The contributions of domain-general and numerical factors to third-grade arithmetic skills and mathematical learning disability

Explanations of the marked individual differences in elementary school mathematical achievement and mathematical learning disability (MLD or dyscalculia) have involved domain-general factors (working memory, reasoning, processing speed and oral language) and numerical factors that include single-digit processing efficiency and multi-digit skills such as number system knowledge and estimation. This study of third graders (N = 258) finds both domain-general and numerical factors contribute independently to explaining variation in three significant arithmetic skills: basic calculation fluency, written multi-digit computation, and arithmetic word problems. Estimation accuracy and number system knowledge show the strongest associations with every skill and their contributions are both independent of each other and other factors. Different domain-general factors independently account for variation in each skill. Numeral comparison, a single digit processing skill, uniquely accounts for variation in basic calculation. Subsamples of children with MLD (at or below 10th percentile, n = 29) are compared with low achievement (LA, 11th to 25th percentiles, n = 42) and typical achievement (above 25th percentile, n = 187). Examination of these and subsets with persistent difficulties supports a multiple deficits view of number difficulties: most children with number difficulties exhibit deficits in both domain-general and numerical factors. The only factor deficit common to all persistent MLD children is in multi-digit skills. These findings indicate that many factors matter but multi-digit skills matter most in third grade mathematical achievement

Calendrical savants: Exceptionality and practice

The exceptionality of the skills of calendrical savants and the role of practice were investigated. Experiment 1 compared four autistic calendrical savants to Professor Conway, a distinguished mathematician with calendrical skills. Professor Conway answered questions over a greater range of years but some savants knew more calendrical regularities. Experiment 2 studied the development of a calendrical savant's ability to answer date questions for very remote future years. He started by making written calculations and progressed to mental calculation. His variation in response time for remote dates was similar to that for near dates. The findings are consistent with the view that calendrical savants develop their skills through practice

The skills and methods of calendrical savants

Calendrical savants are people with considerable intellectual difficulties that have the unusual ability to name the weekdays for dates in the past and sometimes the future. Three criteria are proposed to distinguish savants whose skill depends on memorization from those who calculate: range of years, consistent deviation from the Gregorian calendar, and variation in latency with remoteness from the present. A study of 10 calendrical savants showed 5 met one or both of the criteria concerning range and deviation and 9 met the third criterion. The second study assessed their arithmetical abilities using tests of mental and written arithmetic. This broadly validated the attribution of calculation as the basis for some savants? skills. The results are discussed in relation to views that calendrical savants imply the existence of a modular mathematical intelligence or unconscious integer arithmetic

Why and how people of limited intelligence become calendrical calculators

Calendrical calculation is the rare talent of naming the days of the week for dates in the past and future. Calendrical savants are people with low measured intelligence who have this talent. This paper reviews evidence and speculation about why people become calendrical savants and how they answer date questions. Most savants are known to have intensively studied the calendar and show superior memory for calendrical information. As a result they may answer date questions either from recalling calendars or by using strategies that exploit calendrical regularities. While people of average or superior intelligence may become calendrical calculators through internalising formulae, the arithmetical demands of the formulae make them unlikely as bases for the talents of calendrical savants. We attempt to identify the methods used by a sample of 10 savants. None rely on an internalised formula. Some use strategies based on calendrical regularities probably in conjunction with memory for a range of years. For the rest a decision between use of regularities and recall of calendars cannot be made

Children's understanding of mental states as causes of emotions

Theory of Mind studies of emotion usually focus on children?s ability to predict other people's feelings. This study examined children?s spontaneous references to mental states in explaining others? emotions. Children (4-, 6- and 10-year-olds, n = 122) were told stories and asked to explain both typical and atypical emotional reactions of characters. Because atypical emotional reactions are unexpected, we hypothesized that children would be more likely to refer to mental states, such as desires and beliefs, in explaining them than when explaining typical emotions. From the development of lay theories of emotion, derived the prediction that older children would refer more often to mental states than younger children. The developmental shift from a desire-psychology to a belief-psychology led to the expectation that references to desires would increase at an earlier age than references to beliefs. Our findings confirmed these expectations only partly, because the nature of the emotion (happiness, anger, sadness or fear) interacted with these factors. Whereas anger, happiness and sadness mainly evoked desire references, fear evoked more belief references, even in four-year-olds. The fact that other factors besides age can also play an influential role in children?s mental state reasoning is discussed

Column tessellations

A new class of random spatial tessellations is introduced -- the so-called column tessellations of three-dimensional space. The construction is based on a stationary planar tessellation. Each cell of the spatial tessellation is a prism whose base facet is congruent to a cell of the planar tessellation. Thus intensities, topological and metric mean values of the spatial tessellation can be calculated by suitably chosen parameters of the planar tessellation. A column tessellation is not facet-to-facet.Comment: 18 pages, 3 figure