Pythagoras - Volume 33, Issue 3, January 2012

In this article, we share our combination of analytical concepts drawn from the literature with a set of grounded framing questions for thinking about differences in the nature of coherence and connections in teachers' mathematical discourses in instruction (MDI). The literature-based concepts that we use are drawn from writing focused on transformation activity as a fundamental feature of mathematical activity. Within this writing, the need for connections between stated problems and the representations introduced and subsequently produced through transformation steps are highlighted. Drawing from four empirical episodes located across primary and secondary mathematics teaching, we outline a set of framing questions that explore coherence and connections between these concepts, and the ways in which accompanying explanations work to establish these connections. This combination allows us to describe differences between the episodes in terms of the nature and degree of coherence and connection.

South Africa's performance in international benchmark tests is a major cause for concern amongst educators and policymakers, raising questions about the effectiveness of the curriculum reform efforts of the democratic era. The purpose of the study reported in this article was to investigate the degree of alignment between the TIMSS 2003 Grade 8 Mathematics assessment frameworks and the Revised National Curriculum Statements (RNCS) assessment standards for Grade 8 Mathematics, later revised to become the Curriculum and Assessment Policy Statements (CAPS). Such an investigation could help to partly shed light on why South African learners do not perform well and point out discrepancies that need to be attended to. The methodology of document analysis was adopted for the study, with the RNCS and the TIMSS 2003 Grade 8 Mathematics frameworks forming the principal documents. Porter's moderately complex index of alignment was adopted for its simplicity. The computed index of 0.751 for the alignment between the RNCS assessment standards and the TIMSS assessment objectives was found to be significantly statistically low, at the alpha level of 0.05, according to Fulmer's critical values for 20 cells and 90 or 120 standard points. The study suggests that inadequate attention has been paid to the alignment of the South African mathematics curriculum to the successive TIMSS assessment frameworks in terms of the cognitive level descriptions. The study recommends that participation in TIMSS should rigorously and critically inform ongoing curriculum reform efforts.

In this article I report on research intended to characterise and compare the thinking styles of Grade 10 learners studying Mathematics and those studying Mathematical Literacy in eight schools in the Gauteng West district in South Africa, so as to develop guidelines as to what contributes to their subject choice of either Mathematics or Mathematical Literacy in Grade 10. Both a qualitative and a quantitative design were used with three data collection methods, namely document analysis, interviews and questionnaires. Sixteen teachers participated in one-to-one interviews and 1046 Grade 10 learners completed questionnaires. The findings indicated the characteristics of learners selecting Mathematics and those selecting Mathematical Literacy as a subject and identified differences between the thinking styles of these learners. Both learners and teachers should be more aware of thinking styles in order that the learners are able to make the right subject choice. This article adds to research on the transition of Mathematics learners in the General Education and Training band to Mathematics and Mathematical Literacy in the Further Education and Training band in South Africa.

This article provides an illustration of the explanatory and discovery functions of proof with an original geometric conjecture made by a Grade 11 student. After logically explaining (proving) the result geometrically and algebraically, the result is generalised to other polygons by further reflection on the proof(s). Different proofs are given, each giving different insights that lead to further generalisations. The underlying heuristic reasoning is carefully described in order to provide an exemplar for designing learning trajectories to engage students with these functions of proof.

The challenges inherent in assessing mathematical proficiency depend on a number of factors, amongst which are an explicit view of what constitutes mathematical proficiency, an understanding of how children learn and the purpose and function of teaching. All of these factors impact on the choice of approach to assessment. In this article we distinguish between two broad types of assessment, classroom-based and systemic assessment. We argue that the process of assessment informed by Rasch measurement theory (RMT) can potentially support the demands of both classroom-based and systemic assessment, particularly if a developmental approach to learning is adopted, and an underlying model of developing mathematical proficiency is explicit in the assessment instruments and their supporting material. An example of a mathematics instrument and its analysis which illustrates this approach, is presented. We note that the role of assessment in the 21st century is potentially powerful. This influential role can only be justified if the assessments are of high quality and can be selected to match suitable moments in learning progress and the teaching process. Users of assessment data must have sufficient knowledge and insight to interpret the resulting numbers validly, and have sufficient discernment to make considered educational inferences from the data for teaching and learning responses.

There seems to be paucity of research in South Africa on mathematics teachers' reflective practice. In order to study this phenomenon, the context of lesson study (in an adapted form) was introduced to five mathematics teachers in a rural school in the Free State. The purpose was to investigate their reflective practice whilst they collaboratively planned mathematics lessons and reflected on the teaching of the lessons. Data were obtained through interviews, video-recorded lesson observations, field notes taken during the lesson study group meetings and document analyses (lesson plans and reflective writings). The adapted lesson study context provided a safe space for teachers to reflect on their teaching and they reported an increase in self-knowledge and finding new ways of teaching mathematics to learners. This finding has some potential value for planning professional learning programmes in which teachers are encouraged to talk about their classroom experiences, share their joys and challenges with one another and strive to build a community of reflective practitioners to enhance their learners' understanding of mathematics.

This article engages with the notion of local and global visualisation within the context of figural pattern generalisation. The study centred on an analysis of pupils' lived experience whilst engaged in the generalisation of linear sequences presented in a pictorial context. The study was anchored within the interpretive paradigm of qualitative research and made use of the complementary theoretical perspectives of enactivism and knowledge objectification. A crucial aspect of the analytical framework used was the sensitivity it showed to the visual, phenomenological and semiotic aspects of figural pattern generalisation. A microanalysis of a vignette is presented to illustrate the subtle underlying tensions that can exist as pupils engage with pictorial pattern generalisation tasks. It is the central thesis of this article that the process of objectifying and articulating an appropriate algebraic expression for the general term of a pictorial sequence is complicated when tension exists between local and global visualisation.