Training the teacher educators 1 Introduction

The researcher presented his research plan in front of an audience of over a hundred mathematics teacher educators, representatives of the 39 training colleges for primary school teachers, at the annual conference of the network of experts involved in primary mathematics teaching in the Netherlands. In addition to information about the content of the research, information was given about the conditions for participation in the study, being: the mandatory and conscientious use of the course materials offered, including the course in the regular curriculum, mandatory training for teacher educators, certification, and at least two years of experience as a mathematics teacher educator, the size of student groups, the number of meetings, the number of contact hours and the participation of the teacher educator in a closing interview. These conditions were mentioned again in the flyer that was handed out as well as distributed electronically over the national network. At first, seventeen Pabos responded to the invitation for participation. In the end, twelve met the conditions for participation. The Pabos that participated showed a geographic spread across the Netherlands: they were located in Amsterdam, Breda, ‘s-Hertogenbosch, Eindhoven, Gouda, Hengelo, Leeuwarden, Meppel, Rotterdam and Zwolle. One Pabo dropped out during the study due to organisational problems.

5.3.3.2 Training

A (mandatory) day of training in advance of the study for the teacher educator was organised under supervision of the researcher. The goal of the training was optimizing the analogy in working with students by the various teacher educators. Results from the foregoing development and research were used to inform teacher educators about how to introduce the Guide as a tool for the discourse and for the student teachers’ self- regulated learning, as well as to create an appropriate investigation context for student teachers, help them find reasons to get a successful start, formulate inspiring learning questions, et cetera. The information was described in a teacher educator’s manual (Oonk, 2003; appendix 22). The quotations, given below, from the manual (p. 12) are examples of a general guideline.

General

In the interest of the study it is necessary that the general and specific guidelines are followed. The material offered must be as ‘natural’ as possible for all student teachers, while at the same time it must vary as little as possible between the various locations (...).

The students’ logbook

The term ‘logbook’ is interpreted in a diversity of ways at the Pabos. Therefore this course doesn’t use the term for student teachers. For the purpose of the discussions in meeting 2, 3 and 4, the students’ reflections are aimed at ‘What did you learn?’ (see appendix 5 of the manual).

The students’ individual learning question

It is the intention that all coursework is guided by the students’ individual learning question, which runs through the course like a guideline. The reflective notes from the research into the network for the tables of multiplication and the final assessment are both assumed to be guided by the individual learning question (...).

The manual contains detailed guidelines for each meeting. These guidelines concern the goals of the meeting, the organisation, the subject-specific and course-pedagogical content, suggestions for the students and aspects that are vital for obtaining valid research data, such as the exact instruction for filling in the lists of concepts and handing out the assignments for the assessments. The following quotation from the manual is an example of an instruction for the teacher educators (manual, p. 28; appendix 22).

Instruction final assessment, unit 3

Writing a reflective note for a situation from MILE (‘The suitcase full of balls’). See appendix 9 from the manual.

It was shown in the small scale study that the students, even if they did not yet know MILE, could easily find the fragment in question. Only help the students, if needed, with the procedure, that is to say: if they don’t understand or can’t find the fragment concerned. Do not offer the fragment in a whole-class setting, as students have the understandable tendency to react to the content of the images when viewed in a plenary session. Furthermore, the assignment contains a subtle impulse to stimulate searching at the end – in the italic text; the intention is to find out to what extent students show searching behaviour, partly on the basis of that impulse, and will for instance look to see if there is relevant information to be found before or after the selected video fragments. To remind the students of the focus on their own learning question and to inform the teacher educator, the individual learning questions are noted at the top of the reflective note (see appendix 11, p. 61 in the manual).

The prescribed teaching material has been planned chronologically and in detail in terms of activities for teacher educator and students. During the training day the guidelines were discussed, and crucial interventions by the teacher educator – for instance specific questions – have been practiced in the meetings based on video material from the small scale study. The teacher educators also had tasks, assignments and texts for students that had to follow the letter of the manual pointed out to them. Some guidelines concerned the right moment to show students certain papers or forms, with one of the intentions being to avoid influencing open-minded reflections.

