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-
DOCUMENT RESUME
ED 413 162 SE 055 792
AUTHOR Furinghetti, Fulvia, Ed.TITLE Proceedings of the Conference of the International Group for
the Psychology of Mathematics Education (PME) (15th, Assisi,Italy, June 29-July 4, 1991), Volume 1.
INSTITUTION International Group for the Psychology of MathematicsEducation.
PUB DATE 1991-00-00NOTE 374p.; For volumes 2 and 3, see SE 055 793-794.PUB TYPE Collected Works Proceedings (021)EDRS PRICE MFO1 /PC15 Plus Postage.DESCRIPTORS Affective Behavior; *Algebra; Attitudes; Beliefs;
*Computers; Context Effect; Elementary Secondary Education;Epistemology; *Functions (Mathematics); Geometry; LanguageRole; *Learning Theories; Mathematics Education;Metacognition; Number Concepts; Problem Solving; Proof(Mathematics); Rational Numbers; Social Influences; *SpatialAbility; Student Evaluation; Teacher Education; VisualPerception
IDENTIFIERS Advanced Mathematics; *Mathematical Thinking; Psychology ofMathematics Education; Representations (Mathematics);Situated Learn'ing
ABSTRACTThis document, the first of three volumes, reports on the
15th annual conference of the International Group for the Psychology ofMathematics Education (PME) held in Italy 1991. Plenary addresses andspeakers are: "Social Interaction and Mathematical Knowledge" (B. M.Bartolini); "Meaning: Image Schemata and Protocols" (W. Dorfler); "On theStatus of Visual Reasoning in Mathematics and Mathematics Education" (T.Dreyfus); "The Activity Theory of Learning and Mathematics Education in theUSSR" (T. Gabay). Research reports in this volume include: "Applications ofR-Rules as Exhibited in Calculus Problem Solving" (Amit, M.; Movshovitz-HadarN.); "Effects of Diagrams on the Solution of Problems Concerning theEstimation of Differences" (Antonietti, A.; Angelini, C.); "Le probleme destatut du "millieu" dans un enseignement de la geometric avec supportlogiciel" (Artigue, M.; Belloc, J.; Kargiotakis, G.); "Procedural andRelational Aspects of Algebraic Thinking" (Arzarello, F.); "Hegemony in theMathematics Curricula the Effect of Gender and Social Class on theOrganisation of Mathematics Teaching for Year 9 Students" (Atweh, B.; Cooper,T.); "Instantaneous Speed: Concept Images at College Students' Level and itsEvolution in a Learning Experience" (Azcarate, C.); "Students' MentalPrototypes for Functions and Graphs" (Bakar, M.; Tall, D.); "Illustrations deproblemes mathematiques complexes mettant en jeu un changement au unesequence de changements par des enfants du primaire (Bednarz, N.; Janvier,B.); "The Operator Construct of Rational Number: A Refinement of the Concept"(Behr, M.; Harel, G.; Post, T.; Lesh, R.); "Children's Use of Outside-SchoolKnowledge to Solve Mathematics Problems In-School" (Bishop, A. J.; De Abreu,G.); "Influences of an Ethnomathematical Approach on Teacher Attitudes toMathematics Education" (Bishop, A. J.; Pompeu, G., Jr.); "Gender and theVersatile Learning of Trigonometry Using Computer Software" (Blackett, N.;Tall, D.);\La dimension du travail psychique dans la formation continue desenseignant(e)s de mathematiques" (Blanchard-Laville, C.); "Drawing a Diagram:
-
+++++ ED413162 Has Multi-page SFR---Level=1 +++++Observing a Partially-Developed Heuristic Process in College Students"(Bodner, B. L.; Goldin, G. A.); "The Active Comparison of Strategies inProblem-Solving: an Exploratory Study" (Bondesan, M. G.; Ferrari, P. L.);"Teachers' Conceptions of Students' Mathematical Errors and ConceivedTreatment of Them" (Boufi, A.; Kafoussi, S.); "Children's Understanding ofFractions as Expressions of Relative Magnitude" (Carraher, D. W.; Dias,Schliemann A.); "Dificultad en problemas de estructura multiplicative decomparacion" (Castro, Martinez E.; Rico, Romero L.; Batanero, Bernabeu C.);"Construction and Interpretation of Algebraic Models" (Chiappini G., LemutE.); "Analysis of the Behaviour of Mathematics Teachers in Problem SolvingSituations with the Computer" (Chiappini, G.; Lemut E., Parenti, L.);"Analysis of the Accompanying Discourse of Mathematics Teachers in theClassroom" (Chiocca, C.; Josse, E.; Robert, A.); "Van Hiele Levels ofLearning Geometry" (Clements, D. H.; Battista, M. T.); "Some Thoughts aboutIndividual Learning Group Development, and Social Interaction" (Cobb, P.);"Une analyse des brouillons de calcul d'eleves confrontes a des items dedivisions ecrites" (Conne, F.; Brun, J.); "Brian's Number Line Representationof Fractions" (Davis, R. B.; Alston, A.; Maher, C.); and "Pupils' Needs forConviction and Explanation within the Context of Geometry" (De Villiers, M.).This volume contains the addresses of research report authors. (MKR)
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.
4Le
onar
do d
a V
inci
. Oua
dern
i di a
nato
mia
(in: L
. Vill
ani,
Leon
ardo
a M
ilano
, Mus
umec
i Ed.
)
-
INT
ER
NA
CIO
NA
L G
RO
UP
FO
R T
HE
PS
YC
HO
LOG
YO
Fw
alle
d/ar
sE
DU
CA
TIO
N
PRO
CE
ED
ING
S
EN
TH
PME
CO
NFE
RE
NC
E2R
:.
6
ASS
ISI
(IT
AL
Y)
1991
JUN
E 7
9 -
JUL
Y 4
VO
LU
ME
I
-
Publ
ishe
d by
the
Prog
ram
Com
mitt
ee o
f th
e 15
th P
ME
Con
fere
nce,
Ita
ly.
All
righ
ts r
eser
ved.
Edi
tor:
FU
LV
IA F
UR
ING
HE
TT
ID
ipar
timen
to d
i Mat
emat
ica
dell'
Uni
vers
itk d
i Gen
ova,
Ita
ly
The
pre
sent
vol
ume
is a
n an
asta
tic c
opy
in A
4 fo
rmat
of
the
orig
inal
edi
tion
of 1
991
PRO
CE
ED
ING
S .3
F PM
E 1
5, A
SSIS
I (I
TA
LY
), 1
991
CO
NT
EN
TS
OF
VO
LUM
E 1
Pref
ace
Inte
rnat
iona
l gro
up f
or th
e ps
ycho
logy
of
mat
hem
atic
s ed
ucat
ion
His
tory
and
aim
s of
the
P.M
.E g
roup
Res
earc
h re
port
s cl
assi
fica
tion
Wor
king
gro
ups
Dis
cuss
ion
grou
psPo
ster
sA
ddre
sses
of
the
auth
ors
pres
entin
g re
sear
ch r
epor
ts in
the
conf
eren
cePl
enar
y ad
dres
ses
Res
earc
h re
port
s (c
ontin
ue in
vol
umes
II
and
HI)
9
p.vi
i
p.vi
ii
p.
ix
p.xv
ii
p.xx
viii
p.xx
xvii
p.lx
xxi
p.1
p.57
-
ii
PME
-X
V P
RO
CE
ED
ING
Sed
ited
by F
u !v
ia F
urin
ghet
tiIt
aly
PRE
FAC
E
The
fir
st m
eetin
g of
PM
E to
ok p
lace
in K
arls
rue
in 1
976.
The
reaf
ter
diff
eren
t cou
ntri
es(N
ethe
rlan
ds, G
erm
any,
U.K
., U
.S.A
., Fr
ance
, Bel
gium
, Isr
ael,
Aus
tral
ia, C
anad
a, H
unga
ry,
Mex
ico)
hos
ted
the
conf
eren
ce. I
n 19
91 I
taly
will
pla
y ho
st to
the
PME
. The
con
fere
nce
will
take
pla
ce in
Ass
isi,
a be
autif
ul to
wn
of f
ine
arch
itect
ural
and
pic
tori
cal w
orks
of
the
Ital
ian
Mid
dle
Age
s.
