Orientador: Vinício de Macedo Santos / Banca: Célia Maria Carolino Pires / Banca: José Carlos Miguel / Abstract: The rational numbers are shown as a subject that the students of the Elementary and High School have difficulties to learn. Some of these difficulties are due to the difference established between the daily use of the rational numbers by the student and the way it is taught at the school and, also for the ignorance, on the part of the school, of the multiplicity of their meanings. While the social use is centered in the decimal form, the school use lies more on the fractional form of the rational numbers. It is an undesirable separation that the school practices have accentuated through time. This study tried to characterize the existent dichotomization between it the use and the teaching of the Mathematics, starting from bibliographical research and of documental study that end up being responsible for damages in the students' learning.. This can be verified in the mistakes committed in the official tests (SARESP, SAEB...). It was sought to analyze how that separation has been reinforced in the official documents, by the pedagogic proposals and curricula. It was verified how the different documents and official publications deal with the rational numbers and the articulation among perspectives of the school use and the daily use of the rational numbers. That analysis made possible to understand different types of arguments and justifications for the teaching of the fractions, present in the official curricula, as well as explain the contents and the most appropriate methodologies of the conceptions presented in such documents. All this made possible to know part of the problems that happen with the teaching of fractions and their causes, and so, make suggestions on how these problems can be solved. Although the establishment of relationships between the social use and school use still doesn't happen in an effective way, it is recognized... (Complete abstract, click electronic address below) / Resumo: Os números racionais apresentam-se como conteúdo que os alunos do Ensino Fundamental e Médio têm dificuldades para aprender. Parte dessas dificuldades decorre da diferença instituída entre o uso cotidiano dos números racionais pelo aluno e a maneira como são ensinados na escola e, também pelo desconhecimento, por parte da escola, da multiplicidade dos significados dos racionais. Enquanto o uso social centra-se na forma decimal o uso escolar recai mais sobre a forma fracionária dos números racionais. É uma separação indesejável que as práticas escolares trataram de acentuar ao longo do tempo. A partir de pesquisa bibliográfica e de estudo documental procurou-se caracterizar, nesse trabalho, a dicotomização existente entre o uso e o ensino da Matemática, que acabam sendo responsáveis por prejuízos na aprendizagem dos alunos. Isto pode ser verificado nos erros que os alunos cometeram nas provas oficiais (SARESP, SAEB...). Procurou-se analisar como essa separação vem sendo reforçada nos documentos oficiais, por meio das propostas pedagógicas e curriculares. Verificaram-se como diferentes documentos e publicações oficiais abordam os números racionais e tratam da articulação entre a perspectivas do uso escolar e a do uso cotidiano dos números racionais. Essa análise possibilitou compreender diferentes tipos de argumentações e justificativas para o ensino das frações, presentes nos currículos oficiais, bem como explicitar os conteúdos e metodologias adequadas às concepções apresentadas em tais documentos. Tudo isso possibilitou conhecer parte dos problemas que ocorrem com o ensino de frações e suas causas e por isso sugerir propostas que sinalizam para a sua superação. Embora o estabelecimento de relações entre o uso social e uso escolar ainda não ocorra de maneira efetiva, reconhece-se que aquelas orientações... (Resumo completo, clicar acesso eletrônico abaixo) / Mestre

Techn. Universiẗat, Diplomarbeit--Ilmenau, 2007. / Parallel als Online-Ausg. erschienen unter der Adresse http://www.db-thueringen.de/servlets/DocumentServlet?id=10122.

Made available in DSpace on 2014-06-11T19:24:21Z (GMT). No. of bitstreams: 0 Previous issue date: 2003Bitstream added on 2014-06-13T18:52:14Z : No. of bitstreams: 1 valera_ar_me_mar.pdf: 594283 bytes, checksum: 7fa747413b18f73739f058ca4ea1146e (MD5) / Os números racionais apresentam-se como conteúdo que os alunos do Ensino Fundamental e Médio têm dificuldades para aprender. Parte dessas dificuldades decorre da diferença instituída entre o uso cotidiano dos números racionais pelo aluno e a maneira como são ensinados na escola e, também pelo desconhecimento, por parte da escola, da multiplicidade dos significados dos racionais. Enquanto o uso social centra-se na forma decimal o uso escolar recai mais sobre a forma fracionária dos números racionais. É uma separação indesejável que as práticas escolares trataram de acentuar ao longo do tempo. A partir de pesquisa bibliográfica e de estudo documental procurou-se caracterizar, nesse trabalho, a dicotomização existente entre o uso e o ensino da Matemática, que acabam sendo responsáveis por prejuízos na aprendizagem dos alunos. Isto pode ser verificado nos erros que os alunos cometeram nas provas oficiais (SARESP, SAEB...). Procurou-se analisar como essa separação vem sendo reforçada nos documentos oficiais, por meio das propostas pedagógicas e curriculares. Verificaram-se como diferentes documentos e publicações oficiais abordam os números racionais e tratam da articulação entre a perspectivas do uso escolar e a do uso cotidiano dos números racionais. Essa análise possibilitou compreender diferentes tipos de argumentações e justificativas para o ensino das frações, presentes nos currículos oficiais, bem como explicitar os conteúdos e metodologias adequadas às concepções apresentadas em tais documentos. Tudo isso possibilitou conhecer parte dos problemas que ocorrem com o ensino de frações e suas causas e por isso sugerir propostas que sinalizam para a sua superação. Embora o estabelecimento de relações entre o uso social e uso escolar ainda não ocorra de maneira efetiva, reconhece-se que aquelas orientações...

