Initiatives in many countries to improve learner performances in mathematics in poor communities have been described as largely unsuccessful mainly due to their cursory treatment of curriculum alignment. Empirical evidence has shown that in high achieving countries the notion of coherence was strongly anchored in cognitively demanding mathematics programs. The view that underpins this study is that a cognitively demanding and coherent mathematics curriculum has potential to level the playing field for the poor and less privileged learners. In South Africa beyond 1994, little has been done to understand the potential of such coherent curriculum in the context of the NCSM. This study examined the levels of cognitive demand and alignment between the written, tested and taught NCSM. The study adopted Critical Theory as its underlying paradigm and used a multiple case study approach. Wilson and Bertenthal’s (2005) dimensions of curriculum coherence provided the theoretical framework while Webb’s (2002) categorical coherence criterion together with Porter’s (2004) Cognitive Demand tools were used to analyse curriculum and assessment documents. Classroom observations of lesson sequences were analysed following Businskas’ (2008) model of forms of mathematical connections since connections of different types form the bases for high cognitive demand (Porter, 2002). The results indicated that higher order cognitive skills and processes are emphasized consistently in the new curriculum documents. However, in the 2008 examination papers the first examinations of the new FET curriculum, lower order cognitive skills and processes appeared to be emphasized, a finding supported by Umalusi (2009) and Edwards (2010). Classroom observations pointed to teachers focusing more on rote learning of both concepts and procedures and less on procedural and conceptual understanding. Given the widespread evidence of the tested curriculum impacting on the taught curriculum, this study suggests that this lack of alignment between the advocated curriculum on one hand, the tested and the taught curricula on the other, needs to be investigated further for it endangers the teaching and learning of higher order cognitive skills and processes in the FET mathematics classrooms for the poor and less privileged. Broader evidence suggests that this would work against efforts towards supporting the upward mobility of poor children in the labour market.

With the implementation of No Child Left Behind legislation and a push for reform curricula, prospective teachers must be prepared to facilitate learning at a conceptual level. To address these concerns, an exploratory mixed methods investigation of twenty-eight prospective middle grades teachers’ mathematics knowledge for teaching geometry and mathematical connection-making was conducted at a large public southeastern university. Participants completed a diagnostic assessment in mathematics with a focus on geometry and measurement (CRMSTD, 2007), a mathematical connections evaluation, and a card sort activity. Mixed methods data analysis revealed prospective middle grades teachers’ mathematics knowledge for teaching geometry was underdeveloped and the mathematical connections made by prospective middle grades teachers were more procedural than conceptual in nature.

Die Notwendigkeit einer Untersuchung über Beziehungshaltigkeit und Vernetzungen im Mathematikunterricht ergibt sich einerseits aus den aktuellen bildungspolitischen Forderungen, andererseits aus den reichhaltigen bildungsphilosophischen Traditionen im deutschsprachigem Raum(KMK 2012, 11). Das Ziel der vorliegenden Arbeit besteht vor allem in der Reflexion von Beziehungshaltigkeit und Vernetzungen im Mathematikunterricht. Diese Reflexion ist durch drei Fragen bestimmt: Was kann man als Lehrer über Beziehungshaltigkeit wissen? Wie kann man als Lehrer handeln, so dass die Schüler Beziehungen zwischen mathematischen Inhalten erkennen bzw. selbständig herstellen? Um handeln zu können, muss man die Wirklichkeit oder die Praxis (bzw. Empirie) kennen, in der man handelt. In diesem Sinne ist die vorliegende Arbeit aufgebaut. Dabei wird ein Versuch unternommen, die klassische Aufteilung zwischen Theorie und Empirie bzw. Praxis des Mathematikunterrichts aufzubrechen, um eine Verzahnung zwischen diesen zu verstärken. Das Herzstück der Arbeit bilden zwei ausgearbeitete und in der schulischen Arbeit erprobte Aufgabennetze (Pythagorasbaum und Rund ums Sechseck), die den Rahmen zur Reflexion bieten. / The need for a study on relations sustainability and networks in mathematics stems, on the one hand, from current education policy requirements, and, on the other, from the rich philosophical traditions of education in the German-speaking countries (KMK 2012, 11). The goal of the present work consists, above all, in reflecting on relations sustainability and networks in mathematics lessons. This reflection is guided by three questions: What can one know, as a teacher, about relations sustainability? How can one act a teacher to ensure that students recognise relationships between mathematical content, or independently produce such relations? In order to act, one must know the reality or practice (e.g. empiricism) in which one acts. The project is focused on the development and testing of worked examples of concrete task networks ("Pythagoras’ tree" and "Around the hexagon").