CASCalculators in a centralized national examination

1
CAS-Calculators in a centralized national
examination

• Ministry of Education and Vocational Training
• Luxembourg

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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
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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

4
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

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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 activity?CONFLICT with
  curriculum
• New learning strategies in opposition with
  traditional teaching methods
• Inadequate school books

6
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

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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

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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

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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

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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

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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

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Grade 13 Main focus

• Discovering and interpretation of
• Exponential function
• Logarithmic function
• Discovering and interpretation of integration
• Riemann Darboux sums
• Numerical integration
• Problem solving

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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

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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 ?

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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

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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

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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/questionnai
  re
• What are the principal reflections that the
  student should make for solving the problem

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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 !!)

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Whats 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

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• 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

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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

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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 dont)

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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

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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

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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

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• Contact
• Jos Bertemes
• Professeur chargé de mission
• Ministère de lÉducation nationale et de la
  Formation professionnelle
• L-2926 Luxembourg
• email bertemes_at_men.lu