Ital
y ha
s ha
d a
fine
trad
ition
of
mat
hem
atic
s ed
ucat
ion
sinc
e th
e se
cond
hal
f of
nin
etee
nth
cent
ury
. In
rece
nt y
ears
the
Ital
ian
mat
hem
atic
al c
omm
unity
has
sho
wn
an in
crea
sing
inte
rest
in th
e th
emes
whi
ch c
hara
cter
ize
the
PME
, and
sin
ce 1
988
ther
e ha
s be
en a
sig
nifi
cant
num
ber
otha
lian
rese
arch
rep
orts
.
Inde
ed th
ere
seem
s to
be
a gr
owin
g w
orld
wid
e in
tere
st in
the
conf
eren
ce a
nd it
sth
emes
,with
pap
ers
from
24
coun
trie
s in
clud
ed in
thes
e Pr
ocee
ding
s an
d a
wid
er r
ange
of
dele
gate
s to
the
upco
min
g co
nfer
ence
.
The
aca
dem
ic p
rogr
amm
e of
PM
E in
clud
es:
126
rese
arch
rep
orts
four
ple
nary
add
ress
es
a pa
nel
10 w
orki
ng g
roup
s
8 di
smis
sion
gro
ups
43 p
oste
r pr
esen
tatio
ns.
The
mos
t rel
evan
t inn
ovat
ion
with
res
pect
to th
est
ruct
ure
of th
e pr
evio
us P
ME
Con
fere
nces
is th
e pr
esen
ce o
f a
pane
l spe
cifi
cally
foc
used
on
info
rmat
ion
tech
nolo
gy in
mat
hs
educ
atio
n. T
his
fact
con
stitu
tes
a de
fini
tive
ackn
owle
dgem
ent o
f th
e im
port
ance
of
cons
ider
ing
this
topi
c as
an
inte
grat
ed p
art o
f th
e el
emen
ts to
be
take
n in
to a
ccou
nt in
dis
cuss
ing
the
aspe
cts
of te
achi
ng a
nd le
arni
ng m
athe
mat
ics
and
the
impl
icat
ions
ther
eof.
A n
ote
on th
e re
sear
ch r
epor
ts c
lass
ific
atio
n an
d em
ergi
ng tr
ends
The
res
earc
h re
port
s ar
e pr
esen
ted
in th
ese
Proc
eedi
ngs
in a
lpha
betic
ord
er, w
ith a
fur
ther
clas
sifi
catio
n ac
cord
ing
to th
e fo
llow
ing
them
es s
ugge
sted
by
the
Inte
rnat
iona
l Pro
gram
me
Com
mite
e. A
max
imum
of
thre
e ke
ywor
ds w
as a
ssig
ned
to e
ach
pape
r.
10
1. G
eom
etry
and
spa
tial t
hink
ing
2. C
ompi
lers
and
mat
hem
atic
's le
arni
ng
3. A
lgeb
raic
thin
king
4. F
unct
ions
5. A
dvan
ced
mat
hem
atic
al th
inki
ng
6. F
ract
ions
, dec
imal
s, r
atio
nal n
umbe
rs, p
ropo
rtio
nal r
easo
ning
7. im
ager
y an
d vi
sual
izat
ion
8. E
ady
mat
hem
atic
s le
arni
ng
9. P
roof
10. P
robl
em s
olvi
ng
11i
'
11. P
upils
' con
cept
ions
, bel
iefs
,...
12. T
each
ers'
con
cept
ions
, bel
iefs
,...
13. S
ocia
l and
aff
ectiv
e fa
ctor
s, m
etac
ogni
tion
14. S
ocia
l con
stru
ctio
n of
mat
hem
atic
s kn
owle
dge
and
lingu
istic
s
15. O
ut-
of.s
choo
l mat
hem
atic
s, th
e ro
le o
f co
ntex
t,...
16. A
sses
smen
t, ev
alua
tion
17. T
heor
etic
al a
nd e
pist
emol
ogic
al is
sues
18. C
urri
culu
m m
ater
ials
19. T
each
ers
educ
atio
n
Thi
s cl
assi
fica
tion
and
the
rela
ted
attr
ibut
ion
of a
pap
er to
a g
iven
are
a of
inte
rest
has
to b
e
cons
ider
ed s
ubje
ctiv
e an
d, c
onse
quen
tly, a
s it
happ
ens
to a
ll cl
assi
fica
tions
, may
be
disc
usse
d
and
mod
ifie
d. N
ever
thel
ess,
it s
eem
s us
eful
to o
rgan
ize
the
repo
rted
res
earc
h ac
tiviti
es a
nd to
help
par
ticip
ants
whe
n se
lect
ing
sess
ions
to a
ttend
. Of
cour
se, t
he r
eadi
ng o
f th
e ab
stra
cts
offe
rs m
ost p
erso
nal i
nfor
mat
ion
at th
is r
egar
d.
Thr
ough
this
cla
ssif
icat
ion,
we
have
an
appr
oxim
ate
idea
of
the
curr
ent t
rend
s of
the
rese
arch
in th
e do
mai
n of
PM
E. T
he d
ata
we
colle
cted
con
firm
som
e as
pect
s st
ill o
bser
ved
in
the
prev
ious
edi
tion;
in p
artic
ular
, we
see
that
the
them
es w
ith th
e hi
gher
per
cent
ages
are
(in
decr
easi
ng o
rder
) pr
oble
m s
olvi
ng;
com
pute
rs a
nd m
athe
mat
ics
lear
ning
; pup
ils'
conc
eptio
ns...
; ear
ly m
athe
mat
ics
lear
ning
; soc
ial c
onst
ruct
ion
of m
athe
mat
ics
know
ledg
e an
d
lingu
istic
s; g
eom
etry
and
spa
tial t
hink
ing;
theo
retic
al a
nd e
pist
emol
ogic
al is
sues
; tea
cher
s'
conc
eptio
ns...
.; te
ache
rs' e
duca
tion;
alg
ebra
ic th
inki
ng; f
ract
ions
, dec
imal
s, r
atio
nal n
umbe
rsan
d pm
port
inal
res
onin
g; I
mag
ery
and
visu
aliz
atio
n; s
ocia
l and
aff
ectiv
e fa
ctor
s,m
etac
ogni
don.
The
obs
erva
tions
abo
ut th
e tr
ends
of
curr
ent r
esea
rch
are
conf
irm
ed b
y th
e th
emes
of
the
wor
king
gro
ups
and
of th
e di
scus
sion
gro
ups.
Mor
e th
an in
the
past
the
gene
ral i
nter
est o
f th
e di
ffer
ent a
ctiv
ities
sch
edul
ed in
the
conf
eren
ce s
eem
s fo
cuse
d no
t onl
y on
stu
dent
s' b
ehav
iour
, but
on
the
figu
re o
f th
e te
ache
r an
d
on th
e co
ntex
t and
soc
ial f
acto
rs in
terv
enin
g in
the
teac
hing
lear
ning
pro
cess
es.
-
iv
A n
ote
on th
e re
view
peo
cass
The
hog
szum
e C
onta
tinee
rea
tiver
Iato
ted
of 1
8 I
stse
arde
prop
osek
that
enco
mpa
ssed
a w
ide
vari
ety
of th
eore
tical
and
em
piri
c4ap
prom
ites.
Cka
tty a
n al
to*
obse
rved
for
pre
ceed
ing
rite
etin
gs th
e pr
oces
sor
revi
ewta
g su
ch a
div
anco
llect
ion
of p
apal
; can
not b
e re
duce
d to
an
algo
rith
mic
pro
cedu
re. I
t is
a pr
oces
s th
at c
opes
with
nov
elty
and
dive
rsity
by
rely
ing
on
situ
ated
wis
dom
and
Jud
gem
ent.
Non
ethe
less
, a f
ew g
ener
al r
emar
ks c
anbe
mad
e.
Eac
h pr
opos
al w
as s
ent t
o th
ree
colle
ague
s fo
r re
view
with
the
requ
est
that
com
men
ts b
e
prov
ided
and
that
thes
e w
ould
be
forw
arde
d to
the
autb
x(s)
.The
revi
ew c
ateg
orie
s w
ere:
k de
fini
tivel
y ac
cept
B: a
ccep
t with
res
erva
tions
C: a
ccep
t as
post
er
D: n
iece
The
com
men
ts w
ere
forw
arde
d to
the
auth
ors
alon
g w
ith th
e Pr
ogra
mm
eC
omm
ittee
's d
ecis
ion.
