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Problem-solving and enquiry-based learning are integral in the Mathematics and Science curriculum in South Africa through online-based assessment. Online assessment has rapidly gained recognition because of technology. In this paper, the authors explored the views of online baseline assessment among Senior Phase First-year Mathematics Student Teachers. The researchers adopted a qualitative research method. Data was collected from the participants using an online Google form developed into a questionnaire. The participants in this study were first-year students enrolled for the Bachelor of Education in Mathematics at a university in a rural province of South Africa. The first-year student teachers were exposed to the senior phase baseline assessments through the licensed online Computer Aided Mathematics Instruction (CAMI) tool. One hundred and sixteen (116) senior-phase student teachers completed the online questionnaire. This study used convenience sampling since it was the most appropriate method to conveniently invite the participant. The findings revealed that first-year student teachers were enthusiastic and motivated to write the baseline assessment on computers for the first time. The result also showed that several first-year student teachers experienced difficulties solving mathematics problems using computers. This study recommends more intensive research on student teachers' views on online assessment regarding gender and time.

Pre-service secondary mathematics teachers have a poor command of the exact language of mathematics as evidenced in assignments, micro-lessons and practicums. The unrelenting notorious annual South African National Senior Certificate outcomes in mathematics and the recognition by the Department of Basic Education (DBE) that the correct use of mathematical language in classrooms is problematic is reported in the National Senior Certificate Diagnostic Reports on learner performance (DBE, 2008-2013). The reports further recognise that learners do not engage successfully with mathematical problems that require conceptual understanding. This paper therefore highlights a need for teachers to be taught and master an exact mathematical language that for example, calls an "expression" an "expression" and not an "equation". It must support the call of the DBE to use correct mathematical language that will support and improve conceptual understanding rather than perpetuate rote procedural skills, which are often devoid of thought and reason. The authentic language of mathematics can initiate and promote meaningful mathematical dialogue. Initial teacher education programmes, as in the subject methodologies, affords lecturers this opportunity. The language notions of Vygotskian thought and language, Freirian emancipatory critical consciousness and Habermasian ethical and moral communicative action frame the paper theoretically. Using a grounded approach, after examining examples of student language in a practice based research intervention, the design and development of a repertoire of language categories, literal, algebraic, graphical (Cartesian) and procedural (algorithmic) emerged from three one-year cycles of an action research methodology. The development of these repertoires of language was to assist teachers in communicating about mathematical objects through providing a structured framework within which to think and teach. A course model encompassing small group discussions, an oral examination and a self-study action research project, that helped sustain the teaching of an exact mathematical language, is presented. This is supported by student reflections on the usefulness of implementing them.

This paper analyses student teachers' meta-narratives associated with 'doing' mathematics. The quality of the vocabulary associated with the doing of the mathematical objects of a sample of 56 multilingual third-year mathematics methodology students preparing to be secondary teachers was explored and assessed. Sfard's theoretical concept of 'pronounced foci' and her 'critical categories' for determining if a narrative was mathematical were used to analyse the data. Meta-narratives refer to the written mathematical metacognitive monologues that manifested students' thinking while they were doing the mathematical objects. Students were required to write, in language suitable for teaching, a description of the procedure which led to a solution. The procedures and their associated meta-narratives provided data for a mixed-method analysis in this case study. The absence of a mathematical vocabulary presented difficulties when students engaged in meta-narratives. The data also show that inexact mathematical vocabularies are not culturally specific since the 11 English mother tongue students demonstrated a similar lack of command of mathematical language as their peers. Meta-narratives revealed mathematical misconceptions that may not have been detected without them. For two of the three mathematical objects there is a significant but moderate positive correlation between correct vocabulary and correct procedure. The findings support the need for mathematical vocabulary (and language) to be a teaching focus when preparing teachers for their careers and that the assessment of tasks is not completely achieved by evaluating only the procedural 'doing' of mathematics.

