Almost twenty years ago Texas implemented a functions-based approach (FBA) to teaching algebra. This approach emphasized the use of nearly all multiple representations, use of a graphing calculator to explore graphs, and modeling of linear and quadratic functions. This interpretation of FBA in conjunction with curriculum placing the teaching of simplifying expressions and solving equations close in sequence may contribute to student confounding of the rules for simplifying and solving. The purpose of this exploratory qualitative study was to explore student rating of the usefulness of teacher-provided and function-based approach (FBA) strategies for simplifying expressions and solving equations in Algebra. The subjects of this study were two algebra teachers and their respective algebra students. The teachers, who taught both Algebra 1 and Algebra 2, were at a high school campus located in an urban district. A researcher created survey based on teacher-provided strategies used by participating teachers was administered to 100 students and 22 teachers. The teacher survey results were used as a professional basis for comparing students' results. Descriptive statistics were used to create graphical representations of students by course groups and identify students who confounded rules. Student FBA preferences and course groups were used to identify 18 student interviewees. Student and teacher interviews were used to corroborate survey results. Participating teachers identified and commented on areas of concern from the survey results. Both teachers approved of the low percentages of students rating FBA strategies as useful but were concerned about higher percentages of students (30% or greater) confounding rules or not realizing the usefulness of relevant sub-strategies. Neither teachers nor students were aware of benefits of graphing calculator use in simplifying. Students, regardless of course group or FBA preference, justified the use of teacher-provided strategies with symbolic manipulation and changed FBA ratings to less likely. There were few student references to equivalence and equality that were supported by FBA. These results are important for algebraic instruction in Texas. Texas has mandated use of graphing calculator on 8th grade Mathematics STAAR exam. Recognizing the benefits of a complete FBA along with effective use of graphing technology may prevent this type of confounding. / text

Research indicates that many elementary students do not comprehend that the equal sign is an indication that an equality relation exists between two structures. Instead, they perceive the equal sign as an indication that a particular procedure is to be performed. As students mature, and as their exposure to the equal sign and equality relations in multiple contexts increases, most obtain the ability to interpret the equal sign as an indicator of an equivalence relation. Incorrect usages of the equal sign, however, by post-algebra students indicate a tendency for students to regress back to a comprehension of the equal sign as an operator symbol or to ignore the equal sign altogether. The purpose of this project was to develop an instrument that is relevant to objectives associated with the interpretation of the equals relation, and to perform a test reliability analysis to assess measurement reliability and construct validity for the instrument. The model that was utilized to develop items for the instrument followed a general item development and validity assessment model proposed by Cangelosi. This model requires an iterative process that includes a peer review of objectives and instrument items by a panel of experts and a revision of the items based upon recommendations from the panel. A pilot test was synthesized from the revised items and administered to a group of subjects, and an instrument reliability analysis and an item efficiency analysis were performed. The quantitative and qualitative data obtained from this process were used to create the 18-item instrument entitled, Wheeler Test for Comprehension of Equals. The researcher recommends further validity assessments for the instrument across multiple settings and subject groups.