Submitted by Maria Suzana Diniz (msuzanad@hotmail.com) on 2015-11-30T12:34:27Z No. of bitstreams: 1 arquivototal.pdf: 766531 bytes, checksum: ad20186d0268a15265279ab809f9fd2f (MD5) / Approved for entry into archive by Maria Suzana Diniz (msuzanad@hotmail.com) on 2015-11-30T12:38:24Z (GMT) No. of bitstreams: 1 arquivototal.pdf: 766531 bytes, checksum: ad20186d0268a15265279ab809f9fd2f (MD5) / Made available in DSpace on 2015-11-30T12:38:24Z (GMT). No. of bitstreams: 1 arquivototal.pdf: 766531 bytes, checksum: ad20186d0268a15265279ab809f9fd2f (MD5) Previous issue date: 2014-08-29 / Coordenação de Aperfeiçoamento de Pessoal de Nível Superior - CAPES / This work is about research done Fibonacci's Numbers. Initially it presents a brief account of the history of Leonardo Fibonacci, from his most famous work,The Liber Abaci, to the relationship with other elds of Mathematics. Then we will introduce some properties of Fibonacci's Numbers, Binet's Form, Lucas' Numbers and the relationship with Fibonacci's Sequence and an important property observed by Fermat. Within relationships with other areas of Mathematics, we show the relationship Matrices, Trigonometry and Geometry. Also presents the Golden Ellipse and the Golden Hyperbola. We conclude with Tribonacci's Numbers and some properties that govern these numbers. Made some generalizations about Matrices and Polynomials Tribonacci. / Este trabalho tem como objetivo o estudo dos Números de Fibonacci. Apresenta-se inicialmente um breve relato sobre a história de Leonardo Fibonacci, desde sua obra mais famosa, O Liber Abaci, até a relação com outros campos da Matemática. Em seguida, apresenta-se algumas propriedades dos Números de Fibonacci, a Fórmula de Binet, os Números de Lucas e a relação com a Sequência de Fibonacci e uma importante propriedade observada por Fermat. Dentro das relações com outras áreas da Matemática, destacamos a relação com as Matrizes, com a Trigonometria, com a Geometria. Apresenta-se também a Elipse e a Hipérbole de Ouro. Concluímos com os Números Tribonacci e algumas propriedades que regem esses números. Realizamos algumas generalizações sobre Matrizes e Polinômios Tribonacci.

Note:

Published Article / In the 13th century an Italian mathematician Fibonacci, also known as Leonardo da Pisa, identified a sequence of numbers that seemed to be repeating and be residing in nature (http://en.wikipedia.org/wiki/Fibonacci) (Kalman, D. et al. 2003: 167). Later a golden ratio was encountered in nature, art and music. This ratio can be seen in the distances in simple geometric figures. It is linked to the Fibonacci numbers by dividing a bigger Fibonacci value by the one just smaller of it. This ratio seems to be settling down to a particular value of 1.618 (http://en.wikipedia.org/wiki/Fibonacci) (He, C. et al. 2002:533) (Cooper, C et al 2002:115) (Kalman, D. et al. 2003: 167) (Sendegeya, A. et al. 2007). Artificial Intelligence or neural networks is the science and engineering of using computers to understand human intelligence (Callan R. 2003:2) but humans and most things in nature abide to Fibonacci numbers and the golden ratio. Since Neural Networks uses the same algorithms as the human brain does, the aim is to experimentally proof that using Fibonacci numbers as weights, and the golden rule as a learning rate, that this might improve learning curve performance. If the performance is improved it should prove that the algorithm for neural network's do represent its nature counterpart. Two identical Neural Networks was coded in LabVIEW with the only difference being that one had random weights and the other (the adapted one) Fibonacci weights. The results were that the Fibonacci neural network had a steeper learning curve. This improved performance with the neural algorithm, under these conditions, suggests that this formula is a true representation of its natural counterpart or visa versa that if the formula is the simulation of its natural counterpart, then the weights in nature is Fibonacci values.

