Numerical development and the role of working memory

Make it count!
Numerical development and the
role of working memory
Meijke Kolkman
Cover design Ferdinand van Nispen, Citroenvlinder-dtp.nl, Bilthoven, The Netherlands
Printed by
GVO drukkers & vormgevers B.V. | Ponsen & Looijen, Ede, The Netherlands
ISBN
978-90-393-5999-0
© 2013 M.E. Kolkman
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Make it count!
Numerical development and the
role of working memory
Reken erop! Ontwikkeling van numerieke
vaardigheden en de rol van werkgeheugen
(met een samenvatting in het Nederlands)
Proefschrift
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Meijke Elisabeth Kolkman
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Promotoren:
Prof.dr. P.P.M. Leseman
Prof.dr. H.J.A. Hoijtink
Co-promotor:
dr. E.H. Kroesbergen
This thesis was accomplished with financial support from The Netherlands
Organisation for Scientific Research (NWO)
ŽŶƚĞŶƚƐ
ŚĂƉƚĞƌϭ
General introduction
XX7
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Early numerical development and the role of
non-symbolic and symbolic skills
The role of executive functions in numerical
magnitude skills
X19
Involvement of working memory in longitudinal
development of number-quantity skills
X63
The role of spatial working memory in the
development of number-to-quantity mapping
X83
Summary and general discussion
107
References
123
Samenvatting (Summary in Dutch)
135
Dankwoord (Acknowledgements)
143
About the author
147
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General introduction
1
Chapter 1
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Domain-specific
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Figure 1. Outline of this dissertation.
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Chapter 1
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15
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18
Early numerical
development and the
role of non-symbolic
and symbolic skills
2
Kolkman, M. E., Kroesbergen, E. H., & Leseman, P. P. M. (2013). Early numerical
development and the role of non-symbolic and symbolic skills. Learning and Instruction, 25,
95-103. doi: 10.1016/j.learninstruc.2012.12.001
Chapter 2
Abstract
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20
Non-symbolic and symbolic skills
Numerical Development: Non-symbolic, Symbolic, and Mapping Skills
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21
2
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Non-symbolic and symbolic skills
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Chapter 2
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2
Chapter 2
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26
Non-symbolic and symbolic skills
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2
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QRQQXPHULFDOSDUDPHWHUVVXFKDVGRWVL]HFRYHUHGVXUIDFHGHQVLW\WKUHHFRQGLWLRQV
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Table 2.1
Internal consistency of each task and test-retest correlations in a larger sample (N = 167) within a time
range of six months.
r
p
ɲ
Number-lines NS
.59
< .01
.73
Comparison NS
.24
< .01
.84
Number-naming
.70
< .01
.84
Counting
.39
< .01
.69
Number-lines S
.48
< .01
.79
Comparison S
.61
< .01
.61
Note. NS = Non-symbolic, S = Symbolic
28
Non-symbolic and symbolic skills
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Mapping
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LQGLYLGXDOOLQHDULW\VFRUHVSURYLGHDYDOLGPHDVXUHIRUH[DPLQLQJGHYHORSPHQWLQ
PDSSLQJVNLOOVLQ\RXQJFKLOGUHQ%RRWKDQG6LHJOHUUHSRUWHGFRQVLVWHQF\
29
Chapter 2
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Mathematics
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30
Non-symbolic and symbolic skills
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31
2
Chapter 2
Results
Descriptive Statistics
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Table 2.2
Mean scores (M) and standard deviations (SD) of the numerical tasks at each measurement time.
Time 1
Time 2
Time 3
Number-lines NS
M
00.33
SD
0.27
M
00.61
SD
0.22
M
00.69
SD
00.18
Min.
00.00
Max.
00.98
Comparison NS
11.70
3.10
13.83
0.97
13.90
00.83
05.02
15.00
Number-naming
02.78
1.86
06.32
3.24
09.90
02.78
00.00
12.00
Counting
01.04
0.97
02.97
1.35
03.82
00.85
00.00
05.00
Number-lines S
00.16
0.17
00.38
0.29
00.57
00.27
00.00
00.96
12.77
1.95
14.03
01.06
08.66
15.00
36.07
10.72
12.36
60.32
Comparison S
Math
Note. The symbolic comparison task was not administered at time 1. NS = Non-symbolic; S = Symbolic
32
Non-symbolic and symbolic skills
Structure of Numerical Skills
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SHUIRUPHGLQWKHRULJLQDOVDPSOHWLPH; S &), Table 2.3
Correlations between numerical tasks and math performance.
