Table ronde 3

Sciences expérimentales et mathématiques : quels bénéfices mutuels ?

© www.education.gouv.fr 2008

Table ronde 3: Sciences expérimentales et mathématiques : quels bénéfices mutuels ? - Présentation du modérateur Michèle Artigue ** présentation Michele Artigue ** résumé Michele Artigue - Mathematics and Science : Ideas for a Swedish project ** présentation Ola Helenius ** résumé Ola Helenius - Science learning in a Europe of Knowledge: a perspective from England ** présentation Celia Hoyles ** résumé Celia Hoyles - Innovations in Mathematics Education on European Level: a systemic approach ** présentation Volker Ulm ** résumé Volker Ulm

Table ronde 3 Experimental Sciences and Mathematics : What mutual benefits ?

Michèle Artigue, Université Paris-Diderot – Paris 7

This round table addresses a crucial issue both for mathematics and sciences teaching: that of their mutual relationships. It is usual to stress the specificity of mathematics among the other sciences, by arguing of the abstract nature of its objects and of its specific deductive method of proof. Such visions tend to occult the constitutive role that mathematics have always played and increasingly play in scientific conceptualizations, and conversely the role that problems emerging outside its own domain play in mathematical developments. Such visions also present a very limited vision of mathematical activity which is far from being restricted to the elaboration of deductive proofs. They are very often reinforced by school curricula and practices where each scientific discipline appears as an isolated continent. Such situation serves neither the cause of mathematics, nor that of sciences. As President of the International Commission on Mathematical Instruction (ICMI), I was thus very happy when I was informed that, in this conference focusing on the teaching of sciences, a place would be open for discussing the issue of relationships between mathematics and sciences teaching, and what we can do at the European level for improving the current situation. Many questions immediately arise, and among these the followings:

• What can be expected from improved connections between mathematics and sciences teaching and why?

• How can these expectations be progressively achieved along the school grades, and what is needed for that?

• How can Europe support efficiently such efforts? • What priorities, what agenda could make sense?

There is no doubt that the reflection to be developed does not start from scratch. Educational research has already addressed the issue of relationships between mathematics and sciences teaching from a diversity of perspectives. Many experiments, innovations, institutional actions have already been carried out. What can we learn from these? How to think and manage the up-scaling of the existing successful experiments often of limited scope? Even if the question of relationships between mathematics and sciences education is not new at all, and can be traced in the history of mathematics and sciences education, there is no doubt that the technological evolution affects both our vision of it and the strategies and means at our disposal for addressing it. How can we put the digital world at the service of the required changes? Whatever be the affordances of technology, the key of evolution in that domain as in any educational domain is the teacher. How can teacher initial preparation and continuous professional development support the required changes? The round table is devoted to these questions. We will focus in it on compulsory schooling, having in mind that the mathematics and science teaching we consider aims at being accessible to all students, and make the success of all possible. Four experts have been invited to contribute. I will introduce them now, following the order in which they will speak. Manuel de Leon Rodriguez, who is the current Director of the Institute of Mathematical Sciences in Madrid and vice-President of the International Mathematics Union, will thus speak first, pointing out that a major difficulty in mathematics education consists in making

our students perceive that mathematics is a living discipline, closely connected with the most relevant problems of the modern world. He will advocate that the connection between mathematics and sciences on the one hand, the transposition into primary and secondary schools of mathematical research practices on the other hand, can help us overcome this difficulty. Ola Helenius, who is Deputy Director at the National Center for Mathematics Education, University of Goteborg, will pursue the reflection, relying on ideas from a national project he is involved in, aiming at the improvement of mathematics education and co-operation between education in science, technology and mathematics in compulsory school. He will address the three following issues:

• the role that concrete objects and contexts can play in the emergence of mathematical concepts and how this can be combined with the development of mathematical abstractions; the ways relationships between mathematics and natural sciences can be efficiently transposed into education for the mutual benefits of mathematics and science education, and the students’ diversity made a power not an obstacle.

• the evolution of mathematical tools required by scientific and technological education along the grades.

Celia Hoyles, who is professor at the London Knowledge Laboratory, University of London, and the current Director of the NCETM (National Centre for Excellence in Teaching Mathematics) will rely on her research experience on the use of technology for mathematics learning and on teachers’ preparation and professional development for approaching the issues at stake from the technological and the teacher perspectives. She will stress the potential offered by digital technologies for establishing productive connections between sciences and mathematics teaching, through modelling activities, but also the attention to be given to continuing professional development (CPD) for teachers if one wants this potential become effective, and she will present some ideas of effective. Finally, Volker Ulm who is professor at the University of Augsburg, and Head of the Chair of Didactic of Mathematics, adopting a systemic approach, will address the crucial issue raised by the successful development and subsistence of substantial innovations, pointing out the problems raised by the steering of complex systems such as educational systems are, and making suggestions for overcoming these at the European level inspired by the Pollen and SINUS programmes. After their presentations, the word will be given to the floor, and I invite you to prepare reactions to the contributions, comments and answers regarding the questions at stake, and also raise important points that we could have missed.

