Spatial and verbal routes to number comparison in young children

The ability to compare the numerical magnitude of symbolic numbers represents a milestone in the development of numerical skills. However, it remains unclear how basic numerical abilities contribute to the understanding of symbolic magnitude and whether the impact of these abilities may vary when symbolic numbers are presented as number words (e.g., \u201csix vs. eight\u201d) vs. Arabic numbers (e.g., 6 vs. 8). In the present study on preschool children, we show that comparison of number words is related to cardinality knowledge whereas the comparison of Arabic digits is related to both cardinality knowledge and the ability to spatially map numbers. We conclude that comparison of symbolic numbers in preschool children relies on multiple numerical skills and representations, which can be differentially weighted depending on the presentation format. In particular, the spatial arrangement of digits on the number line seems to scaffold the development of a \u201cspatial route\u201d to understanding the exact magnitude of numerals

The exact (up to infinitesimals) infinite perimeter of the Koch snowflake and its finite area

The Koch snowflake is one of the first fractals that were mathematically described. It is interesting because it has an infinite perimeter in the limit but its limit area is finite. In this paper, a recently proposed computational methodology allowing one to execute numerical computations with infinities and infinitesimals is applied to study the Koch snowflake at infinity. Numerical computations with actual infinite and infinitesimal numbers can be executed on the Infinity Computer being a new supercomputer patented in USA and EU. It is revealed in the paper that at infinity the snowflake is not unique, i.e., different snowflakes can be distinguished for different infinite numbers of steps executed during the process of their generation. It is then shown that for any given infinite number n of steps it becomes possible to calculate the exact infinite number, Nn, of sides of the snowflake, the exact infinitesimal length, Ln, of each side and the exact infinite perimeter, Pn, of the Koch snowflake as the result of multiplication of the infinite Nn by the infinitesimal Ln. It is established that for different infinite n and k the infinite perimeters Pn and Pk are also different and the difference can be infinite. It is shown that the finite areas An and Ak of the snowflakes can be also calculated exactly (up to infinitesimals) for different infinite n and k and the difference An − Ak results to be infinitesimal. Finally, snowflakes constructed starting from different initial conditions are also studied and their quantitative characteristics at infinity are computed

Developmental dyscalculia and low numeracy in Chinese children

Children struggle with mathematics for different reasons. Developmental dyscalculia and low numeracy - two kinds of mathematical difficulties - may have their roots, respectively, in poor understanding of exact non-symbolic numerosities and of symbolic numerals. This study was the first to explore whether Chinese children, despite cultural and linguistic factors supporting their mathematical learning, also showed such mathematical difficulties and whether such difficulties have measurable impact on children's early school mathematical performance. First-graders, classified as dyscalculia, low numeracy, or normal achievement, were compared for their performance in various school mathematical tasks requiring a grasp of non-symbolic numerosities (i.e., non-symbolic tasks) or an understanding of symbolic numerals (i.e., symbolic tasks). Children with dyscalculia showed poorer performance than their peers in non-symbolic tasks but not symbolic ones, whereas those with low numeracy showed poorer performance in symbolic tasks but not non-symbolic ones. As hypothesized, these findings suggested that dyscalculia and low numeracy were distinct deficits and caused by deficits in non-symbolic and symbolic processing, respectively. These findings went beyond prior research that only documented generally low mathematical achievements for these two groups of children. Moreover, these deficits appeared to be persistent and could not be remedied simply through day-to-day school mathematical learning. The present findings highlighted the importance of tailoring early learning support for children with these distinct deficits, and pointed to future directions for the screening of such mathematical difficulties among Chinese children. © 2013 Elsevier Ltd.postprin

ANN-based Innovative Segmentation Method for Handwritten text in Assamese

Artificial Neural Network (ANN) s has widely been used for recognition of optically scanned character, which partially emulates human thinking in the domain of the Artificial Intelligence. But prior to recognition, it is necessary to segment the character from the text to sentences, words etc. Segmentation of words into individual letters has been one of the major problems in handwriting recognition. Despite several successful works all over the work, development of such tools in specific languages is still an ongoing process especially in the Indian context. This work explores the application of ANN as an aid to segmentation of handwritten characters in Assamese- an important language in the North Eastern part of India. The work explores the performance difference obtained in applying an ANN-based dynamic segmentation algorithm compared to projection- based static segmentation. The algorithm involves, first training of an ANN with individual handwritten characters recorded from different individuals. Handwritten sentences are separated out from text using a static segmentation method. From the segmented line, individual characters are separated out by first over segmenting the entire line. Each of the segments thus obtained, next, is fed to the trained ANN. The point of segmentation at which the ANN recognizes a segment or a combination of several segments to be similar to a handwritten character, a segmentation boundary for the character is assumed to exist and segmentation performed. The segmented character is next compared to the best available match and the segmentation boundary confirmed

Why 'scaffolding' is the wrong metaphor : the cognitive usefulness of mathematical representations.

The metaphor of scaffolding has become current in discussions of the cognitive help we get from artefacts, environmental affordances and each other. Consideration of mathematical tools and representations indicates that in these cases at least (and plausibly for others), scaffolding is the wrong picture, because scaffolding in good order is immobile, temporary and crude. Mathematical representations can be manipulated, are not temporary structures to aid development, and are refined. Reflection on examples from elementary algebra indicates that Menary is on the right track with his ‘enculturation’ view of mathematical cognition. Moreover, these examples allow us to elaborate his remarks on the uniqueness of mathematical representations and their role in the emergence of new thoughts.Peer reviewe