How we
use what we learn in Math: An integrative account of the
development of commutativity
Hilde Haider^{a},
Alexandra Eichler^{ a},
Sonja Hansen^{ a},
Bianca Vaterrodt^{ b},
Robert Gaschler^{ c},
Peter A. Frensch^{ b}
^{a}University
of
^{b}HumboldtUniversity
^{c}University
Article received 28 May 2013 /
revised 6 January 2014/ accepted 16 January 2014 / available
online 27 January 2014
Abstract
One crucial issue in mathematics
development is how children come to spontaneously apply
arithmetical principles (e.g. commutativity). According to
expertise research, wellintegrated conceptual and procedural
knowledge is required. Here, we report a method composed of
two independent tasks that assessed in an unobtrusive manner
the spontaneous use of procedural and conceptual knowledge
about commutativity. This allowed us to ask (1) in which grade
students spontaneously apply this principle in different task
formats and (2) in which grade they start to possess an
integrated concept of the commutativity. Procedural and
conceptual knowledge of 8 to 9 year olds (163 second and 180
third graders) as well as 46 adult students was assessed
independently and without any hint concerning commutativity.
Results indicated procedural as well as conceptual knowledge
about commutativity for second graders. However, their
procedural and conceptual knowledge was unrelated. An
integrated relation between the two measures first emerged
with some of the third graders and was further strengthened
for adult students.
Keywords: Conceptual Knowledge;
Procedural Knowledge; Commutativity; Integrated Concept.
http://dx.doi.org/10.14786/flr.v2i1.37
ISSN 22953159
Corresponding author: Hilde Haider, Department of
Psychology, University of Cologne, RichardStraussStr. 2
1. Introduction
One major skill in mathematics is the acquisition
of adaptive expertise. That is, students should be able to
deliberately recognize those constraints that allow to apply a
certain mathematical principle (e.g., Torbeyns, De Smedt,
Ghesquière & Verschaffel, 2009; Verschaffel, Luwel, Torbeyns
& Van Dooren, 2009). For instance, in the
An important question with regard to adaptive
expertise is how students come to recognize that they can use a
certain principle in order to facilitate calculation. Or to put
it in other words, what kind of knowledge underlies the ability
to adaptively apply a mathematical principle spontaneously whenever it facilitates
calculation? In the research on adaptive expertise, it is widely
accepted that this ability is not only based on procedural
knowledge (knowing how to apply a certain strategy), but also on
conceptual knowledge (knowing when and why a certain principle
applies). Procedural and conceptual knowledge should be
integrated and the resulting knowledge base should be abstract
enough to ensure flexibility in knowledge application (e.g.,
Anderson & Schunn, 2000; Baroody, 2003; Gentner &
Toupin, 1986; Haider & Frensch, 1996; Koedinger &
Anderson, 1990; Star, 2005; Verschaffel et al., 2009). As one example of
linking concepts and procedures, this research has revealed that conceptual
knowledge is important to guide attention to task relevant
information in order to solve problems (e.g., Baroody &
Rosu, 2006). An abstract conceptual understanding might also
be of particular importance when knowledge has to be
transferred from one domain to another (e.g., Goldstone &
Sakamoto, 2003; Kaminsky, Sloutsky & Heckler, 2008). It
supports flexible shortcut application when problems to which
a principle applies are presented mixed with problems to which
the principle does not apply. For
instance, Siegler and Stern (1998) have shown that second
graders relied less on inversionbased procedures when inversion
problems (a + b – b) were randomly interspersed with control
problems (a + b – c) compared to blocked presentation. Mixed
presentation of inversion and control problems hindered the use
of inversion shortcuts. This suggests that younger children do
not deliberately recognize the constraints important for
applying the inversion principle. Rather, they simply seem to
know that the strategy applies for a certain class of problems.
Likely they cannot rely on a well integrated, abstract
understanding of the inversion principle (i.e., they have not
yet developed adaptive expertise). But, how and when do children
develop an integrated representation of basic mathematical
principles?
The first goal of the current study was to develop
a method to unobtrusively measure the spontaneous application of
procedural and conceptual knowledge taking the commutativity
principle as a test case. The second goal was to shed some light
on the development of an abstract and well integrated
representation of the commutativity principle. With regard to
the second goal, we pursued to different questions: (1) In which
grade are students able to spontaneously apply commutativity
knowledge in different task formats and (2) in which grade
starts performance expressed in the different tasks to correlate
with oneanother? For this purpose, we investigated the
deliberate use of the commutativity principle in two different
situations. The term “deliberate” means that children did not
receive any hint about the commutativity principle at all (e.g.,
Torbeyns et al., 2009). In the first test children simply solved
addition problems that sometimes allowed for a shortcut based on
the commutativity principle (procedural knowledge). The second
test was aimed at assessing conceptual knowledge. Children were
instructed to mark – without solving the problem – those
problems that they believed could be solved without calculation.
Hence, this task required children to realize that the order of
addends does not change cardinality. The correlation between
these two independent tasks allowed us to gauge how well
integrated children’s knowledge was. Focusing on just two
unobtrusive measures (one procedural and one conceptual), the
current work can potentially lay the ground to develop
multimethod approaches in the same spirit, safeguarding that
multiple testing does not cue participants towards what the test
situation is about.
We focused
on the commutativity principle as it is
one of the most basic properties in mathematics. It refers to
the principle that changing the order of operands in addition
and multiplication does not change the end result. It is known
as a fundamental property of many binary operations. The
commutativity of simple operations, such as the multiplication
and addition of numbers are usually acquired throughout
elementary school. However, many mathematical proofs also depend
on this property.
1.1 Development of
procedural and conceptual knowledge about commutativity
Former research in the field of developmental
psychology has already shown that children acquire informal
knowledge of commutativity as an arithmetic principle long
before they enter school (e.g., Baroody &
Gannon, 1984; Baroody, Ginsburg & Waxman, 1983; Canobi,
Reeve & Pattison, 1998, 2002; Cowan & Renton, 1996;
Resnick, 1992; Siegler & Jenkins, 1989; Sophian, Harley
& Manos Martin, 1995). One potential reason for this early development
is that at least the core property of commutativity, the
orderirrelevance principle, applies to many nonnumerical
situations. For example, children may
experience that some tasks require a certain sequence (e.g.,
putting on one’s clothes), whereas others do not (e.g., laying
the table). Already toddlers have many opportunities to learn
that order does not affect the end result in some situations,
but does in others.
Orderirrelevance is also a core principle for
counting (e.g., Gelman, 1990;
Gelman & Gallistel, 1978). Learning to count requires children to
learn, on the one hand, that the sequence of number words is
relevant. On the other hand, the sequence in which the objects
are counted is irrelevant. Consequently, Briars and Siegler (1984) found that children
need time to understand orderirrelevance in counting.
Furthermore, counting is the dominant skill through which
preschool children learn to map concrete objects to numbers.
Also, counting is one of the important precursors of addition.
Through counting, preschool children can learn
orderirrelevance in a numerical manner before entering school.
They thus do not only have the chance to understand
orderirrelevance in a nonnumerical manner.
However, even though considerable interest in
research on counting and addition principles emerged already in
the 1980s and still continues (e.g., Baroody, 1984;
Baroody & Gannon, 1984; Baroody et al., 1983; Briars &
Siegler, 1984; Canobi, Reeve & Pattison, 1998, 2002, 2003;
Fuson, 1988; Gelman & Gallistel, 1978; Gelman & Meck,
1983; Resnick, 1992; Sophian & Adams, 1987; Starkey &
Gelman, 1982),
the central question has not been solved yet: How
and when do children acquire integrated knowledge
representations, in the sense of true formal arithmetic
principles? For instance, Geary (2006) stated that it is not
clear when children “explicitly understand commutativity as a
formal arithmetical principle” (p. 791).
On the one hand, the difficulties in answering this
question are due to the fact that researchers by no means agree
upon the characteristics of procedural or conceptual knowledge
that must be given in order to conclude that children possess an
abstract mathematical concept (cf., Star, 2004). Concerning procedural knowledge, most
researchers agree that it refers to the ability to apply a
certain strategy when performing a mathematical task (e.g.,
Hiebert & LeFevre, 1986). Conceptual knowledge or
metastrategic competences (Kuhn, GarciaMila, Zohar &
Andersen, 1995) often is assumed to refer to children’s
explicit understanding of a certain principle (i.e., why and