<p>This thesis is based on the recognition that there are similarities and differences between two notions of Christian ministry, that is, a &ldquo / discipleship of equals&rdquo / as defined by Elizabeth Sch&uuml / ssler Fiorenza (1989) and a &ldquo / partnership of both men and women&rdquo / as defined by Mercy Amba Oduyoye (1990).</p> <p><br /> In this thesis, Christian ministry is assessed through the perspectives of both feminist theology and African women&rsquo / s theology. The question which is addressed here is how the similarities and differences between Elizabeth Sch&uuml / ssler Fiorenza&rsquo / s notion of Christian ministry as a &ldquo / discipleship of equals&rdquo / and Mercy Amba Oduyoye&rsquo / s notion of Christian ministry as a &ldquo / partnership of both men and women&rdquo / should be understood and assessed. The main purpose of this thesis is to offer a critical comparison of these two female theologians so as to encourage new visions of Christian ministry in the contemporary church and society.</p> <p><br /> To accomplish this task, a literature survey of books, essays, and articles published on the subject of Christian ministry in feminist and African women&rsquo / s theology by African and Western theologians, between 1960 and 2003, was conducted.<br /> Two notions of how Christian ministry should be understood are offered. Fiorenza emphasizes that both women and men need to be recognized as disciples of Christ who can equally, yet within diversity, minister to God&rsquo / s people by virtue of their baptisms. Oduyoye notes that reciprocity and mutuality is crucial for both men and women who minister to God&rsquo / s people by forming strong partnerships through their respective vocations and ministries, by virtue of their baptisms.</p> <p><br /> This thesis argues that a comparison of the views of Elizabeth Sch&uuml / ssler Fiorenza and Mercy Amba Oduyoye on Christian ministry reveals similarities based on a shared experience of being women within a male-dominated Christian church and differences emanating from the different contexts within which they practice theology, namely, that of Euro-American feminist theology and that of African women&rsquo / s theology. It is argued that there are significant differences between feminist theology and African women&rsquo / s theology, with reference to their context, rhetoric, experiences and modes of expression. A critical assessment of Oduyoye&rsquo / s notion of Christian ministry as a &ldquo / partnership of both men and women&rdquo / reveals that there is an internal problem within African theology. The predicament within which African women theologians find themselves is that they have to struggle not only against patriarchy in church and society but also against the remaining distortions within what is deemed to be a more progressive theology. African women theologians acknowledge their solidarity with African theologians but also identify fundamental flaws within African theology. African women theologians are therefore engaged in a battle on more than one side. They need to unmask and support at the same time. On the other hand, it is clear that Fiorenza is far less critical of the feminist movement, from which she derives the term &ldquo / feminist hermeneutics&rdquo / . She is, of course, quite aware of the various and successive strands of the feminist movement, but she remains at least sympathetic to this movement and seeks to explore its significance for biblical scholarship and especially the Roman Catholic Church, of which she is a member. She recognises the need to complement the (sometimes reductionist) secular manifestations of feminism by highlighting the oppressive but also the potentially liberative role which religious traditions may still have in a secularised civil society.</p> <p><br /> One may therefore conclude that African women&rsquo / s theology remains distinct from other feminist theologies. In a similar way, African-American womanist theologians have insisted that their situation is distinct from that of Euro-American women in what is described as the &ldquo / triple&rdquo / oppression of black women: being women, relatively poor, black and formerly enslaved. This calls for further reflection on the similarities and differences between African-American womanist theology and African women&rsquo / s theology. Important differences which come to mind here are the legacy of slavery, differences in economic status, and military power. African women are often engaged in a struggle to secure a sustainable livelihood in ways that African-American women are not. More importantly, the relationship between American black theology and womanist theology deserves further attention in this regard.</p>

Leadership appointment during mergers has a logical and established precedent when there is a clear target and acquirer. However, in the extraordinary case of Merger Of Equals (MOEs) – where this is relative equal ownership, board representation, earnings contribution etc - this process is much less defined and can often have serious consequences on the merger, both in closing negotiations as well as post-merger performance. Intuition assumes the better performing CEO should and will be appointed. In practice, however, that is often not the case. It is arguable that performance can be defined through objective means (financially & operationally), yet CEO appointment usually is resolved through political negotiations and financial concessions. Building upon previous studies of mergers and employing public information and reported financial data in multivariate statistical analyses, this study examines two research questions pertinent to MOE leadership appointment. First – is prior performance of merger CEOs a significant predictor for leadership selection in MOEs, and second - what metrics of past performance by merger CEOs, if any, are significantly correlated with short-term financial and operational success of the merged company? Determining whether or not a relationship exists between past performance and leadership selection, as well as post-merger success can have significant influence on the leadership appointment process for Mergers of Equals.

