2014. Vol.5, No.2, 127-133
Published Online February 2014 in SciRes (
From Marbles to Numbers—Estimation Influences Looking
Patterns on Arithmetic Problems
Claudia Godau*, Maria Wirth, Sonja Hansen, Hilde Haider, Robert Gaschler
Department of Psychology, Humboldt Universität zu Berlin, Berlin, Germany
Email: *
Received December 19th, 2013; revised January 18th, 2014; accepted February 16th, 2014
Copyright © 2014 Claudia Godau et al. This is an open access article distribu ted under the Creative Commons
Attribution License, which pe rmits unrestricted use, distribu tion, and reproduction in any medium, provided the
original work is properly cited. In accordance of the Creative Commons Attribution License all Copyrights ©
2014 are reserved for SCIRP and the owner of the intellectual property Claudia Godau et al. All Copyright ©
2014 are guarded by law and by SCIRP as a guardian.
Flexibly spotting and applying shortcut options in arithmetic is often a major challenge for children as
well as adults. Recent work has suggests that children benefit in terms of such flexibility from tasks re-
quiring estimation or other operations with quantities that they cannot easily enumerate. Such tasks often
require comparison of quantities by fixation and as such necessitate long-range eye movements, e.g.
across the whole screen. We tested whether fixation patterns account for transfer from estimation to
arithmetic tasks. Conceivably, participants who first solve estimation tasks are more flexible in spotting
and applying shortcuts on later arithmetic tasks, because they stick to scanning the screen with long-range
eye movements (which were necessary for solving the estimation task). To test this account, we mani-
pulated the location of the marbles in an estimation task so that one group of participants had to make
long-range eye movement, whereas another group did not need long-range eye movements to solve the
task. Afterwards participants of both groups solved addition problems that contained a shortcut option
based on the commutativity principle. We tested whether shortcut usage and fixation patterns in the arith-
metic problems were influenced by the variant of the estimation task provided beforehand. The experi-
ment allowed us to explore whether flexibility in spotting and using arithmetic shortcuts can be fostered
by applying a prior task that induces flexible looking patterns. The results suggest that estimation tasks
can indeed influence fixation patterns in a later arithmetic task. While shortcut search and application is
reflected in fixation patterns, we did not obtain evidence for the reverse influence. Changed fixation pat-
terns did not lead to higher shortcut usage. Thus, the results are in line with top-down accounts of strategy
change: fixation patterns reflect rather than elicit strategy change.
Keywords: Mental Arithmetic; Estimation; Fixation Pattern; Transfer; Commuta tiv ity Princip le
For some elementary school children, arithmetic seems to be
a tremendous challenge while for others it’s a child’s play. For
all children it includes the discovery and use of general as well
as abstract principles, which allow them to solve (apparently)
hard problems easily and fast. Unfortunately the connection
between understanding, identifying and applying shortcut op-
tions is at best moderate. Many researchers found large interin-
dividual variability in the children’s solution times and their use
of shortcut strategies. For example, Dubé and Robinson (2010)
found that 1/4 of the children did not use any shortcut at all.
Robinson and Dubé (2012) later argued that children have dif-
ferent attitudes toward accepting strategies that are highly effi-
cient but novel. For instance, from grade three onwards com-
mutativity-based shortcuts are used spontaneously (Gaschler,
Vaterrodt, Frensch, Eichler, & Haider, 2013). Children used a
shortcut when subsequent addition problems contained the
same addends in different order (e.g., 6 + 2 + 3 = ? followed by
2 + 6 + 3 = ?). Furthermore, spontaneous application of two
different commutativity-based shortcuts correlated positively.
However, while it could be documented that commutativity-
based shortcuts were used spontaneously, the rate of usage was
rather low. Apart from difficulties in spontaneously using
shortcuts, past research has documented overgeneralization (e.g.
Siegler & Stern, 1998). Once children started to use one short-
cut, some tended to apply it irrespective of whether the under-
lying mathematical principle was applicable to the current
arithmetic problem or not. This research showed that during
elementary school the flexible use of shortcut strategies is not
very balanced yet.
