A relatively large number of 8th-grade public middle school students in the United States, particularly in urban communities, are not performing at acceptable levels in mathematics. One concept that poses significant difficulty for these students and negatively affects their overall mathematics achievement is fractions. Many researchers have attributed these difficulties primarily to traditional fraction instruction that emphasizes procedural rather than conceptual knowledge. Therefore this study was designed to investigate how students use their computational and conceptual knowledge and fraction representations to solve fraction-related performance-based mathematical tasks. Social constructivism was used as the theoretical framework in examining conceptual knowledge related to learning fractions. This qualitative study was implemented in an urban middle school in the southeast. It involved an initial sample of 37, 8th-grade, African American pre-algebra students who completed a fraction interest questionnaire and two fraction pretests. During the implementation period, 34 students in the researcher’s pre-algebra class completed three performance-based tasks, three reflection logs, and participated in an interview after completing each task. Of the 34 students who completed all tasks, three were purposefully selected as the informants for the study. In addition, observations, field notes, and artifacts (student work) were utilized to facilitate triangulation of the data. The findings of the study indicated the informants could compute fractions with an average of 85% of mastery but could conceptualize fractions only to a small extent. This validated prior findings and led to the conclusion that student deficiency with fractions results primarily from their level of conceptual knowledge. In the investigation of the ways in which 8th-grade students use fraction representations, this study found the informants used representations to develop a visual map of their mathematical thinking and reasoning and to check the accuracy of their computations. Therefore, this study suggests, when students’ mathematical learning experiences relative to fractions have not emphasized the use of representations to develop conceptual knowledge, they may not be comfortable with the accuracy of the solutions demonstrated in their fractions models.

Lorsqu'on étudie les systèmes dynamiques engendrés par une famille de polynômes, il apparait naturellement des courbes algébriques de type cyclotomique, contenant des points périodiques ou prépériodiques. Dans le cas périodique de la famille zd + c, le premier chapitre de cette thèse montre, en collaboration avec Ou, que ces courbes sont toutes lisses et irréductibles, généralisant les résultats connus au cas d=2. Dans le cas prépériodique de la même famille, le deuxième chapitre de la thèse montre, contre tout attendu, que ces courbes sont en général réductibles. En plus, il y contient une caractérisation des composantes irréductibles ainsi que leur relation géométrique et analytique. Le deuxième thème de cette thèse concerne un nouveau sujet développé par W. Thurston, il s'agit d'entropie noyau des polynômes. Thurston a donné un algorithme, sans preuve, pour calculer ces entropies. La thèse contient une preuve rigoureuse de cet algorithme ainsi que des nouvelles méthodes pour étudier la variation de ces entropies en jonglant plusieurs points de vue. Le dernier thème de cette thèse donne une condition nécessaire et suffisante pour qu'une fraction rationnelle possède un compact errant plein dans son ensemble de Julia. On savait que dans le cas particulier des polynômes ce genre de compact ne pouvait pas du tout exister.

