The spelling of many words in English and in other orthographies involves patterns determined by morphology (e.g., ed in past regular verbs). The authors report a longitudinal study that shows that when children first adopt such spelling patterns, they do so with little regard for their morphological basis. They generalize the patterns to grammatically inappropriate words (e.g., sofed for soft). Later these generalizations are confined to the right grammatical category (e.g., keped for kept) and finally to the right group of words (regular verbs). The authors conclude that children first see these spelling patterns merely as exceptions to the phonetic system and later grasp their grammatical significance. The study included two new measures of grammatical awareness, both involving analogies, that predicted success with spelling inflectional morphemes in later sessions.

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Previous research on children's conceptual and procedural understanding of fractions, and other arithmetic skills, has led to contradictory conclusions. Some research suggests that children learn conceptual knowledge before procedural knowledge, some suggests that they learn procedural knowledge before conceptual knowledge, and other research suggests that they learn conceptual knowledge and procedural knowledge in tandem. We propose that these contradictory findings may be explained by considering individual differences in the way that children combine conceptual and procedural knowledge. A total of 318 Grade 4 and 5 students in the United Kingdom (mean age ϭ 9.0 years) completed a measure of fractions understanding, which included subscales of conceptual and procedural knowledge. A cluster analysis identified 5 distinct clusters that differed from each other in terms of their relative success with conceptual and procedural problems. The existence of these clusters suggests that there may be more than one way in which children draw on conceptual and procedural knowledge. Some children rely more on procedural knowledge and others more on conceptual knowledge, but these differences may not be related to developmental processes. The children in the 5 clusters differed in their total fractions performance and in their understanding of intensive quantities. These differences suggest that children who rely on conceptual knowledge may have an advantage compared to those who rely more exclusively on procedural knowledge.

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.

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When children start to leam to read English, they benefit from leaming grapheme-phoneme correspondences. As they become more skilled, they use larger graphophonic units and morphemes in word recognition and spelling. We hypothesized that these 2 types of units in decoding make independent contributions to children's reading comprehension and fluency and that the use of morphological units is the stronger predictor of both measures. In a longitudinal study with a large sample in the United Kingdom, we tested through multiple regressions the contributions that these different units make to the prediction of reading competence (reading comprehension and fluency). The predictors were measured when the children were aged 8-9 years. Reading comprehension and rate were measured concurrentiy, and reading list fluency was measured at 12 and 13 years. After controlling for age and verbal IQ, the children's use of larger graphophonic units and their use of morphemes in reading and spelling made independent contributions to predicting their reading comprehension and reading fluency. The use of morphemes was the stronger predictor in all analyses. Thus, teaching that promotes the development of these different ways of reading and spelling words should be included in policy and practice.The aim of the study reported here is to analyze the effect of different decoding processes on reading competence. We distinguish two types of units that are used by children in decoding: graphophonic units, which carry no meaning, and morphological units, which are units of meaning. We suggest that this analysis of decoding processes can contdbute new insights to the understanding of reading competence. We consider reading competence as assessed by measures of both comprehension and fluency, in view of the fact that fiuency is considered a significant factor connecting

Because the spelling of many words in the English language (and in many other languages as well) depends on their morphemic structure, children have to have some knowledge about morphemes in order to learn to read and write. This raises the possibility that children gain much of their explicit knowledge about morphemes as a direct result of learning to read and to spell. We report two large-scale longitudinal studies that support the idea of this kind of causal connection. In the first study children's success in spelling the inflexion at the end of regular past verbs predicted their performance in two morphological awareness tasks a year later. In the second study the children's consistency in spelling morphemes predicted their ability to define new words on the basis of their morphemic structure. We conclude that the experience of learning to read and write does affect people's knowledge of morphemes, and we argue that the causal relationship between morphemic knowledge and reading and writing is probably a two-way one.

There is clear justification for making a distinction between mathematical reasoning and arithmetic skills. The implication is that schools must plan explicitly to improve mathematical reasoning as well as arithmetic skills.