## Developing Mathematics Teacher Knowledge: The Paradidactic Infrastructure of ‘‘Open Lesson’’ in Japan

Source: *Journal of Mathematics Teacher Education, Vol. 16, No. 3*, June 2013, p. 185-209.

(Reviewed by the Portal Team)

The aim of the current research is twofold:

1. Present a theoretical approach to study mathematics teacher knowledge and the conditions for developing it in direct relation to teaching practice.

The model is based on a well-established theory from the didactics of mathematics.

2. Present and analyse a case of open lesson using this theoretical approach.

In order to conduct their analysis, the authors have developed a new technology about the open lesson practice and some elements of theory as well.

**Open lesson**

The ‘‘open lesson’’ practice is a form of teacher knowledge development which is well established in Japan.

The basic idea of this format is that a large number of teachers from other schools are invited to observe a class taught by a colleague.

The teachers then participate in a discussion session with him on the details of the lesson.

**Data and methods**

The open lesson to be analysed in this paper was taught on 25 June 2009 in a 2nd grade class of the elementary school in Japan.

For this study, the authors collected data from three sources:

• the lesson plan of the teacher (pre-didactic practice) which the participants of the open lesson may study before the lesson;

• the real-time observation of the lesson;

• the discussion after the lesson (post-didactic practice).

They conducted the analysis for each source of data.

The analysis illustrates the essential connection between pre-didactic organization traced in the lesson plan, the observed didactic practice and the post-didactic practice of the discussion.

The open lesson session has been described as a specific form of post-didactic practice related directly to an actual observed lesson, and aiming specifically at elaborating the theoretical aspects of teacher’s didactic practice in the lesson.

The commonness of open lessons in Japan, together with the absence of exterior or material motivation for this practice, strongly indicates that Japanese teachers find this practice useful and professionally rewarding, and it is the nature of this experience that the authors exposed and analysed here.

The study suggests how open lessons, as a shared paradidactic practice, can contribute to elaborate common theoretical blocks related to didactic practices.

The present case study shows that much of the discussion and potential learning for teachers is of a more general nature.

The authors have shown that the discussion relates the lesson to more theoretical aspects of the mathematics curriculum as such, and even to more general pedagogical and societal aims of the school.

This way, the discussion provides a space for developing teacher knowledge that is neither narrowly limited to teaching a particular lesson nor drifting into discussions of teaching philosophies which are more or less detached from the reality of schools and teaching.

And, just as in the case of lesson study, the goal is to develop and strengthen the shared theoretical blocks relating and informing didactic practice in a much wider sense.

In conclusion, the current study indicates why open lessons represent, to Japanese teachers, an attractive element of a professional learning community of teachers, and that the communal learning arising from this practice goes largely beyond what can be said about a single lesson.

The authors hypothesise that the establishment of similar paradidactic infrastructures could be an important contribution to enhance the professional development and status of teachers in many Western countries.