Mathematical thinking in school
FEATURES OF MATHEMATICAL THINKING
AMONG FINNISH STUDENTS IN UPPERSECONDARY SCHOOL
Jo rma Jo utse nlahti
De partme nt o f Te ac he r Educ atio n
Unive rsity o f Tampe re
Finland
Copenhagen 24.4.2008 / JoJo
• Intro duc tio n
• The o re tic al framewo rk
• Aim and pe rspe c tive s
• Re sults
Copenhagen 24.4.2008 / JoJo
Co uld we e xpe c t as
go o d re sults in
mathe matic s also
in Finnish uppe rse c o ndary sc ho o ls?
Finland and Korea, and the
partners Chinese Taipei and Hong
Kong-China, outperformed all
other countries/economies in PISA
2006.
Copenhagen 24.4.2008 / JoJo
• Intro duc tio n
• The o re tic al framewo rk
• Aim and pe rspe c tive s
• Re sults
Copenhagen 24.4.2008 / JoJo
[1 ]
[2 ]
THEORETICAL FRAMEWORK 1 / 4
Copenhagen 24.4.2008 / JoJo
Knowledge
Psychological epistemology
Philosophical epistemology
[4 ],[5 ]
[3 ]
Prosedural
Strategic Knowledge
knowledge
Conceptual
Knowledge
Beliefs
Metacognitions
Strategies
Understanding
[6 ]
Skills
Problem solving
THEORETICAL FRAMEWORK 2 / 4
Copenhagen 24.4.2008 / JoJo
Prosedural
Fluency
Strategic
Competence
Conceptual
Understanding
Adaptive
Reasoning
View of
mathematics
[5 ]
Mathematical proficiency
[7 ]
THEORETICAL FRAMEWORK 3 / 4
Copenhagen 24.4.2008 / JoJo
Anthro po lo gic al
Pe d ago gic al
THEORETICAL
FRAMEWORK 4 / 4
Copenhagen 24.4.2008 / JoJo
• Intro duc tio n
• The o re tic al framewo rk
• Aim and pe rspe c tive s
• Re sults
Copenhagen 24.4.2008 / JoJo
AIM OF THE STUDY
The main pro ble m in the study is to de sc ribe
fe ature s o f the stude nt's mathe matic al thinking
The sub pro ble ms c o nside r what kinds o f
diffe re nc e s exist in the mathe matic al
pro fic ie nc y and in the vie w o f mathe matic s
be twe e n ge nde rs and be twe e n stude nts who
c ho se a co mpulso ry te st o r an o ptio nal te st in
the matric ulatio n e xaminatio n.
Copenhagen 24.4.2008 / JoJo
•Re s ults
•Ite m analysis
Three perspectives
• Te st analysis
• Bac kround
•Que s tionary
•Inte rvie ws
knowle dge
Ac hie ve me nts
Copenhagen 24.4.2008 / JoJo
Societal
perspective
GRADE IN MATRICULATION EXAMINATION
TWO-BY-TWO FREQUENCY TABLE
Student’s
perspective
Teacher’s
perspective
GRADE AT THE END OF UPPER-SECONDARY SCHOOL
Copenhagen 24.4.2008 / JoJo
• Intro duc tio n
• The o re tic al framewo rk
• Aim and pe rspe c tive s
• Re sults
Copenhagen 24.4.2008 / JoJo
L
GRADE IN MATRICULATION EXAMINATION
3%
41%
E
M
R2
C
R3
R1
(5% )
(37% )
R4
24% + 19% = 43%
(26% + 14% = 40% )
B
13%
R3’
19%
A
(18% )
(14% )
I
4
5
6
7
8
9
GRADE AT THE END OF UPPER-SECONDARY SCHOOL
10
Copenhagen 24.4.2008 / JoJo
R1
R2
R4
R3
R3’
Copenhagen 24.4.2008 / JoJo
R1 ''SUCCESSFUL STUDENTS''
Five fe ature s o f the ir mathe matical pro fic ie nc y
are we ll de ve lo pe d.
appre c iate mathe matic s as an impo rtant and
ple ase nt subje c t.
have pe rse ve ranc e to struggle with c o mplex
mathe matical pro ble ms.
bro ad mathe matical thinking
Copenhagen 24.4.2008 / JoJo
R2 ''M ATURE STUDENTS''
The y had diffic ultie s in mathe matic s c o urse s,
but the y studie d hard and the ir mathe matic al
pro fic ie nc y de ve lo pe d just be fo re matric ulatio n
e xaminatio n. (e spe c ially pro c e dural flue nc y)
The y have pe rse ve ranc e to struggle with
c o mple x mathe matics pro ble ms.
