European Journal of Psychology of Education
2002, Vol. XVII, nº 3, 275-291
© 2002, I.S.P.A.
Strategic competence: Applying Siegler’s
theoretical and methodological framework to the
domain of simple addition
Joke Torbeyns
Lieven Verschaffel
Pol Ghesquière
Katholieke Universiteit Leuven, Belgium
In this study we investigated the variability, frequency, efficiency,
and adaptiveness of young children’s strategy use in the domain of
simple addition by means of the choice/no-choice method. Seventyseven beginning second-graders, divided in 3 groups according to
general mathematical ability, solved a series of 25 simple additions in 3
different conditions. In the first condition, children could choose
whatever strategy they wanted to solve each problem. In the second and
third condition, the same children had to solve all problems with one
particular strategy, respectively adding up to 10 and retrieval. The
results demonstrate that second-graders as a whole choose adaptively
between retrieval, decomposition, and counting strategies when solving
simple additions, and that they use these strategies neither equally
frequently nor equally efficiently. Furthermore, our results indicate that
children with different mathematical ability use generally the same
strategies to solve these problems, but differ in the frequency, accuracy
and adaptiveness with which they apply these strategies. Finally, this
study documents the value of the choice/no-choice method to assess the
adaptiveness of young children’s strategy use in the domain of early
arithmetic.
Strategy choice and development: Siegler’s theoretical and methodological
framework
Recent studies on strategy use in cognitive tasks (for an overview, see Siegler, 1996)
have shown that children as well as adolescents and adults have a rich variability of strategies
to their disposal to solve a particular task. Siegler and Jenkins (1989) distinguish two types of
strategies to solve cognitive tasks: “back-up” strategies and “retrieval”. Back-up strategies can
be defined as rather time-consuming, procedural strategies. The speed of execution of a back-up
The authors would like to thank Wim Van den Noortgate for his methodological assistance in the data analysis.
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strategy is strongly influenced by problem-specific characteristics (e.g., it takes more time to
count the answer to 8+3 “1, 2, 3, 4, 5, 6, 7, 8… 9, 10, 11” than to solve 2+3 with the same
strategy “1, 2… 3, 4, 5”). Retrieval refers to the (quasi-)automatic activation of the answer to
the problem in long term memory (e.g., “8+3=11, I know this by heart”). In principle, retrieval
is executed faster than back-up strategies; moreover, the speed of execution of the former
strategy is much less influenced by problem-specific characteristics than the speed of
execution of the latter strategies.
In their “model of strategic change”, Lemaire and Siegler (1995) distinguish four dimensions
to describe developmental changes in strategy use. The first dimension (i.e., strategy repertoire)
involves the different strategies that are used to solve a task. The second dimension (i.e., strategy
distribution) refers to the relative frequency with which each strategy is used. The third
dimension (i.e., strategy efficiency) concerns the accuracy and the speed of strategy execution.
The fourth dimension (i.e., strategy selection) refers to the adaptiveness of strategy choices:
Does the individual choose the cognitively most efficient strategy, i.e. the strategy that leads
the individual fastest to an accurate answer to the problem? Changes in strategy use can thus
occur in at least four different ways: The acquisition of new strategies and the abandonment of
old ones (dimension 1), changes in the relative frequency with which each of the available
strategies is used (dimension 2), changes in the accuracy and the speed of strategy execution
(dimension 3), and changes in the adaptiveness of strategy choices (dimension 4). According to
this model, at the beginning of the learning process, the learner frequently, if not exclusively,
chooses rather primitive back-up strategies (like, for instance, counting), which he or she
executes rather inefficiently (i.e., inaccurately and slowly), and if the repertoire contains different
strategies, the learner is not able to select these different strategies in the most economical way.
With experience, the learner uses more efficient back-up strategies and retrieval, which he or she
executes ever more frequently, more efficiently, and also more adaptively.
