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Item Open Access Big Ideas in primary mathematics: issues and directions(Faculty of Education, University of the Free State, 2013) Askew, MikeThis article is located within the literature arguing for attention to Big Ideas in teaching and learning mathematics for understanding. The focus is on surveying the literature of Big Ideas and clarifying what might constitute Big Ideas in the primary Mathematics Curriculum based on both theoretical and pragmatic considerations. This is complemented by an analysis of the evidence for two Big Ideas in South Africa’s Curriculum and Assessment Policy Statements for Foundation- and Intermediate- Phase Mathematics. This analysis reveals that, while there is some evidence of implicit attention to Big Ideas in the Curriculum, without more explicit attention to these, teachers and, consequently, learners are not likely to develop understanding of Big Ideas and how they connect aspects of mathematics together.Item Open Access Editorial: Primary mathematics: addressing the crisis(Faculty of Education, University of the Free State, 2013) Venkat, Hamsa; Bowie, LynnAbstract not availableItem Open Access Teachers’ understanding of mathematical cognition in childhood: towards a shift in pedagogical content knowledge?(Faculty of Education, University of the Free State, 2013) Henning, ElizabethThis article about the discourse of pedagogy as related to child cognition in mathematics addresses the issue of what constitutes the main disciplinary content and the pedagogical content knowledge (PCK) of foundation-phase teachers. I argue that, unless child cognition itself is the primary disciplinary content of foundationphase teachers’ knowledge, it is likely that they will couch their pedagogical knowledge in teaching methods and materials more than in knowledge of conceptual development of learners and how such knowledge relates to teaching. In this first of a series of case studies, workshop-generated conversational and interview data were analysed qualitatively for discourse. The topics for the workshops were mathematical cognition and training in standardised test administration. The analysis showed that the discourse of teachers’ expressed knowledge about their practice was embedded in the language of policy, curriculum, teaching methods of mathematics, and the omniscience of the annual national assessments (ANAs) in South Africa, with very few discourse markers representing knowledge of child cognition. During the course of the intervention, teachers gradually shifted their talk, expressing some understanding of trends in contemporary developmental cognitive psychology and neuroscience of mathematical cognition. The article recommends a stronger cognitive science focus in teacher professional development initiatives.Item Open Access Raising the quality of primary level mathematics teaching and learning in schools in American Samoa: a model for South Africa(Faculty of Education, University of the Free State, 2013) Muthukrishna, NithiAgainst the background of concerns around teaching and learning outcomes in primary school mathematics in South Africa, this article presents two studies conducted in American Samoa and seeks to draw implications for the teaching and learning of mathematics in South Africa. American Samoa has a very similar educational context to South Africa. The purpose of the two empirical studies was to evaluate the effectiveness of a mathematics intervention with Grade 3 learners, Connecting Math Concepts Comprehensive Edition Level C (CMCCE) curriculum which is framed by a structured and explicit pedagogy. The findings in the two studies indicate that providing teachers who have limited content knowledge and pedagogical content knowledge with explicit and fully developed instructional plans can have an almost immediate and positive effect on children’s mathematics proficiency.Item Open Access Opportunities to develop mathematical proficiency in Grade 6 mathematics classrooms in KwaZulu-Natal(Faculty of Education, University of the Free State, 2013) Ally, Noor; Christiansen, Iben MajIn this article, we propose a rubric for assessing the teacher’s provision of opportunities to develop mathematical proficiency in classrooms. The rubric is applied to data from 30 video recordings of mathematics lessons taught in Grade 6 in one district of KwaZulu-Natal. The results suggest that opportunities to develop procedural fluency are common, but generally of a low quality; that opportunities to develop conceptual understanding are present in about half the lessons, but also are not of a high quality; and that overall opportunities to develop mathematical proficiency are limited, because learners are not engaging in adaptive reasoning, hardly have any opportunities to develop a productive disposition, and seldom are given the opportunity to engage in problem-solving which could develop their strategic competence.Item Open Access Lizzy’s struggles with attaining fluency in multiplication tables(Faculty of Education, University of the Free State, 2013) Bansilal, SarahMany learners struggle to make the transition from addition and subtraction to multiplication and division, which hampers further progress in mathematics. In this article, I present a case study of one learner who struggled to attain