1. Assessing with technology in France New problems and a new tool (Casyopée) Jean-Baptiste LAGRANGE Equipe de Didactique des Mathématiques, Université Paris VII France

2. Technology and (summative) evaluation –Baccalaureate –Teacher employment competition

3. Baccalaureate All calculators allowed An experimental “two parts” exam paper (1998) –Scientific stream –Questions about calculator use –Tasks without the calculator An exam involving spreadsheet knowledge (2000) –Non scientific stream –Traditional written exam A new assessment at the baccalaureate (2006) –Scientific stream –“Practical” assessment with computer

4. Teacher employment competition Secondary teachers –‘Advanced calculators’ at the oral examination (2004) Primary teachers –Possible questions about the use of tools like a spreadsheet or dynamic geometry (2006)

5. An experimental “two parts” exam paper at the baccalaureate (1998)

6. A quite interesting attempt Distinguishing tasks with technology and tasks without Assessing specific calculator knowledge (instrumentation) But… not continued…

7. An exam involving spreadsheet knowledge New curriculum for non scientific stream –Learning about linear and exponentional sequences from ‘real life’ examples –Learning about the spreadsheet supporting Math learning Exploration of sequences’ properties Spreadsheet symbolism prepares Math formalization Implementation by teachers –??? (Lagrange & Erdogan, in preparation)

8. An exam involving spreadsheet knowledge Written exam (without computers) In a class: 50% B2*0,045 (wrong) 50% B2*1,045 (correct) The actual difficulty is not the spreadsheet

9. Teacher employment competition Correct answer??? = I2*8+B6*13 or =I$2*8+$B6*13 School teachers Written exam “ We bought books at 8€ each and others at 13€ each. We paid 150€. What did we buy ? … A solution with a spreadsheet. Question: What is the formula in cell I6?

10. Teacher employment competition Secondary teachers –Calculator more or less compulsory at the oral exams –Function, sequences –Dynamic geometry –‘Advanced calculators’

11. Teacher employment competition

13. The new exam at the baccalaureate Towards integration –(2004) Les technologies de l'information et de la communication dans l'enseignement des mathématiques au collège et au lycée –(2006) in order to perform a real integration, the final evaluation should include the use of computers Math exercises whose solution significantly involves technology Calculators, computers Dynamic geometry, spreadsheet, CAS Specific applications (preferably free)”

14. Rationales Technology use remains marginal –Because of evaluation Exam with calculator more and more problematic –Download whole math textbook –Cheating via wireless communication Important Math proficiencies not evaluated –Conjecturing –Self-inventiveness –Technology use

15. Practically Each item in the bank –A description topic, TICE proficiencies Math proficiencies –The student document –The teacher document Intentions Possible use of technology Comments about evaluation –The evaluation document Exam in school One hour 1/5 of the mark Teachers choose in a “bank of exercises” Teachers attend to 4 students during the exam They fill in an evaluation sheet

16. A description document Optimizing pipes We want to put pipes on the wall of a house to collect rain water. This wall is rectangular. A vertical pipe has to reach the bottom at the middle of the wall. Two other pipes have to collect water from the sides of the roof. We want to use the shortest total length of pipe. Find the position of point M that gives this minimum length. Proficiencies at stake In the use of technology Building a figure using dynamic geometry Using software to transpose a geometrical situation into a graphic. In mathematics Emitting a conjecture from various information: Elaborating a strategy to find the extremum of a function.

17. Student text: conjecturing

18. Student text: proving

19. A critical view Big differences between –Description Open problem “Generic” problem –Student text Particular problem No choice of variable Separation between –Conjecture –Proof Use of software –only for conjecture –numerical approach -> some disappointment Why ? –Acceptance by teachers –Constraints of the evaluation –No adapted tool Positive effects nevertheless –A process of preparing –A new interest for technology (->teacher dev. Courses) –A new way of assessing (evaluation sheet)

20. Conclusion on assessment Assessing is always a compromise –New approaches and Constraints –Innovative views and teacher acceptance Bad and acceptable compromises –Written versus practical A prayer to teachers –Please, please, do not mimic evaluation tasks when teaching with technology.

21. Casyopée Use of numerical tools (spreadsheet, dynamic geometry) is frustrating –Separation between conjecture and proof –No contribution of technology to important activities in mathematics (proving, writing) Standard CAS integration is problematic Our answer: building a new tool –Geometric and algebraic –With help for Exploring, modeling, proving, writing…

22. An example a functional dependency problem –being given: a, b : parameters A(0,a) ; B(0,b) ; C free point on [oB] –Study of the distance EC

23. Creating the objects symbolic objects geometric objects

24. Using Casyopée 3: geometric calculations 4: variable 5: co-variation 6: function7: graph 8: algebraic proof 9: geometric proof 1: parameters 2: figure

25. Generalisation 1: numerical, graphical exploration 3: proof in a numerical case 2: geometrical construction 4: a general proof

26. Experimenting First Experimentations –10th grade students –Variations and equalities of areas –Students took more initiative for modeling –Easy use of computer Algebra facilities –Students dealt with the notion of function interpreting co-variation making sense of the domain linking different representations Cross experimentation (France-Italy) going on.

27. Links About the teacher national competition http://capes.math.jussieu.fr About the “practical exam” at the baccalaureate http://www.igmaths.net/ Debate about assessing mathematical proficiencies especially in the context of technology. http://educmath.inrp.fr/Educmath/en- debat/epreuve-pratique/

28. Links About Casyopée –http://www.irem.univ- rennes1.fr/recherches/groupes/groupe_aide_logici el About ReMath –http://remath.cti.grhttp://remath.cti.gr More on my website soon to open -http://jb.lagrange.free.fr/site