In this paper we offer a theoretical discussion on the mathematical content knowledge development of secondary school mathematics teachers. We discuss relevant studies from literature and use different views and conceptions of school mathematics to propose a model to develop or transform the mathematical content knowledge of secondary school mathematics teachers in such a way that they are able to reconsider their beliefs about the nature of school mathematics. In the model we view mathematics teaching as a kind of problem solving where mathematical content knowledge has to be unpacked and communicated after teachers have reflected on the components of the model, namely different kinds of mathematical knowledge (procedural and conceptual), cognitive processes (problem solving, reasoning, communication, connections and representations) involved in doing mathematics, and working in different problem contexts.

The classroom is a micro-culture where socio and socio-mathematical norms of interactions are fostered by an instructor who represents the mathematics community. Informed by the "emergent approach" developed by Cobb and Yackel (1996) that explains how mathematical growth takes place in the social context of classrooms, I explored the enactment of these norms in small groups' interactions that were not directly influenced by an instructor. I used audio recorded and later transcriptions of groups of undergraduate mathematics students' discussions on class tasks that were analysed using Polkinghorne's (1995) narrative analysis. Using excerpts from the discussions I argue that with a broadened view on socio-mathematical norms, and interpretations of interactions that focus on "learning in the moment", enactments of norms are seen in students' (a) contributions to discussions through initiating discussions, (b) taking responsibility for raising and offering answers, (c) acceptance of arguments that are efficient to the self. (Contains 1 figure and 2 tables.)

Teacher quality in primary schools in South Africa is a serious concern, and how best to prepare teachers in initial teacher education—with regard to mathematics—remains an open and vexing question. There is a confluence of challenges facing the mathematics education community in South Africa: poor conceptual understanding of mathematics amongst teachers; poor attainment in Primary Teacher Education mathematics tests at both first and fourth year levels; and few initial teacher education programmes having evidence of improving knowledge for teaching mathematics. In addition, few lecturers in initial teacher education make explicit the theoretical tenets informing their course design for mathematics. In response to the latter challenge, this paper makes explicit the design choices made in the Emergent Number Sense first year mathematics module in Bachelor of Education degrees, used across six universities in South Africa. It then theorises the choice to commence the B.Ed mathematics programmes with two pathways into number knowledge: (1) a counting pathway, making use of discrete objects or events which can be quantified by counting in ones; and (2) a measurement pathway, making use of continuous extents which can be quantified by introducing a unit of measurement. Each has a distinct metaphor, strategy for the reification of the 10 and supporting representation. This offers the lecturers using the course in their own design trials, and initial teacher educators beyond the Maths4Primary Teachers collective, a conceptual framework for this design choice, which can now be critiqued and further improved. [ABSTRACT FROM AUTHOR]

This paper focuses, empirically, on the developments seen in a group of South African early grades' mathematics teacher educators in the course of a university-provincial education department partnership project. This project sought to support the district Mathematics Subject Advisers to support, in turn, teachers to implement an intervention consisting of a sequence of four lessons focused on multiplicative reasoning. Outcomes based on pre- and post-tests administered by the Subject Advisers suggested substantial pre- to post-test improvement at the student level. Subject Adviser observations and reflections pointed to successes in engendering more dialogic conversations between mathematics teachers and mathematics teacher educators focused on mathematics and its teaching and learning. The 'double move' of increasing overlap between the subject adviser and teacher communities, coupled with evidence that implementing the intervention had enabled the Subject Advisers to develop their mathematical and pedagogical understandings, provides a useful way of considering development in the capacity of mathematics teacher educators to support mathematics teaching and learning. [ABSTRACT FROM AUTHOR]

