Spontaneous partitioning and proportion estimation in children's numerical judgments.

What do numerical estimates tell us about developing an understanding of number? One theory is that bounded number line estimation (NLE) tasks reveal a "representational shift" from logarithmically to linearly organized mental representations of number over development. According to a different theoretical framework, developmental change in estimation reflects changes in children's numerical knowledge and their ability to make appropriate relative judgments. Empirical support for this "proportion estimation" framework includes the fact that quantitative models of proportion estimation describe signature patterns of estimation bias. A recent study argued against this latter theory by suggesting that patterns of curvature in number line placements are simply artifacts of a task procedure in which participants receive explicit information about the location of the numerical midpoint. We tested this claim in two experiments with children aged 6-8 years (Experiment 1: N = 47; Experiment 2: N = 104). Results demonstrated that the proportion estimation framework provides a good explanation of children's number line placement in the absence of explicit midpoint cues, that explicit cues to the midpoint are associated with more frequent use of middle reference points in young children, and that children can use a middle reference point spontaneously in the absence of explicit cues (with this tendency increasing with age). These findings provide novel support for the idea that psychophysical models of proportion estimation successfully account for numerical estimates across development regardless of whether spatial and numerical midpoint cues are provided as part of the NLE task.

Language counts: Early language mediates the relationship between parent education and children's math ability.

Children's early math skills have been hailed as a powerful predictor of academic success. Disparities in socioeconomic context, however, also have dramatic consequences on children's learning. It is therefore critical to investigate both of these distinct contributors in order to better understand the early foundations of children's academic outcomes. This study tests an integrated model of children's developing math ability so as to (1) identify the specific skills and abilities most clearly linked to early math achievement and (2) measure the influence of children's socioeconomic context on each of these skills. We first evaluated the early vocabulary, number word knowledge (knower level), and Approximate Number System (ANS) acuity of a diverse group of preschoolers. Then, approximately 1 year later as they entered Kindergarten, we administered a test of early math achievement. We find that children's early language (general vocabulary and number word knowledge) fully mediates the relationship between parent education and math ability. Additionally, number word knowledge mediates the relationship between ANS acuity and early math. We argue that increased focus on number word knowledge, as well as general vocabulary, may help to minimize disparities in math ability as children enter kindergarten. We also highlight the role of parent education on children's learning and note that this may be an important locus for intervention.

Intuitive proportion judgment in number-line estimation: Converging evidence from multiple tasks.

How children's understanding of numerical magnitudes changes over the course of development remains a key question in the study of numerical cognition. In an ongoing debate about the source of developmental change, some argue that children maintain and access different mental representations of number, with evidence coming largely from common number-line estimation tasks. In contrast, others argue that a theoretical framework based on psychophysical models of proportion estimation accounts for typical performance on these tasks. The current study explored children's (n=71) and adults' (n=27) performance on two number-line tasks: the "number to position" (or NP) task and the inverse "position to number" (or PN) task. Estimates on both tasks are consistent with the predictions of the proportion estimation account and do not support the hypothesis that a fundamental shift in mental representations underlies developmental change in numerical estimation and, in turn, mathematical ability. Converging evidence across the tasks also calls into question the utility of bounded number-line tasks as an evaluation of mental representations of number.

Acquisition of the Cardinal Principle Coincides with Improvement in Approximate Number System Acuity in Preschoolers.

Human mathematical abilities comprise both learned, symbolic representations of number and unlearned, non-symbolic evolutionarily primitive cognitive systems for representing quantities. However, the mechanisms by which our symbolic (verbal) number system becomes integrated with the non-symbolic (non-verbal) representations of approximate magnitude (supported by the Approximate Number System, or ANS) are not well understood. To explore this connection, forty-six children participated in a 6-month longitudinal study assessing verbal number knowledge and non-verbal numerical acuity. Cross-sectional analyses revealed a strong relationship between verbal number knowledge and ANS acuity. Longitudinal analyses suggested that increases in ANS acuity were most strongly related to the acquisition of the cardinal principle, but not to other milestones of verbal number acquisition. These findings suggest that experience with culture and language is intimately linked to changes in the properties of a core cognitive system.

