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This paper presents findings from a case study on the impact of high stakes oral performance assessment on third year mathematics students' approaches to learning (Entwistle and Ramsden, Understanding student learning, 1983). We choose oral performance assessment as this mode of assessment differs substantially from written exams for its dialogic nature and because variation of assessment methods is seen to be very important in an otherwise very uniform assessment diet. We found that students perceived the assessment to require conceptual understanding over memory and were more likely to employ revision strategies conducive to deep learning (akin to conceptual understanding) when preparing for the oral performance assessment than when preparing for a written exam. Moreover, they reported to have engaged and interacted in lectures more than they would have otherwise, another characteristic conducive to deep learning approaches. We conclude by suggesting some implications for the summative assessment of mathematics at university level. [ABSTRACT FROM AUTHOR]

The importance of integrating history into mathematics education is widely recognised in the literature and advocated in curricula worldwide, including in China. However, under the influence of the long-standing centrally designed curricula, teachers in China are accustomed to content- and teacher-centred examination-driven teaching practices. Adopting a life story approach, this paper reports the case of a mathematics teacher who integrated history into her mathematics teaching during the initial two years of her teaching in a Shanghai high school. The agentic perspective adopted in the study allows us to focus on how the teacher's agency was enacted and achieved when engaging in teaching practices. Our findings reveal the roles played by personal qualities, prior experiences, and the structure and culture of schooling in the teacher's agency in integrating history into teaching under a dominant performance-driven context. Implications of the results for integrating history into teaching in restricted contexts are then discussed. [ABSTRACT FROM AUTHOR]

Our research is concerned with teacher’s knowledge, and especially with teacher’s processes of learning, in the classroom, from observing and interacting with students’ work. In the first part of the paper, we outline the theoretical framework of our study and distinguish it from some other perspectives. We argue for the importance of distinguishing a kind of teacher’s knowledge, which we call didactic knowledge. In this paper, we concentrate on a subcategory of this knowledge, namely observational didactic knowledge, which grows from teacher’s observation and reflection upon students’ mathematical activity in the classroom. In modeling the processes of evolution of this particular knowledge in teachers, we are inspired, among others, by some general aspects of the theory of didactic situations. In the second part of the paper, the model is applied in two case studies of teachers conducting ordinary lessons. In conclusion, we will discuss what seems to be taken into account by teachers as they observe students’ activity, and how in-service teacher training can play a role in modifying their knowledge about students’ ways of dealing with mathematical problems. [ABSTRACT FROM AUTHOR]

This article is concerned with an average student's understanding of the notion of function. It presents an analysis of a dialogue held with Kasia, a sixteen-years-old pupil of a general academic secondary school. The conversation concentrates on a non-standard problem – a functional equation. The theoretical framework used here is the PROCEPT THEORY formulated by Gray and Tall (1993, 1994). The article diagnoses Kasia's procept of function revealed during the dialogue. Her difficulties with understanding this notion can be described as based on a very limited procept of function and a misinterpretation of the symbols used in the functional equation. Three kinds of sources of the difficulties are identified: firstly, the intrinsic ambiguities in the mathematical notation, secondly, the restricted context in which some symbols occur in teaching and a limited choice of mathematical tasks at school, and, thirdly, Kasia's idiosyncratic interpretation of mathematical tasks. Moreover, the paper describes several important changes in understanding of symbols and of the concept of function that took place in the course of the dialogue. This article shows that we can use functional equations as both a research and a didactic tool in mathematics education. [ABSTRACT FROM AUTHOR]

This prospective study adopted Vygotsky's concept of perezhivanie to examine students' lived experiences in learning mathematics and how they were related to subject choice plans and actual decisions. Most research has considered students' mathematics subject choice as a decision predictable using important cognitive and social variables. Due attention has not been given to examining how students' lived experiences in learning mathematics relate and contribute to their subject choice plans and decisions over time. Perezhivanie is a psychological structure for understanding dynamic influences derived from personal and contextual sources. In this study, two case studies were crafted using multiple data sources including interviews, observations, surveys and fieldnotes collected over three years on two achieving students who had studied in the same class. Despite this shared context, their perezhivanija differed, revealing complex dynamic interplay between factors derived from personal and external realms. This study described the identified perezhivanija of these two students and explained how they were related to their subject choice plans and their eventual subject choice decisions. The findings revealed that perezhivanija, including both heightened experiences and meta-experiences, were vital personal sources informing subject choice plans and decisions. [ABSTRACT FROM AUTHOR]

Beliefs about mathematics education and their influences on teaching practices have been widely investigated in recent decades. There have been numerous empirical studies on the influences of religions on teachers' and students' beliefs about subjects such as sciences and language. However, the influences of worldviews in general and religions in particular, as one of the major sources of beliefs in relation to mathematics education, are under-researched. The current study is a first step to unpacking the relationship between teachers' religions and their beliefs about mathematics teaching and learning. By means of semi-structured interviews with mathematics teachers of different religious backgrounds, teachers' perceptions on the connection between their personal religious beliefs and their beliefs about teaching and learning are investigated. In-depth analyses of the perceptions of three mathematics teachers reveal the complex relationship between teachers' religious beliefs and their teaching beliefs. First, there are some common values shared by different religions, which influence the beliefs about mathematics teaching and learning as well as education in general. Second, religion is a rich belief system, and the teachers appear to apply only a portion of their religious beliefs to guide their teaching. It is also possible that a teacher is influenced by more than one religion or cultural tradition. Despite its subtleties, our study provides evidence to support the alignment between teachers' personal religious beliefs and their beliefs about mathematics teaching and learning. [ABSTRACT FROM AUTHOR]

This case study illustrates how a 2nd-grade child, Violet, used an ordinal view of number to reason about positive and negative integers and arithmetic involving integers. Violet's ordinal view of number facilitated her ability to reason about and correctly solve some integer-related problems and constrained her solutions to others. We demonstrate how Violet's thinking evolved over time while she extended the properties of whole numbers and addition and subtraction to the integers. Using this case study as a basis, we propose a series of developmental milestones that build toward one's understanding of integers and integer arithmetic in an order-based way. We believe that understanding Violet's order-based reasoning can help us listen to other children. [ABSTRACT FROM AUTHOR]

As a unifying feature of this Special Issue, we have asked proponents of each framework to analyse an empirical classroom account of one student's process of solving a mathematical problem. Here, for the case study of ‘Frank’, we give the main data that were available to all authors. [ABSTRACT FROM AUTHOR]