Numerical and nonnumerical estimation in children with and without mathematical learning disabilities

There are currently multiple explanations for mathematical learning disabilities (MLD). The present study focused on those assuming that MLD are due to a basic numerical deficit affecting the ability to represent and to manipulate number magnitude (Butterworth, 1999 Butterworth, B. 1999. The mathematical brain, London, , United Kingdom: Macmillan.  [Google Scholar], 2005 Butterworth, B. 2005. “Developmental dyscalculia”. In Handbook of mathematical cognition, Edited by: Campbell, J. I. D. 455–467. New York, NY: Psychology Press.  [Google Scholar]; A. J. Wilson &; Dehaene, 2007 Wilson, A. J. and Dehaene, S. 2007. “Number sense and developmental dyscalculia”. In Human behavior, learning, and the developing brain: Atypical development, 2nd, Edited by: Coch, D., Dawson, G. and Fischer, K. 212–237. New York, NY: Guilford Press.  [Google Scholar]) and/or to access that number magnitude representation from numerical symbols (Rousselle &; Noël, 2007 Rousselle, L. and Noël, M. P. 2007. Basic numerical skills in children with mathematics learning disabilities: A comparison of symbolic vs non-symbolic number magnitude processing. Cognition, 102(3): 361–395. [Crossref], [PubMed], [Web of Science ®] , [Google Scholar]). The present study provides an original contribution to this issue by testing MLD children (carefully selected on the basis of preserved abilities in other domains) on numerical estimation tasks with contrasting symbolic (Arabic numerals) and nonsymbolic (collection of dots) numbers used as input or output. MLD children performed consistently less accurately than control children on all the estimation tasks. However, MLD children were even weaker when the task involved the mapping between symbolic and nonsymbolic numbers than when the task required a mapping between two nonsymbolic numerical formats. Moreover, in the estimation of nonsymbolic numerosities, MLD children relied more than control children on perceptual cues such as the cumulative area of the dots. Finally, the task requiring a mapping from a nonsymbolic format to a symbolic format was the best predictor of MLD. In order to explain these present results, as well as those reported in the literature, we propose that the impoverished number magnitude representation of MLD children may arise from an initial mapping deficit between number symbols and that magnitude representation.     

Can conceptual congruency effects between number,time, and space be accounted for by polarity correspondence?

Conceptual congruency effects have been interpreted as evidence for the idea that the representations of abstract conceptual dimensions (e.g., power, affective valence, time, number, importance) rest on more concrete dimensions (e.g., space, brightness, weight). However, an alternative theoretical explanation based on the notion of polarity correspondence has recently received empirical support in the domains of valence and morality, which are related to vertical space (e.g., good things are up). In the present study we provide empirical arguments against the applicability of the polarity correspondence account to congruency effects in two conceptual domains related to lateral space: number and time. Following earlier research, we varied the polarity of the response dimension (left-right) by manipulating keyboard eccentricity. In a first experiment we successfully replicated the congruency effect between vertical and lateral space and its interaction with response eccentricity. We then examined whether this modulation of a concrete-concrete congruency effect can be extended to two types of concrete-abstract effects, those between left-right space and number (in both parity and magnitude judgment tasks), and temporal reference. In all three tasks response eccentricity failed to modulate the congruency effects. We conclude that polarity correspondence does not provide an adequate explanation of conceptual congruency effects in the domains of number and time.     

Manipulating number generation: Loud + long = large?

Humans make numerous choices every day and tend to perceive these choices as free. The present study shows how simple free choices are biased by experiencing unrelated auditory information. In two experiments, participants categorized tones according to their intensity on the dimensions volume and duration on the majority of trials. On some trials, however, they were to randomly generate a number, and we found these choices to be influenced by tone intensity. Particularly, if participants were cued toward volume, loud tones clearly biased participants to generate larger numbers. For tone duration, a similar effect only emerged if spatial information was reinforced by the motor context of the task. The findings extend previous findings relating to the ATOM framework (A Theory of Magnitude) by an explicit focus on auditory magnitude processing. As such, they also constrain ATOM by showing that the connections between different magnitude dimensions vary to a considerable degree.     