<p>This thesis is based on the recognition that there are similarities and differences between two notions of Christian ministry, that is, a &ldquo / discipleship of equals&rdquo / as defined by Elizabeth Sch&uuml / ssler Fiorenza (1989) and a &ldquo / partnership of both men and women&rdquo / as defined by Mercy Amba Oduyoye (1990).</p> <p><br /> In this thesis, Christian ministry is assessed through the perspectives of both feminist theology and African women&rsquo / s theology. The question which is addressed here is how the similarities and differences between Elizabeth Sch&uuml / ssler Fiorenza&rsquo / s notion of Christian ministry as a &ldquo / discipleship of equals&rdquo / and Mercy Amba Oduyoye&rsquo / s notion of Christian ministry as a &ldquo / partnership of both men and women&rdquo / should be understood and assessed. The main purpose of this thesis is to offer a critical comparison of these two female theologians so as to encourage new visions of Christian ministry in the contemporary church and society.</p> <p><br /> To accomplish this task, a literature survey of books, essays, and articles published on the subject of Christian ministry in feminist and African women&rsquo / s theology by African and Western theologians, between 1960 and 2003, was conducted.<br /> Two notions of how Christian ministry should be understood are offered. Fiorenza emphasizes that both women and men need to be recognized as disciples of Christ who can equally, yet within diversity, minister to God&rsquo / s people by virtue of their baptisms. Oduyoye notes that reciprocity and mutuality is crucial for both men and women who minister to God&rsquo / s people by forming strong partnerships through their respective vocations and ministries, by virtue of their baptisms.</p> <p><br /> This thesis argues that a comparison of the views of Elizabeth Sch&uuml / ssler Fiorenza and Mercy Amba Oduyoye on Christian ministry reveals similarities based on a shared experience of being women within a male-dominated Christian church and differences emanating from the different contexts within which they practice theology, namely, that of Euro-American feminist theology and that of African women&rsquo / s theology. It is argued that there are significant differences between feminist theology and African women&rsquo / s theology, with reference to their context, rhetoric, experiences and modes of expression. A critical assessment of Oduyoye&rsquo / s notion of Christian ministry as a &ldquo / partnership of both men and women&rdquo / reveals that there is an internal problem within African theology. The predicament within which African women theologians find themselves is that they have to struggle not only against patriarchy in church and society but also against the remaining distortions within what is deemed to be a more progressive theology. African women theologians acknowledge their solidarity with African theologians but also identify fundamental flaws within African theology. African women theologians are therefore engaged in a battle on more than one side. They need to unmask and support at the same time. On the other hand, it is clear that Fiorenza is far less critical of the feminist movement, from which she derives the term &ldquo / feminist hermeneutics&rdquo / . She is, of course, quite aware of the various and successive strands of the feminist movement, but she remains at least sympathetic to this movement and seeks to explore its significance for biblical scholarship and especially the Roman Catholic Church, of which she is a member. She recognises the need to complement the (sometimes reductionist) secular manifestations of feminism by highlighting the oppressive but also the potentially liberative role which religious traditions may still have in a secularised civil society.</p> <p><br /> One may therefore conclude that African women&rsquo / s theology remains distinct from other feminist theologies. In a similar way, African-American womanist theologians have insisted that their situation is distinct from that of Euro-American women in what is described as the &ldquo / triple&rdquo / oppression of black women: being women, relatively poor, black and formerly enslaved. This calls for further reflection on the similarities and differences between African-American womanist theology and African women&rsquo / s theology. Important differences which come to mind here are the legacy of slavery, differences in economic status, and military power. African women are often engaged in a struggle to secure a sustainable livelihood in ways that African-American women are not. More importantly, the relationship between American black theology and womanist theology deserves further attention in this regard.</p>