Table ronde 3Sciences expérimentales et

mathématiques : quels bénéfices mutuels ?

Michèle ArtigueManuel de León Rodriguez

Ola HeleniusCelia HoylesVolker Ulm

Les relations entre enseignement des mathématiques et des sciences

• Une question importante et complexe du fait : – des liens profonds qui unissent les

mathématiques et les sciences, en tant que champs scientifiques,

– des perspectives et pratiques dominantes dans l’enseignement qui tendent à considérer chaque discipline scientifique comme un continent isolé, et à opposer les mathématiques aux autres champs scientifiques.

Une multiplicité de questions• Que peut-on attendre d’une meilleure

articulation entre enseignement des sciences et des mathématiques, et pourquoi ?

• Comment ces attentes peuvent-elles être satisfaites, progressivement, au fil des niveaux d’enseignement, à quelles conditions et avec quels moyens ?

• Comment l’Europe peut-elle soutenir efficacement de tels efforts ?

• Quelles priorités ? Quel agenda ?

Une réflexion qui ne part pas de rien

• De nombreuses recherches, expériences, innovations, actions institutionnelles ont déjà été développées.

• Quelles leçons peut-on en tirer ? • Comment dépasser le caractère souvent

local des expériences réussies, penser et organiser leur extension à plus grande échelle ?

La technologie

• La question des relations entre l’enseignement des mathématiques et des sciences n’est pas une question nouvelle mais aujourd’hui les avancées technologiques nous la font percevoir différemment et nous donnent de nouveaux moyens pour l’aborder.

• Comment mettre efficacement le monde numérique au service des changements nécessaires ?

Les enseignants

• Comme pour toute question posée dans le domaine de l’éducation, aucune avancée durable ne peut être atteinte sans l’adhésion, la contribution, l’engagement des enseignants.

• Comment la formation des enseignants, formation initiale et formation continue, peut-elle soutenir les évolutions souhaitées ?

Les quatre experts contribuant à la table ronde

• Manuel de León Rodriguez, Directeur de l’Institut de Sciences Mathématiques de Madrid

• Ola Helenius, Directeur du Centre National pour l’Education Mathématique à l’Université de Goteborg

• Celia Hoyles, Professeur au London Knowledge Laboratory, Université de Londres, et Directricedu NCETM

• Volker Ulm, Responsable de la Chaire de Didactique des Mathématiques à l’Universitéd’Augsburg

Mathematics and Science

Ideas from a Swedish project

Ola Helenius, ola.helenius@ncm.gu.seNational Center for Mathematics Education, Göteborg UniversityDepartment of Science and Engineering, Örebro UniversityDepartment of Science Engineering and

I will touch upon three aspects proposed for our

group discussion• the role that “concrete” objects and contexts can

play in the emergence of mathematical concepts, and how this can be combined with the development of mathematical abstractions;

• the ways the productive relationships existing between mathematics and natural sciences can be transposed into education for the mutual benefit of mathematics and sciences education, benefiting from students’ diversity thanks to the development of adequate pedagogical strategies;

• the evolution of mathematical tools required by scientific and technological education as far as this education progresses along the grades;

Mathematics:An abstract and general science for problem solving and method development.

Working with mathematics means using natural sense making powers.

Science:Knowledge about: nature and humanscientific activityhow the knowledge can be used

Inquiry based - but with progression.

Competence/proficiency based descriptions of what it means to know mathematics.

A relevance paradox

Mathematics is effective for solving many problems......but in many distinct situations it is more effective to do it without mathematics (if you do not already know the mathematics).

This is a problem when trying to use science to create relevance in mathematics...

...and maybe even a bigger problem when working inquiry based.

(“Without mathematics”: subtle distinction)

Separation of

•Progression in content

•Progression of scientific thinking and working

•Progression in working with mathematics and in using mathematical tools

Example: Sowing seeds

(a phenomenon)

All four aspects can be varied from pre school level to university level independently of the others in this example.

Biological questions:How many sprout?How fast do they grow?Environmental dependencies?basic (light, water) – advanced (photosynthesis, cell biology)

Scientific progression: How specific questions?How advanced discussions?How sure about results?

Doing mathematics Measuring (using ruCoordinate systemsMean valuesMore advanced stati(significance etc)

Using mathematics: Graphical representa

Summary

• Inquiry based science teaching in three variants:Based around phenomenon, concept or artifact.

• Very adaptive. Opens up for progression in many dimensions. Can handle student diversity and still allow classroom discussions.