NumberComparison NumberCounting
lines NS
NS
naming
r
p
r
p
r
p
r
p
Numberlines S
r
p
Comparison
S
r
p
Time 1
Comparison NS
Numbernaming
-.09
> .24
-.27
> .02
-.20
.06
Counting
-.11
> .19
-.08
.26
.27
> .01
Number-lines S
-.35
< .01
-.01
.48
.29
> .01
-.20
> .06
Math
-.06
> .33
-.14
.13
.19
> .06
-.46
< .01
Comparison NS
Numbernaming
-.23
> .03
-.23
> .03
-.18
.07
Counting
-.14
> .13
-.01
.48
.41
< .01
Number-lines S
-.26
> .02
-.28
.01
.40
< .01
-.26
> .02
Comparison S
Math
-.16
> .15
-.08
.29
.39
< .01
-.18
-.18
> .08
-.05
.34
.21
>.05
-.04
Comparison NS
Numbernaming
-.10
> .22
-.26
> .02
-.17
.09
Counting
-.31
> .01
-.10
.21
.76
< .01
Number-lines S
Comparison S
Math
-.34
< .01
-.04
.38
.30
>.01
-.43
< .01
-.31
-.25
> .01
> .02
-.03
-.16
.41
.10
.28
.19
.01
>.06
-.19
-.17
> .06
> .09
.23
>.03
> .12
.27
> .04
> .38
.35
< .01
-.01
.47
.46
.38
< .01
< .01
-.23
.03
Time 2
Time 3
Note. Non-symbolic tasks = Number-lines NS & Comparison NS; Symbolic tasks = Number-naming
& Counting; Mapping tasks = Number-lines S & Comparison S; NS = Non-symbolic; S = Symbolic
33
2
Chapter 2
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Table 2.4
Fit indices for each model at the different measurement time in the original sample (N=69) and based
on Bootstrap-analyses.
N=69
model 1
Bootstrap-procedure
model 2
model 3
model 1
model 2
model 3
Time 1
2
X
02.71
04.01
07.60
03.15
04.50
08.50
df
04.00
05.00
06.00
04.00
05.00
06.00
p
00.61
00.55
00.27
00.53
00.47
00.20
CFI
01.00
01.00
00.85
01.00
01.00
00.85
RMSEA
00.00
00.00
00.06
00.00
00.00
00.08
AIC
34.71
34.01
35.60
25.15
25.55
26.50
Mean-discrepancy
07.59
09.62
15.09
SD
00.14
00.18
00.23
Time 2
2
X
06.64
09.26
41.79
06.88
09.86
38.79
df
08.00
10.00
11.00
08.00
10.00
11.00
p
00.58
00.51
< .01
00.55
00.45
< .01
CFI
01.00
01.00
00.00
01.00
01.00
00.11
RMSEA
00.00
00.00
00.20
00.00
00.00
00.19
AIC
44.64
43.26
73.79
32.88
31.86
58.79
Mean-discrepancy
14.98
19.78
50.32
SD
00.22
00.25
00.53
Time 3
2
X
25.14
25.15
07.21
25.50
25.51
07.30
df
08.00
10.00
10.00
08.00
10.00
10.00
p
< .01
< .01
00.71
< .01
< .01
00.70
CFI
00.50
00.56
01.00
00.57
00.62
01.00
RMSEA
00.18
00.15
00.00
00.18
00.15
00.00
AIC
63.14
59.15
41.21
51.49
47.51
29.30
Mean-discrepancy
34.15
37.05
17.47
SD
00.35
00.38
00.23
34
Non-symbolic and symbolic skills
The Effect of Non-symbolic and Symbolic Skills on the Development of Mapping
and their Effect on Math Performance
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Discussion
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35
2
Chapter 2
Time 1
Number lines NS
Comparison NS
Number naming
Counting
Nonsymbolic
skills
.77
.15
.39
.70
Number lines S
1.00
Number lines NS
.23
Symbolic
skills
Mapping
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Time 2
Comparison NS
Number naming
Counting
Number lines S
Comparison S
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skills
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.81
Symbolic
skills
.65
Mapping
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.61
Time 3
Number lines NS
.45
Comparison NS
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Number naming
Counting
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.40
Numerical
skills
.77
Number lines S
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Comparison S
Figure 2.1
Best fitting model with the standardized factor loadings at the different measurement times, NS =
non-symbolic, S = symbolic.
