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The teacher and the tool Paul Drijvers Freudenthal Institute Utrecht University The Netherlands cadgme,

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The Freudenthal Institute Aims at developing, investigating and improving mathematics education at primary, secondary and tertiary level Aims at intertwining research, design, professional development and implementation Is the Dutch National Expertise Centre of Mathematics Education Key concepts: Realistic Mathematics Education Design Research

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Outline 1. What’s the problem? 2. Theoretical framework 3. The DME environment 4. Results Four examplary clips Orchestration types and tables Teacher profiles 5. Conclusion problem theory DME results conclusion

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1. What’s the problem? High expectations concerning the integration of technology into mathematics education have not yet come true. Growing awareness that teachers are crucial in the integration of technology into mathematics education: Gueudet & Trouche (2009) Lagrange & Monaghan (2009) Lagrange & Ozdemir Erdogan (2009) Maracci & Mariotti (2009) Pierce & Ball (2009) Ruthven, Deaney & Hennessy (2009) Issue at stake: what do teachers do when they integrate technology in their teaching and why do they do so? problem theory DME results conclusion

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2. Theoretical framework Instrumental orchestration as a model for analysing teacher activity (Trouche, 2004; Drijvers & Trouche, 2008) An instrumental orchestration: the teacher’s intentional and systematic organisation and use of the various artefacts available in a learning environment in a given mathematical task situation, in order to guide students’ instrumental genesis. An instrumental orchestration: a didactical configuration an exploitation mode a didactical performance Didactical configuration Exploitation mode Didactical performance problem theory DME results conclusion

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A didactical configuration is an arrangement of artefacts in the environment, a configuration of the teaching setting and the artefacts involved in it. Cf. the musical metaphor: choosing the musical instruments to include in the orchestra arranging them in space and time to achieve harmony (= convergence of mathematical discourse). Didactical configuration Exploitation mode Didactical performance Didactical configuration problem theory DME results conclusion

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Exploitation mode An exploitation mode is the way the teacher exploits the didactical configuration for her didactical intentions: decisions on the way a task is introduced and worked, on the possible roles of the artefacts, and on the schemes and techniques to be developed. Cf. the musical metaphor: determining the partition for each of the musical instruments involved. Didactical configuration Exploitation mode Didactical performance problem theory DME results conclusion

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Didactical performance A didactical performance involves the ad hoc decisions taken while teaching, on how to actually perform in the chosen didactic configuration and exploitation mode: what question to pose now, how to do justice to (or to set aside) student input, how to deal with an unexpected aspect of the task or the tool? Cf. the musical metaphor: the actual musical performance Didactical configuration Exploitation mode Didactical performance problem theory DME results conclusion

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The orchestration model... has a time dimension: ranging from teachers’ preparation in advance (didactical configuration) to ad hoc decisions while teaching (didactical performance) has a structural – incidental dimension: ranging from teachers’ operational invariants to incidental actualizations Includes two levels of instrumental genesis, namely the students’ level and the teachers’ level, the latter being the focus of this talk problem theory DME results conclusion

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A symphony orchestra conductor? Rather a jazz band leader... Trouche, L. & Drijvers, P. (in press). Handheld technology: Flashback into the future. ZDM, The International Journal on Mathematics Education. problem theory DME results conclusion

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Questions 1. Which types of instrumental orchestration emerge in technology-rich classroom teaching? 2. To what extent are teachers’ repertoires of orchestrations related to their views on mathematics education and the role of technology therein? problem theory DME results conclusion

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Two studies I. ‘Tool Use in an Innovative Learning Arrangement for Mathematics’: NWO-granted small-scale design and evaluation of an instructional sequence on the concept of function in grade 8, using an applet embedded in the electronic learning environment DME See II. ‘EPN-pilot Algebraic Skills through ICT’: Publisher initiated bigger-scale pilot on algebraic skills in grade 12, using applets in the DME, and replacing a textbook chapter See and Both studies use the Digital Mathematics Environment problem theory DME results conclusion

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3. The DME environment problem theory DME results conclusion

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The DME environment FI’s Digital Mathematics Environment DME: Content (applets, SCORM) Authoring environment (including design of feedback) Administration and monitoring system (LMS, CMS, including log facilities) problem theory DME results conclusion

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Examplary feedback Cf yesterday’s presentation by Christian Bokhove problem theory DME results conclusion

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DME qualities DME scores good on feedback facilities Bokhove, C. & Drijvers, P. (2010). Assessing assessment tools for algebra: Design and application of an instrument for evaluating tools for digital assessment of algebraic skills. International Journal of Computers for Mathematical Learning, 15(1). Online First. problem theory DME results conclusion

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4. Results Examplary clips ABC (Study I) and D (Study II) Orchestration types and tables Teacher profiles ABCD problem theory DME results conclusion

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Case Teacher A: grade 8 problem theory DME results conclusion

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Case Teacher A problem theory DME results conclusion

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Case Teacher A [clip]clip problem theory DME results conclusion

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Case Teacher A Orchestration type: Sherpa-at-work (?) Didactical configuration: Whole-class setting, laptop computer with projector, students in ‘regular’ rows Exploitation mode: Have a pair of students explain their work, one talking to the class and the other one operating the computer Didactical performance: Teacher in ‘marginal’ position, but taking care of the process of the student at the laptop supporting the words of his peer problem theory DME results conclusion