Finally, the training day gave attention to collecting the research data and there were opportunities to ask questions.

Based on the experiences during the training days the researcher revised the manual and provided it to the teacher educators who participated.

Theory-enriched practical knowledge in mathematics teacher education

Shortly after the end of the course, every participating teacher educator was interviewed in a session that lasted about an hour. Goal of this interview was to obtain as much information as possible about experiences with the course, particularly information relating to interventions by the teacher educator and striking reactions from students that were related to the use of theory. The teacher educators received a list of nine questions from the researcher not long before the interview, intended for use as a guideline. 5.3.4 The instruments

5.3.4.1 Initial and final assessment

Experiences from the small scale study led to changes in the initial assessment for the large scale study. The change involved a changed set-up for the way in which the reflective note made by the students was done, resulting in a better match between the initial and final assessments (appendix 11 and 12).

The reflective note at the start of the course was intended to test at what level the students used theory within a specific category of the nature of theory use (factual description, interpretation, explain and respond to) (section 5.3.6.3 and 5.3.6.4). The four assignments for four different situations from MILE had been phrased so that they would consecutively evoke these four types of theory use. For instance, in the first assignment the student teachers were asked to observe student Chantal, and then in their own words give a factual description of what occurred in that situation (appendix 11). This part of the initial assessment yielded two types of data: primarily the number of theoretical concepts that each student used in doing the assignments, and also statements by students in which theoretical concepts were used. The four practical situations presented in the video material and the situation for the final assessment were selected from the lessons about learning the tables of multiplication in MILE-grade 2, with the same students and teachers for all situations (appendix 23). The situation that was selected for the final assessment was new to the students. They were given a short explanation about the context of the situation, and where the video clip of the situation could be found in MILE; there was also some advice on writing the reflection (appendix 12).

The large scale of the study necessitated limiting the use of tools and data to those of the written reflections in the initial and final assessments, the numeracy test (section 5.3.4.2) and the questionnaire (section 5.3.4.3).

5.3.4.2 Numeracy test

After the course had ended, the students did a numeracy test (appendix 19). The written test consisted of ten problems, and had been derived from tests for the subject area Mathematics & Pedagogics as used widely at Pabos in the Netherlands at the time (2003) the test was set (Faes et al., 1992; Goffree & Oonk, 2004). Individual numeracy was used as an independent control variable in this study that is marked by the research questions. A positive correlation is supposed between the ability to solve mathematical problems in

students and their level of pedagogical (content) theory use (section 4.3.8 and 4.4.2). Triangulation of the data resulting from the numeracy test with the data produced by the other instruments, had as its goal to generate answers to research question 3b (section 5.2) to the extent of a correlation between the student teachers’ level of numeracy and the nature or the level of their use of theory. To have the problems in the numeracy test target aspects of numeracy even more emphatically, the test as used in the small scale study (appendix 18) has been slightly changed for the benefit of the large scale study (appendix 19). This change involves an extended explanation with the problems in order to evoke reflection. Furthermore, the students could also rate each problem on a five point scale to evaluate how hard they thought a problem was.

A personal evaluation index (PEI) was also determined, to define the relation between the level of difficulty and numeracy score (difficulty total score times two, minus the total score of the numeracy test). The underlying idea is that the index can be a measure for the confidence in one’s own numeracy.

The standards for determining the level of numeracy have been developed in three sessions, with the researcher’s first proposal being discussed with other expert educators37_,

tried out and revised (appendix 20). The second version of the standards was subjected to a random sample (n = 15; appendix 21). The fifteen tests were scored independently by two judges in three sets of five with analysis in between. Independent assessment of the whole random sample yielded an interrater reliability (Cohen’s kappa) of

κ

Section 4.2.3.5 describes the backgrounds, the purpose and the set-up of the questionnaire. The experiences with the questionnaire in the small scale study did not lead to adapting the questionnaire for the large scale study. The fourteen questions relate to the evaluation of the course, and particularly to how the students appreciated the theory as expressed in the course.

Descriptive statistics of the data (mean and std. deviation; appendix 14) have been determined through the use of the computer software SPSS, version 15.0.