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We are four academics teaching mathematics at three different higher education institutions over two continents. Through fortnightly meetings to research how our practices changed as a result of the COVID-19 lockdowns, we formed a community of practice. The process of articulating our experiences and ideas, and reflecting on the recorded meetings and the written homework we set ourselves, was reassuring and motivational. We identified five assessment propositions to guide our mathematics teaching, which we summarize in this paper. A major proposition of assessment we embraced was the use of open book (or open-internet) assessment as a way to test for mathematical understanding. Our community of practice interactions influenced our teaching and assessment practices. We learnt more deeply about assessment by interrogating each other's work, observing and identifying misconceptions or errors (made by ourselves and others), and learning different ways of solving problems through discussion. We noted that sustaining the community of practice required comfort in being confronted and criticized. The unexpected consequence of our community of practice was the push it gave us to think about why we teach what we teach, assess how we assess, and how we can make both more relevant to a changing world.

Mathematical problem-solving is notoriously difficult to teach in a standard university mathematics classroom. The project on which this article reports aimed to investigate the effect of the writing of explanatory strategies in the context of mathematical problem solving on problem-solving behaviour. This article serves to describe the effectiveness of using writing as a tool for deeper engagement with mathematical problems. Students' claims about, and tutor observations of, problem-solving behaviour were analysed through the lens of Piaget's theory of cognitive development. Examples of enhanced problem-solving behaviour are presented as well as reports from student interviews that writing "forces" deeper engagement. The analysis of students' work and their reflections indicated that writing about problem-solving processes potentially resulted in a cognitive perturbation when students were forced to confront their incomplete understanding (and hence their unstable knowledge structures) and therefore had to achieve a deeper level of understanding in order to adequately describe the solution process.

This article draws on data from a larger longitudinal qualitative case study which is tracking the progress of students over the course of their undergraduate degrees at a South African university. For this paper, we used background questionnaires and semi-structured interviews with 62 first-year students from working-class, township schools who were first registered for Extended Degree Programmes in 2009. The article draws on post-structuralist theory on learning and identity to describe and analyse the participants' perspectives on how they negotiated their high school contexts. We analyse the subject positions in which participants invested, as well as how they negotiated their way through social networks and used resources. Our data illustrate the ways in which students had to carry the burden of negotiating their way through home, school and neighbourhood spaces that were generally not conducive to learning. Nevertheless, participants consciously positioned themselves as agents. They were resilient, motivated and took highly strategic adult decisions about their learning. We argue that a focus on how successful students negotiate their environments challenges the pathologising paradigm of "disadvantage" that characterises research and debates in higher education. It also offers an additional lens for admissions processes and for providing appropriate intervention strategies in the tertiary setting.

In this paper we propose that learning in engineering involves taking on the discourse of an engineering community, which is intimately bound up with the identity of being a member of that community. This leads to the notion of discursive identity, which emphasises that students' identities are constituted through engaging in discourse. This view of learning implies that success in engineering studies needs to be defined with particular reference to the sorts of identities that students develop and how these relate to identities in the world of work. In order to achieve successful learning in engineering we need to recognise the multiple identities held by our students, provide an authentic range of engineering-related activities through which students can develop engineering identities and make more explicit key aspects of the discourse of engineering of which lecturers are tacitly aware. We include three vignettes to illustrate how some of the authors of this paper (from across three different institutions) have applied this perspective of learning in their teaching practice.

