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Mathematical Concepts (mathematical + concept)
Selected AbstractsTransfer of Mathematical Knowledge: The Portability of Generic InstantiationsCHILD DEVELOPMENT PERSPECTIVES, Issue 3 2009Jennifer A. Kaminski Abstract, Mathematical concepts are often difficult to acquire. This difficulty is evidenced by failure of knowledge to transfer to novel analogous situations. One approach to this challenge is to present the learner with a concrete instantiation of the to-be-learned concept. Concrete instantiations communicate more information than their abstract, generic counterparts and, in doing so, they may facilitate initial learning. However, this article argues that extraneous information in concrete instantiations may distract the learner from the relevant mathematical structure and, as a result, hinder transfer. At the same time, generic instantiations, such as traditional mathematical notation, can be learned by both children and adults and can, in turn, allow for transfer, suggesting that generic instantiations result in a portable knowledge representation. [source] Analysis of scattering from polydisperse structure using Mellin convolutionJOURNAL OF APPLIED CRYSTALLOGRAPHY, Issue 2 2006Norbert Stribeck This study extends a mathematical concept for the description of heterogeneity and polydispersity in the structure of materials to multiple dimensions. In one dimension, the description of heterogeneity by means of Mellin convolution is well known. In several papers by the author, the method has been applied to the analysis of data from materials with one-dimensional structure (layer stacks or fibrils along their principal axis). According to this concept, heterogeneous structures built from polydisperse ensembles of structural units are advantageously described by the Mellin convolution of a representative template structure with the size distribution of the templates. Hence, the polydisperse ensemble of similar structural units is generated by superposition of dilated templates. This approach is particularly attractive considering the advantageous mathematical properties enjoyed by the Mellin convolution. Thus, average particle size, and width and skewness of the particle size distribution can be determined from scattering data without the need to model the size distributions themselves. The present theoretical treatment demonstrates that the concept is generally extensible to dilation in multiple dimensions. Moreover, in an analogous manner, a representative cluster of correlated particles (e.g. layer stacks or microfibrils) can be considered as a template on a higher level. Polydispersity of such clusters is, again, described by subjecting the template structure to the generalized Mellin convolution. The proposed theory leads to a simple pathway for the quantitative determination of polydispersity and heterogeneity parameters. Consistency with the established theoretical approach of polydispersity in scattering theory is demonstrated. The method is applied to the best advantage in the field of soft condensed matter when anisotropic nanostructured materials are to be characterized by means of small-angle scattering (SAXS, USAXS, SANS). [source] Framing French Success in Elementary Mathematics: Policy, Curriculum, and PedagogyCURRICULUM INQUIRY, Issue 3 2004FRANCES C. FOWLER ABSTRACT For many decades Americans have been concerned about the effective teaching of mathematics, and educational and political leaders have often advocated reforms such as a return to the basics and strict accountability systems as the way to improve mathematical achievement. International studies, however, suggest that such reforms may not be the best path to successful mathematics education. Through this qualitative case study, the authors explore in depth the French approach to teaching elementary mathematics, using interviews, classroom observations, and documents as their data sets. They apply three theoretical frameworks to their data and find that the French use large-group instruction and a visible pedagogy, focusing on the discussion of mathematical concepts rather than on the completion of practice exercises. The national curriculum is relatively nonprescriptive, and teachers are somewhat empowered through site-based management. The authors conclude that the keys to French success with mathematics education are ongoing formative assessment, mathematically competent teachers, policies and practices that help disadvantaged children, and the use of constructivist methods. They urge comparative education researchers to look beyond international test scores to deeper issues of policy and practice. [source] Understanding waiting lists as the matching of surgical capacity to demand: are we wasting enough surgical time?ANAESTHESIA, Issue 6 2010J. J. Pandit Summary If surgical ,capacity' always matched or exceeded ,demand' then there should be no waiting lists for surgery. However, understanding what is meant by ,demand', ,capacity' and ,matched' requires some mathematical concepts that we outline in this paper. ,Time' is the relevant measure: ,demand' for a surgical team is best understood as the total min required for the surgery booked from outpatient clinics every week; and ,capacity' is the weekly operating time available. We explain how the variation in demand (not just the mean demand) influences the analysis of optimum capacity. However, any capacity chosen in this way is associated with only a likelihood (that is, a probability rather than certainty) of absorbing the prevailing demand. A capacity that suitably absorbs the demand most of the time (for example, > 80% of weeks) will inevitably also involve considerable waste (that is, many weeks in which there is spare, unused capacity). Conversely, a level of capacity chosen to minimise wasted time will inevitably cause an increase in size of the waiting list. Thus the question of how to balance demand and capacity is intimately related to the question of how to balance utilisation and waste. These mathematical considerations enable us to consider objectively how to manage the waiting list. They also enable us critically to analyse the extent to which philosophies adopted by the National Health Service (such as ,Lean' or ,Six Sigma') will be successful in matching surgical capacity to demand. [source] Facilitating student understanding of buffering by an integration of mathematics and chemical concepts,BIOCHEMISTRY AND MOLECULAR BIOLOGY EDUCATION, Issue 2 2004Robert Curtright Abstract We describe a simple undergraduate exercise involving the titration of a weak acid by a strong base using a pH meter and a micropipette. Students then use their data and carry out graphical analyses with a spreadsheet. The analyses involve using mathematical concepts such as first-derivative and semi-log plots and provide an opportunity for collaboration between biochemistry and mathematics instructors. By focusing on titration data, rather than the titration process, and using a variety of graphical transformations, we believe that students achieve a deeper understanding of the concept of buffering. [source] | ||||||||||