1

• The mandate of the working party “Reform of

mathematics”

• The philosophy of the reform and the basic decisions

2

• The form of the syllabus • Basic exercise versus problem solving

3

• Teaching mathematics with the support of technology • The use of technology during the assessment • Characteristics and description of the new technology

THE MANDATE OF THE WORKING PARTY

     

To adapt the European syllabuses to the changes in the national syllabuses over the last 20 years.

To consider the enlargement of the EU from 10 to 27 member countries.

To integrate the changes induced in the teaching of mathematics by the development of technology.

To reconsider the syllabuses and the Baccalaureate with regard to the changes implemented at the universities in member states of the EU.

To find a compromise between the different teaching cultures and assessment methods of teachers coming from all the member states.

To create an in-service training structure for the teachers to guarantee a high level of teaching and to insure common standards for examinations.

THE PHILOSOPHY OF THE REFORM

     

The teaching of mathematics cannot be replaced by the simple use of technology. Technology is a tool and not the content of the syllabus.

The teaching of mathematics must be opened to more exploration, problem solving, reasoning and reflection.

The basic concepts and skills in mathematics must be well understood before using technology.

The classrooms have to be transformed into laboratories of mathematics in order to change the teaching methods, to encourage exploration and discussion.

Teaching of mathematics needs a larger differentiation between strong and weak pupils, a higher motivation of the pupils and a larger interdisciplinary approach.

The use of the technical support must be the same for all pupils, at all the levels and guarantee equal conditions at examinations.

THE BASIC DECISIONS OF THE WORKING PARTY



The syllabuses define the competences, the knowledge and the skills the student must be able to handle.



All syllabuses are presented in 3 columns.

1. Subjects 2. Knowledge & Skills 3. Use of Technology the knowledge and the skills the student must be able to handle

without any technological tool

the skills in using the tool to deepen notions, to explore problems and/or to search for solutions

THE BASIC DECISIONS OF THE WORKING PARTY

 

Evaluations are always based partly on examinations without any support of a technical tool and partly on examinations with this tool.

All students in all European schools use exactly the same technical tool in class, at home and during all their examinations.

THE FORM OF THE SYLLABUS AN EXAMPLE FROM THE S4 SYLLABUS

1. Subjects Linear dependency and proportionality 1 st degree functions and equations 2. Knowledge & Skills

Pupils must be able to / understand:

• • • • • •

recognise that one value depends on another value and define a function accordingly know and recognise a linear function y = mx + p transform an equation ax+by=c in the form y=mx+p and the converse recognise that the graphical representation of ax+by=c ( inclu ding when b=0 or a=0 ) is a straight line and the converse (with and without a calculator) understand the meaning of m and p define geometrically m and p 3. Use of Technology

Pupils must be able to / understand:

     

use of the Graph option (variation of m and p) use of the Window (grab and move) use of Intersection and Slope draw the graph of a linear function use cursors to vary m and p use Function table to plot a set of (x, y) values and the graph of a linear function

BASIC EXERCICE VERSUS PROBLEM SOLVING AN EXAMPLE OF A CLASSICAL EXERCISE

Consider the family of functions f



with



defined by : real and strictly positive and noting that C



f



in the orthonormal plane.

is the curve representing 1. Determine the real value of 2. Show that all the curves C when the range of

  

so that the curve passes through the origin.

have a minimum and express the coordinates of this minimum as a function of



.

3. Determine the equation of the curve upon which these minimums lie is real and strictly positive.

In the remainder of this question take



=2 noting for simplicity f 2 =f and C 2 =C.

4. Determine for this value of same diagram.



: The zeros of the function f; The asymptotes of the curve C; The regions of x for which f increases and for which it decreases, and any extrema of f; The point of inflexion of C; The equation of the tangent to C at the point of inflexion.

5. Sketch the curve C and the tangent at the point of inflexion in the 6. Let A(t) be the area defined by C, the horizontal asymptote and the line with equation y = t where t is real and strictly negative. Calculate A(t).

7. Calculate: lim A(t) when t



-∞

BASIC EXERCISES – REAL PROBLEMS

It is clearly written what to do and often the techniques to use are indicated in the text.

There is no doubt about the strategy to follow to answer the question.

The solution is obtained by applying skills and techniques learned previously.

It is possible beforehand to estimate the time needed to find the solution.

Easy to solve if the level of required skills is reached or higher, impossible to solve if this level is not attained.