Bro ad mathe matic al thinking
Copenhagen 24.4.2008 / JoJo
R3 ''´ JUST DOING´ STUDENTS''
c o nce ntrate d o n pro c e dural flue nc y.
fe lt that the re was to o fast te mpo in
mathe matic s le sso ns and the re fo re the y didn’t
unde rstand ne w c o nc e pts de e ply.
se lf-c o nfide nc e in mathe matic s was we ak
Narro w mathe matic al thinking
Copenhagen 24.4.2008 / JoJo
R3 ’ ''LOSERS‘ ’
Mo st stude nts ín the ”lo se rs” – gro up c ho ic e d
mathe matic s as an o ptio nal te st and the y had
c o nc e ntrate d o n c o mpulso ry te sts (c alc ulatio n:
o ptimize re sult, minimize wo rk)
quite go o d base (grade 6 o r 7 ) in mathe matic s to
de ve lo pe the ir mathe matic al pro fic ie nc y if the y just
had studie d mo re syste matic ally
se lf-c o nfide nc e in mathe matic s was we ak
narro w mathe matic al thinking
Copenhagen 24.4.2008 / JoJo
R4 ''DISAPPOINTED STUDENTS''
have to kno w in advanc e fo r suc c e ssful so lving
to what are a o f mathe matic s the pro ble m
be lo ngs (unde ve lo pe d me tac o gnitive skills in
mathe matic s)
manage d we ll in c o urse s whic h c o nsist o f
spe c ific are a o f mathe matic s and the y fe e l
the mse lve s go o d in mathe matic s (be fo re
matric ulatio n examinatio n)
c o nte nts bo und mathe matic al thinking
Copenhagen 24.4.2008 / JoJo
REFERENCE
[1 ] Ste rnbe rg, R. 1 9 9 6 . What is mathe matic al thinking? In R. Ste rnbe rg & T. Be nZe ev (e ds.) The nature o f mathe matic al thinking. Mahwah (NJ): Erlbaum, 3 0 3 -31 8 .
[2 ] Be re ite r, C. 2 0 0 2 . Educ atio n and mind in the kno wle dge age . Mahwah (NJ):
Erlbaum.
[3 ] Hie be rt, J. & Le fe vre , P. 1 9 8 6 . Co nc e ptual and pro c e dural kno wle dge in
mathe matic s: an intro duc to ry analysis. Te o kse ssa J. Hie be rt (e ds.) Co nc e ptual and
pro c e dural kno wle dge : the c ase o f mathe matic s. Hillsdale (NJ): Lawre nc e Erlbaum,
1 -27.
[4 ] Mc Le o d, D. B. 1 9 9 2 . Re se arc h o n affe c t in mathe matic s e duc atio n: a
re c o nc e ptualisatio n. In D. A. Gro ws (e ds.) Handbo o k o f re se arc h mathe matics
te ac hing and le arning. Lo ndo n: Mc millan Publishing Co , 57 5 -5 9 6 .
[5 ] Pe hko ne n, E. 1 9 9 8 . On the c o nc e pt ''mathe matic al be lie f''. In E. Pe hko ne n & G.
Tö rne r (e ds .) The s tate -o f-art in mathe matic s -re late d be lie f re s e arc h. Re s ults o f the
MAVI ac tivitie s. Unive rsity o f He lsinki. De partme nt o f Te ac he r Educ atio n. Re se arc h
re po rt 1 9 5 , 37 -7 2 .
[6 ] Sc ho e nfe ld, A. 1 9 8 5 . Mathe matic al pro ble m so lving. Orlando : Ac ade mic Pre ss.
[7 ]Kilpatric k, J., Swaffo rd, J. & Finde ll, B. (e ds.) 2 0 0 1 . Adding it up. Washingto n DC:
Natio nal Ac ade my Pre ss.