One potential way to foster adaptive flexibility in strategy
usage (Verschaffel, Luwel, Torbeyns, & Dooren, 2009) might
be to employ estimation tasks. By this, the mathematical prin-
ciples and the corresponding shortcuts might be conveyed at
early age. This might provide a head start for tackling the re-
spective shortcuts in mental arithmetic. Several studies on com-
mutativity have shown that children have at least some under-
standing of the concept of commutativity before entering school
(Canobi, Reeve, & Pattison, 2003; Cowan & Renton, 1996;
Resnick, 1992; Wilkins, Baroody, & Tiilikainen, 2001). Fur-
*Corresponding authors.
thermore, there is evidence for that toddlers develop an infor-
mal understanding of relations between objects in the real
world before entering school (Baroody & Gannon, 1984; Ba-
roody, Ginsburg, & Waxman, 1983; Canobi, Reeve, & Pattison,
2002; Gallistel & Gelman, 1992; Gilmore, McCarthy, & Spelke,
2010; Resnick, 1992; Siegler & Jenkins, 1989; Sophian, Harley,
& Martin, 1995). This idea is based on Resnicks (1992) model
of mathematical thinking, in which on the first level, the level
of protoquantities, mathematical thinking is strictly object-
Moreover, Sherman & Bisanz (2009) showed that working
with non-symbolic material can encourage subsequent strategy
use in symbolic equivalence problems. In this context, research
hints at that estimation might positively influence exact calcu-
lation. Gilmore, McCarthy and Spelke (2007) found that even
preschoolers were able to solve symbolic problems as long as
they were instructed to just estimate the results, rather than to
calculate the exact result. Along these lines, Hansen and col-
leagues (submitted) tested whether children would profit from
an estimation task involving commutative arithmetic problems
on later arithmetic problems. They confirmed the assumption
that symbolic estimation increased the spontaneous spotting
and applying of commutativity-based shortcuts in a later arith-
metic task. Surprisingly, the positive effect seemed to be con-
fined to actual usage of commutativity-based shortcuts (proce-
dural knowledge). In a task measuring conceptual understand-
ing of the commutativity principle, children showed liberaliza-
tion in the response criterion rather than improved or main-
tained performance when before confronted with a commuta-
tivity-based est imation task .
Apparently, symbolic as well as non-symbolic estimation
tasks support the use of commutativity-based shortcuts—but
not by activating conceptual knowledge of the mathematical
principle. This suggests to explore potential ways of transfer
between the task formats that side-track conceptual knowledge.
One potential account for the incoherent transfer results re-
ported by Hansen et al. (submitted) and Sherman and Bisanz
(2009) might be a transfer of eye movement patterns. Partici-
pants might profit from the estimation task in spotting and ap-
plying shortcuts in later arithmetic problems, because (a) long-
range eye movements are helpful in both contexts and are (b)
transferred from the estimation task to the arithmetic task. Spe-
cifically, short cut strategies that entail comparing addends
across subsequent addition problems should necessitate un-
usually long eye movements. Potentially, such looking patterns
triggered by an estimation task—would still be present when
later faced with an addition task and raise the chance that a
child spots and applies the shortcut.
Our hypothesis was that a variant of an estimation task that
requires long-range eye movements, would lead to a larger
amount of shortcut usage in a later arithmetic task (as compared
to an estimation task not requiring long-range eye movements).
To test this hypothesis, we actively manipulated the eye move-
ments in order to investigate the influence of long-range eye
movements on the detection and application of possible short-
cuts. Ey e movement patterns can influence spatial reasoning by
means of an implicit eye-movement-to-cognition link. For in-
stance, Thomas and Lleras (2007) showed an implicit compati-
bility between spatial cognition and the eye movement. How-
ever, until now the influence of eye movement patterns (i.e.,
those triggered by an estimation task) on arithmetic problems
presented later on has been neglected. Therefore, we contrasted
(a) one experimental condition starting with an estimation task
necessitating long-range eye movements (scattered group) with
(b) another group of primary school children starting with an
estimation task without such demands (centered group). As
dependent measure we tested the extent to which children saved
calculation effort by a commutativity-based shortcut in a mental
arithmetic task presented afterwards. Comparing addends in
subsequent arithmetic problems, one could avoid calculation on
problems that presented the same addends as the predecessor
problem (e.g., 6 + 2 + 3 = ? followed by 2 + 6 + 3 = ?), this
shortcu t we called addends compare strategy.
