Recent research on mathematics teachers’ expertise has extended its perspective beyond teachers’ professional knowledge (Shulman, 1986; Kuntze, 2012) and focuses additionally on how teachers make use of their professional knowledge in situation contexts: “Noticing” in the sense of “selective attention and knowledge-based reasoning” (Sherin et al., 2011) or teachers’ “Usable Knowledge” (Kersting et al., 2012) are constructs describing teacher expertise in relationship with specific classroom situations. Such approaches assume that the teachers’ professional knowledge might be more extensive than the knowledge they actually use in specific noticing or analysing processes, for example when being confronted with classroom situations. However, due to their mainly phenomenological perspective, the approaches do not answer the question how teachers’ noticing or analysing of classroom situations is started, and what guides and triggers these processes of noticing or analysing. Against this background, qualifying professional knowledge as “usable” or “non-usable” appears as less appropriate than examining in more detail why teachers use specific knowledge in their analysis of classroom situations. There is hence a need for theoretical models and for empirical research which can explain the use of professional knowledge in teachers’ analysing.

With the aim to address this research need, we concentrate on teachers’ analysing (Kuntze, Dreher & Friesen, 2015; Friesen & Kuntze, 2016; cf. Seidel et al., 2011; Schneider et al., 2016). This notion builds on core elements of the above-mentioned approaches (Kersting et al., 2012; Sherin et al., 2011; Berliner, 1991; Sherin et al., 2011) and is understood as “an awareness-driven, knowledge-based process which connects the subject of analysis with relevant criterion knowledge and is marked by criteria-based explanation and argumentation” (Kuntze et al., 2015, p. 3214). In a process model (Kuntze & Friesen, 2016; Kuntze & Friesen, submitted), we have described teachers’ analysing as a circular and adaptive process which links observation and knowledge-based interpretation. According to our definition (see above; Kuntze et al., 2015), we consider the process as awareness-driven. This means that (possibly simultaneous) awareness for (possibly different) specific criteria continuously supports the knowledge-based interpretation, connection with professional knowledge and validation. In short, criterion awareness keeps the analysis cycle moving, comparable to a computer stand-by, which fully activates the corresponding explicit knowledge-based analysis cycle in case a criterion appears as useful for describing aspects of a classroom situation.

However, very different criteria (including teachers’ individual criteria based on teachers’ views) can be used for analysing classroom situations (e.g., quality criteria discussed by Clausen, Reusser & Klieme, 2003). For example, in a related qualitative study (Kuntze & Dreher, 2015) we observed that teachers’ criterion awareness for students’ motivation can be predominant over criterion awareness for students’ understanding of mathematical representations and impede teachers’ corresponding analysis. Consequently, criterion awareness related to different criteria might coexist in a competing relationship. It could thus make sense to investigate rather profiles of awareness criteria than awareness related to single criteria only.

Building on prior research, we use an existing assessment instrument for teachers’ analysis of how representations of mathematical objects are dealt with in classroom situations (e.g., Friesen & Kuntze, 2016). Moreover, we concentrate on teachers’ awareness related to how representations are used in classroom situations. To our knowledge, teachers’ awareness regarding the use of representations in relation with other criteria such as students’ motivation, attention, or prior knowledge has not been explored yet.

Consequently, the study aims at answering the following research questions:

(1) How do teachers see their criterion awareness related to the use of representations, to students’ motivation, attention, or prior knowledge?

(2) Is teachers’ competence of analysing the use of representations in classroom situations related with their reported awareness?

The first research question focuses on the teachers’ reported criterion awareness. In a corresponding questionnaire , teachers were asked about their awareness related to four different criteria, which were reflected in the scales “reported awareness related to students’ motivation”, “reported awareness related to students’ attention”, “reported awareness related to the use of representations”, and “reported awareness related to students’ prior knowledge” (4-point Likert scales, 1: strong disapproval, 4: strong approval). The items express what criteria teachers might draw on when observing that students have difficulties in the process of solving a task (e.g., “If a student does not advance with a task, I check whether s/he is currently not very motivated” is a sample item for the scale “reported awareness related to students’ motivation”). Given that there were two items per scale, the reliability values ranging from .67 to .85 were acceptable. The teachers’ competence of analysing the use of representations was assessed with a test comprising of eight vignettes, in which the teachers had to analyse the use of representations in eight classroom situations (cf. Friesen & Kuntze, 2016). Core parts of the competence test instrument have been presented in prior papers (Friesen & Kuntze, 2016; cf. also Kuntze & Friesen, 2016). As explained in detail in these papers, in all the classroom situations shown in the vignettes, the vignette teachers were asked for help by students and reacted with a change of representation of a mathematical object, which was unnecessary and which was lacking of a connection with the initial students’ representations. The test scores resulting from the test-takers’ analyses of these vignettes can be described empirically using a one-dimensional Rasch model. In this way, we were able to measure the teachers’ competence of analysing in a specific domain, which corresponded to one of the awareness criteria, namely awareness of the use of representations. The sample of this study consists of N=125 German mathematics teachers at the beginning of their induction phase at secondary schools (81 female, 44 male, mean age 26.9 years; SD=4.1 years). We chose this sample as these teachers had finished their pre-service teacher education phase recently and did not differ very much with respect to their teaching experience. Moreover by selecting this sample, we will be able to track their professional development regarding their competence of analysing and their awareness in a follow-up study.

The teachers’ mean reported criterion awareness regarding the use of representations was positive (M=3.02; SD=.64). None of the other awareness criteria (awareness of students’ motivation (M=2.51; SD=.64), awareness of students’ attention (M=2.24; SD=.70), awareness of students’ prior knowledge (M=3.02; SD=.66)) received higher approval. Significant correlations between the scales (in particular motivation/attention rpearson=.52; p<.01; and use of representations/prior knowledge rpearson=.42; p<.01) indicate that there are two correlated pairs of variables. In order to further explore profiles of teachers’ awareness criteria, a cluster analysis (Ward method) was carried out on the base of the four variables which yielded two clusters. Both clusters (cluster 1 and 2) did not differ significantly in their positive ratings of the awareness criteria related to the use of representations and related to prior knowledge. However, the teachers in cluster 1 (n=63) evaluated their awareness regarding the criteria of motivation and attention on average negatively, their counterparts in cluster 2 (n=62) gave rather positive ratings for all awareness criteria. Addressing the second research question, the clusters were compared with respect to the teachers’ competence of analysing The over-all mean score was 1.26 (SD=0.63). We observed a significantly higher competence score for cluster 1 (T=2.33; df=123; p=.02; d=0.42). This means that the teachers with a stronger relative focus on the use of representations showed a higher competence of analysing the use of representations. The findings suggest that different awareness criteria might compete with each other in the analysis process. Of course these results should be interpreted with care as they are non-representative. The evidence needs replication, also across different cultural settings, e.g. on the international level. Such follow-up research can help to find out whether in the future, teachers’ criterion awareness merits more focused attention also in teacher professional development – in European countries and beyond.

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