36
Non-symbolic and symbolic skills
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37
2
38
Mapping skills 1
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Mapping skills 1
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Symbolic skills 1
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Non-symbolic skills 1
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Symbolic skills 1
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Non-symbolic skills 2
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.50*
.09
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Numerical skills 3
.39*
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Numerical skills 3
Figure 2.2
Fit índices, standardized regression coefficients and covariances (boxes with borders) for models A and B. Note. * p < .05
Bootstrap
2
ʖ = 19.26
df = 15
p = .20
CFI = .96
RMSEA = .07
Model B
2
ʖ = 19.93
df = 15
p = .18
CFI = .95
RMSEA = .07
Bootstrap
2
ʖ = 7.14
df = 5
p = .21
CFI = .98
RMSEA = .08
Model A
2
ʖ = 7.35
df = 5
p = .20
CFI = .95
RMSEA = .07
Chapter 2
Non-symbolic and symbolic skills
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39
2
Chapter 2
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41
2
3
The role of executive
functions in numerical
magnitude skills
Kolkman, M. E., Hoijtink, H. J. A., Kroesbergen, E. H., & Leseman, P. P. M. (2013). The role of
executive functions in numerical development. Learning and Individual Differences, 24, 145151. doi: 10.1016/j.lindiff.2013.01.004
Chapter 3
Abstract
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Introduction
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45
3
Chapter 3
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Numerical magnitude skills
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46
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47
3
Chapter 3
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48
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49
Chapter 3
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51
3
Chapter 3
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YDOXHRIDK\SRWKHVLVE\WKH303YDOXHRIWKHXQFRQVWUDLQHGK\SRWKHVLVZKLFKUHVXOWV
LQDYDOXHVKRZLQJWKHHYLGHQFHLQIDYRURIRQHK\SRWKHVLVFRPSDUHGZLWKWKH
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Table 3.1
Descriptive Statistics of the Executive Functions and Numerical Estimation Skills
Differences
Control group
Experimental group
between
groups
N
M
SD
N
M
SD
Range
t
df
Pre-test
Updating
24 15.92
5.45 16 17.88 4.29
0 - 28 -1.21* 38
Performance
above chance
level
t
df
-
-
Shifting
24
11.79
4.76
16
11.56
5.06
0 - 18
-0.15*
38
15.46*
43
Inhibition
24
14.29
1.90
16
14.31
3.63
0 - 16
0-.02*
38
30.73*
43
Comparison
24
08.17
1.61
16
08.81
1.11
0 - 10
-1.40*
38
15.08*
43
Categorization
24
07.08
4.19
16
06.38
3.32
0 - 22
-0.57*
38
03.84*
43
Number lines
Post-test
Comparison
24
00.49
0.28
16
00.53
0.21
0 - 10
-0.47*
38
-
-
22
08.45
1.26
16
08.88
1.03
0 - 10
-1.10*
36
Categorization
22
09.36
5.11
16
12.75
4.57
0 - 22
-2.11*
36
Number lines
Gain
Comparison
22
00.54
0.30
16
00.57
0.27
0 - 10
-0.32*
36
20
00.28
1.39
19
00.07
1.18
0 - 22
-0.24*
37
Categorization
20
02.20
4.19
19
05.84
5.53
0 - 10
-2.35*
37
Number lines
20
00.03
0.16
19
0.09
0.15
0 - 10
-1.30*
37
52
ǆĞĐƵƟǀĞĨƵŶĐƟŽŶƐŝŶŶƵŵĞƌŝĐĂůƐŬŝůůƐ
Results
Descriptive statistics
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VHH7DEOH
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VLJQLILFDQWFRUUHODWLRQVZHUHIRXQGEHWZHHQWKHJDLQVFRUHVDQGWKHH[HFXWLYH
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Table 3.2
Correlations (1-tailed) between the executive functions and the numerical tasks at pre-test (N
= 40) and the gain scores within the experimental group (N = 19) and control group (N = 18)
Both groups
Pre-test
Comp.
Updating
Shifting
Inhibition
Cat.
Experimental group
Gain
Control group
Gain
Nl.
Cat.
Nl.
Cat.
Nl.
†
.22 *
-.38**
.28*
.64**
-.07
.59**
-.26
-.10
-.27**
†
.40**
-.32
.01**
-.26
-.03
-.07
.31**
-.03
-.11
.01*
†
.15 *
††
.26 *
†
Note. ** p < .01; * p < .05; p < .10; Comp. = Comparison, Cat. = Categorization, Nl. = Number
lines
53
3
Chapter 3
Relation between executive functions and numerical magnitude skills.
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Table 3.3
PMP-values and Bayes-factor for each hypothesis under consideration for the pre-test scores in the
control and experimental group (N = 40) and for the gain scores in the experimental group (N = 19)
Control & experimental group
Pre-test
Pre-test
Pre-test
Comp.
Cat.
Nl.
Experimental group
Gain
Gain
Cat.
Nl.
PMP-values
H1
unconstrained
hypothesis
Upd. = Shi. = Inh.
H2
Upd.> Shi. = Inh.
.32
.27
.36
.14
.26
H3
Upd.> Shi. > Inh.
.31
.14
.20
.12
.11
H4
Upd. > Inh. > Shi.
.09
.16
.24
.06
.39
Hu
.12
.12
.12
.35
.16
.16
.30
.08
.34
.08
Bayes-factor
H1
Upd. = Shi. = Inh.
1.33
2.50
0.67
0.97
0.50
H2
Upd. > Shi. = Inh.
2.67
2.25
3.00
0.40
1.63
H3
Upd. > Shi. > Inh.