when it is allowed to a apply a certain strategy; e.g.,
Baroody, Feil & Johnson, 2007; Hiebert & LeFevre, 1986;
RittleJohnson, Siegler & Alibali, 2001).
On the other hand, there is no consensus how best
to assess procedural and conceptual knowledge. One frequently
used approach to measure procedural knowledge is to ask children
to solve addition problems and afterwards have them explain
their strategies (e.g., Baroody & Gannon, 1984; Baroody,
Ginsburg & Waxman, 1983; Bisanz & LeFevre, 1992; Canobi
et al., 1998, 2002, 2003; Cowan & Renton, 1996). Conceptual
knowledge in these studies has, for example, been assessed by
letting children observe a puppet solving problem pairs (see,
e.g. Baroody et al., 1983; Canobi et al, 1998). If a child on
enquiry stated that the puppet could know the answer to the
second problem from looking at the previous one, he or she was
asked for reasons and eventually prompted for more detailed
explanations. This form of assessment implies that children are
being informed about the underlying arithmetic principle – at
least they are made aware that different efficient strategies
are applicable. Such procedural and the conceptual knowledge
tests might guide children’s attention to the taskrelevant
information. Also, it is conceivable that they look at the
problems more attentively when they are asked to verbalize their
strategies. Consequently, conclusions concerning the question
whether a child possesses abstract conceptual knowledge may vary
depending on the tests that were applied.
Based on the abovementioned forms of assessment,
the empirical research on commutativity suggests that conceptual
and procedural knowledge in this domain are moderately related (e.g., Baroody et al.,
1983; Canobi, 2004; Canobi et al., 1998). However, the findings do
not allow to exclude that the acquired conceptual knowledge of
first or even second graders is still domainspecific rather
than akin to an abstract concept representing the formal
arithmetic principle of commutativity (e.g., Bisanz, Watchhorn,
Piatt & Sherman, 2009; Geary, Hoard, ByrdCraven &
DeSoto, 2004; LeFevre et al., 2006). Therefore, investigating
the spontaneous application of commutativity knowledge would
complement and broaden this research.
In summary,
the goal of our study was twofold: First, we
aimed to develop a method to unobtrusively test for spontaneous application of
procedural and conceptual commutativity knowledge. Our second
goal was to investigate the degree of integration of this
spontaneously expressed procedural and conceptual knowledge of
second and third graders. Additionally, for means of comparison,
we also tested adult students. As described above, knowledge
about a mathematical principle like, for instance, the
commutativity principle, can be said to represent an integrated
or abstract concept in the sense of a true formal mathematic
principle when learners are able to apply their knowledge whenever task constraints
permit. That is, learners should be able to deliberately
recognize task properties allowing them to apply the
mathematical principle irrespectively of task context.
2. Method
2.1.
General Method
We investigated the commutativity principle with
threeelement addition problems (i.e., 5+3+7 = ?)[1].
These threeelement problems are unfamiliar at least for younger
students. Since we wanted to investigate whether or not students
would recognize the applicability of the commutativity principle
without any further information about this principle, we needed
less familiar problems. Therefore, we accepted that
threeelement problems implicitly presuppose knowledge about
associativity (e.g., Canobi et al., 1998).
The threeelement problems were always presented in
blocks, one problem beneath the other. Unbeknownst to the
participating students, some problems consisted of identical
addends in a different order as the preceding problem, and thus
could be solved without calculation (commutative problems,
hereafter). Students received two different and completely
independent task formats.
The first task, the arithmetic task, consisted of
two blocks. One block contained interspersed commutative
problems, the other one did not. Participants did not receive
any information about the existence of these commutative
problems. Rather, they were simply asked to solve the two blocks
of addition problems as fast and accurately as possible. If
students are faster when working on the block that includes
threeelement commutative problems as compared to the block
which does not contain such shortcut options, they can be said
to possess procedural knowledge about commutativity.
In the second task, the socalled judgment task,
students were instructed to identify those problems which they
believed need no calculation. They explicitly were told to
refrain from calculating any problems. If students understand
that the order of identical addends does not change the
cardinality, they should be able to correctly mark the
commutative problems. By virtue of this second task type, we
were able to assess conceptual knowledge about commutativity
without cueing the concept.
Importantly and in contrast to former experiments
(e.g., Baroody et al., 1983; Canobi, 2005), participants in our
experiment did not receive any hint about the existence of
commutative problems in either task. That is, they were not
instructed to further explain their strategies. The rationale
behind this procedure was that any instruction to think about
the strategies used to solve the problems might trigger active
search for regularities, thereby making it impossible to assess
the spontaneously activated concept
of commutativity. If students possess an abstract understanding
of the commutativity principle, they should be able to recognize
and rely on the relevant task characteristics in any task
context and without any hint (e.g., Bisanz et al., 2009; Prather
& Alibali, 2009).
To the
extent that children have acquired an abstract concept of
commutativity, performance should correlate between both of our
two tasks reflecting knowledge about this principle. Likely
procedural and conceptual knowledge about commutativity becomes
iteratively more integrated in the first years of primary
school. We should thus find that the relation between procedural
and conceptual knowledge is stronger in third graders as
compared to second graders (e.g., LeFevre et al. 2006).
2.2 Participants
Overall, 163
second graders (79 girls) with a mean age of 8 years 1 month (SD
= 7 months), 180 third graders (91 girls) with a mean age of 9
years 1 month (SD = 8 months) participated in the study. As a
control condition, we also collected data of 46 students of the
University of Cologne (37 women) with a mean age of 23.6 years
(SD = 5.2). Children were recruited from six different
elementary schools located in middle socioeconomic status
suburbs of Cologne. All children had their parents’ or
guardians’ permission to participate in the study.
2.3 Procedure and
Materials
The study consisted of two parts. In the first
part, participants received the arithmetic task: one block with
interspersed repetitions of addends in changed order in
consecutive problems (commutativity block) and one block without
such repetitions (control block). In the second part,
participants were administered the judgment task. Both tasks
were designed as paperpencil tests and children and adult
students were tested in groups of up to 25 participants in a
classroomlike setting.
We generated
three sets of 30 arithmetic problems with three addends between
2 and 9 (e.g., 3 + 6 + 8 =
?; maximum
result was 24; 1 as an addend was not included). The problems in
all three sets yielded at least approximately the same totals
and within a problem each numeral could only occur once.
The 30
problems of each of the two blocks were distributed over five
pages with six problems on each page. In the commutativity
block, each page contained two pairs of commutative problems
(i.e., one problem and its repetition with a different order of
addends). In the control block no such commutative pairs
occurred. Instead participants received pairs of control
problems which yielded the same results but were composed of
different addends. In both blocks, participants were instructed
to calculate the problems page by page from top to bottom.
The judgment
task consisted of overall 30 problems with 10 problems per page.
On each page, three pairs were commutative pairs and the
remaining four problems were filler problems. The first page was
for practice only. Participants were instructed to first solve
all 10 problems from top to bottom on the page. Afterwards they
were asked to mark those problems that needed no calculation on
that page. In particular, they were told that some of the
problems need no calculation and that they should figure out for
which of these problems they could have written down the result
without calculation. After this practice
page, participants were instructed to only judge on the next pages
whether or not they needed to calculate the result for a problem
without actually attempting to solve it. Therefore, all problems
on pages 2 and 3 were presented without equal sign. Instead,
there was a circle to the right of each problem and participants
were told to mark this circle when they believed they did not
need to calculate the result. Again, students were instructed to
work on the problems from the top to the bottom of each page.
Table 1 depicts examples of the problems in each of the two
arithmetic blocks and the judgment task.
Table 1
Examples of the problems presented on one page in
the two arithmetic blocks (commutativity and control block)
and the judgment task
Arithmetic task 
Judgment task 

