic-ruler representation was not found. Although young children were more likely to show overestimation of small numbers than older children, this developmental trend was small and not significant.     

Consistency of Response Patterns in Different Estimation Tasks

The current study aimed at addressing two issues concerning children's estimation performance: (1) to investigate whether the log-to-linear framework or the proportional judgment framework provided a better explanation of children's estimation patterns, and (2) to examine the consistency of response patterns in different estimation tasks. A sample of 179 Chinese first graders was assessed on their arithmetic performance and estimation skills (including numerosity naming, numerosity production, and number line). The log-to-linear framework was suggested to provide a better framework in explaining children's estimation patterns. Under this framework, we identified both common features and uniqueness of children's response patterns in different estimation tasks. Furthermore, different estimation skills uniquely contributed to children's arithmetic performance. The theoretical and practical implications of the findings are discussed.     

About why there is a shift from cardinal to ordinal processing in the association with arithmetic between first and second grade

Digit comparison is strongly related to individual differences in children's arithmetic ability. Why this is the case, however, remains unclear to date. Therefore, we investigated the relative contribution of three possible cognitive mechanisms in first and second graders’ digit comparison performance: digit identification, digit–number word matching and digit ordering ability. Furthermore, we examined whether these components could account for the well‐established relation between digit comparison performance and arithmetic. As expected, all candidate predictors were related to digit comparison in both age groups. Moreover, in first graders, digit ordering and in second graders both digit identification and digit ordering explained unique variance in digit comparison performance. However, when entering these unique predictors of digit comparison into a mediation model with digit comparison as predictor and arithmetic as outcome, we observed that whereas in second graders digit ordering was a full mediator, in first graders this was not the case. For them, the reverse was true and digit comparison fully mediated the relation between digit ordering and arithmetic. These results suggest that between first and second grade, there is a shift in the predictive value for arithmetic from cardinal processing and procedural knowledge to ordinal processing and retrieving declarative knowledge from memory; a process which is possibly due to a change in arithmetic strategies at that age. A video abstract of this article can be viewed at: https://youtu.be/dDB0IGi2Hf8     

Children's expectations about training the approximate number system

Humans possess a developmentally precocious and evolutionarily ancient approximate number system (ANS) whose sensitivity correlates with uniquely human symbolic arithmetic skills. Recent studies suggest that ANS training improves symbolic arithmetic, but such studies may engender performance expectations in their participants that in turn produce the improvement. Here, we assessed 6‐ to 8‐year‐old children's expectations about the effects of numerical and non‐numerical magnitude training, as well as states of satiety and restfulness, in the context of a study linking children's ANS practice to their improved symbolic arithmetic. We found that children did not expect gains in symbolic arithmetic after exercising the ANS, although they did expect gains in ANS acuity after training on any magnitude task. Moreover, children expected gains in symbolic arithmetic after a good night's sleep and their favourite breakfast. Thus, children's improved symbolic arithmetic after ANS training cannot be explained by their expectations about that training.     

Children's Understanding of Second‐Order False Belief: Comparisons of Content and Method of Assessment

This research examined children's performance on second‐order false belief tasks as a function of the content area for the belief and the method of assessing understanding. A total of 70 kindergarten and first‐grade children responded to four second‐order stories. On two stories, the task was to judge a belief about a belief, and on two, the task was to judge a belief about an emotion. On one trial within each group, the task was to predict the target's belief, and on one trial, the task was to explain the belief. Older children outperformed younger children on the prediction measure. Differences as a function of content area and method of assessment were limited; when they did occur, performance was generally better with belief than with emotion as the target, and better with prediction than with explanation as the response criterion. Finally, there was no relation between number of siblings and performance.     

Contributions of linguistic,quantitative, and spatial attention skills to young children's math versus reading: Same,different, or both?

The study used Bayesian and Frequentist methods to investigate whether the roles of linguistic, quantitative, and spatial attention skills are distinct in children's acquisition of reading and math. A sample of 175 Chinese kindergarteners was tested with measures of linguistic skills (phonological awareness and phonological memory), quantitative knowledge (number line task, symbolic digit comparison, and non-symbolic number estimation), spatial attention skills (visual span, mental rotation, and visual search), word reading, and calculation. After statistically controlling for age and nonverbal intelligence, phonological awareness and digit comparison performance explained unique variance in both math and reading. Moreover, number line estimation was specifically important for math, while phonological memory was specifically essential for reading. These findings highlight the possibility of developing early screening tools with different cognitive measures for children at risk of learning disabilities in reading and/or math.     