2.58
1.17
1.67
0.34
0.69
H4 Upd. > Inh. > Shi.
0.75
1.33
2.00
0.17
2.44
Note. The PMP-values printed in bold indicate the hypothesis with the highest support in the data;
Upd. = Updating, Shi. = Shifting, Inh. = Inhibition, Comp. = Comparison, Cat. = Categorization, Nl.=
Number lines
54
ǆĞĐƵƟǀĞĨƵŶĐƟŽŶƐŝŶŶƵŵĞƌŝĐĂůƐŬŝůůƐ
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Role of executive functions within a training focusing on improving numerical
magnitude skills.
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ZHUHODUJHUWKDQWKHRWKHU303YDOXHVLQGLFDWLQJWKDWIRUWKHJDLQVFRUHVRQWKH
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QRQHRIWKHK\SRWKHVL]HGLQHTXDOLW\FRQVWUDLQWVZHUHVXSSRUWHGE\WKHGDWD2QWKH
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LQGLFDWLQJWKDWWKHLQHTXDOLW\FRQVWUDLQWVVSHFLILHGLQWKHK\SRWKHVHVZHUHVXSSRUWHG
E\WKHGDWD
Table 3.4
Unconstrained parameter estimates for the predictor variables at pre-test (N = 40) and in the gain
scores (N = 19)
Parameter
Control & experimental group
Pre-test
Pre-test
Pre-test
Comp.
Cat.
Nl.
Experimental group
Gain
Gain
Cat.
Nl.
0
-.00
.00
-.00
-.00
-.00
1 Updating
-.33
.23
-.62
-.34
-.64
2 shifting
-.08
.07
-.01
-.45
-.22
3 Inhibition
-.04
.11
-.19
-.01
-.20
Note. Comp. = Comparison, Cat. = Categorization, Nl.= Number lines
55
3
Chapter 3
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Discussion
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OHGWRWZRLPSRUWDQWFRQFOXVLRQVWKHWKUHHH[HFXWLYHIXQFWLRQVVHHPWRFRQWULEXWH
GLIIHUHQWO\WRSHUIRUPDQFHRQQXPHULFDOPDJQLWXGHWDVNVDQGWKHLPSRUWDQFHRI
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7KHUHVXOWVVKRZHGWKDWXSGDWLQJPLJKWEHDPRUHLPSRUWDQWSUHGLFWRURIQXPHULFDO
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DQGQXPEHUOLQHWDVNDQGDOWKRXJKWKHHYLGHQFHRQWKHFDWHJRUL]DWLRQWDVNIRUWKH
SUHGRPLQDQWUROHRIXSGDWLQJZDVOHVVFRQYLQFLQJWKHSDUDPHWHUHVWLPDWHVVKRZHG
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Table 3. 5
ANCOVA analyses for the gain in categorization and number lines.
Gain Categorization
Gain Number lines
2
Source
F
df
p
ɻ
Intercept
Group
Updating
Shifting
Inhibition
Group * Upd.
Group * Shi.
Group * Inh.
0.20
0.49
0.54
0.58
0.18
3.03
0.89
0.07
1.00
1.00
1.00
1.00
1.00
1.00
1.00
1.00
.66
.49
.47
.45
.67
.09
.35
.79
.01
.02
.02
.02
.01
.09
.03
.00
56
F
df
p
ɻ2
0.97
2.53
0.83
0.53
2.29
4.38
2.99
1.54
1.00
1.00
1.00
1.00
1.00
1.00
1.00
1.00
.33
.12
.37
.47
.14
.04
.09
.22
.03
.08
.03
.02
.07
.12
.09
.05
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57
3
Chapter 3
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3
Chapter 3
Appendix
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61
3
4
Involvement of
working memory in
longitudinal
development of
number-quantity skills
Kolkman, M. E., Kroesbergen, E. H., & Leseman, P. P. M. (2013). Involvement of working
memory in longitudinal development of number-quantity skills. Manuscript submitted for
publication.
Chapter 4
Abstract
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Chapter 4
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67
4
Chapter 4
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9663DQGWKHFHQWUDOH[HFXWLYH&(DQGWKHLULQYROYHPHQWLQQXPEHUPDJQLWXGH
VNLOOV
Method
Participants and Procedure
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68
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$WWKHILUVWDQGODVWPHDVXUHPHQWWLPHFKLOGUHQFRPSOHWHGIRXUWDVNVDLPHGDW
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6RIWZDUH7RROVKWWSZZZSVWQHWFRP
Measures
Number-magnitude skills.