A dissertation presented to the Faculty of Health, Durban University of Technology, for the Masters Degree in Technology: Chiropractic, 2009. / Background: The International Classification of Diseases (ICD) provides codes to classify diseases in such a manner, that every health condition is assigned to a unique category. Some of the most common diagnoses made by chiropractors are not included in the ICD-10 coding system, as it is mainly medically orientated and does not accommodate these diagnoses. This can potentially lead to reimbursement problems for chiropractors in future and create confusion for medical aid schemes as to what conditions chiropractors actually diagnose and treat. Aim: To determine the level of compliancy of chiropractors, in South Africa, to the ICD-10 coding procedure and the factors that may influence the use of correct ICD-10 codes. As well as to determine whether the ICD-10 diagnoses chiropractors commonly submit to the medical aid schemes, reflect the actual diagnoses made in practice. Method: The study was a retrospective survey of a quantitative nature. A self-administered questionnaire was e-mailed and posted to 380 chiropractors, practicing in South Africa. The electronic questionnaires were sent out four times at two week intervals for the duration of eight weeks; and the postal questionnaires sent once. A response rate of 16.5% (n = 63) was achieved. Raw data was received from the divisional manager of the coding unit of Discovery Health (Pty) Ltd. in the form of an excel spreadsheet containing the most common ICD-10 diagnoses made by chiropractors in South Africa, for the period June 2006 to July 2007, who had submitted claims to the Medical Scheme. The spreadsheet also contained depersonalised compliance statistics of chiropractors to the ICD-10 system from July 2006 to October 2008. SPSS version 15 was used for descriptive statistical data analysis (SPSS Inc., Chicago, Ill, USA). Results: The age range of the 63 participants who responded to the questionnaire was 26 to 79 years, with an average of 41 years. The majority of the participants were male (74.6%, n = 47). KwaZulu-Natal had 25 participants (39.6%), Gauteng 17 (26.9%), Western Cape 12 (19%), Eastern Cape four (6.3%), Free State and Mpumalanga two (3.1%) each and North West one (1.5%). The mean knowledge score for ICD-10 coding was 43.5%, suggesting a relatively low level of knowledge. The total percentage of mistakes for electronic claims was higher for both the primary and unlisted claims (3.93% and 2.18%), than for manual claims iv (1.57% and 1.59%). The total percentage of mistakes was low but increased marginally each year for both primary claims (1.43% in 2006; 1.99% in 2007; 2.33% in 2008) and unlisted claims (0% in 2006; 2.61% in 2007; 3.07% in 2008). CASA members were more likely to be aware of assistance offered, in terms of ICD-10 coding through the medical schemes and the association (p = 0.131), than non-members. There was a non-significant trend towards participants who had been on an ICD-10 coding course (47.6%; n = 30), having a greater knowledge of the ICD-10 coding procedures (p = 0.147). Their knowledge was almost 10% higher than those who had not been on a course (52.4%; n = 33). Most participants (38.1%; n = 24) did not use additional cause codes when treating cases of musculoskeletal trauma, nor did they use multiple codes (38.7%; n = 24) when treating more than one condition in the same patient. Nearly 70% of participants (n = 44) used the M99 code in order to code for vertebral subluxation and the majority (79.4%; n = 50) believed the definition of subluxation used in ICD-10 coding to be the same as that which chiropractors use to define subluxation. According to the medical aid data, the top five diagnoses made by chiropractors from 2006 to 2007 were: Low back pain, lumbar region, M54.56 (8996 claims); Cervicalgia, M54.22 (6390 claims); Subluxation complex, cervical region, M99.11 (2895 claims); Other dorsalgia, multiple sites in spine, M54.80 (1524 claims) and Subluxation complex, sacral region, M99.14 (1293 claims). According to the questionnaire data, the top five diagnoses (Table 4.24) were: Lumbar facet syndrome, M54.56 (25%); Lumbar facet syndrome, M99.13 (23.3%); Cervical facet syndrome, M99.11 (21.7%); Cervicogenic headache, G44.2 (20%) and Cervicalgia, M54.22 (20%). Conclusion: The sample of South African chiropractors were fairly compliant to the ICD-10 coding system. Although the two sets of data (i.e. from the medical aid scheme and the questionnaire) regarding the diagnoses that chiropractors make on a daily basis correlate well with each other, there is no consensus in the profession as to which codes to use for chiropractic specific diagnoses. These chiropractic specific diagnoses (e.g. facet syndrome) are however, the most common diagnoses made by chiropractors in private practice. Many respondents indicated that because of this they sometimes use codes that they know will not be rejected, even if it is the incorrect code. For more complicated codes, the majority of respondents indicated that they did not know how to or were not interested in submitting the correct codes to comply with the level of specificity required by the medical aid schemes. The challenge is to make practitioners aware of the advantages of correct coding for the profession.

