Richard Cowan

Number Skills and Knowledge in Children With Specific Language Impairment

The number skills of groups of 7- to 9-year-old children with specific language impairment (SLI) attending mainstream or special schools were compared with an age and nonverbal reasoning matched group (age control [AC]) and with a younger group matched on oral language comprehension. The SLI groups performed below the AC group on every skill. They also showed lower working memory functioning and had received lower levels of instruction. Nonverbal reasoning, working memory functioning, language comprehension, and instruction accounted for individual variation in number skills to differing extents depending on the skill. These factors did not explain the differences between SLI and AC groups on most skills.

Do They Know What They Are Doing? Children's Use of Economical Addition Strategies and Knowledge of Commutativity

Abstract Childrens methods of solving addition problems progress towards strategies that reverse the order of addends. The role knowledge of commutativity plays in this development is not clear. Previous studies suggest some use the strategies without believing addend order is irrelevant to addition. The numbers of such children are very small and the tests of commutativity may be inaccurate. Two studies using new tasks compare strategy use and knowledge. In the first, 47/48 children between 6 and 10 years old predicted order of addends would not alter quantity in some contexts. They also predominantly used strategies that reversed addend order. In the second study, 5‐year‐olds (18/24) were more likely to know commutativity than to use a strategy that reversed addend order or to answer sums correctly. Knowing commutativity precedes using strategies that presuppose it and does not derive from doing sums. Representing the addends with objects induced more arithmetic errors. Children used less accurate stra...

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 multidigit skills such as number system knowledge and estimation. This study of 3rd graders (N = 258) finds both domain-general and numerical factors contribute independently to explaining variation in 3 significant arithmetic skills: basic calculation fluency, written multidigit computation, and arithmetic word problems. Estimation accuracy and number system knowledge show the strongest associations with every skill, and their contributions are independent of both 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 multidigit skills. These findings indicate that many factors matter but multidigit skills matter most in 3rd grade mathematical achievement.

Do calendrical savants use calculation to answer date questions? A functional magnetic resonance imaging study

Calendrical savants can name the weekdays for dates from different years with remarkable speed and accuracy. Whether calculation rather than just memory is involved is disputed. Grounds for doubting whether they can calculate are reviewed and criteria for attributing date calculation skills to them are discussed. At least some calendrical savants possess date calculation skills. A behavioural characteristic observed in many calendrical savants is increased response time for questions about more remote years. This may be because more remote years require more calculation or because closer years are more practised. An experiment is reported that used functional magnetic resonance imaging to attempt to discriminate between these explanations. Only two savants could be scanned and excessive head movement corrupted one savants mental arithmetic data. Nevertheless, there was increased parietal activation during both mental arithmetic and date questions and this region showed increased activity with more remote dates. These results suggest that the calendrical skills observed in savants result from intensive practice with calculations used in solving mental arithmetic problems. The mystery is not how they solve these problems, but why.

When Do Children Trust Counting as a Basis for Relative Number Judgments

Abstract There are several explanations of why children make relative number judgments inconsistent with their counting. Their counting may be unreliable or restricted to single sets. They may forget the results of counting when they judge or not know which number is bigger. In four experiments childrens judgments were studied in conditions which controlled for these explanations. Whether the child or the experimenter counted made no difference. which implies that children did not ignore count information because they mistrusted their counting. Very few competent counters less than 6 years old judged consistently with their counting when conflicting length cues were present. Trusting counting over length to tell whether two rows were the same preceded trusting counting to decide which row had more. The results are discussed in relation to childrens judgments when using matching and D. Klahr and J. G. Wallaces (1973) . Cognitive Psychology. 4, 301–327) account of number development.

Calendrical calculation and intelligence

Naming the days of the week for dates in the past and future is a rare talent observed in people with low measured intelligence. The talent and other savant skills are more common in the autistic population, suggesting features of autistic cognition such as obsessive preoccupation and weak central coherence may facilitate development of savant skills. This study describes the date calculation skills and performance on other calendar tasks by 10 calendrical savants whose WAIS IQs range from 50 to 97. Their Block Design scores were unexceptional, contrary to the weak central coherence explanation. Accuracy in date calculation and knowledge of calendrical regularities correlated with full scale IQ, indicating that the talent depends on intelligence. Accuracy, range and latency of date calculation and latency for other calendrical tasks showed marked associations with Digit Symbol subscale scores.

Children as Apprentices to Number.

Preschool childrens understandings and use of numerals are investigated in a series of tasks constructed to draw upon common experiences at home. Most 3and 4‐year‐olds explained the purpose of numerals on telephones and birthday cards. However fewer noticed when these were missing from pictures of these or other familiar objects. Children rarely said number words in giving fast‐food orders. They used several methods to represent number on notes for the milkman, party invitations and labels. Many children showed knowledge of the language used in specifying dates, times, addresses or telephone numbers. Across a range of tasks, understanding preceded use of numerals.

The emergence of additive composition of number

Two competences have been used to infer a grasp of additive composition of number, They are rearranging quantities to solve arithmetic word problems and constructing amounts with coins of different denominations in shopping tasks. The use of a counting-on strategy to solve addition problems has been suggested to be a precursor of additive composition. Counting-on presupposes an ability to continue counting from a number greater than one. In the present study, 152 children between 4 and 7 years old were tested on story problems, shopping tasks and continuation of counting. Use of the counting-on strategy was sought in a variety of arithmetical tasks. Each child was tested on three occasions during a school year. Children were more likely to succeed on the shopping task than on arithmetic word problems. Only children who could continue counting succeeded on these tasks. In contrast, several passed the shopping task without using counting-on to solve arithmetical problems. While both counting-on and the ability to combine coins of different denominations develop from continuation of counting, there is no necessary link between them.

Why and how people of limited intelligence become calendrical calculators

Abstract 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 internalizing 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.

Social Context Effects on Children's Performance of Number Conservation Tasks: plausibility and evidence

Abstract Recent controversy over Piagetian tasks concerns whether the experimenters actions result in children misunderstanding the question. Evidence for this has been derived from differences in performance of conservation tasks according to whether the transformation is intentional or accidental. Studies reporting large differences according to transformation type are claimed to be flawed procedurally. Including a preliminary talk and restricting questions to one domain result in little or no transformation type effect. Whether childrens correct judgments in accidental versions are reliable indications of their knowledge has also been disputed. They may reflect childrens ignoring of the transformation or covert counting. Using an identity conservation pretest 192 Yemeni children were selected to be tested on number conservation tasks varying in transformation type, numerosity and equality. The results were analysed using a model‐fitting procedure. Identity nonconservers were affected by transformati...