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1XPEHUWRSRVLWLRQWDVNFKLOGUHQZHUHDVNHGWRHVWLPDWHWKHSRVLWLRQRIDJLYHQ
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69
4
Chapter 4
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PHDVXUHWKHVDPHFRQVWUXFW.RONPDQHWDOZHFRPELQHGWKHWZRVFRUHVLQWR
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WKDWWKHFRUUHODWLRQVEHWZHHQERWKWDVNVLQWKHFXUUHQWVWXG\UDQJHGEHWZHHQU DQGU VHH7DEOH
Working memory skills.
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ZHUHXVHGWRDVVHVVZRUNLQJPHPRU\VNLOOV7HVWUHWHVWUHOLDELOLW\RIWKHVHWDVNV
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PHDVXUHRIWKH3/
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70
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ILQDOVFRUH
Analyses
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EHORZRUDERYHWKHPHDQZHUHFRQYHUWHGLQWRVFRUHVDWDVWDQGDUGGHYLDWLRQRIRU
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71
4
Chapter 4
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WKH9663IDFWRU:RUGUHFDOOEDFNZDUGVDQGRGGRQHRXWDWERWKWLPHSRLQWVORDGHG
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YDULDEOHVLQRXUPRGHOFRQVLGHULQJWKHUHODWLYHO\VPDOOVDPSOHVL]HSURYLGHDPRUH
YDOLGPHDVXUHRIZRUNLQJPHPRU\E\LQFOXGLQJPHDVXUHVIURPGLIIHUHQWWLPHSRLQWV
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ORDGLQJVVWDQGDUGL]HGVFRUH
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Table 4.1
Standardized factor loadings as a result of the CFA on working memory skills at measurement time 1
and time 3.
VSSP
PL
CE
*
Dot Matrix T1
.46
Odd One Out T1
.48
*
*
.08
*
Word recall forward T1
.71
Word recall backward T1
.03
*
*
Dot Matrix T3
.46
Odd One Out T3
.18
*
*
.62
*
Word recall forward T3
.63
Word recall backward T3
Note. * p < .05
.09
72
*
.52
*
*
.44
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PRGHOVWKHGHYHORSPHQWRIDVSHFLILFVNLOOE\HVWLPDWLQJWKHEHVWJURZWKFXUYHIRU
HDFKFKLOG7KHVORSHDQGWKHLQWHUFHSWDUHHVWLPDWHGDVODWHQWYDULDEOHVZLWKWKH
VFRUHVDWHDFKPHDVXUHPHQWWLPHDVREVHUYHGLQGLFDWRUV7KHORDGLQJVRIWKHLQWHUFHSW
DUHIL[HGDWVLQFHWKHLQWHUFHSWGRHVQRWFKDQJHRYHUWLPH7KHORDGLQJVRIWKHVORSH
UHIOHFWWKHJURZWK6LQFHZHZDQWHGWRPRGHOWKHEHVWILWWLQJJURZWKFXUYHWKHVORSH
ORDGLQJVIRUWKHVXFFHVVLYHPHDVXUHPHQWWLPHVZHUHIL[HGDWIRUWLPHDQGIL[HGDW
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RIWKHEHVWILWWLQJVORSHORDGLQJDSURFHGXUHGHVFULEHGE\:HOFK)RUERWKWKH
LQWHUFHSWDQGWKHVORSHPHDQVDQGYDULDQFHVDUHHVWLPDWHGUHIOHFWLQJWKHDYHUDJH
LQWHUFHSWDQGVORSHDQGLQGLYLGXDOGLIIHUHQFHV$VLJQLILFDQWFRYDULDQFHEHWZHHQ
LQWHUFHSWDQGVORSHUHIOHFWVWKDWFKLOGUHQZLWKKLJKHUOHYHOGHYHORSIDVWHULIWKH
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DQG&(RQWKHVORSHDQGLQWHUFHSWRIQXPEHUPDJQLWXGHVNLOOV,QWKHQH[WVHFWLRQZH
4
ZLOOSUHVHQWWKHILWLQGLFHVRIWKLVPRGHOFKLVTXDUH;ZLWKLWVSYDOXH&),DQG
506($&KLVTXDUHLVDGLVFUHSDQF\PHDVXUHEHWZHHQWKHFXUUHQWPRGHODQGWKH
VDWXUDWHGPRGHODQGVKRXOGEHDVORZDVSRVVLEOHZLWKDSYDOXHWKDWLVDVKLJKDV
SRVVLEOH&),FRPSDUHVWKHILWRIWKHPRGHOWRWKHLQGHSHQGHQFHPRGHODQGLV
FRQVLGHUHGJRRGLI!DQGDFFHSWDEOHLI!506($LVDPHDVXUHRISDUVLPRQ\
IDYRULQJVLPSOHUPRGHOVDQGLVFRQVLGHUHGJRRGLIDQGDFFHSWDEOHLI
