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The development of urban students' mathematical proving ability is a goal of several curricula frameworks, including some located in the southern hemisphere. However, in achieving this goal, most curriculum frameworks do so from a Western worldview, which is characterized by competition and the role of the individual. The purpose of this study was to use the emancipatory lens to critique the use of a quantitative methodology in favor of the Ubuntu worldview, a methodology grounded in indigenous African epistemologies, particularly storytelling. To this end, I analyzed data drawn from the administration of a survey questionnaire to a conveniently selected sample of 135 11th-grade students enrolled in three separate high schools from ethnically and socioeconomically diverse communities in the eThekwini metropolitan area of South Africa. The context for the argument in this study was provided by correlating students' understanding of functions of proof (verification, explanation, communication, discovery, and systematization) with their argumentation ability, two variables often considered as the key limiting factors for meaningful learning of mathematical proofs. The poor results obtained from a quantitative analysis of data using Western perspectives highlight the emerging need for finding postcolonial methodologies that are sensitive to ethnic issues in addition to language and gender issues. In addition, the inadequacy of the current mathematics curriculum to serve the linguistic and gender needs of urban African students became apparent. This increases the need for sub-Saharan instructors to have knowledge to pursue emancipatory instruction. The key contribution of this study to the field is that it sheds light on the marginalization of African students in learning mathematical proof and related concepts from Western perspectives rather than conducting instructional practices in the Global South's terms; the scope of the effort may explain why research efforts in this line of work have not been documented extensively in literature.

It is widely recognized that students encounter difficulties with proof across all grades and beyond, yet standardized instruments related specifically to students' perceived self-efficacy for mathematical proof have not been readily available. The purpose of this study was to develop and investigate preliminary validity evidence for a new instrument for measuring self-efficacy for mathematical proof that can be of importance to the field. The new Perceived Self-Efficacy for Proof (PSEP) questionnaire is a self-administered, 8-item questionnaire that quantifies experimentation, conjecturing, inductive reasoning, justification, and validation. To validate the PSEP, two studies with 260 eleventh grade students--recruited from three Dinaledi schools in EThekwini metropolitan area, South Africa--were conducted. In Study 1 (n=128), face and content validity were evaluated, and an exploratory factor analysis (EFA) was performed. In Study 2 (n=132), a confirmatory factor analysis (CFA) was conducted and external validity was investigated. In both samples, the PSEP was found to possess good internal consistency reliability with relatively high factor loadings on a single component. Although the findings in this report represent preliminary validation evidence, it can be concluded that the PSEP is a valid, reliable and sensitive measure of 11th grade students' perceptions of their ability to construct a proof and may serve as a meaningful outcome in mathematical proof research and classroom proof education.

The research presented in this article reports on the results of a case study examining the classroom practice of one early career Grade 4 teacher (Nox, pseudonym) as she teaches equivalent fractions. The goal was to explore the ways and extent to which her instruction reflected a dialogical teaching approach, defined as a pedagogical approach underpinned by five specific principles that can be enacted through a range of possible talk strategies to achieve sustained participation of learners and thus enhance meaningful learning. I provide a pedagogical activity to illustrate to teachers how, by instigating and developing classroom talk in the primary classes, a dialogic teaching sequence may be implemented. However, the majority of the existing literature on dialogic teaching stems from studies conducted in Asian, European, and North American countries, whereas systematic research on dialogic teaching across international contexts remains limited. Nox was interviewed after obtaining observational data to seek clarity on some of the observed instructional practices. Analysis of transcripts using the notion of dialogic teaching as a theoretical lens revealed that there was little evidence of Nox's attempts to use talk to make learning of equivalent fractions a cumulative process. In addition, time constraint was the most significant factor in Nox's teaching of equivalent fractions: she considered the curriculum too congested. Implications are drawn for evaluating dialogic teaching in primary mathematics classrooms. Future, larger studies may shed light on the extent of these results and, if need be, a significant investment on initial teacher training may be necessary to underscore the value of dialogic teaching in enhancing meaningful learning of, at least, equivalent fractions.