Commutativity Block 
Control Block 


3 + 5 + 4 = 4 + 9 + 8 = 4 + 8 + 9 = 6 + 2 + 5 = 9 + 7 + 2 = 2 + 7 + 9 = 
5 + 3 + 4 = 8 + 9 + 4 = 6 + 7 + 8 = 5 + 2 + 6 = 2 + 7 + 9 = 9 + 4 + 5 = 
2 + 7 + 9 9 + 5 + 4 2 + 6 + 5 6 + 5 + 2 8 + 7 + 5 3 + 5 + 6 6 + 5 + 3 2 + 9 + 5 6 + 7 + 9 9 + 6 + 7 
š š š š š š š š š š 
Problems
in bold indicate the commutative pairs of the respective task
Each of the two arithmetic blocks was administered
as a separate booklet, as was the judgment task. Students only
worked with a pencil and were not allowed to use an eraser.
Rather, to increase the reliability of the timing measure, they
were told to cross out any errors and to write the correct
answer right beside the problem. An experimenter instructed all
participants in the classroom.
The experiment started with six arithmetic practice
problems with three addends. The only goal of this phase was to
familiarize the children with the task requirements. Students
were given 2 minutes to solve these six warmup problems. Then,
the first of the two arithmetic blocks was presented.
Approximately half of the children (second graders and third
graders) and all adults in the control condition received the
commutativity block first, followed by the control block. The
remaining participants started with the control block and
subsequently received the commutativity block. The time limit
was set to 3 minutes per block (1 minute for adult students)
with a 1minute break between blocks.
One minute
after having finished the second arithmetic block, the judgment
task was presented. Participants were allowed 2 minutes (adult
students again 1 minute) to calculate the problems on the
practice page. Afterwards they had the same amount of time for
marking those problems they believed they could have solved
without calculation. After the practice phase, the same time
limit was applied for the two subsequent pages, so that time did
not suffice to calculate the problems and to concurrently mark
those problems requiring no calculation. In addition, up to four
additional experimenters observing small groups of children (up
to six) ensured that they were not calculating the problems.
After this last block, all children received some sweets. Adult
students in the control condition were debriefed about the
study.
2.4. Design
Independent
variables were grade (second versus third graders) and block
type (commutativity versus control block in the arithmetic
task). Dependent variables in the arithmetic problem blocks were
calculation time per problem in each of the two arithmetic
blocks, as well as the number of correct results. Calculation
time was computed separately for each participant and each of
the two arithmetic problem blocks by dividing the individual
number of completed problems by the total time given for the
block in the respective age group (three minutes for second and
third graders; 1 minute for adults). For the judgment task, the
dependent variables were relative number of hits (correctly
identified commutative problems) and false alarms (problems
incorrectly identified as commutative problems), as well as the
sensitivity index d’ from signal detection theory (i.e., the
difference between ztransformed hit rate and false alarms
rate).
2.5 Splithalf
Reliability
In order to
check if our measures are reliable, we computed splithalf
reliabilities for each task type. That is, for each of the two
age groups and the adults, we calculated correlations between
the two arithmetic blocks (control and commutativity block) and
between the second and third pages of the judgment task (the
practice page of the judgment task was excluded). Table 2 shows
the SpearmanBrown corrected correlation coefficients separately
for each age group and each task format (arithmetic task and
judgment task).
Table 2
SpearmanBrown corrected correlation coefficients
for the arithmetic task and the judgment task for all
participants and separately for the three age groups
(Arithmetic task: correlation between the amount of computed
problems in the commutativity block and the control block;
Judgment task: correlation between correct responses on the
first and on the second pages of the test)
Arithmetic
task 
Amount
of computed commutative problems 

All
participants 
Grade
2 
Grade
3 
Adults 

Amount
of computed control problems 
.90 
.88 
.88 
.95 
Judgment
task 
Amount
of correct judgments on the first page 

All
participants 
Grade
2 
Grade
3 
Adults 

Amount
of correct judgments on the second page 
.82 
.83 
.78 
.86 
As can be
seen from Table 2, the correlation coefficients in each age
group ranged between r = .78 and r = .95. Thus, the two tasks
used to assess participants’ procedural and conceptual knowledge
seem to be reliable measures.
3. Results
Second or
third graders were excluded from further analyses if they
completed less than 16 problems across the two arithmetic
problem blocks (i.e., 2 standard deviations below the group
means; 15 second graders, 12 third graders, and 1 adult). They
were also excluded from further analyses if they solved all 30
problems in the control and the commutativity block, as
calculation times could not be calculated for these participants
(2 second graders, 23 third graders, and 8 adults). This led to
146 second graders, 145 third graders, and 37 adult students in
the control condition. The following result section is divided
into three parts. We first describe the results for the
arithmetic problem blocks. Second, we report
the performance in the judgment task. Lastly, we analyze the
relation between these two tasks.
3.1 Arithmetic Task
As a
preliminary analysis did not reveal substantial effects of the
order of presentation (commutative problem first followed by
control problem or vice versa), we collapsed the data for all
participants within the groups of second and third graders.
Table 3 depicts the calculation times for problems in the
commutativity and the control blocks per age group. Mean
calculation times suggest that second and third graders
benefitted from the commutative problems whereas adult students
did not.
Table 3
Calculation times per task in the commutativity
and the control block for each age group. The table holds the
means and standard deviations for the different age group in
seconds as well as lower and upper limit of the 95%
confidence interval (CI; Loftus & Masson, 1994)
Age group 
Commutativity block 
Control block 
N 

M (SD) 
M±95%CI 
M (SD) 
M±95%CI 

Grade 2 
12.33 (3.74) 
12.09 
13.28 (4.39) 
12.95 
146 
Grade 3 
9.55 (2.51) 
9.34 
9.93 (3.12) 
9.72 
145 
Adults 
3.79 (1.04) 
3.66 
3.63 (1.04) 
3.50 
37 
A 2 (Age group) X 2 (Block type: commutativity
block vs. control block) analysis of variance (ANOVA) with
calculation time as dependent variable revealed significant main
effects of Age group (F[1, 289] = 66.23, MSe = 20.64, p < .01, ²
= .23), and of Block type (F[1,
289] = 15.94, MSe = 4.04, p < .01, ² = .06). The interaction between
Age group and Block type was close to significance (F[1,
289] = 2.98, MSe = 4.04, p = .088). Planned
contrasts revealed that only second graders significantly
profited from the commutative problems (second graders: F[1, 289] =
16.32, MSe = 4.04, p < .01, ² = .06; third graders: F[1,289] =
2.59, MSe = 4.04, p = .108). A separate
ttest with Block type as withinparticipants variable revealed
that the adult control group did not show a significant benefit
from commutative problems (t < 1).
In addition,
we also analyzed the percentage of correct responses in the
commutativity and the control blocks. Table 4 presents the
percentage of correct responses in the two age groups for these
two types of problems. As can be seen from Table 4, percentage
of correct responses was higher for third as compared to second
graders. Accordingly, the 2 (Age group) X 2 (Block type) ANOVA
yielded a significant main effect of Age group (F[1, 289]
= 3.8, MSe = 118.27, p < .05, η² = .01). No other effect was
significant.
Table 4
Mean percent correct responses in the three age
groups for the commutativity and control blocks. Also depicted
are standard deviations (in parentheses) and the lower and
upper limit of the 95% confidence interval (CI; Loftus &
Masson, 1994)
Age group 
Commutativity block 
Control block 
N 

