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 role of general and number-specific order processing in adults’ arithmetic performance

Digit order processing is highly related to individual differences in arithmetic performance. To examine whether serial scanning or associative mechanisms underlie order processing, order tasks (i.e. deciding whether three digits were presented in an order or not) were administered in two experiments. In the first experiment, digits were presented in different directions namely ascending, descending and non-ordered. For each direction, close and far distance sequences were presented. Results revealed reversed distance effects for ordered sequences, but ascending sequences elicited faster performance and stronger reversed distance effects than descending sequences, suggesting that associative mechanisms underlie order processing. In the second experiment, it was examined to which extent the relation between order processing and arithmetic is number-specific by presenting order tasks with digits, letters and months. In all order tasks similar distance effects were observed and similar relations with arithmetic were found, suggesting that both general associative mechanisms and number-specific mechanisms contribute to arithmetic.     

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.     

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.     

How to trigger elaborate processing? A comment on Kunde, Kiesel, and Hoffmann (2003)

Recently, [Kunde, W., Kiesel, A., & Hoffmann, J. (2003). Conscious control over the content of unconscious cognition. Cognition, 88, 223-242] used a masked priming paradigm to argue that neither the 'elaborate processing' or the 'evolving automaticity' view can account for the processing of unconscious numerical stimuli. In our Experiment 1 we replicated [Kunde, W., Kiesel, A., & Hoffmann, J. (2003). Conscious control over the content of unconscious cognition. Cognition, 88, 223-242] Experiment 4 and show that with a less demanding mask than that used by Kunde et al., 'elaborate processing' can explain priming results given that there are side conditions to trigger elaborate processing of unconscious stimuli. The second experiment further explores this influence of the masks by increasing the relevance of the symbols by which the mask is composed. The results show that an increase in relevance of the mask is accompanied by a decrease in the priming effect, though there was no significant change in conscious awareness of the prime.     

Semantic priming in number naming

The issue of semantic and non-semantic conversion routes for numerals is still debated in numerical cognition. We report two number-naming experiments in which the target numerals were preceded by another numeral (prime). The primes and targets could be presented either in arabic (digit) notation or in verbal (alphabetical) notation. The results reveal a semantically related distance effect: Latencies are fastest when the prime has the same value as the target and increase when the distance between prime and target increases. We argue that the present results are congruent with the idea that the numerals make access to an ordered semantic number line common to all notations, as the results are the same for within-notation priming (arabic-arabic or verbal- verbal) and between-notations priming (arabic-verbal or verbal-arabic). The present results also point to a rapid involvement of semantics in the naming of numerals, also when the numerals are words. As such, they are in line with recent claims of rapid semantic mediation in word naming.     

The interplay between nonsymbolic number and its continuous visual properties

To date, researchers investigating nonsymbolic number processes devoted little attention to the visual properties of their stimuli. This is unexpected, as nonsymbolic number is defined by its visual characteristics. When number changes, its visual properties change accordingly. In this study, we investigated the influence of different visual properties on nonsymbolic number processes and show that the current assumptions about the relation between number and its visual characteristics are incorrect. Similar to previous studies, we controlled the visual cues: Each visual cue was not predictive of number. Nevertheless, participants showed congruency effects induced by the visual properties of the stimuli. These congruency effects scaled with the number of visual cues manipulated, implicating that people do not extract number from a visual scene independent of its visual cues. Instead, number judgments are based on the integration of information from multiple visual cues. Consequently, current ways to control the visual cues of the number stimuli are insufficient, as they control only a single variable at the time. And, more important, the existence of an approximate number system that can extract number independent of the visual cues appears unlikely. We therefore propose that number judgment is the result of the weighing of several distinct visual cues. (PsycINFO Database Record (c) 2012 APA, all rights reserved).     

Generating nonsymbolic number stimuli

Studies investigating nonsymbolic numbers (e.g., dot arrays) are confronted with the problem that changes in numerosity are always accompanied by changes in the visual properties of the stimulus. It is therefore debated whether the visual properties of the stimulus rather than number can explain the results obtained in studies investigating nonsymbolic number processing. In this report, we present a program (available at ; note that the program is designed to work with the Psychophysics Toolbox in MATLAB) that exports information about the visual properties of stimuli that co-vary with number (area extended, item size, total surface, density, and circumference). Consequently, insight into the relation between the visual properties of the stimulus and numerical distance can be achieved, and post hoc analyses can be conducted to directly reveal whether numerical distance or (some combinations of) the visual properties of a stimulus could be the most likely candidate underlying the results. Here, we report data that demonstrate the program’s usefulness for research on nonsymbolic number stimuli.     

Are Arabic numerals processed as pictures in a Stroop interference task?

In a picture-word interference task, picture naming is interfered by an incongruent word, but word naming is hardly hindered by the presence of an incongruent picture. In this study, we investigated whether Arabic digits are processed more like pictures or like words. We report two experiments in which Arabic digits and verbal numerals were confronted in a Stroop task. Arabic digit naming is interfered by the presence of an incongruent verbal numeral, while naming the verbal numeral is not influenced by the presence of an incongruent Arabic digit. In a second experiment, we excluded the hypothesis that the results are due to ignoring the Arabic digits: interferences from an incongruent distracter were similar for both notations in a semantic classification task. It seems that an asemantic conversion for Arabic digits is too slow to influence naming times, and that Arabic digit naming, like picture naming, is semantically mediated.     

Children’s representation of symbolic magnitude: The development of the priming distance effect

The comparison distance effect (CDE), whereby discriminating between two numbers that are far apart is easier than discriminating between two numbers that are close, has been considered as an important indicator of how people represent magnitudes internally. However, the underlying mechanism of this CDE is still unclear. We tried to shed further light on how people represent magnitudes by using priming. Adults have been shown to exhibit a priming distance effect (PDE), whereby numbers are processed faster when they are preceded by a close number than when they are preceded by a more distant number. Surprisingly, there are no studies available that have investigated this effect in children. The current study examined this effect in typically developing first, third, and fifth graders and in adults. Our findings revealed that the PDE already occurs in first graders and remains stable across development. This study also documents the usefulness of number priming in children, making it an interesting tool for future research.