36,639 research outputs found
Dimensions and values for legal CBR
We build on two recent attempts to formalise reasoning with dimensions which effectively map dimensions into factors. These enable propositional reasoning, but sometimes a balance between dimensions needs to be struck, and to permit trade offs we need to keep the magnitudes and so reason more geometrically. We discuss dimensions and values, arguing that values can play several distinct roles, both explaining preferences between factors and indicating the purposes of the law
Number sense : the underpinning understanding for early quantitative literacy
The fundamental meaning of Quantitative Literacy (QL) as the application of quantitative knowledge or reasoning in new/unfamiliar contexts is problematic because how we acquire knowledge, and transfer it to new situations, is not straightforward. This article argues that in the early development of QL, there is a specific corpus of numerical knowledge which learners need to integrate into their thinking, and to which teachers should attend. The paper is a rebuttal to historically prevalent (and simplistic) views that the terrain of early numerical understanding is little more than simple counting devoid of cognitive complexity. Rather, the knowledge upon which early QL develops comprises interdependent dimensions: Number Knowledge, Counting Skills and Principles, Nonverbal Calculation, Number Combinations and Story Problems - summarised as Number Sense. In order to derive the findings for this manuscript, a realist synthesis of recent Education and Psychology literature was conducted. The findings are of use not only when teaching very young children, but also when teaching learners who are experiencing learning difficulties through the absence of prerequisite numerical knowledge. As well distilling fundamental quantitative knowledge for teachers to integrate into practice, the review emphasises that improved pedagogy is less a function of literal applications of reported interventions, on the grounds of perceived efficacy elsewhere, but based in refinements of teachers' understandings. Because teachers need to adapt instructional sequences to the actual thinking and learning of learners in their charge, they need knowledge that allows them to develop their own theoretical understanding rather than didactic exhortations
Fractal analysis of the galaxy distribution in the redshift range 0.45 < z < 5.0
Evidence is presented that the galaxy distribution can be described as a
fractal system in the redshift range of the FDF galaxy survey. The fractal
dimension was derived using the FDF galaxy volume number densities in the
spatially homogeneous standard cosmological model with ,
and H_0=70 \; \mbox{km} \; {\mbox{s}}^{-1} \;
{\mbox{Mpc}}^{-1}. The ratio between the differential and integral number
densities and obtained from the red and blue FDF
galaxies provides a direct method to estimate , implying that and
vary as power-laws with the cosmological distances. The
luminosity distance , galaxy area distance
and redshift distance were plotted against
their respective number densities to calculate by linear fitting. It was
found that the FDF galaxy distribution is characterized by two single fractal
dimensions at successive distance ranges. Two straight lines were fitted to the
data, whose slopes change at or depending on
the chosen cosmological distance. The average fractal dimension calculated
using changes from to for all galaxies, and decreases as
increases. Small values of at high mean that in the past galaxies were
distributed much more sparsely and the large-scale galaxy structure was then
possibly dominated by voids. Results of Iribarrem et al. (2014,
arXiv:1401.6572) indicating similar fractal features with in the far-infrared sources of the Herschel/PACS evolutionary
probe (PEP) at are also mentioned.Comment: LaTex, 15 pages, 28 figures, 4 tables. To appear in "Physica A
Doing and Showing
The persisting gap between the formal and the informal mathematics is due to
an inadequate notion of mathematical theory behind the current formalization
techniques. I mean the (informal) notion of axiomatic theory according to which
a mathematical theory consists of a set of axioms and further theorems deduced
from these axioms according to certain rules of logical inference. Thus the
usual notion of axiomatic method is inadequate and needs a replacement.Comment: 54 pages, 2 figure
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