M (SD) 
M±95%CI 
M (SD) 
M±95%CI 

Grade 2 
92.66 (9.85) 
91.64 
93.11 (9.13) 
92.0894.13 
146 
Grade 3 
94.33 (8.62) 
91.64 
94.82 (8.31) 
94.0495.60 
145 
Adults 
96.66 (9.08) 
94.8198.51 
95.94 (12.55) 
94.0997.78 
37 
Overall, the
results up to this point show that third graders were faster and
less error prone as compared to second graders. Furthermore and
more importantly, second graders showed a substantial benefit of
commutative problems. Third graders in tendency also profited
from these problems, but for them the effect was not
significant. Adults, by comparison, did not show such a benefit,
probably due to a floor effect based on the simplicity of the
problems (for similar results, see Robinson & Dubé, 2009;
Robinson & Ninowski, 2003).
3.2 Judgment Task
For each
student, we individually computed the hit rate, false alarms
rate, and the sensitivity index (d’) from signal detection
theory[2].
Table 5 depicts the means for these dependent measures
separately for each of the two age groups and the adult
students. As expected, hit rate was higher than false alarms
rate in all age groups. Accordingly, the sensitivity index d’
differed significantly from chance (all ts > 2.5, ps <
.01). This suggests that students were able to correctly
identify at least some of the commutative problems. In addition,
d’ was substantially higher in third as compared to second
graders (t[289] = 2.18, p < .05, η² =
.02).
Table 5
Rate of hits, false alarms and d’ in each of the
three age groups in the judgment task. Standard deviants are
given in parentheses.
Age group 
Judgment task 

Hits 
False alarms 
d’ 

Grade 2 
.70 (.28) 
.36 (.33) 
1.81 (2.35) 
Grade 3 
.81 (.22) 
.35 (.32) 
2.41 (2.32) 
Adults 
.82 (.25) 
.11 (.18) 
3.92 (2.33) 
Low sensitivity could result from two different
sources: the difficulty to identify commutative problems (hit
rate) or a tendency to mark other than commutative problems
(false alarm rate). Therefore, we additionally analyzed the hit
and false alarm rates in the two age groups. These analyses revealed that
higher sensitivity in grade 3 as compared to grade 2 was mainly
due to a higher hit rate. Second graders were less able to
identify the commutative problems than third graders
(t[289] =
3.44, p < .01, h² = .04). The false alarm
rate did not differ significantly between these two age groups (t < 1). By
contrast, as can be seen from Table 5, the higher sensitivity in
adults as compared to third graders resulted from a lower false
alarm rate in adults, whereas hit rate was almost identical in
these two age groups.
Thus, older participants were better able to
discriminate between commutative and control problems than
younger participants. This finding from our crosssectional
agecomparison suggests a progress in conceptual knowledge with
increasing age (as cohort differences are unlikely).
3.3 Relation between
Procedural and Conceptual Knowledge
The results reported up to this point are somewhat
counterintuitive. Even though third graders were better able to
identify commutative problems in the judgment task, they seemed
to rely less on a commutativitybased shortcut during
calculation than second graders. In addition, for adults we
found no benefit of commutative problems in the arithmetic task.
Thus, it seems that either the willingness to use more efficient
arithmetic strategies or procedural knowledge of commutativity
itself decreases (while conceptual knowledge increases with
age).
The last analyses of the relationship between
procedural and conceptual knowledge might help to reconcile this
picture. These analyses will answer the research question
whether or not participants possess an integrated concept of
commutativity. If so, we should find significant positive
correlations between the use of the commutativebased shortcut
in the arithmetic task (procedural knowledge) and the ability to
correctly identify the commutative problems in the judgment task
(conceptual knowledge).
For procedural knowledge, we used for each participant the average calculation time per problem in the control and the commutativity block of the arithmetic task as well as the difference between these two measures (i.e., savings; with positive values indicating shorter calculation times in the commutativity block). For conceptual knowledge, we used hit rate, false alarms rate, and the sensitivity measure d’. In a first analysis we calculated correlations across second and third graders. Second, we calculated correlations within the two age groups and for the adults. Table 6 depicts the correlation between procedural and conceptual knowledge.
Table 6
Correlation coefficients between procedural and
conceptual knowledge depicted separately for all second and
third graders as well as for the three age groups. Procedural
knowledge is indicated by calculation times in seconds for
commutative problems, control problems, and in addition for
savings. Hit rate, false alarms, and d’ indicate conceptual
knowledge


Arithmetic
tasks 



Judgment task 
Commutative problems 
Control problems 
Savings 
N 
Second and third graders 
Hits 
0.23^{**} 
0.17^{**} 
0.001 
291 
False Alarms 
0.09 
0.09 
0.02 


d’ 
0.05 
0.02 
0.01 







Grade 2 
Hits 
0.23^{**} 
0.18^{**} 
0.04 
146 
False Alarms 
0.13 
0.15 
0.03 


d’ 
0.09 
0.04 
0.008 







Grade 3 
Hits 
0.02 
0.06 
0.11 
145 
False Alarms 
0.06 
0.04 
0.01 


d’ 
0.05 
0.09 
0.06 







Adults 
Hits 
0.23 
0.03 
0.39^{*} 
37 
False Alarms 
0.15 
0.24 
0.14 


d’ 
0.11 
0.13 
0.36^{*} 
^{** }p
< .01; ^{*} p < .05
As can be seen from Table 6, hit rate for the
entire group correlated negatively with calculation time for
commutative and control problems.
That is, the faster second and third graders solved the
arithmetic problems, the better they were able to identify
commutative problems in the judgment task. Savings in solution
time due to commutative problems were not related to the ability
to identify commutative problems, suggesting that their
knowledge about commutativity was not very well integrated.
A closer
look at the different age groups revealed, however, that adults showed the expected
positive correlation between savings and sensitivity. Adults who
applied the commutativitybased shortcut in the arithmetic
blocks were also those who were better able to identify the
commutative problems in the judgment task. This correlation
suggests that the tested adults do possess an integrated
knowledge representation of the commutativity principle.
In contrast, second and third graders’ procedural
and conceptual knowledge were only weakly related at best. As
Table 6 additionally reveals, second graders’ hit rate
correlated negatively with calculation time. Again, this
correlation suggests that the faster second graders solved the
arithmetic problems the better they were able to identify the
commutative problems in the judgment task. Thus, second graders’
ability to discriminate between commutative and control problems
was linked to more general calculation competencies rather than
to their procedural knowledge about using the
commutativitybased shortcut. Third graders, by contrast, did
not show any significant correlation between calculation
performance and discrimination.
Overall, these findings suggest that only adults’
spontaneous application of commutativity knowledge is based on
an integrated concept of the commutativity principle. In
contrast, procedural and conceptual knowledge seem to be only
weakly related in second and third graders. Note that alternatively, one also could argue that our
assessments of procedural and conceptual knowledge are not
sufficiently reliable (but, see Table 2). In order to further
rule out this latter argument and to better understand the
missing correlations between savings in the arithmetic task
(procedural) and the sensitivity index in the judgment task
(conceptual knowledge), we conducted a final fine grained
analysis for second and third graders.
In the judgment task, false alarms rate of second
and third graders was rather high (approximately 40%; Table 5)
and differed largely between participants in both age groups.
Presumably, children with a high false alarms rate might have
correctly recognized the commutative problems in the judgment
task, but at the same time might have believed that also easy to
calculate problems (i.e., those with comparatively small
addends) needed no calculation. This might have inflated false
alarms rate and thus might have reduced the correlations between
procedural and conceptual knowledge within second and third
graders.
Table 7
Mean hit rates for second and third graders with
no (FA = 0), medium (FA ≤ 50%), or high (FA > 50%) false
alarms rate in the judgment task

