Development of numerical estimation in Chinese preschool children

The development of number line estimation in children at risk of mathematics learning difficulties: A longitudinal study

Children with mathematics learning difficulties (MLD) show poorer performance on the number line task, but how performance on this task relates to other mathematical skills is unclear. This study examined the association between performance on the number line task and mathematical skills during the first 2 years of school for children at risk of MLD. Children (N = 100; Mage = 83.63 months) were assessed on four occasions on the number line task and other mathematical skills (math fluency, numerical operations, and mathematical reasoning). Estimation patterns were analyzed based on the representational shift and proportional judgment accounts separately. More consistent longitudinal trends and stronger evidence for differences in mathematical skills based on estimation patterns were found within the representational shift account. Latent growth curve models showed accuracy on the number line task as a predictor of growth in some mathematical skills assessed. We discuss impacts of methodological limitations on the study of estimation patterns.

Development of number line estsimation in Chinese preschoolers: a comparison between numerical and non-numerical symbols

To examine the level of number line estimation (NLE) in Chinese children with respect to representations of both numerical (Arabic numerals) and non-numerical symbols (dots), a total of 192 Chinese preschoolers aged between 4 and 5 years participated in four different NLE tasks. These tasks were paired to evaluate the accuracy and patterns of children's estimations in both numerical and non-numerical symbol contexts. Our findings indicate that, for Chinese preschoolers, relatively precise numerical symbol representations begin to emerge as early as 4 years of age. The accuracy of number line estimates for both 4- and 5-year-old children gradually increases in tasks involving both numerical and non-numerical symbols. Additionally, the development and patterns observed in the number line estimates of 4- and 5-year-old Chinese preschoolers are similar in both numerical symbol and non-numerical symbol tasks. These results indicate that the initiation of relatively precise numerical symbol representation and the turning point in the developmental trajectory, where the relatively precise representation for numerical symbols surpasses that of non-numerical ones, occur earlier in Chinese children than in their Western counterparts.

Longitudinal relationship between number line estimation and other mathematical abilities in Chinese preschool children

Many cross-sectional studies have shown that number line estimation is associated with other mathematical skills, but there has been limited longitudinal research. To systematically examine such associations longitudinally at the earliest stage of mathematical learning, the current study tested 40 Chinese preschoolers (mean age = 4.97 years, SD = 0.18) and followed them up 8 months later. For both waves of data collection, children were administered six tasks: number line estimation, dot counting, comparison of two dot arrays, comparison of triple dot arrays, symbolic number comparison, and simple addition. Results of two-wave cross-lagged panel analysis showed that (a) dot counting and non-symbolic numerical comparison at Time 1 had significant longitudinal associations with number line estimation at Time 2, (b) number line estimation had bidirectional associations with symbolic number comparison, and (c) number line estimation at Time 1 had a marginally significant longitudinal association with simple addition at Time 2. These results extend the small but accumulating literature on the longitudinal relations between number line estimation and other mathematical skills and specify the important role of number line estimation in the early development of mathematical skills.

Cognitive mediators of US-China differences in early symbolic arithmetic

Chinese children routinely outperform American peers in standardized tests of mathematics knowledge. To examine mediators of this effect, 95 Chinese and US 5-year-olds completed a test of overall symbolic arithmetic, an IQ subtest, and three tests each of symbolic and non-symbolic numerical magnitude knowledge (magnitude comparison, approximate addition, and number-line estimation). Overall Chinese children performed better in symbolic arithmetic than US children, and all measures of IQ and number knowledge predicted overall symbolic arithmetic. Chinese children were more accurate than US peers in symbolic numerical magnitude comparison, symbolic approximate addition, and both symbolic and non-symbolic number-line estimation; Chinese and U.S. children did not differ in IQ and non-symbolic magnitude comparison and approximate addition. A substantial amount of the nationality difference in overall symbolic arithmetic was mediated by performance on the symbolic and number-line tests.

Number line development of Chilean children from preschool to the end of kindergarten

Children's performance on number line tasks reflects their developing number system knowledge. Before 5 years of age, most children perform poorly on even the simplest number lines (i.e., 0-10). Our goal was to understand how number line skills develop before formal schooling. Chilean preschoolers attempted a 0-10 number line task three times over 2 years: at the beginning of pre-kindergarten (M = 4:7 [years;months]; Time 1), at the end of pre-kindergarten (M = 5:0; Time 2), and at the end of kindergarten (M = 5:10; Time 3). We used latent class analysis to group children according to their patterns of performance across number targets. At Time 1, 86% of children had error patterns indicating that they randomly placed estimates on the line. At Time 2, 56% of children continued to respond randomly. At Time 3, 56% of children showed competent performance across the number line, 23% were accurate only near the endpoints, and 21% were accurate only for low target numbers near the origin. Latent transition analyses showed that the transition from less to more proficient estimation classes was predicted by children's number identification skills. Thus, number line performance changed dramatically from 4 to 6 years of age as children began to develop the cognitive and numerical skills necessary to accurately estimate numbers on a number line.

