This article reports on a study that investigated the extent to which 57 Grade 6 students at a particular school have misconceptions regarding equality, with the equal sign as focus. It also investigated this school’s three Grade 5 and 6 teachers’ mathematical knowledge for teaching regarding equality, again focusing on the equal sign. The research took the form of a case study in which a qualitative approach was used. Data collection methods included teacher questionnaires and a focus group interview with teachers. Students answered a set of questions after which six students were purposefully selected, based on their responses in the questionnaire, and interviewed. To gauge students’ level of understanding of the equal sign, the construction map for knowledge of the equal sign developed by Matthews et al. was used, while the Mathematical Knowledge for Teaching model proposed by Ball et al. was employed to interpret teachers’ knowledge. Results indicate that few students had a well-developed relational conception of the equal sign. The majority of students could not describe the meaning of the equal sign correctly. The same group of students that applied ‘closure’ was inclined to ‘use all the numbers in an equation’. Grade 5 and 6 teachers’ Mathematical Knowledge for Teaching of the equal sign indicated that in general they lacked the knowledge and skills to identify, prevent, reduce or correct students’ misconceptions about the equal sign.

"Acknowledgments." African Journal of Research in Mathematics, Science and Technology Education, 21(1), p. 107

This paper reports on a study that investigated teacher code switching consistency and precision in multilingual secondary school mathematics classrooms in South Africa. Data was obtained through interviewing and observing five lessons of each of three mathematics teachers purposively selected from three township schools in the Eastern Cape Province. Elements of Gumperz and Mercer’s work on lesson categories and Dowling’s Domains of Mathematical Practice were used to analyse data. Results showed that code switching frequency in general was inconsistent across different lessons for the same teacher. Code switching frequency by all teachers was, however, consistently highest during questioning and explaining when teaching. All participating teachers used code switching strategies most consistently in the public domain and least consistently in the esoteric domain. Some formal isiXhosa translations of mathematical terms were consistently and precisely used and some were not. Two major forms of code switching emerged, namely borrowing code switching and transparent code switching. Very little transparent code switching, which is critical for supporting students’ understanding and thinking in mathematics, was evident in teacher language. Teachers were consistent in the use of borrowed terms. We conclude that consensual understanding of best practices for code switching is required to promote code switching that is precise, consistent, transparent and thus supportive of teaching for conceptual understanding of mathematics in secondary schools.

Drawing on a literature review of classifications developed by each of Riley, Verschaffel and Carpenter and their respective research groups, a refined typology of additive relations word problems is proposed and then used as analytical tool to classify the additive relations word problems in South African Curriculum and Assessment Policy Standard (CAPS) for Foundation Phase. Inconsistencies and errors within the CAPS are exposed. Two different typologies are presented in the CAPS: first ‘compare, combine and compare’ and then ‘change, compare and equalise’ problem types are presented. For the latter typology ‘join’ and ‘separate’ actions are imposed onto static situations. Rather than referring to the ‘whole–part–part’ structure which is common across all problem types, a generalised ‘start–change–result’ structure is imposed. This is both incorrect (static situations do not have this structure) and inconsistent with the articulated distinction between take-away and difference models for subtraction expressed elsewhere in the CAPS. In light of this analysis it is recommended that adapted curriculum guidelines are developed and circulated to Foundation Phase teachers, teacher educators and curriculum advisors. The refined typology of additive relations word problems may be used to underpin professional development initiatives for Foundation Phase teachers, and for structuring the setting of the annual diagnostic assessments at this level.

This study investigated and compared the pedagogical orientations of physical sciences teachers in Malawi and South Africa towards inquiry or direct methods of science teaching. Pedagogical orientation has been theorized as a component of pedagogical content knowledge. Orientations were characterized along a spectrum of two variants of inquiry and two variants of direct instruction. Teacher preferences for pedagogy were measured using case-based assessment items which depicted actual scenarios for teaching particular physical sciences topics and provided four alternative teaching method options. Item topics were common to both the Malawian and South African science syllabuses. Response data were obtained from 164 Malawian high school physical sciences teachers at community, conventional and grant-aided schools, for comparison with previous South African data. Analyses produced pedagogical orientation profiles for each teacher and item in terms of the distribution of choices between the four options and these were aggregated by school type and compared with the corresponding profiles for South African teachers on the same items. Responses to items covered the whole spectrum of approaches, although relative proportions depended on teacher, topic, type of school and country. Aggregated responses for both countries tended more toward variants of inquiry than of direct instruction. In some cases, Malawian teachers had an unexpectedly strong direct didactic preference. A direct didactic orientation was mainly prevalent amongst teachers from disadvantaged Malawian community schools and South African rural schools. South African teachers from suburban schools exhibited a stronger guided inquiry orientation than their Malawian counterparts from grant-aided schools. In contrast, teachers at Malawian conventional schools had a stronger preference to adopt variants of inquiry (guided and open) than South African teachers at corresponding township schools. Teachers at more privileged schools, that is, Malawian grant-aided and South African suburban schools, showed stronger guided- and open-inquiry orientations than teachers at less privileged schools. School type and contextual circumstances affected teachers’ pedagogy choices in both countries, having a substantial influence on the approaches that teachers would adopt in their own situations.

