1. The design of a mixed media learning arrangement for the concept of function in grade 8 Paul Drijvers

2. Outline 1. The project 2. The concept 3. The tools 4. The educational arrangement 5. The data and analysis 6. The conclusions

3. Outline 1. The project 2. The concept 3. The tools 4. The educational arrangement 5. The data and analysis 6. The conclusions

4. 1 The research project: fact sheet  “Tool Use in Innovative Learning Arrangements for Mathematics“  Granted by the Netherlands Organisation for Scientific Research  Timeline: 2006 – 2008  Research team: Peter Boon, programmer / researcher0.2 fte Michiel Doorman, researcher 0.4 fte Paul Drijvers, PI / researcher 0.4 fte Sjef van Gisbergen, teacher / researcher0.2 fte Koeno Gravemeijer, supervisorPM  www.fi.uu.nl/nl/tooluse www.fi.uu.nl/nl/tooluse

5. Research Questions 1. How can applets be integrated in an instructional sequence for algebra, so that their use fosters the learning? 2. How can teachers orchestrate tool use in the classroom community?

6. Applets For the design products see:  www.fi.uu.nl/wisweb/en/ (primary) www.fi.uu.nl/wisweb/en/  www.fi.uu.nl/rekenweb/en/ www.fi.uu.nl/rekenweb/en/ (secondary)  So far: a lot of design / development of games / applets than research on their use

7. Theoretical framework  Realistic Mathematics Education  Vygotsky and others on tool use  Instrumental approach to tool use  Cultural-historical activity theory

8. Research setup and methodology  Design research (Gravemeijer): design of an educational arrangement including Student worksheets (paper) Applet with tasks embedded in an electronic learning environment Teacher guide with scenario’s of use  Three cycles: PilotOne class, 2006 First cycleThree classes, 2007 Second cycleScaling up, 2008  Data: both qualitative (video, screenvideo) and quantitive (student work saved in ELE)

9. The project: Concretisation  Mathematical subject: the notion of fonction  Tools: an applet embedded in an electronic learning environment  Target group: mid – high achieving students in grade 8 (14 year olds)  Teaching sequence: 7 lessons of 50 minutes

10. Outline 1. The project 2. The concept 3. The tools 4. The educational arrangement 5. The data and analysis 6. The conclusions

11. 2 The concept of function  The function: Lower secondary level (13 – 15 year olds): a way to describe a calculation process, a relation, a ‘machine’ Upper secondary level (16 – 18 year olds): a mathematical object, with several faces, which one can consider as membre of a family, or that can be submitted to a higher level procedure such as differentiation.  Intention: bridge the gap between the two, facilitate the transition and promote a rich conception of the notion of function.

12. 2 The concept of function  The function is a special kind of dependence, that is, between variables which are distinguished as dependent and independent. (...) This - old fashioned - definition stresses the phenomenologically important element: the directedness from something that varies freely to something that varies under constraint. (Freudenthal, 1983, p. 496).

13. 2 The concept of function Three aspects of the notion of function:  A directed relation from input to output  A dynamical process of co-variation  A mathematical object with several representations Mathematical phenomenology, but how about the didactical phenomenology?

14. Outline 1. The project 2. The concept 3. The tools 4. The educational arrangement 5. The data and analysis 6. The conclusions

15. 3 The tools  The applet AlgebraPijlen (“AlgebraArrows”): chains of operations, connected by arrows, with tables, graphs and formulas.AlgebraPijlen

16. 3 The tools  The Digital Mathematics Environment (EDM) : Author: design tasks and activities, ‘Digital textbook’ Student: work, look back, improve, continue, ‘Digital worksheet’ Teacher: prepare, comment, assess, ‘Collection of digital worksheets’ Researcher: observe, analyse the digital results, ‘Digital database’

17. The tools and the function concept  The function as a directed relation from input to output: construct and use chains

18. The tools and the function concept  The function as a dynamical process of co- variation: change input values to study the effect, use trace (graph) and scroll (input/table)

19. The tools and the function concept The function as a mathematical object with several representations: compose chains, construct inverse chains, link representations and study families of functions