• Many different types of connections between science and mathematics(math as tool - conceptual connections - working aspects (inquiry) - relevance).

• Takes the relevance problem of mathematics seriously.

Short summary Grenoble 15 min talk, Ola Helenius By taking some examples from an ongoing science*-mathematics collaboration project in Sweden I will discuss primarily the three first points proposed for our panel, namely the relation between concrete objects and mathematical abstraction, the relationship between mathematics and science and the co-progression of science and mathematics through the school system (grades). We characterize between a few different types of connections between mathematics and science, some related to the “content” and some related to what it means to work with the subjects. In an inquiry based approach, we identify three different ways of working: phenomenon centered, concept centered or artifact centered, that can be used for specific purposes. In an example from biology I will indicate how we can separate between four dimensions: content, “scientific thinking”, usage of mathematical tools and working with mathematical objects. In the same basic example, it is possible to vary each of these aspects from pre school level to university level. This does not only open up for possibilities to address pupils’ diversity while maintaining a base for classroom communication. I will also indicate how I think this can help in handling the relevance problem that mathematics is often plagued with. *Science is used in the same way as in the Rocard report, ie to mean the physical sciences, life sciences, computer science and technology.

Science Learning in a Europe of Knowledge

Grenoble 8-9th Oct

Professor Celia HoylesDirector of the NCETM

uniqueness of Mathematicsmultliple faces of mathematics

• core skill for all• subject in its own right • service subject for science,

technology & engineering (STEM) & • ... more and more subjects & careers

each face has different demands for mathematics in terms of

• content & skill• language & structure• pedagogy & trajectory of learning

issues in teachingmathematics …and STEM

reputation as compared to other subjects ● more difficult ● higher risk

setting means tendency for high expectations only of top set

negative attitude: dislike, boring & irrelevant

stereotypes of success & limits on expectations

girls are

• less likely to be confident & take risks

• stress enjoyment & coping rather than usefulness

• continue if they feel encouraged

more issues for STEM

invisibility of mathematics in STEM subjects

mathematics is ‘just a tool”

yet

mathematics is the enzyme that catalyses STEM investigation & activity

les mathématiques agissent commeenzyme pour les matières scientifiques

Potential of digital technologies

can make it easier to connect with ● learners’ agendas & culture

●goals in outside world

● the STEM agenda– explore a situation– build a model & – share, discuss, improve model

Modelling for STEMexamples

● energy & movement– rolling marbles down a ramp, what for what angle does

marble travel furthest? is it true for all marbles? predict & test for different marbles

● population growth● predator/prey models● disease● poverty● living graphs

modelling can be interesting, challenging & relevant for each component of STEM

but

need to agree the vision in STEM community● joint planning● iterative design ● joint evaluation

ICT invariably serves as a catalyst for this collaborative engagement

but tools need to be learned

importance of time and space for professional development for teachers

The National Centre for Excellence in the Teaching of Mathematics (NCETM)the Centre promotes a blended approach to

professional learning through a combination of ● funded by Government●face-to-face national & regional activity ● interactions with NCETM’s on line portal

see www.ncetm.org.uk

NCETM’s Professional Learning Framework

Resources

Courses and Events

Professional Self evaluationResearch

Mathemapedia

Communities

Blogs

Personal and professional

learning space

ResourcesNCETM portal Micro-sites

Teachers Talking Theory in Action

Learning Outside the Classroom

Maths at Work: video clips “What mathematics would be involved in the work you have just watched?”

other initiatives in England

national network of Further Mathematics Centres. http://www.fmnetwork.org.uk/

every elementary school will have a mathematics specialist by 2012

NCETM community“Through the NCETM I have a sense that a real

mathematical community is starting to be developed, nurtured and appreciated. As a maths teacher for over 25 years I now have access to external support and dialogue, peer support, opportunities for learning and to build on my own expertise as a leader of CPD within my department.” Head of Mathematics in school

Can we foster a European community around Mathematics in STEM?

thank you

merci

- 1 -

Science learning in a Europe of Knowledge: a perspective from England

Professor Celia Hoyles,

London Knowledge Lab, Institute of Education, University of London, U.K. Director of the National Centre for Excellence in the Teaching of Mathematics