Numerical predictors of arithmetic success in grades 1–6

Math relies on mastery and integration of a wide range of simpler numerical processes and concepts. Recent work has identified several numerical competencies that predict variation in math ability. We examined the unique relations between eight basic numerical skills and early arithmetic ability in a large sample (N = 1391) of children across grades 1–6. In grades 1–2, children's ability to judge the relative magnitude of numerical symbols was most predictive of early arithmetic skills. The unique contribution of children's ability to assess ordinality in numerical symbols steadily increased across grades, overtaking all other predictors by grade 6. We found no evidence that children's ability to judge the relative magnitude of approximate, nonsymbolic numbers was uniquely predictive of arithmetic ability at any grade. Overall, symbolic number processing was more predictive of arithmetic ability than nonsymbolic number processing, though the relative importance of symbolic number ability appears to shift from cardinal to ordinal processing.     

Parents' Estimations of Preschoolers' Number Skills Relate to at‐Home Number‐Related Activity Engagement

According to Hunt's match hypothesis, the accuracy of parents' beliefs about their children's abilities can influence the nature of the early learning experiences they provide. The present study examined the accuracy of parents' beliefs about their preschoolers' number development and relations to parent‐reported frequency of engaging children in number related experiences at home. Parents reported engaging their preschoolers more frequently in conventional numeracy activities, (i.e. counting and identifying numbers) than advanced number‐related activities (e.g. arithmetic) at home, though the frequency of advanced activities increased with the development of children's advanced number skills. Parents were most uncertain about their children's advanced number skills, though they demonstrated an overall tendency to overestimate their children's abilities across number tasks. Increased rates of overestimation and decreased rates of underestimation were associated with increased incidences of advanced activity engagement at home. Thus, results suggest guiding parents to understand their own children's numerical understanding in a wide range of number domains could promote more advanced at‐home number‐related activity engagement. Copyright © 2016 John Wiley & Sons, Ltd.     

Developmental dynamics between mathematical performance,task motivation,and teachers' goals during the transition to primary school

Background. It has been suggested that children's learning motivation and interest in a particular subject play an important role in their school performance, particularly in mathematics. However, few cross‐lagged longitudinal studies have been carried out to investigate the prospective relationships between academic achievement and task motivation. Moreover, the role that the classroom context plays in this development is largely unknown. Aims. The aim of the study was to investigate the developmental dynamics of maths‐related motivation and mathematical performance during children's transition to primary school. The role of teachers' pedagogical goals and classroom characteristics on this development was also investigated. Sample. A total of 196 Finnish children were examined four times: (0) in October during their preschool year; (1) in October and (2) April during their first grade of primary school; and (3) in October during their second grade. Method. Children's mathematical performance was tested at each measurement point. Task motivation was examined at measurement points 2, 3, and 4 using the Task‐value scale for children. First‐grade teachers were interviewed in November about their pedagogical goals and classroom characteristics. Results and conclusions. The results showed that children's mathematical performance and related task motivation formed a cumulative developmental cycle: a high level of maths performance at the beginning of the first grade increased subsequent task motivation towards mathematics, which further predicted a high level of maths performance at the beginning of the second grade. The level of maths‐related task motivation increased in those classrooms where the teachers emphasized motivation or self‐concept development as their most important pedagogical goal.     

Association between basic numerical abilities and mathematics achievement

Various measures have been used to investigate number processing in children, including a number comparison or a number line estimation task. The present study aimed to examine whether and to which extent these different measures of number representation are related to performance on a curriculum‐based standardized mathematics achievement test in kindergarteners, first, second, and sixth graders. Children completed a number comparison task and a number line estimation task with a balanced set of symbolic (Arabic digits) and non‐symbolic (dot patterns) stimuli. Associations with mathematics achievement were observed for the symbolic measures. Although the association with number line estimation was consistent over grades, the association with number comparison was much stronger in kindergarten compared to the other grades. The current data indicate that a good knowledge of the numerical meaning of Arabic digits is important for children's mathematical development and that particularly the access to the numerical meaning of symbolic digits rather than the representation of number per se is important.     