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WHVWRXUK\SRWKHVLV $OODQDO\VHVZHUHSHUIRUPHGXVLQJ,%063666WDWLVWLFVIRU:LQGRZV9HUVLRQ
DQG$026YHUVLRQ
Results
Descriptive statistics
7DEOHSUHVHQWVWKHPHDQVFRUHVDQGVWDQGDUGGHYLDWLRQVIRUWKHPHDVXUHVXVHGLQ
WKLVVWXG\)RUWKHQXPEHUWRSRVLWLRQWDVNOLQHDUILWVFRUHVDUHSUHVHQWHGZKHUHDVIRU
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73
74
**
**
**
Note. * p < .05; ** p < .01
Word recall backward T3
-.07
**
0.07
Word recall forward T3
**
**
0.05
Odd One Out T3
†*
**
0.05
Dot Matrix T3
**
**
0.03
Word recall backward T1
†*
**
0.15
Word recall forward T1
**
0.10
Odd One Out T1
**
0.06
**
**
Dot Matrix T1
**
**
0.36
Comparison T3
**
0.48
Number lines T3
.09
.07
.32
.23
.47
.23
.36
.52
.27
.42
.42
**
0.30
Comparison T2
**
.31
**
0.40
**
†
Comparison T1
**
-.24
Number lines T2
Comparison T1
Number lines T1
Table 4.3
Correlations between all tasks (1-tailed)
Number lines T2
**
**
0.03
**
0.03
**
0.27
**
0.23
*
-.05
**
0.08
**
0.23
-.08
*
†*
0.18
**
0.61
0.33
Comparison T2
**
0.08
*
-.06
**
0.26
**
0.18
**
0.10
*
-.01
**
0.18
**
0.15
Comparison T3
Number lines T3
**
.54
†*
.16
**
.00
**
.37
**
.12
**
.13
**
.24
**
.25
**
.14
**
.26
**
.08
**
.32
**
.20
**
.27
†*
.17
†*
.19
**
.21
**
**
0.11
*
-.05
**
0.23
**
0.20
**
0.23
**
0.02
0.30
Working memory tasks
**
0.31
**
0.38
Dot Matrix T1
Number-magnitude tasks
Odd One Out T1
**
**
0.14
*
-.12
**
0.34
**
0.24
**
0.24
0.12
**
.14
**
.46
**
.13
**
.00
†*
Word recall forward
T1
.16
**
.25
**
.08
**
.36
†
Word recall
backward T1
.20
Dot Matrix T3
**
0.02
-.05
*
**
0.32
**
.38
**
Odd One Out T3
.10
**
Word recall forward
T3
.15
Chapter 4
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Table 4.2
Descriptive statistics of working memory and numerical measures.
Time 1
Number lines
(linear fit scores)
Comparison
(proportion scores)
Dot matrix
(sum of accuracy)
Odd One Out
(sum of accuracy)
Word Recall Forward
(sum of accuracy)
Word Recall
Backward
Time 2
Time 3
Empirical
M
SD
M
SD
M
SD
Min.
Max.
00.28
00.25
.38
.29
00.56
00.27
00.00
00.96
00.73
0.13
.89
.08
00.92
0.07
0.48
01.00
09.21
03.03
11.79
02.75
02.84
17.86
09.39
02.23
11.68
02.09
04.80
16.22
13.21
02.10
14.69
01.90
09.00
18.74
04.23
01.81
05.49
01.59
00.36
09.06
(sum of accuracy)
75
4
Chapter 4
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Table 4.4
Estimates for the growth models of the combined number-magnitude model, the number line model
and the comparison-only model.
Number-toComparison-only
Combined model
position -only
model
model
**
.37
**
-.82 0
**
Unstandardized factor loading Time 2
-.44
Intercept Mean
-.42
.28
**
-.22 0
**
.03
**
-.001
**
.28
**
-.06 0
**
.04
**
-.001
**
.19
**
-.94 0
Intercept Variance
-.04
Slope Mean
-.33
Slope Variance
-.05
Covariance intercept & slope
Note. * p < .05; ** p < .01
-.73
76
**
**
**
**
**
* *
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Table 4.5
Estimates for the regression coefficient and covariance in the combined number-magnitude model, the
number line-only model and the comparison only model.
