DESIGN AND REDESIGN OF AN IN-SERVICE COURSELisser Rye Ejersbo

Learning Lab Denmark, DPU, Aarhus UniversityIn this paper, I describe how an in-service course based on design research and meta-didactical transposition produced new knowledge about how to communicate and reflect when teaching mathematics. A case study is used to show the meta-didactical transposition of a theoretical concept – the virtual monologue – into a reflective tool for practitioners on an in-service course. The transposition is effected through what has elsewhere been called an aesthetical learning process. The researcher, the teacher educator, and the teachers in the workshop may have different agendas and different practices, but they all work towards their separate goals by reflecting on their practices – they are all reflective practitioners. The virtual monologue, which can be used as a reflection tool at any level, serves to bring out those commonalities. Through the design process, a set of guiding principles for preparing and running an in-service course was shaped.

THE BENEFIT OF AN IN-SERVICE COURSEIn this paper, I draw on my many years of experience as a teacher educator, who has conducted in-service training, and on my research in the same field. In Denmark, teachers are qualified to teach four subjects in grades 1-10, and may enrol on in-service courses if they need to gain skills for their professional work. The in-service course, described here, is for certified teachers, who are specializing in mathematics teaching. The goal of the course is to develop communication and reflection when teaching mathematics with open problems. For evaluation of an in-service course, we often rely on the participating teachers’ oral and written evaluations, and this actual course generally received a very positive feedback from the teachers. However, when I started to make observations in the teachers’ classrooms, I realised that my intentions with the course were less than fully implemented in their teaching. This is particularly true of the important but difficult skills of communication and reflection, the core topic of the course. This discrepancy between the teachers’ (Ts) evaluation of the course and the teacher educator’s (TE) evaluation was the engine that drove my research, and is the topic of this paper. The two conflicts that became evident were the mismatch between what happened in the course and what happened in the classroom and the Ts’ goals for the course versus my goals as TE.There may be many reasons why these conflicts exist; so I decided to concentrate on redesigning ‘my’ course. The subsequent development of the in-service course went through various cycles of instructional design, inspired by specific theoretical concepts from the literature, and ‘transposed’ by me into instructional practice. My focus was on the redesign of the instruction and the Ts’ reactions when they worked with the new design.

Unfortunately, it can be difficult for practice and theory to benefit directly from each other. In the redesign, where I used design research as the methodology, I try my best to inspire practice by new theoretical ideas. This resulted in the development of three guiding principles; meant to guide and inspire those who conduct in-service education.

OBSERVATIONS IN CLASSROOMSThe particular in-service course that I analysed was divided into two. The first part contained the task to plan a course for the Ts’ own classes, and this course should be run and evaluated before the second part of the in-service course. My observations in the classrooms took place after the whole course had finished, but the teachers I visited used their plan from the in-service course when I observed them. They all expressed satisfaction with the in-service course and felt that it had inspired them to change some of their habits. What I observed was that the Ts indeed used ideas from the in-service course on open problem-solving, which they participated, but they lacked knowledge and skills of how to communicate and reflect upon this communication, and of inclusive listening. The problems I noticed were that the classroom discourse involved more explanatory than questioning. The Ts in general found it difficult to listen to pupils’ mathematical communication and suggested often solutions without listening. To ask powerful questions that would take the pupils further in their understanding of mathematics seemed to cause trouble. I observed that the Ts had difficulties with their mathematical skills as well. However, despite this playing a crucial role, it was not the core issue of the course, and therefore it only plays a minor role in the redesigns.My next observation was of the in-service course itself. At the time we finished the in-service course, I was not aware of what, if anything, each T took home with them. Based on the evaluation and the Ts’ expression of satisfaction, I expected that they were prepared to teach with open problems in such a way that their pupils would gain from it. From observing the school classrooms, I found that teaching mathematics through open problem-solving led to more trouble than I had expected; therefore I analysed how I had taught on the course, in particular the parts which should help the Ts to communicate and reflect. I saw that what I did most on the course was set up activities. These activities were not always connected to a certain theory; instead they were often practiced on ‘practice-level 1’ (Handal & Lauvås, 2000). I thought that this would prepare the Ts for their daily teaching, because the activities were combined with discussions about constructing educational programs. Because of my beliefs in the importance of activities in the learning process, I ‘forgot’ to teach the Ts how to communicate or reflect; I did not give them any specific analytical tools they could use to reflect upon their communication or how to improve it. I took for granted that they could reflect on the activities if they only had time and few

guidelines for doing it. I was wrong. Confusing the Ts’ good evaluation of the course with my expectation of their benefit was indeed naïve.

