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Understanding conceptual relationships is an important aspect of learning arithmetic. Most studies of arithmetic, however, do not distinguish between children's understanding of a concept and their ability to identify situations in which it might be relevant. We compared 8- to 9-year-old children's use of a computational shortcut based on the inverse relationship between addition and subtraction, in problems where it was transparently applicable (e.g. 17+11-11=) and where it was not (e.g. 15+11-8-3=). Most children were able to construct inverse transformations and apply the shortcut in at least some situations, although they used the shortcut more for problems where it was transparently applicable. There were individual differences in the relationship between children's understanding of the inverse relationship and computational skill that have implications for theories of mathematical development. [ABSTRACT FROM AUTHOR]

In learning mathematics, children must master fundamental logical relationships, including the inverse relationship between addition and subtraction. At the start of elementary school, children lack generalized understanding of this relationship in the context of exact arithmetic problems: they fail to judge, for example, that 12+9-9 yields 12. Here, we investigate whether preschool children's approximate number knowledge nevertheless supports understanding of this relationship. Five-year-old children were more accurate on approximate large-number arithmetic problems that involved an inverse transformation than those that did not, when problems were presented in either non-symbolic or symbolic form. In contrast they showed no advantage for problems involving an inverse transformation when exact arithmetic was involved. Prior to formal schooling, children therefore show generalized understanding of at least one logical principle of arithmetic. The teaching of mathematics may be enhanced by building on this understanding. [ABSTRACT FROM AUTHOR]

Since the first aeroplane flight more than 100 years ago, aeroplanes have been propelled using moving surfaces such as propellers and turbines. Most have been powered by fossil-fuel combustion. Electroaerodynamics, in which electrical forces accelerate ions in a fluid 1,2 , has been proposed as an alternative method of propelling aeroplanes-without moving parts, nearly silently and without combustion emissions 3-6 . However, no aeroplane with such a solid-state propulsion system has yet flown. Here we demonstrate that a solid-state propulsion system can sustain powered flight, by designing and flying an electroaerodynamically propelled heavier-than-air aeroplane. We flew a fixed-wing aeroplane with a five-metre wingspan ten times and showed that it achieved steady-level flight. All batteries and power systems, including a specifically developed ultralight high-voltage (40-kilovolt) power converter, were carried on-board. We show that conventionally accepted limitations in thrust-to-power ratio and thrust density 4,6,7 , which were previously thought to make electroaerodynamics unfeasible as a method of aeroplane propulsion, are surmountable. We provide a proof of concept for electroaerodynamic aeroplane propulsion, opening up possibilities for aircraft and aerodynamic devices that are quieter, mechanically simpler and do not emit combustion emissions.

In jurisdictions including the US and the EU ground transportation and marine fuels have recently been required to contain lower concentrations of sulfur, which has resulted in reduced atmospheric SO(x) emissions. In contrast, the maximum sulfur content of aviation fuel has remained unchanged at 3000 ppm (although sulfur levels average 600 ppm in practice). We assess the costs and benefits of a potential ultra-low sulfur (15 ppm) jet fuel standard ("ULSJ"). We estimate that global implementation of ULSJ will cost US$1-4bn per year and prevent 900-4000 air quality-related premature mortalities per year. Radiative forcing associated with reduction in atmospheric sulfate, nitrate, and ammonium loading is estimated at +3.4 mW/m(2) (equivalent to about 1/10th of the warming due to CO(2) emissions from aviation) and ULSJ increases life cycle CO(2) emissions by approximately 2%. The public health benefits are dominated by the reduction in cruise SO(x) emissions, so a key uncertainty is the atmospheric modeling of vertical transport of pollution from cruise altitudes to the ground. Comparisons of modeled and measured vertical profiles of CO, PAN, O(3), and (7)Be indicate that this uncertainty is low relative to uncertainties regarding the value of statistical life and the toxicity of fine particulate matter.

Geometrical concepts are critical to a host of human cognitive achievements, from maps to measurement to mathematics, and both the development of these concepts, and their variation by gender, have long been studied. Most studies of geometrical reasoning, however, present children with materials containing both geometric and non-geometric information, and with tasks that are open to multiple solution strategies. Here we present kindergarten children with a task requiring a focus on geometry: navigation in a small-scale space by a purely geometric map. Children spontaneously extracted and used relationships of both distance and angle in the maps, without prior demonstration, instruction, or feedback, but they failed to use the sense information that distinguishes an array from its mirror image. Children of both genders showed a common profile of performance, with boys showing no advantage on this task. These findings provide evidence that some map-reading abilities arise prior to formal instruction, are common to both genders, and are used spontaneously to guide children's spatial behavior., (© 2011 Blackwell Publishing Ltd.)

