Paul Drijvers Freudenthal Institute for Science and Mathematics Education Utrecht University

1. An ICT-rich learning arrangement for the concept of function in grade 8: student perspective and teacher perspective Paul Drijvers Freudenthal Institute for Science and Mathematics Education Utrecht University Universität Köln, 20.01.09 www.fi.uu.nl

2. Outline • The project • The function concept • The ICT tools • Learning arrangement • Some results on learning • Some results on teaching • Conclusion

3. Outline • The project • The function concept • The ICT tools • Learning arrangement • Some results on learning • Some results on teaching • Conclusion

4. 1 The project • Project name: Tool Use in Innovative Learning Arrangements for Mathematics • Granted by the Netherlands Organisation for Scientific Research NWO • Timeline: 2006 – 2008 • Research team: • Peter Boon, programmer / researcher • Michiel Doorman, researcher • Paul Drijvers, PI / researcher • Sjef van Gisbergen, teacher / researcher • Koeno Gravemeijer, supervisor • Helen Reed, master student • www.fi.uu.nl/tooluse/en

5. Project theme: math & technology • Integrating technology in mathematics education seems promising • But optimistic claims are not always realized! • Technology for ‘drill & practice’ or also for conceptual development? • If yes, how to achieve this?

6. 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?

7. Applets For collections of applets see: • www.fi.uu.nl/wisweb/en/(primary) • www.fi.uu.nl/rekenweb/en/ (secondary) • So far: rather much design / development of games / applets than research on their use in the classroom

8. Project concretisation • Mathematical subject: the concept 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: 7-8 lessons of 50 minutes

9. Outline • The project • The function concept • The ICT tools • Learning arrangement • Some results on learning • Some results on teaching • Conclusion

10. 2 The function concept Two quotes: • “The very origin of function is stating and producing dependence (or connection) between variables occurring in the physical, social, mental world (i.e. in and between these worlds).” (Freudenthal, 1982) • “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)

11. Function definitions • "a quantity composed in any of [a] variable and constant" (Bernoulli, 1718) • an "analytic expression" (Euler, 1747) • f is a function from a set A to a set B if f is a subset of the Cartesian product of A (the domain) and B (the range), so that for each a in A there exists exactly one b in B with (a, b) in f. (Dirichlet-Bourbaki, 1934) How useful are these definitions for lower secondary mathematics education?

12. The ‘function gap’ • Lower secondary level (SI, 13 – 15 year olds): a way to describe a calculation process, an input-output ‘machine’ for numerical values. • Upper secondary level (SII, 16 – 18 year olds): a mathematical object, with several representational faces, which one can consider as membre of a family, or that can be submitted to a higher level procedure such as differentiation.

13. Intentions and didactical ideas Intentions: • To bridge the gap between the two, facilitate the transition and promote a rich conception of the notion of function including both the process and the object view. Relevant ideas from mathematics didactics: • Vinner (1983), Vinner & Dreyfus (1989): Concept definition and concept image • Janvier (1987): Multiple representations – formula, graph, table • Sfard (1991): Process – object duality • Malle (2000): Function as assignment and as co-variation

14. Proces-object duality (Sfard, 1991): • Operational conception: processes • Structural conception: objects • In the process of concept formation, operational conceptions precede the structural

15. Three aspects of the notion of function: • Dependency relation from input to output • Dynamical process of co-variation • Mathematical object with several representations Mathematical phenomenology or didactical phenomenology?

16. Outline • The project • The function concept • The ICT tools • Learning arrangement • Some results on learning • Some results on teaching • Conclusion

17. 3 The ICT tools (1) • Freudenthal (1983) mentions activities with arrow chains as one means to approach the function concept • ICT tool: The applet AlgebraPijlen(“AlgebraArrows”): chains of operations, connected by arrows, with tables, graphs and formulas.