Temos, nagrinėjamos šioje disertacijoje, buvo doktorantūros studijų Vilniaus universiteto Matematikos ir informatikos fakultete objektas. Pateikti tyrimai yra susiję su sudėtinių skaičių egzistavimu tokiose sekose kaip fiksuoto skaičiaus laipsnių sveikųjų dalių seka bei tiesinė rekurentinė seka, sudaryta iš sveikųjų skaičių. Disertaciją sudaro įvadas, 3 skyriai, išvados ir literatūros sąrašas. Pirmame skyriuje nagrinėjami sudėtiniai skaičiai racionaliųjų skaičių laipsnių sveikųjų dalių sekoje bei yra įrodoma, kad sekoje [ξ(5/4)^n], n=1,2,..., kur ξ yra bet koks teigiamas skaičius, yra be galo daug sudėtinių skaičių. Be to, įrodoma, kad yra be galo daug tokių natūraliųjų skaičių n, kad ([ξ(5/4)^n]; 6006)>1, čia 6006 = 2•3•7•11•13. Įrodoma panašių rezultatų pastumtoms kai kurių kitų racionaliųjų skaičių sekoms. Pavyzdžiui, tas pats įrodoma sveikųjų skaičių, esančių arčiausiai ξ(5/3)^n bei ξ(7/5)^n, n=1,2,..., sekoms. Vėlgi nurodomos atitinkamos galimų daliklių aibės. Antrame skyriuje nagrinėjami sudėtiniai skaičiai antros eilės tiesinėse rekurentinėse sekose bei įrodoma, kad kiekvienai tokiai sveikųjų skaičių porai (a; b), kad b≠0 ir (a; b)≠(±2; -1), egzistuoja tokie du natūralieji tarpusavyje pirminiai skaičiai x_1, x_2, kad sekoje, apibrėžtoje lygtimi x_{n+1}=ax_n+bx_{n-1}, n=2,3,..., visų narių moduliai yra sudėtiniai skaičiai. Trečiame skyriuje egiptietiškų trupmenų kontekste nagrinėjamos skaičių, užrašomų tam tikru tiesiniu pavidalu, aibės. Ieškoma, kokie skaičiai... [toliau žr. visą tekstą] / The topics examined in this thesis were the subject of my research as a PhD student at the Faculty of Mathematics and Informatics of Vilnius University. The presented investigation concerns the existence of composite numbers in some special sequences, such as the sequence of integer parts of powers of a fixed number and a linear recurrence sequence consisting of integer numbers. The thesis consists of the introduction, 3 sections, conclusions and bibliography. In Section 1 we consider composite numbers in the sequences of integer parts of powers of rational numbers and prove that the sequence [ξ(5/4)^n], n=1,2,..., where ξ is an arbitrary positive number, contains infinitely many composite numbers. Furthermore, it is shown that there are infinitely many positive integers n such that ([ξ(5/4)^n]; 6006)>1, where 6006 = 2•3•7•11•13. Similar results are obtained for shifted powers of some other rational numbers. In particular, the same is proved for the sets of integers nearest to ξ(5/3)^n and to ξ(7/5)^n, n=1,2,.... The corresponding sets of possible divisors are also described. In Section 2 we consider composite numbers in the binary linear recurrence sequences and prove that for every pair of integer numbers (a; b), where b≠0 and (a; b)≠(±2; -1), there exist two positive relatively prime composite integers x_1, x_2 such that the sequence given by x_{n+1}=ax_n+bx_{n-1}, n=2,3,..., consists of composite terms only, i.e., the absolute value of each term is a composite integer... [to full text]