How feedback improves children's numerical estimation.

Developmental change in children's number-line estimation has been thought to reveal a categorical logarithmic-to-linear shift in mental representations of number. Some have claimed that the broad and rapid change in estimation patterns that occurs with corrective feedback provides strong evidence for this shift. However, quantitative models of proportion judgment may provide a better account of children's estimation patterns while also predicting broad and rapid change following feedback. Here we test the hypothesis that local corrective feedback provides children with additional reference points, rather than catalyzing a shift to a different mental representation of number. We tested 117 children from several second-grade classrooms in a number-line feedback study. Data indicate that the proportion-judgment framework accounts for individual differences in estimation patterns, and that the effects of feedback are consistent with the unique quantitative predictions of the framework. They do not provide evidence supporting the representational shift hypothesis or, more broadly, for the proposal that cognitive change can occur rapidly at the level of entire mental representations.

Spatial estimation: a non-Bayesian alternative.

A large collection of estimation phenomena (e.g. biases arising when adults or children estimate remembered locations of objects in bounded spaces; Huttenlocher, Newcombe & Sandberg, 1994) are commonly explained in terms of complex Bayesian models. We provide evidence that some of these phenomena may be modeled instead by a simpler non-Bayesian alternative. Undergraduates and 9- to 10-year-olds completed a speeded linear position estimation task. Bias in both groups' estimates could be explained in terms of a simple psychophysical model of proportion estimation. Moreover, some individual data were not compatible with the requirements of the more complex Bayesian model.

Connecting numbers to discrete quantification: a step in the child's construction of integer concepts.

The present study asks when young children understand that number words quantify over sets of discrete individuals. For this study, 2- to 4-year-old children were asked to extend the number word five or six either to a cup containing discrete objects (e.g., blocks) or to a cup containing a continuous substance (e.g., water). In Experiment 1, only children who knew the exact meanings of the words one, two and three extended higher number words (five or six) to sets of discrete objects. In Experiment 2, children who only knew the exact meaning of one extended higher number words to discrete objects under the right conditions (i.e., when the problem was first presented with the number words one and two). These results show that children have some understanding that number words pertain to discrete quantification from very early on, but that this knowledge becomes more robust as children learn the exact, cardinal meanings of individual number words.

Developmental change in numerical estimation.

Mental representations of numerical magnitude are commonly thought to undergo discontinuous change over development in the form of a "representational shift." This idea stems from an apparent categorical shift from logarithmic to linear patterns of numerical estimation on tasks that involve translating between numerical magnitudes and spatial positions (such as number-line estimation). However, the observed patterns of performance are broadly consistent with a fundamentally different view, based on psychophysical modeling of proportion estimation, that explains the data without appealing to discontinuous change in mental representations of numerical magnitude. The present study assessed these 2 theories' abilities to account for the development of numerical estimation in 5- through 10-year-olds. The proportional account explained estimation patterns better than the logarithmic-to-linear-shift account for all age groups, at both group and individual levels. These findings contribute to our understanding of the nature and development of the mental representation of number and have more general implications for theories of cognitive developmental change.

Find the picture of eight turtles: a link between children's counting and their knowledge of number word semantics.

An essential part of understanding number words (e.g., eight) is understanding that all number words refer to the dimension of experience we call numerosity. Knowledge of this general principle may be separable from knowledge of individual number word meanings. That is, children may learn the meanings of at least a few individual number words before realizing that all number words refer to numerosity. Alternatively, knowledge of this general principle may form relatively early and proceed to guide and constrain the acquisition of individual number word meanings. The current article describes two experiments in which 116 children (2½- to 4-year-olds) were given a Word Extension task as well as a standard Give-N task. Results show that only children who understood the cardinality principle of counting successfully extended number words from one set to another based on numerosity-with evidence that a developing understanding of this concept emerges as children approach the cardinality principle induction. These findings support the view that children do not use a broad understanding of number words to initially connect number words to numerosity but rather make this connection around the time that they figure out the cardinality principle of counting.