Lis
t of
the
PNII
I-X
V r
evie
wer
s:A
dds
Jam
e, F
ranc
eA
rtig
' ue
Mic
hele
, Fra
nce
Bal
ache
ff N
icol
as ,
Fran
ce
Bat
oody
Art
hur,
USA
Bar
ra M
ark)
, It
alia
Beh
r M
erly
n, U
SAB
ell A
ka, U
.K
Ber
geso
n Ja
cque
s, C
anad
a
Bis
hop
Ala
n; U
KB
ooke
r G
eorg
e, A
ustr
alia
Car
raha
Dav
id, B
razi
lC
obb
Paul
, USA
Coo
per
Mar
tin, A
ustr
alia
Dav
enpo
rt L
inda
, USA
Doe
rfle
r W
illib
ald,
Aus
tria
Dou
ady
Reg
ina,
Fra
nce
Dre
yfus
Tom
my,
Isr
ael
Eis
enbe
rg T
ed, I
srae
lD
avis
Bob
, "U
SA
Fasa
no M
argh
erita
, har
tsFe
rnan
dez
Dom
ingo
, Por
tuga
l
12
Ferr
ari M
ario
, Ita
lia;
Fillo
y Y
ague
Eug
enio
, Mex
ico
Furi
nghe
tti F
ulvi
a, I
talia
Gau
lin C
laud
e, C
anad
a
Gol
din
Ger
ald,
USA
Gut
ierr
ez A
ngel
, Spa
inH
anna
Gila
, Can
ada
Har
t Kat
h, U
KH
art L
ynn,
USA
Hcr
scov
ics
Nic
olas
, Can
ada
Hie
bert
Jam
es, U
SA
Hitt
Fer
nand
o, M
exic
o
Hoy
les
Cel
ia,
U.K
.
Jaw
orsk
i Bar
bara
, UK
Jens
en U
lla, D
enm
ark
Kap
lan
Roc
helle
, USA
Kap
ut J
ames
, USA
Kie
ran
Car
olyn
, Can
ada
Klu
mm
eheu
r G
otz,
Ger
man
y
Lab
orde
Col
ette
, Fra
nce
Led
er G
ilah,
Aus
tral
ia
Lei
tao
Rod
rigu
ez A
na M
aria
, Por
tuga
l
Les
h D
ick,
USA
Les
ter
Fran
k , U
SA
Lin
Fou
Lai
, Tai
wan
Mah
er C
arol
yn, U
SA
Mal
aya
Nic
olin
a, I
talia
Mat
os J
oao,
Por
tuga
l
Mat
os J
ose,
Por
tuga
l
Mei
ssne
r H
artw
ig, G
erm
any
Men
dicu
ti T
eres
a, M
exic
o
Noh
da N
obuh
iko,
Jap
anN
oss
Ric
hard
, UK
Nun
es T
erez
inha
, Bra
zil
Pim
m D
avid
, UK
Piri
e Su
san,
UK
Pont
e Jo
ao P
edro
, Por
tuga
l 13
-
Pull;
Lul
l, Sp
ain
Roj
ano
Ter
esa,
Mex
ico
Schl
iesn
ann
Ana
lucl
a D
in, B
razi
l
Sim
on M
artin
, U
SA
Sper
anza
Fra
nces
co, I
taly
Sidl
e L
eslie
, USA
Swee
tland
Lke
n, N
ethe
rlan
ds
Tir
osh
Din
a, I
srae
l
Ver
gnan
d G
erar
d, F
ranc
e
Ver
bage
Hel
en, N
ethe
rlan
ds
dinn
er S
hlom
o, I
srae
l
Woo
d T
erry
, USA
14
vivi
i
INT
ER
NA
TIO
NA
L G
RO
UP
FOR
TH
E P
SYC
HO
LO
GY
OF
MA
TH
EM
AT
ICS.
ED
UC
AT
ION
TH
E P
RE
SEN
T O
FFIC
ER
S O
F T
HE
GR
OU
P A
RE
:-
Pres
iden
tK
ath
Har
t (U
.K.)
- V
ice-
Pres
iden
tsG
ilah
Led
er (
Aus
tral
ia),
Fou
Lai
Lin
(T
aiw
an)
Secr
etar
yM
artin
Coo
per
(Aus
tral
ia)
- T
reas
urer
Fran
k L
este
r (U
.S.A
.)
OT
HE
R M
EM
BE
RS
OF
TH
E I
NT
ER
NA
TIO
NA
L C
ON
LM
ITT
E A
RE
:
Mic
hele
Art
igue
(Fr
ance
)-
Bar
bara
Jaw
orsk
i (U
.K.)
- Pa
ul C
obb
(U.S
.A.)
- C
olet
te L
abor
de (
Fran
ce)
- T
heod
ore
Eis
enbe
rg (
Isra
el)
- N
obuh
iko
Noh
da (
Japa
n)
- C
laud
e G
aulin
(C
anad
a)-
Dav
id P
imm
(U
.K.)
- A
ngel
Gut
ierr
ez (
Spai
n)-
Joao
Pon
te (
Port
ugal
)
- G
ila H
anna
(C
anad
a)-
Din
a T
iros
h (I
srae
l)
PME
-XV
PR
OG
RA
MM
E C
OM
MIT
TE
E:
- Fe
rdin
ando
Arz
arel
lo (
Ital
y)
- Pa
olo
Boe
ro (
Ital
y)-
Elis
a G
allo
(It
aly)
- G
ila H
anna
(C
anad
a)
- C
olet
te L
abor
de (
Fran
ce)
- T
eres
a N
. de
Men
dicu
ti (M
exic
o)
- M
iche
le P
elle
rey
(Ita
ly)
- D
avid
Pim
m (
U.K
.)
- Jo
ao P
onte
(Po
rtug
al)
- D
ina
Tir
osh
(Isr
ael)
- V
inic
io V
illan
i (It
aly)
PME
-XV
LO
CA
L O
RG
AN
IZIN
G C
OM
MIT
TE
E:
- Pa
olo
Boe
ro (
Gen
oa)
- Fr
ance
sca
Con
ti (P
erug
ia)
- Fu
lvia
Fur
ingh
etti
(Gen
oa)
15
-
viii
ix
HIS
TO
RY
AN
D A
IMS
OF
TH
E P
.M.E
. GR
OU
PR
ESE
AR
CH
RE
POR
T C
LA
SSIF
ICA
TIO
N
At t
he T
hird
Int
erna
tiona
l Con
gres
s on
Mat
hem
atic
al E
duca
tion
(IC
ME
3, K
arls
ruhe
, 197
6) P
rofe
ssor
E.
Fish
bein
of
the
Tel
Avi
v U
nive
rsity
,Isr
ael,
inst
itute
d a
stud
ying
gro
up b
ring
ing
toge
ther
peo
ple
wor
king
in th
e ar
ea o
f th
e ps
ycho
logy
of
mat
hem
atic
s ed
ucat
ion.
PM
E is
aff
iliat
ed w
ith th
e In
tern
atio
nal
Com
mis
sion
for
Mat
hem
atic
al I
nsttu
ctio
n (I
CM
I). I
ts p
ast p
resi
dent
s ha
ve b
een
Prof
. Efr
aim
Fis
hbei
n ,
Prof
. Ric
hard
R. S
kem
p of
the
Uni
vers
ity o
f W
arw
ick,
Dr.
Ger
ard
Ver
gnau
d of
the
Cen
tre
Nat
iona
l de
la R
eche
rche
Sci
entif
ique
(C
.N.R
.S.)
in P
aris
, Pro
f. K
evin
F. C
ollis
of
the
Uni
vers
ity o
f T
asm
ania
,
Prof
. Pav
ia N
eshe
r of
the
Uni
vers
ity o
f H
aifa
, Dr.
Nic
olas
Bal
ache
ff, C
.N.R
.S. -
Lyo
n.
The
maj
or g
oals
of
the
Gro
up a
re:
- T
o pr
omot
e in
tern
atio
nal c
onta
cts
and
the
exch
ange
of
scie
ntif
icin
form
atio
n in
the
psyc
holo
gy o
f
mat
hem
atic
s ed
ucat
ion;
- T
o pr
omot
e an
d st
imul
ate
inte
rdis
cipl
inar
y re
sear
ch in
the
afor
esai
d ar
ea w
ith th
e co
oper
atio
n of
psyc
holo
gist
s, m
athe
mat
icia
ns a
nd m
athe
mat
ics
teac
hers
;
- T
o fu
rthe
r a
deep
er a
nd b
ette
r un
ders
tand
ing
of th
e ps
ycho
logi
cal a
spec
ts o
fte
achi
ng a
nd le
arni
ng
mat
hem
atic
and
the
impl
icat
ions
ther
eof.