Syftet med denna studie är att undersöka och bidra med kunskap om hur undervisningen om likhetstecknets innebörd kan se ut i ett svenskt klassrum. Det som undersöks är hur en lärare skapar möjlighet för eleverna att urskilja kritiska aspekter av likhetstecknets innebörd genom variationsmönster. Datainsamlingen genomfördes genom stimulated recall i en åk 2 med en lärare som har tio års arbetslivserfarenhet som lärare vid ett undervisningstillfälle. Analysen av datamaterialet gjordes med inspiration från variationsteorin. Resultatet visade att fyra kritiska aspekter gjordes möjliga för eleverna att urskilja. Detta gjordes genom olika variationsmönster som skapades genom användandet av öppna utsagor och olika representationsformer. / The aim of this study is to research and bring knowledge about how the meaning of the equality sign can be taught in a Swedish classroom. The research focuses on how a teacher creates possibilities for the students to discern critical aspects of the meaning of the equality sign through patterns of variation. The data was collected by using stimulated recall during one lesson in a year 2 classroom with a teacher who has been working as a teacher for ten years. The collected material was analysed with inspiration from the variation theory. The result shows that four critical aspects were made visible to the students. This was done through dif- ferent patterns of variation that was created by the use of different forms of representation.

Syftet med denna studie är att undersöka och bidra med kunskap om hur undervisningen om likhetstecknets innebörd kan se ut i ett svenskt klassrum. Det som undersöks är hur en lärare skapar möjlighet för eleverna att urskilja kritiska aspekter av likhetstecknets innebörd genom variationsmönster. Datainsamlingen genomfördes genom stimulated recall i en åk 2 med en lärare som har tio års arbetslivserfarenhet som lärare vid ett undervisningstillfälle. Analysen av datamaterialet gjordes med inspiration från variationsteorin. Resultatet visade att fyra kritiska aspekter gjordes möjliga för eleverna att urskilja. Detta gjordes genom olika variationsmönster som skapades genom användandet av öppna utsagor och olika representationsformer. / The equality sign means that it should be the same, end of discus-sion. -A study about how students are given possibilities to discern critical aspects of the meaning of theequality sign The aim of this study is to research and bring knowledge about how the meaning of the equality sign can be taught in a Swedish classroom. The research focuses on how a teacher creates possibilities for the students to discern critical aspects of the meaning of the equality sign through patterns of variation. The data was collected by using stimulated recall during one lesson in a year 2 classroom with a teacher who has been working as a teacher for ten years. The collected material was analysed with inspiration from the variation theory. The result shows that four critical aspects were made visible to the students. This was done through dif-ferent patterns of variation that was created by the use of different forms of representation.

The purpose of this study was to determine if the use of the Equals mathematics curriculum had any statistically significant impact on the 2012 Florida Alternate Assessment mathematics scores of students with disabilities in six Florida school districts when comparing the scores of those who received mathematics instruction via the curriculum to the scores of students with disabilities in six other Florida school districts who did not receive mathematics instruction via the curriculum. This study further examined the perceptions of and attitudes about the Equals mathematics curriculum that exist among Exceptional Student Education (ESE) teachers who teach mathematics to students with disabilities participating in the Florida Alternate Assessment (FAA). The study utilized a mixed methods approach to research that included both quantitative and qualitative methodologies. The quantitative portion of the study was conducted using archival data obtained from the 2012 Florida Alternate Assessment Data Book. A Mann-Whitney U Test was conducted through SPSS at a significance level of α = .05 to test for relative differences in performance between the treatment (Equals) and control (non-Equals) groups. Based on the results, the Equals mathematics curriculum did not have any statistically significant impact on the 2012 FAA mathematics scores for students with disabilities in the treatment school districts at any of the tested grade groupings (elementary, middle, high, and overall). The qualitative method of data collection utilized an online teacher survey. The results were analyzed using the researcher-coded results and assisted through summary iv tables provided by Survey-Monkey. Differences and similarities among the survey question responses were explored. Common terms and themes were noted and compared. Data triangulation was used by surveying teachers from five of the six treatment counties. This promoted generalizability for study replication since the treatment counties train teachers and utilize the curriculum in different manners. The results of the qualitative analysis indicated that many teachers were not satisfied with the training they received and felt overwhelmed by the curriculum itself, specifically in the areas of lesson planning and delivery. Based on the quantitative and qualitative results, it was concluded that further research needs to be completed to determine the effectiveness of the Equals mathematics curriculum when used with true fidelity.

