Individual differences in children's mathematical competence are related to the intentional but not automatic processing of Arabic numerals

In recent years, there has been an increasing focus on the role played by basic numerical magnitude processing in the typical and atypical development of mathematical skills. In this context, tasks measuring both the intentional and automatic processing of numerical magnitude have been employed to characterize how children’s representation and processing of numerical magnitude changes over developmental time. To date, however, there has been little effort to differentiate between different measures of ‘number sense’. The aim of the present study was to examine the relationship between automatic and intentional measures of magnitude processing as well as their relationships to individual differences in children’s mathematical achievement. A group of 119 children in 1st and 2nd grade were tested on the physical size congruity paradigm (automatic processing) as well as the number comparison paradigm to measure the ratio effect (intentional processing). The results reveal that measures of intentional and automatic processing are uncorrelated with one another, suggesting that these tasks tap into different levels of numerical magnitude processing in children. Furthermore, while children’s performance on the number comparison paradigm was found to correlate with their mathematical achievement scores, no such correlations could be obtained for any of the measures typically derived from the physical size congruity task. These findings therefore suggest that different tasks measuring ‘number sense’ tap into different levels of numerical magnitude representation that may be unrelated to one another and have differential predictive power for individual differences in mathematical achievement.     

Number versus continuous quantity in numerosity judgments by fish

A fundamental question in human cognition is how people reason about space. We use a computational model to explore cross-cultural commonalities and differences in spatial cognition. Our model is based upon two hypotheses: (1) the structure-mapping model of analogy can explain the visual comparisons used in spatial reasoning; and (2) qualitative, structural representations are computed by people’s visual systems and used in these comparisons. We apply our model to a visual oddity task, in which individuals are shown an array of two-dimensional images and asked to the pick the one that does not belong. This task was previously used to evaluate understanding of geometric concepts in two disparate populations: North Americans, and the Mundurukú, a South American indigenous group. Our model automatically generates representations of each hand-segmented image and compares them to solve the task. The model achieves human-level performance on this task, and problems that are hard for the model are also difficult for people in both cultures. Furthermore, ablation studies on the model suggest explanations for cross-cultural differences in terms of differences in spatial representations.     

Squeezing, striking, and vocalizing: Is number representation fundamentally spatial?

Numbers are fundamental entities in mathematics, but their cognitive bases are unclear. Abundant research points to linear space as a natural grounding for number representation. But, is number representation fundamentally spatial? We disentangle number representation from standard number-to-line reporting methods, and compare numerical estimations in educated participants using line-reporting with three nonspatial reporting conditions (squeezing, bell-striking, and vocalizing). All three cases of nonspatial-reporting consistently reproduced well-established results obtained with number-line methods. Furthermore, unlike line-reporting—and congruent with the psychophysical Weber–Fechner law—nonspatial reporting systematically produced logarithmic mappings for all nonsymbolic stimuli. Strikingly, linear mappings were obtained exclusively in conditions with culturally mediated elements (e.g., words). These results suggest that number representation is not fundamentally spatial, but builds on a deeper magnitude sense that manifests spatially and nonspatially mediated by magnitude, stimulus modality, and reporting condition. Number-to-space mappings—although ubiquitous in the modern world—do not seem to be rooted directly in brain evolution but have been culturally privileged and enhanced.     

Restructuring partitioned knowledge: the role of recoordination in category learning

Knowledge restructuring refers to changes in the strategy with which people solve a given problem. Two types of knowledge restructuring are supported by existing category learning models. The first is a relearning process, which involves incremental updating of knowledge as learning progresses. The second is a recoordination process, which involves novel changes in the way existing knowledge is applied to the task. Whereas relearning is supported by both single- and multiple-module models of category learning, only multiple-module models support recoordination. To date, only relearning has been directly supported empirically. We report two category learning experiments that provide direct evidence of recoordination. People can fluidly alternate between different categorization strategies, and moreover, can reinstate an old strategy even after prolonged use of an alternative. The knowledge restructuring data are not well fit by a single-module model (ALCOVE). By contrast, a multiple-module model (ATRIUM) quantitatively accounts for recoordination. Low-level changes in the distribution of dimensional attention are shown to subsequently affect how ATRIUM coordinates its modular knowledge. We argue that learning about complex tasks occurs at the level of the partial knowledge elements used to generate a response strategy.     