Mem
bers
hip
Mem
bers
hip
is o
pen
to p
eopl
e in
volv
ed in
act
ive
rese
arch
con
sist
ent w
ith th
e G
roup
's a
ims,
or
prof
essi
onal
ly in
tere
sted
in th
e re
sults
of
such
res
earc
h.
Mem
bers
hip
is o
pen
on a
n an
nual
bas
is a
nd d
epen
ds o
n pa
ymen
t of
the
subs
crip
tion
for
the
curr
ent y
ear
(Jan
uary
to D
ecem
ber)
.
The
sub
scri
ptio
n ca
n be
pai
d to
geth
er w
ith th
e co
nfer
ence
fee
.
I. G
eom
etry
and
spa
tial t
hink
ing
M.A
rtig
ue, J
. Bel
loc,
G.K
argi
otak
is;
D.A
.Cle
men
ts, M
.T.B
attis
ta:
M. D
e V
illie
rs;
L.D
. Edw
ards
:
A. G
utie
rrez
, A. J
aim
e, J
.M. S
haug
hnes
sy, W
.F. B
urge
r:
K. K
rain
er;
C. K
ynig
os;
F. L
opez
- R
eal ;
S. M
agin
a, C
.Hoy
les;
M.A
.Mar
iotti
;
A.L
.Mes
quita
:
R. N
unez
Err
azur
iz;
I.O
sta;
F.Pl
uvin
age,
J.C
. Rau
sche
r, C
. Dup
uis;
R. S
tras
ser,
B. C
appo
ni;
G.H
. Whe
atle
y, A
.Rey
nold
s;
0.Z
asla
ysky
2. C
ompu
ters
and
mat
hem
atic
al le
arni
ngM
.Art
igue
, J. B
ello
c, G
.Kar
giot
akis
;
N. B
lack
ett,
D. T
all ;
G.C
hiap
pini
, E.L
emut
, L.P
aren
ti;
L.D
. Edw
ards
;
N.H
ativ
a, O
.Pom
eran
z, S
. Her
shko
vitz
, I. M
echm
anda
rov;
L. H
ealy
, S. P
ozzi
, C.H
oyle
s;
R. H
ersh
kovi
tz, A
. Fri
edla
nder
, T. D
reyf
us;
C.H
oyle
s, R
.Sut
herl
and,
R. N
oss;
JJ. K
aput
, C. H
anco
ck;
G. K
rum
mhe
uer;
C. K
ynig
os;
T. L
emer
ise,
B. C
ote;
R. L
esh,
A.E
.Kel
ly;
N.A
.Mal
ara,
R.G
arut
i;
R. N
emir
ovsk
y, A
. Rub
in;
1617
-
L N
evi l
e;
R. N
oss,
C.H
oyle
s;
1.()
sta;
M A
egea
n' ;
f .1
Roj
ano,
R.S
uthe
rlan
d;
B.S
chw
arz,
T.D
reyr
us;
I. S
teve
nson
, R. N
oss;
R. S
tras
ser,
B. C
appo
ni;
M. Y
erus
halm
y, R
. Gaf
ni;
E. W
enze
lbur
ger
Gut
tenb
erge
r ,
3. A
lgeb
raic
thin
king
F. A
rzar
ello
;
G. c
hlap
pini
, E. L
emut
;
E.F
illoy
-Vag
ue ;
N. H
ersc
ovic
s, L
.Lin
chev
ski;
C.K
ienm
;
L.L
inch
evsk
i, A
. Sfa
rd;
C.M
argo
linas
;
T.C
. O'B
rien
;I.
Pele
d;
T.R
ojan
o, R
.Sut
heri
and;
D.T
iros
h, R
. Had
ass,
N. M
ovsh
ovitz
-Had
ar;
S. W
ain'
:
M.A
. Wol
ters
;
4. F
unct
ions
Md
Nor
Bak
er, D
. Tal
l;
R. E
ven,
Z. M
arko
vitz
;
G.A
.Gol
din,
N. H
ersc
ovic
s;
G. H
arel
, E. D
ubin
sky;
A. v
an S
trom
;
E. W
enze
lbur
ger
Gut
tenb
erge
r;
5. A
dvan
ced
mat
hem
atic
al th
inki
ngM
Am
it, N
. Mov
shov
itz-H
adar
;
C.A
zcsr
ate;
18
xi
Edw
atst
s;
G.A
.Gol
din,
N. H
ersc
ovic
s;
J.J.
Kap
ut, C
. Han
cock
;F.
L.L
in;
R. N
emir
ovsk
y, A
. Rub
in;
G. S
ham
a, T
. Dre
yfus
;
6. F
ract
ions
, dec
imal
s, r
atio
nal n
umbe
rs, p
ropo
rtio
nal r
easo
ning
M.B
ehr,
G.H
arel
, T.P
ost,
R.L
esh;
D.W
. Car
rahe
r, A
.Dia
s Sc
hlie
man
n;
R.B
. Dav
is, A
. Als
ton,
C. M
aher
:
L.G
rugn
etti,
C. M
ured
du T
orre
s;
G. H
ard,
M.B
ehr,
T.P
ost,
R. L
esh;
C.H
oyle
s, R
.Sut
herl
and,
R. N
oss;
T. L
emer
ise,
B. C
ote;
F.L
Lin
;S.
Llin
ares
Cis
car,
M. V
. San
chez
Gar
cia;
D.N
eum
an;
S. O
hlss
on, N
.Bee
;A
.Sae
nz-L
udlo
w;
7. I
mag
ery
and
visu
aliz
atio
nA
. Ant
onie
tti,C
.Ang
elin
i ;
Md
Nor
Bak
ar, D
. Tal
l;
B.L
. Bod
ner,
G.A
. Gol
din;
R. H
ersh
kovi
tz, A
. Fri
edla
nder
, T. D
reyf
us;
F.L
opez
-Rea
l
M.A
. Mar
iotti
;
T.C
. O'B
rien
;
N.C
. Pre
smeg
;
G. S
ham
a, T
. Dre
yfus
;
M. Y
erus
halm
y, R
. Gaf
ni;
8. E
arly
mat
hem
atic
al le
arni
tigD
.W.T
arra
her,
A.D
ias
Schl
iem
ann;
E.C
astr
o M
artin
ez, L
.Ric
o R
omer
o, C
.Bat
aner
o B
erns
heu,
E. C
astr
o M
artin
ez;
F. C
onne
, J. B
run;
19
-
xii
K. H
asem
ann;
N. H
ersc
ovic
s, L
.Lin
chev
ski;
K. I
to;
G.C
. Led
er;
C.A
.Mah
er, A
.M. M
artin
o;
T. N
unes
, P. L
ight
, J. M
ason
;
LM
orga
do;
H.M
urra
y, A
. Oliv
ier,
P. H
uman
;
D. N
eum
an;
T.C
. O'B
rien
;
M. O
rozc
o H
orm
aza;
I.Pe
led;
LPo
irie
r, N
.Bed
narz
;
A.S
aerr
z-L
udlo
w;
G.B
. Sax
e;
G.H
. Whe
atle
y, A
.Rey
nold
s;
M. A
. Wol
ters
;
9. P
roof
M. D
e V
illie
rs;
F. F
urin
gbet
ti,D
.Pao
la;
M.M
iyaz
ald;
N.M
ovsh
ovitz
-Had
ar;
10. P
robl
em s
olvi
ngA
.J.B
isho
p,G
.de
Abr
eu ;
M.A
mit,
N. M
ovsh
ovitz
-Had
ar;
A. A
nton
ietti
,C.A
ngel
ini;
N.B
edna
tz, B
. Jan
vier
;
B.L
. Bod
ner,
G.A
. Gol
din;
M.G
.Bon
desa
n, P
.L.F
erra
ri;
E.C
astr
o M
artin
ez, L
.Ric
o R
omer
o, C
.Bat
aner
o B
erna
beu,
E. C
astr
o M
artin
ez;
B.G
reer
, M. M
cCan
n;
U. G
revs
muh
l;
LG
rugn
etti,
C. M
iare
ddu
Tor
res;
J.L
. Gur
tner
,B.V
itale
;
N.H
adva
, 0.P
omer
anzi
S. H
ersh
kovi
te, I
. Mee
hman
daro
v;
G.C
. Led
er;
P. N
eshe
r, S
. Her
skov
iu; G
. Kru
mm
heue
r;
N.A
.Mal
ara,
R.G
arut
i;
LM
orga
do;
H.M
urra
y, A
. Oliv
ier,
P. H
uman
;
P.N
eshe
r, S
. Her
shko
vitz
;
M. O
rozc
o H
orm
aza;
L.P
oiri
er, N
.Bed
narz
;
M.R
eggi
ani;
T.R
ojan
o, R
.Sut
heda
nd;
K. S
chul
tz;
K.S
tace
y;
R. S
tavy
, D. T
iros
h;
A. v
an S
treu
n;
T. W
ood;
11. P
upils
' con
cept
ions
, bel
iefs
,...