A META-DIDACTICAL TRANSPOSTIONThe difficulties of using theory in practice have often been discussed in the research literature (Strauss, 2001; Skott, 2004; Rasmussen, 2004). Why is it so hard, and how can we build better bridges between theory and practice so that practitioners will be able, so to speak, to ‘walk the talk’? One of the bridges between theory and practice is the didactical transposition, which Brousseau (1997, p. 35) describes as follow:

To teach it, then, a teacher must reorganize knowledge so that it fits this description, this “epistemology”. This is the beginning of the process of modification of knowledge that changes its organization, its relative importance, its presentation and its genesis, following the needs of the didactical contract. We called this transformation didactical transposition.

Brousseau referred to the transposition of mathematical knowledge to suit the audience of mathematics students. The didactical transposition has two steps, from the scientific discipline to the curriculum and from there to the classroom. The two steps in this transposition are performed by different people (Chevallard, 1985). I propose to extend the idea to the didactical transposition of didactical knowledge itself – the theory – to suit the audience of mathematics teachers, and I call this a ‘meta-didactical transposition’, which has only one step; from the research to the in-service classroom, done by the TE. Often, the theory presented to Ts is ‘just theory’ for them and it is not implemented in their practice; but when a theory is transposed into a workshop and is experienced as an emotional event, as will be demonstrated below, the participants can then reflect on the event, analyse it, and eventually use it in their practice.Aesthetical learning processesBesides being specialised in mathematics, I was trained in art teaching. Adopting terminology from art education I was inspired by the theoretical framework ‘Aesthetical Learning Processes’ (Hohr & Pedersen, 1996). The word ‘aesthetic’ is not intended to connote beauty. Rather, it is used as the ancient Greek ‘aisthesis’, meaning ‘knowledge which comes through the senses’. The method is designed to shape a space for expressing experiences that are still not formulated in a linear way or completely understood by the individuals. The idea is to create an emotional experience from the beginning, and from that platform to let the event be a personal experience, a need that precedes analysis. ‘Aesthetical Learning Processes’ can be depicted as a tripartite model for the process of experiencing a new conception:

Figure 1: ‘Aesthetical learning processes’

A conception arises at the moment feelings find a conscious form, and this form later becomes an experience that can be analysed. From the arts, where this method has its roots, the expression of the impression is again a ‘non-linear’ form without words such as a painting, a sculpture or a performance. In this study it was different: The expression had to end up in a linear analysis, expressed through words and with argumentation. Inspired by ‘aesthetical learning processes’, I used the part for teaching mathematics education, in which we perceived some common events in a kind of activity, which engaged and engendered emotions in a ‘safe setting’.Design researchI chose ‘design research’ as methodology for the cycles of redesigns of the in-service course. Design research is a continuous process of instructional design and classroom-based analysis, where the researcher formulates a hypothetical learning trajectory. Development of theory is one of the primary goals of design research and theory must do real design work in generating, selecting and validating design alternatives at the level at which they are consequential for learning (Cobb et al., 2003). The design can be materials, a form or a program – in my case an educational program for an in-service course - where the instructional design is of interest. Figure 2 illustrates how design research works (ibid). In this case, the question was how the analyses of the classroom-based observations influence designing new instruction.

Figure 2: Design research is a continuous process of design and analysis. The arrows symbolise the process from preparation of the design to a realisation.

