The teacher and the tool

1. The teacher and the tool Paul Drijvers Freudenthal Institute Utrecht University The Netherlands cadgme, 30-06-2010 www.fi.uu.nl/~pauld

2. 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 • www.fi.uu.nl/en/

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

4. problem •theory • DME• results •conclusion 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?

5. Didactical performance Exploitation mode Didactical configuration problem •theory • DME• results •conclusion 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

6. Didactical performance Exploitation mode Didacticalconfiguration problem •theory • DME• results •conclusion Didactical configuration • 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).

7. Didactical performance Exploitation mode Didacticalconfiguration problem •theory • DME• results •conclusion 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.

8. Didactical performance Exploitation mode Didacticalconfiguration problem •theory • DME• results •conclusion 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

9. problem •theory • DME• results •conclusion 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

10. problem •theory • DME• results •conclusion 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.

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

12. problem •theory • DME• results •conclusion Two studies • ‘Tool Use in an Innovative Learning Arrangement for Mathematics’: • NWO-grantedsmall-scale design and evaluation of an instructionalsequence on the concept of function in grade 8, using an applet embedded in the electronic learning environment DME • Seewww.fi.uu.nl/tooluse/en • ‘EPN-pilot AlgebraicSkillsthrough ICT’: • Publisher initiatedbigger-scale pilot on algebraicskillsin grade 12, using applets in the DME, and replacing a textbookchapter • Seehttp://www.fi.uu.nl/dwo/gr-pilot/ andhttp://www.epn.nl/wps/portal/epn/getalenruimte/ictpilot#vwo Both studies use the Digital Mathematics Environment

13. problem •theory • DME• results •conclusion 3. The DME environment http://www.fi.uu.nl/dwo/en/

14. problem •theory • DME• results •conclusion 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)

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

16. problem •theory • DME• results •conclusion 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.

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

18. problem •theory • DME• results •conclusion Case Teacher A: grade 8

19. problem •theory • DME• results •conclusion Case Teacher A

20. problem •theory • DME• results •conclusion Case Teacher A [clip]

21. problem •theory • DME• results •conclusion 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

22. problem •theory • DME• results •conclusion Case Teacher B: grade 8

23. problem •theory • DME• results •conclusion Case Teacher B

24. problem •theory • DME• results •conclusion Case Teacher B [clip]

25. problem •theory • DME• results •conclusion 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

26. problem •theory • DME• results •conclusion Case Teacher C: grade 8

27. problem •theory • DME• results •conclusion Case Teacher C [clip]

28. problem •theory • DME• results •conclusion 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

29. problem •theory • DME• results •conclusion Case Teacher D: grade 12

30. problem •theory • DME• results •conclusion Case Teacher D

31. problem •theory • DME• results •conclusion Case Teacher D [clip]

32. problem •theory • DME• results •conclusion 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

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

34. problem •theory • DME• results •conclusion Orchestration table from study I

35. problem •theory • DME• results •conclusion 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.

36. problem •theory • DME• results •conclusion 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”

37. problem •theory • DME• results •conclusion 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”

38. problem •theory • DME• results •conclusion 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”

39. problem •theory • DME• results •conclusion 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.”

40. problem •theory • DME• results • conclusion Conclusion • Which types of instrumental orchestration emerge in technology-rich classroom teaching? Seven orchestration types are identified, as well as their frequencies. • 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.

41. problem •theory • DME• results • conclusion 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?

42. problem •theory • DME• results • conclusion 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. 363-392). Charlotte, NC: Information Age. • http://www.fi.uu.nl/dwo/gr-pilot/ • http://www.fi.uu.nl/dwo/en/ • http://www.fi.uu.nl/tooluse/en/ • http://www.epn.nl/wps/portal/epn/getalenruimte/ictpilot#vwo • http://ec.europa.eu/dgs/education_culture/eve/alfresco/n/browse/workspace/SpacesStore/1fe0b2bc-d45d-11de-b89b-09f36ec598be

43. Thank you! Paul Drijvers p.drijvers@fi.uu.nl www.fi.uu.nl/tooluse/en/