To obtain unbiased information about the efficiency of strategy use and the adaptiveness
of individual strategy choices, Siegler and Lemaire (1997) propose the use of the “choice/nochoice” method1. This method requires testing each subject under two types of conditions. In
the choice condition, subjects can freely choose which strategy they use to solve a series of
problems from a given task domain. In the no-choice condition, the researcher forces them
experimentally to solve all problems with one particular strategy. The number of no-choice
conditions can vary according to the number of strategies available to the subject, research
interests, technical possibilities, etc. Comparison of the data about the accuracy and the speed
of the different strategies as gathered in the no-choice conditions, with the strategy choices
made in the choice condition, allows the researcher to assess the adaptiveness of individual
strategy choices in the choice condition accurately: Does the subject (in the choice condition)
solve each problem by means of the strategy that leads fastest to an accurate answer to this
problem, as evidenced by the information obtained in the no-choice conditions? Siegler and
Lemaire (1997; Lemaire & Lecacheur, 2001a) studied the adaptive nature of adults’ strategy
choices in multiplication and currency conversion tasks successfully using the choice/nochoice method. Furthermore, Lemaire and Lecacheur (2001b, 2002) applied this method to
describe developmental differences in 9- and 11-year olds’ strategy use in the domain of
computational estimation and spelling. Although Geary, Hamson, and Hoard (2000) and
Jordan and Montani (1997) offered their subjects (second- and third-grade children,
respectively) simple addition and subtraction problems in a choice as well as a forced retrieval
condition, the choice/no-choice method has thus far not been used systematically to study the
adaptiveness of young children’s strategy choices in the domain of early arithmetic.
Previous work on strategy use in the domain of simple addition
Previous studies in the domain of simple addition (e.g., Fuson, 1992; Geary, BowThomas, Liu, & Siegler, 1996; Geary & Wiley, 1991; LeFevre, Sadesky, & Bisanz, 1996;
STRATEGIC COMPETENCE: SIMPLE ADDITION
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Siegler, 1987, 1988; Siegler & Jenkins, 1989; Svenson, 1985; Svenson & Sjöberg, 1983;
TAL-team, 2001) revealed that children as well as adolescents and adults use a rich diversity
of back-up strategies and retrieval to solve simple additions up to 20. The children who
participated in Svenson and Sjöberg’s (1983) longitudinal study (during the first three years of
primary education) for instance solved these problems with multiple counting strategies, such
as counting fingers and verbal counting with steps greater than one unit (“7+6=7, 9, 11, 13”).
Moreover, they used a variability of decomposition strategies, such as adding up to 10
(“7+6=7+3+3=10+3=13”) and the tie strategy (“7+6=6+6+1=12+1=13”), and they retrieved
the answer from long term memory (“7+6=13”). Likewise, Geary and Wiley (1991) observed
that the adults who participated in their study, applied multiple back-up strategies, such as
counting and decomposition strategies, as well as retrieval to solve simple additions up to 20.
The results of the previously cited studies also demonstrate that children and adults use
all available strategies neither equally frequently nor equally efficiently. Furthermore, with
experience, the frequency of the most efficient counting and decomposition strategies and of
retrieval increases, while the frequency of less efficient counting strategies decreases.
Likewise, as experience increases, the accuracy and the speed of strategy execution grow too.
Previous research further indicated that people adapt their strategy choices to problemspecific characteristics, such as problem difficulty: Children as well as adults prefer retrieval
to solve easier problems; the more difficult the problem, the more frequently they apply a
back-up strategy.
Research on strategic competence in children with mathematical problems (MD; Geary,
1990, 1993; Geary & Brown, 1991; Geary, Brown, & Samaranayake, 1991; Jordan & Hanich,
2000; Jordan & Montani, 1997) finally showed that their mathematical development is,
compared to the development of their normally progressing peers, not only marked by a
developmental delay, but also by a more fundamental deficit. Children with MD use the same
repertoire of back-up strategies and retrieval as their peers without MD, but differ in the
frequency and the accuracy with which they execute these strategies. Children with MD solve
simple additions up to 20 more frequently with counting strategies than their normally
achieving peers, and execute these strategies less accurately than the latter. This difference in
the frequency and the accuracy of counting strategies decreases as practice increases, which
indicates that the development of children with MD is delayed in comparison with the
development of their peers without MD. Furthermore, children with MD use retrieval less
often and also less accurately than their normally achieving peers. This difference in the
frequency and the accuracy of retrieval does not decrease as practice increases, which
evidences that the development of children with MD is not only delayed, but also qualitatively
different from the development of children without MD. Finally, Geary (1990) concluded that
first-graders with MD make “rather poor strategy choices” (p. 374) in comparison with their
normally achieving peers. Geary et al. (1991) and Geary and Brown (1991) did not observe
any difference in the adaptiveness of individual strategy choices at older ages (second-grade
and third- and fourth-grade children, respectively): Both children with MD and children
without MD preferred counting strategies to solve more difficult problems, and solved the
easier ones most frequently with retrieval.