What I needed for the design process was relevant theoretical concepts about communication and reflection. Schön (1983) talks about various professionals (such as architects, artists, teachers and baseball players), when he introduces the concepts knowing-in-practice, reflecting-in-practice, and reflecting-on-practice. His ideas are

Classroom-based Analyses

Instructional Design

relevant, not only to the teachers, but for the students, teacher educators and researchers in mathematics education as well. Of particular relevance is his distinction between reflection-in-action (e.g., by a teacher during an intensive classroom activity) and reflection-on-action (e.g., the same teacher reflecting on her classroom activity after school hours).

A CASE STUDYI used several different theoretical concepts for the meta-didactical transpositions. The case study took place in an in-service course and the theme for the transposition was communication and reflection. The basis for this choice is the notion of the ‘reflective practitioner’ (Schön, 1983), which unites all level of practice and theory in mathematics education. I see teaching and learning processes as crucially involving reflection in and on practice (ibid), though the specific practices may vary according to the kind of learner: The pupil is a reflective practitioner when she learns mathematical ideas through reflection in and on her mathematical problem-solving or investigations; the teacher is a reflective practitioner when he consolidates his mathematical and didactical knowledge through reflection in and on his teaching practices; the teacher educator learns by reflecting in and on her practice in designing and conducting workshops for teachers; and the math education researcher gains his insights by reflecting in and on all the above practices.Several tools may aid the reflective practitioner, and in this paper I will focus on one such tool: the virtual monologue. Leron & Hazzan (1997) introduce the ‘virtual monologue’ (VM), where an experienced teacher or researcher uses the narrative mode (Bruner, 1985; Bruner & Haste, 1987), to vividly convey his or her view of the student’s mental processes. Thus, the VM is one tool that helps the reflective practitioner move from practice to theory through reflection. The VM can be a powerful tool for reflection, but like all such tools it should be used with care and with respect for its limitations and shortcomings. One obvious limitation is the subjective and ambiguous nature of any particular VM created in a particular situation by a particular person. As will be seen later, this particular limitation can sometimes be turned into an advantage by building on the variety of VMs produced in a group. A second and perhaps more serious weakness is the fact that the VM is using a verbal medium for describing an essentially non-verbal phenomenon – the student’s mental state. A more thorough discussion of the tool’s strengths and weaknesses can be found in Leron & Hazzan (1997).I developed a novel use for the VM and used it extensively in my practice as TE. In fact, I effected a meta-didactical transposition of the theoretical idea into practice, where the Ts experienced an emotional event that was then used for an analysis of the theory and for reflection on their own practice.

Reflection on practice is clearly a vital task for teachers, but nonetheless, one that many find difficult. Teaching situations that will help teachers reflect on their actions, beliefs and norms are therefore necessary. The practitioner in actionThe following took place half-way through the first part of the course. Twenty-four teachers were enrolled in the course. The following first-person narrative is an abridged and edited version of parts of the diary I kept about what I considered to be reflection-on-action. In my preparation of the course, I wanted to draw the Ts’ attention to the possible interpretations of a VM, both from the student’s and the teacher’s perspective. I was excited to adopt a coping perspective, to take an empathic attitude. I assumed that the Ts’ values would be evident through the way they expressed their empathy. Furthermore, this practice would highlight individual differences in how they view the pupil’s mental processes during the solution process. Unlike how L&H did it in their original article, I would try to imagine what kind of mental processes took place in the teacher’s (or researcher’s) head. It is easy to criticise the teacher while empathising with the students, but here was a challenge in my course to focus on both the student’s and the teacher’s inner voices. In the design of the actual teaching, I wanted to work with various kinds of reflections; the first step was selecting the article on VM.Workshop diary (part I): cognitive perspective. Or, how the theoretical ideas in L&H were transposed into my practice. I translated the main part of Section 2.2 of L&H into Danish: the task on linear equations with a parameter, the researcher’s expectations, the interview with the student (Dina) and its original interpretation, and finally, the authors’ interpretation, as seen through their virtual monologue. I reproduce three parts of this material for the reader; this material is needed to understand my story. The task, the ‘expectations’, and the Dina interview are taken from Sfard & Linchevsky, 1994, pp. 218-220 (henceforth abbreviated to S&L). For the complete discussion, including the VM analysis of the Dina interview, cf. L&H, Section 2.2.