The zone of proximal development (ZPD) is a well-known and frequently used notion in both educational research and practice with a wide and diverse range of interpretations. My aim in writing this theoretical article is not to provide a critical examination or an extensive literature review of the ZPD, but rather to highlight some significant issues surrounding the use of the notion in the mathematics education literature. Extending a series of questions contributed by Del Río and Álvarez, I have added an element about theoretical space enabling researchers who are not in psychology to use the questions to situate their own use of the ZPD within their theoretical perspective and to unpack their assumptions about each of the questions. More importantly, the paper exemplifies how the use of an organising framework assists in achieving some clarity on the different ways the notion is conceptualised in the educational literature. I propose that this framework could enable both researchers and educators to locate their own conceptualisation of the ZPD within the broader, complex space and to assist in achieving coherence between the theoretical and methodological perspectives they work with.

We describe and use an analytical framework to document mathematics discourse in instruction (MDI), and interpret differences in mathematics teaching. MDI is characterised by four interacting components in the teaching of a mathematics lesson: exemplification (occurring through a sequence of examples and related tasks), explanatory talk (talk that names and legitimates what comes to count as mathematics in a particular lesson), learner participation (interaction between teacher and learners and amongst learners) and the object of learning (the lesson goal). MDI is grounded empirically in mathematics teaching practices in South Africa, and theoretically in sociocultural theoretical resources. The MDI framework allows for nuanced descriptions of mathematics teaching and interpretations of differences in what is mathematically made available to learn.

Science teachers in South Africa and globally experience difficulties with the integration of indigenous knowledge into their science lessons—a requirement of many science curricula. One of the reasons for this may relate to the views teachers hold about indigenous knowledge. Such views can form a barrier against successful inclusion of indigenous knowledge in the science classroom. It was, therefore, deemed useful to investigate teachers’ views on indigenous knowledge. This article reports on the development of a theoretical framework, and a questionnaire derived from it, to investigate teachers’ views on indigenous knowledge. The researchers were informed by the framework developed by Lederman, Abd-El-Khalick, Bell and Schwartz regarding the nature of science (NOS) and their views-on-the-NOS questionnaire. A qualitative study was done to develop and validate the views-on-the-nature-of-indigenous-knowledge instrument (VNOIK). The findings indicate that the VNOIK instrument is suitable to determine a wide range of views of science teachers on the nature of indigenous knowledge. We found that South African science teachers held mainly a partially informed view on the nature of indigenous knowledge. The new instrument can be used to measure the effect of a short learning programme and to identify further development needs of science teachers in addressing the tenets of science and indigenous knowledge effectively in the classroom.

With evolutionary biology relatively recently introduced into the South African school curriculum, the need arose to explore practising teachers' knowledge of the subject. A number of anticipated as well as unanticipated cognitive and affective barriers to the understanding of evolutionary biology were identified from a questionnaire with open-ended items and administered at the start of a short learning programme on evolution education. The questionnaire was face validated and piloted. Teachers were conveniently sampled and comprised all course participants (n = 57). Responses to the questions were analysed using inductive and deductive coding. These codes were organised into themes, guided by the research question. These themes indicated poor content knowledge, essentialist and teleological reasoning, creationist objections and a perceived racist agenda to the teaching of this subject.

Through analysing a critical incident where a small group of pre-service secondary mathematics teachers work together on an annuities problem, we gain insight into the ways in which students make use of timelines and attend to time in their talk. Drawing on Lave and Wenger's notion of transparency, I argue that it was only when time became visible for the students, that they were able to resolve the impasse they had reached in working with the timeline. I propose the notion of pre-visible to describe the ways in which a newcomer makes use of a resource in an intuitive way when s/he still lacks awareness of how the resource is typically used in the practice. I discuss five issues pertaining to the use of timelines in working with annuities: the changing role of the timeline; attending to discrete points in time vs intervals; explicit attention to time on the timeline; explicit attention to time in students’ talk; and potential problems associated with the use of month-names on a timeline. I make recommendations for teaching based on these findings.