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

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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

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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

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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.

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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.

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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

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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.

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

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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)

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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.

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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 teachers 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.

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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.

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Some elements for comparison

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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

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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

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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?

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The instrumental approach: the institutional dimension Michèle Artigue Université Paris 7.

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