NCETM In thinking about the role of mathematics in science learning it is important to consider all the different roles that mathematics has to perform: as a core skill, as a subject in its own right and as a service subject for science. engineering and technology - as well of course for many other subjects. Each role places constraints on mathematics and the way it is taught. There are other issues that make teaching mathematics complex, for example, its reputation as being more difficult than other subjects, the stereotypes of success and the limits placed on expectations for example through setting, and the negative attitudes often held towards the subject. All these factors have led to some groups of students not persisting with mathematics, a trend widely noticed among girls; even girls who achieve highly tend to express lack of confidence in their mathematics ability and drop out as soon as they can. Most concerned with science, would acknowledge the importance of fluency in mathematics but not an appreciation of the subject itself: in general mathematics is just invisible if it can be ‘done’, it is ‘just a tool’, and little attention paid therefore to how best to introduce relevant mathematical expertise in science settings. I suggested one avenue that might usefully be explored in interdisciplinary teams is through modelling with joint design, planning and evaluation. But for this initiative to have any chance of success, teachers must have time and recognition for professional development. I am Director of the National Centre for Excellence in the Teaching of Mathematics (NCETM). Earlier this year (2008), it was announced that in the latest Comprehensive Review of Government Spending, there would be £140m available over the next three years (2008- 2011) to improve mathematics and science teaching, an amount that includes continued funding for the NCETM.

Figure 1: The NCETM’s Professional Learning Framework

Resources

Courses and Events

Self-evaluation

Research

Blogs

Personal learning space

Mathemapedia

Communities

- 2 -

This long term funding is evidence of Government support for mathematics as at the heart of so much of education across all phases, and recognition of the importance of professional development for teachers of. mathematics The NCETM promotes a blended approach to continuing professional development (CPD) though face-face-face activities and through interaction on our portal www.ncetm.org.uk. I showed and illustrated some of the parts of the NCETM’s Professional Learning framework (see Fig 1) through which we are seeking to build a community of mathematics teachers across the country. And I ended with a plea that we together foster a European community around Mathematics in Science.

Innovations in Mathematics Education on European Level

–A Systemic Approach

Volker Ulm, University of Augsburg

1. Deficiencies

2. Innovation: Invention and Implementation

3. How to change complex systems

4. Conclusion

Steering complex systems

on the meta-level

incremental-evolutionary

analytic-constructive

on the object level

Innovations in complex systems

on the meta-level

incremental-evolutionary

analytic-constructive

on the object level

4. What should be done?

• aiming at teachers• very large European programme• main areas of activity• aiming at the meta-level• networks of teachers• strong leading consortium• processes take time

1

Inno

vatio

ns in

Mat

hem

atic

s Ed

ucat

ion

on E

urop

ean

Leve

l –

A S

yste

mic

App

roac

h

Con

fere

nce

“L’A

ppre

ntis

sage

des

Sci

ence

s da

ns l’

Euro

pe d

e la

Con

nais

sanc

e”

Gre

nobl

e 08

./09.

Oct

ober

200

8

Prof

. Dr.

Vol

ker U

lm

Uni

vers

ity o

f Aug

sbur

g, C

hair

of D

idac

tics o

f Mat

hem

atic

s,

8613

5 A

ugsb

urg,

Ger

man

y, u

lm@

mat

h.un

i-aug

sbur

g.de

1.

B

ackg

roun

d: D

efic

ienc

ies o

f Mat

hem

atic

s Edu

catio

n In

tern

atio

nal

stud

ies

like

TIM

SS a

nd P

ISA

hav

e re

veal

ed s

erio

us d

efic

ienc

ies

of

com

mon

mat

hem

atic

s ed

ucat

ion.

Too

man

y st

uden

ts la

ck in

the

abili

ty to

act

on

thei

r ow

n or

to s

olve

pro

blem

s co

oper

ativ

ely.

The

ir un

ders

tand

ing

for m

athe

mat

ics

is ra

ther

su

perf

icia

l and

thei

r mat

hem

atic

al k

now

ledg

e is

qui

te in

cohe

rent

. The

y fa

il in

tack

ling

mat

hem

atic

al s

ituat

ions

tha

t re

quire

cre

ativ

ity o

r th

orou

gh u

nder

stan

ding

of

basi

c co

ncep

ts. T

o so

me

exte

nt s

tude

nts

are

train

ed to

imita

te w

hat t

hey

are

show

n by

thei

r te

ache

r an

d ar

e ex

pect

ed to

mem

oriz

e an

d ap

ply

give

n pr

oced

ures

. So

inno

vatio

ns in

m

athe

mat

ics e

duca

tion

seem

to b

e ne

cess

ary.

2.

In

nova

tion

as In

vent

ion

and

Impl

emen

tatio

n Th

e O

ECD

def

ines

an

inno

vatio

n as

the

im

plem

enta

tion

of a

new

or

sign

ifica

ntly

im

prov

ed p

rodu

ct,

proc

ess

or m

etho

d (O

ECD

, Eu

rost

at,

2005

, p.

46)

. Th

us a

n in

nova

tion

requ

ires b

oth

an in

vent

ion

and

the

impl

emen

tatio

n of

the

new

idea

. In

the

edu

catio

nal

syst

em w

e ar

e in

a s

ituat

ion

whe

re l

ots

of c

once

pts,

met

hods

and

to

ols

have

bee

n de

velo

ped

for

subs

tant

ial

impr

ovem

ents

of

teac

hing

and

lea

rnin

g.