A Kondo lattice Hamiltonian for arbitrary total angular momentum J is formulated using a pseudofermion representation and without addition of RKKY interaction terms. An Hartree-Fock treatment is applied, and both variational and Green's function methods are used to calculate physical quantities from the linearized Hamiltonian. The Kondo phase is represented by finite hybridization. Magnetic ordering is examined via ordering vectors, but coexistence with the Kondo phase is not allowed. Phase diagrams are produced in S=1/2 and J=3/2 with second-order transitions at Kondo-paramagnetic and magnetic-paramagnetic boundaries, and first order transitions between Kondo and magnetic phases. Various coupling strengths are explored. Magnetic phases found include antiferromagnetism, ferromagnetism, and spin-density wave ordering of both commensurate and incommensurate varieties. In S=1/2, the magnetic phase exhibits a spike in critical temperature at half-filling. In J=3/2, the Kondo phase is reentrant at weaker coupling but not at stronger coupling. / Physics

Elevers uppfattning av likhetstecknet har länge diskuterats inom forskning och ansetts varakritiskt, då förståelsen för tecknets relationella innebörd ofta saknas. Syftet med denna textär att undersöka hur likhetstecknet framställs i svenska läromedel, och diskutera hur det kantänkas påverka elevers uppfattning av symbolen. Detta genom att besvara frågeställningarna;på vilket sätt uppmuntrar läroböckerna i matematik till en relationell förståelse avlikhetstecknet och på vilket sätt uppmuntrar läroböckerna i matematik till en operationellförståelse av likhetstecknet?Studien grundar sig i variationsteorins idéer och resultatet har analyserats kopplat tillvariationsmönster och kritiska aspekter. Kopplingar har även gjorts mellan traditionellaekvationer och en operationell syn på likhetstecknet, samt icke-traditionella ekvationer ochen relationell syn på tecknet. Metoden som använts är en kvantitativ innehållsanalys där sexolika svenska läromedel i matematik, riktade till årskurs 1, analyserats för att få syn på hur devarierar framställningen av likhetstecknet. Resultatet pekar på bristande variation i form avkontrast, generalisering och fusion, då det i störst grad är traditionella ekvationer som fårutrymme i böckerna. Detta kan tänkas bidra till att elever utvecklar en operationell syn pålikhetstecknet snarare än en relationell, vilket i längden kan försvåra övergången frånaritmetik till algebra i matematikundervisningen. / Students' perception of the equals sign has for a long time been debated in research andconsidered critical, as the understanding of the relational meaning of the equals sign is oftenlacking. The purpose of this text is to examine how the equals sign is presented in Swedishtextbooks, and to discuss how it might affect students' perception of the symbol. This is doneby addressing the following questions: in what way do mathematics textbooks encourage arelational understanding of the equals sign, and in what way do they encourage anoperational understanding of the equals sign?The study is based on the ideas of variation theory, and the results are analyzed in relation topatterns of variation and critical aspects. Connections have also been made betweentraditional equations and an operational view of the equals sign, as well as non-traditionalequations and a relational view of the sign. The method used is a quantitative contentanalysis where three different Swedish mathematics textbooks, aimed at grade 1, are analyzedto observe how they vary in the presentation of the equals sign. The results indicate a lack ofvariation in terms of contrast, generalization, and fusion, as the textbooks predominantlyfocus on traditional equations. This may contribute to students developing an operationalview of the equals sign rather than a relational one, which could ultimately complicate thetransition from arithmetic to algebra in mathematics education.