The formulation is open and it is not always clear from the beginning what the final outcome is.

At the beginning, it is not necessarily clear, which way to chose to solve the problem.

Developing strategies, deepening the problem, formulating hypothesis or questioning the chosen way are necessary steps to find the solution.

It is difficult and sometimes impossible to estimate in advance the time needed.

Different level of skills and knowledge allow to find the solution, not only the level reached is determinant .

BASIC EXERCISE VERSUS PROBLEM SOLVING AN EXAMPLE OF AN REAL PROBLEM

Proof, that if you put the five points on the right inside the triangle, at least 2 of them are not separated by more then 1 cm.

Here the proof And now the 5th point?

2 cm 1 cm 1 cm <1cm 1 cm Paloma or Dirichlet principle

TEACHING MATHEMATICS WITH THE SUPPORT OF TECHNOLOGY

Mathematics will always need knowledge, skills and techniques on the one hand, reflection, strategies and proofs on the other.

The new syllabuses aim to strengthen these two foundations of mathematical thinking by introducing a new balance in the way these two basic aspects of mathematics are taught in the European schools.

TEACHING MATHEMATICS WITH THE SUPPORT OF TECHNOLOGY

The aims of the syllabuses will be reached by: 1.

Giving a clear definition of the basic knowledge and skills our students must acquire, handle and apply without any technical support or tool.

2.

And by introducing in all our syllabuses a technological tool freeing the pupils and the teachers from simple training activities and liberating thus the time necessary for mathematical thinking and problem solving.

TEACHING MATHEMATICS WITH THE SUPPORT OF TECHNOLOGY

2010 S4 MAT-4 S4 MAT-6 2010 S6 MAT-3 S6 MAT-5 2011 S6 MAT-A 2011 S5 MAT-4 S7 MAT-3 S5 MAT-6 S7 MAT-5 S7 MAT-A  

The use of the technology will begin in the 4 th and 6 th of the secondary cycle and grow steadily up to the baccalaureate. year The syllabuses and the assessments take into account the age of the students and the level they have chosen to study mathematics.

THE USE OF TECHNOLOGY DURING THE ASSESSMENT

All examinations evaluate the students on



their basic knowledge and skills without the use of technology, and



on their competences in real problem solving with a technological tool available if required.

The assessment of the students in these 2 fundamental aspects in the teaching of mathematics is done according to the following scheme: ELEMENTARY LEVEL STANDARD LEVEL ASSESSMENT 4 th 5 th 6 th 7 th YEAR YEAR YEAR YEAR BACCALAUREATE WITHOUT 50% 50% 33% 33% 33% WITH 50% 50% 67% 67% 67% WITHOUT 50% 33% 25% 25% 25% WITH 50% 67% 75% 75% 75%

THE USE OF TECHNOLOGY DURING THE ASSESSMENT

70% 60% 50% 40% 30% 20% 10% 0%

ELEMENTARY LEVEL

50% S4 without with S5 45% 40% 35% 30% 25% 20% 15% 10% 5% S6 0% S7

FURTHER LEVEL

80%

STANDARD LEVEL

without without 70% with with choice 60% S6 50% 40% 30% 20% 10% 0% S7 S4 S5 S6 S7

THE USE OF TECHNOLOGY DURING THE ASSESSMENT

WRITTEN BACCALAUREATE

without with 80% 70% 60% 50% 40% 30% 20% 10% 0% 33% 67% 25% 75%

ELEMENTARY LEVEL STANDARD LEVEL

CHARACTERISTICS OF THE TECHNOLOGY PART I

The technology introduced with the new syllabuses must be usable on multiple platforms: computers, smart boards, overhead projectors and as a normal calculator in classrooms without any specific equipment .

The technology must be linked to a software with a document structure in order to allow easy upgrades, the use of modern communication technologies for the exchange of data, files, teaching units etc. between the pupils and the teachers of all the European schools.

All the possibilities offered by the software must be available on a separate tool, called handheld, working independently from computers or other specific equipment in a classroom. This characteristic is essential for the examinations and the baccalaureate.

CHARACTERISTICS OF THE TECHNOLOGY PART II

The technology must integrate in a dynamic, interactive way and with one interface: analysis, algebra, geometry, probability, statistics, spreadsheets, graphs, formal calculus and a text edito r.

A reset function, without the loss of data is necessary for the examinations.

