28,875 research outputs found
Student perspectives on the relationship between a curve and its tangent in the transition from Euclidean Geometry to Analysis
The tangent line is a central concept in many mathematics and science courses. In this paper we describe a model of students’ thinking – concept images as well as ability in symbolic manipulation – about the tangent line of a curve as it has developed through students’ experiences in Euclidean Geometry and Analysis courses. Data was collected through a questionnaire administered to 196 Year 12 students. Through Latent Class Analysis, the participants were classified in three hierarchical groups representing the transition from a Geometrical Global perspective on the tangent line to an Analytical Local perspective. In the light of this classification, and through qualitative explanations of the students’ responses, we describe students’ thinking about tangents in terms of seven factors. We confirm the model constituted by these seven factors through Confirmatory Factor Analysis
The principle of pointfree continuity
In the setting of constructive pointfree topology, we introduce a notion of
continuous operation between pointfree topologies and the corresponding
principle of pointfree continuity. An operation between points of pointfree
topologies is continuous if it is induced by a relation between the bases of
the topologies; this gives a rigorous condition for Brouwer's continuity
principle to hold. The principle of pointfree continuity for pointfree
topologies and says that any relation which induces
a continuous operation between points is a morphism from to
. The principle holds under the assumption of bi-spatiality of
. When is the formal Baire space or the formal unit
interval and is the formal topology of natural numbers, the
principle is equivalent to spatiality of the formal Baire space and formal unit
interval, respectively. Some of the well-known connections between spatiality,
bar induction, and compactness of the unit interval are recast in terms of our
principle of continuity.
We adopt the Minimalist Foundation as our constructive foundation, and
positive topology as the notion of pointfree topology. This allows us to
distinguish ideal objects from constructive ones, and in particular, to
interpret choice sequences as points of the formal Baire space
On Salem numbers, expansive polynomials and Stieltjes continued fractions
A converse method to the Construction of Salem (1945) of convergent families
of Salem numbers is investigated in terms of an association between Salem
polynomials and Hurwitz quotients via expansive polynomials of small Mahler
measure. This association makes use of Bertin-Boyd's Theorem A (1995) of
interlacing of conjugates on the unit circle; in this context, a Salem number
is produced and coded by an m-tuple of positive rational numbers
characterizing the (SITZ) Stieltjes continued fraction of the corresponding
Hurwitz quotient (alternant). The subset of Stieltjes continued fractions over
a Salem polynomial having simple roots, not cancelling at , coming from
monic expansive polynomials of constant term equal to their Mahler measure, has
a semigroup structure. The sets of corresponding generalized Garsia numbers
inherit this semi-group structure.Comment: 35 pages, Journal de Th{\'e}orie des nombres de Bordeaux, Soumissio
The age of imagination: imagining play and invention: implications for creative development
This paper presents findings from The Irish Neighbourhood Play Study; a national, cross-border research project which recorded children’s play patterns in Ireland during 2012. The study incorporated 1688 families across 240 communities. Data was established on the play choices of children aged from birth to 14 years. Multiple differentials were explored including socio-economic and geographical environments.
This paper focuses on the findings within imaginary play patterns for the full cohort. As such, it presents the play patterns for imaginary play in children aged birth-14 years. The findings are discussed in the context of developmental patterns with particular emphasis on the relationship between imaginary play and the development of creativity.
Creativity is a key concept within contemporary education. Its central nexus is problem solving in the face of uncertainty. Within a rapidly changing world, it is a key skill requirement for today’s children as they grow towards efficacy within instability. The relationship between the development of creativity and children’s engagement with imaginary play practices are explored in this paper.
©IATED (2017). Reproduced in Research Online with permission
Multiresolution hierarchy co-clustering for semantic segmentation in sequences with small variations
This paper presents a co-clustering technique that, given a collection of
images and their hierarchies, clusters nodes from these hierarchies to obtain a
coherent multiresolution representation of the image collection. We formalize
the co-clustering as a Quadratic Semi-Assignment Problem and solve it with a
linear programming relaxation approach that makes effective use of information
from hierarchies. Initially, we address the problem of generating an optimal,
coherent partition per image and, afterwards, we extend this method to a
multiresolution framework. Finally, we particularize this framework to an
iterative multiresolution video segmentation algorithm in sequences with small
variations. We evaluate the algorithm on the Video Occlusion/Object Boundary
Detection Dataset, showing that it produces state-of-the-art results in these
scenarios.Comment: International Conference on Computer Vision (ICCV) 201
Children\u27s choices: the technology choices that children make within their free time. Influences and implications
The Irish Neighbourhood Play Research Project included almost 1700 families and 240 communities throughout Ireland. Using parental surveys and naturalistic observation, data was secured on how children in modern Ireland aged 0-14 are spending their free time. An all-island approach was taken incorporating cities, towns and rural areas across a variety of socio-economic groupings. Interesting findings arose from the data relating to the choices that children are making within their free time. This paper focuses on the choices they are making within technology use. Data on the children’s technological engagement will be presented and discussed through a child development lens. The positive and negative implications for both learning and development are raised. This generation of children will be the first to emerge into adulthood without ever experiencing a world without technology. For them, it will always have been central to their existence. What does this mean for the next generation of humanity?
©IATED (2016). Permission granted by IATED for inclusion in ResearchOnline@N
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