Towards a local theory of integrating theory and practice

The results of the study and the analysis of the student teachers’ activities in the course of the four parts of the study, provides the basis for reflection to a local theory for learning to integrate theory and practice by student teachers. On the one side this theory involves the student teachers’ process of learning to integrate, on the other it involves the learning environment that is intended to support that process with teaching materials

and targeted interventions by the teacher educator. Both components are presented mainly in an integrated manner in the following description.

Below, first a description is given of the context in which the intended learning by the student teacher and the support of that learning process by the teacher educator took place, providing an overview of the ingredients of the local theory. After that, the theory will be further elaborated and finally presented in summary.

The first confrontation student teachers had with theory within this study, was the moment that the theoretical framework was presented as a multifunctional list of theoretical key concepts that would come up in the learning environment. At first this list functioned as an advance organizer. The students could indicate which concepts were (un)known to them in the context of a practice story, and the source of that story (own practice, literature, MILE, lectures and workshops). At this stage it was likely that for most of the students the theory of the domain in question was a disjointed collection of concepts, parts of which were, as separate elements, related to narratives of practice. The stories were not always meaningful to the students, sometimes they even turned out to be linked to concepts that were thought to be meaningful on the basis of misconceptions. A number of students indicated in the evaluation of the study that certain concepts had gained a different, or more, meaning for them during the course than their original ideas. The intention of the course was to evoke, in several ways, meaningful use of the concepts by the students, to expand and deepen their repertoire, with the highest goal attaining a cognitive network of ‘theory-enriched practical knowledge.’ The most important sources were theory-laden ‘practice stories’ from MILE and The Guide, and the ‘research stories’ from the students’ own practice. The theoretical reflections by the teacher educator that were related to those narratives and the reflective notes in The Guide functioned as mirror and sounding board in the discourse and during individual study.

Multimedia learning environments as used in this study, give student teachers the opportunity to observe ‘practice’ alone or together, to discuss and study it, without being distracted by having to keep order or all kinds of organisational problems. The experience and identity of student teachers do place specific demands on that learning environment. Opinions about teaching and learning that students have acquired, also by earlier experiences, can easily lead to critical judgements and a focus on cut-and-dried answers in analysing practical situations. It requires extensive coaching to put the students on the investigative trail, and any approach must lead students towards an attitude that is marked by being prepared to ask questions of oneself and pronouncing cautious suspicions and preliminary conclusions. In such a learning environment, including sophisticated coaching, students can learn to integrate theory and practice. The variety of data collected from both the small and large scale studies has shown how

Theory-enriched practical knowledge in mathematics teacher education

students made connections at different levels between theory and practical situations. In the second part of the course on offer, the focus of student activities shifted more and more towards constructing a cognitive network of theoretical concepts. Examples are the reflections on the investigations about childrens’ knowledge of tables of multiplication in the student teachers’ teaching practice, the activities related to the game of concepts, the concept-map activities and the concluding ‘collaborative lecture’ in which the knowledge and experience that had been gained were positioned in the stages of the multiplication course under the teacher educator’s supervision.

The search for answers to the student teachers’ individual learning questions could lead to a more profound ‘ownership’ of the enriched practical knowledge. At the end of each meeting, students were invited to think, respectively become aware of, the theory- enriched practical knowledge they had gained, using the motto: “What (else) did I learn?” The practical knowledge that was gained could be further deepened and widened by writing reflective notes at some points during the course.

At that stage, the list of concepts gained two new functions, that of giving support and providing an overview, and providing an insight into progress with acquiring theory. In the final assessment, students could show the theory-enriched practical knowledge they had gained by writing a reflective note based on observation of a teaching situation from MILE that had not been brought up in the course.

The theoretical character of the course showed itself in the number of theoretical concepts that students used and their ability to meaningfully relate theoretical concepts to each other. In Dutch mathematics teacher education student teachers are faced with

subject specific theory, with the realistic mathematics domain-specific instructional theory in that area (RME; e.g., section 2.6 and 3.2) and with general pedagogical

theories. That complexity of teaching mathematics (Lampert, 2001) was reflected on a small scale in the study, through the learning environment, the theory in the list of fifty- nine theoretical concepts that were central to the course, together with the theory laden

practice narratives. The study showed large differences in the way in which the students involved these theoretical concepts in their arguments. Two dimensions were distinguished, the nature and the level of theory use. The nature of theory use relates to four ways of using theory: factual description, interpretation, explanation and ‘responding to.’

