table ronde 3 sciences expérimentales et...
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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, [email protected] 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?
- 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