It has often been claimed that children's mathematical understanding is based on their ability to reason logically, but there is no good evidence for this causal link. We tested the causal hypothesis about logic and mathematical development in two related studies. In a longitudinal study, we showed that (a) 6-year-old children's logical abilities and their working memory predict mathematical achievement 16 months later; and (b) logical scores continued to predict mathematical levels after controls for working memory, whereas working memory scores failed to predict the same measure after controls for differences in logical ability. In our second study, we trained a group of children in logical reasoning and found that they made more progress in mathematics than a control group who were not given this training. These studies establish a causal link between logical reasoning and mathematical learning. Much of children's mathematical knowledge is based on their understanding of its underlying logic.Logical relations lie at the heart of many fields of inquiry. Think of the relation known as transitivity, for example, if A ¼ B and B ¼ C, then A ¼ C. This relation is pertinent to the number of objects in a set, to length, to volume, to colour, to shape, to people's intelligence, to the matching of photographs of fingerprints, to the taste of orange juice etc. Transitivity is significant in the domains of number and measurement, but it is neither number nor measurement. Not all the relations are transitive: if A is the father of B and B is the father of C, it does not follow that A is the father of C. For this reason, Piaget (1950) argued that the pertinence of a logical relation is given meaning in a domain through experience.A similar argument was advanced by Simon and Klahr (1995;Klahr, 1982) about conservation. Transformations, they argued, have different effects on different quantities, and children learn about these differences by experience. If we add a jar of water at 208C to another jar of water also at 208C, the quantity of water (the * Correspondence should be addressed to Terezinha Nunes, 15 Norham Gardens, Oxford OX2 6PY, UK Reproduction in any form (including the internet) is prohibited without prior permission from the Society extensive quantity) increases, but the temperature (an intensive quantity) stays the same. Children must learn the domains where different logical relations (or axioms) apply.Children almost certainly need to understand the logical relations between quantities in order to learn how to represent numbers and arithmetic. These relations are not the same as arithmetic or the numeration system, but are relevant to them. One such relation is correspondence: if two sets contain the same number of objects, then the objects in one set are in one-to-one correspondence with those in the other. If set B is in two-to-one correspondence with set A, and C is in two-to-one correspondence with A, then B and C are equivalent.
Each February, competitors convene in Big Lake, Alaska, to participate in the "Iditasport Human Powered Ultra-Marathon". Who would attempt this challenging race? Personality might be one factor predicting participation. Iditasport represents a unique athletic event with a distinctive social and psychological climate that might be reflected in the personalities of the participants in many ways. This study was designed to identify the personality profile of Iditasport athletes when compared to normative populations and to explore differences between athletes competing in different race divisions.
Multiplicative reasoning is required in different contexts in mathematics: it is necessary to understand the concept of multipart units, involved in learning place value and measurement, and also to solve multiplication and division problems. Measures of hearing children's multiplicative reasoning at school entry are reliable and specific predictors of their mathematics achievement in school. An analysis of deaf children's informal multiplicative reasoning showed that deaf children under-perform in comparison to the hearing cohorts in their first two years of school. However, a brief training study, which significantly improved their success on these problems, suggested that this may be a performance, rather than a competence difference. Thus, it is possible and desirable to promote deaf children's multiplicative reasoning when they start school so that they are provided with a more solid basis for learning mathematics.
Congenitally, profoundly deaf children whose fi rst language is British Sign Language (BSL) and whose speech is largely unintelligible need to be literate to communicate effectively in a hearing society. Both spelling and writing skills of such children can be limited, to the extent that no currently available assessment method offers an adequate appraisal of their competence. Our aim was to create such an instrument to aid assessment and to support teachers in setting objectives for their deaf students' writing development.Writing samples describing the same four-picture story were collected from 29 congenitally, profoundly deaf 10-year-old users of BSL. Six experienced teachers of the deaf ranked their writing productions in fi ve levels; the correlations between their ranks were high and signifi cant. This indicates that the children's texts were classifi ed reliably into categories, which could then be used for further descriptive analysis. The texts in each category were analysed qualitatively to provide descriptive profi les for each level. An indication of the concurrent validity of the profi les was obtained through signifi cant correlations with reading comprehension measures. Future research should ascertain further the reliability and validity of this instrument and its usefulness in setting goals for improving deaf children's writing ability.
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