CAS-Calculators in a centralized national examination Ministry of Education and Vocational Training Luxembourg CAS-Calculators and national examination MENFP 2006 In Luxembourg Luxembourg's Education system Pre-Primary education starts at the age of 4 ( optional at age of 3) 1 3 Primary education lasts for six years. 2 (grades 1 – 6) 3 Lower technical secondary education Lower general secondary education (grades 7 – 9) (grades 7 – 10) Upper general secondary education 1 (grades 11 – 13) 2 Upper technical secondary education Vocational education 3 (grades 10 – 13) Projet CAS-Calculators concerns about 50 % of students in upper secondary education CAS-Calculators and national examination In Luxembourg Starting point of the project 1999-2000 : Internet-Hype in education Luxembourg : project ELABOTIC 3 Use of ICT for teaching and learning Grades 7 - 10 Languages and science Mathematical tools : Cabri, Excel and DERIVE Model lessons and training courses MENFP 2006 CAS-Calculators and national examination In Luxembourg First conclusions 4 After 3 years, project evaluation brought the following conclusions : Some interesting, but isolated pedagogical hot spots No general acceptance A lot of problems MENFP 2006 CAS-Calculators and national examination In Luxembourg What were the problems ? Computers in separate media-rooms 5 ICT in math lessons = unnatural situation Technical problems / no assistance Inadequate student equipment at home difficult acceptance by reluctant teachers ICT = time-consuming activity CONFLICT with curriculum New learning strategies in opposition with traditional teaching methods Inadequate school books MENFP 2006 CAS-Calculators and national examination In Luxembourg 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 MENFP 2006 CAS-Calculators and national examination In Luxembourg MENFP 2006 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 CAS-Calculators and national examination In Luxembourg MENFP 2006 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 CAS-Calculators and national examination In Luxembourg MENFP 2006 Progression scheme Constraints : The use of the V200 had to be build around/into the “old” currciculum Division of the examination in 2 tests : 9 Without V200 : main focus geometry and probability With V200 : analysis and problemsolving CAS-Calculators and national examination In Luxembourg MENFP 2006 Grade 11 : Main focus Understanding the notion of function 10 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 CAS-Calculators and national examination In Luxembourg Grade 12 : Main focus Discovering and interpretation of “Reverse function engineering” (Steckbriefaufgaben) Problem solving and modelisation (using function study) 11 Function derivation curvature Extremum problem Tangent problems “Realistic” situations MENFP 2006 CAS-Calculators and national examination In Luxembourg Grade 13 : Main focus Discovering and interpretation of Discovering and interpretation of integration 12 Exponential function Logarithmic function Riemann – Darboux sums Numerical integration Problem solving MENFP 2006 CAS-Calculators and national examination In Luxembourg New aspects The classical study of functions is extended to new aspects The problems do not necessary have a unique solution 13 Reverse function engineering Presentation of realistic situations 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 MENFP 2006 CAS-Calculators and national examination In Luxembourg MENFP 2006 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 CAS-Calculators and national examination In Luxembourg MENFP 2006 New tool – new problems ! No adapted school books Evaluation problems 15 What do we evaluate if manipulation of expressions is dealt with the CAS-calculator ? How do we evaluate now ? Organisational problems CAS-Calculators and national examination In Luxembourg 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 16 Resolution of equations (both rational and irrational) Numerical evaluation of primitives Linearization of trigonometric expressions Manipulation of rational and irrational expressions MENFP 2006 CAS-Calculators and national examination In Luxembourg 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 17 What are the objectives of the problem/questionnaire What are the principal reflections that the student should make for solving the problem MENFP 2006 CAS-Calculators and national examination In Luxembourg Communication strategy Public website for students (v200.myschool.lu) Closed internet community for teachers 18 Presents the relevant documents for the grades 11 -13 Presents sample examination questionnaires Sample lessons for grades 11 - 13 Questionnaires with evaluation model Technical documents Event calendar – discussion forum (not used !!) MENFP 2006 CAS-Calculators and national examination In Luxembourg What’s new in the final exam Part New form I : organisation (180 minutes) Part II : (240 minutes) According to the curriculum, problem solving without V200 with V200 tasks address the following student skills Contents : Understand the problemContents : Develop a plan 3 classical analysis Execute the plan questions Geometry Communicate in an appropriate way 1 problem solving Reflect on the found solution 4 questions Probability question 19 MENFP 2006 CAS-Calculators and national examination In Luxembourg In the exam we do not test real problem solving 20 MENFP 2006 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 CAS-Calculators and national examination In Luxembourg MENFP 2006 Lessons from a first experience No organisational problem at the final exam : It takes time to integrate the CAS-calculator into learning and teaching mathematics 21 No cheating No maintenance problem time for the teachers to “accept” the machine time for finding a meaningful use CAS-Calculators and national examination In Luxembourg Lessons from a first experience Different reactions from the students 22 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) MENFP 2006 CAS-Calculators and national examination In Luxembourg Lessons from a first experience The use of the V200 does not solve old problems A lot of student do not use the potential of the V200 especially in the “classical” part of the exam 23 Lack of motivation for mathematics Inappropriate communication of results (illegible manuscripts, no care on presentation, no justification..) They do not use the V200 to foresee results They do not use the V200 to crosscheck results This is also a teaching problem MENFP 2006 CAS-Calculators and national examination In Luxembourg 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 24 They think that they have to show that they master the calculator This is also a teaching problem MENFP 2006 CAS-Calculators and national examination In Luxembourg Previsions for the next years Short Term : Medium Term : Evaluate the impact with the teachers Collect reactions from students More emphasis on the notion of parameter Long Term : New subdivision in the exam (not based on content) 25 One test without V200 covering basic calculation techniques One test with V200 addressing problem solving and mathematical modeling MENFP 2006 CAS-Calculators and national examination In Luxembourg MENFP 2006 Contact Jos Bertemes Professeur chargé de mission Ministère de l’Éducation nationale et de la Formation professionnelle L-2926 Luxembourg email : [email protected] 26

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# CAS-Calculators in a centralized national exam