The task (S&L): Is it true that the following system of linear equations k – y = 2 x + y = k has a solution for every value of k?

The expectations of the researcher (S&L): In a problem like this one, the objects that the students are supposed to consider are not just numbers – they are functions. To understand the question, one must realize that each of the equations, [...] represents a whole family of linear functions [...].The interview with Dina (S&L):

(Dina (D) is a tenth-grade student, working on the above task, I is the interviewer)

D: [reads the question silently] “... has a solution...for every values of k…”

I: What does it mean ‘has a solution’?

D: That we can put a number instead of k and it will come out true.

I: When we say that the system has a solution for every value of k, what is the meaning of the word ‘solution’? Is it a number or what?

D: Yes, it’s a number.

I: One number?

D: Yes, it’s the number that when you put instead of k, then the system is true.

[...]

I: This word ‘solution’ here, to what does it refer? Solution of what?

D: Of the equations, k - y = 2 and x + y = k.

I: What is a solution of these equations?

D: When we substitute numbers...

I: Instead of what?

D: ... instead of x, y, and k, and it comes out true.

I: So, once more, what are the solutions we are talking about in the question [points to the words ‘has a solution’]?

D: I think ... I think that I need three numbers: x, y, and k.

I presented the translated materials to the participants on OHP transparencies, together with L&H’s first interpretation (the cognitive perspective; not included here due to space limitations). The teachers saw the task on the OHP, at this stage without any discussion, but with enough time to read and think about how to solve it. I assumed that some of them would have difficulties understanding the task just like Dina had; this was a part of my expectations. After a dramatization of the communication we looked at the interpretation of Dina’s helplessness and confusion and then I started a discussion with the following question: what is your opinion on the interview and its interpretation?Some reacted quickly and said:

- Dina’s answers seem relatively rational and the interviewer seemed to stress her in a way that made it difficult for her to think.

- The interviewer plays the usual teacher’s game ‘Guess what the teacher is thinking’. They clearly expressed understanding and sympathy for Dina. After a while I asked them:Could we guess what was going on in Dina’s head during the interview?One of the teachers said to me with an angry voice:- It irritates me that you ask that question ‘what goes on in her head’. I am not able to know what is going on in the heads of my 24 students. I was a little surprised, but before I could answer her, one of the other teachers said to her:- Why does it irritate you? Don’t we all guess when we communicate with the students? How do you listen to them? After a little discussion about communication and listening, I ended with a quotation from Covey (1989): “Try to understand before you want to be understood.” Workshop diary (part 2): VM and coping perspective.Now I presented OHP transparencies to the teachers, with the translation of Dina’s virtual monologue from L&H (pp. 271-2; The italicized phrases are taken from the actual interview with Dina, as quoted above from S&L):

What do I have here? A system of equations... Oh, well, I know how to do that. You just have to solve it. It does look a bit different, but I can just do the usual solution.

D: [reads the question silently] “... has a solution... for every value of k...”

I don’t understand this phrase. Why don’t they just say ‘solve’ as they always do? I don’t think we had this question before. So how can I solve it? What am I going to do? I really feel I am groping in the dark here.

I: What does it mean ‘has a solution’?

I am not sure, but usually solution means

D: …that we can put a number instead of k and it will come out true.

I: When we say that the system has a solution for every value of k, what is the meaning of the word ‘solution’? Is it a number or what?

I really don’t know. I don’t even understand the question. What was the question? “Is it a number?” well, what else could it be? I don’t know. Oh, well... [performing a leap of faith]

D: Yes, it’s a number.

I: One number?

Of course, what else? I wish I knew where these questions are leading, I am getting more and more confused. But at least it seems from the question that I was right – it is a number.

D: Yes, it’s the number that when you put instead of k, then the system is true.

[...]

I: This word ‘solution’ here, to what does it refer? Solution of what?

What do you mean ‘solution of what’? When we do equations in class we never have such questions. We just need to know how to solve them. What was the question? Solution of what?

D: Of the equations, k - y = 2 and x + y = k, what else could it be?

I: What is a solution of these equations?

D: When we substitute numbers...

I: Instead of what?

What are the letters here?

