Constanta Olteanu

Enhancing mathematics communication using critical aspects and dimensions of variation

This article deals with two prominent topics in the field of mathematics education: the communication in mathematics and its teaching and learning and the continuous professional development of mathematics teachers. In this article, a framework is proposed for analysing the effectiveness of communication in mathematics classrooms. The presentation is based on data collected, during a 3-year period, while different objects of learning is presented in nine classes, and it includes 22 teachers and 884 students. Among other things, the data consist of the students’ tests, the teachers’ lessons plan and reports of the lessons’ instructions. In the analysis, concepts relating to variation theory have been used as analytical tools. The results show that effective communication occurs in the classroom if it has the real critical aspects in student learning as its starting point. Also, the results show that teachers develop new strategies to present the contents by having the focus to open up dimensions of variation.

Differences in success in solving second-degree equations due to differences in classroom instruction

The point of departure in this study is that differences in solving second-degree equations result from differences in what is presented to the students in a learning situation. However, the specified aim is to describe what kind of differences seem to be crucial for learning, and what does seem fruitless. The study is carried out in an upper secondary school in Sweden. Data consist of video-recorded lessons and tests. Two teachers and 45 students participated in this study. The analysis has been made with the aid of variation theory. The results show that the teachers focused on different aspects of a second-degree equation, and that this implied different learning outcomes as some of the aspects were not critical to promoting learning while others were.

Reflection and the object of learning

– The purpose of this paper is to present a model for supporting and increasing teachers’ reflection process. , – The model is grounded in two concepts from variation theory: critical aspects and dimensions of variation. Those concepts are related to the concepts of reflection in-, on- and for-action. Analysis of data allows for determination of what kind of reflection is used in teachers’ professional development (PD) and the teachers’ perceptions of the relevance and usefulness of the PD. , – The results show that, using the concepts of critical aspects and dimensions of variation, teachers practiced reflection in-, on- and for-action in a non-dualistic way. , – The preliminary results confirm that the teachers change their behaviours, through reflection processes, if they have the opportunity to focus on the object of learning and to use research results found in mathematics education in order to open up dimensions of variation in the identified critical aspects. , – The design of algebra modules, according to concepts from variation theory and the reflection process, is the main result of this study. The model used in creating the modules has the potential for supporting and increasing teachers’ reflection process. , – A conceptual model of work practice evolution is proposed. The model is conceived as an intermediary between models of reflection, action at work and models of organisational teaching in the classroom. It considers work practices as evolving through confrontations between critical aspects, action in context, reflections in-, on- and for-action. The author advocate that the use of the presented methodology is not only inevitable but also vital to improve teachers’ PD.

Learning study : promoting and hindering factors in mathematics teaching

ABSTRACT This article focuses on presenting success factors for a group of teachers in carrying out a learning study in mathematics at their school. The research questions are: what are the actions of the school teaching community during development projects? What factors enable a group of teachers to carry out a learning study at their school? Activity theory provides a holistic framework to investigate relationships among the components present in a learning study. The results are based on analysis of interviews with teachers, students, principal organizers of schools and project coordinators, videotaped lessons, students’ tests and minutes taken at meetings of mathematics projects. The results show that the skills of facilitators, the time devoted to collaborative work, the link to learning theory and avoiding overly comprehensive content when teaching lessons are important promoting factors in mathematics teaching. The findings raise important questions about the way in which teacher work within universities.