In this thesis we prove several different results about the number of primes represented by linear functions. The Brun-Titchmarsh theorem shows that the number of primes which are less than x and congruent to a modulo q is less than (C+o(1))x/(phi(q)log{x}) for some value C depending on log{x}/log{q}. Different authors have provided different estimates for C in different ranges for log{x}/log{q}, all of which give C>2 when log{x}/log{q} is bounded. We show in Chapter 2 that one can take C=2 provided that log{x}/log{q}> 8 and q is sufficiently large. Moreover, we also produce a lower bound of size x/(q^{1/2}phi(q)) when log{x}/log{q}>8 and is bounded. Both of these bounds are essentially best-possible without any improvement on the Siegel zero problem. Let k>1 and Pi(n) be the product of k linear functions of the form a_in+b_i for some integers a_i, b_i. Suppose that Pi(n) has no fixed prime divisors. Weighted sieves have shown that for infinitely many integers n, the number of prime factors of Pi(n) is at most r_k, for some integer r_k depending only on k. In Chapter 3 and Chapter 4 we introduce two new weighted sieves to improve the possible values of r_k when k>2. In Chapter 5 we demonstrate a limitation of the current weighted sieves which prevents us proving a bound better than r_k=(1+o(1))klog{k} for large k. Zhang has shown that there are infinitely many intervals of bounded length containing two primes, but the problem of bounded length intervals containing three primes appears out of reach. In Chapter 6 we show that there are infinitely many intervals of bounded length containing two primes and a number with at most 31 prime factors. Moreover, if numbers with up to 4 prime factors have `level of distribution' 0.99, there are infinitely many integers n such that the interval [n,n+90] contains 2 primes and an almost-prime with at most 4 prime factors.

We introduce decompositions determined by Galois field and degree of the space of algebraic numbers modulo torsion and the space of algebraic points on an elliptic curve over a number field. These decompositions are orthogonal with respect to the natural inner product associated to the L² Weil height recently introduced by Allcock and Vaaler in the case of algebraic numbers and the inner product naturally associated to the Néron-Tate canonical height on an elliptic curve. Using these decompositions, we then introduce vector space norms associated to the Mahler measure. For algebraic numbers, we formulate L[superscript p] Lehmer conjectures involving lower bounds on these norms and prove that these new conjectures are equivalent to their classical counterparts, specifically, the classical Lehmer conjecture in the p=1 case and the Schinzel-Zassenhaus conjecture in the p=[infinity] case. / text

The purpose of this study was to determine whether fourth-, sixth-, and eighth-grade students used rational number reasoning to solve comparison of area tasks, and whether the tendency to use such reasoning increased with grade level. The areas to be compared were not similar and therefore, could not directly be compared in a straightforward manner. The most viable solution involved comparing the part-whole relationships inherent in the tasks. Rational numbers in the form of fractional terms could be used to express the part-whole relationships. The use of fractional terms provided a means for students to express the areas to be compared in an abstract manner and thus free themselves from the perceptual aspects of the tasks. The study examined how students solve unique problems in a familiar context where rational number knowledge could be applied. It also noted the effect of introducing fraction symbols into the tasks after students had indicated how they would solve the problems without any reference to fractions. Data were gathered through individual task-based interviews which consisted of 21 tasks, conducted with 36 elementary and junior high school students (12 students each in the fourth, sixth, and eighth grades). Each interview was video and audio taped to provide a record of the students' behavioral and verbal responses. The student responses were analyzed to determine the strategies the students used to solve the comparison of area tasks. The student responses were classified into 11 categories of strategies. There were four Part-Whole Categories, one Part-Whole/Direct Comparison Combination category and six Direct Comparison categories. The results of the study indicate that the development of rational number instruction should include: learning sequences which take students beyond the learning of a set of fraction concepts and skills, attention to the interaction of learning and the visual aspects of instructional models, and the careful inclusion of different types of fractions and other rational number task variables. This study supports the current national developments in curriculum and evaluation standards for mathematics instruction which stress the ability of students to problem solve, communicate, and reason.