D: ... instead of x, y, and k, and it comes out true.

I: So, once more, what are the solutions we are talking about in the question [points to the words ‘has a solution’]?

D: I think ... I think that I need three numbers: x, y, and k.

I then asked:What are your comments and opinions about this version? And why? The responses again came immediately: - No, it is not what she thinks, she thinks… Different suggestions now filled the air:- I have three unknowns here but only two equations, strange.- What does the k do here?- The k must be a letter like x and y – then I just have to find the value.- Why does she ask that way? I am sure she wants me to say something special. What

could it be? It seemed like their own difficulties made them identify with Dina. The plenary discussion focused on what might have been going through Dina’s head. After a while I turned to asking how the task could have been thought out originally, why the interviewer asked the way she did, and how they would have asked, if they had been the interviewer. Now they faced some difficulties. It was easy for them to identify with Dina, the student, but much harder to identify with the teacher (here the interviewer), even though it should have been natural for them to think like a teacher.

It was easier to criticize the teacher than to understand her. Maybe they felt resistance to the interviewer because they themselves had difficulties in solving the problem. Eventually, of course, we took the time to solve the task for ourselves.Workshop diary (part 3): A VM of their own. The whole group was very engaged, even though some hesitated initially. It was easy for everybody to join the discussion. For the next part, I chose another transcript. My choice here was a discussion between a teacher and two students at a Danish oral examination concerning percentages – an area they all felt safe with. It is a dialogue where it is easy to laugh a little at the teacher.The participants were split into four smaller groups: Two groups would create a VM for the student and the other two for the teacher. They were given 20 minutes to do this task. Then the ‘teacher-groups’ and the ‘student-groups’ presented their VMs at the plenary meeting, followed by lively questions and discussion. It gave some new insights for all of us. Instead of only judging how the teacher asked questions and how he confused the students, they tried to understand and identify with him. The questions they now asked were: - How was he caught in that trap?- How could he come out of it without confusing the student?- What kind of questions or comments could he make instead?- Furthermore they started to reflect on their own way of asking, like- How do I ask questions myself and what kind of answers do I expect? - Why are my questions like they are?One of the ‘students-groups’ suggested that the students had a clever strategy for asking the teacher questions without answering anything themselves, a strategy they had not noticed before, but in retrospect could now recognize in their own communication with their students. Working with the VM in this way gave them the time and the possibility to become aware of many more details. They were guided by their own emotional involvement and by the communication in the group. The discussion became different from what went on before: it was more balanced and contained more understanding and less criticism of the teacher. Finally, I presented the article with other highlights, and they could all have a copy of it if they wanted. In this session, they were allowed to discuss the whole process or ask any questions as well.The conclusion of my reflection on that lesson was that we all had a new experience in reflection because the situation was authentic for all of us. When we talked about how the course had affected their teaching in the second part of the course, I specifically asked about the influence of the VM workshop. Some teachers answered that it had influenced their way of listening to themselves; they were more aware of how they asked and listened to the students; they paid more attention to the

communication in the classroom; things they didn’t notice before became clearer to them. But at the same time, they also became more uncertain. What they had been doing automatically before, now, all of a sudden, seemed questionable, and they had not yet developed an alternative behaviour. Even though this has not been an easy process, I value it as a first step in learning how to reflect on communication in action. Workshop diary (part 4): concluding reflections on the practice. The way the teachers experienced the idea of VM has created for them an emotional event for them. The teachers became involved with their feelings, both positive and negative, and they experienced this before we did any analyzing or theorizing. In reflecting on the workshop and what it has achieved, I was aided by the theoretical framework Aesthetical Learning Processes, mentioned above.The task of creating a VM, or trying to express what Dina was thinking and feeling, is an open problem that does not have only one solution, nor even a best one. It has brought up in the Ts’ many feelings and ideas, and has given them the opportunity to discuss what came up. It was easy for them to express what they thought she might have been thinking, rather than having to learn an abstract and detached theory. It started from their knowledge, from their understanding, from what they knew best and felt safe with. They were able to use experiences from their daily school life. They have acquired a tool for reflection in and on their practice. Developing knowledge in action would come if and when they are ready to use it.This part was more or less as I planned: I wanted them to be emotionally involved and to reflect in action, and, through that let their beliefs come into view. What I couldn’t foresee was what kind of discussion would emerge. This is where I had to be a reflective practitioner and intensively reflect in action myself. Working in this way, the workshop facilitator may experience a loss of control from, having to deal with so many voices from the participants, and from being the one that needs to decide what kind of feedback to give, what kind of summary to make, what will be the next step, and what take-home problems to give the participants. The energy comes from all the participants, but the facilitator has to give the direction.