Despite proof being fundamental to the mathematics discipline and its role as a means to convey mathematical content, little is known about the effect of resources on influencing students' proof construction ability. The purpose of this study was to compare two didactic environments, one regarded as resourced (favored) and the other under-resourced (disadvantaged), in relation to the construction of a mathematical proof. Motivated by the discrepancies in the literature on the influence of school resources on students' performance and the unfortunately prevalent view that the sole function of proof in mathematics is verification by using confirmatory cases, this study sought to examine the differences (if any) between resourced and under-resourced classrooms in relation to students' proof construction ability. To this end, data were drawn from a proof-related task performed by 78 Grade 11 students in the Ethekwini Metropolitan area, South Africa. A modified version of the Proof Construction Assessment tool showed that students in resourced schools significantly performed better than those in under-resourced schools in relation to proof construction. In addition, there was an observable and noticeable effect of this. Specifically, at an alpha = 0.05, the t-test for independent means revealed a significant difference between the two groups, t(76) = 2.749, p < 0.01, d = 0.624 SD. The practical significance of the results emphasizes the importance of taking into account the role of resources when investigating the learning and teaching of proofs. Further, preliminary results also suggested that most students struggled to even begin to prove the proposition. Recommendations and implications for the students' careers and future research are raised and discussed.

This study used a modified mathematical knowledge for teaching (MKT) framework to compare the "common content knowledge" (CCK) of two groups of purposively selected pre-service primary mathematics teachers (n = 86) in relation to spatial visualization. Specifically, thirty-five pre-service teachers (n[subscript 1] = 35) with a pure school mathematics background and fifty-one pre-service teachers (n[subscript 2] = 51) with a high school mathematical literacy background worked on problems that involved spatial objects, their properties and relationships. The responses of pre-service teachers with a mathematical literacy content background to the "Surface Development Test" (SDT) were significantly different from their counterparts' which suggested that their CCK in relation to spatial visualization was thin. Analysis of the "Probes" questionnaire which inquired about the pre-service teachers' views on the concept of spatial visualization showed that misconceptions hindered their performance. Future research directions are discussed.

Learners' difficulties with proof have been ascribed to their lack of understanding of functions that proof performs in mathematics, namely, verification, explanation, communication, discovery, and systematization. However, the extant mathematics education literature on validation of instruments designed to measure learners' beliefs about the functions of proof is scant. The purpose of this mixed methods study was to use the "Standards for Educational and Psychological Testing" (as discussed by the American Educational Research Association/American Psychological Association/National Council on Measurement in Education (AERA/APA/NCME), 2014) as a theoretical and methodological platform to support arguments for the validity of the learners' beliefs about the functions of proof (BAFP) instrument. Scale items were generated from de Villiers' model and a review of the literature, panel of experts, and key informants. The resulting instrument, comprising 28 Likert-scale items and 5 open questions that assessed the five functions of proof, was administered to 87 grade 11 learners in one high school in the Pinetown Education District, KwaZulu-Natal, South Africa. The results put the validity of the scores on the BAFP instrument into question. The study was an exploration of validity arguments based on the evidence from various data sources. Confirmatory factor analysis with a larger sample in a different context needs to be conducted to warrant the use of the BAFP instrument. The key contribution of this study to the field is that it sheds light on the complexities of instrument validation; the scope of the effort may explain why comprehensive validation efforts have not been documented extensively in literature.

The purpose of this quasi-experimental study was to investigate the effect of science, technology, engineering, and mathematics (STEM) problem-based learning (PBL) intervention on students' problem-solving beliefs (PSB). To this end, the PSB questionnaire was administered to a group of eighty-six 10th graders across different socio-economic spectra working with an expert facilitator in a rural school that received curricular support and resources to specifically develop STEM teaching in eastern South Africa. The sample participated in two time periods (pre- vs. postintervention) in which problem-based activities were utilized. A quantitative evaluation of the impact of an intervention on students' subsequent beliefs as well as qualitative analysis of interviews with a sample of fourteen purposively selected students is presented. Results showed that participants increased their mathematical PSB (< .001, = .50). The implications of these findings speak to the potential for teachers to utilize the results to provide opportunities for students to experience PBL activities. [ABSTRACT FROM AUTHOR]