The Use of a Novel Term Helps Preschoolers Learn the Concept of Angle: An Intervention Study With Chinese Preschool Children

Angle is an important concept in geometry. Young children have difficulty separating angle size from other dimensions such as the length of angle sides, perhaps due to whole-object bias in word learning. The present study used the pre-test-training-post-test design to investigate the effectiveness of two ways of separating angle from angle size in 3-6-year-old Chinese preschoolers. A total of 228 children were given a pre-test and 219 of them failed the crucial test. 168 of the 219 children were present at school during the training phase and were randomly assigned to three groups: the "toma" group (n = 57), which received training to call the whole angle figure as "toma" and angle size as angle size; the "angle/angle size" group (n = 56), which received the training of separating "angle" from "angle size"; and the control group (n = 55), which used "angle size" alone to represent both the overall angle figure and angle size. Results showed that the "toma" group improved significantly more than the other two groups, the latter of which did not differ from each other. These results suggest that it is insufficient to have two separate words/phrases (angle and angle size) for children to learn to differentiate angle from angle size, perhaps due to their shared usage of the word angle. Instead, the use of a novel term is necessary and sufficient to improve learning. Implications for preschool education are discussed.

Gender differences in the development of semantic and spatial processing of numbers

This study recruited kindergarteners and first graders to investigate gender and grade differences in semantic and spatial processing of number magnitude. Results based on the Bayesian statistics showed that (1) there was extreme evidence in favour of grade differences in both semantic processing and spatial processing; (2) there were no gender differences in semantic processing; and (3) boys developed earlier than girls in spatial processing of numbers, especially for the more difficult task. These results are discussed in terms of gender differences in cognitive mechanisms underlying semantic and spatial processing of number magnitude.

Development of number-space associations: SNARC effects and spatial attention in 7- to 11-year-olds

The spatial numerical association of response codes effect, referred to as the SNARC effect, reveals that small numbers elicit faster left than right responses, and conversely, large numbers elicit faster right responses. Here, we explored the development of this number-space association by assessing how 7-, 9-, 11-year-olds, and adults differed in spatial orienting of attention on Posner' paradigm. Compared with the previous research, we examined how the cues would affect the level and strength of the SNARC effect in children under the different attentional conditions. Subjects made parity decisions for endogenous attention (Experiment 1) and exogenous attention (Experiment 2). The results showed that adults displayed the SNARC effect in both experiments, relatively speaking, 7- to 11-year-old Chinese children's ability of spatial numerical association progressed gradually. With endogenous attention, the SNARC effect appeared in all age groups except for 7-year-olds for invalid cues. Compared with the endogenous attention condition, the SNARC effect was more significantly affected by cues in the exogenous attention condition. This result might be owing to the fact that the SNARC effect was not demonstrated in 7-year-olds with either valid or invalid cues. Our results suggest that the differences in the spatial orienting of attention are based on the cognitive load associated with processing number information and that this process can be affected by cues. Further, there may be cross-cultural influences on the SNARC effect, as early family training may explain the results seen in this sample of Chinese 7-year-olds. Thus, reaction times decreased with increasing age in the parity judgment task, and reaction times for valid cues were faster than for invalid cues regardless of the age group in both experiments. The SNARC effect was only present for 7-year-olds for valid cues, for endogenous attention.

Benchmark-based strategies in whole number line estimation

In this study, we used verbal protocols to identify whether adults spontaneously apply quartile-based strategies or whether they need additional external support to use these strategies when solving a 0-1,000 number line estimation (NLE) task. Participants were assigned to one of three conditions based on the number of external benchmarks provided on the number line. In the bounded condition only the origin and endpoint were indicated, the mid-point condition included an additional external benchmark at 50%, and in the quartile condition three additional external benchmarks at 25%, 50%, and 75% were specified. Firstly, participants in the bounded condition reported to spontaneously apply quartile-based strategies to calibrate their estimates. Moreover, participants frequently relied on the external benchmarks for creating internal benchmarks at the mid-point, quartiles, and even octiles of the number line. Secondly, overall estimation accuracy improved as the number of external benchmarks increased, and target numbers close to external benchmarks were estimated more accurately and with less variability. Thirdly, the use of a larger variety in benchmark-based strategies was positively related to NLE accuracy. In summary, this study provides evidence that the NLE task induces more sophisticated strategy use in participants than initially anticipated.

The Development of Symbolic and Non-Symbolic Number Line Estimations: Three Developmental Accounts Contrasted Within Cross-Sectional and Longitudinal Data

Three theoretical accounts have been put forward for the development of children's response patterns on number line estimation tasks: the log-to-linear representational shift, the two-linear-to-linear transformation and the proportion judgment account. These three accounts have not been contrasted, however, within one study, using one single criterion to determine which model provides the best fit. The present study contrasted these three accounts by examining first, second and sixth graders with a symbolic and non-symbolic number line estimation task (Experiment 1). In addition, first and second graders were tested again one year later (Experiment 2). In case of symbolic estimations, the proportion judgment account described the data best. Most young children's non-symbolic estimation patterns were best described by a logarithmic model (within the log-to-lin account), whereas those of most older children were best described by the simple power model (within the proportion judgment account).