The topics examined in this thesis were the subject of my research as a PhD student at the Faculty of Mathematics and Informatics of Vilnius University. The presented investigation concerns the existence of composite numbers in some special sequences, such as the sequence of integer parts of powers of a fixed number and a linear recurrence sequence consisting of integer numbers. The thesis consists of the introduction, 3 sections, conclusions and bibliography. In Section 1 we consider composite numbers in the sequences of integer parts of powers of rational numbers and prove that the sequence [ξ(5/4)^n], n=1,2,..., where ξ is an arbitrary positive number, contains infinitely many composite numbers. Furthermore, it is shown that there are infinitely many positive integers n such that ([ξ(5/4)^n]; 6006)>1, where 6006 = 2•3•7•11•13. Similar results are obtained for shifted powers of some other rational numbers. In particular, the same is proved for the sets of integers nearest to ξ(5/3)^n and to ξ(7/5)^n, n=1,2,.... The corresponding sets of possible divisors are also described. In Section 2 we consider composite numbers in the binary linear recurrence sequences and prove that for every pair of integer numbers (a; b), where b≠0 and (a; b)≠(±2; -1), there exist two positive relatively prime composite integers x_1, x_2 such that the sequence given by x_{n+1}=ax_n+bx_{n-1}, n=2,3,..., consists of composite terms only, i.e., the absolute value of each term is a composite integer... [to full text] / Temos, nagrinėjamos šioje disertacijoje, buvo doktorantūros studijų Vilniaus universiteto Matematikos ir informatikos fakultete objektas. Pateikti tyrimai yra susiję su sudėtinių skaičių egzistavimu tokiose sekose kaip fiksuoto skaičiaus laipsnių sveikųjų dalių seka bei tiesinė rekurentinė seka, sudaryta iš sveikųjų skaičių. Disertaciją sudaro įvadas, 3 skyriai, išvados ir literatūros sąrašas. Pirmame skyriuje nagrinėjami sudėtiniai skaičiai racionaliųjų skaičių laipsnių sveikųjų dalių sekoje bei yra įrodoma, kad sekoje [ξ(5/4)^n], n=1,2,..., kur ξ yra bet koks teigiamas skaičius, yra be galo daug sudėtinių skaičių. Be to, įrodoma, kad yra be galo daug tokių natūraliųjų skaičių n, kad ([ξ(5/4)^n]; 6006)>1, čia 6006 = 2•3•7•11•13. Įrodoma panašių rezultatų pastumtoms kai kurių kitų racionaliųjų skaičių sekoms. Pavyzdžiui, tas pats įrodoma sveikųjų skaičių, esančių arčiausiai ξ(5/3)^n bei ξ(7/5)^n, n=1,2,..., sekoms. Vėlgi nurodomos atitinkamos galimų daliklių aibės. Antrame skyriuje nagrinėjami sudėtiniai skaičiai antros eilės tiesinėse rekurentinėse sekose bei įrodoma, kad kiekvienai tokiai sveikųjų skaičių porai (a; b), kad b≠0 ir (a; b)≠(±2; -1), egzistuoja tokie du natūralieji tarpusavyje pirminiai skaičiai x_1, x_2, kad sekoje, apibrėžtoje lygtimi x_{n+1}=ax_n+bx_{n-1}, n=2,3,..., visų narių moduliai yra sudėtiniai skaičiai. Trečiame skyriuje egiptietiškų trupmenų kontekste nagrinėjamos skaičių, užrašomų tam tikru tiesiniu pavidalu, aibės. Ieškoma, kokie skaičiai... [toliau žr. visą tekstą]

The aims and objectives: The aims and objectives of this study were to characterize and compare the airborne particulate matter in the tankhouse and crusher areas of a base metal refinery sampled with two separate methods, in terms of mass concentration, nickel content, and particle size distribution. Methods: Area sampling was conducted in the two areas. Two methods were applied to collect particulate samples. The first is a multi–stage virtual impactor, the Respicon, which was used to determine the three critical particle fractions (inhalable, thoracic and respirable). The NIOSH 7300 method determined the particle concentration and nickel percentage present in each fraction. Using formulas provided by the manufacturers two additional particle–size fractions (extra–thoracic and trachea–bronchial) could be calculated. The second was based on the standard NIOSH 0500 method, which determined particle size distribution depicted as cumulative percentages. The samples were analyzed using laser scattering instrumentation. Results: In the tankhouse the highest level of exposure was to particles bigger than 10 um, with the highest nickel percentage also falling into this range. However, high nickel percentages were present in all three cut–off sizes (4 um, 10 um and > 10 um). The particle concentration for the crusher area was the highest for particulates bigger than 10 um, with the highest nickel percentage present in this fraction. After comparing the tankhouse and crusher areas, it is clear that the particle concentration is much higher in the crusher area according to all sampling methods used. The nickel content present in the analysis of these areas is of great concern. Conclusion: With the knowledge obtained through this research one hopes to establish a basis for particle size sampling in the platinum mining industry. This may lead to the development of health based OEL's and reflect a more accurate evaluation of workers particulate exposure. This information will give a greater understanding of health risks workers are exposed to. / Thesis (M.Sc. (Occupational Hygiene))--North-West University, Potchefstroom Campus, 2011.

