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This contribution aimed at developing an understanding of student teachers' conceptions of guided discovery teaching approaches. A cross-sectional survey design involving eleven secondary mathematics teachers who had enrolled for an in-service mathematics education degree was used to address the research question: What are undergraduate student teachers' conceptions of deductive and inductive teaching approaches? Task-based interviews were used in conjunction with oral interviews as settings for unravelling students' conceptions of the two teaching approaches. Drawing in part from Ausbel's learning theory and Tall's notion of a met-before, the study also aimed at assessing the students' level of grasp of fundamental limitation of empirical explorations despite many benefits associated with them such as helping in identifying patterns, use in formulation and communicating of conjecture, and providing insights on what needs to be solved. Verbatim transcriptions from follow up interviews and textual data from task based interviews were subjected to directed content analysis to infer meaning about students' conceptions of guided teaching approaches. Qualitative data analysis using in part Robert Moore's notion of concept usage uncovered conceptual limitations that include inconsistencies in student teachers' definitions of the teaching approaches, use of specific examples instead of arbitrary mathematical objects in illustrating analytic teaching. Limitations identified should be given attention by mathematics educators in order to increase understanding of the approaches among teachers. Research studies into factors contributing to limitations identified can go a long way in improving the teaching and learning of school mathematics.

The aim of the study is to develop an understanding of the kinds and sources errors and misconceptions that characterise students' learning of school algebra. Systematic random sampling was used to draw sixty-five participants from a population of two hundred and twenty-three form three students. A cross sectional survey design was employed to collect data using written tests, a structured questionnaire and interviewing of the students from one high school in Zimbabwe. Content analysis technique was applied to textual data from three sources in order to determine the types of errors and misconceptions. The main findings are that both procedural and conceptual errors were prevalent that errors and misconceptions can be explained in terms of the students' limited understanding of the nature of algebra; in particular their fragile grasp of the notion of a variable. Sources of misconceptions could be explained in terms of the abstract nature of algebra Mathematics educators should embrace errors and misconceptions in their teaching and should not regard them as obstacles to learning but rather engage with them for better understanding of algebraic concepts by students. Future studies can be carried on systematic errors as one of the ways of improving students' understanding school mathematics.

Intending to improve the teaching and learning of the notion of mathematical proof this study seeks to uncover the kinds of flaws in postgraduate mathematics education student teachers. Twenty-three student teachers responded to a proof task involving the concepts of transposition and multiplication of matrices. Analytic induction strategy that drew ideas from the literature on evaluating students’ proof understanding and Yang and Lin’s model of proof comprehension applied to informants’ written responses to detect the kinds of flaws in postgraduates’ proof attempts. The study revealed that the use of empirical verifications was dominant and in situations. Whereby participants attempted to argue using arbitrary mathematical objects, the cases considered did not represent the most general case. Flawed conceptualizations uncovered by this study can contribute to efforts directed towards fostering strong subject content command among school mathematics teachers.

Despite its central place in the mathematics curriculum the notion of mathematical proof has failed to permeate the curriculum at all scholastic levels. While the concept of mathematical proof can serve as a vehicle for inculcating mathematical thinking, studies have revealed that students experience serious difficulties with proving that include (a) not knowing how to begin the proving process, (b) the proclivity to use empirical verifications for tasks that call for axiomatic methods of proving, and (c) resorting to rote memorization of uncoordinated fragments of proof facts. While several studies have been conducted with the aim of addressing students' fragile grasp of mathematical proof the majority of such studies have been based on activities that involve students reflecting and expressing their level of convincement in arguments supplied by the researchers, thereby compromising the voice of the informants. Further, research focus has been on the front instead of the back of mathematics. Hence, there is a dearth in research studies into students' thinking processes around mathematical proof that are grounded in students' own proof attempts. Therefore current investigations should aim at identifying critical elements of students' knowledge of the notion of proof that are informed by students' actual individual proof construction attempts.