The second exploratory research

3.8.1 Research question and method

The second exploratory study was designed to explore the nature of relating theory to practice. The research question in this context was: Which signals of utilizing theory do student teachers show in their reflections on studied practices of MILE?

MILE has been expanded with video records of mathematics activities in Kindergarten, group discussions in grade 1, and interviews and mathematics lessons in grades 2, 4 and 6, to an amount of 70 gigabytes.

Research at the teacher training college at Helmond was designed to find out how prospective teachers make connections between theory and practice in MILE. This research involved two classes, each with 25 student teachers. The teacher educator gave his students a list of 150 key theoretical concepts from previous courses, to serve as a theoretical framework to help student teachers to value their previous theoretical knowledge when they start the new MILE course (‘The Foundation’) for second year student teachers (Dolk et al., 2000). Ten two-hour meetings were held. Following the method of triangulation (Maso & Smaling, 1998), four pairs of student teachers were observed and interviewed, and a participating study of the group work with two student teachers was conducted.

3.8.2 Identifying theory in action

Schön (1983) has demonstrated that theory in action is primarily implicit, so the researchers generated a list of possible signals of theory in action to support the observations of student teachers at work (see appendix 1). Following are some examples, in which each signal is coupled with a representative case of theory in action and references to its sources:

- While observing practical situations, student teachers can refer to the theory that comes to mind. Example: student teacher points to a teacher who interprets the product of 2 x 5 and, in doing so, employs the rectangle model (Treffers & De Moor, 1990, p. 75).

- Theory is used to explain (as a means to understand) what occurred in the practical situation observed. Example: student teacher explains the method employed by the pupil who is using MAB (base ten) material as a working model (Gravemeijer, 1994, p. 57).

iii _{This section has been published in adapted form as: Oonk, W. (2001). Putting theory into}

practice: Growth of appreciating theory by student teachers. In M. van den Heuvel-Panhuizen (Ed.), Proceedings of the 25th conference of the International Group for the Psychology of Mathematics Education 4 (pp. 17-24).

Theory-enriched practical knowledge in mathematics teacher education

- The student interprets the intention of the teacher or pupil(s) with the help of theory. Example: student teacher points out the ‘mirroring technique’ applied by the teacher as a means to help the pupil reflect his own actions (Van Eerde, 1996, p. 143).

- The theory generates new practical questions. Example: student teacher wonders at which level (stage) of learning multiplication the pupils are (Goffree, 1994, p. 280). 3.8.3 Theory in action. An example

Student teachers indeed showed certain predicted signals of utilizing theory (Oonk, 2001). However, these signals were generally rather weak and ambiguous to localize, and it was hard to determine how and when utilized theoretical knowledge had been acquired.

We will portray the essentials of the research in three steps: the characterization of a fragment in MILE, the theory the researcher/teacher educator linked to that practice, and what student teachers had to say about the fragment.

- The MILE fragment shows a pupil, Fadoua, and her teacher, Minke, sitting at the instruction table during an independent working session in grade 2. Using a diagnostic interview, Minke seeks to identify the thinking behind Fadoua’s mistake (18 - 6 = 11) in her seatwork. It appears that Fadoua counts backwards starting from 18 and in the process also skips two numbers (12 and 14).

- The theory that makes this practical situation more comprehensible is a result of research into subtraction strategies employed by young children, in particular the method of counting backwards. Initial errors, counting mistakes, and counting too far are well-known problem areas. To avoid problems in the transition from manipulative to mental calculations when learning to shorten procedures, structural models based on visualising ‘fives structures’ can be employed to learn to subtract numbers to twenty (Gravemeijer, 1994; Van den Heuvel-Panhuizen, et al., 1998).

- After watching and analysing the video, the student teachers Denise and Marieke discuss the most appropriate way to assist Fadoua. Denise initially suggests solving the problem using 18 blocks (units). Marieke rejects this idea however, because she believes that it doesn’t solve Fadoua’s counting problem. She also rejects a second suggestion – using the number line – for the same reasons. Marieke ultimately agrees with Denise and suggests using the reckon rack. Denise’s foremost argument is that the fives structure of the reckon rack can help Fadoua to address the problem by directly subtracting 6. “And that doesn’t involve counting anymore,” she says.

In this discussion between the student teachers Denise and Marieke we see theory in action. They compare, face, and consider, on the basis of theoretical perspectives, which material or model is (or is not) appropriate and why. A similar process occurs when they design an explanatory approach for pupil Fadoua, partially on the basis of theoretical considerations (the reckon rack teaching method).

3.8.4 Some results

The results of this study revealed that student teachers used theory as a means to understand and explain practical situations. The frame of reference of second-year student teachers appeared somewhat diffuse and fragmented. It remained difficult to separate practical wisdom from pedagogical theorizing (cf. Shulman, 1987; Sockett, 1989; Fenstermacher & Richardson, 1993; Pendlebury, 1995). The ability to articulate observations of and reflections on practical situations in theoretical terms remained largely undeveloped. The culture of teacher training colleges also seemed to hamper this development. As a result, there is a real danger that student teachers will hang on to their personal (subjective) theories instead of learning to integrate theory and practice to attain Theory-Enriched Practical Knowledge (EPK). To improve the educative character of MILE it was concluded that accommodations in two directions were necessary. On the one hand, theory should be stated more explicitly in the learning environment. On the other hand, in mathematics teacher training more time should be spent on discourse and tutoring, less at the level of ‘coaching at a distance.’

In the next section we will go into the intended accomodation of the learning environment for the following research.

3.9 Practice based professionalization and enriched practical knowledge