All above-mentioned studies provide empirical support for the validity of Siegler’s
theoretical ideas about developmental changes in strategy repertoire (first dimension of the
model of strategic change) and strategy frequency (second dimension) in the domain of simple
addition up to 20. However, the results about the efficiency of strategy execution (third
dimension) and the adaptiveness of individual strategy choices (fourth dimension) are less
convincing.
First of all, none of these studies applied the choice/no-choice methodology to describe
(developmental changes in) the accuracy and the speed of strategy use; the efficiency of
strategy execution was determined on the basis of data gathered in one free-choice condition.
As argued by Siegler and Lemaire (1997), the data about the speed and the accuracy of
strategies obtained in the choice condition can be biased by selection effects: A strategy that is
used mainly to solve easy problems, or primarily applied by the most able subjects will seem
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more efficiently than a strategy that is almost exclusively used to solve the most difficult
problems, respectively employed most frequently by the least able subjects.
Secondly, one can question the way these researchers defined and operationalised the
adaptiveness criterion in their studies. They focussed on subjects’ choices between, on the one
hand, retrieval, and, on the other hand, all available back-up strategies. Subjects chose
adaptively between the two types of strategies if they preferred retrieval to solve the easier
problems, and a back-up strategy to solve the more difficult ones. Objective data (like, for
instance, the size of the given numbers in the problems, or the mean accuracy rate of the
problems as gathered in a large group of subjects being asked to solve the problems in one
free-choice condition), were used to specify problem difficulty. The latter definition and
operationalisation of adaptive strategy choices does not fit exactly with Siegler’s idea of
choosing the strategy that leads the individual fastest to an accurate answer to the problem.
Indeed, Siegler’s definition of adaptive strategy choices does not just refer to deciding whether
to state a retrieved answer or use a back-up strategy; it also implies choosing which of all
available back-up strategies you will use, in other words, choosing between the different backup strategies. Furthermore, as stated in the definition and discussed in several publications
(see, among others, Siegler & Shipley, 1995; Siegler, 1996; Siegler & Lemaire, 1997),
strategy efficiency plays a major role in deciding which strategy to use; strategy choices are
not merely determined by problem difficulty, but also by strategy performance characteristics.
As mentioned earlier, the use of the choice/no-choice method is required to obtain reliable
data about these strategy performance characteristics and thus the adaptive nature of
individual strategy choices.
In line with these criticisms, the major goal of our study was to analyse, by using the
choice/no-choice method, the strategies young children with different mathematical ability use
to solve simple additions with the bridge over 10 in terms of the four dimensions of Siegler’s
model of strategic change, with special attention for the fourth dimension of this model,
namely the adaptiveness of individual strategy choices.
Method
Participants
Subjects were 77 beginning second-graders from two different mixed-sex schools for
primary education in Flanders. Children from three intact classes were involved in the study.
Both sexes were equally represented in the sample (41 boys, 36 girls), and the mean age of the
children was 89 months (range=82-104 months). Based on their overall scores for
mathematics at the end of the first grade, children were divided in three groups according to
general mathematical ability. The group of strong pupils consisted of the seven strongest
pupils of each class with respect to general mathematical ability (N=21), the group of weak
pupils of the seven weakest children of each class with respect to general mathematical ability
(N=21), and the group of moderate pupils of the other 35 children from the three classes.
All children were tested in the month of November. At that moment, they all had learned
how to solve simple additions with the bridge over 10. As is typically the case in elementary
mathematics education in Flanders (Feys, 1995), their teachers had heavily focussed on
the mastery of the strategy adding up to 10 (which involves the decomposition of one
addend into two parts: One part which adds the other addend up to 10, and a rest part; e.g.,
“7+6=7+3+3=10+3=13”), rather than on the flexible use of a rich variety of clever calculation
strategies for sums up to 20, such as the reversal strategy (which means that one changes the
order of the addends before starting to count or calculate the answer to the problem; e.g.,
“6+7=7+6”) and the tie strategy (referring to the use of an automatised tie sum to answer the
problem; e.g., “7+6=6+6+1=12+1=13”).