Influences of problem format and SES on preschoolers’ understanding of approximate addition

Recent studies suggest that 5-year-olds can add and compare large numerical quantities through approximate representations of number. However, the nature of this understanding and its susceptibility to environmental influences remain unclear. We examined whether children's early competence depends on the canonical problem format (i.e., arithmetic operations presented on the left side). Sixty children from middle-to-high-SES backgrounds (Experiment 1) and 47 children from low-SES backgrounds (Experiment 2) viewed events that required them to add and compare large numbers. Events were shown in a canonical or noncanonical format. Children from both SES backgrounds performed above chance on the approximate addition tasks, but children from middle-to-high-SES backgrounds performed significantly better. Moreover, children from middle-to-high SES backgrounds performed better when problems were presented in the canonical format, whereas children from low-SES backgrounds did not. These results suggest that children's understanding of approximate number is affected by some of the same environmental factors that affect performance on exact arithmetic tasks.     

Risky visuomotor choices during rapid reaching in childhood

Many everyday actions are implicit gambles because imprecisions in our visuomotor systems place probabilities on our success or failure. Choosing optimal action strategies involves weighting the costs and gains of potential outcomes by their corresponding probabilities, and requires stable representations of one's own imprecisions. How this ability is acquired during development in childhood when visuomotor skills change drastically is unknown. In a rewarded rapid reaching task, 6‐ to 11‐year‐old children followed ‘risk‐seeking’ strategies leading to overly high point‐loss. Adults' performance, in contrast, was close to optimal. Children's errors were not explained by distorted estimates of value or probability, but may reflect different action selection criteria or immature integration of value and probability information while planning movements. These findings provide a starting point for understanding children's risk‐taking in everyday visuomotor situations when suboptimal choices can be dangerous. Moreover, children's risky visuomotor decisions mirror those reported for non‐motor gambles, raising the possibility that common processes underlie development across decision‐making domains.     

Is understanding regret dependent on developments in counterfactual thinking?

Children's understanding of counterfactual emotions such as regret and relief develops relatively late compared to their ability to imagine counterfactual worlds. We tested whether a late development in counterfactual thinking: understanding counterfactuals as possibilities, underpinned children's understanding of regret. Thirty 5‐ and 6‐year‐olds completed tasks assessing counterfactual thinking and understanding regret. Performance on the counterfactual task was better than that on the regret task. We suggest that thinking about counterfactuals as possibilities is a necessary but not sufficient cognitive development in children's understanding of regret. We discuss how other developments in counterfactual thinking may underpin children's emotional understanding.     

Individual differences in children's understanding of inversion and arithmetical skill

Background and aims. In order to develop arithmetic expertise, children must understand arithmetic principles, such as the inverse relationship between addition and subtraction, in addition to learning calculation skills. We report two experiments that investigate children's understanding of the principle of inversion and the relationship between their conceptual understanding and arithmetical skills. Sample. A group of 127 children from primary schools took part in the study. The children were from 2 age groups (6–7 and 8–9 years). Methods. Children's accuracy on inverse and control problems in a variety of presentation formats and in canonical and non‐canonical forms was measured. Tests of general arithmetic ability were also administered. Results. Children consistently performed better on inverse than control problems, which indicates that they could make use of the inverse principle. Presentation format affected performance: picture presentation allowed children to apply their conceptual understanding flexibly regardless of the problem type, while word problems restricted their ability to use their conceptual knowledge. Cluster analyses revealed three subgroups with different profiles of conceptual understanding and arithmetical skill. Children in the ‘high ability’ and ‘low ability’ groups showed conceptual understanding that was in‐line with their arithmetical skill, whilst a 3rd group of children had more advanced conceptual understanding than arithmetical skill. Conclusions. The three subgroups may represent different points along a single developmental path or distinct developmental paths. The discovery of the existence of the three groups has important consequences for education. It demonstrates the importance of considering the pattern of individual children's conceptual understanding and problem‐solving skills.     

Arithmetic knowledge from the spontaneous attention to relations

Children's knowledge of arithmetic principles is a key aspect of early mathematics knowledge. Knowledge of arithmetic principles predicts how children approach solving arithmetic problems and the likelihood of their success. Prior work has begun to address how children might learn arithmetic principles in a classroom setting. Understanding of arithmetic principles involves understanding how numbers in arithmetic equations relate to another. For example, the Relation to Operands (RO) principle is that for subtracting natural numbers (A ? B = C), the difference (C) must be smaller than the minuend (A). In the current study we evaluate if individual differences in arithmetic principle knowledge (APK) can be predicted by the learners' spontaneous attention to relations (SAR) and if feedback can increase their attention to relations. Results suggest that participants’ Spontaneous Attention to Number (SAN) does not predict their knowledge of the RO principle for symbolic arithmetic. Feedback regarding the attention to relations did not show a significant effect on SAR or participants’ APK. We also did not find significant relations between reports of parent talk and the home environment with individual differences in SAN. The amount of parent's talk about relations was not significantly associated with learner's SAR and APK. We conclude that children's SAR with non‐symbolic number does not generalize to attention to relations with symbolic arithmetic.     

