An online community of practice for pre-service and beginning teachers of secondary mathematics

The aim of this study was to investigate how a community of practice focused on learning to teach secondary mathematics was created and sustained by pre-service and beginning teachers. Bulletin board discussions of one pre-service cohort are analysed in terms of Wenger’s (1998) three defining features of a community of practice: mutual engagement, joint enterprise, and a shared repertoire. The study shows that the emergent design of the community contributed to its sustainability in allowing the pre-service teachers to define their own professional goals and values. Sustainability was also related to how the participants expanded, transformed, and maintained the community during the pre-service program and after graduation.

Decompositions of complete tripartite graphs into triangles with an edge attached

Let K(r, s, t) denote the complete tripartite graph with partite sets of size r, s and t, where r less than or equal to s less than or equal to t. Let D be the graph consisting of a triangle with an edge attached. We show that K(r, s, t) may be decomposed into copies of D if and only if 4 divides rs + st + rt and t less than or equal to 3rs/(r + s).

Existence of complementary pairs of proportionally balanced designs

Proportionally balanced designs (pi BDs) were introduced by Gray and Matters in response to a need for the allocation of markers of the Queensland Core Skills Test to have a certain property. Subsequent papers extended the theoretical results relating to such designs and provided further instances and general constructions. This work focused on designs comprising blocks of precisely two sizes, and when each variety occurs with one of precisely two possible frequencies. Two designs based on the set V of varieties are complementary if, whenever B is a block of one, then its complement with regard to the set V is a block of the other. Here we present necessary conditions for the existence of complementary pairs of such pi BDs and provide lists of some restricted parameter sets satisfying these necessary conditions. The lists are arranged according to the number of blocks. We demonstrate that not all of these parameter sets give rise to designs. However we establish by construction of the sets of blocks that, for every feasible number of blocks less than or equal to 100, with the possible exception of 63, there exists at least one pair of complementary pi BDs. We also investigate the conditions under which the complementary design can be isomorphic to the original design, and again provide a list of feasible parameters for pairs of such designs with at most 400 blocks.

Strategy-making process and firm performance in small firms

This paper argues that individual small firms just like large firms, place differing emphasis on strategy-making and may employ different modes of strategy-making. It offers a typology of the different modes of strategy-making that seem most likely to exist in small firms, and hypothesises how this typology relates to performance. It then describes the results of an empirical study of the strategy-making processes of small firms. The structural equation analysis of the data from 477 small firms with less than 100 employees indicates among other results that the simplistic, adaptive, intrapreneurial and participative modes of strategy-making exist in these small firms. Of these modes, the simplistic mode exhibits the strongest relationship with firm performance.