Mental representations of arithmetic facts: Evidence from eye movement recordings supports the preferred operand-order-specific representation hypothesis

There are three main hypotheses about mental representations of arithmetic facts: the independent representation hypothesis, the operand-order-free single-representation hypothesis, and the operand-order-specific single-representation hypothesis. The current study used electrical recordings of eye movements to examine the organization of arithmetic facts in long-term memory. Subjects were presented single-digit addition and multiplication problems and were asked to report the solutions. Analyses of the horizontal electrooculograph (HEOG) showed an operand order effect for multiplication in the time windows 150–300 ms (larger negative potentials for smaller operand first problems than for larger operand first ones). The operand order effect was reversed in the time windows from 400 to 1,000 ms (i.e., larger operand first problems had larger negative potentials than smaller operand first problems). For addition, larger operand first problems had larger negative potentials than smaller operand first in the series of time windows from 300 to 1,000 ms, but the effect was smaller than that for multiplication. These results confirmed the dissociated representation of addition and multiplication facts and were consistent with the prediction of the preferred operand-order-specific representation hypothesis.     

The format of representation of recognized words in infants' early receptive lexicon

Eleven-month-olds can recognize a few auditorily presented familiar words in experimental situations where no hints are given by the intonation, the situation, or the presence of possible visual referents. That is, infants of this age (and possibly somewhat younger) can recognize words based on sound patterns alone. The issue addressed in this article is what is the type of mental representations infants use to code words they recognize. The results of a series of experiments with French-learning infants indicate that word representations in 11-month-olds are segmentally underspecified and suggest that they are all the more underspecified when infants engage in recognizing words rather than merely attending to meaningless speech sounds. But underspecification has limits, which were explored here with respect to word-initial consonants. The last two experiments show the way to investigating further these limits for word-initial consonants as well as for segments in other word positions. In French, infants' word representations are flexible enough to allow for structural changes in the voicing or even in the manner of articulation of word-initial consonants. Word-initial consonants must be present, however, for words to be recognized. In conclusion, a parallel is proposed between the emerging capacities to ignore variations that are irrelevant for word recognition in a “lexical mode” and to ignore variations that are phonemically irrelevant in a “neutral mode” of listening to native speech.     

Empirical evaluation of a model of global psychophysical judgments: IV. Forms for the weighting function

Understanding the psychological interpretation of numerals is of both practical and theoretical interest. In classical magnitude estimation, respondents match numerals to sensations and in magnitude production they select sensations that stand in a prescribed numerical ratio to a given standard. The present work focusses on evaluating several possible, and related, forms for the function W formulating the distortion of numerals. The main form, of which a power function is a special case, is the Prelec exponential/power representation. Behavioral equivalents to power and to Prelec functions are formulated, tested, and rejected. It is argued that either the mathematical form or the assumption W(1)=1 is wrong. Whereas, the axiomatic literature has focussed exclusively on the former inference, we explore the alternate that W(1)≠1. Behavioral axioms are formulated in each case and experimentally tested. We conclude that most respondents satisfy a general power function and that those who do not, satisfy the general Prelec function.     

How lexical is the lexicon? Evidence for integrated auditory memory representations

Previous research has shown that lexical representations must include not only linguistic information (what word was said), but also indexical information (how it was said, and by whom). The present work demonstrates that even this expansion is not sufficient. Seemingly irrelevant information, such as an unattended background sound, is retained in memory and can facilitate subsequent speech perception. We presented participants with spoken words paired with environmental sounds (e.g., a phone ringing), and had them make an “animate/inanimate” decision for each word. Later performance identifying filtered versions of the words was impaired to a similar degree if the voice changed or if the environmental sound changed. Moreover, when quite dissimilar words were used at exposure and test, we observed the same result when we reversed the roles of the words and the environmental sounds. The experiments also demonstrated limits to these effects, with no benefit from repetition. Theoretically, our results support two alternative possibilities: (1) Lexical representations are memory representations, and are not walled off from those for other sounds. Indexical effects reflect simply one type of co-occurrence that is incorporated into such representations. (2) The existing literature on indexical effects does not actually bear on lexical representations – voice changes, like environmental sounds heard with a word, produce implicit memory effects that are not tied to the lexicon. We discuss the evidence and implications of these two theoretical alternatives.     

Measuring the approximate number system

Recent theories in numerical cognition propose the existence of an approximate number system (ANS) that supports the representation and processing of quantity information without symbols. It has been claimed that this system is present in infants, children, and adults, that it supports learning of symbolic mathematics, and that correctly harnessing the system during tuition will lead to educational benefits. Various experimental tasks have been used to investigate individuals' ANSs, and it has been assumed that these tasks measure the same system. We tested the relationship across six measures of the ANS. Surprisingly, despite typical performance on each task, adult participants' performances across the tasks were not correlated, and estimates of the acuity of individuals' ANSs from different tasks were unrelated. These results highlight methodological issues with tasks typically used to measure the ANS and call into question claims that individuals use a single system to complete all these tasks.     

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.