Michal Pinhas

Comparing two-digit numbers: The importance of being presented together

The effect of presentation mode on magnitude comparisons of two-digit (2D) numbers was examined using the stimuli set developed by Nuerk, Weger, and Willmes (2001). In Experiment 1, only number pairs from difference decades were presented either simultaneously or sequentially. In the former case there was evidence for the parallel processing of both the units and decades digits and for a components representation, consistent with previous findings. In contrast, in the latter case there was evidence for the processing of mainly the decades digits. In Experiment 2, within-decade number pairs were added to make both digits task relevant. The results from the simultaneous condition were again consistent with a components representation, while results from the sequential presentation were in line with a holistic representation, in line with Zhang and Wangs (2005) research. Results therefore suggest that the processing of 2D numbers depends on the way they are presented.

Heed the signs: Operation signs have spatial associations

Mental arithmetic shows systematic spatial biases. The association between numbers and space is well documented, but it is unknown whether arithmetic operation signs also have spatial associations and whether or not they contribute to spatial biases found in arithmetic. Adult participants classified plus and minus signs with left and right button presses under two counterbalanced response rules. Results from two experiments showed that spatially congruent responses (i.e., right-side responses for the plus sign and left-side responses for the minus sign) were responded to faster than spatially incongruent ones (i.e., left-side responses for the plus sign and right-side responses for the minus sign). We also report correlations between this novel operation sign spatial association (OSSA) effect and other spatial biases in number processing. In a control experiment with no explicit processing requirements for the operation signs there were no sign-related spatial biases. Overall, the results suggest that (a) arithmetic operation signs can evoke spatial associations (OSSA), (b) experience with arithmetic operations probably underlies the OSSA, and (c) the OSSA only partially contributes to spatial biases in arithmetic.

Estimating linear effects in ANOVA designs: The easy way

Research in cognitive science has documented numerous phenomena that are approximated by linear relationships. In the domain of numerical cognition, the use of linear regression for estimating linear effects (e.g., distance and SNARC effects) became common following Fias, Brysbaert, Geypens, and d’Ydewalle’s (1996) study on the SNARC effect. While their work has become the model for analyzing linear effects in the field, it requires statistical analysis of individual participants and does not provide measures of the proportions of variability accounted for (cf. Lorch & Myers, 1990). In the present methodological note, using both the distance and SNARC effects as examples, we demonstrate how linear effects can be estimated in a simple way within the framework of repeated measures analysis of variance. This method allows for estimating effect sizes in terms of both slope and proportions of variability accounted for. Finally, we show that our method can easily be extended to estimate linear interaction effects, not just linear effects calculated as main effects.

Holistic representation of negative numbers is formed when needed for the task

Past research suggested that negative numbers are represented in terms of their components—the polarity marker and the number (e.g., Fischer & Rottmann, 2005; Ganor-Stern & Tzelgov, 2008). The present study shows that a holistic representation is formed when needed for the task requirement. Specifically, performing the numerical comparison task on positive and negative numbers presented sequentially required participants to hold both the polarity and the number magnitude in memory. Such a condition resulted in a holistic representation of negative numbers, as indicated by the distance and semantic congruity effects. This holistic representation was added to the initial components representation, thus producing a hybrid holistic-components representation.

Expanding on the Mental Number Line: Zero Is Perceived as the "Smallest"

The representation of 0 in healthy adults was studied with the physical comparison task. Automatic processing of numbers, as indicated by the size congruity effect, was used for detecting the basic numerical representations stored in long-term memory. The size congruity effect usually increases with numerical distance between the physically compared numbers. This increase was attenuated for comparisons to 0 or 1 (but not to 2) when they were perceived as the smallest number in the set. Furthermore, the size congruity effect was enlarged in these cases. These results indicate an end effect in automatic processing of numbers and suggest that 0, or 1 in the absence of 0, is perceived as the smallest entity on the mental number line. The implications of these findings are discussed with regard to models of number representation.

