More linear than log? Non-symbolic number-line estimation in 3- to 5-year-old children

The number-line estimation task has become one of the most important methods in numerical cognition research. Originally applied as a direct measure of spatial number representation, it became also informative regarding various other aspects of number processing and associated strategies. However, most of this work and associated conclusions concerns processing numbers in a symbolic format, by school children and older subjects. Symbolic number system is formally taught and trained at school, and its basic mathematical properties (e.g., equidistance, ordinality) can easily be transferred into a spatial format of an oriented number line. This triggers the question on basic characteristics of number line estimation before children get fully familiar with the symbolic number system, i.e., when they mostly rely on approximate system for non-symbolic quantities. In our three studies, we examine therefore how preschool children (3–5-years old) estimate position of non-symbolic quantities on a line, and how this estimation is related to the developing symbolic number knowledge and cultural (left-to-right) directionality. The children were tested with the Give-a-number task, then they performed a computerized number-line task. In Experiment 1, lines bounded with sets of 1 and 20 elements going left-to-right or right-to-left were used. Even in the least numerically competent group, the linear model better fit the estimates than the logarithmic or cyclic power models. The line direction was irrelevant. In Experiment 2, a 1–9 left-to-right oriented line was used. Advantage of linear model was found at group level, and variance of estimates correlated with tested numerosities. In Experiment 3, a position-to-number procedure again revealed the advantage of the linear model, although the strategy of selecting an option more similar to the closer end of the line was prevalent. The precision of estimation increased with the mastery of counting principles in all three experiments. These results contradict the hypothesis of the log-to-linear shift in development of basic numerical representation, rather supporting the linear model with scalar variance. However, the important question remains whether the number-line task captures the nature of the basic numerical representation, or rather the strategies of mapping that representation to an external space.

The Use of Local and Global Ordering Strategies in Number Line Estimation in Early Childhood

A lot of research has been devoted to number line estimation in primary school. However, less is known about the early onset of number line estimation before children enter formal education. We propose that ordering strategies are building blocks of number line estimation in early childhood. In a longitudinal study, children completed a non-symbolic number line estimation task at age 3.5 and 5 years. Two ordering strategies were identified based on the children's estimation patterns: local and global ordering. Local ordering refers to the correct ordering of successive quantities, whereas global ordering refers to the correct ordering of all quantities across the number line. Results indicated a developmental trend for both strategies. The percentage of children applying local and global ordering strategies increased steeply from 3.5 to 5 years of age. Moreover, children used more advanced local and global ordering strategies at 5 years of age. Importantly, level of strategy use was related to more traditional number line estimation outcome measures, such as estimation accuracy and regression fit scores. These results provide evidence that children use dynamic ordering strategies when solving the number line estimation task in early stages of numerical development.

A general number-to-space mapping deficit in developmental dyscalculia

Previous research on developmental dyscalculia (DD) suggested that deficits in the number line estimation task are related to a failure to represent number magnitude linearly. This conclusion was derived from the observation of logarithmically shaped estimation patterns. However, recent research questioned this idea of an isomorphic relationship between estimation patterns and number magnitude representation. In the present study, we evaluated an alternative hypothesis: impairments in the number line estimation task are due to a general deficit in mapping numbers onto space. Adults with DD and a matched control group had to learn linear and non-linear layouts of the number line via feedback. Afterwards, we assessed their performance how well they learnt the new number-space mappings. We found irrespective of the layouts worse performance of adults with DD. Additionally, in case of the linear layout, we observed that their performance did not differ from controls near reference points, but that differences between groups increased as the distance to reference point increased. We conclude that worse performance of adults with DD in the number line task might be due a deficit in mapping numbers onto space which can be partly overcome relying on reference points.

Longitudinal changes in young children's 0-100 to 0-1000 number-line error signatures

We use a latent difference score (LDS) model to examine changes in young children's number-line (NL) error signatures (errors marking numbers on a NL) over 18 months. A LDS model (1) overcomes some of the inference limitations of analytic models used in previous research, and in particular (2) provides a more reliable test of hypotheses about the meaning and significance of changes in NL error signatures over time and task. The NL error signatures of 217 6-year-olds' (on test occasion one) were assessed three times over 18 months, along with their math ability on two occasions. On the first occasion (T1) children completed a 0-100 NL task; on the second (T2) a 0-100 NL and a 0-1000 NL task; on the third (T3) occasion a 0-1000 NL task. On the third and fourth occasions (T3 and T4), children completed mental calculation tasks. Although NL error signatures changed over time, these were predictable from other NL task error signatures, and predicted calculation accuracy at T3, as well as changes in calculation between T3 and T4. Multiple indirect effects (change parameters) showed that associations between initial NL error signatures (0-100 NL) and later mental calculation ability were mediated by error signatures on the 0-1000 NL task. The pattern of findings from the LDS model highlight the value of identifying direct and indirect effects in characterizing changing relationships in cognitive representations over task and time. Substantively, they support the claim that children's NL error signatures generalize over task and time and thus can be used to predict math ability.

Spatial Associations in Numerical Cognition—From Single Digits to Arithmetic

The literature on spatial associations during number processing is dominated by the SNARC (spatial–numerical association of response codes) effect. We describe spatial biases found for single digits and pairs of numbers, first in the “original” speeded parity task and then extending the scope to encompass different tasks, a range of measures, and various populations. Then we review theoretical accounts before surveying the emerging evidence for similar spatial associations during mental arithmetic. We conclude that the mental number line hypothesis and an embodied approach are useful frameworks for further studies.