Two didactical scenarios for the same problem

1. Two didactical scenarios for the same problem Comparing students’ mathematical activity IREM PARIS 7 BERLIN, MAY 2008

2. General framework The E-colab group: A collaboration between INRP, three IREMs (Montpellier-Lyon-Paris 7) and Texas Instruments . The IREM Paris 7 team: one researcher and two senior high school teachers. I

3. The specific concerns of the IREM Paris 7 team • Instrumental genesis • Didactical scenarios and optimization of the mathematical responsibility given to the students • Collective exploitation of the students’ work through classroom discussions orchestrated by the teacher and institutionalization

4. Instrumental genesis • Instrumentalization: How to combine productively instrumentalization processes and the development of mathematical knowledge in the classroom? • Interaction between applications: How to make students really benefit from the new potential offered by TI-nspire in terms of interactions between applications? • The crucial role played by the choice of mathematical situations and the design of scenarios.

5. Scenarios • The development and progressive evolution of didactical scenarios as a central point in the collaborative work of the E-colab group • What evolutions do we focus on and why? Trying to optimize the mathematical responsibility given to the students, in realistic contexts • Identifying for that purpose didactical variables related to the mathematical tasks or their management, on which one can play, and testing their effects.

6. Collective exploitation of the students’ work • The importance attached to: • the collective discussion of germs of knowledge and issues emerging from the students’ autonomous work • the interaction between the mathematical and technological dimensions of these discussions • Analyzing the role of the teacher

7. Methodology and data collection • The same mathematical task experimented with different scenarios • Two observers for each classroom session, audio-recording of the groups of students observed, and of the teacher during group-work sessions and collective discussions • Screen-capture for two students observed working with TI-nspire CAS. • Collection of all students’ drafts and research narratives and of all students’ tns files.

8. The problem • The task • General aims: • State of instrumental genesis. • Knowledge about functions. • Work in groups of 4 pupils.

9. Potential offered by the calculator • Exploitation of the dynamic geometry potential • Exploitation of the spreadsheet potential • Exploitation of the graphical potential • Exploitation of the CAS potential • Interaction geometry-spreadsheet through data capture (not exploited)

10. The first scenario • Three successive explorations with a variable but imposed order: dynamic geometry, spreadsheet, graphics. • Algebraic solving with the calculator. • Paper and pencil solution. • A written production in terms of research narrative.

11. A second scenario. Why? • Contradiction between the first scenario choices and its didactic goals. • Students treat each exploration as a separate task, coherence issues, interactions between applications remain teacher’s responsibility and are essentially dealt with in the collective discussion. • Difficulties with the spreadsheet use. • Difficulties with the questions asking is the different results obtained are coherent.

12. Second scenario • An open problem and only one question: asking for the position of M ensuring the equality of areas. • Students are asked: • to try to find a solution, using at least two different applications among those listed, • to say if their solution is exact or approximate (explanation of the distinction is given in the collective presentation of the task), • to say if the results provided by the different explorations are coherent or not (explanation is also given in the collective presentation of the task). • Some instrumental (regarding spreadsheet use) and mathematical (regarding the paper and pencil solving of the equation) hints are prepared and accessible on demand.

13. Some elements for comparison

14. Commonalities • Dynamic geometry: first exploration, conviction that the solution exists but cannot be accessed through DG. • Graphics: the conviction that the coordinates of the intersecting point provide an exact solution to the problem, all the more as the number of decimals is reduced; a good management of the second intersection point. • Spreadsheet: some instrumental and math difficulties • CAS: no particular problem, interesting link between exact and approximate, and with paper and pencil resolution • Many interesting and similar germs for a productive collective discussion

15. Differences • Spontaneous interaction between applications with the second scenario • Different orders, but priority to CAS and graphics after a first geometrical exploration, spreadsheet often added by the teacher • The devolution of the « exact-approximate » issue and of the coherence issue successful for many students • Diversity in effective autonomy

16. Launching the debate • Improving one initial scenario or developing a diversity of scenarios attached to the same mathematical task? • What are the didactic variables that one can play with, how to identify them and analyzing their effects? • What is gained and what is lost passing from one scenario to another one? • What can be reasonable aims in terms of students’ autonomy? • The role of memory and analogy with previous situations in students activity: potential and limits? And how the characteristics of the TI-nspire influence it? • What role for the teacher? What can be planed in advance, and what cannot?