Upper bounds for Heilbronn's triangle problem in higher dimensions

We develop a new simple approach to prove upper bounds for generalizations of the Heilbronn's triangle problem in higher dimensions. Among other things, we show the following: for fixed $d \ge 1$, any subset of $[0, 1]^d$ of size $n$ contains: - $d+1$ points which span a simplex of volume at most $C_d n^{-\log d+ 6}$, - $1.3 d$ points whose convex hull has volume at most $C_d n^{-1.1}$, - $k + 1\ge 4\sqrt{d}$ points which span a $k$-dimensional simplex of volume at most $C_d n^{-\frac{k}{d} - \frac{k^2}{8d^2}}$.Comment: 10 pages, a significant update with new result

Pedagogic incongruities: a case of Initial Teacher Education and speaking skills in modern foreign languages

This research focused on achieving greater understanding of the teaching of speaking within modern foreign languages (MFL) and of Initial Teacher Education (ITE). It is a case study with elements of action research, including an intervention in my own practice and in two classrooms. However, the intervention was not the sole or primary or focus of the research. As a practitioner researcher, my aim was to generate knowledge which might improve practice in schools but could also be applied to my own practice. The thesis addresses the research questions: To what extent can focused Initial Teacher Education improve speaking skills in secondary Modern Foreign Language classrooms? How do MFL trainees and secondary school students experience the teaching of speaking skills? How do trainees plan for input and practice, including target language? To what extent is MFL subject-specific pedagogic knowledge valued and utilised in secondary schools? The research consisted of three elements: First a study of three cohorts’ work in the ITE MFL course, including documents generated by the trainees supplemented by group interviews with the trainees. Second, a study of an intervention within the ITE MFL course, involving changes to its curriculum, pedagogy and assessment which were intended to raise the profile of speaking in trainees’ preparation for classroom practice. Third, a study of a classroom intervention in which two trainees prepared and conducted a group talk activity with their Year 8 classes. The lessons and students’ comments on speaking in MFL lessons were recorded and analysed. The over-arching theoretical framework of the thesis was pragmatism, drawing on the work of Biesta (2010) and Dewey (1936), and the analytical framework was based on Engeström’s (2007) Activity Theory. The data were analysed thematically as part of Quantitative Content Analysis (Silverman, 2011). Students’ language during the classroom intervention was analysed using Halliday’s (1973) linguistic functions and Ellis’ (2005) principles of instructed language learning were used as an evaluative framework for trainees’ lesson plans. The literature review compares key elements of both the Key Stage Three Framework for MFL and the GCSE assessment framework for speaking in MFL with theories of second language acquisition. The data analysis suggests that subject-specific pedagogy is dominated by generic pedagogy in trainees’ academic writing and in their feedback from school-based subject mentors. This is attributed, in part, to an over-emphasis on measurable outcomes in current objectives-based educational policies. The qualified success of the group talk intervention suggests that incorporating a task-based language teaching approach into school schemes of work would be beneficial, accommodating the meta-cognitive benefits of assessment for learning within an established model of language teaching. An analysis of the Initial Teacher Education partnership using Activity Theory indicates that structural constraints allow limited scope for innovation in the classroom practice of either teachers or trainees. Students expressed anxiety about making errors and appearing foolish to their peers. However, trainees also commented that teachers’ anxieties about poor behaviour prevented them conducting pair work or small group work with some of their classes. In conclusion, using wider professional content knowledge could avoid an over-emphasis on short-term performance goals when complying with policy initiatives and external assessment frameworks concerning linguistic and professional knowledge. Trainees need to “fit in” with the culture of the host department by adopting its rules and tools but changes in the division of labour to allow increased collaborative work including trainees, mentors and tutors could support innovation. MFL pedagogy should provide sufficient input for the foreign language to be learned, thus enabling speaking in the target language, rather than using speaking as an aspect of performativity