D.W
. Car
rahe
r, A
.Dia
s Sc
hlie
man
n;
F. C
onne
, J. B
run;
R.B
. Dav
is, A
. Als
ton,
C. M
aher
;
M. D
e V
illie
rs;
F. F
urin
ghet
ti, D
. Pao
la;
B.G
reer
,M.M
c C
ann;
L.G
rugn
etti,
C. M
ured
dn T
orre
s;
J.L
. Gur
tner
,B.V
itak;
K. H
asem
ann;
K. I
to;
K. K
lein
er;
E.L
ove,
C. S
hiu;
S. M
agin
a, C
.Hoy
les;
C.A
.Mah
er, A
.M. M
artin
o;
T. N
unes
, P. L
ight
, J. M
ason
;
R. N
emir
ovsk
y, A
. Rub
in;
D. N
eum
an;
M.J
. Per
rin-
Glo
rian
;
T.L
Schr
oede
r;
K.S
tace
y;
2021
-
.x iv
O. Z
asla
ysky
12. T
each
ers'
con
cept
ions
, bel
iefs
,...
A.J
.Bis
hop,
G.P
ompe
u;
A.B
oufi
, S.K
afou
ssi;
G.C
hiap
pini
, E.L
imut
, LPa
rent
i;
R. E
ven,
Z. M
arko
vitz
;
G. H
ard,
E. D
ubin
sky;
B.J
awor
ski;
M. J
urda
k;
R. L
esh,
A.E
.Kel
ly;
S. L
linar
es C
isca
r, M
. V. S
anch
ez G
arci
a;
C.M
orei
ra;
J.A
. Mou
sley
;R
ausc
her,
C. D
upui
s;
N.C
.Pre
smei
;
T.L
Schr
o2de
r;
MA
. Sim
on;
D.T
iros
h, R
. Had
ass,
N. M
ovsh
ovitz
-Had
ar;
13. S
ocia
l and
aff
ectiv
e fa
ctor
s, m
etae
ogni
tion
B. A
twel
l, T
. Coo
per
;
N.B
lack
ett,D
.Tal
l;
T. E
isen
berg
;
J.T
.Eva
ns;
U. G
revs
muh
l;
LC
. Har
t;C
.K
anes
:
S. L
erm
an, R
. Sco
tt-H
odge
tts;
M.J
. Per
rin-
Glo
rian
;
D.P
imm
;
14. S
ocia
l con
stru
ctio
n of
mat
hem
atic
al k
now
ledg
e an
d lin
guis
tics
M.G
.Bon
desa
n, P
.L.F
erra
ri;
P.C
obb;
N.F
. Elle
rton
;
U. G
revs
muh
l;
X-V
C. C
lioce
a,E
. Jes
se, A
._6t
obet
r4
L. M
aly,
S. P
ozzi
, C.H
oyle
a;
C. K
anes
;
G. K
rum
mhe
uer;
L. L
inde
skov
;
F. L
opez
-Rea
l;
C.A
.Mah
er, A
.M. M
artin
o;
C.F
. New
man
,
D.P
imm
;
S. P
ixie
;
S.Pi
rie,
T. K
iere
n;
H. S
akon
idis
,J.B
liss;
E. Y
acke
l;
1. W
ood;
15. O
ut-o
f-sc
hool
mat
hem
atic
s, th
e ro
le o
f co
ntex
t,...
A.J
.Bis
hop;
A.J
.Bis
hop,
G.P
ompe
u;
G. d
e A
breu
;
J.T
. Eva
ns;
L.L
inde
skov
;
G.B
. Sax
e;
A. v
an S
treu
n;
16. A
sses
smen
t, ev
alua
tion
D.A
.Cle
men
ts, M
.T.B
attis
ta;
A. G
utie
rrez
, A. J
aim
e, J
.M. S
haug
hnes
sy, W
.F. B
urge
r;
E. L
ove,
C.S
hiu;
J.A
. Mou
sley
;
F.Pl
uvin
age,
J.C
. Rau
sche
r, C
. Dup
uis;
B.S
chw
arz,
T.D
reyf
us;.
17. T
heor
etic
al a
nd e
pist
emol
ogic
al is
sues
M.A
rtig
ue, J
. Bel
loc,
G.K
argi
otak
is;
R.B
. Dav
is, A
. Als
ton,
C. M
aher
,
P. E
rnes
t;
G.A
.Gol
din,
N. H
ersc
ovic
s;
-
XV
I
E. G
ray,
D. T
all;
J.L
. Gur
tner
,B.V
itale
;
N. H
all;
B.J
awor
ski;
M.J
urda
k;L
Lin
chev
iki,
A. S
fard
;
L L
inde
skov
;N
.Mov
shov
itz-H
adar
;
C. M
argo
line
s;
C.F
. New
man
, S.
Pir
ie;
R. N
unez
Err
azur
iz;
S.Pi
rie,
T. K
iere
n;
K. S
chuh
z;
13. C
urri
culu
m m
ater
ials
K. S
tace
y;
19. T
each
ers
educ
atio
nC
. Bla
ncha
rd-L
avill
e;
A.B
oufl
, S.K
afou
ssi;
G.C
hiap
pini
, E.L
emut
, LPa
rend
;
C. C
hioc
ca.E
. Jos
se, A
. Rob
ert;
R. E
ven,
Z. M
arko
witz
:
G. H
arel
, E. D
ubin
sky:
LC
. Har
t;
S. L
erm
an, R
. Sco
tt-H
odge
tts;
R. L
ash,
A.E
.Kel
ly:
S. L
linar
es a
scar
, M. V
. San
chez
Gar
cia;
LN
evik
;R
. Nos
s, C
.Hoy
les;
M.A
.Sim
on;
D.T
iros
h, R
. Hat
less
. N. M
ovsh
ovIt
z-H
adar
;
M. Y
oush
alm
y, R
. Gaf
ni:
24
xvii
Wor
king
gro
ups
BISHOP A. J. -
Soda
! ps
ycho
logy
of
mat
hent
atic
s ed
ucat
ion
DAWSON A. J. (S); JAWORSKI B; WOOD T. - Psychology ofinservice-
education ofm
athe
mad
cs te
ache
rs -
a r
esea
rch
pers
pect
ive
ELLERTON N.
- R
esea
rch
onthe
psyc
holo
gyof
mat
hem
atic
s te
ache
rde
velo
pmen
t
ERVYNCK G.; TALL D.
- A
dvan
ced
mat
hem
atic
al th
inki
ng (
A.M
.T.)
GOIDIN G.
A. -
Rep
rese
ntat
ions
LE
RM
AN
S.;
SCO
TT
- H
OD
GE
TT
S R
. - T
each
ers
asre
sear
cher
s In
mat
hem
atic
s ed
ucat
ion
LIN
F. L
; HA
RTK. M.; BERGERON J. C.
- R
atio
and
pro
port
ion
MANSFIELD H. - Geometry
SUM
MA
ND
R. -
Alg
ebra
ic p
roce
sses
and
str
uctu
re
VAN DEN BRINK J.
- C
lass
mom
res
earc
h
25
p.xv
iii
p.xi
x
p.xx
p.xx
i
p.xx
ii
p.ra
dii
P.xx
iv
p.xx
v
p.xx
vi
p.xx
vii
-
xviii
Mah
n fi
lm I
n =
grid
Psy
chol
ogy
fabl
iau:
Wig
: Edu
catio
n"
All
mat
hem
atic
s le
arni
ng ta
kes
plac
e in
a s
ocia
l set
ting
and
part
icul
arly
with
in th
e PM
E
com
mun
ity, w
e ne
ed to
be
able
to in
terp
ret,
and
theo
rise
abo
ut, m
athe
mat
ics
lear
ning
.
kggr
pers
olut
Tly
as
wel
l as
jogp
erso
ruill
y.M
athe
mat
icsI
Min
ing
in it
s ed
ucat
iona
l con
text
..
..
..
cann
ot b
e fu
lly in
tesp
rete
d as
an
intia
pers
ixia
l phe
nom
enon
bec
ause
of
Me
soci
al C
onte
xt in
.
whi
ch it
occ
urs.