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Case Teacher B: grade 8 problem theory DME results conclusion

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Case Teacher B problem theory DME results conclusion

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Case Teacher B [clip]clip problem theory DME results conclusion

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Case Teacher B Orchestration type: Link-screen-board Didactical configuration: Teacher computer + data projector, positioned next to the ‘regular’ blackboard Exploitation mode: Show students’ work on screen; use the blackboard to transform applet notation and technique into conventional paper-and-pencil equivalents Didactical performance: Interactively use students’ input problem theory DME results conclusion

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Case Teacher C: grade 8 problem theory DME results conclusion

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Case Teacher C [clip]clip problem theory DME results conclusion

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Case Teacher C Orchestration type: Technical-demo Didactical configuration: Teacher computer connected to the interactive whiteboard. Students in ‘traditional setting’ Exploitation mode: Students listen and watch the teacher’s demonstration of applet technique Didactical performance: Teachers remains seated and does not use the facilities of the IAW problem theory DME results conclusion

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Case Teacher D: grade 12 problem theory DME results conclusion

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Case Teacher D problem theory DME results conclusion

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Case Teacher D [clip]clip problem theory DME results conclusion

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Case Teacher D Orchestration type: Work-and-walk-by Didactical configuration: Individual netbook computers in a wifi network, digital module including textbook files Exploitation mode: Students work on their own, teacher assists and answers questions Didactical performance: Taking neighbour’s netbook to compare findings; both technical and mathematical problems; student and teacher on different tracks problem theory DME results conclusion

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Orchestration types Seven orchestration types identified through theory- driven (1,3,6) as well as bottom-up analysis (2,4,5,7): 1. Technical-demo 2. Explain-the-screen 3. Link-screen-board 4. Discuss-the-screen 5. Spot-and-show 6. Sherpa-at-work 7. Work-and-walk-by teacher-centred: teacher dominates communication student-centred: interactive communication, student input problem theory DME results conclusion

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Orchestration table from study I problem theory DME results conclusion

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Orchestration table from study II Drijvers, P. (submitted). Teachers transforming resources into orchestrations. In Gueudet, G., Pepin, B., & Trouche, L. (Eds.), Mathematics Curriculum Material and Teacher Development: from text to lived resources? (pp. - ). New York/Berlin: Springer. problem theory DME results conclusion

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Teacher profiles Teacher A: prioritizes student-centred orchestrations sees learning as an interactive process in which students should have a voice sees technology as a means to achieve this ”…so you can have discussions with the students using the images that you saw on the screen, … that makes it more lively” problem theory DME results conclusion

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Teacher profiles Teacher B: prioritizes teacher-centred orchestrations finds mathematical content to be paramount sees technology as a means to teach this attention to the links between the DME-work and paper- and-pencil or blackboard mathematics. ”… take distance from the specific ICT-environment; otherwise the experience remains too much linked to the ICT ” problem theory DME results conclusion

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Teacher profiles Teacher C: prioritizes teacher-centred orchestrations finds clear explanations and instructions important has a concern for student difficulties when learning mathematics, and when using technology ”As a teacher, one has to tell students clearly what they should do with ICT ” problem theory DME results conclusion

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Teacher profiles Teacher D: prioritizes student-centred orchestrations leaves much initiative to the students does not see a technology-rich lesson as a specific one to prepare “Well, I don’t know much about it myself. I did not invest time in preparation. […] I prefer the kids act, and raise questions based on their actions.” problem theory DME results conclusion

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Conclusion 1. Which types of instrumental orchestration emerge in technology-rich classroom teaching? Seven orchestration types are identified, as well as their frequencies. 2. To what extent are teachers’ repertoires of orchestrations related to their views on mathematics education and the role of technology therein? The four teacher profiles suggest a clear relationship between orchestrations and views. problem theory DME results conclusion

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Reflections How generalizable are the orchestration types, to other topics, to other types of technology, to other forms of teaching, to other teachers … ? Why such a preference for individual orchestrations in Study II? Do teachers feel that they should step back as soon as technology enters the classroom? What are possible implications for teachers’ professional development concerning ‘TPACK’? What does the model of instrumental orchestration offer? A fruitful framework for analysing teachers’ practices when teaching mathematics with technological tools? problem theory DME results conclusion

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Literature & Links Drijvers, P., Boon, P., Doorman, M., Van Gisbergen, S., Gravemeijer, K., & Reed, H. (in press). The teacher and the tool: whole-class teaching behavior in the technology-rich mathematics classroom. Educational Studies in Mathematics. Drijvers, P., & Trouche, L. (2008). From artifacts to instruments: A theoretical framework behind the orchestra metaphor. In G. W. Blume & M. K. Heid (Eds.), Research on technology and the teaching and learning of mathematics: Vol. 2. Cases and perspectives (pp ). Charlotte, NC: Information Age. space/SpacesStore/1fe0b2bc-d45d-11de-b89b-09f36ec598be space/SpacesStore/1fe0b2bc-d45d-11de-b89b-09f36ec598be problem theory DME results conclusion

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Thank you! Paul Drijvers

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