A memory-based model of Hick's law

We propose and evaluate a memory-based model of Hick’s law, the approximately linear increase in choice reaction time with the logarithm of set size (the number of stimulus–response alternatives). According to the model, Hick’s law reflects a combination of associative interference during retrieval from declarative memory and occasional savings for stimulus–response repetitions due to non-retrieval. Fits to existing data sets show that the model accounts for the basic set-size effect, changes in the set-size effect with practice, and stimulus–response-repetition effects that challenge the information-theoretic view of Hick’s law. We derive the model’s prediction of an interaction between set size, stimulus fan (the number of responses associated with a particular stimulus), and stimulus–response transition, which is subsequently tested and confirmed in two experiments. Collectively, the results support the core structure of the model and its explanation of Hick’s law in terms of basic memory effects.     

Behavioral intervention to treat selective mutism across multiple social situations and community settings

We evaluated a behavioral intervention for a 9-year-old girl with selective mutism. The intervention consisted of role play and video self-modeling. The frequency of spoken initiations, responses to questions, and communication breakdowns was measured during three social situations (i.e., ordering in a restaurant, meeting new adults, and playing with new children) and in three community settings. Results demonstrated increases in spoken initiations and responses and decreases in communication breakdowns across all situations and settings.     

Using multiple-exemplar training to teach a generalized repertoire of sharing to children with autism

The current study examined the utility of multiple-exemplar training to teach children with autism to share. Stimuli from 3 of 4 categories were trained using a treatment package of video modeling, prompting, and reinforcement. Offers to share increased for all 3 children following the introduction of treatment, with evidence of skill maintenance. In addition, within-stimulus-category generalization of sharing was evident for all participants, although only 1 participant demonstrated across-category generalization of sharing. Offers to share occurred in a novel setting, with familiar and novel stimuli, and in the presence of novel adults and peers for all participants during posttreatment probes.     

Strategic adaptation to performance objectives in a dual-task setting

How do people interleave attention when multitasking? One dominant account is that the completion of a subtask serves as a cue to switch tasks. But what happens if switching solely at subtask boundaries led to poor performance? We report a study in which participants manually dialed a UK‐style telephone number while driving a simulated vehicle. If the driver were to exclusively return his or her attention to driving after completing a subtask (i.e., using the single break in the xxxxx‐xxxxxx representational structure of the number), then we would expect to see a relatively poor driving performance. In contrast, our results show that drivers choose to return attention to steering control before the natural subtask boundary. A computational modeling analysis shows that drivers had to adopt this strategy to meet the required performance objective of maintaining an acceptable lateral position in the road while dialing. Taken together these results support the idea that people can strategically control the allocation of attention in multitask settings to meet specific performance criteria.     

The Importance of Community Context for Young Women’s Occupational Aspirations

The effects of community context on occupational aspirations are examined in a national sample of young women in high school in the USA in 1979 (n = 2,210). Multilevel statistical models indicate that young women living in counties with a lower divorce rate, a lower percentage of women working, and more people employed in the wholesale and retail industrial sector tended to be less likely to aspire to paid work than young women living in areas with a higher divorce rate, a higher percentage of women working, and fewer people employed in the wholesale and retail industrial sector. Community context does not affect the level of young women’s occupational aspirations as predicted by prior scholarship. An earlier version of this paper was presented at the 2006 Southern Sociological Society Annual Meetings. The author thanks John Reynolds, Patricia Yancey Martin, Irene Padavic, Robert Kunovich, Jennifer Keene, Mary Guy, and the editor and reviewers at Sex Roles for their helpful comments on previous drafts of this paper.