1. Putting theory into practice: The evolution of an in-service course for mathematics teachers Lisser Rye Ejersbo Haifa, November, 2008 Lisser Rye Ejersbo, Learning Lab Denmark, School of Education, University of Aarhus

2. Two Conflicts What happened on the course The teacher’s goals What happened in the classrooms My goals, as the TE Lisser Rye Ejersbo, Learning Lab Denmark, School of Education, University of Aarhus

3. Design and Redesign of an In-service Course: The Interplay of Theory and Practice in Learning to Teach Mathematics with Open Problems Lisser Rye Ejersbo, Learning Lab Denmark, School of Education, University of Aarhus

4. Classroom Problems Analytical Tools Reflection Skills Mathematical Communication Lisser Rye Ejersbo, Learning Lab Denmark, School of Education, University of Aarhus

5. To what extent and in what way can a meta-didactical transposition be incorporated into the successive stages of a redesign of the in-service course, and how effective is this redesign, as evaluated by the participating teachers’ reactions on the course? Lisser Rye Ejersbo, Learning Lab Denmark, School of Education, University of Aarhus My Research Question

6. Lisser Rye Ejersbo, Learning Lab Denmark, School of Education, University of Aarhus Meta-Didactical Transposition A ‘didactical transposition’ denotes the process of transferring mathematics as a scientific discipline into an educational subject at school. [Chevallard, 1985] Scientific Mathematics Teaching Tools Mathematics Education Research In-service Teaching Tools

7. Design Research Design research is a continuous process between instructional design and classroom based analysis, where the researcher, when preparing for the design experiment, formulates a hypothetical learning trajectory. Developing theory should be 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, diSessa, Gravemeijer) In this case, the classroom is an in-service classroom and the arrows symbolise the process from design and preparation to a realisation and redesigns. Classroom-based analyses Instructional design Lisser Rye Ejersbo, Learning Lab Denmark, School of Education, University of Aarhus

8. Four Theoretical Concepts Sfard & Kieran’s ‘Interaction Flowchart’ (2001) Steinbring’s ‘Epistemological Triangle’ (1998) Leron & Hazzan’s ‘Virtual Monologue’ (1997) Yackel & Cobb’s ‘Socio-mathematical norms’ (1996) Lisser Rye Ejersbo, Learning Lab Denmark, School of Education, University of Aarhus

9. “Is the researchers’ view of the student’s state of mind, reproduced in a monologue in the student’s own voice, incorporating the original data. (…) Note that it is not essential that our interpretation be actually correct (…) We aim to demonstrate that this is a viable way to interpret students’ productions in general.” [Leron & Hazzan (1996): The world according to Johnny: A Coping Perspective in Mathematics Education. ESM, 32] Lisser Rye Ejersbo, Learning Lab Denmark, School of Education, University of Aarhus Virtual Monologue

10. An Algebra Task Is it true that the following system of linear equations k – y = 2 x + y = k has a solution for every value of k? [Sfard & Linchevsky, (1994): The gains and the pitfalls reification – the case of Algebra. ESM, 26.] Lisser Rye Ejersbo, Learning Lab Denmark, School of Education, University of Aarhus

11. Interview with a tenth-grade student, Dina, working on the above task. [Sfard & Linchevsky, (1994)] D: [reads the question silently] “... has a solution...” 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. Lisser Rye Ejersbo, Learning Lab Denmark, School of Education, University of Aarhus

12. Interpretation of this Interview “Dina was helpless when faced with the problem. She asked the interviewer what she was supposed to do. The question was obviously not clear to her at all. After a minute or two of looking at the problem she said, ‘I am groping in the dark’. She was unable to interpret the question in a meaningful, consistent way. It left her confused and helpless. Thus, when asked what she was supposed to look for, she had no choice but to ‘shoot at random’ with pieces of standard statements which had worked in the past.” [Sfard & Linchevsky, (1994)] Lisser Rye Ejersbo, Learning Lab Denmark, School of Education, University of Aarhus

13. Interpretation via a Virtual Monologue: [Leron & Hazzan, 1996] 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. Lisser Rye Ejersbo, Learning Lab Denmark, School of Education, University of Aarhus

14. Interview with a tenth-grade student, Dina, working on the above task. [Sfard & Linchevsky, (1994)] D: [reads the question silently] “... has a solution...” 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. Lisser Rye Ejersbo, Learning Lab Denmark, School of Education, University of Aarhus

15. Succeeding discussions New proposals for a VM for Dina Proposals for a VM for the teacher (I) The teachers’ own examples of mathematical communication How to collect data An eye-opener to one’s own communication Reflection of why: habits and changes Lisser Rye Ejersbo, Learning Lab Denmark, School of Education, University of Aarhus

16. New questions The teachers faced difficult and much more authentic questions. ”Such hard questions cause a deep level of probing into the reasons for actions, interactions, activities, decisions, responses – all the elements which contribute to teaching and learning approaches in a mathematical classroom. The questions are hard because they challenge the fabric and philosophy of a teacher’s mode of operation.” B. Jaworski, 1998: Mathematics teacher research: process, practice and the development og teacing. JMTE, 1: 3-31 Lisser Rye Ejersbo, Learning Lab Denmark, School of Education, University of Aarhus

17. The Design and Redesigns Transposing the Virtual Monologue from a research tool to an in-service teaching tool. A Virtual Monologue for the teacher as well as for the student. Reflection upon mathematical communication. Using the teachers’ own mathematical communication. Working with mathematical communication in the teachers’ own classes. Lisser Rye Ejersbo, Learning Lab Denmark, Schools of Education, University of Aarhus

18. Virtual Monologue as a Tool The teachers’ perspective The teacher educator’s perspective The researcher’s perspective Lisser Rye Ejersbo, Learning Lab Denmark, School of Education, University of Aarhus

19. The Guiding Principles On the in-service course, each teaching session is based on a particular theoretical concept from the research literature. The theoretical concept is transformed into practical activities so that –the theoretical concept is made into a tool; –the teachers can use the method in their own classes; –the teachers’ tacit knowledge becomes apparent; –room for reflection is shaped along with the activities. In the preparation phase, theory precedes activities; in the practical phase, activities precede theory. Lisser Rye Ejersbo, Learning Lab Denmark, School of Education, University of Aarhus

20. A Concluding Thought In-service teaching only works if the participants become aware of their own habits and beliefs, and from that point of conscious awareness decide what kind of competences it is necessary to work on improving in mathematics teaching. Lisser Rye Ejersbo, Learning Lab Denmark, School of Education, University of Aarhus