fluency in multiplication by seven. The purpose of this study was to identify and explain how previous non-encapsulations related to addition and subtraction number bonds hampered her efforts in attaining fluency in the multiplication tables. Data for the study were generated from observations and interactions with the child, Lizzy, over a period of two years. The study employed a narrative analysis technique, while drawing upon the APOS (action, process, object, schema) framework together with the notion of conceptual embodiment. The findings suggest that Lizzy’s struggles were due to a non-encapsulation of the various number bond strings, which did not allow her to see patterns in addition by seven. By using an alternative intervention, Lizzy was able to use this strategy as a conceptual embodiment leading her to greater fluency in the seven-times table.Item Open Access Research tensions with the use of timed numeracy fluency assessments as a research tool(Faculty of Education, University of the Free State, 2013) Stott, Debbie; Graven, MellonyIn this paper, we describe how we came to use timed fluency activities, along with personal learner reflections on those activities, in our after-school maths club as a complementary research and developmental tool for assessing the changing levels of learners’ mathematical proficiency over time. We use data from one case-study after-school maths club. Not only did the activities provide us as researchers, and mentors, with a quick way of tracking, evaluating, encouraging and valuing learner progress, but also with a mechanism for the learners to practise the fluency they were developing through other activities of the club. More importantly, the use of learner reflections assisted learner buy-in and reduced the stress related to such timed assessments. This alleviated, to some extent, our ethical unease with the use of such instruments. We have subsequently extended this research and development tool in all clubs that we run and continue to research their affect and effect in order to gain deeper understanding of the research and development opportunities enabled by such activities.Item Open Access The relationship between teachers’ instructional practices and their learners’ level of geometrical thinking(Faculty of Education, University of the Free State, 2013) Bleeker, Cheryl; Stols, Gerrit; Van Putten, SonjaThis case study describes and investigates the instructional practices of Grades 1 to 5 teachers and the levels of geometry thinking of the learners, according to the Van Hiele model, with a view to determining whether there is a match between the instructional practice and the learners’ level of thinking. The instructional practices of the teachers were observed and analysed, and their learners’ levels of geometry thinking were accessed through a Van Hiele test. The results suggest that there is not a simple relationship between the phases of learning, as described by Crowley in 1987, and geometric development in terms of the Van Hiele levels. It is, however, possible to explain the geometric development to a limited extent in terms of the Van Hiele levels of the observed teaching activities. Although the presence of activities on an appropriate level does not guarantee growth in terms of the Van Hiele model, the absence thereof results in stagnation. The instructional practices in primary schools in all Grades should span geometry experiences on all the levels, because the previsualisation level and Van Hiele Level 1 thinking are still evident up to Grade 5.Item Open Access Focusing on the object of learning and what is critical for learning: a case study of teachers’ inquiry into teaching and learning mathematics(Faculty of Education, University of the Free State, 2013) Runesson, UllaDuring the last decade, Lesson Study, a Japanese professional development model involving a community of learners, has been spreading to countries around the world. Lesson Study is a systematic process of inquiry into classroom practices where teachers collaborate in planning, implementing, observing and revising lessons. The approach used in the study reported in this paper involves a version of Lesson Study, namely Learning Study. Whereas the underpinning theory in Lesson Study is often implicit, Learning Study has an explicit theory that can help teachers to pedagogically theorise about their practice, students’ learning and the object of learning. Through one Swedish Learning Study in mathematics, I demonstrate what teachers can learn by exploring what is critical for their students’ learning and how teaching can be improved to enhance learning.Item Open Access Perspectives on pre-service teacher knowledge for teaching early algebra(Faculty of Education, University of the Free State, 2013) McAuliffe, Sharon; Lubben, FredThis paper examines a pre-service teacher’s content knowledge for teaching early algebra from two perspectives, i.e. using Rowland’s Knowledge Quartet theory and Ball’s framework for Mathematical Knowledge for Teaching (MKfT). The study intends to examine the differences between the inferences using each framework and to reflect on how they may help us to better understand the knowledge needed for teaching early algebra. Both perspectives are used to interpret selected episodes from a videoed Grade 3 patterns lesson. The findings