Mathematics instruction is required to not only support the acquisition of routines in a teacher-centred and small-stepped fashion, but also to stimulate the fostering of a deeper understanding of the learning contents (De Corte, 2004). The instructional action of teachers is closely linked to their beliefs about teaching and learning. Beliefs do not only have an influence on teaching style and teaching practices, but also have effects on the achievement of the students (Staub & Stern, 2002). In addition, the integration of beliefs is decisive for successful teacher training and further development. The belief-system consist of, inter alia, general beliefs on the one hand and situational beliefs related to concrete subjects, contents and instructional situations and actions on the other (Calderhead, 1996). As a condensed form of knowledge, pedagogical content knowledge (Shulman, 1987) overlaps with beliefs. However, within the belief system of a teacher, conflicting beliefs can be found, such as constructivist ideas on teaching, and beliefs that correspond to traditional teaching ideas. Furthermore, these different beliefs can be equally pronounced and exist alongside one another (Philipp, 2007). The aim of this analysis is therefore a) to record teachers’ pedagogical content beliefs on the one hand on a general level, and on the other hand in relation to a particular content (three introductory lessons to Pythagorean theorem) b) to capture the associationist and the constructivist dimensions of the teachers beliefs in a differentiated way (Leuchter et al., 2006) c) to relate the teachers’ beliefs to the instructional actions observable in the video data (Leuchter et al., 2008), and d) to compare the teachers’ beliefs with the students’ achievements and perceptions of the teaching style. The perceptions of the teaching style of a teacher on the one hand from the point of view of the pupils as participants in the instruction and on the other hand from the perspective of researchers as video observers demonstrate the extent to which the teachers’ beliefs are reflected in the perceived instructional practices. The influence of teachers’ pedagogical content beliefs on pupils’ achievements indicate the importance of the integration of beliefs for successful measures of teacher training. The analysis is part of a Swiss-German study on teaching and mathematical understanding in German and Swiss 8th and 9th grades (N=40 classes) (Klieme & Reusser, 2003). Video recordings of the three introductory lessons to Pythagorean theorem are combined with complex surveys of teachers and students throughout the course of a whole school year. The general pedagogical content beliefs were measured through a questionnaire. To record situational pedagogical content beliefs, a semi-structured interview referred to selected occurrences of the three videotaped lessons. The interview was coded by means of content analysis. The coding system allows the teacher statements to be quantified in order to categorize the teachers’ belief patterns on the instructional interaction. The standardized contents of the videotaped lesson units enable a microanalysis of the instruction with the greatest possible comparability. The coding of the videotaped lessons was carried out using developed codings following the manual used in the TIMSS (Stigler, Gallimore & Hiebert, 2000). The students’ perceptions (N=980) were recorded using questionnaires. The analysis of the students’ achievements is based on two tests, before and after the three introductory lessons to Pythagorean theorem. Statistical analysis enabled a comparison of the student, the teacher and the video data, an examination of the interaction of the three perspectives and the impact of the teachers’ beliefs on the students’ achievement.

By means of frequency analysis and cluster analysis, it is possible to categorize the teachers into different groups which can be distinguished according to the belief pattern on the instructional interaction. Frequency analysis and cluster analysis enables us as well to categorize the teachers instructional action. The interrelation of the teachers’, students’ and outsiders’ perspective is examined by means of correlation analysis. By means of hierarchical linear models, it is possible to analyze the impact of the teachers’ beliefs and teachers’ action on the student’s achievement and their perception of the teaching style.

The representation of differentiated pedagogical content beliefs of teachers enables an insight to be gained into the highly specific and structured knowledge of teachers. The comparison with the students’ achievement and the perception of the instruction by the researchers provides additional indications of the link between the teachers’ beliefs, their observable actions in the classroom and the achievement of the students. The results also represent an important contribution with regard to teacher training. The integration of specific, subject-related pedagogical beliefs is important for successful measures of teacher training. This study gives the opportunity to formulate targeted requirements in this area.