Equ
ally
, int
erpe
rson
al o
rsoc
iolo
gica
l con
stru
cts
wff
l be
Inad
equa
te a
lone
sin
ce
it is
alw
ays
the
indi
vidu
al le
as n
et w
ho m
ust m
ake
sens
e an
d m
eani
ng in
the
mat
hem
atic
s.'
The
refo
re, i
t is
vita
lly im
port
ant t
o re
sear
ch th
e w
ays
this
intr
kint
erpe
rson
al c
ompl
emen
tari
ty
infl
uenc
es th
e hi
nd o
f m
athe
mat
ical
kno
wle
dge
acqu
ired
by
pupi
ls in
cla
ssro
oms.
In
orde
r to
purs
ue th
is r
esea
rch
it is
ther
efor
e ne
cess
ary
to a
naly
se a
nd d
evel
op b
oth
theo
retic
al c
onst
ruct
s
and
met
hodo
logi
cal t
ools
.
Thi
s is
wha
t the
SPM
E w
orki
ng g
roup
has
con
cent
rate
d on
. At P
ME
10,
the
firs
t off
icia
l,
mee
ting
cif
the
grou
p, w
e tr
ied
out v
ario
us s
mal
l gro
up ta
sks
amon
gst o
usse
lves
and
dis
cuss
ed
thei
r va
lue
as r
esea
rch
'site
s' a
nd a
lso
teac
hing
situ
atio
ns. A
t PM
E 1
1, w
e m
oved
to o
ther
soci
al d
eter
min
ants
of
mat
hem
atic
s le
arni
ng p
artic
ular
ly th
inki
ng a
bout
Inf
luen
ces
of o
ther
pupi
ls a
nd o
f th
e te
ache
r A
t PM
E 1
2, w
e fo
cuss
ed o
n th
e id
ea o
f "b
ring
ing
soci
ety
into
die
clas
sroo
m"
and
the
issu
es o
f ju
stif
ying
res
earc
h w
hich
mig
ht c
onfl
ict w
ith w
hat "
soci
ety"
,
cons
ider
s ed
ucat
ion
shou
ld b
e do
ing
At P
ME
13,
we
wor
ked
on tw
o ar
eas,
fir
stly
the
way
s in
whi
ch th
e co
nstr
uct "
mat
hem
atic
s" is
soc
ially
med
iate
d in
the
clas
sroo
m, a
nd s
econ
dly,
the
use
of v
ideo
s of
cla
ssro
om in
tera
ctio
ns, a
nd th
eir
anal
yses
. At P
ME
14,
we
cons
ider
ed th
e si
tuat
ion
of b
i-cu
ltura
l lea
rner
s, th
e so
cial
set
ting
of th
e nu
rser
y-sc
hool
, and
the
lear
ning
val
ues
of
cocr
peta
tive
gem
At P
ME
15,
we
will
con
side
r th
e fo
llow
ing:
(1)
bi-c
ultu
ral l
earn
ers
- pa
rtic
ular
ly id
eas
floo
r th
e ev
iden
ce o
f G
olds
de
Abr
eu f
rom
Bra
zil;
(2)
the
rela
tions
hips
bet
wee
n th
e so
cial
con
text
s of
mat
hem
atic
s an
d th
e ch
ild d
evel
opm
ent
mod
el, l
ed b
y L
eo R
oger
s fr
om U
K;
(3)
soci
al is
sues
of
asse
ssm
ent,
let b
y L
ucia
na B
azzi
ni a
nd L
ucia
Gru
gnet
ti fr
om I
taly
, and
(4)
aspe
cts
of c
ultu
ral a
nd s
ocia
l 'di
ffer
ence
' whi
ch m
ay b
e of
sig
nifi
canc
e in
mat
hem
atic
s
lear
ning
26
Ala
n J
Bis
hop
is is
PM
E X
V W
orki
ng G
roup
Psy
chol
ogy
of in
serv
ice-
educ
atto
n of
Mat
hem
atic
sT
each
ers
- a
Res
earc
h P
ersp
ectiv
eSa
ndy
Daw
son
- an
us F
rase
r U
nive
rsity
, Van
couv
er, C
anad
a
Bar
bara
Jaw
cs-
Uni
vers
ity o
f 13
irm
ingh
ana,
UK
Ter
ry W
ood
- Pu
rdue
Uni
vers
ity, I
ndia
na, U
SA
Thi
s gr
oup
is c
once
rned
with
the
deve
lopm
ent o
f m
athe
mat
ics
teac
hing
. Its
wor
k so
far
may
be
sum
mar
ised
as
follo
ws:
1. D
iscu
tsbn
of
part
icul
ar in
serv
ice
proj
ects
with
mat
tem
atic
s le
ache
rs in
whi
ch P
ME
part
icip
ants
are
, or
have
bee
n, e
ngag
ed, l
eadi
ng to
iden
tific
atio
n of
que
stio
ns a
ndis
sues
bot
h of
inse
rvic
e its
elf,
or
of r
esea
rch
rela
ting
to it
2. A
ttem
ptin
g to
ref
ine
ques
tions
and
issu
es r
aise
d ha
(1)
with
reg
ard
to e
mer
gent
theo
retic
al f
ram
ewor
ks. C
onso
lidat
ing
grou
p un
ders
tand
ing
and
ratio
nalis
ing
haliv
Idua
l per
spec
tives
.-
3. L
ooki
ng to
the
futu
re. W
hat w
ait m
ay b
e un
dert
aken
at a
nd b
etw
eten
PM
Em
eetin
gs?
Mem
bers
dec
lare
thei
r M
aly
invo
lvan
eat w
ith im
am r
aise
d an
d ;p
ossi
bly
agre
e to
wri
te p
aper
s fo
r dr
adat
ion.
Fta
ibild
lea
incl
ude
assi
dera
tion
of r
esea
rch
met
hodo
logi
es a
nd p
oten
tial t
heor
eeca
l fra
mes
:a:b
.
Issu
es r
aise
d ha
ve in
clud
ed:
1E
xpec
tatio
ns -
pos
sibl
e m
ism
atch
es o
f ex
pect
atio
ns b
etw
een
lead
ers
and
part
icip
ants
in I
NSE
T e
vent
s or
cou
rses
. How
can
the
degr
ee o
f co
mpa
tibili
ty o
fex
pect
atio
ns o
f co
mae
par
ticip
ants
with
the
aim
s of
the
cour
se a
ffec
t wha
t occ
urs?
For
exam
ple,
how
mig
ht a
pro
vide
r's p
ositi
on, p
erha
ps in
tam
s of
not
giv
ing
'ans
vies
s', e
ncou
ragi
ng s
eff
deve
lopm
ent a
nd th
e 'a
ctiv
e' le
arne
r, b
e m
arri
ed to
som
e pa
rtic
ipan
ts' p
ositi
ons,
per
haps
of
wan
ting
answ
era
lei,
me
wha
t to
de, t
hepa
ssiv
e le
arne
r, th
e pr
ovid
er a
s 'e
xper
t? H
ow d
o pa
rtic
ipan
ts p
erce
ptio
ns o
fm
athe
mat
ics
and
its te
achi
ng in
flue
nce
expe
ctat
ions
?
2 T
he te
ache
r-ed
ucat
or's
dile
mm
a -
flex
ibili
ty v
ersu
s co
ntro
l. H
ow o
pen
is it
poss
ible
to le
ave
activ
ities
, whi
le s
till h
opin
g to
ach
ieve
obj
ectiv
es. I
f ob
ject
ives
are
dire
ctly
sou
ght,
can
they
be
achi
eved
?Is
it p
ossi
be to
sha
re o
bjec
tives
with
part
icip
ants
? C
an p
artp
ants
und
erst
and
a le
ader
's o
bjec
tives
bef
ore
havi
ng g
aine
dth
e aw
aren
ess
or e
xper
ienc
e w
hich
the
cour
se s
ets
out t
o en
gend
er?
3Im
plem
enta
tion
- th
e da
nger
that
'not
hing
hap
pens
' aft
er r
etur
n to
mur
al w
ork.
How
mig
ht p
artic
ipan
ts b
e su
ppor
ted
in th
e im
plem
entin
g of
idea
s an
d aw
aren
ess
deve
lope
d as
par
t of
the
cour
se?
Whi
ch c
anes
fir
st -
enta
tion
orch
ange
? D
o te
ache
rs n
eed
to s
tart
to m
odif
y th
eir
1-.
base
.M
iloc6
reP
hica
ltheyc
an m
ake
chan
ges
to th
eir
prac
tice?