The aims and objectives: The aims and objectives of this study were to characterize and compare the airborne particulate matter in the tankhouse and crusher areas of a base metal refinery sampled with two separate methods, in terms of mass concentration, nickel content, and particle size distribution. Methods: Area sampling was conducted in the two areas. Two methods were applied to collect particulate samples. The first is a multi–stage virtual impactor, the Respicon, which was used to determine the three critical particle fractions (inhalable, thoracic and respirable). The NIOSH 7300 method determined the particle concentration and nickel percentage present in each fraction. Using formulas provided by the manufacturers two additional particle–size fractions (extra–thoracic and trachea–bronchial) could be calculated. The second was based on the standard NIOSH 0500 method, which determined particle size distribution depicted as cumulative percentages. The samples were analyzed using laser scattering instrumentation. Results: In the tankhouse the highest level of exposure was to particles bigger than 10 um, with the highest nickel percentage also falling into this range. However, high nickel percentages were present in all three cut–off sizes (4 um, 10 um and > 10 um). The particle concentration for the crusher area was the highest for particulates bigger than 10 um, with the highest nickel percentage present in this fraction. After comparing the tankhouse and crusher areas, it is clear that the particle concentration is much higher in the crusher area according to all sampling methods used. The nickel content present in the analysis of these areas is of great concern. Conclusion: With the knowledge obtained through this research one hopes to establish a basis for particle size sampling in the platinum mining industry. This may lead to the development of health based OEL's and reflect a more accurate evaluation of workers particulate exposure. This information will give a greater understanding of health risks workers are exposed to. / Thesis (M.Sc. (Occupational Hygiene))--North-West University, Potchefstroom Campus, 2011.

This paper combines data from two different international research studies that used problem posing in analyzing elementary teacher candidates’ understanding of rational numbers. In 2007, a mathematics educator from the United States and a mathematician from Northern Ireland collaborated to investigate their respective elementary teacher candidates’ understanding of addition and division of fractions. A year later, the same US mathematics educator collaborated with a mathematics educator from South Africa on a similar research project that focused solely on the addition of fractions. The results of both studies show that elementary teacher candidates from the three different continents share similar misconceptions regarding the addition of fractions. The misconceptions that emerged were analyzed and used in designing teaching strategies intended to improve elementary teacher candidates’ understanding of rational numbers. The research also suggests that problem posing may improve their understanding of addition of fractions.

Diss. (sammanfattning) Linköping : Linköpings universitet, 2005. / Härtill 4 uppsatser.