Malleable conjoint partworths: How the breadth of response scales alters price sensitivity

In one laboratory study and one field study conducted with a large, representative sample of respondents, we show that seemingly innocuous questions that precede a conjoint task, such as demographic and usage-related screening questions can alter the price sensitivities recovered from the main conjoint task. The findings demonstrate that whether these prior questions use broad response categories (i.e., few scale points) or narrow response categories (i.e., many scale points) systematically influences consumers' price sensitivity in a CBC (Choice Based Conjoint) study. We suggest that this may occur because the narrow (vs. broad) response categories in the prior questions lead to consideration of a greater (vs. fewer) number of attributes during the key conjoint task. Since both groups of consumers readily consider the naturally salient price attribute, responding to previous questions with narrow (vs. broad) response categories leads to a greater (vs. fewer) number of non-price attributes being considered, and consequently, decrease the weight afforded to price and reduce price sensitivity.     

Holistic or compositional representation of two-digit numbers? Evidence from the distance, magnitude, and SNARC effects in a number-matching task

Whether two-digit numbers are represented holistically (each digit pair processed as one number) or compositionally (each digit pair processed separately as a decade digit and a unit digit) remains unresolved. Two experiments were conducted to examine the distance, magnitude, and SNARC effects in a number-matching task involving two-digit numbers. Forty undergraduates were asked to judge whether two two-digit numbers (presented serially in Experiment 1 and simultaneously in Experiment 2) were the same or not. Results showed that, when numbers were presented serially, unit digits did not make unique contributions to the magnitude and distance effects, supporting the holistic model. When numbers were presented simultaneously, unit digits made unique contributions, supporting the compositional model. The SNARC (Spatial-Numerical Association of Response Codes) effect was evident for the whole numbers and the decade digits, but not for the unit digits in both experiments, which indicates that two-digit numbers are represented on one mental number line. Taken together, these results suggested that the representation of two-digit numbers is on a single mental number line, but it depends on the stage of processing whether they are processed holistically or compositionally.     

Numeric comparison in a visually-guided manual reaching task

Nearly all studies on perception and cognition have used discrete responses to infer internal cognitive processes. In the current study, we demonstrate that visually-guided manual reaching can provide new opportunities to access internal processes over time. In each trial, participants were required to compare a single digit Arabic number presented on the center square with the standard, 5. Participants were asked to reach and touch one of three squares on the screen with their index finger while their hand movement trajectories were recorded: the left square for 1-4, the center for 5, and the right for 6-9. Direct evidence for an analogue representation of numbers was found in early as well as in later portions of hand trajectories, showing systematic shifts in position for small differences in numerical magnitude.     

Number as a cognitive technology: evidence from Pirahã language and cognition

Does speaking a language without number words change the way speakers of that language perceive exact quantities? The Pirah? are an Amazonian tribe who have been previously studied for their limited numerical system [Gordon, P. (2004). Numerical cognition without words: Evidence from Amazonia. Science 306, 496-499]. We show that the Pirah? have no linguistic method whatsoever for expressing exact quantity, not even "one." Despite this lack, when retested on the matching tasks used by Gordon, Pirah? speakers were able to perform exact matches with large numbers of objects perfectly but, as previously reported, they were inaccurate on matching tasks involving memory. These results suggest that language for exact number is a cultural invention rather than a linguistic universal, and that number words do not change our underlying representations of number but instead are a cognitive technology for keeping track of the cardinality of large sets across time, space, and changes in modality.     

How counting represents number: what children must learn and when they learn it

This study compared 2- to 4-year-olds who understand how counting works (cardinal-principle-knowers) to those who do not (subset-knowers), in order to better characterize the knowledge itself. New results are that (1) Many children answer the question "how many" with the last word used in counting, despite not understanding how counting works; (2) Only children who have mastered the cardinal principle, or are just short of doing so, understand that adding objects to a set means moving forward in the numeral list whereas subtracting objects mean going backward; and finally (3) Only cardinal-principle-knowers understand that adding exactly 1 object to a set means moving forward exactly 1 word in the list, whereas subset-knowers do not understand the unit of change.     

Developing access to number magnitude: a study of the SNARC effect in 7- to 9-year-olds

The SNARC (spatial-numerical association of response codes) effect refers to the finding that small numbers facilitate left responses, whereas larger numbers facilitate right responses. The development of this spatial association was studied in 7-, 8-, and 9-year-olds, as well as in adults, using a task where number magnitude was essential to perform the task and another task where number magnitude was irrelevant. When number magnitude was essential, a SNARC effect was found in all age groups. But when number magnitude was irrelevant, a SNARC effect was found only in 9-year-olds and adults. These results are taken to suggest that (a) 7-year-olds represent number magnitudes in a way similar to that of adults and that (b) when perceiving Arabic numerals, children have developed automatic access to magnitude information by around 9 years of age.