False Alarm rate 

No false alarms 
Medium FArate 
High FArate 

Hit rate 
N 
Hit rate 
N 
Hit rate 
N 

Grade 2 
.81 
41 
.52 
61 
.85 
44 
Grade 3 
.88 
44 
.73 
60 
.84 
41 
As can be
seen from Table 7, for second and third graders hit rate was
high when either the false alarm rate was low or when the false
alarm rate was high (i.e., some children marked only the
commutative problems while others marked the commutative and
many other problems). This might have caused the overall low
correlations between procedural and conceptual knowledge within
these two age groups. Therefore, we reanalyzed the correlation
between hit rates and d’ and arithmetic abilities separately for
these three groups within second and third graders. Table 8
presents these correlations. In both age groups, only those
participants who produced high hit rates without incorrectly
marking the filler problems also showed substantial
correlations. However, second and third graders differed
qualitatively with regard to these correlations.
Table 8
Correlations between procedural and conceptual
knowledge for second and third graders with no, medium, or
high false alarms rate. Procedural knowledge is indicated by
calculation times in seconds for commutative and control
problems as well as savings. Hit and false alarms rate (FA)
indicate conceptual knowledge

Grade
2 

No
false alarms (N
= 41) 
Medium
FArate (N
= 61) 
High
FArate (N
= 44) 

Hits 
FA 
Hits 
FA 
Hits 
FA 

Commutative 
.50^{**} 
 
.11 
.07 
.01 
.01 
Control 
.39^{**} 
 
.11 
.10 
.02 
.04 
Savings 
.03 
 
.11 
.15 
.04 
.04 

Grade
3 


No
false alarms (N
= 44) 
Medium
FArate (N
= 60) 
High
FArate (N
= 41) 


