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The purpose of this paper is to present a model for supporting and increasing teachers’ reflection process.

The model is grounded in two concepts from variation theory: critical aspects and dimensions of variation. Those concepts are related to the concepts of reflection in-, on- and for-action. Analysis of data allows for determination of what kind of reflection is used in teachers’ professional development (PD) and the teachers’ perceptions of the relevance and usefulness of the PD.

The results show that, using the concepts of critical aspects and dimensions of variation, teachers practiced reflection in-, on- and for-action in a non-dualistic way.

The preliminary results confirm that the teachers change their behaviours, through reflection processes, if they have the opportunity to focus on the object of learning and to use research results found in mathematics education in order to open up dimensions of variation in the identified critical aspects.

The design of algebra modules, according to concepts from variation theory and the reflection process, is the main result of this study. The model used in creating the modules has the potential for supporting and increasing teachers’ reflection process.

A conceptual model of work practice evolution is proposed. The model is conceived as an intermediary between models of reflection, action at work and models of organisational teaching in the classroom. It considers work practices as evolving through confrontations between critical aspects, action in context, reflections in-, on- and for-action. The author advocate that the use of the presented methodology is not only inevitable but also vital to improve teachers’ PD.

Efforts to promote equity in mathematics education are becoming more widespread. The National Council of Teachers of Mathematics elevated its position on equity from one of societal need to one of six guiding principles required to ensure excellence in mathematics education. I investigate how efforts to promote educational equity in mathematics education are informed by democratic views. In particular, I examine the intersection of course taking, equity, mathematics, and democracy in the context of a case study conducted in one high school mathematics department. This case study was conducted over one academic year, and data included a focus survey, interviews, field notes from department meetings and class observations, and school documents. In this study, I investigated the curricular redesign process undertaken by one high school mathematics department addressing high failure rates in low-level courses disproportionately enrolled with students of color designed to increase access to more advanced mathematics courses. I discuss how those changes impacted access, influenced equity, and aligned with democratic aims and their relationship to mathematics education.

Carrying on the arithmetization of Euler diagrams by George Boole, with refinements by Augustus De Morgan and others, the full‐scale incorporation of binary strings into the scheme is discussed, with emphasis on the usefulness of the operator EXOR and its consequent EXORcism of what had remained unclear. Examples and applications are given. By this powerful extension of Boolean algebra Euler/Venn diagrams of class relations can be digitally computed. Its uses to system theory and cybernetics are manifold.

The purpose of this paper is to provide an introduction to alysidal algebra. Deontical impure systems are systems whose object set is formed by an s‐impure set, whose elements are perceptuales significances (relative beings) of material and/or energetic objects (absolute beings) and whose relational set is freeways of relations, formed by sheaves of relations going in two‐way directions. Objects and freeways form chains.

The paper looks at the mathematical and logical development of human society structure.

Existence of relations with positive imperative modality (obligation) would constitute the skeleton of the system. Negative imperative modality (prohibition) would be the immunological system of protection of the system. Modality permission the muscular system, that gives the necessary flexibility to the system, in as much to the modality faculty its neurocerebral system, because it allows him to make decisions. Transactions of energy, money, merchandise, population, etc. would be the equivalent one to the sanguineous system. These economic transactions and inferential relations, depend, as well, on the existence of a legislative body with their obligations, prohibitions and permissions that regulate them.

The concepts of alysidal set are introduced, whose elements are chains and coupling function between alysidal sets. Environment is formed as well by different DIS. That is to say, by an alysidal set whose elements are simultaneously systems. Specific interchanges (stimulus‐response) leave certain nodes and act on certain nodes of the alysidal sets (stimulus environment‐DIS‐response environment). The paper defines a special coupling function, denominated gnorpsic function, that can be used for algebraic operations between alysidal sets.

Because teacher training is an important component of high-quality early care and education (ECE), states are employing various efforts to increase the credentials of teachers in private ECE centers. In New Jersey, teachers who serve disadvantaged students in the state’s community-based Abbott preschools are under a court mandate to obtain a Bachelor’s degree and Preschool – Grade 3 certification by September 2004 or lose their jobs. This chapter describes a phenomenological study of five teachers’ experiences in attempting to meet that mandate, and offers implications for policymakers to consider when evaluating the overall success of this reform effort.

This study applies complexity theory to propose and empirically examine asymmetric case conditions of antecedents and outcome models of high (low) willingness-to-engage in workplace romance (WEWR). This study focuses on constructing complex antecedent conditions that accurately indicate which employees, and under what conditions, employees are high in WEWR.