The level relates to the degree to which the concepts are expressed meaningfully and in relation to each other in the statements and notes of the students. The highest level (3) is reached when students express a meaningful relationship between two or more theoretical concepts in a written (meaningful) unit. In such level 3 units, the transition from the second to the third level can often be seen. A first or second sentence will contain statements using a theoretical concept, while the following sentences will

contain different concepts that correlate meaningfully with the foregoing concepts. There are also rises in level within the third level. One such rise in level has for instance been observed in student Anne, when she showed a tendency towards hypothetical thinking and reasoning (section 4.3.2 and 4.3.3). This could be defined as a fourth level of ‘responding to situations’ (D4), something that Ruthven (2001) might call ‘practical theorizing’ (section 2.7.1) and Simon (1995) as the start of developing a ‘hypothetical learning trajectory’ (HLT) (section 2.7.1). A rise in level from D3 to ‘D4’ also occurs when Anne reflects at a higher level than the level of the network of theoretical concepts, by reasoning about the relationships within that network (section 4.3.4). Section 4.4.2 argues that these rises in level seem related to the kind of level-rise that Van Hiele (1973) describes in his theory on levels in mathematical thinking. That level theory has influenced many scientists both within and outside the Netherlands, among other things in the development of theory about mathematical learning processes in students. For instance, Gravemeijer (2007) describes rises in level within the framework of the design heuristics of emergent modelling as the development of a network of mathematical relations. And this is in fact what student Anne did, to construct abstraction by reflection on the relationships she distinguished. In section 4.4.2 this has been interpreted as the transition from horizontal to vertical didacticizing (Freudenthal, 1991).

The study has shown that the role of the teacher educator regarding the stimulation of rises in level is crucial. The teacher educator has the expertise to theorise, to evoke theory use and to stimulate it, among other things by selecting adequate video fragments, asking challenging questions, making use of differences in argumentations, presenting confronting situations (Piaget, 1974; section 2.7.1) and inspiring ‘pedagogical conflicts,’ sharpening the discourse with theory-laden summaries or by stimulating hypothetical thinking. It is exactly the combination of these ingredients that can lead student teachers to adopt theory (section 2.6.4) and construct EPK. The narratively oriented learning environment (Pendlebury, 1995) provides the EPK with a lasting meaning. The ‘theory in narratives’ leaves a lasting impression and can be recalled.

Taking the above considerations and their relation to the results of the study as its starting point, a local theory of integrating theory and practice in mathematics teacher education has been formulated, based on the concepts theory, practice and the relationship between theory and practice as they have been described in the sections 2.3 up to 2.7. There, theory is defined as a collection of descriptive concepts that show cohesion, with that cohesion being supported by reflection on ‘practice.’ For the acquisition of theory-practice relationships by students, the first step is to look for a connection with theoretical notions that students already have. This is done by making connections between theoretical

Theory-enriched practical knowledge in mathematics teacher education

concepts with multiple definitions (definitions, notes, contexts; list of concepts) and practical situations students themselves have experienced. Afterwards practical knowledge is made explicit and theory-enriched through cycles of observation, analysis of theory- enriched practical situations and ‘responding’ to them. That enrichment occurs in the discourse, led by the teacher educator, in collective work, during individual study and by writing reflective notes. Impulses for enrichment are: the ‘narrativised’ theoretical framework of concepts, adequate literature, the learning and investigation assignments, confrontational situations, reflective conversations, challenging questions, reflection on successes, (collaborative) lectures, and reflective notes.

To some degree, the cycles of observing, analysing and ‘responding,’ are the detailed elaboration of the cyclical process that for example was observed in The Pioneers in the first exploratory research project (section 3.5.5). The ‘theory-enriched practical knowledge’ that student teachers acquire, contains the key insights in relation to learning and teaching mathematics.

The connections between theory and practice that students themselves make, become visible in the nature and level of theory use. A rise in level is caused by practical reasoning and reflection; it leads to an extension and refinement of the ‘theory-enriched practical knowledge’ network.

The reflection-analysis instrument can be used as a guidance or (self)assessment tool to establish the degree to which students are competent to integrate theory and practice. In summary, and in line with what has been described about the definition of theory, it can be established that the local theory is determined by three main components, the

formulated concepts of theory, practice and the relationship between theory and practice, the theoretical knowledge base of the learning environment for student teachers and the guidelines for teacher educators, to support the learning and developmental processes of students.

These lead to the theory gaining a function as an orientation basis for reflection on practice. The coherence of the descriptive concepts that was mentioned in the definition of theory, is determined by the learning and teaching theory of realistic mathematics education and the concepts for nature and level of the use of theory.

The research into theory use by student teachers has provided the reason in this study to design a learning environment that is optimized with respect to the possibilities for students to use theory. The research questions could be answered in this learning environment. The fact that the development of the learning environment was guided by theory, and that there are guarantees that the development can be traced, makes it possible to do a similar study in other domains and other subjects in teacher education. The design of the learning environment can be considered as a paradigmatic case of a broader class of phenomena (Cobb & Gravemeijer, 2008). The trackability also involves

the reflection analysis instrument that is part of this study and which can be used as a guidance and assessment tool.