Thre

e ex

ampl

es: C

urre

nt p

edag

ogic

al th

eorie

s em

phas

ize

self-

orga

nise

d, in

divi

dual

and

co

oper

ativ

e in

quiry

-bas

ed l

earn

ing.

The

re e

xist

dat

a ba

ses

and

othe

r co

llect

ions

of

mat

eria

l fo

r te

achi

ng a

nd l

earn

ing

in a

con

stru

ctiv

ist

man

ner.

A l

arge

var

iety

of

softw

are

and

othe

r too

ls is

ava

ilabl

e fo

r the

inte

grat

ion

of IC

T in

edu

catio

nal p

roce

sses

. B

ut fo

r rea

l inn

ovat

ions

thes

e pr

omis

ing

theo

ries

and

prod

ucts

hav

e to

be

impl

emen

ted

in th

e ed

ucat

iona

l sys

tem

in E

urop

e. L

et’s

thin

k at

the

thre

e ex

ampl

es: T

each

ers

shou

ld

teac

h ac

cord

ing

to c

urre

nt p

edag

ogic

al c

once

pts.

The

prop

osed

new

task

cul

ture

sho

uld

beco

me

stan

dard

in

ever

yday

les

sons

. A

nd I

CT

shou

ld b

e us

ed a

s co

mm

on t

ool

for

expl

orin

g m

athe

mat

ics.

So fo

r sub

stan

tial i

nnov

atio

ns w

e ne

ed c

hang

es in

the

notio

ns o

f tea

chin

g an

d le

arni

ng

proc

esse

s, in

th

e at

titud

es

tow

ards

m

athe

mat

ics

and

in

the

belie

fs

conc

erni

ng

educ

atio

nal

proc

esse

s in

sch

ool.

Hen

ce t

he c

ruci

al q

uest

ion

is:

How

can

sub

stan

tial

inno

vatio

ns

in

the

com

plex

sy

stem

of

m

athe

mat

ics

educ

atio

n be

in

itiat

ed

and

mai

ntai

ned

succ

essf

ully

? 3.

H

ow to

cha

nge

com

plex

syst

ems

3.1

Wha

t is a

com

plex

syst

em?

In

theo

ries

of c

yber

netic

s a

syst

em is

cal

led

“com

plex

”, if

it c

an p

oten

tially

be

in s

o m

any

stat

es th

at n

obod

y ca

n co

gniti

vely

gra

sp a

ll po

ssib

le s

tate

s of

the

syst

em a

nd a

ll

2

poss

ible

tran

sitio

ns b

etw

een

the

stat

es. E

xam

ples

are

the

bios

pher

e, a

nat

iona

l par

k, th

e ec

onom

ic sy

stem

or m

athe

mat

ics e

duca

tion

in E

urop

e (M

alik

199

2, V

este

r 199

9).

Com

plex

sys

tem

s us

ually

are

net

wor

ks o

f mul

tiply

con

nect

ed c

ompo

nent

s. O

ne c

anno

t ch

ange

a

com

pone

nt

with

out

influ

enci

ng

the

char

acte

r of

th

e w

hole

sy

stem

. Fu

rther

mor

e re

al c

ompl

ex sy

stem

s are

in p

erm

anen

t exc

hang

e w

ith th

eir e

nviro

nmen

t.

May

be th

is c

hara

cter

izat

ion

of c

ompl

ex s

yste

ms

seem

s a

bit f

uzzy

. But

, nev

erth

eles

s, it

is o

f co

nsid

erab

le m

eani

ng.

Let

us r

egar

d th

e op

posi

te:

If a

sys

tem

is

not

com

plex

, so

meo

ne c

an o

verv

iew

all

poss

ible

sta

tes

of th

e sy

stem

and

all

trans

ition

s be

twee

n th

e st

ates

. So

thi

s pe

rson

sho

uld

be a

ble

to s

teer

the

sys

tem

as

an o

mni

pote

nt m

onar

ch

lead

ing

it to

“go

od”

stat

es.

In c

ontra

st,

com

plex

sys

tem

s do

not

allo

w t

his

way

of

stee

ring.

3.

2 St

eeri

ng c

ompl

ex sy

stem

s Th

e fu

ndam

enta

l pro

blem

of m

anki

nd d

ealin

g w

ith c

ompl

ex s

yste

ms

is h

ow to

man

age

the

com

plex

ity, h

ow t

o st

eer

com

plex

sys

tem

s su

cces

sful

ly a

nd h

ow t

o fin

d w

ays

to

soun

d st

ates

.