Thirty-four elementary school children (16 females and 18
males) with an age range between 6 and 11 years took part in
the experiment. The mean age of the children in the scattered
group was 8.78 years (SD = .92), and in the centered group
8.47 years (SD = 1.02). The children attended second to fifth
grade of various Berlin elementary schools, most of them in the
fourth grade (50%). The children were randomly assigned to
either the scattered group (18 children) or the centered group
(16 children).
The experiment comprised two parts. In the first part, the
children were presented with an estimation task that differed
between the groups (scattered/centered). In the second part,
both groups were presented with an arithmetic task. All materi-
al was computerized.
For the estimation task four sets of eight estimation problems
were designed, depicting different quantities of marbles. The
two sets for the centered group consisted of one quantity of
marbles, which belonged to one fictional character (either “Tim”
or “Lisa”). In order to provoke a small spatial range and low
number of saccades in the centered group, the quantity of mar-
bles was presented centrally. Children were asked to estimate if
the character owned few or many marbles (see Figure 1). The
two sets for the scattered group included two different quanti-
ties of marbles of which one belonged either to “Tim” or “Lisa”.
To trigger a larger spatial range and higher number of saccades
(compared to the centered group), the quantities of marbles
were presented at right and left edge of the presentation frame.
The children had to indicate which of the characters owned
more marbles or if both own the same amount of marbles.
The 12 arithmetic problems were presented in two groups of
six simultaneously depicted problems on two consecutive
screens (see Table A1). We presented six problems in black on
grey background simultaneously on the screen. Digits were
approximately .5 cm wide and 1 cm tall. The distance both
between the lines and columns of digits was 5 cm. Each addi-
tion problem consisted of three addends between 2 and 9 (e.g.,
6 + 2 + 3 = ?). Each number occurred only once in a problem.
Small and large numbers were balanced across the different
problems. Each screen contained two commutative problem-
pairs, in which the addends compare strategy could be used
one problem and its repetition with a different order of the same
addends. All other problems were filler problems, with no
shortcut option.
Figure 1.
Example of the estimation task for the scattered group (left) and the centered group (right).
Each child was tested individually in the laboratory of the
Department of General Psychology at Humboldt-University,
Berlin. The children were seated in front of a 22 inch TFT
computer monitor, which was equipped with the SMI RED
stationary eye tracking system recording at 250 Hz. Children
were asked to find a comfortable position (about 60 cm from
the screen) and reminded to sit as still as possible. After five-
point calibration, they received the instruction of the estimation
task, which introduced them to fictional characters “Tim” and
“Lisa” as owners of different quantities of marbles. They were
then presented with an example problem. The children were
reminded that the task did not require counting but only estima-
tion and that they should answer as fast and as correct as possi-
In the estimation task, eight different estimation problems
were presented consecutively. In a between subject design we
wanted to manipulate the range and the number of the saccades
during the estimation task to investigate the influence on dis-
covering and using the commutativity shortcut strategy on later
addition problems. For the centered group the marble quantities
were presented centrally and the child had to decide if the pic-
tured character had few or many marbles. For the scattered
group the marble quantities were presented on the right and left
edges of the screen and the child had to decide if one of the
pictured characters had more marbles or if both had the same
amount. After presenting each estimation problem for two
seconds, the screen went blank. The time limit ensured that
children had to rely on estimation. The experimenter entered
the verbal answer of the child and started the next trial of the
estimation task.
After completing the estimation task, the children calculated
simple addition problems (arithmetic task). The children were
instructed to work through the problems (six per screen) in
strict order from top to bottom and not to leave out any. The
experimenter entered the given answers so it was directly visi-
ble and remained visible when working on subsequent prob-
lems. After completing the first six problems, the experimenter
started the second and last screen presenting the next six prob-
lems. Overall the procedure took about 10 minutes.
