Math in actions: Actor mode reveals the true arithmetic abilities of French-speaking 2-year-olds in a magic task

Our previous studies provide some evidence of between-language effects on arithmetic performance in 2-year-olds. French-speaking children were especially biased by the use of the word un as a cardinal value and as an article in the singular/plural opposition (1 vs. the set 2, 3, …). Here we evaluated the ability of a new action-based assessment method to avoid this bias. A total of 80 French-speaking 2- and 3-year-olds were confronted with impossible (1 + 1 = 1 or 1 + 1 = 3) and possible (1 + 1 = 2) addition problems that triggered the bias. The problems were either presented to the children by the experimenter (onlooker mode) or realized by themselves (actor mode). The 2-year-olds performed better in the actor mode than in the onlooker mode. A subtraction control with no language ambiguity (2-1 = 2 or 1) was conducted with 80 other children; both modes elicited comparable performances regardless of age. These data indicate that the actor mode is effective for assessing arithmetic ability in French-speaking 2-year-olds.     

Operational momentum in large-number addition and subtraction by 9-month-olds

Recent studies on nonsymbolic arithmetic have illustrated that under conditions that prevent exact calculation, adults display a systematic tendency to overestimate the answers to addition problems and underestimate the answers to subtraction problems. It has been suggested that this operational momentum results from exposure to a culture-specific practice of representing numbers spatially; alternatively, the mind may represent numbers in spatial terms from early in development. In the current study, we asked whether operational momentum is present during infancy, prior to exposure to culture-specific representations of numbers. Infants (9-month-olds) were shown videos of events involving the addition or subtraction of objects with three different types of outcomes: numerically correct, too large, and too small. Infants looked significantly longer only at those incorrect outcomes that violated the momentum of the arithmetic operation (i.e., at too-large outcomes in subtraction events and too-small outcomes in addition events). The presence of operational momentum during infancy indicates developmental continuity in the underlying mechanisms used when operating over numerical representations.     

Tactile Enumeration and Embodied Numerosity Among the Deaf

Representations of the fingers are embodied in our cognition and influence performance in enumeration tasks. Among deaf signers, the fingers also serve as a tool for communication in sign language. Previous studies in normal hearing (NH) participants showed effects of embodiment (i.e., embodied numerosity) on tactile enumeration using the fingers of one hand. In this research, we examined the influence of extensive visuo-manual use on tactile enumeration among the deaf. We carried out four enumeration task experiments, using 1–5 stimuli, on a profoundly deaf group (n = 16) and a matching NH group (n = 15): (a) tactile enumeration using one hand, (b) tactile enumeration using two hands, (c) visual enumeration of finger signs, and (d) visual enumeration of dots. In the tactile tasks, we found salient embodied effects in the deaf group compared to the NH group. In the visual enumeration of finger signs task, we controlled the meanings of the stimuli presentation type (e.g., finger-counting habit, fingerspelled letters, both or neither). Interestingly, when comparing fingerspelled letters to neutrals (i.e., not letters or numerical finger-counting signs), an inhibition pattern was observed among the deaf. The findings uncover the influence of rich visuo-manual experiences and language on embodied representations. In addition, we propose that these influences can partially account for the lag in mathematical competencies in the deaf compared to NH peers. Lastly, we further discuss how our findings support a contemporary model for mental numerical representations and finger-counting habits.     

多词素词的通达表征:分解还是整体

有关通达表征的结构主要有三种观点词素、整词和混合的观点.分解存储的通达表征认为,在通达表征层次中只有不可再分的词素,而没有彼此独立的词条.也就是说,词语是以词素分解形式存储在通达表征中的.整词存储的通达表征认为,在通达表征中存储的都是整词,每一个词都有其独立的词条.混合的通达表征认为词素和整词都有可能是通达表征中的单元.它具有更大的灵活性.综合考虑已有的文献和进一步的理解思考,作者提出影响词语通达表征的两个因素语义透明度和词频.     

Disentangling the Mechanisms of Symbolic Number Processing in Adults’ Mathematics and Arithmetic Achievement

A growing body of research has shown that symbolic number processing relates to individual differences in mathematics. However, it remains unclear which mechanisms of symbolic number processing are crucial—accessing underlying magnitude representation of symbols (i.e., symbol‐magnitude associations), processing relative order of symbols (i.e., symbol‐symbol associations), or processing of symbols per se. To address this question, in this study adult participants performed a dots‐number word matching task—thought to be a measure of symbol‐magnitude associations (numerical magnitude processing)—a numeral‐ordering task that focuses on symbol‐symbol associations (numerical order processing), and a digit‐number word matching task targeting symbolic processing per se. Results showed that both numerical magnitude and order processing were uniquely related to arithmetic achievement, beyond the effects of domain‐general factors (intellectual ability, working memory, inhibitory control, and non‐numerical ordering). Importantly, results were different when a general measure of mathematics achievement was considered. Those mechanisms of symbolic number processing did not contribute to math achievement. Furthermore, a path analysis revealed that numerical magnitude and order processing might draw on a common mechanism. Each process explained a portion of the relation of the other with arithmetic (but not with a general measure of math achievement). These findings are consistent with the notion that adults’ arithmetic skills build upon symbol‐magnitude associations, and they highlight the effects that different math measures have in the study of numerical cognition.     