AMONG FINNISH STUDENTS IN UPPERSECONDARY SCHOOL
Jo rma Jo utse nlahti
De partme nt o f Te ac he r Educ atio n
Unive rsity o f Tampe re
Finland
Copenhagen 24.4.2008 / JoJo
• Intro duc tio n
• The o re tic al framewo rk
• Aim and pe rspe c tive s
• Re sults
Copenhagen 24.4.2008 / JoJo
Co uld we e xpe c t as
go o d re sults in
mathe matic s also
in Finnish uppe rse c o ndary sc ho o ls?
Finland and Korea, and the
partners Chinese Taipei and Hong
Kong-China, outperformed all
other countries/economies in PISA
2006.
Copenhagen 24.4.2008 / JoJo
• Intro duc tio n
• The o re tic al framewo rk
• Aim and pe rspe c tive s
• Re sults
Copenhagen 24.4.2008 / JoJo
[1 ]
[2 ]
THEORETICAL FRAMEWORK 1 / 4
Copenhagen 24.4.2008 / JoJo
Knowledge
Psychological epistemology
Philosophical epistemology
[4 ],[5 ]
[3 ]
Prosedural
Strategic Knowledge
knowledge
Conceptual
Knowledge
Beliefs
Metacognitions
Strategies
Understanding
[6 ]
Skills
Problem solving
THEORETICAL FRAMEWORK 2 / 4
Copenhagen 24.4.2008 / JoJo
Prosedural
Fluency
Strategic
Competence
Conceptual
Understanding
Adaptive
Reasoning
View of
mathematics
[5 ]
Mathematical proficiency
[7 ]
THEORETICAL FRAMEWORK 3 / 4
Copenhagen 24.4.2008 / JoJo
Anthro po lo gic al
Pe d ago gic al
THEORETICAL
FRAMEWORK 4 / 4
Copenhagen 24.4.2008 / JoJo
• Intro duc tio n
• The o re tic al framewo rk
• Aim and pe rspe c tive s
• Re sults
Copenhagen 24.4.2008 / JoJo
AIM OF THE STUDY
The main pro ble m in the study is to de sc ribe
fe ature s o f the stude nt's mathe matic al thinking
The sub pro ble ms c o nside r what kinds o f
diffe re nc e s exist in the mathe matic al
pro fic ie nc y and in the vie w o f mathe matic s
be twe e n ge nde rs and be twe e n stude nts who
c ho se a co mpulso ry te st o r an o ptio nal te st in
the matric ulatio n e xaminatio n.
Copenhagen 24.4.2008 / JoJo
•Re s ults
•Ite m analysis
Three perspectives
• Te st analysis
• Bac kround
•Que s tionary
•Inte rvie ws
knowle dge
Ac hie ve me nts
Copenhagen 24.4.2008 / JoJo
Societal
perspective
GRADE IN MATRICULATION EXAMINATION
TWO-BY-TWO FREQUENCY TABLE
Student’s
perspective
Teacher’s
perspective
GRADE AT THE END OF UPPER-SECONDARY SCHOOL
Copenhagen 24.4.2008 / JoJo
• Intro duc tio n
• The o re tic al framewo rk
• Aim and pe rspe c tive s
• Re sults
Copenhagen 24.4.2008 / JoJo
L
GRADE IN MATRICULATION EXAMINATION
3%
41%
E
M
R2
C
R3
R1
(5% )
(37% )
R4
24% + 19% = 43%
(26% + 14% = 40% )
B
13%
R3’
19%
A
(18% )
(14% )
I
4
5
6
7
8
9
GRADE AT THE END OF UPPER-SECONDARY SCHOOL
10
Copenhagen 24.4.2008 / JoJo
R1
R2
R4
R3
R3’
Copenhagen 24.4.2008 / JoJo
R1 ''SUCCESSFUL STUDENTS''
Five fe ature s o f the ir mathe matical pro fic ie nc y
are we ll de ve lo pe d.
appre c iate mathe matic s as an impo rtant and
ple ase nt subje c t.
have pe rse ve ranc e to struggle with c o mplex
mathe matical pro ble ms.
bro ad mathe matical thinking
Copenhagen 24.4.2008 / JoJo
R2 ''M ATURE STUDENTS''
The y had diffic ultie s in mathe matic s c o urse s,
but the y studie d hard and the ir mathe matic al
pro fic ie nc y de ve lo pe d just be fo re matric ulatio n
e xaminatio n. (e spe c ially pro c e dural flue nc y)
The y have pe rse ve ranc e to struggle with
c o mple x mathe matics pro ble ms.