The aim was to evaluate whether different eye movement
patterns induced by different estimation tasks (scattered and
centered) lead to different eye movement patterns and/or in-
creased use of the addends-compare strategy in the arithmetic
Eye Movement Data
The eyetracking data of one child were not recorded cor-
rectly so we present the data of 33 children. In the analysis of
the eye movements we focused on saccade distances, which
were computed as Euclidean distance in angular degree. The
saccade distances from commutative problems were compared
to all other (non-commutative) addition problems. Figure 2
depicts a bimodal distribution of saccade distances for the cen-
tered and scattered group. The bimodal character seemed more
pronounced for commutative problems and especially so for the
centered group.
For a more detailed analysis, we differentiated between hori-
zontal and vertical saccades. To test if the two groups (scattered
and centered group) differ in the saccade distances in the arith-
metic task, we conducted ANOVAs for horizontal and vertical
saccades separately. The results showed a significant difference
between both groups for the horizontal saccade distances in the
commutative problems, F(1, 31) = 13.57, p = .001. Surprisingly,
the horizontal saccade distance for commutative problems was
lower for the scattered as compared to the centered group (see
Table 1). There were neither differences for the horizontal
saccade distances for the non-commutative problems, F(1, 31)
= 1.33, p = .26, nor between-group differences for the vertical
saccade distances for non-commutative problems, F(1, 31)
= .65, p = .43, and the commutative problems, F(1, 31) = .780,
p = .38.
Additionally, we conducted a 2 (horizontal versus vertical
saccades) × 2 (commutative versus non-commutative problems)
× 2 (condition: scattered versus centered) ANOVA. As sug-
gested by Table 1, the horizontal saccade distances were longer
than the vertical saccade distances (main effect for horizontal
versus vertical saccade distances, F(1, 31) = 12605.84, p < .001;
ηp2 = .998). The ANOVA did not show a significant overall
difference in the saccade distance for the commutative versus
non-commutative problems, F(1, 31) = 1.35, p = .25; ηp2 = .042.
However, we found a significant interaction of commutative
versus non-commutative problems and condition, F(1, 31) =
10.64, p = .01; ηp2 = .255. This indicates that condition had an
impact on the difference of saccade distances for commutative
Figure 2.
Distribution in percent of saccade distances in angular degree for commutative and non-commutative problems for scattered
group (left) and centered group (right).
Table 1.
Mean saccade distance horizontal and vertical for commutative and non-commutative problems analysed for condition.
Condition Saccade distance horizontal Saccade distance vertical
Non-commutative problems Commutative problems Non-commutative problems Commutative problems
Scattered 4.38 4.11 2.28 2.24
Centered 4.24 4.41 2.32 2.3
Total 4.31 4.24 2.30 2.27
and non-commutative problems. Furthermore, this impact was
different for horizontal and vertical saccade distances, because
the 2-fold interaction was significant, too, F(1, 31) = 15.28, p
< .001; ηp2 = .330. We did not obtain a main effect for condi-
tion, F < 1. In conclusion we manipulated the eye-moveme nt in
the estimation task (scattered and centered) and found different
eyemovement patterns in the arithmetic task presented after-
wards. For conclusions concerning the strategy use we present
now the results of the solving times in the arithmetic task.
Solving Times
The solving time comprises the time between responding to
one arithmetic problem and the time of the key press of the
experimenter entering the verbalized answer to the subsequent
problem (first key in entering the child’s answer). Figure 3
suggests that children in both experimental conditions bene-
fitted from the addends-compare strategy. In line with exploit-
ing the commutativity principle, children were faster when
faced with the same addends in altered order for a second time
on subsequent addition problems. The prior problem needed to
be calculated conventionally, whereas the second problem of
the commutativity pair consisted of the same addends in a dif-
ferent order. Supposed a child first calculated 6 + 3 + 2 and
then 6 + 2 + 3 it would not need to calculate when confronted
with the second problem—if it used the addends compare
strategy. Across experimental conditions, a pairwise compari-
son of these two problems showed a significant benefit in solv-
ing times for commutative compared to their preceding non-
commutative problems t(32) = 3.37, p = .01. Note that mean
solving times of the filler problems were higher, 9.82 sec for
the scattered group and 10.04 sec for the centered group. The
comparison we focused on can thus be regarded as a conserva-
tive estimate of the usage of the addends compare strategy.