18. 3 The ICT tools (2) The Digital Mathematics Environment (DME) : • 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’

19. The tools and the function concept a. The function as a dependency relation from input to output: construct and use chains

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

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

22. Outline • The project • The function concept • The ICT tools • Learning arrangement • Some results on learning • Some results on teaching • Conclusion

23. 4 Learning 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 trajectory

24. Learning arrangement: lesson 1 • Group work on three central problems

25. Learning arrangement: lesson 2 • Posters, presentations and ‘living chains’

26. Learning arrangement: lesson 3 • First work in pairs with the applet after introduction

27. Learning arrangement: lesson 4 • Second work in pairs with the applet after plenary homework review

28. Learning arrangement: lesson 5 • Group work on the ‘matching’ of representations

29. Learning arrangement: lesson 6 • Third applet session in pairs after plenary discussion

30. Learning arrangement: lesson 7 (+8) • Final work with the applet and reflections on the concept of function and its notation

31. Outline • The project • The function concept • The ICT tools • Learning arrangement • Some results on learning • Some results on teaching • Conclusion

32. 5 Some results on learning • Difficulties to express the reasoning • Mixed media approach fruitful (paper-pencil <-> applet) • Form-function shift as a model for describing conceptual change in ICT-rich learning

33. ADifficulties to express the reasoning Students explaining dynamic co-variation: • “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”

34. B Mixed media approach fruitful

35. C Form-function shift • Form-function shift as a model for describing conceptual change in ICT-rich learning • Example: task 1.6

36. The work of two girls • Their work ‘real time’: Atlas (clip 59:9) • Their final product:

37. Hypothesis: form-function shift (1) A form-function shift (Saxe, 1991) takes place concerning the functions that arrow chains have for the student: • Initially, the arrow chain represents a calculation process, and is a means to calculate the output value once the input value is given. The arrow chain helps to organize the calculation process. • Evidence: students make new chains for the same calculation:

38. Hypothesis: form-function shift (2) • Later, the arrow chains become object-like entities that represent functional relationships and can be compared and reasoned with.

39. Verfication of the hypothesis: Task 4.1

40. The work of the two girls

41. Results of three classes

42. Chain of operations Theoretical interest • Form-function shift here might be a suitable construct to explain conceptual change when there is little technical development in the use of the ICT tool. • Instrumental genesis, which was one of the points of departure of this study, seems to be more appropriate for more versatile technological tools. Formula Table of independent input values Table of dependent output values Graphic representation

43. Outline • The project • The function concept • The ICT tools • Learning arrangement • Some results on learning • Some results on teaching • Conclusion

44. 6 Some results on teaching • Different whole-class orchestrations • Relations with teachers’ views on teaching and learning • Interaction teacher – student

45. ADifferent whole-class orchestrations Main orchestrations observed: • Technical demo • Explain the screen • Link screen board • Discuss the screen • Spot and show: example • Sherpa at work: example

46. Orchestrations by teacher

47. B Relations with teachers’ views on teaching and learning • Teacher A: “…so you could discuss it with the students using the images that you say on the screen, […] it makes it more lively…” • Teacher B:“I use the board to take distance from the specific ICT-environment, otherwise the experience remains too much linked to the ICT” • Teacher C:“I am a typical teacher for mid-ability students, and these students need clear demonstrations and explanations”

48. C Interaction teacher – student Different types of interactions: • Content of interaction: • Mathematical meaning • Technical meaning • Situational meaning • Interaction-meaning-technical • Form of interaction: • Revoicing • Questioning • Answering

49. Outline • The project • The function concept • The ICT tools • Learning arrangement • Some results on learning • Some results on teaching • Conclusion

50. 7 Conclusion on learning • How can applets be integrated in an instructional sequence for algebra, so that their use fosters the learning? • Global learning trajectory works, but which problem does the function concept solve for the students? • Mixed media approach fruitful Subtle relation between applet technique and concept development (instrumentation, FFS) • Form-function shift as a model for describing conceptual change in ICT-rich learning