This extended review of the literature on pupil grouping includes an analysis and synthesis of current and yet to be published research to identify types of grouping suited to particular pupils, the range of organisational policies regarding pupil grouping within schools that are related to different levels of performance and subjects suited to particular types of grouping. The review also considers how type of grouping may affect pupil learning and how the transition from primary to secondary school may be affected by various pupil groupings. This review of the literature draws upon studies undertaken in primary and secondary schools. The literature review draws together school-based information on ‘organisational’ and ‘within-class’ grouping of pupils, as well as theoretical background and practical implementation issues. The methodology adopted used systematic procedures that include electronic and hand searching, mapping the research territory and quality-assuring the studies. This review identifies issues in the study of grouping, theories underlying grouping initiatives, the role of grouping practices in school transfer and the importance of teaching pupils to work in groups.

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Given n points in a d dimensional Euclidean space, the Minimum Enclosing Ball (MEB) problem is to find the ball with the smallest radius which contains all n points. We give two approximation algorithms for producing an enclosing ball whose radius is at most ɛ away from the optimum. The first requires O(ndL / √ ɛ) effort, where L is a constant that depends on the scaling of the data. The second is a O ∗ (ndQ / √ ɛ) approximation algorithm, where Q is an upper bound on the norm of the points. This is in contrast with coresets based algorithms which yield a O(nd/ɛ) greedy algorithm. Finding the Minimum Enclosing Convex Polytope (MECP) is a related problem wherein a convex polytope of a fixed shape is given and the aim is to find the smallest magnification of the polytope which encloses the given points. For this problem we present O(mndL/ɛ) and O ∗ (mndQ/ɛ) approximation algorithms, where m is the number of faces of the polytope. Our algorithms borrow heavily from convex duality and recently developed techniques in non-smooth optimization, and are in contrast with existing methods which rely on geometric arguments. In particular, we specialize the excessive gap framework of Nesterov [19] to obtain our results. 1