Hits 
FA 
Hits 
FA 
Hits 
FA 
Commutative 
.25 
 
.04 
.12 
.04 
.22 
Control 
.01 
 
.07 
.08 
.04 
.18 
Savings 
.31^{*} 
 
.05 
.03 
.02 
.04 
^{** }p
< .01; ^{*} p < .05
4.
Discussion
With the current study we aimed at presenting an
approach to unobtrusively measure the spontaneous usage of
procedural and conceptual knowledge of the commutativity
principle. Apart from providing a basis to develop the method
further (see below), the second goal of our study was to
investigate the relation between procedural and conceptual
knowledge about the commutativity principle in second and third
graders. For this we asked (1) at which grade the different
forms of commutativity knowledge can be detected and (2) at
which grade they start to correlate with oneanother.
Overall, our study yielded three main results:
First, as expected, third graders showed higher general
calculation proficiency (procedural knowledge) and more
conceptual knowledge about commutativity than second graders.
Second, a solution time benefit based on the procedural use of
the commutativity principle was only found for second graders.
They calculated commutative problems faster than control
problems. Neither calculation times of third graders nor of
adult students reflected significant profit from interspersed
commutative problems. Third, the correlation between (a) the
benefit resulting from a commutativitybased shortcut and (b)
conceptual knowledge of commutativity was rather weak in second
and third graders. The relation seems to arise in some of the
third graders. The link was also present in the control group
(adults). The second and third findings merit some further
discussion before we come to the theoretical and practical
implications.
The second finding (i.e., that only second graders’
calculation performance reflected the exploitation of
commutativity whereas that of third graders and adults did not)
was somewhat surprising. Interestingly however, Gaschler,
Vaterrodt, Frensch, Eichler, and Haider (2013) found similar
patterns of results with the identical arithmetic task.
Therefore, we assume that this finding is not due to a sample
artefact. Nevertheless, it does not fit the general claim that
with experience, children become faster and more accurate at
solving addition problems and also tend to use more
sophisticated strategies, such as orderirrelevant,
decomposition, and retrieval strategies (Baroody et al., 1983;
Canobi et al., 1998, 2002; 2003; Geary, Brown &
Samaranayake, 1991; Goldman, Mertz & Pellegrino, 1989;
Resnick, 1992; RittleJohnson & Siegler, 1998; Siegler,
1987; but see, McNeil, 2007; Robinson & Dubé, 2009; Robinson
& Ninowski, 2003; Torbeyns et al., 2009). It also seems to
contradict the results of Baroody et al. (1983), which show that
approximately 80% of their third graders applied the
commutativitybased shortcut to solve arithmetic problems (see also, Canobi et al.,
2003).
One obvious reason for these divergent findings
might be that we used threeelement addition problems which
probably were hard for second graders but (due to the rather
small addends) easy for third graders and adults. This may have
caused second graders to rely on the more efficient
commutativitybased shortcut strategy, whereas third graders and
adults were fast in solving the problems anyway, so that they
did not consider any gain through using the shortcut strategy.
For instance, Siegler and Araya (2005) mentioned that
participants are more likely to adopt solution strategies if
they contribute to significant performance advantages. This
argument is further supported by the results of Gaschler et al.
(2013) who found larger benefits when presenting problems with
large rather than with small addends.
A second reason might be that in former studies
(e.g., Baroody et al., 1983; Canobi et al., 1998;
FarringtonFlint, Canobi, Wood & Faulkner, 2010) students
were instructed to explain their strategy immediately after
responding. By contrast, our participants received no hint about
the existence of commutative problems. While in our study the
use of any shortcut strategy was spontaneous, it is possible
that the explanation required in the Baroody et al.’s study
might have triggered students to apply the commutativitybased
strategy. For instance, Torbeyns et al. (2009) found less
strategy application when students could spontaneously apply
different strategies during calculation than when they were
instructed to do so. In a similar vein, a yet unpublished study
from our labs revealed that second and third graders as well as
adults substantially benefitted from instruction (compared to a
noninstructed group). Participants reminded of the
commutativity principle and alerted to the fact that commutative
problems might occur, showed a larger solution time advantage on
commutative problems as compared to control problems. As
students of all three age groups relied on the commutativity
principle after being instructed accordingly, it seems justified
to conclude that (with the exception of second graders) students
in our study indeed did not profit much from spontaneously
applying the commutativitybased shortcut strategy.
Concerning our second research question, we found
that the second graders’ understanding of commutativity was
unrelated to their use of commutativitybased shortcut
strategies. First signs of an integrated concept (assessed by
the correlation of procedural and conceptual knowledge measures)
occurred in a small group of third graders and were substantial
only for adult students. Thus, the integration of procedural and
conceptual knowledge seems to increase with age. However, it
also suggests that second graders may have used the shortcut
strategy without entirely understanding the commutativity
principle. This finding seems at odds with the early onset
assumption of, for example, Baroody and Gannon (1984). Furthermore, Canobi et
al. (1998; 2002) had found that
second graders’ conceptual (assessed by an explanation task) and
procedural knowledge (assessed by solving addition problems)
correlated moderately. However, as already discussed concerning
Baroody et al.’s (1983) findings, Canobi
(2009; Canobi et al., 1998, 2002) assessed conceptual
knowledge by asking participants to explain their strategies
after they had solved an addition problem. Thus, even though
Canobi et al. used different tasks for assessing procedural and
conceptual knowledge, the knowledge assessed by their addition
task might have resulted from a mixture of procedural and
conceptual competencies (see also, Robinson &
Dubé, 2009).
This might have led to a higher correlation between both tests
compared to a variant were spontaneous application of procedural
and conceptual knowledge is independently assessed.
Alternatively, one might suspect that our measures
of procedural and conceptual knowledge were unreliable. However,
this is not likely as we did find satisfying splithalf
reliabilities for all age groups and both task formats (see,
Table 2). In addition, our results showed significant
correlations (a) for all second graders between the calculation
time and percentage of hits and (b) for at least some third
graders and all adult students between savings due to the use of
commutativitybased shortcuts and hits in the judgment task.
Therefore, it seems worthwhile to ask for further theoretical
causes concerning our third finding.
4.1 Theoretical
Implications
At first glance, the results seem to fit with a
proceduralfirst development of commutativity (e.g., Baroody et al.,
2007; Briars & Siegler, 1984; Siegler & Stern, 1998). That is, second graders
use the commutativitybased shortcut before having acquired an
abstract understanding of the principle. It therefore appears
that the development of conceptual knowledge progresses more
slowly than that of procedural knowledge – at least as measured
in this study and for commutativity (cf. Canobi, 2004; Canobi
et al., 1998).
However, in their review about relations between children’s
understanding of mathematical concepts and their ability to
execute arithmetic procedures, RittleJohnson and Siegler (1998) provided ample
evidence that with regard to commutativity, children first
acquire conceptual knowledge before then applying corresponding
strategies.
In order to reconcile this conflict, we refer to
the iterative model of the development of conceptual and
procedural knowledge (e.g., Resnick, 1992; RittleJohnson et
al., 2001). Our findings suggest that second graders possess at
least rudimentary conceptual knowledge of the commutativity
principle, but their conceptual representation of the
commutativity principle is less well integrated (with procedural
knowledge) than that of third graders or adult students. This is
in line with many findings in the field of mathematic
development showing that already second graders possess
conceptual knowledge about commutativity (for a review, see
RittleJohnson & Siegler, 1998). Also, our sensitivity
index d’ indicated such knowledge. However, the consistent use
of this knowledge may still be reduced; that is, their
competency to identify the relevant task properties for applying
a certain shortcut strategy has not fully developed yet.
Therefore, they may need a certain external trigger in order to
activate their knowledge about commutativity and the
corresponding strategies. Our instructions for the arithmetic
and the judgment task did not provide any such trigger which
probably made it rather difficult, particularly for second
graders and also for most of the third graders, to realize that
they should rely on the commutativity principle in both tasks.
Consequently, it may be that some participants applied the
commutativitybased shortcut strategy to solve the arithmetic
problems, but did not use it in the judgment tasks or vice
versa. This does not imply that they first learn procedures
before they acquire conceptual knowledge. Rather, we assume that
such a finding mainly reflects that children’s conceptual
knowledge is not sufficiently integrated to spontaneously
recognize that they could rely on the commutativity principle.
In a similar vein, research
on expertise (e.g.,
Anderson & Schunn, 2000; Gentner & Toupin, 1986; Haider
& Frensch, 1996; Koedinger & Anderson, 1990) also shows that
wellintegrated and thus abstract conceptual knowledge is
required to identify task
relevant information in order to solve problems and to flexibly
transfer knowledge from one task domain to another (see
also e.g., Sloutsky & Fisher, 2008; Star & Seifert,
2006).
To
summarize, we assume that the divergent findings concerning the
development of an abstract understanding of the commutativity