show that the two perspectives are simultaneously complementary and distinct in both purpose and outcomes. The Knowledge Quartet identifies and describes ways in which the teacher’s content knowledge for teaching is enacted in the classroom including those aspects that need reinforcement. The MKfT scheme identifies situations and tasks in teaching where content knowledge emerges including instances where a broader base of knowledge for teaching is required. The results of the analysis show how each perspective emphasises different aspects of teacher content knowledge and, together, provide a more holistic account of what happens in developing knowledge for teaching patterns. They highlight the difficulty for teachers in helping learners to shift from focusing solely on number pattern to a simultaneous focus on function, a transition central to the teaching of early algebra.Item Open Access Assessing early number learning: how useful is the Annual National Assessment in Numeracy?(Faculty of Education, University of the Free State, 2013) Weitz, Maria; Venkat, HamsaAnnual National Assessment (ANA) performance in Mathematics across the primary grades in South Africa indicates a decrease in mean performance across Grades 1–6. In this paper, we explore the apparently high performance in Grade 1 through a comparative investigation of learner responses on two assessments: the Grade 1 ANA taken in February 2011 by Grade 2 learners and a diagnostic oral interview test drawn from the work of Wright et al. (2006), administered at the same time. Our findings point to a predominant pattern of high performance on the ANA and low performance on Wright et al.’s tests. In-depth analysis of the responses of two learners in this group indicates that this discrepancy is due to acceptance in the ANA of correct answers produced through highly rudimentary counting strategies. The diagnostic test, in contrast, awards lower marks when correct answers are produced in inefficient ways. We conclude with concerns that acceptance of low-level counting strategies in the ANA may well work against persuading Grade 1 and 2 teachers to work towards more sophisticated strategies.Item Open Access Selecting and sequencing mathematics tasks: seeking mathematical knowledge for teaching(Faculty of Education, University of the Free State, 2013) Galant, JaamiaIn this article, I present an initial analysis of an empirical study that was undertaken in an attempt to elicit what subject-matter knowledge, pedagogic content knowledge and curriculum knowledge teachers bring to bear on decisions for teaching. The analysis is based on interview data with 46 Grade 3 teachers, who were presented with two mathematical tasks taken from the 2010 NDBE Grade 2 and Grade 3 Numeracy Workbooks. Teachers were required to justify the selection and sequencing of the two mathematical tasks for teaching multiplication. In so doing, they provide some indication of what they know or do not know about the mathematical concepts in the tasks; about the connections between mathematical concepts; about the representations of those concepts, and about how learners learn those concepts. Teachers’ responses varied from an articulation of the pedagogic and mathematical intentions of the tasks, to the use and consequences of pictorial representations in the tasks and how learners would respond to the tasks. The variation in responses reflected different criteria that teachers used to justify the selection and sequencing of the tasks. The analysis raises critical questions regarding the interplay between teachers’ subject-matter knowledge, their pedagogic content knowledge and curricular knowledge, which they bring to bear on pedagogic decisions. The analysis raises further critical questions concerning the pedagogic and mathematical explicitness of tasks in the NDBE Numeracy Workbooks. The analysis suggests that careful consideration must be given to the construction of mathematical tasks in Grade 3, and probably the Foundation Phase, to ensure that the mathematical purpose of tasks is explicit, and that ‘contextual noise’ is not introduced that distracts from the pedagogic and mathematical intent of the tasks.Item Open Access Unveiling the South African official primary mathematics teacher pedagogic identity(Faculty of Education, University of the Free State, 2013) Pausigere, Peter; Graven, MellonyThis article is theoretically informed by Bernstein’s (2000) notion of pedagogic identity, supplemented by Tyler’s (1999) elaboration of Bernstein’s theory into an analytical framework that describes four possible identity positions relating to classification and framing properties. The article analyses key primary mathematics curriculum policy documents to investigate the official primary mathematics teacher identity as constructed by both previous and current South African education curricula. The article reveals that the first post-apartheid curriculum, Curriculum 2005 (C2005), projected a ‘therapeutic’ primary mathematics teacher identity with symbolic pedagogical intentions. The recent South African curriculum policy changes to a common curriculum framework (Curriculum and Assessment Policy, CAPS) and universal primary learner tests (Annual National Assessments, ANA) construct and promote a ‘market’ (Bernstein 2000) primary mathematics teacher identity.