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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.

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.

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. [ABSTRACT FROM AUTHOR]

Multiplicative understanding is essential for mathematics learning and is supported by models for multiplication, such as equal groups and rectangular area, different calculations and arithmetical properties, such as distributivity. We investigated two students’ multiplicative understanding through their connections between models for multiplication, calculations and arithmetical properties and how their connections changed during the school years when multiplication is extended to multi-digits and decimal numbers. The case studies were conducted by individual interviews over five semesters. The students did not connect calculations to models for multiplication, but showed a robust conceptualisation of multiplication as repeated addition or equal groups. This supported their utilisation of distributivity to multi-digits, but constrained their utilisation of commutativity and for one student to make sense of decimal multiplication [ABSTRACT FROM AUTHOR]

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]