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An appreciation of variation is central to statistical thinking, but very little research has focused directly on students' understanding of variation. In this exploratory study, four students from each of grades 4, 6, 8, and 10 were hlterviewed individually on aspects of variation present in three settings. The first setting was an isolated random sampling situation, whereas the other two settings were real world sampling situations. Four levels of responding were identified and described in relation to developing concepts of variation. Implications for teaching and future research on variation are considered.

A key element in developing ideas associated with statistical inference involves developing concepts of sampling. The objective of this research was to understand the characteristics of students' constructions of the concept of sample. Sixty-two students in Grades 3, 6, and 9 were interviewed using open-ended questions related to sampling; written responses to a questionnaire were also analyzed. Responses were characterized in relation to the content, structure, and objectives of statistical literacy. Six categories of construction were identified and described in relation to the sophistication of developing concepts of sampling. These categories illustrate helpful and unhelpful foundations for an appropriate understanding of representativeness and hence will help curriculum developers and teachers plan interventions.

Abstract. A recent form of the Todd-Coxeter algorithm, known as the lookahead algorithm, is described. The time and space requirements for this algorithm are shown experimentally to be usually either equivalent or superior to the Felsch and HaselgroveLeech-Trotter algorithms. Some findings from an experimental study of the behaviour of Todd-Coxeter programs in a variety of situations are given.1. Introduction. The Todd-Coxeter algorithm [20] (TC algorithm) is a systematic procedure for enumerating the cosets of a subgroup H of finite index in a group G, given a set of defining relations for G and words generating H. At the present time, Todd-Coxeter programs represent the most common application of computers to group theory. They are used for constructing sets of defining relations for particular groups, for determining the order of a group from its defining relations, for studying the structure of particular groups and for many other things.As an example of the use of the algorithm, consider the following family of defining relations, Men(n), due to Mennicke:

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In this article, we report on the use of a teacher profiling instrument with 62 middle school teachers at the start of a 3-year professional learning programme. The instrument was designed to assess the aspects of teachers' knowledge identified by Shulman (1987) refined by Ball et al. (2008) and extended to include teachers' confidence to use and teach various topics in the middle school mathematics curriculum and their beliefs about mathematics teaching and learning. Based on a hierarchical coding of items, the application of the partial credit Rasch model revealed that the profile items were measuring a single underlying construct and suggested that the various facets of teacher knowledge develop together. We describe the characteristics of four levels of the hierarchical construct measuring teacher knowledge and understanding for teaching mathematics in the middle years of schooling, and discuss the unique affordances of a holistic view of teacher knowledge in contrast to considerations of multiple knowledge categories.

The aim of this study was, first, to provide evidence to support the notion of statistical literacy as a hierarchical construct and, second, to identify levels of this hierarchy across the construct. The study used archived data collected from two large-scale research projects that studied aspects of statistical understanding of over 3000 school students in grades 3 to 9, based on 80 questionnaire items. Rasch analysis was used to explore an hypothesised underlying construct associated with statistical literacy. The analysis supported the hypothesis of a unidimensional construct and suggested six levels of understanding: Idiosyncratic, Informal, Inconsistent, Consistent non-critical, Critical, and Critical mathematical: These levels could be used by teachers and curriculum developers to incorporate appropriate aspects of statistical literacy into the existing curriculum.  First published November 2003 at Statistics Education Research Journal: Archives