Eye gaze patterns reveal how we reason about fractions

ABSTRACT

Fractions are defined by numerical relationships, and comparing two fractions’ magnitudes requires within-fraction (holistic) and/or between-fraction (componential) relational comparisons. To better understand how individuals spontaneously reason about fractions, we collected eye-tracking data while they performed a fraction comparison task with conditions that promoted or obstructed different types of comparisons. We found evidence for both componential and holistic processing in this mixed-pairs task, consistent with the hybrid theory of fraction representation. Additionally, making within-fraction eye movements on trials that promoted a between-fraction comparison strategy was associated with slower responses. Finally, participants who performed better on a non-numerical test of reasoning took longer to respond to the most difficult fraction trials, which suggests that those who had greater facility with non-numerical reasoning attended more to numerical relationships. These findings extend prior research and support the continued investigation into the mechanistic links between numerical and non-numerical reasoning.     

The link between logic,mathematics and imagination: evidence from children with developmental dyscalculia and mathematically gifted children

This study examined performance on transitive inference problems in children with developmental dyscalculia (DD), typically developing controls matched on IQ, working memory and reading skills, and in children with outstanding mathematical abilities. Whereas mainstream approaches currently consider DD as a domain‐specific deficit, we hypothesized that the development of mathematical skills is closely related to the development of logical abilities, a domain‐general skill. In particular, we expected a close link between mathematical skills and the ability to reason independently of one's beliefs. Our results showed that this was indeed the case, with children with DD performing more poorly than controls, and high maths ability children showing outstanding skills in logical reasoning about belief‐laden problems. Nevertheless, all groups performed poorly on structurally equivalent problems with belief‐neutral content. This is in line with suggestions that abstract reasoning skills (i.e. the ability to reason about content without real‐life referents) develops later than the ability to reason about belief‐inconsistent fantasy content.A video abstract of this article can be viewed at http://www.youtube.com/watch?v=90DWY3O4xx8     

An exploration into the influence of academic and social values, procrastination, and perceived school belongingness on academic performance

The results of a structural equation model showed that a tendency to procrastinate, assessed early in college students’ first term, was positively related to social values, assessed as concerns over social exclusion, but was negatively related to academic task values and grade goal-setting. The results suggest that procrastination may be a partial mediator of the influence of academic and social values on perceived school belongingness. In addition, the tendency to procrastinate had a direct negative relationship with self-regulatory self-efficacy and perceived school belongingness and was positively related to perceived stress near the end of the term. There was also a statistically significant negative total effect of procrastination on performance-approach and mastery-approach goal orientation and end-of-term grade point average (GPA). Statistically significant positive total effects of performance-approach and mastery-approach achievement goal orientations on GPA were found. However, while a statistically significant direct positive effect of performance-approach orientation on GPA was found, a direct relationship between mastery-approach orientation and GPA was not statistically significant.     

Saccadic eye movements as indicators of cognitive function in older adults

Older adults appear to have greater difficulty ignoring distractions during day-to-day activities than younger adults. To assess these effects of age, the ability of adults aged between 50 and 80 years to ignore distracting stimuli was measured using the antisaccade and oculomotor capture tasks. In the antisaccade task, observers are instructed to look away from a visual cue, whereas in the oculomotor capture task, observers are instructed to look toward a colored singleton in the presence of a concurrent onset distractor. Index scores of the Repeatable Battery for the Assessment of Neuropsychological Status (RBANS) were compared with capture errors, and with prosaccade errors on the antisaccade task. A higher percentage of capture errors were made on the oculomotor capture tasks by the older members of the cohort compared to the younger members. There was a weak relationship between the attention index and capture errors, but the visuospatial/constructional index was the strongest predictor of prosaccade error rate in the antisaccade task. The saccade reaction times (SRTs) of correct initial saccades in the oculomotor capture task were poorly correlated with age, and with the neurospsychological tests, but prosaccade SRTs in both tasks moderately correlated with antisaccade error rate. These results were interpreted in terms of a competitive integration (or race) model. Any variable that reduces the strength of the top-down neural signal to produce a voluntary saccade, or that increases saccade speed, will enhance the likelihood that a reflexive saccade to a stimulus with an abrupt onset will occur.     

Elevation easier than plan for sighted and early-blind adults in a perspective-taking task

Plans show shapes of objects from above, and represent both their left–right order and their order in the z-dimension (the distance of the objects). Elevations show only the vertical shapes of objects arranged from left to right. Plans, having more spatial information, may be more difficult for participants to construct. Results from a study with sighted, sighted-blindfolded, and early-blind participants on Piaget's perspective-taking three-mountain task support this hypothesis. The plan task was judged more difficult than the elevation task even when participants performed with the same level of accuracy on both tasks. In visual and tactile tasks, amount of spatial-order information may determine difficulty, rather than plan versus elevation per se.     

The influence of parafoveal typographical errors on eye movements in reading

Three experiments are reported, examining the effects of a typographical error in parafoveal vision on aspects of foveal inspection time and saccade targeting. All the experiments involved reading for comprehension. A contingent presentation procedure ensured that typographical errors were restored to their correct form before they were viewed in foveal vision: They were never available for foveal processing. In Experiment 1, the error was formed by replacing the first letter of the target word with a second occurrence of its second letter, producing an illegal nonword. This manipulation had no significant effect on foveal inspection time, but lowered the probability that a short word (“de” or “du”) prior to the target would be skipped. In Experiment 2 the familiarity of the target's initial letters was maintained constant across conditions. This manipulation removed the target 1 skipping effect, suggesting that the outcome of Experiment 1 was due to orthographic rather than lexical illegality, but revealed shorter foveal inspection times as a function of the presence of the error. Experiment 3 manipulated lexical and sublexical properties of the parafoveal typing error. Properties of the parafoveal error again influenced prior foveal inspection times. The pattern of results suggested that the determining properties were sublexical rather than lexical. The results as a whole are incompatible with a view of information processing in reading in which foveal processing remains immune from concurrent parafoveal influences.     

Mechanisms of theory formation in young children

Research suggests that by the age of five, children have extensive causal knowledge, in the form of intuitive theories. The crucial question for developmental cognitive science is how young children are able to learn causal structure from evidence. Recently, researchers in computer science and statistics have developed representations (causal Bayes nets) and learning algorithms to infer causal structure from evidence. Here we explore evidence suggesting that infants and children have the prerequisites for making causal inferences consistent with causal Bayes net learning algorithms. Specifically, we look at infants and children's ability to learn from evidence in the form of conditional probabilities, interventions and combinations of the two.     

Children's understanding of the pretence–reality distinction: a review of current theory and evidence

This paper provides an update on the current status of theory and evidence relating to children's understanding of the pretence–reality distinction. The paper starts by highlighting the striking paradox between children's early competence in pretence and their experiences of pretence–reality confusions as late as middle childhood. This is followed by a detailed review of various theories that have been offered to explain this phenomenon. Specifically, theories attributing the paradoxical findings to methodological differences between studies are reviewed and dismissed before considering the transmigration and availability hypotheses ( Harris, Brown, Marriott, Whittall & Harmer, 1991 ; Johnson & Harris, 1994 ), the role of context and the child's emotional involvement in the pretence, and the pretence continuation account ( Golomb & Galasso, 1995 ). It is argued that none of these theories alone can explain pretence–reality confusions and that these are best explained in terms of the combined influences of cognitive availability, empirical evidence of reality, context, affect and individual differences. Further research is necessary to fully explore the nature, cause and developmental trajectory of individual differences in this domain.