Combined model
Number-to-position
-only model
Comparison-only
model
Regression weights
ɴ
p
ɴ
p
ɴ
p
CE
ї
Intercept
-.19
< .260
-.24
- .270
-.11
- .53
PL
ї
Intercept
-.13
- .390
-.13
- .540
-.19
- .25
VSSP
ї
Intercept
-.33
- .050
-.28
- .210
-.58
< .01
Raven
ї
Intercept
-.01
- .940
-.14
- .560
-.12
- .50
CE
ї
Slope
-.36
- .030
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< .010
-.04
- .85
PL
ї
Slope
-.01
- .950
-.12
- .490
-.18
- .36
VSSP
ї
Slope
-.17
- .310
-.05
- .810
-.56
< .01
Raven
ї
Slope
-.06
- .750
-.29
- .160
-.14
- .51
Covariances
r
P
r
p
r
p
Intercept
ў
Slope
-.74
< .010
-.28
- .480
-.93
< .010
CE
ў
PL
-.24
- .020
-.24
-.020
-.24
- .020
CE
ў
VSSP
-.56
< .001
-.56
< .001
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< .001
Raven
ў
CE
-.42
< .010
-.42
< .010
-.42
< .010
Raven
ў
PL
-.39
< .010
-.39
< .010
-.39
< .010
Raven
ў
VSSP
-.44
< .010
-.44
< .010
-.44
< .010
77
4
Chapter 4
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Discussion
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Chapter 4
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81
4
The role of spatial
working memory in
the development of
number-to-quantity
mapping
5
Kolkman, M. E., Kroesbergen, E. H., & Leseman, P. P. M. (2013). The role of spatial working
memory in the development of number-to-quantity mapping. Manuscript submitted for
publication.
Chapter 5
Abstract
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Chapter 5
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89
Chapter 5
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Numerical skills.
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Chapter 5
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Chapter 5
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5
Tabel 5.1
Means and standard deviations of the working memory tasks and all the numerical measures.
Time 1
Time 2
Time 3
Time 4
M
SD
M
SD
M
SD
M
SD
Odd one out
7.58
2.64
8.73
2.83
9.65
2.90
10.37
2.70
Word recall backward
2.97
2.28
3.94
1.88
4.64
1.59
5.25
1.49
Comparison
0.80
0.14
0.86
0.09
0.89
0.07
0.91
0.07
Quantity-to-position
0.44
0.27
0.43
0.27
0.54
0.27
0.57
0.25
Number naming
0.25
0.16
0.36
0.23
0.52
0.29
0.61
0.30
Counting
0.35
0.18
0.36
0.27
0.55
0.24
0.61
0.30
Number-to-position
0.16
0.16
0.26
0.23
0.38
0.30
0.44
0.30
95
Chapter 5
Results
Descriptive statistics
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Table 5.2
Correlations among all variables (part 1)
2
3
4
Number-toposition
1
Odd one out
5
6
7
8
10
11
Word recall
backward
9
Quantity-to-position
12
13
14
15
16
Comparison
17
18
19
20
Number naming
21
22
23
24
Counting
25
26
27
28
*
Time
2
Time
3
Time
4
Time
1
Time
2
Time
3
Time
4
Time
1
Time
2
Time
3
Time
4
Time
1
Time
2
Time
3
Time
4
Time
1
Time
2
Time
3
Time
4
Time
1
Time
2
Time
3
Time
4
Time
1
Time
2
Time
3
Time
4
2
4
5
6
7
8
9
10
11
12
-.21**
+
-.13+*
.20**
-.03**
.26**
.47**
-.06**
.10**
.15**
.34**
-.01**
.11+*
.23**
.25**
.52**
-.05**
.18**
.28**
.19**
.37**
.53**
-.03**
.05**
.32**
.24**
.43**
.45**
.59**
-.20**
.20**
.30**
.14**
.28**
.36**
.53**
.42**
-.06**
.13+*
.31**
.20**
.30**
.39**
.34**
.36**
.48**
-.12+*
.24**
.21**
.23**
.22**
.16**
.32**
.27**
.36**
.31**
-.02**
.12+*
.25**
.07**
.12+*
.15**
.28**
.24**
.35**
.27**
.27**
-.08**
.05**
.20**
.14**
.28**
.17**
.29**
.21**
.21**
.25**
.06**
.05**
-.15**
.26**
.33**
.23**
.25**
.39**
.39**
.32**
.33**
.39**
.11+*
.15**
-.04**
.08**
.26**
.17**
.30**
.30**
.27**
.40**
.24**
.28**
.03**
.15**
-.07**
.24**
.24**
.26**
.27**
.26**
.34**
.40**
.21**
.28**
.08**
.15**
-.04**
.05**
.14**
.17**
.24**
.21**
.36**
.22**
.26**
.04**
.09**
.16**
-.10**
.34**
.21**
.22**
.20**
.18**
.25**
.23**
.29**
.26**
.17**
.19**
-.09**
.18**
.31**
.13+*
.15**
.06**
.27**
.19**
.25**
.12+*
.33**
.31**
-.08**
.15**
.26**
.14**
.06**
.09**
.18**
.17**
.17**
.04**
.10**
.13+*
-.12+*
.30**
.31**
.28**
.30**
.39**
.34**
.24**
.30**
.21**
.15**
.25**
-.25**
.37**
.30**
.38**
.27**
.34**
.37**
.22**
.43**
.22**
.24**
.18**
-.25**
.36**
.44**
.40**
.18**
.22**
.29**
.22**
.31**
.09**
.30**
.31**
-.14**
.31**
.47**
.45**
.23**
.31**
.33**
.29**
.27**
.14**
.34**
.31**
-.21**
.20**
.19**
.24**
.28**
.29**
.24**
.21**
.35**
.18**
.25**
.01**
-.19**
.27**
.34**
.37**
.31**
.34**
.40**
.21**
.37**
.34**
.30**
.20**
-.18**
.12+*
.33**
.27**
.34**
.32**
.35**
.36**
.33**
.20**
.25**
.26**
-.03**
.17**
.38**
.28**
.13+*
.21**
.25**
.31**
.21**
.24**
.23**
.24**
Note. p < .05; p < .10 (1-tailed)
96
3
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Table 5.2
Correlations among all variables (part 2)
14
15
16
Quantity-toposition
13
18
19
Comparison
17
21
22
23
24
Number naming
20
26
27
Counting