Zooming in and out from the mental number line: Evidence for a number range effect

The representation of numbers is commonly viewed as an ordered continuum of magnitudes, referred to as the mental number line. Previous work has repeatedly shown that number representations evoked by a given task can be easily altered, yielding an ongoing discussion about the basic properties of the mental number line and how malleable they are. Here we studied whether the resolution of the mental number line is fixed or depends on the relative magnitudes that are being processed. In 2 experiments, participants compared the same number pairs under 2 conditions that differed in terms of the overall range of numbers present. We report a novel number range effect, such that comparisons of the same number pairs were responded to faster under the smaller versus larger number range. This finding is consistent with the idea that the resolution of the mental number line can be adjusted, as if a unit difference is perceived as larger in smaller ranges.

Addition goes where the big numbers are: evidence for a reversed operational momentum effect.

Number processing evokes spatial biases, both when dealing with single digits and in more complex mental calculations. Here we investigated whether these two biases have a common origin, by examining their flexibility. Participants pointed to the locations of arithmetic results on a visually presented line with an inverted, right-to-left number arrangement. We found directionally opposite spatial biases for mental arithmetic and for a parity task administered both before and after the arithmetic task. We discuss implications of this dissociation in our results for the task-dependent cognitive representation of numbers.

Exploring the mental number line via the size congruity effect.

To address the ongoing debate about the origins of the size effect (faster comparison time for smaller than larger numbers, given a fixed intrapair distance), an indication of automatic number processing was searched for. Participants performed a quantity comparison task in which they had to decide which of two sketched cups contained more liquid, while ignoring the number superimposed on each cup. In the congruent condition, the larger number appeared on the cup containing more liquid, while in the incongruent condition the larger number appeared on the cup containing less liquid. The size effect was found in a numerical comparison task, while in the quantity comparison task the size congruity effect decreased as the magnitude of the irrelevant numbers increased. Together, these results suggest that the size effect reflects a basic feature of the mental number line.

Heuristics and biases in mental arithmetic: revisiting and reversing operational momentum

ABSTRACT Mental arithmetic is characterised by a tendency to overestimate addition and to underestimate subtraction results: the operational momentum (OM) effect. Here, motivated by contentious explanations of this effect, we developed and tested an arithmetic heuristics and biases model that predicts reverse OM due to cognitive anchoring effects. Participants produced bi-directional lines with lengths corresponding to the results of arithmetic problems. In two experiments, we found regular OM with zero problems (e.g., 3+0, 3−0) but reverse OM with non-zero problems (e.g., 2+1, 4−1). In a third experiment, we tested the prediction of our model. Our results suggest the presence of at least three competing biases in mental arithmetic: a more-or-less heuristic, a sign-space association and an anchoring bias. We conclude that mental arithmetic exhibits shortcuts for decision-making similar to traditional domains of reasoning and problem-solving.

The Neural Signatures of Processing Semantic End Values in Automatic Number Comparisons

The brain activity associated with processing numerical end values has received limited research attention. The present study explored the neural correlates associated with processing semantic end values under conditions of automatic number processing. Event-related potentials (ERPs) were recorded while participants performed the numerical Stroop task, in which they were asked to compare the physical size of pairs of numbers, while ignoring their numerical values. The smallest end value in the set, which is a task irrelevant factor, was manipulated between participant groups. We focused on the processing of the lower end values of 0 and 1 because these numbers were found to be automatically tagged as the “smallest.” Behavioral results showed that the size congruity effect was modulated by the presence of the smallest end value in the pair. ERP data revealed a spatially extended centro-parieto-occipital P3 that was enhanced for congruent versus incongruent trials. Importantly, over centro-parietal sites, the P3 congruity effect (congruent minus incongruent) was larger for pairs containing the smallest end value than for pairs containing non-smallest values. These differences in the congruency effect were localized to the precuneus. The presence of an end value within the pair also modulated P3 latency. Our results provide the first neural evidence for the encoding of numerical end values. They further demonstrate that the use of end values as anchors is a primary aspect of processing symbolic numerical information.