With

ref

eren

ce t

o th

eorie

s of

cyb

erne

tics

two

dim

ensi

ons

of

stee

ring

com

plex

sy

stem

s ca

n be

di

stin

guis

hed

(Mal

ik

1992

). Th

e fir

st o

ne c

once

rns

the

man

ner,

the

seco

nd o

ne th

e ta

rget

leve

l of

stee

ring

mea

sure

s. Th

e m

etho

d of

an

alyt

ic-c

onst

ruct

ive

stee

ring

need

s a

cont

rolli

ng a

nd g

over

ning

au

thor

ity th

at d

efin

es o

bjec

tives

for

the

syst

em a

nd d

eter

min

es w

ays

for

reac

hing

the

aim

s. H

iera

rchi

cal-a

utho

ritar

ian

syst

ems

are

foun

ded

on t

his

prin

cipl

e. H

owev

er,

fund

amen

tal

prob

lem

s ar

e ca

used

jus

t by

the

com

plex

ity o

f th

e sy

stem

. In

com

plex

sy

stem

s no

one

has

the

chan

ce to

gra

sp a

ll po

ssib

le s

tate

s of

the

syst

em c

ogni

tivel

y. S

o th

e an

alyt

ic-c

onst

ruct

ive

appr

oach

pos

tula

tes

the

avai

labi

lity

of in

form

atio

n ab

out t

he

syst

em th

at c

anno

t be

reac

hed

in re

ality

.

In c

ontra

st in

crem

enta

l-evo

lutio

nary

ste

erin

g is

bas

ed o

n th

e as

sum

ptio

n th

at c

hang

es

in c

ompl

ex sy

stem

s re

sult

from

nat

ural

gro

win

g an

d de

velo

ping

pro

cess

es. T

he st

eerin

g ac

tiviti

es tr

y to

influ

ence

thes

e sy

stem

ic p

roce

sses

. The

y ac

cept

the

fact

that

com

plex

sy

stem

s ca

nnot

be

stee

red

entir

ely

in a

ll de

tails

and

they

aim

at i

ncre

men

tal c

hang

es in

pr

omis

ing

dire

ctio

ns. T

he fo

cus

on li

ttle

step

s is

ess

entia

l, si

nce

revo

lutio

nary

cha

nges

ca

n ha

ve u

npre

dict

able

con

sequ

ence

s w

hich

may

end

ange

r th

e so

undn

ess

or e

ven

the

exis

tenc

e of

the

who

le sy

stem

. Th

e se

cond

dim

ensi

on d

istin

guis

hes

betw

een

the

obje

ct a

nd th

e m

eta-

leve

l. Th

e ob

ject

le

vel c

onsi

sts

of a

ll co

ncre

te o

bjec

ts o

f th

e sy

stem

. In

the

scho

ol s

yste

m s

uch

obje

cts

are

e. g

. tea

cher

s, st

uden

ts, b

ooks

, com

pute

rs, b

uild

ings

etc

. Cha

nges

on

the

obje

ct le

vel

take

pla

ce if

new

boo

ks a

re b

ough

t or

if a

new

com

pute

r la

b is

fitt

ed o

ut. O

f co

urse

su

ch c

hang

es a

re su

perf

icia

l with

out r

each

ing

the

subs

tant

ial s

truct

ures

of t

he sy

stem

.

incr

emen

tal-

evol

utio

nary

an

alyt

ic-

cons

truc

tive on

the

obje

ct le

vel

on th

e m

eta-

leve

l

Stee

ring

com

plex

syst

ems

3

The

met

a-le

vel

com

preh

ends

e.

g. o

rgan

izat

iona

l st

ruct

ures

, so

cial

rel

atio

nshi

ps,

notio

ns o

f th

e ta

sks

of s

yste

m e

tc. I

n th

e sc

hool

sys

tem

e. g

. not

ions

of

the

natu

re o

f di

ffer

ent s

ubje

cts a

nd b

elie

fs c

once

rnin

g te

achi

ng a

nd le

arni

ng a

re in

clud

ed.

3.3

Inno

vatio

ns b

y in

crem

enta

l-evo

lutio

nary

cha

nges

on

the

met

a-le

vel

How

can

sub

stan

tial

inno

vatio

ns i

n th

e co

mpl

ex s

yste

m “

mat

hem

atic

s ed

ucat

ion

in E

urop

e” b

e in

itiat

ed s

ucce

ssfu

lly?

The

theo

ry o

f cyb

erne

tics

depi

cted

in 3

.2 g

ives

us

eful

hin

ts:

A

ttem

pts

of

anal

ytic

-con

stru

ctiv

e st

eerin

g w

ill f

ail

in t

he l

ong

term

, si

nce

they

ig

nore

th

e co

mpl

exity

im

man

ent i

n th

e sy

stem

.