It must be possible to use the same technology in other subjects, e.g. physics, biology, chemistry or in geography and economics.

CHARACTERISTICS OF THE TECHNOLOGY PART III

The software and the user manual must be available in most of the official languages of the European Union.

The new syllabuses seek not only to give the students a new tool. The use of the tool by the teacher is equally important and will generate a renewal of the teaching methods for mathematics.

From the 4 th year to the European Baccalaureate, the students keep the same tool.

Upgrades are done through the software and not by buying a new tool.

CHARACTERISTICS OF THE TECHNOLOGY PART IV

The syllabuses are independent from the use of a specific technological tool.

The functional requirements of the tool are part of the syllabus .

Based on these requirements, a group of experts from the European Schools chooses the tool to be used.

The choice of the tool will be reconsidered by the experts according to technological innovations.

. A permanent structure for training of teachers has to be created.

For the implementation of the reform in 2010 the working party recommends the TI-Nspire CAS technology from Texas instruments.

Different Approaches for Different Learning styles Dynamic linked multi representations Gain a better insight in Maths Unique Mathematical Spreadsheet More Active Student Engagement All In One = One Interface (PC & HH) Improve Skills Document based



Saving of Investigations Calculations Reasoning

Multiple Representations of Concepts Algebraic

Representation

Graphical

Representation

Numerical

Representation OPTIMIZATION

Geometric

Representation

Written

Representation

Multiple Representations of Concepts Algebraic

Representation

Graphical

Representation

Numerical

Representation

Geometric

Representation

Written

Representation

Graphs & Geometry: Geometry - Objects



Point



Line



Segment



Circle Arc



Ray



Tangent



Vector



Circle



Triangle



Rectangle



Polygon



Regular Polygon

Graphs & Geometry: Geometry - Tools



Measurement Tools



Length



Area

  

Text Slope Angle



Coordinates & Equations



Calculate



Scale

Geometry – Constructions



Perpendicular



Parallel



Perpendicular Bisector



Angle Bisector



Midpoint



Locus



Compass



Measurement Transfer

Geometry – Transformations



Symmetry



Reflection



Translaton



Rotation



Dilatation

Graphs & Geometry Views Graphing View Plane Geometry View Plane Geometry View With Analytic Window

Four Types of Graphs

Function Scatter Plot Polar Parametric

to see an example of the interactivity Click the graph Polar Roses

Representation of Families of Graphs

Grab, Move and Graph Quadratic functions Observe the role of the parameters a, b and c.

to see the example Click the graph

Calculator Algebra Statistics Matrices Calculus Complex Numbers

Polynomial Tools Calculator



Remainder of Polynomial



Quotient of Polynomial



Greatest Common Divisor



Coefficients of Polynomial



Degree of Polynomial

Series Calculator

  

Enhanced Summation & Taylor Approximation Dominant Term of Series Asymptotic Series

Two Other Important Characteristics of the Tool 1.

The potential use in other subjects 2.

Some connectivity features

Data Collection Sensor Data Logging Easy Link CBR 2

Data Collection

25-g Accelerometer Low-g Accelerometer Barometer Charge Sensor Conductivity Probe Current Probe Differential Voltage Probe Dual-Range Force Sensor Electrode Amplifier EKG Sensor Extra Long Temperature Probe Flow Rate Sensor Force Plate Gas Pressure Sensor Hand Dynamometer Infrared Thermometer Compatible Sensors Light Sensor Magnetic Field Sensor O2 Gas Sensor ORP Sensor pH Sensor Relative Humidity Sensor Salinity Sensor Soil Moisture Sensor Sound Level Meter Stainless Steel Temperature Probe Surface Temperature Sensor Thermocouple Turbidity Sensor UVA Sensor UVB Sensor Voltage Probe

Connectivity Handheld to Handheld

 

Document Transfer Operating System ( OS software ) Transfer



Document Transfer



File Browser



Backup/Restore



Operating System OS Installation



Screen Capture Connectivity Computer to Handheld

Connectivity Computer to Classroom



Document Transfer



Send & Collect



Redistribute



Delete



Operating System OS Installation

Mr. Wolfgang Fruhauf Secretary European school of Alicante [email protected]

Mr. Luc Blomme ICT coordinator European school of Brussels III [email protected]

Members of the working party in the European Schools Mr. Pierre Brzakala Secondary inspector, responsible for mathematics, physics and ICT