Using an experimental design, 162 employees were assigned one of nine hypothetical vignettes describing different workplace romance contexts including three discrete policies regarding workplace romances (i.e. strictly forbidden, moderate, vs no policy), two motivations for the workplace romance (i.e. job vs love), and two organizational positions of the romance (i.e. hierarchical vs lateral). Participants then reported WEWR responses. Participants also provided demographic, behavioral, and psychological work-related information. This study assesses and supports recipes (i.e. algorithms) of case and organizational structure conditions to identify cases high (low) in WEWR accurately and consistently.

The results provide clarity of which and when employees are willing vs unwilling to engage in workplace romances – and the contextualized impacts of organizational bans on WEWR. The study’s results are useful for estimating for whom specific workplace policies are effective or not by specific workplace contexts.

In highlighting the role of varying antecedent conditions in predicting WEWR, this research will assist organizations and practitioners in understanding the context in which workplace romances are more likely to occur, providing insight as to when employees are likely to comply with workplace romance policies.

This paper is the first in the workplace romance literature to examine unique combinations of antecedent conditions on WEWR, adding nuance to the current understanding of the behavior.

Program efficacy and outcomes can often be determined through an examination of collaborative activities between and among inner city school districts with high dropout rates and private, public, and nonprofit organizations. Kettl (2004) adeptly describes additional collaborative practice trends that not only transform governance structures, but blurs the line between and among sectors. These trends illuminate the need for governmental agencies to collaborate with nonprofit and for-profit organizations to address “wicked problems” where no single organization has sufficient resources and the consequences are enormous. The paper aims to discuss theses issues.

Utilizing a quantitative approach, this research compares the efficacy of a newly developed collaborative alternative education program to existing programs in New Jersey's Newark Public Schools during 2008-2009.

The results indicate that the overall performance of the students enrolled in the new research models is significantly higher than in the existing program due to incentives and not administrative collaboration.

Implications for future research include: first, the need for studies to reveal enduring, universal effects of collaboration; second, longitudinal studies of the effects of collaboration on alternative education issues; and third, an evaluation of the effectiveness of collaborative training.

This research intends to contribute to the literature concerning these distinctive types of partnerships – specifically the integration of three very different systems into a collaborative service. This single case study presents support of how these services subsist within four settings and what force they have on special services for students in alternative education in the public schools.

This study aims to understand the asset and success factors that contributed to college completion of African American males who persisted through college. Only a dismal 22 per cent of African American males receive bachelor’s degrees compared to 41 per cent of White males (Kena et al., 2015).

The data were analyzed using interpretive phenomenological analysis. The authors interviewed two first-generation African-American males from rural backgrounds to capture their experiences of their process to college completion.

Themes, based in cultural capital theory, that impacted their college persistence were identified within their pre-college experiences, college experiences and post-college perceptions. Recommendations for helping rural African-American males attend and persist through college are offered.

Only two participants from one predominately white institution in the southeastern USA were interviewed. Rural students from other geographical areas might have different backgrounds, challenges, assets and successes. Although the interview questions were based on relevant literature, they may not have covered all key aspects of the participants’ experiences. As in any qualitative study, biases of the researchers and research team may have influenced the results, although these were identified and shared before reading any of the transcripts and then discussed several times during the data analysis process.

Educators not only should try to address the cultural capital limitations of these men but also highlight and build on their cultural assets. These assets include familial and platonic individuals who see their potential for success and encourage them to attend college to become something better than what they see in their community, reverse role models who encourage youth to make different choices than they did, media-based examples of successful Black students, cultural messages of strength and determination (e.g. Million Man March) and the exhortation to be an example that other African-American boys could look up to.

This paper addresses the need for K-12 and higher education institutions to understand how to assist first-generation, rural African-American males in getting admitted to college, matriculating through college and graduating from college.

The aims of this paper is to prove that every semisimple Jordan algebra bundle is locally trivial and establish the decomposition theorem for locally trivial Jordan algebra bundles using the decomposition theorem of Lie algebra bundles.

Using the decomposition theorem of Lie algebra bundles, this paper proves the decomposition theorem for locally trivial Jordan algebra bundles.

Findings of this paper establish the decomposition theorem for locally trivial Jordan algebra bundles.

To the best of the author’s knowledge, all the results are new and interesting to the field of Mathematics and Theoretical Physics community.