Task- and age-dependent effects of visual stimulus properties on children's explicit numerosity judgments.

Researchers investigating numerosity processing manipulate the visual stimulus properties (e.g., surface). This is done to control for the confound between numerosity and its visual properties and should allow the examination of pure number processes. Nevertheless, several studies have shown that, despite different visual controls, visual cues remained to exert their influence on numerosity judgments. This study, therefore, investigated whether the impact of the visual stimulus manipulations on numerosity judgments is dependent on the task at hand (comparison task vs. same-different task) and whether this impact changes throughout development. In addition, we examined whether the influence of visual stimulus manipulations on numerosity judgments plays a role in the relation between performance on numerosity tasks and mathematics achievement. Our findings confirmed that the visual stimulus manipulations affect numerosity judgments; more important, we found that these influences changed with increasing age and differed between the comparison and the same-different tasks. Consequently, direct comparisons between numerosity studies using different tasks and age groups are difficult. No meaningful relationship between the performance on the comparison and same-different tasks and mathematics achievement was found in typically developing children, nor did we find consistent differences between children with and without mathematical learning disability (MLD).

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.

Children's representation of symbolic and nonsymbolic magnitude examined with the priming paradigm.

How people process and represent magnitude has often been studied using number comparison tasks. From the results of these tasks, a comparison distance effect (CDE) is generated, showing that it is easier to discriminate two numbers that are numerically further apart (e.g., 2 and 8) compared with numerically closer numbers (e.g., 6 and 8). However, it has been suggested that the CDE reflects decisional processes rather than magnitude representation. In this study, therefore, we investigated the development of symbolic and nonsymbolic number processes in kindergartners and first, second, and sixth graders using the priming paradigm. This task has been shown to measure magnitude and not decisional processes. Our findings revealed that a priming distance effect (PDE) is already present in kindergartners and that it remains stable across development. This suggests that formal schooling does not affect magnitude representation. No differences were found between the symbolic and nonsymbolic PDE, indicating that both notations are processed with comparable precision. Finally, a poorer performance on a standardized mathematics test seemed to be associated with a smaller PDE for both notations, possibly suggesting that children with lower mathematics scores have a less precise coding of magnitude. This supports the defective number module hypothesis, which assumes an impairment of number sense.

The approximate number system is not predictive for symbolic number processing in kindergarteners.

The relation between the approximate number system (ANS) and symbolic number processing skills remains unclear. Some theories assume that children acquire the numerical meaning of symbols by mapping them onto the preexisting ANS. Others suggest that in addition to the ANS, children also develop a separate, exact representational system for symbolic number processing. In the current study, we contribute to this debate by investigating whether the nonsymbolic number processing of kindergarteners is predictive for symbolic number processing. Results revealed no association between the accuracy of the kindergarteners on a nonsymbolic number comparison task and their performance on the symbolic comparison task six months later, suggesting that there are two distinct representational systems for the ANS and numerical symbols.

Concurrent validity of approximate number sense tasks in adults and children.

Reasoning with non-symbolic numerosities is suggested to be rooted in the Approximate Number System (ANS) and evidence pointing to a relationship between the acuity of this system and mathematics is available. In order to use the acuity of this ANS as a screening instrument to detect future math problems, it is important to model ANS acuity over development. However, whether ANS acuity and its development have been described accurately can be questioned. Namely, different tasks were used to examine the developmental trajectory of ANS acuity and studies comparing performances on these different tasks are scarce. In the present study, we examined whether different tasks designed to measure the acuity of the ANS are comparable and lead to related ANS acuity measures (i.e., the concurrent validity of these tasks). We contrasted the change detection task, which is used in infants, with tasks that are more commonly used in older children and adults (i.e., comparison and same-different tasks). Together, our results suggest that ANS acuity measures obtained with different tasks are not related. This poses serious problems for the comparison of ANS acuity measures derived from different tasks and thus for the establishment of the developmental trajectory of ANS acuity.

Automatic number priming effects in adults with and without mathematical learning disabilities.

This study examined automatic number processing in adults with mathematical learning disabilities (MLDs). The performance of adults with MLD during an automatic symbolic and non-symbolic priming task was compared to gender-, age-, and IQ-matched controls. No difference in the priming distance effect was found between the adults with and without MLD, suggesting that adults with MLD have an intact magnitude representation. Moreover, the adults with MLD did not have problems in processing the numerical symbols 1-9, suggesting that this basic deficit which is experienced by children with MLD is resolved by adulthood.

Numerical matching judgments in children with mathematical learning disabilities.

Both deficits in the innate magnitude representation (i.e. representation deficit hypothesis) and deficits in accessing the magnitude representation from symbols (i.e. access deficit hypotheses) have been proposed to explain mathematical learning disabilities (MLD). Evidence for these hypotheses has mainly been accumulated through the use of numerical magnitude comparison tasks. It has been argued that the comparison distance effect might reflect decision processes on activated magnitude representations rather than number processing per se. One way to avoid such decisional processes confounding the numerical distance effect is by using a numerical matching task, in which children have to indicate whether two dot-arrays or a dot-array and a digit are numerically the same or different. Against this background, we used a numerical matching task to examined the representation deficit and access deficit hypotheses in a group children with MLD and controls matched on age, gender and IQ. The results revealed that children with MLD were slower than controls on the mixed notation trials, whereas no difference was found for the non-symbolic trials. This might be in line with the access deficit hypothesis, showing that children with MLD have difficulties in linking a symbol with its quantity representation. However, further investigation is required to exclude the possibility that children with MLD have a deficit in integrating the information from different input notations.

What can the same-different task tell us about the development of magnitude representations?

We examined the development of magnitude representations in children (Exp 1: kindergartners, first-, second- and sixth graders, Exp 2: kindergartners, first-, second- and third graders) using a numerical same-different task with symbolic (i.e. digits) and non-symbolic (i.e. arrays of dots) stimuli. We investigated whether judgments in a same-different task with digits are based upon the numerical value or upon the physical similarity of the digits. In addition, we investigated whether the numerical distance effect decreases with increasing age. Finally, we examined whether the performance in this task is related to general mathematics achievement. Our results reveal that a same-different task with digits is not an appropriate task to study magnitude representations, because already late kindergarteners base their responses on the physical similarity instead of the numerical value of the digits. When decisions cannot be made on the basis of physical similarity, a similar numerical distance effect is present over all age groups. This suggests that the magnitude representation is stable from late kindergarten onwards. The size of the numerical distance effect was not related to mathematical achievement. However, children with a poorer mathematics achievement score seemed to have more difficulties to link a symbol with its corresponding magnitude.

The reliability of and the relation between non-symbolic numerical distance effects in comparison, same-different judgments and priming.

The development of number processing is generally studied by examining the performance on basic number tasks (comparison task, same-different judgment, and priming task). Using these tasks, so-called numerical distance effects are obtained. All these effects are generally explained by assuming a magnitude representation related to a mental number line: magnitudes are represented from left to right with partially overlapping representations for nearby numbers. In this study, we compared the performance of adults on these different tasks using non-symbolic stimuli. First, we investigated whether the effects obtained in these behavioral tasks are reliable. Second, we examined the relation between the three different effects. The results showed that the observed effects in the case of the comparison task and the same-different task proved to be reliable. The numerical distance effect obtained in the priming task, however, was not reliable. In addition, a correlation was found between the distance effects in the comparison task and the same-different task. The priming distance effect did not correlate with the other two effects. These results suggest important differences between distance effects obtained under automatic and intentional task instructions regarding the use of them as indices of mathematical ability.