Cha

nges

on

the

obje

ct l

evel

do

not

nece

ssar

ily c

ause

stru

ctur

al c

hang

es

of th

e sy

stem

. A

ccor

ding

to

th

e th

eory

of

cy

bern

etic

s it

is

muc

h m

ore

prom

isin

g to

in

itiat

e in

crem

enta

l-evo

lutio

nary

cha

nges

on

the

met

a-le

vel.

They

are

in

acco

rd w

ith t

he

com

plex

ity o

f th

e sy

stem

and

do

not

enda

nger

its

exi

sten

ce.

Nev

erth

eles

s, th

ey c

an

caus

e su

bsta

ntia

l ch

ange

s w

ithin

the

sys

tem

by

havi

ng e

ffec

ts o

n th

e m

eta-

leve

l, es

peci

ally

whe

n th

ey w

ork

cum

ulat

ivel

y.

4.

Con

clus

ion:

Wha

t sho

uld

be d

one?

A

sho

rt in

term

edia

te s

umm

ary:

We

have

see

n th

at f

or s

ubst

antia

l in

nova

tions

in

the

educ

atio

nal s

yste

m th

ere

is n

o la

ck o

f ge

nera

l ide

as, p

edag

ogic

al c

once

pts

or d

idac

tic

tool

s. B

ut th

ere

is a

wid

e ga

p be

twee

n th

eore

tical

kno

wle

dge

and

prac

tice

in s

choo

l. So

w

e ha

ve to

dev

elop

stra

tegi

es to

brid

ge th

is g

ap. T

he th

eorie

s of

cyb

erne

tics

give

use

ful

hint

s ho

w t

hat

can

be r

each

ed:

Act

iviti

es a

re m

ost

prom

isin

g, i

f th

ey a

im a

t in

crem

enta

l-evo

lutio

nary

cha

nges

on

the

met

a-le

vel

of b

elie

fs a

nd a

ttitu

des

of a

ll pe

rson

s rel

ated

to th

e ed

ucat

iona

l sys

tem

– e

spec

ially

teac

hers

and

stud

ents

. H

ow c

an th

is b

e do

ne c

oncr

etel

y? T

here

is a

lread

y ex

perie

nce

e. g

. fro

m th

e Eu

rope

an

prog

ram

me

Polle

n an

d th

e G

erm

an p

rogr

amm

e SI

NU

S.

(1) T

he k

ey p

erso

ns fo

r inn

ovat

ions

in sc

hool

are

the

teac

hers

. The

ir be

liefs

, mot

ivat

ion

and

abili

ties a

re c

ruci

al fo

r eve

ryda

y te

achi

ng a

nd le

arni

ng in

scho

ol. O

ne sh

ould

set u

p a

very

lar

ge E

urop

ean

prog

ram

me

whi

ch d

irect

ly a

ims

at t

each

ers’

wor

k. I

n al

l co

untri

es i

n a

first

pha

se a

crit

ical

mas

s of

at

leas

t 8

% o

f al

l sc

hool

s sh

ould

be

invo

lved

. In

the

long

term

one

shou

ld re

ach

at le

ast 4

0 - 5

0 %

of t

he sc

hool

s. (2

) A

s de

velo

pmen

ts s

houl

d be

incr

emen

tal o

ne s

houl

d de

fine

mai

n ar

eas

of a

ctiv

itiy

like:

Dev

elop

ing

a ta

sk c

ultu

re, a

uton

omou

s le

arni

ng, p

rom

otin

g st

uden

t coo

pera

tion,

cu

mul

ativ

e le

arni

ng

and

secu

ring

basi

c kn

owle

dge,

le

arni

ng

from

m

ista

kes,

inte

rdis

cipl

inar

y le

arni

ng, p

rom

otin

g gi

rls a

nd b

oys,

expl

orin

g m

athe

mat

ics w

ith IC

T.

incr

emen

tal-

evol

utio

nary

an

alyt

ic-

cons

truc

tive on

the

obje

ct le

vel

on th

e m

eta-

leve

l

Inno

vatio

ns in

com

plex

syst

ems

4

Parti

cipa

ting

scho

ols

shou

ld fi

rst c

once

ntra

te o

n on

e or

a fe

w a

reas

. It i

s no

t pro

mis

ing

to a

im a

t tot

al c

hang

es o

f m

athe

mat

ics

educ

atio

n –

beca

use

of th

e co

mpl

exity

of

the

syst

em.

How

ever

, su

ch b

ound

ed f

ield

s of

act

ivity

allo

w t

each

ers

to b

egin

with

su

bsta

ntia

l cha

nges

with

out t

he ri

sk o

f los

ing

thei

r pro

fess

iona

l com

pete

nce

in c

lass

. (3

) Sin

ce d

evel

opm

ents

on

the

met

a-le

vel o

f bel

iefs

and

atti

tude

s are

env

isag

ed it

is n

ot

suff

icie

nt o

nly

to d

istri

bute

gui

delin

es o

r mat

eria

l. Te

ache

rs s

houl

d ch

ange

thei

rs w

ays

of te

achi

ng a

nd b

ehav

ing.