To test the difference between the scattered group and the
centered group in solving times, we conducted a 2 (commuta-
tivity: commutative vs. non-commutative problems) × 2 (con-
dition: scattered vs. centered group) ANOVA. As suggested by
Figure 3, we obtaine d a significant main effect of commutativ-
ity, F(1, 31) = 10.74, p = .01, ηp2 = .26, but no effect of condi-
tion and no interaction effect of commutativity and condition
(Fs < 1). The error rate for the twelve arithmetic problems was
10.2% and individual number of errors ranged from 0 to 6
(mean = 1.18; SD = 1.29). Note that children of different age
seemed to profit to a similar extent from the addends compare
strategy. We obtained no correlation between age and the bene-
fit in solving times on commutative as compared to non-com-
mutative problems.
Prior work (Obersteiner et al., 2013; Hansen et al., submitted)
indicates that estimation can positively influence subsequent
calculation tasks. Our assumption was that this influence might
in part be based on transfer of eye movement patterns. In an
eyetracking experiment with primary school children, we con-
trasted a variant of an estimation task that did necessitate long-
range eye movements with one that required central fixations.
In particular, we were interested in how these two variants of an
estimation task would influence later spontaneous usage of
Figure 3.
The mean solving times in seconds per arithmetic problem for non-
commutative (prior) problems (dark grey) in comparison to the com-
mutative problems (light grey) f or the scattered and th e centered group.
The error bar displays the 95% confidence interval of the comparison
with each other.
commutativity-based shortcuts in mental arithmetic. While we
did find an effect of estimation task variant on eye movement
patterns in the arithmetic task, we did not obtain any effect on
shortcut usage. Children of both experimental groups profited
from the commutative addition problems to the same extent.
Surprisingly, the effect of condition on eye movement s in the
arithmetic task was opposite to our expectations. At present we
can only state that we did see an effect of the estimation task on
eye movements in a later calculation task and speculate about
the reasons for its unexpected direction. The estimation tasks in
the scattered and centered group were designed to elicit long-
range eye movements to a different extent. To minimize these
saccades in the centered group we presented only one character
(Tim/Lisa) centrally and slightly changed the question to,
whether Tim/Lisa has many or few marbles. In comparison, the
scattered group was asked in the estimation task whether Tim
or Lisa has more marbles. Both tasks required some form of
estimation, but the children in the centered group had to com-
pare the centrally depicted marbles with their own concept of
few or many. To the scattered group, however, the comparison
array was presented simultaneously on the other side of the
computer screen. Presumably, in the centred group one addi-
tional step of processing was needed (i.e., “What do I consider
as few ore many marbles?”). Accordingly, one limitation of the
current research is that the two variants of the estimation task
not only differed considerably in the type of saccades de-
manded, but also in other cognitive processes involved in gen-
erating an answer. The requirements in the centered condition
might have supported flexibility in thinking in this group, be-
cause the criterion to decide between few and many can adap-
tively change between the problems (e.g. in one problem a child
might consider 8 marbles as many, but after seeing 20 marbles,
8 seem to be only a few). Presumably, such demanding com-
parisons might lead to more comparisons between numbers on
the screen once the addition problems are being presented. This
in turn, might be the reason for the centred group executing
longer saccades. Yet, these differences in fixation patterns did
not lead to differences in spotting and applying shortcut options.
This is coh erent with models emphasizing the role of top-down
decisions on strategy change in skill acquisition. For instance,
Haider and Frensch (1999) suggested that changes in fixation
patterns in a skill acquisition task involving a shift towards a
more efficient strategy reflect the voluntary decision to change
the strategy. Changes in fixation patterns might often reflect
rather than cause changes in processing strategy. This is in line
with the concept of adaptive expertise (Verschaffel et al., 2009),
according to which learners need to autonomously regulate
whether (a) to solve an arithmetic problem in a standard way or
to (b) search for/apply a shortcut.
