Implications of the small scale study for the large scale study

The previous section 4.4 gave a description of the findings regarding the answers to the research questions of the small scale study. Below the most important implications of the small scale study for the large scale study will be described. As mentioned in section 4.3.9 and 4.4, the small scale study provided a new insight into the character of the use of theory by students. Two dimensions were distinguished, namely nature and level of theory use (section 4.3.9). On the one hand the use of theory appears to manifest in the way students describe situations with the help of theory; this we refer to as the nature of theory use. This can take place by factual description, interpreting or explaining a situation, and by responding to a situation. On the other hand the use of theory can be expressed in the presence or absence of a meaningful connection between the theoretical

Theory-enriched practical knowledge in mathematics teacher education

concepts used by students, the level of theory use. These new insights were reason to focus the research questions for the large scale study on nature and level of theory use and on the connection between these two dimensions. A related implication was that the reflection-analysis tool had to be developed further and tested on validity and reliability. We assumed that the learning environment of the small scale research would prompt student teachers to engage in practical reasoning to a greater degree than the earlier versions of the learning environment. That hypothesis appeared to be confirmed by the small scale study. The results showed a large variety in theory use, though the conjecture arose that theory use was insufficiently evoked in some of the students. The study gave indications for a further optimisation of the learning environment. The ideas arose from the analysis of the experiences, based on focal points described in the theoretical framework, particularly in sections 2.6, 2.7 en 3.9. Below we will briefly describe five of the desired changes. Each will mention the most important motivation for the proposed change, followed by the desired design activity and (between brackets) the relevant focal point for theory use (see section 2.6.4).

- Some of the students felt uncertain about the tasks they were set, and the requirements attached to them, particularly the ones relating to independent study. Design activity: designing a detailed manual for teacher educators, with hints and tools for both students and teacher educators, such as criteria for good learning questions with examples and guidelines for support from teacher educators (focal point for theory use: finding confidence).

- After two meetings the researcher and the teacher educators felt the need to make student teachers even more explicitly aware of their own learning, respectively learning yield. Design activity: designing a ‘logbook activity’ called ‘What (else) was learned in this meeting?’ (focal point for theory use: gaining awareness of one’s own views and develop sensitivity towards theory use).

- Within the time available for the course, a choice had to be made from several student activities that had taken place within this study, but also in some cases in the preceding try-out. Regarding the use of theory, more was expected of the concept game than from the activities relating to ‘the theorem.’ Design activity: decision to further work out the ‘concept game’ to the detriment of ‘the theorem’ (focal point for theory use: underpinning opinions).

- The output of the questionnaire and statements by students during meetings and interviews showed that the students appreciated the lists of concepts. Some of the students did however report that they could not indicate in the list that a concept had gained more meaning as a result of the course. Others thought at first that they did know a concept and could give meaning to it in a narrative of practice, but became aware during the course that their earlier interpretation had been incorrect

(appendix 2A,B). These experiences implied that it would be necessary in the large scale study to change the text in the list of concepts for the final assessment. The experience with the second part of the initial assessment, the reflective note for a narrative from ‘The Guide,’ led to another idea for changing the assessment. Students were in the opportunity to choose the narrative themselves, which was intended to give them a motivating first orientation on the learning environment. This approach would be inefficient for the large scale study, though, and endanger unambiguousness within the data. Design activity: adapting the lists of concepts and designing a new assessment (focal point for theory use: from subjective concepts to general applicability).

- In this study, the interventions by the teacher educator turned out to be of crucial importance for stimulating the use of theory by students, particularly in relation to raising the level of that theory use. Design activity: organize a one-day training for teacher educators and write a teacher educators’ manual containing detailed guidelines for the treatment, containing examples of key insights, key questions and pitfalls (focal points for theory use: e.g., particularly ‘developing sensitivity for the use of theory’ and ‘the function of persuasiveness, justifiability and usefulness of theory’).

The assumption was that the proposed changes would optimize the use of theory by students, and that their ‘enriched practical knowledge’ could be mapped and analyzed systematically.