Bro ad mathe matic al thinking
Copenhagen 24.4.2008 / JoJo
R3 ''´ JUST DOING´ STUDENTS''
c o nce ntrate d o n pro c e dural flue nc y.
fe lt that the re was to o fast te mpo in
mathe matic s le sso ns and the re fo re the y didn’t
unde rstand ne w c o nc e pts de e ply.
se lf-c o nfide nc e in mathe matic s was we ak
Narro w mathe matic al thinking
Copenhagen 24.4.2008 / JoJo
R3 ’ ''LOSERS‘ ’
Mo st stude nts ín the ”lo se rs” – gro up c ho ic e d
mathe matic s as an o ptio nal te st and the y had
c o nc e ntrate d o n c o mpulso ry te sts (c alc ulatio n:
o ptimize re sult, minimize wo rk)
quite go o d base (grade 6 o r 7 ) in mathe matic s to
de ve lo pe the ir mathe matic al pro fic ie nc y if the y just
had studie d mo re syste matic ally
se lf-c o nfide nc e in mathe matic s was we ak
narro w mathe matic al thinking
Copenhagen 24.4.2008 / JoJo
R4 ''DISAPPOINTED STUDENTS''
have to kno w in advanc e fo r suc c e ssful so lving
to what are a o f mathe matic s the pro ble m
be lo ngs (unde ve lo pe d me tac o gnitive skills in
mathe matic s)
manage d we ll in c o urse s whic h c o nsist o f
spe c ific are a o f mathe matic s and the y fe e l
the mse lve s go o d in mathe matic s (be fo re
matric ulatio n examinatio n)
c o nte nts bo und mathe matic al thinking
Copenhagen 24.4.2008 / JoJo
REFERENCE
[1 ] Ste rnbe rg, R. 1 9 9 6 . What is mathe matic al thinking? In R. Ste rnbe rg & T. Be nZe ev (e ds.) The nature o f mathe matic al thinking. Mahwah (NJ): Erlbaum, 3 0 3 -31 8 .
[2 ] Be re ite r, C. 2 0 0 2 . Educ atio n and mind in the kno wle dge age . Mahwah (NJ):
Erlbaum.
[3 ] Hie be rt, J. & Le fe vre , P. 1 9 8 6 . Co nc e ptual and pro c e dural kno wle dge in
mathe matic s: an intro duc to ry analysis. Te o kse ssa J. Hie be rt (e ds.) Co nc e ptual and
pro c e dural kno wle dge : the c ase o f mathe matic s. Hillsdale (NJ): Lawre nc e Erlbaum,
1 -27.
[4 ] Mc Le o d, D. B. 1 9 9 2 . Re se arc h o n affe c t in mathe matic s e duc atio n: a
re c o nc e ptualisatio n. In D. A. Gro ws (e ds.) Handbo o k o f re se arc h mathe matics
te ac hing and le arning. Lo ndo n: Mc millan Publishing Co , 57 5 -5 9 6 .
[5 ] Pe hko ne n, E. 1 9 9 8 . On the c o nc e pt ''mathe matic al be lie f''. In E. Pe hko ne n & G.
Tö rne r (e ds .) The s tate -o f-art in mathe matic s -re late d be lie f re s e arc h. Re s ults o f the
MAVI ac tivitie s. Unive rsity o f He lsinki. De partme nt o f Te ac he r Educ atio n. Re se arc h
re po rt 1 9 5 , 37 -7 2 .
[6 ] Sc ho e nfe ld, A. 1 9 8 5 . Mathe matic al pro ble m so lving. Orlando : Ac ade mic Pre ss.
[7 ]Kilpatric k, J., Swaffo rd, J. & Finde ll, B. (e ds.) 2 0 0 1 . Adding it up. Washingto n DC:
Natio nal Ac ade my Pre ss.