Past work (e.g. Gaschler et al., 2013; Godau et al., submitted)
showed that flexible strategy use is mirrored in fixation patterns
and that different commutativity-based shortcuts are mirrored
in different fixation patterns. For the use of the addends-com-
pare strategy, the eye movement pattern showed that the child-
ren looked back to the preceding problem featuring the same
addends in different order. While using a shortcut strategy is
reflected in the corresponding fixation patterns, influencing
fixation patterns does not necessarily lead to changes in arith-
metic strategies. Future studies might test more direct ways to
trigger flexibility in calculation strategies by inducing flexibili-
ty in fixation patterns. Visual cues can help the learner attend to
and notice relevant information in the problem, which they
previously may have ignored (Madsen, Rouinfar, Larson, Los-
chky, & Rebello, 2013). In their study, Madsen and colleagues
(2013) found that inducing participants to look at helpful areas
in physics problems and to ignore distracting areas when no
visual cues are present is possibly a first step to support them to
reason correctly about the problem. They also found transfer
effects in that students could successfully answer and reason
about related but different problems without cues (Madsen et al.,
2013). Guiding attention to areas with regard to contents is one
point; we, on the contrary, wanted to focus more on the eye
movement as such, without special attention on content. For
further investigations concerning a direct intervention to sup-
port flexible strategy use, the eye movement pattern could also
be manipulated in a calculation task. During solving the arith-
metic problems, children who have more long distance sac-
cades (e.g. by presenting distractors in changing corners of the
screen) maybe recognize the shortcut strategy more often.
In summary, the current results suggest that estimation prob-
lems can indeed influence fixation patterns in a later arithmetic
task. While shortcut search and application is reflected in fixa-
tion patterns, we did not obtain evidence for the reverse influ-
ence. Changed fixation patterns did not lead to higher shortcut
usage. Thus, the results are in line with top-down accounts of
strategy change: fixation patterns reflect rather than elicit strat-
egy change (cf. Haider & Frensch, 1999).
Author Note
The first two authors shared first authorship and contributed
equally to this work. Claudia Godau, Humboldt-Universität zu
Berlin, Germany; Maria Wirth, Universität Leipzig, Germany;
Sonja Hansen and Hilde Haider, Universität zu Köln, Germany;
Robert Gaschler, Universität Koblenz-Landau, Germany. This
work was supported by Grant FR 1471/12-1 from the Deutsche
Forschungsgemeinschaft (DFG, as well as by the
Berlin Cluster of Excellence Image Knowledge Gestaltung
Baroody, A. J., & Gannon, K. E. (1984). The development of the co m-
mutativity principle and economical addition strategies. Cognition
and Instruction, 1, 321-339.
Baroody, A. J., Ginsburg, H. P., & Waxman, B. (1983). Ch ildren’s use
of mathematical structure. Journal for Research in Mathematics
Education, 14, 156-168.
Canobi, K. H., Reeve, R. A., & Pattison, P. E. (2002). Young children’s
understanding of addition concepts. Educational Psychology, 22, 513-
Canobi, K. H., Reeve, R. A., & Pattison, P. E. (2003). Patterns of knowl-
edge in children’s addition. Developmental Psychology, 39, 521-534.
Cowan, R., & Renton, M. (1996). Do they know what they are doing?
Children’s use of economical addition strategies and knowledge of
commutativity. Educational Psychology, 16, 407-420.
Dubé, A. K., & Robinson, K. M. (2010). The relationship between adults’
conceptual understanding of inversion and associativity. Canadian
Journal of Experimental Psychology/Revue Canadienne de Psy-
chologie Expérimentale, 64, 60-66.
Gallistel, C. R., & Gelman, R. (1992). Preverbal and verbal counting and
computation. Cognition, 44, 43-74.
Gaschler, R., Vaterrodt, B., Frensch, P. A., Eichler, A., & Haider, H.
(2013). Spontaneous usage of different shortcuts based on the com-
mutativity principle. PLoS ONE, 8, Article ID: e74972.
Gilmore, C. K., McCarthy, S. E., & Spelke, E. S. (2007). Symbolic
arithmetic knowledge without instruction. Nature, 447, 589-591.