Although buoyed by the induction of a democratic government, and the high ideals of our constitution, the South African education system has in many ways not met the expectations of its people, in this case, the mathematics education community. With the birth of an expansive intended curriculum came the monitoring of the outcomes through systemic type testing, the so-called attained curriculum. In time, it became clear that the inevitable 'teaching to the test' would constitute a narrowing of the implemented curriculum. Two possible constraining influences of the systemic test were identified, namely, a narrowing of the curriculum, and reliance on only one source of external monitoring. To counteract these, a project titled Assessment Enhanced Teaching and Learning (AETL) has been initiated, involving Grade Nine mathematics teachers. The aim of the project is to provide intermittent markers of progress to the teachers and learners at strategic points throughout the year. Teacher involvement in the design and implementation of these formative assessment tasks is thus central to the project. In this paper, we report on a case study at one school in the Pretoria region, and explore the use of structured assessment tasks as an approach to professional development. The question posed here is, "How does the implementation of strategically designed assessment tasks support professional development and enable professional agency?" Our findings indicate a strong sense of agency motivated by the need to excel in systemic type testing. [ABSTRACT FROM AUTHOR]

The study reported on here was conducted to explore four Grade 7 mathematics teachers' understanding of the relationship between the objectives indicated in the CAPS for teaching different topics/concepts and instructional time stipulated for doing so. Seven lessons were studied in 4 primary schools in the Vhembe district of the Limpopo province, South Africa. A constructivist philosophical approach was used. The data collection methods included interviews and classroom observations. The main finding indicates teachers' understanding of instructional time as either a regulatory or a teaching tool. The ability to translate instructional time into social time depends on the level of teachers' curriculum literacy. Consideration of how CAPS objectives could be fulfilled based on the topic/concept to be taught and the stipulated instructional time, influenced the learners' conceptual understanding, procedural fluency, and competency in mathematics. The conclusion underscores the importance of instructional time as both an essential authoritative regulatory tool and signal of a pedagogic device and communication required to advance the instructional objectives. [ABSTRACT FROM AUTHOR]

Participating in classroom mathematics conversations, particularly engaging with the opinions of others, can improve learners' mathematical comprehension. Teachers can use a variety of invitation and follow-up moves to encourage student engagement. The study aimed to explore teacher follow-up on learners' initial responses to teacher questions and building on learner ideas in Grade 9 mathematics classrooms. A qualitative approach and descriptive case study design were utilised. Data were collected from three schools in the same circuit in South Africa through classroom lesson observations, face-to-face teacher interviews and researchers' field notes. Three Grade 9 mathematics teachers, one from each school, were purposively selected, and the selected teachers' already established Grade 9 classes, with a maximum of 30 learners each (owing to the COVID-19 health protocol), were automatically involved. Cognitively guided instruction, a learner-centred approach that uses social constructivism as a research paradigm, guided this study. The exploration was supplemented by Brodie's conceptual framework on teacher follow-up moves. The findings revealed that the three teachers followed up on learner contributions and used different follow-up moves, namely insert, elicit, press, maintain and confirm. The study showed hybrid teacher moves. Although more teacher moves were skewed towards elicit and insert, which are closer to traditional practices, there is an indication of moving towards the press, maintain and confirm, which are reform-oriented moves. In addition to these moves identified by Brodie, acknowledging or praising the learner for responding was another move that emerged in this study. [ABSTRACT FROM AUTHOR]

The aim of this paper is to discuss the barriers that learners encounter when learning mathematics and to explore an approach that will simplify the teaching of mathematics in Foundation Phase. A qualitative research design within the interpretive paradigm was used. Five professionally qualified Grade 3 mathematics teachers were sampled from five primary schools in the Tshwane South District in South Africa. The data was collected through lesson observation, interviews and focus groups. The data was analysed by reading transcripts to identify patterns that emerged from the data; these patterns were developed into categories. The participants in this study included Grade 3 mathematics learners. Grade 3 learners were selected because they have already done all the mathematics computations. The research revealed that the main barriers to understanding mathematics relate to language, an inability to solve word sums, being unable to carry over to the tens/hundreds, and confusing multiple signs and addition signs. Recommendations were that the teachers should not handle too many aspects at any given time. Until the teachers are sure that the learners have mastered one element or function, such as addition, they should not move to the next section. [ABSTRACT FROM AUTHOR]