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Materials
Taking into account the literature on strategy use in the domain of simple addition up to
20, we made a distinction between three levels of strategies children can use to solve simple
additions with the bridge over 10, namely retrieval, decomposition strategies, and counting
strategies. Furthermore, we distinguished two clever strategies to solve simple additions with
the bridge over 10, namely the reversal strategy and the tie strategy.
All children were asked to solve a series of 25 simple additions up to 20. The problems
were constructed from the 49 possible pair-wise combinations of the integers 3 to 9. On the
basis of our distinction between three levels of strategy use and between two clever strategies,
we selected five different problem types, with five problems in each type: One type of simple
additions up to 10 (buffer items; e.g.: 3+4), and four types of additions with the bridge over 10
(experimental items). The latter problem types can be described as follows: (a) L+s problems,
with a large first addend and a small second addend (e.g., 8+3=.), which can be solved with
strategies at the three levels of strategy use, but do not favour the use of either of the two
clever strategies; (b) s+L problems, with a small first addend and a large second addend (e.g.,
3+8=.), which favour the use of the reversal strategy; (c) tie sums (e.g., 6+6=.), which are
generally known to be memorised earlier than non-tie sums, and thus were expected to be
solved more often with the retrieval strategy; and (d) almost-tie sums (e.g., 6+7=.), which
elicit the use of the tie strategy.
Conditions
All children solved the series of 25 simple additions in three different conditions. In the
first condition, the choice condition, children could solve each problem by using their
preferential strategy. Immediately after solving each problem, children were asked to report
verbally which strategy they had used to solve the problem. The identification of the strategies
was based on the verbal reports of the children in combination with the experimenter’s
observations of the behaviour of the children during problem solving.
In the second condition, the adding up to 10 condition, children were experimentally
forced to solve all problems with the adding up to 10 strategy. We forced them to use this
strategy by means of the instruction given as well as the presentation format of the items (see
below). The children were not allowed to use the reversal strategy in this condition: They had
to decompose the second addend of each problem. Note that, as a consequence of the
obligatory use of the adding up to 10 strategy, the five simple additions up to 10 were not
administered in this condition.
In the third condition, the retrieval condition, children were experimentally forced to
retrieve the answer to all problems by means of the instruction given as well as the
presentation time of the items. Indeed, all items were offered during maximum two seconds,
which made it very difficult, if not impossible for these young children to use a strategy other
than retrieval2.
Procedure
The children were tested individually. In each condition, problems were presented one at
a time in the middle of a computer screen. Children were asked to verbally state their answer
as soon as they knew it. The experimenter then entered the answer by means of the numerical
path of the keyboard. After the experimenter had entered the answer, the next problem
appeared on the screen. The computer registered automatically the reaction time (i.e., the time
needed to solve the problem, operationalised as the time between problem appearance and the
beginning of the verbal statement of the answer) and the answer to each problem. In the
choice and the retrieval condition, problems were presented as follows: X+Y=. In the adding
up to 10 condition, the presentation format was as follows: X+Y=X+(.+.)=.; children were
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asked to explain how they split the second addend of each problem in the latter condition,
immediately after they had stated their answer.
All children solved the series of problems in the same order in the three conditions. The
order of the problems was fixed on the basis of the following constraints: (a) No repetition of
problem type was permitted across consecutive problems, and (b) L+s problems and tie sums
were not allowed to be either preceded or followed by s+L problems respectively almost-tie
sums. In each condition, the series of problems was preceded by three practice trials, which
familiarized the children with the presentation format of the items as well as the testing
procedure. All children first solved the problems in the choice condition. Half of the children
continued with the adding up to 10 condition, and ended with the retrieval condition. For the
other half of the children, the order of the two no-choice conditions was reversed. For each
child, the consecutive conditions were separated in time for at least one day and at most two
days.
Results
Results are presented for the four dimensions of Siegler’s model of strategic change 3. We
start with a description of the different types of strategies the children used to solve simple
additions with the bridge over 10 in the choice condition, as well as the relative frequency
with which they applied these strategies (first and second dimension). Next, we discuss the
accuracy and the speed of strategy execution (third dimension). Finally, we present our results
about the adaptiveness of individual strategy choices (fourth dimension).