THE GUIDING PRINCIPLEThe gap between researchers and practitioners has often been noted and criticised. It has been my repeated experience that theoretical ideas and research papers can be powerful tools in the professional development of teachers, but only after a substantial meta-didactical transposition. I created a set of guiding principles to address this issue (Ejersbo, 2007). The set of guiding principles is meant as a kind of rules or scaffolding, which I developed from my own teaching combined with theories from the pedagogical

literature. The guiding principles (GP) for the rules are the framework for the meta-didactical transposition. They are as follows:1.

On the in-service course, each teaching session is based on a particular theoretical concept from the research literature.

2.The theoretical concept is transformed into practical activities so that:

a. the theoretical concept is made into a tool;b. the teachers can use the methods in their own classes;c. the teachers’ tacit knowledge becomes apparent;d. room for reflection is shaped along with the activities.

3. In the preparation phase, theory precedes activities; in the practice phase, activities precede theory.

These guiding principles (GP) represent one way to perform a meta-didactical transposition. The principles are formed in such a way that different content could be used in the transposition. In this case, the content was mathematical communication and reflection when teaching with open problem solving. If the Ts became aware of their own (inappropriate) habits and at the same time were offered a theoretical framework that could help them improve that particular habit, it would have a pronounced beneficial effect on the learning processes.The case study presented in this paper represents an example of how the principles were realised on an in-service course, and of how the Ts responded. I will present a brief argument for each principle and explain how and why they play a role in the meta-didactical transposition. The first GP concerns preparation. It means that the course objective should be guided by the identification of a number of relevant articles from the research in the area. The research in the articles should be in such a form that it could be transposed into pedagogical practice by the TE. In the actual case study it concerned mathematical communication and reflection, and the particular theoretical framework was the virtual monologue (Leron & Hazzan, 1997).The second GP concerns activities and how they should serve as a vehicle between the theoretical part and the practical application of the theories. My urge to design these activities came from years of experience as a TE combined with pedagogical research that demonstrate when and how reflection takes place (Horh & Pedersen, 1996; Dewey, 1933; Mezirow, 1990; Schön, 1983, among others). The design of the activity for teachers is one example of a transposition of theoretical ideas into

practice. In existing courses, the TE often only lectures on the research ideas and leaves it to the T to transpose them into practice. The TE might even lecture on how the theory could be transposed. With GP-teaching, the TE does not leave this difficult transfer to the teachers alone. In this case, I translated and prepared questions to ask the Ts about how to work with the VM. The third GP concerns the order of activities and theory. The third GP is not only about the order, but also a wish to present the original research articles and at the same time overcome the learning paradox (Bereiter, 1985); that you have to know about the topic to learn about it. The point is that before the lecture, the Ts know from their activities about their own strengths and weaknesses on the topic. Analysis and evaluation of this tool could as well take place during this presentation. My GP is developed through design research, and in that sense, the case study was only the first redesign that should be followed by new cycles. For each case of the design process, an analysis of how the effectiveness of the teaching design was compared with the hypothetical learning trajectory was conducted. The GP itself was the result of finding a theory that came from analysis of the meta-didactical transpositions.