The left digit effect in a complex judgment task: Evaluating hypothetical college applicants

A left digit effect has been broadly observed across judgment and decision-making contexts ranging from product evaluation to medical treatment decisions to number line estimation. For example, $3.00 is judged to be a much greater cost than $2.99, and “801” is estimated strikingly too far to the right of “798” on a number line. Although the consequences of the effects for judgment and decision behavior have been documented, the sources of the effects are not well established. The goal of the current work is to extend investigations of the left digit effect to a new complex judgment activity and to assess whether the magnitude of the effect at the individual level can be predicted from performance on a simpler number skills task on which the left digit effect has also recently been observed. In three experiments (N = 434), adults completed a judgment task in which they rated the strength of hypothetical applicants for college admission and a self-paced number line estimation task. In all experiments, a small or medium left digit effect was found in the college admissions task, and a large effect was found in number line estimation. Individual-level variation was observed, but there was no relationship between the magnitudes of the effects in the two tasks. These findings provide evidence of a left digit effect in a novel multiattribute judgment task but offer no evidence that such performance can be predicted from a simple number skills task such as number line estimation.     

Young children's psychological explanations and their relationship to perception‐ and intention‐understanding

The present study examined two key aspects of young children's ability to explain human behaviour in a mentalistic way. First, we explored desires that are of a level of difficulty comparable with that of false beliefs. For this purpose, the so‐called ‘alternative desires’ were created. Second, we examined how children's psychological explanations are related to their understanding of perception and intention. A perception‐understanding task, an intention‐understanding task and a psychological‐explanation task were administered to 80 three‐year‐olds. Results offer support for the thesis that the level of difficulty of belief and desire explanations is comparable. Moreover, children's psychological explanations are related to their understanding of perception and intention. The results lend support to the idea that mentalistic explanations are an explicit manifestation of children's level of theory of mind. Copyright © 2008 John Wiley & Sons, Ltd.     

Parental Guidance and Children's Executive Function: Working Memory and Planning as Moderators During Joint Problem‐Solving

Cognitive aspects of children's executive function (EF) were examined as moderators of the effectiveness of parental guidance on children's learning. Thirty‐two 5‐year‐old children and their parents were observed during joint problem‐solving. Forms of guidance geared towards cognitive assistance were coded as directive or elaborative, and children's responses were recorded. Children were then assessed on an independent version of the same task. A parent‐rated composite of working memory and planning was used as a measure of EF. Directive guidance by parents was associated with more child errors during the joint activity, whereas elaborative guidance was associated with better performance. Parent‐rated EF moderated the relation, such that the relation between elaborative guidance and better performance was only significant for children with low EF. During the independent task, EF again moderated the relation between parent guidance and children's performance, such that children with low EF did worse when parents had provided more directive guidance; for children with high EF, directive guidance was associated with better independent performance. These findings suggest that the extent to which children's performance relates to different forms of parents' guidance varies, and elaborative assistance may be more helpful for children with low EF. Copyright © 2016 John Wiley & Sons, Ltd.     

Children's cognitive representation of the mathematical number line

Learning of the mathematical number line has been hypothesized to be dependent on an inherent sense of approximate quantity. Children's number line placements are predicted to conform to the underlying properties of this system; specifically, placements are exaggerated for small numerals and compressed for larger ones. Alternative hypotheses are based on proportional reasoning; specifically, numerals are placed relative to set anchors such as end points on the line. Traditional testing of these alternatives involves fitting group medians to corresponding regression models which assumes homogenous residuals and thus does not capture useful information from between‐ and within‐child variation in placements across the number line. To more fully assess differential predictions, we developed a novel set of hierarchical statistical models that enable the simultaneous estimation of mean levels of and variation in performance, as well as developmental transitions. Using these techniques we fitted the number line placements of 224 children longitudinally assessed from first to fifth grade, inclusive. The compression pattern was evident in mean performance in first grade, but was the best fit for only 20% of first graders when the full range of variation in the data are modeled. Most first graders' placements suggested use of end points, consistent with proportional reasoning. Developmental transition involved incorporation of a mid‐point anchor, consistent with a modified proportional reasoning strategy. The methodology introduced here enables a more nuanced assessment of children's number line representation and learning than any previous approaches and indicates that developmental improvement largely results from midpoint segmentation of the line.