principle reflect the fact that after students have acquired
some procedural and conceptual knowledge in this domain, this
knowledge needs to be integrated. This integration of knowledge,
we suspect, is done in an iterative way which means that
procedures are applied, which then refine the conceptual
knowledge (RittleJohnson et al.,
2001). The
conceptual knowledge is then used to guide children’s attention
to information which is needed to adaptively apply efficient
strategies.
4.2
Further Improvements of the
Measurement of Spontaneous Application of a Mathematical
Principle
With the current study, we took a first step to
measure spontaneous usage of commutativity knowledge in a
nonreactive way. Participants worked on the paperandpencil
tasks in a setting very similar to other tests in the classroom.
In the arithmetic blocks, we asked for fast and correct
solutions to the arithmetic problems and did not mention that
regularities in the task material might be exploited for
efficient task processing. We inferred procedural knowledge of
the commutativity principle from the performance benefits on
material containing identical addends in changed order in
consecutive problems (as compared to material that did not
contain such pairs of problems). Probing for conceptual
knowledge, we asked participants to indicate in which cases
calculation was not necessary – again without hinting that it
might be the commutativity principle that made calculation
superfluous. The rationale behind this procedure was that
participants who had well integrated knowledge of the
commutativity principle should recognize the respective
arithmetic problems and consequently should be able to relate it
to the task demand (marking problems where calculation was not
necessary). Agerelated changes in hits and false alarms in the
judgment task suggested that this was indeed the case. Similar
indirect approaches to measure knowledge have been developed in
order to measure insight (cf. Haider & Rose, 2007). When
investigating insight, it is not feasible either to directly ask
participants again and again if they already have discovered the
regularity in the task material – without providing them with a
strong hint that such a regularity exists.
One way to further improve the method would be to
include control problems which also feature the same numbers as
their predecessor problems in changed order but do not allow to
apply the commutativity principle (i.e. subtractions). Such
interspersed control problems could help to rule out superficial
matching strategies (i.e. “same numbers = same result”) that do
not capture the essence of commutativity. First explorations in
our labs indicate that second graders do not confuse subtraction
problems containing the same digits as a preceding addition with
genuine commutative problems. Furthermore, it would be
interesting to implement our instruments within a multimethod
approach, administering multiple measurements per person and
construct (cf. Prather & Alibali, 2009). As a first step one
would have to estimate to what extend repeated testing of
spontaneous usage of a mathematical principle induces
participants to recognize and use the principle – and by this
spoils the possibility to assess spontaneous usage.
Paperandpencilbased
testing in the classroom has the advantage that the test
situation is similar to other tests the students take. In a
parallel line of research, we have started to employ eyetracking
to obtain process measures related to commutativity knowledge
(e.g., Gaschler et al. 2013; Godau, Wirth, Hansen, Haider, &
Gaschler, in press). For instance, it is possible to quantify
the extent to which a child searches for repetitions of addends
in subsequent addition problems. However, when they are tested
individually with an eyetracking system, children are aware that
the measurement is about where they look and how they calculate.
Classbased testing in computer labs within schools might offer
the possibility to obtain process measures while keeping up the
character of the assessment as allowing to measure spontaneous
application of the principle knowledge.
4.3 Theoretical
Conclusions and Practical Implications
Recently, Prather and Alibali (2009; see also,
Bisanz et al., 2009; Schneider & Stern, 2010) called for
multifaceted knowledge assessment in the context of arithmetic
development. Our use of the arithmetic and judgment tasks in
order to independently assess procedural and conceptual
knowledge can be seen as a first step in this direction.
Complementing earlier work on commutativity knowledge (e.g.,
Baroody & Gannon, 1984; Canobi et al., 2002), our findings
suggest that, when second graders and third graders are not
alluded to rely on the commutativity principle, second graders
and most of the third graders show but weak signs of spontaneous
application of commutativity and interrelation of different
forms of commutativity knowledge. This suggests that they do not
possess wellintegrated knowledge about commutativity in the
sense of an abstract formal mathematical principle.
Our results suggest that even if children use
procedures that suggest integrated conceptual knowledge about
commutativity, the learning process has by far not reached an
endpoint. Rather, it still progresses, before leading to a
wellintegrated, abstract representation of the mathematical
principle as with our measures found in adults. As long as
children do not possess such an abstract representation, they
will not be able to flexibly and adaptively use the
commutativity principle in different task contexts. Accordingly,
we suspect that increasing experience in the field of
mathematics is needed in order to better integrate conceptual
knowledge about various arithmetic principles. This might
explain why transfer of knowledge from one context to another is
often found to be rather weak (e.g., Frensch & Haider, 2008;
Kaminsky, Sloutsky & Heckler, 2008; Siegler & Stern,
1998; Sloutsky & Fisher, 2008). Therefore, helping students
to develop wellintegrated knowledge concepts should be one of
the most important tasks education has to fulfill (see, e.g.,
Geary et al., 2008; Prather & Alibali, 2009; Verschaffel et
al. 2009).
In more
practical terms, if children are taught the commutativity
principle in the context of addition, they seem to learn that
they can use this principle to avoid unnecessary labor. However,
our results suggest that this does not mean that they
concurrently acquire an idea of the abstract principle of
cardinality. We suspect that many children only acquire a
procedure (or a strategy) that they can easily apply for
twoelement addition problems. In order to help students to
understand the abstract principle of commutativity, it might be
worthwhile to activate students’ prior knowledge of this
principle, such as the orderirrelevance principle they already
use in counting. When introducing the commutativity principle in
addition (or multiplication) it might be helpful to tell
students that they already have used this principle in other
contexts and explain how and why it works in all these different
situations. This then might help them to understand the
commutativity principle in a more abstract manner and probably
also to understand task properties needed to correctly apply
this principle. Further,
it may help to support children in recognizing the
consequences of using alternative strategies in order to
ensure representational redescription (e.g., Baroody &
Gannon, 1984).
Keypoints
Procedural
and conceptual commutativity knowledge increase with increasing
age.
Second
graders show no signs of an integrated concept of commutativity.
First
signs of an integrated concept of commutativity emerge in grade
three.
Acknowledgements
This research was supported by the German Research Foundation (DFG; HH 1471/121). Some of the results were presented at the Kongress der Deutschen Gesellschaft für Psychologie 2010 in Bremen, Germany. We thank Annette Bräutigam, Yvonne Radermacher, Pia Blase and Ester Jung for help with data collection.
References
Anderson, J. R. & Schunn, C. D. (2000). Implications
of the actr learning theory: No magic bullets. In R. Glaser
(Ed.), Advances
in instructional psychology: Educational design and cognitive
science (Vol.
5, pp. 133). Mahwah, NJ: Lawrence Erlbaum Associates
Publishers.
Baroody, A. J. (1984). More precisely
defining and measuring the orderirrelevance principle. Journal of
Experimental Child Psychology, 38, 3341.
doi:10.1016/00220965(84)900171
Baroody, A. J. (2003).
The Development of Adaptive Expertise and Flexibility: The
Integration of Conceptual and Procedural Knowledge. In A.
Baroody & A. Dowker (Eds.), The Development
of Arithmetic Concepts and Skills. Constructing Adaptive
Expertise (1st
ed., pp. 1–34). Mahwah, NJ: Lawrence Erlbaum Associate
Publishers.
Baroody, A. J., Feil, Y. & Johnson, A.
R. (2007). An alternative reconceptualization of procedural and
conceptual knowledge. Journal
for Research in Mathematics Education, 38, 115131.
Baroody, A. J. & Gannon, K. E. (1984).
The development of the commutativity principle and economical
addition strategies. Cognition
& Instruction, 1, 321339. doi:10.1207/s1532690xci0103_3
Baroody, A. J., Ginsburg, H. P. &
Waxman, B. (1983). Children's use of mathematical structure. Journal for
Research in Mathematics Education, 14, 156168. doi:10.2307/748379
Baroody, A.J., & Rosu, L. (2006). Adaptive
expertise with basic addition and subtraction combinations 
The number sense view. Paper
presented at theAnnual Meeting of the American Educational
Research Association (April) San Francisco, CA.
Bisanz, J., & LeFevre, J. (1992).
Understanding elementary mathematics. In J. D. Campbell (Ed.) , The nature and
origins of mathematical skills (pp. 113136).
Oxford England: NorthHolland. doi:10.1016/S01664115(08)608857
Bisanz, J., Watchorn, R. P. D., Piatt, C.,
& Sherman, J. (2009). On “Understanding” Children's
Developing Use of Inversion. Mathematical
Thinking and Learning, 11,
10–24. doi:10.1080/10986060802583907
Briars, D. & Siegler, R. S. (1984). A
featural analysis of preschoolers' counting knowledge. Developmental
Psychology, 20, 607618. doi:10.1037/00121649.20.4.607
Canobi, K. H. (2004). Individual
differences in children's addition and subtraction knowledge. Cognitive
Development, 19, 8193. doi:10.1016/j.cogdev.2003.10.001
Canobi, K. H. (2005). Children's profiles
of addition and subtraction understanding. Journal of
Experimental Child Psychology, 92, 220246.
doi:10.1016/j.jecp.2005.06.001
Canobi, K. H. (2009). Conceptprocedure
interactions in children's addition and subtraction. Journal Of
Experimental Child Psychology, 102(2),