25
28
*
Time
2
Time
3
Time
4
Time
1
Time
2
Time
3
Time
4
Time
1
Time
2
Time
3
Time
4
Time
1
Time
2
Time
3
Time
4
-.26
14
15
16
17
18
19
20
21
22
23
24
25
26
27
**
5
-.21**
.36**
-.21**
.45** .59**
-.18**
.07** .16**
.30**
-.14**
.30** .23**
.25**
.16**
-.11+*
.12+* .19**
.18**
.24**
.28**
-.07**
.24** .24**
.21**
.11**
.26**
.40**
-14**
.25** .17**
.19**
.28**
.13+*
.13+*
.19**
.19**
.29** .16**
.12**
.30**
.29**
.18**
.15**
.68**
.09**
.21** .15**
.09**
.30**
.20**
.29**
.25**
.62**
.64**
.15**
.21** .15**
.15**
.30**
.15**
.25**
.17**
.50**
.58**
.70**
.18**
.25** .15**
.15**
.20**
.16**
.03**
.08**
.39**
.46**
.36**
.41**
-18**
.32** .20**
.23**
.22**
.33**
.20**
.12+*
.40**
.61**
.45**
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.33**
-23**
.21** .19**
.14**
.22**
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.15**
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.46**
.46**
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.32**
.46**
-08**
.21** .20**
.27**
.17**
.04**
.25**
.09**
.23**
.21**
.35**
.33**
.10**
.36**
.26**
+
Note. p < .05; p < .10 (1-tailed)
97
Chapter 5
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Table 5.3
Fit indices of the univariate growth models and multivariate growth models.
2
ʖ /df
CFI
RMSEA
OOO
2.28
0.98
.09
WRB
0.52
1.00
.00
Non-symbolic skills
2.01
0.97
.08
Symbolic skills
1.18
0.99
.03
Mapping
0.04
1.00
.00
Model 1
1.37
0.95
.05
Model 2
1.20
0.97
.04
Model 3
2.17
0.88
.09
Table 5.4
Parameter estimates for the univariate growth models.
UnStandardpmodel
StandardSE
ized
value
ized
UnStandardStandardized
ized
factor loadings time 2
pvalue
SE
factor loadings time 3
OOO
0.410
.26
< .001
0.070
0.780
.49
< .001
0.070
WRB
0.440
.25
< .001
0.060
0.760
.50
< .001
0.060
Non-symbolic skills
0.300
.24
< .001
0.080
0.700
.53
< .001
0.060
Symbolic skills
0.210
.14
< .001
0.040
0.760
.55
< .001
0.050
Mapping
0.350
.37
< .001
0.060
0.780
.65
< .001
0.080
Intercept variance
Slope variance
OOO
4.190
< .001
0.900
3.190
< .001
1.300
WRB
2.700
< .001
0.610
1.150
< .070
0.780
Non-symbolic skills
0.007
< .010
0.002
0.014
< .001
0.004
Symbolic skills
0.020
< .001
0.002
0.030
< .001
0.005
Mapping
0.010
< .050
0.005
0.060
< .001
0.002
98
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skills
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skills
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*
**
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skills
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skills
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skills
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Standardized regression coefficients and covariances (boxes with borders) for model 2.
*
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Chapter 5
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skills
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skills
0.02
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Slope
Symbolic
skills
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skills
0.05***
Intercept
Slope
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WM
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WM
0.01*
0.08***
-4.22***
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*
**
***
Note. p < .05, p < .01, p < .001
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Chapter 5
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Summary and general
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6
Chapter 6
Introduction
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Chapter 6
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110
Summary and general discussion
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111
6
Chapter 6
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112
Summary and general discussion
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113
6
Chapter 6
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114
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115
6
Chapter 6
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116
Summary and general discussion
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6
Domain-specific
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Domain-general ability
Verbal working memory
Figure 6.1
Reconsideration of the theoretical model proposed in the introduction of this dissertation
117
Chapter 6
Practical implications
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Chapter 6
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Dankwoord
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About the author
Curriculum vitae
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