Making sense of the relation between number sense and math

While several studies have shown that the performance on numerosity comparison tasks is related to individual differences in math abilities, others have failed to find such a link. These inconsistencies could be due to variations in which math was assessed, different stimulus generation protocols for the numerosity comparison task, or differences in inhibitory control. This within-subject study is a conceptual replication tapping into the relation between numerosity comparison, math, and inhibition in adults (N = 122). Three aspects of math ability were measured using standardized assessments: Arithmetic fluency, calculation, and applied problem solving skills. Participants’ inhibitory skills were measured using Stroop and Go/No-Go tasks with numerical and non-numerical stimuli. Finally, non-symbolic number sense was measured using two different versions of a numerosity comparison task that differed in the stimulus generation protocols (Panamath; Halberda, Mazzocco & Feigenson, 2008, https://doi.org/10.1038/nature07246; G&R, Gebuis & Reynvoet, 2011, https://doi.org/10.3758/s13428-011-0097-5). We find that performance on the Panamath task, but not the G&R task, related to measures of calculation and applied problem solving but not arithmetic fluency, even when controlling for inhibitory control. One possible explanation is that depending on the characteristics of the stimuli in the numerosity comparison task, the reliance on numerical and non-numerical information may vary and only when performance relies more on numerical representations, a relation with math achievement is found. Our findings help to explain prior mixed findings regarding the link between non-symbolic number sense and math and highlight the need to carefully consider variations in numerosity comparison tasks and math measures.

Counting and the ontogenetic origins of exact equality

Humans are unique in their capacity to both represent number exactly and to express these representations symbolically. This correlation has prompted debate regarding whether symbolic number systems are necessary to represent exact number. Previous work addressing this question in innumerate adults and semi-numerate children has been limited by conflicting results and differing methodologies, and has not yielded a clear answer. We address this debate by adapting methods used with innumerate populations (a “set-matching” task) for 3- to 5-year-old US children at varying stages of symbolic number acquisition. In five studies we find that children’s ability to match sets exactly is related not simply to knowing the meanings of a few number words, but also to understanding how counting is used to generate sets (i.e., the cardinal principle). However, while children were more likely to match sets after acquiring the cardinal principle, they nevertheless demonstrated failures, compatible with the hypothesis that the ability to reason about exact equality emerges sometime later. These findings provide important data on the origin of exact number concepts, and point to knowledge of a counting system, rather than number language in general, as a key ingredient in the ability to reason about large exact number.

Disentangling the Effects of SFON (Spontaneous Focusing on Numerosity) and Symbolic Number Skills on the Mathematical Achievement of First Graders. A Longitudinal Study

Research has established that Spontaneous Focusing on Numerosity (SFON) and symbolic number skills (e.g., counting out loud, counting objects, linking small magnitudes and numbers) are predictors of mathematical achievement in primary school. However, little is known about the relationship between SFON and symbolic number skills, or whether one of these factors is more influential on a child’s subsequent mathematical achievement. This study investigated the effect of SFON and symbolic number skills on mathematical achievement at the end of Grade 1 by controlling for first language, gender, working memory and nonverbal IQ. Participants were N = 1,279 first graders. SFON, symbolic number skills and control variables (first language, gender, working memory, and nonverbal IQ) were measured at the beginning of Grade 1. SFON was assessed with a verbally-based task. Data on mathematical achievement was collected at the end of Grade 1. Descriptive statistics demonstrated that the children’s SFON was relatively low at the beginning of Grade 1. Structural equation modeling was used to examine the relationship between SFON, symbolic number skills and mathematical achievement at the end of Grade 1. The results revealed a weakly significant correlation between SFON and symbolic number skills. SFON and symbolic number skills were significant predictors of mathematical achievement at the end of Grade 1. However, the effect of symbolic number skills on mathematical achievement was greater than the effect of SFON. It is therefore concluded that numerical skills are more important than SFON for predicting mathematical achievement over the course of first grade.

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.

The Influence of the Place Value System on Symbolic Number Perception

Past research on numerical cognition has suggested that both symbolic and non-symbolic numbers are mapped onto the same compressed mental analogue representation. However, experiments using magnitude estimation tasks show logarithmic compression of symbolic numbers while the compression of non-symbolic numbers has a power-function shape. This warrants closer inspection of what differentiates the two processes. In this study, we hypothesized that estimates of symbolic numbers are systematically shaped by the format in which they are represented, namely, the place value system. To investigate this, we tested adults (N = 188) on a magnitude estimation task with unfamiliar base-26 and base-5 scales. Results reveal that adults showed systematic, logarithmic-looking underestimation on both scales, indicating that the place value system itself can cause the pattern. Additionally, the observed shape of participants’ estimates on both scales could be well explained with a simple model that assumed insufficient understanding of exponential growth (i.e., a characteristic of place value systems). Taken together, our results suggest that the discrepancy between symbolic and non-symbolic number compression can be explained by taking the effect of the place value system into account.