This study examined numerical magnitude processing in first graders with severe and mild forms of mathematical difficulties, children with mathematics learning disabilities (MLD) and children with low achievement (LA) in mathematics, respectively. In total, 20 children with MLD, 21 children with LA, and 41 regular achievers completed a numerical magnitude comparison task and an approximate addition task, which were presented in a symbolic and a nonsymbolic (dot arrays) format. Children with MLD and LA were impaired on tasks that involved the access of numerical magnitude information from symbolic representations, with the LA children showing a less severe performance pattern than children with MLD. They showed no deficits in accessing magnitude from underlying nonsymbolic magnitude representations. Our findings indicate that this performance pattern occurs in children from first grade onward and generalizes beyond numerical magnitude comparison tasks. These findings shed light on the types of intervention that may help children who struggle with learning mathematics., (Copyright © 2010 Elsevier Inc. All rights reserved.)

Children take years to learn symbolic arithmetic. Nevertheless, non-human animals, human adults with no formal education, and human infants represent approximate number in arrays of objects and sequences of events, and they use these capacities to perform approximate addition and subtraction. Do children harness these abilities when they begin to learn school mathematics? In two experiments in different schools, kindergarten children from diverse backgrounds were tested on their non-symbolic arithmetic abilities during the school year, as well as on their mastery of number words and symbols. Performance of non-symbolic arithmetic predicted children's mathematics achievement at the end of the school year, independent of achievement in reading or general intelligence. Non-symbolic arithmetic performance was also related to children's mastery of number words and symbols, which figured prominently in the assessments of mathematics achievement in both schools. Thus, non-symbolic and symbolic numerical abilities are specifically related, in children of diverse socio-economic backgrounds, near the start of mathematics instruction.

When children learn to count and acquire a symbolic system for representing numbers, they map these symbols onto a preexisting system involving approximate nonsymbolic representations of quantity. Little is known about this mapping process, how it develops, and its role in the performance of formal mathematics. Using a novel task to assess children's mapping ability, we show that children can map in both directions between symbolic and nonsymbolic numerical representations and that this ability develops between 6 and 8 years of age. Moreover, we reveal that children's mapping ability is related to their achievement on tests of school mathematics over and above the variance accounted for by standard symbolic and nonsymbolic numerical tasks. These findings support the proposal that underlying nonsymbolic representations play a role in children's mathematical development.

Symbolic arithmetic is fundamental to science, technology and economics, but its acquisition by children typically requires years of effort, instruction and drill. When adults perform mental arithmetic, they activate nonsymbolic, approximate number representations, and their performance suffers if this nonsymbolic system is impaired. Nonsymbolic number representations also allow adults, children, and even infants to add or subtract pairs of dot arrays and to compare the resulting sum or difference to a third array, provided that only approximate accuracy is required. Here we report that young children, who have mastered verbal counting and are on the threshold of arithmetic instruction, can build on their nonsymbolic number system to perform symbolic addition and subtraction. Children across a broad socio-economic spectrum solved symbolic problems involving approximate addition or subtraction of large numbers, both in a laboratory test and in a school setting. Aspects of symbolic arithmetic therefore lie within the reach of children who have learned no algorithms for manipulating numerical symbols. Our findings help to delimit the sources of children's difficulties learning symbolic arithmetic, and they suggest ways to enhance children's engagement with formal mathematics.

Unlabelled: Background and aims. In order to develop arithmetic expertise, children must understand arithmetic principles, such as the inverse relationship between addition and subtraction, in addition to learning calculation skills. We report two experiments that investigate children's understanding of the principle of inversion and the relationship between their conceptual understanding and arithmetical skills., Sample: A group of 127 children from primary schools took part in the study. The children were from 2 age groups (6-7 and 8-9 years)., Methods: Children's accuracy on inverse and control problems in a variety of presentation formats and in canonical and non-canonical forms was measured. Tests of general arithmetic ability were also administered., Results: Children consistently performed better on inverse than control problems, which indicates that they could make use of the inverse principle. Presentation format affected performance: picture presentation allowed children to apply their conceptual understanding flexibly regardless of the problem type, while word problems restricted their ability to use their conceptual knowledge. Cluster analyses revealed three subgroups with different profiles of conceptual understanding and arithmetical skill. Children in the 'high ability' and 'low ability' groups showed conceptual understanding that was in-line with their arithmetical skill, whilst a 3rd group of children had more advanced conceptual understanding than arithmetical skill., Conclusions: The three subgroups may represent different points along a single developmental path or distinct developmental paths. The discovery of the existence of the three groups has important consequences for education. It demonstrates the importance of considering the pattern of individual children's conceptual understanding and problem-solving skills.