Or
can
star
ting
toch
ange
str
igge
r aw
aren
ess,
and
thus
pro
atot
e a
chan
ge o
f ph
ikec
sphy
? O
r do
the
two
thin
gsoc
cur
sim
ulta
neou
sly?
How
mig
ht a
nsw
ers
to th
ese
ques
tions
aff
ect t
he w
ayco
urse
s m
e st
ruct
ured
?
4. W
hat m
etho
dolo
gies
- p
roce
sses
, atm
:Wm
tech
nic/
pea
of I
NSE
T d
o w
eem
Plor
(For
mor
e in
form
atio
n pl
ease
con
tact
Bar
bara
Jaw
orsk
i: Sc
hool
cif
Edu
catio
n, U
nive
rsity
of
Bir
min
gham
, Bir
min
gham
B15
2T
T, U
K)
LE27
-
RE
SEA
RC
H O
N T
HE
PSY
CH
OL
OG
Y O
F M
AT
HE
MA
TIC
ST
EA
CH
ER
DE
VE
LO
PME
NT
he W
orki
ng G
roup
Res
earc
h on
the
Psyc
holo
gy o
f M
athe
mat
ics
Tea
cher
Dev
elop
men
t was
fir
st
3nve
ned
as a
Dis
cuss
ion
Gro
up a
t PM
E X
in L
ondo
n in
198
6, a
nd c
ontin
ued
in th
is f
orm
at u
ntil
le W
orki
ng G
roup
was
for
med
in 1
990
at P
ME
XIV
in M
exic
o. T
his
year
, at P
ME
XV
, we
have
le o
ppor
tuni
ty to
bui
ld o
n th
e fo
unda
tion
of s
hare
d un
ders
tand
ings
that
hav
e. b
eene
stab
lishe
d.
.bus
of
the
Wor
king
Gro
upT
he d
evel
opm
ent,
com
mun
icat
ion,
and
exam
inat
ion
of p
arad
igm
s an
d fr
amew
orks
for
res
earc
h in
it ps
ycho
logy
of
mat
hem
atic
s te
ache
r de
velo
pmen
t
The
col
lect
ion,
dev
elop
men
t, di
scus
sion
and
cri
tiqui
ng o
f to
ols
and
met
hodo
logi
es f
or3n
duct
ing
natu
ralis
tic a
nd in
terv
entio
n re
sear
ch s
tudi
es o
n th
e de
velo
pmen
t of
mat
hein
atic
s
ache
rs' k
now
ledg
e, b
elie
fs, a
ctio
ns a
nd th
inki
ng
The
impl
emen
tatio
n of
col
labo
rativ
e re
sear
ch p
roje
cts
The
fos
teri
ng a
nd d
evel
opm
ent o
f co
mm
unic
atio
n be
twee
n pa
rtic
ipan
ts
The
pro
duct
ion
of a
join
t pub
licat
ion
on r
esea
rch
fram
ewor
ks a
nd m
etho
dolo
gica
l iss
ues
with
in
cis
rese
arch
dom
ain.
: ese
arch
' Que
stio
nsbe
fol
low
ing
are
exam
ples
of
the
type
s of
res
earc
h qu
estio
ns th
at w
ill b
e ad
dres
sed
at P
ME
XV
:
Wha
t are
the
impl
icat
ions
of
ethn
omat
hem
atic
s/ev
eryd
ay c
ogni
tion
rese
arch
for
pre
serv
ice
teac
her
educ
atio
n an
d fo
r pr
ofes
sion
al d
evel
opm
ent p
rogr
ams
for
teac
hers
?
To
wha
t ext
ent d
o m
athe
mat
ics
educ
ator
s ap
ply
cons
truc
tivis
t ten
ets
in th
eir
ever
yday
*pra
ctic
e of
teac
her
educ
atio
n? I
s th
ere
a m
ism
atch
bet
wee
n th
eir
idea
ls a
nd th
eir
prac
tice?
Wha
t sho
uld
be d
one
to h
elp
youn
g pr
e-se
rvic
e st
uden
ts w
ho la
ck c
onfi
denc
e in
thei
r
mat
hem
atic
al a
bilit
y an
d in
thei
r ab
ility
to te
ach
mat
hem
atic
s ?
Why
is it
that
mos
t you
ng m
athe
mat
ics
teac
hers
bec
ome
prog
ress
ivel
y m
ore
cons
erva
tive
as th
ey g
ain
clas
sroo
m e
xper
ienc
e?
'ropo
aed
Out
com
es o
f th
e W
orki
ng G
roup
at P
ME
XV
. Col
labo
rativ
e R
esea
rch
Proj
ects
:M
embe
rs o
f th
e W
orki
ng G
roup
hav
e ov
erla
ppin
g
esea
rch
inte
rest
s, a
nd it
is h
oped
that
col
labo
rativ
e re
sear
ch p
roje
cts
can
be m
ount
ed. T
his
wou
ld
tave
the
bene
fit n
ot o
nly
of e
nric
hing
the
data
ava
ilabl
e, p
artic
ular
ly w
here
suc
h co
llabo
rativ
e
esea
rch
coul
d be
car
ried
out
in d
iffe
rent
cou
ntri
es, b
ut a
lso
of a
cces
sing
a w
ider
exp
erie
nce
base
or th
e de
velo
pmen
t of
suita
ble
inst
rum
ents
and
for
sub
sequ
ent a
naly
sis.
!.Pu
blic
atio
n of
an
Edi
ted
Col
lect
ion
of R
esea
rch
Pape
rs: I
t is
prop
osed
that
the
Wor
king
Gro
up p
ublis
h a
colle
ctio
n of
res
earc
h pa
pers
that
add
ress
a s
ingl
e th
eme
(one
of
the
tbov
e re
sear
ch q
uest
ions
, for
exa
mpl
e).
Ner
ida
Elle
rton
, Con
veno
r)
IS
xxi
WO
RK
ING
GR
OU
P on
AD
VA
NC
ED
MA
TH
EM
AT
ICA
L T
HIN
KIN
G (
A.M
.T.)
Org
anis
ers:
Gon
trtu
i Erv
ynek
, Dav
id T
all
SESS
ION
Ir
TH
E R
EL
AT
ION
SHIP
BE
TW
EE
N P
RO
CE
SS, C
ON
CE
PT A
ND
DE
FIN
ITIO
N. I
nitia
tor:
Dav
id T
all (
U.K
.).
We
have
not
ed th
e w
ay in
whi
ch s
ome
stud
ents
clin
g to
pro
cess
es (
e.g.
cou
ntin
g in
ari
thm
etic
)
whi
ch d
o no
t bec
ome
enca
psul
ated
as
conc
epts
whi
ch c
an th
en b
e m
anip
ulat
ed a
t a h
ighe
r
leve
l. W
e ha
ve f
ound
that
the
sym
bolis
m is
ver
y im
port
ant b
ecau
se th
roug
hout
mat
hem
atic
s
ther
e ar
e in
stan
ces
of th
e sa
me
nota
tion
bein
g us
ed f
or p
roce
ss a
nd o
utpu
t of
the
proc
ess:
e.g
.
1/4;
1(4
; Km
(o,)
; etc
. The
mat
hem
atic
ian
uses
the
sam
e sy
mbo
lism
to r
epre
sent
bot
h pr
oces
s
and
outp
ut c
once
pt. I
t is
this
am
bigu
ity a
nd f
lexi
bilit
y th
at m
akes
mat
hem
atic
al th
inki
ng s
o
pow
erfu
l. W
e ha
ve c
oine
d th
e te
rm 'p
roce
pe to
be
the
com
billa
tion
of p
roce
ss a
nd c
once
pt w
hich
are
repr
esen
ted
by th
e sa
me
sym
bolis
m. W
e ar
e at
pre
sent
wor
king
on
the
cons
eque
nces
at a
ll
leve
ls o
f m
athe
mat
ical
dev
elop
men
t, in
clud
ing
AM
T.
SESS
ION
Ifs
TH
E T
RA
ININ
G O
F R
ESE
AR
CH
ER
S IN
MA
TH
EM
AT
ICS
ED
U-
CA
TIO
N. I
nitia
tone
Gon
tzan
Erv
ynek
(B
elgi
um),
Ted
Eis
enbe
rg (
Isra
el).
