Action Research and Lesson Study are common approaches in professional teacher development, in several countries (Elliot, 1991; Lewis et al, 2009; Stigler & Hiebert, 1999). A similar approach is Learning Study which also has proved to be promising as a means to improve both student and teacher learning (Lo & Pong, 2005).

In a Learning Study, teachers jointly work in an iterative process to gain knowledge about how their teaching practice in the classroom affects student learning. Focus lies on the object of learning, in this study a mathematical relationship, and how students understand and learn the same relationship. In this approach, teachers formulate the research question based on their students misconceptions or reflections. In this case a student question “How can the result of division sometimes be greater than the dividend?”  The aim of this study is to explore the necessary conditions for students to learn that the quotient in division may be greater than the dividend. The purpose of this presentation is to report on a Learning Study and how teachers explore the object of learning, in particular which features are critical for students to learn.

In this study, variation theory (Marton & Tsui, 2004) has been used as a framework to understand the relation between what is taught and what is learned. More specifically, it is a theory used in practice to analyse and design lessons and students’ learning outcomes. Variation theory has its roots in phenomenographic research (Marton & Booth, 1997) which implies that learning is always directed at “something”, a phenomenon, a situation, a concept or the object of learning. Learning is also explained as a result of seeing this “something” in a new and more powerful way.

In variation theory, learning is analysed using three concepts: discernment, variation and simultaneity (Marton & Tsui, 2004). Discernment is what the learner sees or notices, some features of the object of learning are in the learner’s focal awareness while other features are not. To see the object of learning in a certain way presupposes that the learner is able to discern the critical features. Variation does not refer to the organisation in the classroom, such as different teaching activities, different skills or different interactive tools. We learn from discerning variation of the aspects of a particular object of learning. In this study, learning possibilities are analysed with the focus on these dimensions of variation in the classrooms. Simultaneity implies that the analysis is directed to the possibilities for the learner to experience one or several aspects at the same time.

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