The unsatisfactory mathematics performance of South African learners has long been a concern. South Africa is ranked among the lowest performing countries in mathematics, calling for a rethinking of the way the subject is being taught. Mathematics needs to be contextually relevant and interesting for learners to be motivated to learn. Ethnomathematical games provide a means of making mathematics meaningful to learners because of the potential mathematical knowledge embedded in them. Playing gamesis a fun way of learning, and indigenous games are an integral part of learner culture. The aim of this article is to explore the games played by community elders to determine how the games can be used in the classroom. We employed observations and interviews to collect data from community elders on the indigenous games. This article is framed by the following research question: which indigenous games, with mathematical relevance are played by community elders, and what mathematical skills and knowledge are identifiable in the games? The study offers four (4) main findings that are relevant to address the crisis in mathematics teaching and learning. Thisstudy identified various ethnomathematical games and various mathematical operations embedded in them. Based on these findings, recommendations are made to use ethnomathematics to improve the mathematics performance of learners in the intermediate phase. A further recommendation is that mathematics teachers need to be trained in alternate methods of mathematics instruction such as the use of indigenous games. Future research comprises the piloting of indigenous games in intermediate phase mathematics classrooms. [ABSTRACT FROM AUTHOR]

Algebra is a key course in the secondary mathematics curriculum. As a prerequisite for post-secondary mathematics, science and engineering courses, it is considered a gateway course. Since mathematical knowledge for teaching involves more than content knowledge algebra, teachers must be able to do more than successfully complete algebraic problems. They should understand algebra deeply and flexibly so that they are able to mediate students’ ideas, make choices about representations of content and modify curriculum materials. Yet, little is known about how certain groups view algebra and its teaching. In order to determine the beliefs held by a group of South African mathematics teachers with historically disadvantaged backgrounds teaching in schools that were previously segregated through Apartheid, this paper reports on what the group felt are the most important features of algebra. [ABSTRACT FROM PUBLISHER]

This study, conducted in one high-achieving secondary school in a disadvantaged area in the Western Cape province of South Africa, asks whether mathematics teachers alter their "talk" across the face-to-face and computer-based lessons. Data were gathered through video recordings of teacher/student interactions across ten face-to-face and computer-based mathematics lessons conducted by two teachers. Findings indicate that the teachers studied significantly altered their "talk" across the two contexts, with a medium effect size. Further, a significant association at a medium effect level between any two of the three variables—location of lesson, "talk" type and scale of interaction—was found. In particular, teachers used significantly less mathematical talk in the computer lessons. As language is the primary mediator of conceptual acquisition (Vygotsky, 1978), this calls into question what mathematical concepts are learnt in a computer-based lesson. [ABSTRACT FROM AUTHOR]

This article draws on the notion of epistemic agency to illustrate how the enactment of shared epistemic agency is enabled or hindered in a mathematics classroom for student teachers. I used data from an ethnographic study that was conducted in a third-year class of mathematics for educators. Data was drawn from verbal interactions that took place between mathematics student teachers, and between student teachers and myself as their lecturer, as they worked on class tasks. A narrative mode of analysing data was employed to synthesise data from critical incidents, while excerpts of these incidents represented data. From this analysis, I argue that, with extended wait time by a facilitator, students' disposition of epistemic virtues and vices may enable or hinder enactment to epistemic agency, respectively. In article, those vices presented themselves as epistemic stinginess, prejudices against answers by peers and epistemic disclaimer. The epistemic virtues were presented as impartiality and intellectual courage. I further suggest that the recommended wait time of 1–3 during classroom questioning and answering should be extended for classes where epistemic practices are encouraged. [ABSTRACT FROM AUTHOR]