We executed multilevel analyses to describe the frequency, accuracy, speed, and
adaptiveness of strategy execution. We used hierarchical linear models with two levels: The
higher-mentioned characteristics of strategy use (level 1) are “nested” within children (level
2). We analysed the data about the frequency and speed of strategy use by means of the
so-called “proc mixed” of the computer program SAS (Littell, Milliken, & Stroup, 1996),
which assumes the data to be normally distributed. Taking into account the binomial
distribution of the accuracy and adaptiveness data, we analysed the latter data by means of the
so-called “glimmix macro” of the same statistical program. All reported results are statistically
significant at an alpha level of .05, unless otherwise noted.
Strategy repertoire and strategy frequency in the choice condition
In the first analysis, we predicted the frequency of strategy use in the choice condition by
means of the variables level of strategy use (retrieval, decomposition, counting), group
(strong, moderate, weak), and problem type (L+s, s+L, tie, almost-tie). We added the variable
level of strategy use as a random coefficient to the model, which means that the influence of
the latter variable on the frequency of strategy use differs from child to child. Table 1
describes the relative frequency of retrieval, decomposition, and counting strategies in the
choice condition per group and per problem type (least-squares means, i.e. predicted
population means).
Table 1 shows that the children spontaneously applied retrieval, decomposition and
counting strategies to solve simple additions with the bridge over 10. Children retrieved the
answer to some problems, used a rich diversity of decomposition strategies, such as the adding
up to 10 strategy, the tie strategy, the reversal strategy, and the “one less than 10” strategy
(e.g., “9+5=10+5–1=15-1=14”), and applied multiple counting strategies, such as verbal
counting and counting fingers. This diversity in strategy use (retrieval, decomposition, and
counting strategies) was observed for the three ability groups and for the four problem types.
The results in Table 1 also indicate that children did not use retrieval, decomposition, and
counting strategies equally frequently (F(2,148)=48.70, p<.0001). They preferred the use of
decomposition strategies, and especially the strategy adding up to 10. The latter strategy was
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281
applied on 87% of the items that they solved by means of a decomposition strategy. Retrieval
was used more often than counting strategies.
Table 1
Relative frequency of strategy use in the choice condition per group and per problem type
(Least-Squares Means)
Retrieval
Decomposition
Counting
Group
Strong
Moderate
Weak
All
42.84
31.29
25.83
33.32
56.05
60.29
37.44
51.26
11.11
18.43
36.73
15.42
Problem type
L+s
s+L
Tie
Almost-tie
28.40
24.59
57.62
22.67
55.62
57.88
30.97
60.57
15.98
17.53
11.41
16.76
Note. Relative frequency is expressed in percentage; L+s=Problems with a large first and a small second addend, like
8+3=.; s+L=Problems with a small first and a large second addend, like 3+8=.; Tie=Tie sum, like 6+6=.; Almosttie=Almost-tie sum, like 6+7=.
Furthermore, we found strong group differences in the frequency of strategy use
(F(6,666)=8.96, p<.0001). Children with strong mathematical abilities retrieved the answer to
the problems more often than children with moderate and weak mathematical abilities.
Children with weak mathematical abilities solved the problems less frequently with
decomposition strategies than the other children, and used counting strategies more frequently
than the other children.
The results in Table 1 finally reveal that the frequency of strategy use differed between
the four problem types (F(9,666)=26.92, p<.0001). This difference was due to the fact that tie
sums were solved more often with retrieval, and less often with a decomposition strategy, than
the other problem types.
Strategy accuracy and strategy speed in the choice and no-choice conditions
We first derived the accuracy and the speed of strategy execution in the choice condition
from, respectively, the score and the reaction time in this condition. We predicted the accuracy
of strategy execution by means of the variables level of strategy use (retrieval, decomposition,
counting), group (strong, moderate, weak), and problem type (L+s, s+L, tie, almost-tie). We
predicted the speed of strategy execution by means of the same variables, with level of
strategy use and problem type as random coefficients. Table 2 describes the accuracy and the
speed of strategy execution in the choice condition per group and per problem type (leastsquares means).
Table 2 demonstrates that the children did not execute all strategies equally accurately in the
choice condition (F(2,1425)=8.86, p=.0001). They solved the problems more accurately with
retrieval or a decomposition strategy than by means of a counting strategy. The children
answered the problems almost always correct when they chose freely to use retrieval or a
decomposition strategy. We observed group differences in the accuracy of strategy execution in
the choice condition (F(4,1425)=2.21, p=.0660; border significant). Children with strong
mathematical abilities executed retrieval more accurately than children with moderate
mathematical abilities, who, at their turn, applied this strategy more accurately than children with
weak mathematical abilities. The latter group of children executed decomposition strategies less
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accurately than children with strong and moderate mathematical abilities. The accuracy of
strategy execution did not differ between the four problem types (F(6,1425)=1.39, p=0.2151).