Conselho Nacional de Desenvolvimento Científico e Tecnológico / In spite of the great importance of the soil organic matter for the sustainability of the natural systems and of agricultural production, its study remains incipient in the area of the Edge of the Rio Grande do Sul Plateau, where different research have been reporting negative aspects of the inadequate soil use. In that sense, the carbon distribution evaluation among the chemically separated fractions can supply subsidies to analyze the effects of land use on those sites. Besides that, it is important to know the role carried out by the different geomorphological features present in the area, once that the variation of the soil attributes can be different in each segment of the landscape. The purpose of this research was to evaluate the influence of the land use on the soil organic matter fractions, chemical attributes and aggregates distribution in different soil depths and landscape positions, analyzing three topossequences of soils of hillsides areas of the Edge of the Plateau. Soil characteristics such morphological, environmental, chemistries and physical, were analyzed for each soil horizon and layer. The chemical fractionation of the soil organic matter was realized to quantificate the carbon amount of the humic fractions and to determinate the amount of Fe+3 and Al3+ co-extracted with the humic substances. The results allowed to verify that the chemical and physical attributes of the soil were effected by the land use, and the position of the profile in the landscape has an important effect in the distribution of the materials in the studied topossequences. The forest showed the largest indexes of aggregation of the soil in the superficial layer and in the top of the landscape, where the larger amount of total organic carbon was verified. The amount of carbon in the fractions extracts of the soil organic matter was different for each land uses. The acid fraction carbon showed low amount in relation to the other extracts. The humina fraction corresponded to the most sensitive indicator to the change of use of the soil. Most of the soil organic matter was distributed in the alkaline-soluble fractions. However, higher values were observed at the forest due to the soil litter. Among the alkaline-soluble fractions, the humic acids were predominant, but they showed variations according to the position in the landscape. This fraction associates preferentially with Fe+3, while Al+3 showed larger affinity for the fulvic acids. / Apesar da grande importância da matéria orgânica do solo para a sustentabilidade dos sistemas naturais e de produção agrícola, seu estudo ainda é incipiente na região do Rebordo do Planalto (RS), onde diferentes trabalhos têm relatado aspectos negativos da utilização inadequada destes solos. Nesse sentido, a avaliação da distribuição do carbono entre as frações separadas quimicamente pode fornecer subsídios para analisar os efeitos de uso empregado nestes locais. Aliado a isto, é importante conhecer o papel desempenhado pelas diferentes feições geomorfológicas presentes na região, já que a variação dos atributos do solo pode ser diferente em cada segmento da paisagem. Desta maneira, o presente trabalho visou avaliar a influência do uso da terra sobre as frações da matéria orgânica, atributos químicos e distribuição de agregados em diferentes profundidades do solo e posições na paisagem, analisando três topossequências de solos de áreas de encostas do Rebordo do Planalto (RS). Para isso, foram determinadas características morfológicas, ambientais, químicas e físicas dos horizontes e camadas do solo. Nesta última, procedeu-se o fracionamento químico da matéria orgânica do solo para quantificação dos teores de carbono das frações húmicas e determinação dos teores de Fe+3 e Al+3 co-extraídos com as substâncias húmicas. Os resultados permitiram verificar que os atributos químicos e físicos do solo foram afetados pelo uso da terra, tendo a posição do perfil na paisagem importante papel na distribuição dos materiais nas topossequências estudadas. A floresta apresentou os maiores índices de agregação do solo na camada superficial e na posição de topo do relevo, onde se verificou maior teor de C total. O teor de carbono nos extratos das frações da matéria orgânica do solo foi diferente para os três usos avaliados. O carbono da fração ácida apresentou baixos teores em relação aos demais extratos avaliados. A fração humina correspondeu ao indicador mais sensível à mudança de uso do solo. Nas três topossequências, a maior parte da matéria orgânica do solo distribuiu-se nas frações alcalino-solúveis. Entretanto, valores mais elevados foram observados na floresta em decorrência da liteira. Dentre as frações alcalino-solúveis, os ácidos húmicos foram predominantes, mas apresentaram variações conforme a posição do perfil na paisagem. Esta fração se associou preferencialmente com o Fe+3, enquanto o Al+3 mostrou maior afinidade pelos ácidos fúlvicos.

[pt] Os processos de destilação atmosférica e a vácuo do petróleo têm como sub-produtos mais pesados os resíduos atmosféricos e de vácuo, respectivamente. Modelos encontrados na literatura não descrevem bem a dependência da viscosidade com a temperatura para estas frações. Há grandes diferenças entre valores experimentais e calculados e no comportamento da curva temperatura X viscosidade. As viscosidades dinâmicas de sete diferentes amostras de resíduos foram determinadas experimentalmente. As amostras foram caracterizadas através de ensaios de: densidade, SARA, concentração de asfaltenos e destilação simulada. Com base em suas características físicas são propostos modelos de viscosidade de frações pesadas de petróleo. / [en] Atmospheric and vacuum residua are the heaviest byproducts of the distillation processes. Literature models don't describe well the viscosity dependency on temperature of these fractions. There are large differences between experimental and calculated values and at the temperature X viscosity curve. Dynamic viscosities of seven different samples were experimentally determined. The samples were characterized by: density, SARA, asphalthene concentration and simulated distillation. Based on their physical characteristics, viscosity models for heavy fractions of oil are proposed.