For

that

teac

hers

nee

d

re

gula

r and

syst

emat

ic te

ache

r edu

catio

n of

fers

,

stru

ctur

es to

wor

k co

oper

ativ

ely

with

col

leag

ues i

n re

gion

al n

etw

orks

,

poss

ibili

ties

to

exch

ange

ex

perie

nces

w

ith

colle

ague

s on

na

tiona

l an

d in

tern

atio

nal l

evel

and

the

free

dom

to tr

y ne

w id

eas i

n th

eir o

wn

scho

ol.

In re

cent

pro

ject

s lik

e SI

NU

S an

d Po

llen

netw

orks

of t

each

ers

on d

iffer

ent l

evel

s ha

ve

prov

en t

o be

foc

al p

oint

s of

pro

fess

iona

l de

velo

pmen

t. Th

ey c

an b

e re

gard

ed a

s “l

earn

ing

envi

ronm

ents

” fo

r te

ache

rs.

Teac

hers

ge

t ac

quai

nted

w

ith

curr

ent

peda

gogi

cal i

deas

, exc

hang

e ex

perie

nce

and

mak

e st

eps

tow

ards

sys

tem

ic in

nova

tions

co

oper

ativ

ely.

M

oreo

ver,

netw

orks

of

teac

hers

and

sch

ools

are

ess

entia

l m

eans

for

dis

sem

inat

ion

proc

esse

s (s

ee 1

). Ex

perie

nced

tea

cher

s co

ach

colle

ague

s fr

om s

choo

ls s

tarti

ng w

ith

inno

vatio

n ac

tiviti

es.

(4)

Of

cour

se s

uch

far

reac

hing

eff

orts

on

Euro

pean

lev

el n

eed

a st

rong

lea

ding

co

nsor

tium

whi

ch c

ombi

nes

the

expe

rtise

fro

m d

iffer

ent

stak

ehol

ders

. In

par

ticul

ar

scie

ntis

ts h

ave

to c

arry

and

dis

sem

inat

e in

nova

tive

idea

s an

d pr

ovid

e th

eore

tical

ba

ckgr

ound

. A

dmin

istra

tors

sho

uld

crea

te l

inks

to

the

polit

ical

lev

el;

they

hav

e to

en

sure

the

org

aniz

atio

nal

func

tioni

ng o

f al

l ac

tiviti

es a

nd t

heir

inte

grat

ion

in t

he

com

mon

edu

catio

nal s

yste

m. H

ere

we

have

to k

eep

in m

ind

that

edu

catio

nal a

ffai

rs a

re

regu

late

d on

nat

iona

l or

eve

n re

gion

al l

evel

s. So

fro

m e

ach

(par

ticip

atin

g) E

urop

ean

coun

try s

ever

al re

pres

enta

tives

hav

e to

be

inte

grat

ed in

ord

er to

ach

ieve

acc

epta

nce

and

influ

ence

of t

he in

nova

tions

in th

e di

ffer

ent s

choo

l sys

tem

s.

(5) A

s dee

p sy

stem

ic d

evel

opm

ents

are

evo

lutio

nary

, all

thes

e pr

oces

ses t

ake

time.

Thi

s se

ems

to b

e ra

ther

triv

ial.

But

one

can

not e

xpec

t rea

lly s

ubst

antia

l inn

ovat

ions

in th

e Eu

rope

an e

duca

tiona

l sys

tem

with

in th

e us

ual p

roje

ct fu

ndin

g pe

riods

of 2

– 3

yea

rs. A

m

ore

real

istic

tim

e-fr

ame

cove

rs 1

0 –

15 y

ears

. But

for

that

we

need

pro

spec

tive

and

fore

sigh

ted

polit

ical

dec

isio

ns to

fund

such

long

-term

inno

vatio

n pr

ogra

mm

es.

5.

R

efer

ence

s M

alik

, F. (

1992

): St

rate

gie

des M

anag

emen

ts k

ompl

exer

Sys

tem

e, P

aul H

aupt

, Ber

n

OEC

D

&

Euro

stat

(2

005)

: Pr

opos

ed

guid

elin

es

for

colle

ctin

g an

d in

terp

retin

g te

chno

logi

cal i

nnov

atio

n da

ta, O

slo

Man

ual (

2nd

ed.),

Eur

osta

t, Pa

ris

Ves

ter,

F. (1

999)

: Die

Kun

st v

erne

tzt z

u de

nken

, Ide

en u

nd W

erkz

euge

für e

inen

neu

en

Um

gang

mit

Kom

plex

ität,

Deu

tsch

e V

erla

gs-A

nsta

lt, S

tuttg

art