36,634 research outputs found

    Dimensions and values for legal CBR

    Get PDF
    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

    Get PDF
    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

    Get PDF
    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 DD was derived using the FDF galaxy volume number densities in the spatially homogeneous standard cosmological model with Ωm0=0.3\Omega_{m_0}=0.3, ΩΛ0=0.7\Omega_{\Lambda_0}=0.7 and H_0=70 \; \mbox{km} \; {\mbox{s}}^{-1} \; {\mbox{Mpc}}^{-1}. The ratio between the differential and integral number densities γ\gamma and γ∗\gamma^\ast obtained from the red and blue FDF galaxies provides a direct method to estimate DD, implying that γ\gamma and γ∗\gamma^\ast vary as power-laws with the cosmological distances. The luminosity distance dLd_{\scriptscriptstyle L}, galaxy area distance dGd_{\scriptscriptstyle G} and redshift distance dzd_z were plotted against their respective number densities to calculate DD 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 z≈1.3z \approx 1.3 or z≈1.9z \approx 1.9 depending on the chosen cosmological distance. The average fractal dimension calculated using γ∗\gamma^\ast changes from ⟨D⟩=1.4−0.6+0.7\langle D \rangle=1.4^{\scriptscriptstyle +0.7}_{\scriptscriptstyle -0.6} to ⟨D⟩=0.5−0.4+1.2\langle D \rangle=0.5^{\scriptscriptstyle +1.2}_{\scriptscriptstyle -0.4} for all galaxies, and DD decreases as zz increases. Small values of DD at high zz 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 ⟨D⟩=0.6±0.1\langle D \rangle =0.6 \pm 0.1 in the far-infrared sources of the Herschel/PACS evolutionary probe (PEP) at 1.5≲z≲3.21.5 \lesssim z \lesssim 3.2 are also mentioned.Comment: LaTex, 15 pages, 28 figures, 4 tables. To appear in "Physica A

    Doing and Showing

    Get PDF
    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
    • …
    corecore