We
are
conc
erne
d ab
out t
he tr
aini
ng o
f re
sear
cher
s in
mat
hem
atic
s ed
ucat
ion
vers
us d
octo
ral p
ro-
gram
s. W
hat s
houl
d w
e re
quir
e, n
ot o
nly
in th
e w
ay o
f re
sear
ch b
ut a
lso
in th
e w
ay o
f co
urse
wor
k ?
The
wor
ld o
f m
athe
mat
ics
is c
hang
ing,
but
man
y re
cent
gra
duat
es d
on't
seem
to b
e aw
are
of th
ese
chan
ges.
A s
hift
in th
e un
ders
tand
ing
of th
e co
ncep
ts (
crea
tion
of n
ew, a
ppro
pria
te c
on-
cept
imag
es)
is n
eede
d bu
t oft
en in
exis
tent
. The
y ar
e no
t res
earc
hers
in m
athe
mat
ics,
but
they
shou
ld d
evel
op a
sui
tabl
e ps
ycho
logi
cal a
ppro
ach
to th
e su
bjec
t. W
e th
ink
it w
ould
be
nice
to a
t
leas
t dis
cuss
the
issu
e. U
ltim
atel
y, w
e w
ould
not
min
d se
eing
a r
ecom
men
ded
curr
icul
um a
s th
e
resu
lt of
a d
iscu
ssio
n in
the
AM
T G
roup
.
SESS
ION
III
: TH
E P
RO
CE
SS B
Y W
HIC
H S
TU
DE
NT
S C
OM
E T
O A
PPR
E-
CIA
TE
TH
E N
EE
D F
OR
RIG
OR
AN
D F
OR
MA
LIT
Y. P
ropo
sed
by S
teve
Will
iam
s
(U.S
.A.)
.
Thi
s di
ffer
s fr
om 'p
roof
' in
that
con
cept
s an
d pr
oced
ures
may
als
o be
add
ress
ed w
ith f
orm
ality
and
rigo
r (e
.g. t
he f
orm
al d
efin
ition
of
limit)
. Thi
s to
pic
may
als
o in
volv
e a
disc
ussi
on o
f st
u-
dent
s' v
iew
s of
wha
t con
stitu
tes
mat
hem
atic
s, th
eir
epis
tem
olog
ieal
com
mitt
men
ts, t
heir
bel
iefs
abou
t gen
eral
ity. W
e se
e th
is a
s an
impo
rtan
t com
plem
ent t
o cu
rren
t eff
orts
to m
ake
mat
hem
atic
s
intu
itive
and
con
text
ual;
whi
le 'p
ract
ical
" m
athe
mat
ics
may
be
dida
ctic
ally
impo
rtan
t, it
is a
lso
impo
rtan
t to
stud
y ho
w th
e m
ovem
ent t
owar
d ri
gor
is m
ade
and
mat
hem
atic
al m
atur
ity is
gai
ned.
SESS
ION
IV
: TH
E W
OR
KIN
G O
P T
HE
GR
OU
P IN
TH
E F
UT
UR
E.
In th
e la
st s
essi
on w
e w
ill d
iscu
ss a
s us
ual p
lans
for
the
futu
re w
ork
of th
e G
roup
. In
part
icul
ar
sugg
estio
ns a
nd v
olun
teer
s fo
r an
AM
T-p
rogr
am f
or n
ext y
ear
are
wel
com
e.
29
-
The
IGP
ME
Wor
king
Gro
up o
n R
epre
sent
atio
ns
Ger
ald
A. G
oldi
n, C
ente
r fo
r M
athe
mat
ics,
Sci
ence
, and
Com
pute
r E
duca
tion,
Rut
gers
Uni
vers
ity, N
ew B
runs
wic
k, N
ew J
erse
y 08
903,
US
A
The
Wor
king
Gro
up o
n R
epre
sent
atio
ns o
f the
Inte
rnat
iona
l Gro
up fo
r th
eP
sych
olog
y of
Mat
hem
atic
s E
duca
tiorr
was
gui
ded
durin
g Its
firs
t sev
eral
yea
rs b
y F
ranc
es L
owen
thal
(M
ons,
Bel
gium
).I t
ook
on th
e re
spon
sibi
lity
of c
oord
inat
ing
it at
the
end
of th
e 19
89 m
eetin
g in
Par
is,
and
Cla
ude
Janv
ier
(Mon
trea
l, C
anad
a) p
rovi
ded
valu
able
ass
ista
nce
In o
rgan
izin
g ou
rse
ssio
ns a
t
the
1990
mee
ting
in M
exic
o, w
here
we
had
42 p
artic
ipan
ts.
In th
e la
st y
ear
our
focu
s ha
s be
enbr
oade
ned
subs
tant
ially
, to
incl
ude
a nu
mbe
r of
diff
eren
t int
erpr
etat
ions
that
hav
e be
en g
iven
toth
e te
ns 'r
epre
sent
atio
n" in
con
nect
ion
with
mat
hem
atic
s se
amin
g,. t
each
ing,
and
dev
elop
men
t
(a)
Ext
orne
l phy
sica
l em
bodi
men
ts (
incl
udin
g co
mpu
ter
envi
ronm
ents
):A
n: e
xter
nal,
stru
ctur
ed p
hysi
cal s
ituat
ion
or s
et o
f situ
atio
ns th
at c
an b
e de
scrib
ed m
athe
mat
ical
ly o
r se
en a
sem
body
ing
a m
athe
mat
ical
idea
e.g.
. a n
umbe
r lin
e, d
raw
n an
d la
bele
d; a
con
figur
atio
n of
peg
son
a p
eg-b
oard
pro
vidi
ng a
n ar
ray
mod
el fo
r m
ultip
licat
ion,
or
mor
e br
oadl
yth
e pe
g-bo
ard
appa
rate
s N
ast a
cal
cula
tor-
or
com
pute
r-en
viro
nmen
t whe
re c
onst
ruct
s su
ch a
s fu
nctio
ns o
rgr
aphs
can
Ise
disp
laye
d an
d m
anip
ulat
ed; e
tc.
(b)
Ling
uist
ic e
mbo
dim
ents
: Ver
bal,
synt
actic
.an
d M
eted
sem
antic
asp
ects
of t
he la
ngua
ge in
whi
ch p
robl
ems
are
pose
d an
d M
athe
mat
ics
isdi
scus
sed.
(c)
For
mal
mat
hem
atic
al c
onst
ruct
s: A
diff
eren
t mea
ning
. of '
repr
esen
tatio
n'. s
till
with
-em
phas
is o
n a
prob
lem
env
ironm
ent e
xter
nal t
o th
e in
divi
dual
, is
that
of a
form
alst
ruct
ural
or
mat
hem
atic
al a
naly
sis
of a
'situ
atio
n or
. set
of '
situ
atio
nse.
g., s
tate
-spa
cere
pres
ente
tions
of p
robl
eths
or
gam
es s
uch
is th
e T
ower
of H
anoi
,- N
ita, e
tc.;
repr
esen
tatio
ns o
fm
athe
rnal
ka9
entit
ies,
suc
h as
gro
ups,
rin
gs, f
unct
ions
, etc
., by
mea
ns o
f oth
er m
athe
mat
ical
entit
ies,
suc
h as
ope
rato
rs o
n ve
ctor
spa
ces.
gra
phs.
etc
. Alth
ough
ther
e is
.a s
ense
in w
hich
all
mat
hem
atic
s ca
n be
reg
arde
d as
'Int
erne
t,' th
e em
phas
is h
ere
is're
pres
enta
tion"
as
anan
alyt
ical
tool
for
form
aliz
ing
or m
akin
g pr
ecis
e m
athe
mat
ical
-id
eas
or m
athe
mat
ical
beh
avio
r.(d
) W
arne
d co
gniti
ve r
epre
sent
atio
ns: V
ery
impo
rtan
t em
phas
es in
clud
e ..s
tude
nts'
inte
rnal
.in
divi
thse
l mpr
esen
tatio
n(s)
for
mat
hem
atic
al id
eas
such
as
'are
as,
func
tions
% e
tc..
as w
ell a
ssy
arai
ns o
f cog
nitiv
e re
pres
enta
tion
in a
bro
ader
sen
se th
at c
an d
eric
ribe
the
proc
esse
s of
hum
ante
emin
g en
d pr
oble
m s
olvi
ng in
mat
hem
atic
s.T
hese
and
oth
er m
eani
ngs
of th
e te
rm*r
epre
asen
taffo
n! w
ere
addr
esse
d In
at l
east
a p
relim
inar
y w
ay In
the
book
, Pro
blem
s of
Rep
rese
ntat
ion
in th
e T
each
ing
and
Lear
nin