This article reports on an activity in a first-year engineering mathematics class designed to strengthen students' personal identities as novice engineers. The literature on identity suggests that students are more likely to be retained in an engineering degree programme if they develop an identity as an engineer and that development of such an identity is encouraged and supported if students can see the relevance of their studies to future studies or their future career. The mathematics encountered at a first-year level is often of an unrealistic nature due to its largely algebraic content as well as to the fact that real-world engineering problems are often intractable without using more advanced mathematics than is accessible in a first-year course. The activity described in this article endeavoured to build empirically on the theoretical issues raised in the literature by asking "Can students identify the presence of classroom mathematics in real-world engineering texts and does this recognition encourage the development of identity as a novice engineer?" Sixty-six students studying first-year engineering mathematics in an academic development programme at a South African university took part in the activity. Data consisted of the students' written assignments and their responses to a Likert-style questionnaire. The written assignments were graded on the strength of their alignment with the task's mathematical requirements. Specifically within the course topic of "Applications of Differentiation," the students were required to use resources from the library and the internet to find examples in real-world engineering where differentiation is used for practical purposes. The examples that the students investigated were necessarily expressed in the discourse of engineering, yet drew on mathematics the students had recently encountered in the classroom. This evident trajectory of knowledge from pure classroom practice to real-world engineering use allowed the students ready access to the discourse of engineering and ideally fostered development of identity as an active novice participant in the world of real engineering. A minority of students did not succeed in the task requirements, but the bulk of the students found the task interesting and informative. Several students expressed surprise and pleasure that they were able to understand what they were reading, revealing to them that they were already participants in the engineering community with some fluency in the discourse.

In the transition years, Grades 7 to 9, the shift from natural numbers to rational numbers and the associated multiplicative concepts prove challenging for many learners. The new concepts, operations and notation must be mastered if the student is to thereafter rise to meet the challenges of algebra and more advanced and powerful mathematics. The multiplicative conceptual field (MCF) groups together such concepts as fraction, ratio, rate, percentage and proportion, all of which are related yet subtly distinct from one another, each with its own challenges. Rasch analysis allows us to compare the difficulty of mathematical problems located within the MCF while, on the same scale, locating the degree to which individual learners have mastered the necessary skill set. Such location of problems and learners on the same unidimensional scale allows for fine-grained analysis of which aspects of the problems being analysed make one problem more difficult than another. Simultaneously the scale gives the teacher clear evidence of which students have mastered which concepts and skills and which have not, thereby allowing more targeted assistance to the class and individual learners. This paper illustrates the process involved in such analysis by reporting on results located within a larger study. It is suggested that implementing Rasch analysis within the school classroom on appropriately designed assessment instruments would provide clarity for the teacher on the locations of difficulty within the problems used in the assessment and the relative degree to which individual learners are achieving success at mastering the targeted concepts.

In this paper, we propose that learning in engineering involves taking on the discourse of an engineering community, which is intimately bound up with the identity of being a member of that community. This leads to the notion of discursive identity, which emphasises that students' identities are constituted through engaging in discourse. This view of learning implies that success in engineering studies needs to be defined with particular reference to the sorts of identities that students develop and how these relate to identities in the world of work. In order to achieve successful learning in engineering, we need to recognise the multiple identities held by our students, provide an authentic range of engineering-related activities through which students can develop engineering identities and make more explicit key aspects of the discourse of engineering of which lecturers are tacitly aware. We include three vignettes to illustrate how some of the authors of this paper (from across three different institutions) have applied this perspective of learning in their teaching practice.

This article draws on data from a larger longitudinal qualitative case study which is tracking the progress of students over the course of their undergraduate degrees at a South African university. For this paper, we used background questionnaires and semi-structured interviews with 62 first-year students from working-class, township schools who were first registered for Extended Degree Programmes in 2009. The article draws on post-structuralist theory on learning and identity to describe and analyse the participants' perspectives on how they negotiated their high school contexts. We analyse the subject positions in which participants invested, as well as how they negotiated their way through social networks and used resources. Our data illustrate the ways in which students had to carry the burden of negotiating their way through home, school and neighbourhood spaces that were generally not conducive to learning. Nevertheless, participants consciously positioned themselves as agents. They were resilient, motivated and took highly strategic adult decisions about their learning. We argue that a focus on how successful students negotiate their environments challenges the pathologising paradigm of "disadvantage" that characterises research and debates in higher education. It also offers an additional lens for admissions processes and for providing appropriate intervention strategies in the tertiary setting. [ABSTRACT FROM AUTHOR]