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Within the commognitive perspective, ritual and explorative routines are used in a very particular way to distinguish students' routines according to whether they are driven by social reward or by generating a substantiated narrative. Explorative routines in this theorisation may refer not to inquiry-based activity but to the result of a student's routine moving from being process-oriented to becoming outcome-oriented, a deritualisation. Choice of tasks as well as a teacher's moves offer students different opportunities to engage in rituals, explorative routines and deritualisations. Through nuancing the space spanned by opportunities to engage in rituals and explorative routines respectively, we describe and contrast classroom practices in three lessons from three contexts. The lessons share a commonality in encouraging explorative routines as a starting point, yet being adapted towards ritual activity through decreased openings for student agentivity, fewer invitations for students' own substantiations or both. We argue that such adaptations are driven by the teachers' commitment to reach mathematical closure in a lesson, to balance considerations of the classroom community and individual students and to meet curricular requirements. Our model helps interrogate the nature and relevance of hybrids of explorative routines and rituals.

This study investigates the relationship between approaches to studying and course completion in two online preparatory university courses in mathematics and computer programming. The students participating in the two courses are alike in age, gender, and approaches to learning. Four hundred and ninety-three students participating in these courses answered the short version of the Approaches and Study Skills Inventory for Students (ASSIST). Results show that students demonstrating a deep approach to learning in either course are more likely to complete. In the mathematics course, a combination of deep and strategic approaches correlates positively with course completion. In the programming course, students who demonstrate a surface approach are less likely to complete. These results are in line with the intentions of the course designers, but they also suggest ways to improve these courses. Furthermore, the study demonstrates that ASSIST can be used to evaluate course design.

This combined interview and survey study explored the relationship between interview data and data from an inventory describing engineering students' ratings of their approaches to studying. Using the 18-item Approaches and Study Skills Inventory for Students (ASSIST) students were asked to rate their approaches to studying in relation to particular statements. A subsample of nine first-year engineering students participated in subsequent interviews exploring their experiences of studying and learning. The students' views were examined and interpreted into inventory scores which were compared to the students' actual ratings. The interviews confirmed the scales measured in the inventory and provided illustrations to them. While students who were extreme in either approach were easier to interpret, others provided a good example of the complex combination of approaches that can exhibit itself in one individual. The study illustrates how combined data sets can contribute to achieve a holistic understanding of student learning in its context.

Analyzing and designing productive group work and effective communication constitute ongoing research interests in mathematics education. In this article we contribute to this research by using and developing a newly introduced analytical approach for examining effective communication within group work in mathematics education. By using data from 12 to 13-year old students playing a dice game as well as from a group of university students working with a proof by induction, the article shows how the link between visual mediators and technical terms is crucial in students' attempts to communicate effectively. The critical evaluation of visual mediators and technical terms, and of links between them, is useful for researchers interested in analyzing effective communication and designing environments providing opportunities for students to learn mathematics.

Research on student learning in higher education suggests that threshold concepts within various disciplines have the capacity to transform students' understanding. The present study explores students' understanding in relation to particular threshold concepts in mathematics--integral and limit--and tries to clarify in what sense developing an understanding of those threshold concepts involves a transformation of understanding in relation to ways of thinking in mathematics. Drawing on data collected in interviews with students taking a basic course on calculus the analysis offers an initial characterization of students' understandings as algorithmic. It then proceeds to construct a more fine-grained theoretical account for how these understandings develop in the course of the interview, suggesting that the transformative aspects of threshold concepts may be conceptualized in terms of shifts in students' contextualisations allowing the development of conceptions at different levels of abstraction simultaneously interacting to shape students' awareness of the ways of thinking and practicing in the subject.

The study explores the nature of students' conceptual understanding of calculus. Twenty students of engineering were asked to reflect in writing on the meaning of the concepts of limit and integral. A sub-sample of four students was selected for subsequent interviews, which explored in detail the students' understandings of the two concepts. Intentional analysis of the students' written and oral accounts revealed that the students were expressing their understanding of limit and integral within an "algorithmic context," in which the very "operations" of these concepts were seen as crucial. The students also displayed great confidence in their ability to deal with these concepts. Implications for the development of a conceptual understanding of calculus are discussed, and it is argued that developing understanding within an algorithmic context can be seen as a stepping stone towards a more complete conceptual understanding of calculus. (Contains 3 tables and 1 note.)

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The study explores the nature of students' conceptual understanding of calculus. Twenty students of engineering were asked to reflect in writing on the meaning of the concepts of limit and integral. A sub-sample of four students was selected for subsequent interviews, which explored in detail the students' understandings of the two concepts. Intentional analysis of the students' written and oral accounts revealed that the students were expressing their understanding of limit and integral within an algorithmic context, in which the very 'operations' of these concepts were seen as crucial. The students also displayed great confidence in their ability to deal with these concepts. Implications for the development of a conceptual understanding of calculus are discussed, and it is argued that developing understanding within an algorithmic context can be seen as a stepping stone towards a more complete conceptual understanding of calculus. [ABSTRACT FROM AUTHOR]