On the geometry of the smallest circle enclosing a finite set of points

A number of numerical codes have been written for the problem of finding the circle of smallest radius in the Euclidean plane that encloses a finite set P of points, but these do not give much insight into the geometry of this circle. We investigate geometric properties of the minimal circle that may be useful in the theoretical analysis of applications. We show that a circle C enclosing P is minimal if and only if it is rigid in the sense that it cannot be translated while still enclosing P. We show that the center of the minimal circle is in the convex hull of P. We use this rigidity result and an analysis of the case of three points to find sharp estimates on the diameter of the minimal circle in terms of the diameter of P.     

School Reform in a Vacuum: Demographic Change,Social Policy,and the Future of Children

In their search for magic bullets to fix failing schools, policymakers seldom directly address powerful ecological factors impacting schooling. This article identifies several major demographic, societal, economic, and educational changes and trends in U.S. society over the past several years; analyzes their impact on schoolchildren; and offers a series of policy recommendations for public sector reform initiatives that show promise in ameliorating these conditions, which in turn would improve the educational prospects for all children, particularly those most at risk.     

Examining preservice teacher content and pedagogical content knowledge needed to teach reading in elementary school

Recently, increased attention has been placed on the importance of having knowledgeable and skilled teachers in order to influence reading achievement. Yet many international reports suggest that large numbers of children are not learning to read. How can we better prepare teachers with the necessary knowledge to teach reading? The current study examined the reading instructional knowledge of elementary education preservice teachers (N = 87) from two teacher education programmes. Programme A required five reading methods courses and Programme B required two methods courses. Findings indicate that the preservice teachers presented fairly robust levels of knowledge in the areas of phonological awareness, phonics, comprehension and vocabulary. However, we found statistically significant differences in knowledge based on the programme attended and the number of courses completed. Suggestions for improving reading teacher education are provided.     

Teacher job satisfaction: lessons from the TSW Pathfinder Project

Government policy assumes that modernization and remodelling will be effective as external intervention mechanisms to improve job satisfaction. Based on data collected as part of the evaluation of the ‘Transforming the School Workforce Pathfinder Project’, an argument is presented here which suggests that internal management models may be more effective in improving teacher job satisfaction. By comparing the responses of teachers within primary and special schools with those from secondary schools, internal factors are identified which may be more relevant than externally imposed measures.     

From the Editors: Math Achievement and Early Childhood Teacher Preparation

The researchers conducting this study investigated the relationship between early childhood teachers' educational levels and their beliefs about appropriate and inappropriate practices by examining the differences between lead teachers and teacher assistants in publicly funded prekindergarten classrooms. For this purpose, the researchers analyzed self-reported teachers' beliefs using analysis of variance and examined 26 individual items. Findings of this study suggest that lead teachers endorsed developmentally appropriate practices more strongly than did teacher assistants. Teacher assistants were more likely than lead teachers to agree with developmentally inappropriate practices. Despite the significant differences between the two groups of teachers, it is important to note that there was a general tendency for both lead teachers and teacher assistants to endorse developmentally appropriate practices. Thus, although there appears to be a link between teachers' educational levels and teacher beliefs, teachers' educational levels alone may not be sufficient for understanding their beliefs and practices in the classroom. Implications for practice and future research in teacher education are also discussed.     

‘Quicksand of Hate’: Experiences of Islamophobia and Poetic Resistance

How might English classrooms be spaces where Muslim students are empowered to recognise and resist Islamophobia? This qualitative study draws on theories of critical consciousness and Islamophobia to show how Bassim, a Muslim student, makes sense of Islamophobic encounters through the poetic and analytic writing he creates in his public high school English classroom. When examined intertextually, Bassim’s words offer a glimpse of what Islamophobic experiences are like for some contemporary Muslim students. The findings suggest that through his English class, Bassim is able to use poetic and analytic writing as the means by which he recognises and resists Islamophobia. Further, this study indicates that more work should take up the implications of students’ religious beliefs and the ways those beliefs shape their writing – particularly when they are members of a marginalised religion.