Comma N' cents in pricing: The effects of auditory representation encoding on price magnitude perceptions

Numbers and prices can be processed and encoded in three different forms: 1) visual [based on their written form in Arabic numerals (e.g., 72)], 2) verbal [based on spoken word-sounds (e.g., “seventy” and “two”), and 3) analog (based on judgments of relative “size” or amount (e.g., more than 70 but less than 80)]. In this paper, we demonstrate that including commas (e.g., $1599 vs. $1599) and cents (e.g., $1599.85 vs. $1599) in a price's Arabic written form (i.e., how it is perceived visually) can change how the price is encoded and represented verbally in a consumer's memory. In turn, the verbal encoding of a written price can influence assessments of the numerical magnitude of the price. These effects occur because consumers non-consciously perceive that there is a positive relationship between syllabic length and numerical magnitude. Three experiments are presented demonstrating this important effect.     

How are number words mapped to approximate magnitudes?

How do we map number words to the magnitudes they represent? While much is known about the developmental trajectory of number word learning, the acquisition of the counting routine, and the academic correlates of estimation ability, previous studies have yet to describe the mechanisms that link number words to nonverbal representations of number. We investigated two mechanisms: associative mapping and structure mapping. Four dot array estimation tasks found that adults' ability to match a number word to one of two discriminably different sets declined as a function of set size and that participants' estimates of relatively large, but not small, set sizes were influenced by misleading feedback during an estimation task. We propose that subjects employ structure mappings for linking relatively large number words to set sizes, but rely chiefly on item-by-item associative mappings for smaller sets. These results indicate that both inference and association play important roles in mapping number words to approximate magnitudes.     

Comparing performance in discrete and continuous comparison tasks

The approximate number system (ANS) theory suggests that all magnitudes, discrete (i.e., number of items) or continuous (i.e., size, density, etc.), are processed by a shared system and comply with Weber's law. The current study reexamined this notion by comparing performance in discrete (comparing numerosities of dot arrays) and continuous (comparisons of area of squares) tasks. We found that: (a) threshold of discrimination was higher for continuous than for discrete comparisons; (b) while performance in the discrete task complied with Weber's law, performance in the continuous task violated it; and (c) performance in the discrete task was influenced by continuous properties (e.g., dot density, dot cumulative area) of the dot array that were not predictive of numerosities or task relevant. Therefore, we propose that the magnitude processing system (MPS) is actually divided into separate (yet interactive) systems for discrete and continuous magnitude processing. Further subdivisions are discussed. We argue that cooperation between these systems results in a holistic comparison of magnitudes, one that takes into account continuous properties in addition to numerosities. Considering the MPS as two systems opens the door to new and important questions that shed light on both normal and impaired development of the numerical system.     

Strategies for written additions in adults

Studies about strategies used by adults to solve multi-digit written additions are very scarce. However, as advocated here, the specificity and characteristics of written calculations are of undeniable interest. The originality of our approach lies in part in the presentation of two-digit addition problems on a graphics tablet, which allowed us to precisely follow and analyse individuals’ solving process. Not only classic solution times and accuracy measures were recorded but also initiation times and starting positions of the calculations. Our results show that adults largely prefer the fixed columnar strategy taught at school rather than more flexible mental strategies. Moreover, the columnar strategy is executed faster and as accurately as other strategies, which suggests that individuals’ choice is usually well adapted. This result contradicts past educational intuitions that the use of rigid algorithms might be detrimental to performance. We also demonstrate that a minority of adults can modulate their strategy choice as a function of the characteristics of the problems. Tie problems and additions without carry were indeed solved less frequently through the columnar strategy than non-tie problems and additions with a carry. We conclude that the working memory demand of the arithmetic operation influences strategy selection in written problem-solving.     

Spatial associations in relational reasoning: Evidence for a SNARC-like effect

Relational reasoning (A > B, B > C, therefore A > C) shares a number of similarities with numerical cognition, including a common behavioural signature, the symbolic distance effect. Just as reaction times for evaluating relational conclusions decrease as the distance between two ordered objects increases, people need less time to compare two numbers when they are distant (e.g., 2 and 8) than when they are close (e.g., 3 and 4). Given that some remain doubtful about such analogical representations in relational reasoning, we determine whether numerical cognition and relational reasoning have other overlapping behavioural effects. Here, using relational reasoning problems that require the alignment of six items, we provide evidence showing that the subjects' linear mental representation affects motor performance when evaluating conclusions. Items accessible from the left part of a linear representation are evaluated faster when the response is made by the left, rather than the right, hand and the reverse is observed for items accessible from the right part of the linear representation. This effect, observed with the prepositions to the left of and to the right of as well as with above and below, is analogous to the SNARC (Spatial Numerical Association of Response Codes) effect, which is characterized by an interaction between magnitude of numbers and side of response.