CONCLUDING REMARKSAt the beginning of this paper, I mentioned two conflicts: the mismatch between the school classroom and the in-service classroom, and the difference between their goals. My research concerned the redesigns of the successive courses and the Ts’ reactions during the courses to face these conflicts. The research findings revolve around the effectiveness of the meta-didactical transpositions realised through the GP-teaching and investigated on the basis of the teachers’ response to the GP-activities and the use of various educational research concepts. The GPs are my answers to what extent I could design a meta-didactical transposition by aesthetical learning processes, and how effective it was related to the aims of the teaching.As for the effectiveness, it was only measured through the Ts’ reactions on the course; through their comments and behaviour. This means that most of the interpretation of their reactions passes through my ‘filter’. I documented as much as possible with video and audio recording, logs and other materials. As for importance, it is my hope that the transposition of research concepts should not only work on the courses. I hope the chosen concept, in this case the VM, will work as a catalyst for the Ts’ own practice after the courses in reflecting their communication.The research concepts were not developed specifically for the Ts, but rather as research directed at other researchers. Yet, in the transposition of the research ideas/concepts into teaching activities on in-service courses, they became practicable for the Ts, which made the concepts available first in the in-service classroom and eventually also in the school classrooms. The transformation gave the concept a new

perspective, but at the same time in a way changed it, which was not mentioned or included in the original research concept. The research was now transformed into a tool for teaching, where the real challenge was to keep the main idea of the concept intact. It is an open question whether we can still call it the same idea. It could be seen as a criticism that the concept, which was meant for other researchers, was now transposed into teaching (maybe beyond recognition?), but it could also be seen as a strength that the transposition brought the research from theory into practice and in that way obtained another value. It depends on how much the original idea was changed in the transposition. A transposition will always change a concept; the question is how much and for what purpose.My intention was that the teaching on the courses should help the maths teachers understand concepts from mathematics education journals/articles. For that purpose I needed the concepts to be more ‘user-friendly’. In the final lecture, I gave a copy of the original journal/article to the teachers. The guiding principles were meant to address the Ts’ problems, to help them understand what kinds of skills they needed and to facilitate their access to theoretical concepts that contains some answers and analytical tools for these issues. My hope was that the Ts afterwards would read the articles and be able to use the source directly. This failed; none of the Ts read the articles. On closer inspection I admit that it was a naïve hope. When I ask my research colleges, even they admit how difficult it is to find time to read articles after a conference, so how could I expect Ts to assume that practice? The Ts’ reaction to the redesigned course can be divided into two groups: defensive and exploratory. It is reasonable to assume that the defensive reactions occurred when the Ts felt threatened in their habitual routines, where they normally felt successful. It seems quite natural to defend habits that normally work. Yet, the learning process was made harder with this behaviour. I see the Ts’ defence mechanism as an attempt to avoid being confronted with their didactical competencies. This is a glimpse into the frustrations often encountered in the learning process. The Ts and the TE have the common goal that the course should benefit the Ts’ daily lives and influence their teaching in a positive way, but the way to do it and the prise to pay for it were maybe not so clear for each of the Ts. I will point out that this is not to blame the Ts; it is hard to avoid falling into the trap of just mastering the pupils’ curriculum. If pre-service teaching does not connect mathematics with didactics, pedagogy or education, nor teaches students how to work with mathematical concepts, then the Ts do have little to lean on. This means that the Ts, when they start their practice after pre-service education, mostly rely on tasks taken from the text-books and experiences from their own past. In the school system, the main aim is to teach the pupils to solve tasks as the final exam requires. The criteria for success are that the pupils get high marks, which seems to be the same as to master the tasks, in which the mathematical concepts are hidden. Making these mathematical ideas apparent depends on the T’s skills and competencies. The T should be aware of mathematical concepts and possess the skills to ask questions that demonstrate the concept so that the pupils

become motivated to work with it. These competencies are not what I see most, when I observe Ts working. The Ts who reacted with exploratory actions had the courage to examine their own behaviour and to reflect upon patterns in it, which gave rise to fruitful discussions. On the courses, there were many of these exploratory participants, but the defensive participants had a tendency to dominate the process.The validity and interpretation of my data can of course be discussed. The investigations I made gave me new insight into an area that I am very familiar with, but had never considered before in terms of how the teachers managed to communicate and reflect when they teach mathematics with open problems. As mentioned earlier, I used both my intuition and academic methods in my study. It is difficult to document intuition in detail, but still I hope my work will convince the readers.

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