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## CAS-Calculators in a centralized national examination

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**CAS-Calculators in a centralized national examination**Ministry of Education and Vocational Training Luxembourg**Luxembourg's Education system**Pre-Primary education starts at the age of 4 ( optional at age of 3) Primary education lasts for six years. (grades 1 – 6) Lower general secondary education (grades 7 – 10) Lower technical secondary education (grades 7 – 9) Upper technical secondary education (grades 10 – 13) Vocational education Upper general secondary education (grades 11 – 13) Projet CAS-Calculators concerns about 50 % of students in upper secondary education 2**Starting point of the project**• 1999-2000 : Internet-Hype in education • Luxembourg : project ELABOTIC • Use of ICT for teaching and learning • Grades 7 - 10 • Languages and science • Mathematical tools : Cabri, Excel and DERIVE • Model lessons and training courses 3**First conclusions**• After 3 years, project evaluation brought the following conclusions : • Some interesting, but isolated pedagogical hot spots • No general acceptance • A lot of problems 4**What were the problems ?**• Computers in separate media-rooms • ICT in math lessons = unnatural situation • Technical problems / no assistance • Inadequate student equipment at home • difficult acceptance by reluctant teachers • ICT = time-consuming activityCONFLICT with curriculum • New learning strategies in opposition with traditional teaching methods • Inadequate school books 5**A new direction in 2002**Reasons for reorientation : • One Laptop-school : 1200 laptops for every teacher and student model for the other schools • Contacts with curriculum foreign researchers and curriculum developers • Starting reflection about the purpose of math education setting up a mathematical competency model 6**A fresh impetus : V200**Reasons for the choice of the V200 • Handy tool during lessons (immediate startup) • Integration of the aspects from the Elabotic project : CABRI, DERIVE, Spreadsheet • Not as expensive as a laptop • (nearly) No technical problems 7**Implementation of the V200**Missions of the project group (math teachers) • definition of curricular progression for grades 11 to 13 • Elaboration of documents for teachers and students • Continuous training sessions for teachers • Elaboration of examination questionnaires • Development of an evaluation scheme • Expertise of questionnaires in the exam of 2006 8**Progression scheme**Constraints : • The use of the V200 had to be build around/into the “old” currciculum • Division of the examination in 2 tests : • Without V200 : main focus geometry and probability • With V200 : analysis and problemsolving 9**Grade 11 : Main focus**• Understanding the notion of function • Fostering the graphical apprehension • Problem-solving (extremum problems without analytic study) • Notion of equation / inequation (visual aspect) • Linking the three representational aspects of functions ( graphical, numerical, algebraic) • Sequences and series 10**Grade 12 : Main focus**• Discovering and interpretation of • Function derivation • curvature • “Reverse function engineering” (Steckbriefaufgaben) • Problem solving and modelisation (using function study) • Extremum problem • Tangent problems • “Realistic” situations 11**Grade 13 : Main focus**• Discovering and interpretation of • Exponential function • Logarithmic function • Discovering and interpretation of integration • Riemann – Darboux sums • Numerical integration • Problem solving 12**New aspects**• The classical study of functions is extended to new aspects • Reverse function engineering • Presentation of realistic situations • The problems do not necessary have a unique solution • Discussion of the validity of a solution • Emphasis on the interpretation of models • Argumentation and communication of mathematical concepts • The look on analysis is changing 13**New tool – new problems**Questions from hesitating teachers : • To what extend do students still have to be able to do “paper-pencil” calculation ? • What can be delegated to the machine ? • Do the students still learn when the use the V200? • What do they learn then ? 14**New tool – new problems !**• No adapted school books • Evaluation problems • What do we evaluate if manipulation of expressions is dealt with the CAS-calculator ? • How do we evaluate now ? • Organisational problems 15**On the way to the national exam**• Description of what V200 techniques the students should master at the end of the different grades • Description of what a student can delegate to the V200 in the final exam • Resolution of equations (both rational and irrational) • Numerical evaluation of primitives • Linearization of trigonometric expressions • Manipulation of rational and irrational expressions 16**On the way to the national exam**• Description of a standard procedure to empty the memory of the V200 • Description of a standard procedure for controlling the status of the memory at the beginning of the exam • Development of guidelines for examination questions • What are the objectives of the problem/questionnaire • What are the principal reflections that the student should make for solving the problem 17**Communication strategy**• Public website for students (v200.myschool.lu) • Presents the relevant documents for the grades 11 -13 • Presents sample examination questionnaires • Closed internet community for teachers • Sample lessons for grades 11 - 13 • Questionnaires with evaluation model • Technical documents • Event calendar – discussion forum (not used !!) 18**Part I : (180 minutes)**without V200 Contents : 4 questions Geometry Probability Part II : (240 minutes) with V200 Contents : 3 classical analysis questions 1 problem solving question What’s new in the final exam • New organisation form • According to the curriculum, problem solving tasks address the following student skills • Understand the problem • Develop a plan • Execute the plan • Communicate in an appropriate way • Reflect on the found solution 19**In the exam we do not test real problem solving**• Because of time constraints (1 hour dedicated to problem solving questions) • Problem solving is fostered throughout the grades 11 – 13 • “Problem solving” questions in the exam are limited to a certain number of predefined categories • Questions concentrate as well on “argumentation” and “communication” skills 20**Lessons from a first experience**• No organisational problem at the final exam : • No cheating • No maintenance problem • It takes time to integrate the CAS-calculator into learning and teaching mathematics • time for the teachers to “accept” the machine • time for finding a meaningful use 21**Lessons from a first experience**• Different reactions from the students • Some “refuse” to use the V200 • Some use it all the time • Some use it not enough • To get a more deep impact of the calculator, other subjects (chemistry, physics, economics) should also use the V200 (which, for the moment, they don’t) 22**Lessons from a first experience**• The use of the V200 does not solve old problems • Lack of motivation for mathematics • Inappropriate communication of results (illegible manuscripts, no care on presentation, no justification..) • A lot of student do not use the potential of the V200 especially in the “classical” part of the exam • They do not use the V200 to foresee results • They do not use the V200 to crosscheck results This is also a teaching problem 23**Lessons from a first experience**• In the problem solving part of the exam it became obvious that a lot of student are not aware of the role of the calculator • They think that they have to show that they master the calculator • This is also a teaching problem 24**Previsions for the next years**• Short Term : • Evaluate the impact with the teachers • Collect reactions from students • Medium Term : • More emphasis on the notion of parameter • Long Term : • New subdivision in the exam (not based on content) • One test without V200 covering basic calculation techniques • One test with V200 addressing problem solving and mathematical modeling 25**Contact**Jos Bertemes Professeur chargé de mission Ministère de l’Éducation nationale et de la Formation professionnelle L-2926 Luxembourg email : bertemes@men.lu 26