Gilmore, C. K., McC arth y, S. E., & Spelke, E. S. (2010). Non-symbolic
arithmetic abilities and achievement in the first year of formal
schooling in mathematics. Cognition, 115, 394-406.
Godau, C., Haider, H., Hans en, S., Vaterrodt, B. , Schubert, T., Frensch ,
P. A., & Gaschler, R. (2013). Increasing the usage of an arithmetic
shortcut by offering an easy-to-find shortcut based on the same
mathematical principle. Manuscript submitted for publication.
Haider, H., & Frens ch, P. A. (1999). Eye movement during skill acqui-
sition: More evidence for the information-reduction hypothesis.
Journal of Experimental Psychology: Learning, Memory, and Cogni-
tion, 25, 172-190.
Hansen, S., Haider, H., Eichler, A., Gaschler, R., Godau, C., & Frensch,
P. A. (2013). Fostering formal commutativity knowledge with ap-
proximate arithmetic. Manuscript submitted for publication.
Madsen, A., Rou infa r, A., Larso n, A. M., Loschk y, L. C., & R ebello , N.
S. (2013). Can short duration visu al cues influence students’ reason-
ing and eye movements in physics problems? Physical Review Spe-
cial Topics—Physics Education Research, 9, Article ID: 020104.
Obersteiner, A., Reiss, K., & Ufer, S . (2013). How training on exact or
approximate mental representations of number can enhance first-
grade students’ basic number processing and arith metic skills . Learn-
ing and Instruction, 23, 125-135.
Resnick, L. B. (1992). From protoquantities to operators: Building mathe-
matical competence on a foundation of everyday knowledge. In G.
Leinhardt, R. P utnam, & R. A. Ha ttrup (Eds.), Ana lysis of arithmetic
for mathematics teaching. Hillsdale, NJ: L. Erlbaum Associates.
Robinson, K. M., & Dubé, A. K. (2012). Children’s use of arithmetic
shortcuts: The role of attitu des in strategy choice. Child Develop ment
Research, 2012, 10.
Sherman, J., & Bisanz, J. (2009). Equivalence in symbolic and non-
symbolic contexts: Benef its of solving problems with manipulativ es.
Journal of Educational Psychology, 101, 88-100.
Siegler, R. S., & Jenkins, E. (1989). How children discover new stra-
tegies (Vol. xiv). Hillsdale, NJ: Lawre n ce Erlbaum Asso ciates, Inc.
Siegler, R. S., & Stern, E. (1998). Conscious and unconscious strategy
discoveries: A microgenetic analysis. Journal of Experimental Psy-
chology: General, 127, 377-397.
Sophian, C., Harley, H., & Martin, C. S. M. (1995). Relational and
representational aspects o f early number development. Cognition and
Instruction, 13, 253-268.
Thomas, L. E., & Lleras, A. (2007). Moving eyes and moving thought:
On the spatial compatibility between eye movements and cognition.
Psychonomic Bulletin & Review, 14, 663-668.
Verschaffe l, L., Luwel, K., Torb eyns , J ., & Do o ren , W. V . (2 0 09 ). C o n -
ceptualizing, investigating, and enhancing adaptive expertise in ele-
mentary mathematics educatio n. European Journal of Psychology of
Education, 24, 335-359.
Wilkins, J. L., Bar oo dy, A. J. , & Tiilikain en, S. (20 01). Kindergartn ers ’
understanding of additive commutativity within the context of word
problems. Journal of Experimental Child Psychology, 79, 23-36.
Table A1.
The 12 arithmetic problems (co mmutative and non-co mmutative prob-
lems). The results printed in italics had to be calculated by the partici -
Screen 1
3 + 5 + 4 = 12
4 + 9 + 8 = 21
4 + 8 + 9 = 21
6 + 2 + 5 = 13
9 + 7 + 2 = 18
2 + 9 + 7 = 18
Screen 2
6 + 3 + 2 = 11
6 + 2 + 3 = 11
8 + 9 + 6 = 23
7 + 2 + 6 = 15
6 + 7 + 2 = 15
7 + 4 + 8 = 19