Teachers writing resources integrating dynamic geometry: a professional development opportunity

1. Teachers writing resources integrating dynamic geometry: a professional development opportunity Colette Laborde University of Grenoble, France Teacher Education Institute Colette.Laborde@imag.fr

2. Introducing myself • Researcher in mathematics education • Involved in preservice teacher education • Research and development projects of resources for teachers integrating dynamic geometry

3. Focus on three projects • A research project in the frame of a PhD work (Seden Tapan, 2006) • How does teacher preparation for integrating dynamic geometry into their teaching impact their ability of adapting and designing tasks making use of DG? • Two R&D projects with teams of teachers, teacher educators and researchers in maths education

4. Two R&D projects • Writing and experimenting teaching scenarios meant for high school mathematics teachers for using Cabri-geometry (software and application of TI92) • Local project • Book published in 2001 • Writing and experimenting teaching scenarios meant for primary school and middle school for using Cabri-geometry software • National project • Design of a CD-Rom in progress

5. Common features of the projects • Designing tasks • Within the French curriculum • Making use of a specific technology (i.e. Dynamic Geometry) • Designing tasks is a professional situation • But designing technology based tasks is out of the range of the ordinary situations faced by teachers • Even if in France the use of technology and in particular of DG is compulsory, only about 20 to 30% of teachers really integrate technology

6. Dynamic geometry: major specific features • Dragging is one of the distinctive features of DG • Geometric tools for constructing figures preserving their properties when dragged • Three possible uses of dragging • Dragging for seeing • Dragging for conjecturing • Dragging for validating/invalidating

7. Preservice teachers as a window on the complexity of the design of tasks • Three preparation sessions • First session • introduction to the use of Cabri • and presentation of many examples of situations commented by the teacher educator • Second session: How to use Cabri from a pedagogical perspective • They must solve situations proposed by the teacher educator • They must analyze them from a pedagogical perspective: what is the contribution of DG to learning in the task? And learning what?

8. 13 Schedule of observations of preservice teachers designing tasks Session 1Initiation to Cabri Session 2Pedagogical use of DG Session 3Didactique [ ] [ ] [ ] StudentTeachers adapt or create tasks - 2 Student Teachers adapt or create tasks - 3 Student Teachers adapt or create tasks - 1

9. Adaption or creation of tasks • Involving the notion of reflection and axial symmetry • After the first session, student teachers • proposed in Cabri exactly the same task as proposed in paper and pencil • Very little use of dragging in their tasks • After the second session • Dragging for validating/invalidating and for conjecturing planned in their tasks • Contribution of DG contrasted with paper and pencil tasks • They started from reference tasks (given either in paper and pencil or Cabri) to create new tasks: very seldom new tasks

10. Preservice teachers Conclusions • Presenting a variety of tasks to them is not enough for preparing teachers • They must themselves manipulate and analyze the tasks to take advantage of them When they design tasks • The full pedagogical use of dragging requires time • The creation of really new tasks is very rare • Evolution over time and after the second session in the design of tasks

11. Resorting to several types of knowledge • Three kinds of knowledge strongly intertwined in the design of tasks • Mathematical knowledge • Knowledge of Cabri • Pedagogical content knowledge • about mathematics teaching • and about the use of Cabri for fostering learning

12. Activity of designing tasks • is complex • Requires to coordinate various types of knowledge about maths, pupils, technology, learning • As such it may be a tool for professional development provided that some conditions are fulfilled

13. R&D projects of writing teaching scenarios in a team • We assumed that feedback to the design of scenarios was critical for their evolution • Two kinds of feedback were planned • Experimenting the tasks in classroom • Working in team and discussing the tasks in a team

14. Teams • Various profiles in the team of each project • Teachers • Experienced teachers familiar with the use of technology • Experienced teachers novice with the use of technology • Novice teachers with different degree of familiarity with technology • Teacher educators • Researchers

15. Several phases in the design of the scenario • First phase: one member wrote a scenario project experimented in a class • Second and further phases: discussion in the team, possibly re-experimentation, and modification

16. First phase: a contrast • The proposed tasks written by inexperienced teachers • Series of disconnected tasks at the border of the curriculum • Not full use of the software facilities • Minimal perturbation in their practice • The experienced teachers wrote complete scenarios aimed at introducing pupils to new notions at the core of the curriculum (vectors, geometric transformations) and to provide a complete teaching of these notions • Cabri was used at critical moments • Extensive use of the Cabri tools and of the dragging: dragging for exploring, conjecturing, validating

17. Three categories of tasks • Cabri as facilitating the task while not changing it conceptually (visual amplifier, provider of numerical data) • Cabri modifies the ways of solving the task • The task takes its meaning from Cabri

18. An example: First version of scenario “Dilation” 1/2 • In Cabri mark a point I; create by means of the tool Polygon any quadrilateral ABCD. • 1) Select the tool Number and type 3 • Construct the image of ABCD by using the tool Dilation in the following way: point out successively the quadrilateral, point I and number 3. Label A’, B’, C’ and D’ the corresponding vertices of the new quadrilateral. • Compare vectors • IA and IA’, • IB and IB’, • AB and AB’, • BC and BC’, • area (ABCD) and area (A’B’C’D’) • Which equality is valid for vectors IA and IA’? For vectors IB and IB’?

19. An example: First scenario “Dilation” 2/2 • 2) Modify number 3 into -0.5 and answer again questions of activity 1 • 3) Do several trials by changing the position of I and then point A

20. First version : Cabri provider of data • Provider of static diagrams and data • No use of continuous drag of points and updating of displayed measurements • Questions about numerical relationships • No qualitative questions • Strong guidance of pupils • Elements to be compared and to be changed were given • Task as such possible in paper and pencil environment • Minimizing uncertainty for the teacher

21. Scenario Dilation: Second version 1/2 • In the toolbox Transformation of Cabri, in addition to reflection and point symmetry, there is the tool Dilation. You will study this transformation. • Create a point I, edit a number k by using as starting value 2.5, create by means of the tool Polygon a quadrilateral ABCD and construct its image through the dilation with centre I and ratio k (tool Dilation, point the quadrilateral, then the centre I and number k). Characterize the obtained image.

22. Scenario Dilation: Second version 2/2 • Do not hesitate to drag polygon ABCD, points A, B, C and D, centre I and to modify number k; do not forget that you can display measures with tool “Distance and length” and that a calculator is available in Cabri. • Give to k a negative value (choose in a first step -0.5) and complete the previous characterisation. Do not hesitate to vary k.

23. Changes from version 1 to 2 • Central place to dragging including for numbers • More open ended questions • Qualitative exploration made possible • Reference to a larger number of tools • Task as such impossible in paper and pencil environment • Larger variety of possible answers, more potential questions encountered by pupils when manipulating • More uncertainty for the teacher

24. Type of tasks introduced later • Inverse problem: an unknown dynamic object is given, reconstruct the object • After two years, Dilation was introduced in this way in a third version of scenario Dilation • The task takes its meaning from the possibility of dragging

25. Reasons for evolution • Perturbation introduced by technology • Activity of writing scenarios calls for knowledge about maths, pupils, learning teaching • Two levels • Action • Discussion and reflection • Two kinds of feedback • Team work: when presenting the scenario to the team, the author had to justify it • The product is object of reflections • Different experiences, different beliefs: making them explicit, exchange and confrontation • Experimenting in classroom: testing his/her own ideas as well as proposals of the others

26. Questions • Which professional activities other than designing tasks could be object of professional development and why? • More autonomy of pupils is likely to increase uncertainty of teachers in the classroom. What could be the contribution of professional development to increasing the confidence of teachers?