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)
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3
Primary education lasts for six years.
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(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
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1999-2000 : Internet-Hype in education
Luxembourg : project ELABOTIC
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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
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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 ?
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Computers in separate media-rooms
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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
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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
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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
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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 :
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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
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Understanding the notion of function
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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
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Discovering and interpretation of
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“Reverse function engineering”
(Steckbriefaufgaben)
Problem solving and modelisation (using
function study)
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Function derivation
curvature
Extremum problem
Tangent problems
“Realistic” situations
MENFP
2006
CAS-Calculators and national examination
In Luxembourg
Grade 13 : Main focus
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Discovering and interpretation of
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Discovering and interpretation of integration
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Exponential function
Logarithmic function
Riemann – Darboux sums
Numerical integration
Problem solving
MENFP
2006
CAS-Calculators and national examination
In Luxembourg
New aspects
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The classical study of functions is extended to
new aspects
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The problems do not necessary have a unique
solution
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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 ?
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CAS-Calculators and national examination
In Luxembourg
MENFP
2006
New tool – new problems !
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No adapted school books
Evaluation problems
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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
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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
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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
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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
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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
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Public website for students (v200.myschool.lu)
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Closed internet community for teachers
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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
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4 questions
Probability
question
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MENFP
2006
CAS-Calculators and national examination
In Luxembourg
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In the exam we do not test real problem solving
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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
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No organisational problem at the final exam :
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It takes time to integrate the CAS-calculator into
learning and teaching mathematics
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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
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Different reactions from the students
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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
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The use of the V200 does not solve old
problems
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A lot of student do not use the potential of the
V200 especially in the “classical” part of the
exam
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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
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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
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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
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Short Term :
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Medium Term :
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Evaluate the impact with the teachers
Collect reactions from students
More emphasis on the notion of parameter
Long Term :
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New subdivision in the exam (not based on content)
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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]
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CAS-Calculators in a centralized national exam