131149. doi:10.1016/j.jecp.2008.07.008
Canobi, K. H., Reeve, R. A. & Pattison,
P. E. (1998). The role of conceptual understanding in children's
addition problem solving. Developmental
Psychology, 34, 882891. doi:10.1080/01443410903473597
Canobi, K. H., Reeve, R. A. & Pattison,
P. E. (2002). Young children’s understanding of addition
concepts. Educational
Psychology, 22, 513532. doi:10.1080/0144341022000023608
Canobi, K. H., Reeve, R. A. & Pattison,
P. E. (2003). Patterns of knowledge in children’s addition. Developmental
Psychology, 39, 521–534. doi:10.1037/00121649.39.3.521
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, 407420. doi:10.1080/0144341960160405
FarringtonFlint, L., Canobi, K.H., Wood,
C. & Faulkner, D. (2010). Children’s patterns of reasoning
about reading and addition concepts. British Journal
of Developmental Psychology, 28, 427–448.
doi:10.1348/026151009X424222
Frensch, P. A. & Haider, H. (2008). Transfer
and Expertise: The Search for Identical Elements. In H. L.
Roediger, III (Ed.), Cognitive
Psychology of Memory. Vol. [2] of Learning and Memory: A
Comprehensive Reference (pp.
579596) Oxford: Elsevier. doi:10.1016/B9780123705099.001777
Fuson, K. C. (1988). Children's
counting and concepts of number. New
York, NY: SpringerVerlag Publishing.
Gaschler, R., Vaterrodt, B., Frensch, P. A.,
Eichler, A. & Haider, H. (2013). Spontaneous
usage of different shortcuts based on the commutativity
principle. PLoS
ONE 8(9): e74972. doi:10.1371/journal.pone.0074972
Geary, D. C. (2006). Development of
Mathematical Understanding. In W. Damon, R. M. Lerner, & N.
Eisenberg (Eds.), Handbook
of child psychology: Social, emotional, and personality
development (Vol.
3, pp. 777–810). John Wiley and Sons.
Geary, D. C., Boykin, A. W., Embretson, S.,
Reyna, V., Siegler, R., Berch, D. B., et al. (2008). Report of
the task group on learning processes. In National
mathematics advisory panel, reports of the task groups and
subcommittees (pp. 41–4211).
Geary, D. C., Brown, S. C. &
Samaranayake, V. A. (1991). Cognitive addition: A short
longitudinal study of strategy choice and speedofprocessing
differences in normal and mathematically disabled children. Developmental
Psychology, 27, 787797. doi:10.1037/00121649.27.5.787
Geary, D. C., Hoard, M. K., ByrdCraven, J.
& DeSoto, M. C. (2004). Strategy choices in simple and
complex addition: Contributions of working memory and counting
knowledge for children with mathematical disability. Journal of
Experimental Child Psychology, 88, 121–151.
doi:10.1016/j.jecp.2004.03.002
Gelman, R. (1990). First principles
organize attention to and learning about relevant data: Number
and the animateinanimate distinction as examples. Cognitive
Science, 14, 79106.
Gelman, R. & Gallistel, C. R. (1978).
The child's understanding of number. In. Cambridge, MA: Harvard
University Press.
Gelman, R. & Meck, E. (1983).
Preschoolers' counting: Principles before skill. Cognition, 13,
343359. doi:10.1016/00100277(83)900148
Gentner, D. & Toupin, C. (1986).
Systematicity and surface similarity in the development of
analogy. Cognitive
Science, 10, 277300. doi:10.1207/s15516709cog1003_2
Godau, C., Wirth, M., Hansen, S., Haider,
H., & Gaschler, R. (in press). From marbles to numbers 
Estimation influences looking patterns on arithmetic problems. Psychology.
Goldman, S. R., Mertz, D. L. & Pellegrino,
J. W. (1989). Individual
differences in extended practice functions and solution
strategies for basic addition facts. Journal of
Educational Psychology, 81, 481496.
doi:10.1037/00220663.81.4.481
Goldstone, R. L., & Sakamoto, Y.
(2003). The transfer of abstract principles governing complex
adaptive systems. Cognitive
Psychology, 46, 414466.
doi:10.1016/S00100285(02)005194
Haider, H. & Frensch, P. A. (1996). The
role of information reduction in skill acquisition. Cognitive
Psychology, 30, 304337. doi:10.1006/cogp.1996.0009
Haider, H. & Rose, M. (2007). How to
investigate insight: A proposal. Methods, 42, 49–57.
doi: 10.1016/j.ymeth.2006.12.004
Hannula, M. M., & Lehtinen, E. (2005).
Spontaneous focusing on numerosity and mathematical skills of
young children. Learning
and Instruction, 15,
237256. doi:10.1016/j.learninstruc.2005.04.005
Hannula, M. M., Lepola, J., & Lehtinen,
E. (2010). Spontaneous focusing on numerosity as a
domainspecific predictor of arithmetical skills. Journal of
Experimental Child Psychology, 107,
394406. doi:10.1016/j.jecp.2010.06.004
Hiebert, J., & Lefevre, P. (1986).
Conceptual and procedural knowledge in mathematics: An
introductory analysis. In J. Hiebert (Ed.), Conceptual and
procedural knowledge: The case of mathematics (pp. 127).
Hillsdale, NJ: Erlbaum.
Kaminski, J. A.,
Sloutsky, V. M., & Heckler, A. F. (2008). Learning Theory:
The Advantage of Abstract Examples in Learning Math. Science, 320,
454–455.
Koedinger, K. R. & Anderson, J. R.
(1990). Abstract planning and perceptual chunks: Elements of
expertise in geometry. Cognitive
Science, 14, 511550.
doi:10.1207/s15516709cog1404_2
Kuhn, D., GarciaMila, M., Zohar, A., &
Andersen, C. (1995). Strategies
of knowledge acquisition. Society
for Research in Child Development Monographs, 60 (4), Serial no.
245.
LeFevre, J.A., SmithChant, B. L., Fast,
L., Skwarchuk, S.L., Sargla, E., Arnup, J. S., et al. (2006).
What counts as knowing? The development of conceptual and
procedural knowledge of counting from kindergarten through grade
2. Journal of
Experimental Child Psychology, 93, 285303.
doi:10.1016/j.jecp.2005.11.002
McMullen, J.A., HannulaSormunen, M.M.,
& Lehtinen, E. (2011). Young children’s spontaneous focusing
on quantitative aspects and their verbalizations of their
quantitative reasoning. In Ubuz, B. (Ed.). Proceedings of
the 35th Conference of the International Group for the
Psychology of Mathematics Education, 3, pp.
217224. Ankara, Turkey: PME.
McNeil, N. M. (2007). Ushaped development
in math: 7yearolds outperform 9yearolds on equivalence
problems. Developmental
Psychology, 43, 687–695. doi:10.1037/00121649.43.3.687
Prather, R. W., & Alibali, M. W.
(2009). The development of arithmetic principle knowledge: How
do we know what learners know? Developmental
Review, 29,
221–248. doi:10.1016/j.dr.2009.09.001
Resnick, L. B. (1992). From protoquantities
to operators: Building mathematical competence on a foundation
of everyday knowledge. In G. Leinhardt, R. T. Putnam & R. A.
Hattrup (Eds.), Analysis
of arithmetic for mathematics teaching (pp. 373429).
Hillsdale, NJ, UK: Lawrence Erlbaum Associates, Inc.
RittleJohnson, B. & Siegler, R. S.
(1998). The relation between conceptual and procedural knowledge
in learning mathematics: A review. In C. Donlan (Ed.), The development
of mathematical skills (pp.
75110)). East Sussex, UK: Psychology Press.
RittleJohnson, B., Siegler, R. S. &
Alibali, M. W. (2001). Developing conceptual understanding and
procedural skill in mathematics: An iterative process. Journal of
Educational Psychology, 93, 346362.
doi:10.1037/00220663.93.2.346
Robinson,
K. M. & Dubé, A. K. (2009). Children’s understanding of
addition and subtraction concepts. Journal of
Experimental Child Psychology, 103, 532–545.
doi:10.1016/j.jecp.2008.12.002
Schneider,
M. &. Stern, E. (2010). The Developmental Relations Between
Conceptual and Procedural Knowledge: A Multimethod Approach. Developmental
Psychology, 46,
178–192. doi:10.1037/a0016701
Siegler, R. S. (1987). The perils of
averaging data over strategies: An example from children's
addition. Journal
of Experimental Psychology: General, 116, 250264.
doi:10.1037/00963445.116.3.250
Siegler, R. S. & Araya, R. (2005). A
computational model of conscious and unconscious strategy
discovery. In R. V. Kail (Ed.), Advances in
child development and behavior (vol. 33) (pp. 142). Oxford,
UK: Elsevier.
Siegler, R. S. & Jenkins, E. (1989). How children
discover new strategies. Hillsdale, NJ, UK: Lawrence
Erlbaum Associates, Inc.
Siegler, R. S. & Stern, E. (1998).
Conscious and unconscious strategy discoveries: A microgenetic
analysis. Journal
of Experimental Psychology: General, 127, 377397.
doi:10.1037/00963445.127.4.377
Sloutsky, V. M. & Fisher, A. V. (2008).
Attentional learning and flexible induction: How mundane
mechanisms give rise to smart behaviors. Child
Development, 79, 639651.
doi:10.1111/j.14678624.2008.01148.x
Sophian, C. & Adams, N. (1987).
Infants' understanding of numerical transformations. British Journal
of Developmental Psychology, 5, 257264.
doi:10.1111/j.2044835X.1987.tb01061.x
Sophian, C., Harley, H. & Manos Martin,
C. S. (1995). Relational and representational aspects of early
number development. Cognition
& Instruction, 13, 253268.
doi:10.1207/s1532690xci1302_4
Star, J.R. (2004, April). The development
of flexible procedural knowledge in equation solving. Paper presented at
the annual meeting of the American Educational Research
Association, San Diego.
Star, J. R. (2005). Reconceptualizing
procedural knowledge. Journal
for Research in Mathematics Education, 36, 404411.
Star, J. R. & Seifert, C. (2006). The
development of flexibility in equation solving. Contemporary
Educational Psychology, 31, 280300.
doi:10.1016/j.cedpsych.2005.08.001
Starkey, P. & Gelman, R. (1982). The
development of addition and subtraction abilities prior to
formal schooling in arithmetic. In T. P. Carpenter, J. M. Moser
& T. A. Romberg (Eds.), Addition
and subtraction: A cognitive perspective (pp. 99–116).Hillsdale,
NJ: Erlbaum.
Torbeyns, J., De Smedt, B., Ghesquière, P.
&Verschaffel, L. (2009). Acquisition
and use of shortcut strategies by traditionally schooled
children. Educational
Studies in Mathematics, 71(1), 117.
Verschaffel, L.,
Luwel, K., Torbeyns, J. & Van Dooren, W. (2009). Conceptualizing,
investigating, and enhancing adaptive expertise in elementary
mathematics education. European
Journal of Psychology of Education, 24(3), 335359.
doi:10.1007/BF03174765
[1] Some
researchers use the term associativity instead
of commutativity when an
addition or multiplication problem has more than two addends
or factors (Geary
et al., 2008). Other researchers (Canobi,
et al., 1998) refer
to commutativity as
the property that problems containing the same terms in a
different order have the same answer independent of the
number of terms, whereas associativity is the
property that problems in which terms are decomposed and
recombined in different ways have the same answer [(a + b) + c
= a + (b + c)].
[2] Separately for each student within the
respective age groups, we computed zscore of his or her hit
and false alarms rate. Then, we individually computed the
sensitivity index d’ from signal detection theory by
subtracting the ztransformed false alarms rate from the
ztransformed hit rate.