20. Outline 1. The project 2. The concept 3. The tools 4. The educational arrangement 5. The data and analysis 6. The conclusions

21. 4 The educational arrangement Main ideas:  Mixture of working formats: group work, individual work, work in pairs with the computer, plenary teaching and discussion  Mixture of tools: paper – pencil, posters, cards, applet, DME, both in school and at home  First step: a hypothetical learning trajectoryhypothetical learning trajectory

22. The educational arrangement: lesson 1  Group work on three central problems

23. The educational arrangement: lesson 2  Posters, presentations and ‘living chains’

24. The educational arrangement: lesson 3  First work in pairs with the applet after introduction

25. The educational arrangement: lesson 4  Second work in pairs with the applet after plenary homework review

26. The educational arrangement: lesson 5  Group work on the ‘matching’ of representations

27. The educational arrangement: lesson 6  Third applet session in pairs after plenary discussion

28. The educational arrangement: lesson 7  Final work with the applet and reflections on the concept of function and its notation

29. Outline 1. The project 2. The concept 3. The tools 4. The educational arrangement 5. The data and analysis 6. The conclusions

30. 5 The data and analysis 1. Data on the use of applets and learning: Videos of the group work du travail en groupe Screen video of pairs of students Students’ work (digital, paper, posters) 2. Data on the use of applets and teaching: Videos of plenaries in the classroom Videos of student - teacher interactions  Data analysis ongoing

31. Digital student work

32. Coding in Atlas ti CODES 13 15 16 17 ---------------------------------------- docent-huishoudelijk 10 11 11 7 39 docent-inhoudelijkeh 5 10 8 8 31 docent-proceshulp 17 8 3 3 31 docent-reflectiehulp 4 10 10 6 30 docent-technischehul 8 4 3 4 19 functie-dynamiek-cov 5 1 2 4 12 functie-globale-in-u 0 6 7 7 20 functie-locale-in-ui 3 2 2 1 8 functie-object-repre 1 3 7 2 13 functie-variabele 2 4 4 2 12 techniek-in-uitzoome 3 5 5 1 14 techniek-ketting-mak 3 5 2 0 10 techniek-ketting-rec 1 0 0 0 1 techniek-scrollen-ta 1 3 1 3 8 techniek-variatie-in 2 5 3 4 14 techniek-verbinden 1 1 0 1 3 ---------------------------------------- Totals 68 93 87 62 310

33. Some prelimenary results A. Difficulties to phrase the reasoning B. Interaction paper-pencil / applet fruitful C. Some lasting conceptual difficulties D. Interaction teacher – student crucial

34. A Difficulties to phrase the reasoning  Goes up sidewards  Straigt line  Further and further away from 0  All equally steep  With the same jumps  The point is always moving  It goes up steeper and steeper  It gets higher and higher

35. B Interaction paper-pencil / applet fruitful

36. C Some lasting conceptual difficulties The work of Juliette, who insists on representing a function with several chains.  Atlas Atlas  Online Online

37. D Interaction teacher – student crucial  Either in individual contact and in plenary discussions  The teacher plays an important role while orchestrating the discussion on, the reflection on, and the convergence of techniques and thinking within the class

38. Outline 1. The project 2. The concept 3. The tools 4. The educational arrangement 5. The data and analysis 6. The conclusions

39. 6 Provisional conclusions 1. How can applets be integrated in an instructional sequence for algebra, so that their use fosters the learning?  Global learning trajectory OK, but which problem does the function concept solve for the students?  Dynamic did not work -> change applet  More insight needed into the relation between applet technique and concept development

40. 6 Provisional conclusions 2. How can teachers orchestrate tool use in the classroom community?  Technical class management time consuming and not self-evident!  Mixture of scenario’s and media (ICT and P&P) led to a fruitful arrangement that fosters transfer between media  Demonstration/presentation/class discussion important for reflection and collective instrumental genesis  DME offers means to monitor the learning  Teacher interactions that focus both on concept and technique are effective.

41. 6 Provisional conclusions Theoretical questions:  Is the framework of instrumental genesis, with its stress on the relation between technical and conceptual development, is useful in case the tool is as ‘simple’ as an applet?  How to observe and to analyse the orchestration by the teacher?