Table 2
Accuracy and speed of strategy execution in the choice condition per group and per problem
type (Least-Squares Means)
Retrieval
Decomposition
Counting
Accuracy
Speed
Accuracy
Speed
Accuracy
Speed
Group
Strong
Moderate
Weak
All
0.99
0.96
0.86
0.96
2.90
2.72
3.91
3.18
0.98
0.97
0.90
0.96
5.85
5.82
7.84
6.50
0.81
0.82
0.79
0.81
10.29
17.94
11.62
19.95
Problem Type
L+s
s+L
Tie
Almost-tie
0.96
0.97
0.97
0.95
3.27
3.32
2.65
3.46
0.97
0.96
0.94
0.96
5.45
5.46
7.52
7.60
0.87
0.84
0.81
0.68
16.17
16.90
11.53
15.21
Note. Accuracy is expressed in proportion correct, speed in seconds; L+s=Problems with a large first and a small
second addend, like 8+3=.; s+L=Problems with a small first and a large second addend, like 3+8=.; Tie=Tie sum,
like 6+6=.; Almost-tie=Almost-tie sum, like 6+7=. Accuracy (A) and speed (S) of responding in the choice
condition per group and per problem type: (a) Strong: A=0.97, S=6.35; Moderate: A=0.94, S=5.49; Weak:
A=0.86, S=7.79; (b) L+s: A=0.95, S=4.96; s+L: A=0.94, S=5.23; Tie: A=0.93, S=7.23; Almost-tie: A=0.91, S=8.75
Furthermore, the children did not execute all strategies equally fast in the choice condition
(F(2,93)=51.85, p<.0001). Problems were solved fastest with retrieval, and slowest with counting
strategies. We observed no group differences in the speed of strategy execution in the choice
condition (F(4,1098)=1.09, p=.3587). However, the speed of strategy execution differed between
the four problem types (F(6,1098)=6.14, p<.0001). The children executed decomposition and
counting strategies faster on L+s and s+L problems than on tie and almost-tie sums.
Next, we derived the accuracy of the adding up to 10 and the retrieval strategy from the
scores in, respectively, the adding up to 10 condition and the retrieval condition4. We predicted
the accuracy of strategy execution in both no-choice conditions by means of the variables
strategy (adding up to 10, retrieval), group (strong, moderate, weak), and problem type (L+s,
s+L, tie, almost-tie). The variable strategy was added to the model as random coefficient.
Table 3 presents the accuracy of strategy execution in the adding up to 10 and the retrieval
condition per group and per problem type (least-squares means).
Table 3 reveals that the children did not execute the adding up to 10 and the retrieval
strategy equally accurately (F(1,74)=396.21, p<.0001). The children made less errors when
they had to apply the adding up to 10 strategy than when they were forced to use retrieval.
Moreover, we observed clear group differences in the accuracy of strategy execution in the nochoice conditions (F(2,2914)=13.95, p<.0001). Children with strong mathematical abilities
applied the adding up to 10 strategy more accurately in the respective no-condition than
children with moderate mathematical abilities. Children with weak mathematical abilities
executed both the adding up to 10 and retrieval strategy less accurately than children with
strong and moderate mathematical abilities. The accuracy of strategy execution in the nochoice conditions also differed between the four problem types (F(3,2914)=12.86, p<.0001).
The adding up to 10 strategy was executed least accurately on s+L problems, and most
accurately on L+s problems and tie sums. Children made least retrieval errors on tie sums;
they answered L+s problems more accurately with retrieval than s+L problems (t(2914)=1.85,
p=.0639; border significant) and almost-tie sums.
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283
Table 3
Accuracy of strategy execution in the adding up to 10 and the retrieval condition per group
and per problem type (Least-Squares Means)
Adding up to 10 condition
Retrieval condition
Group
Strong
Moderate
Weak
All
0.98
0.95
0.87
0.95
0.33
0.25
0.07
0.18
Problem type
L+s
s+L
Tie
Almost-tie
0.98
0.90
0.97
0.94
0.15
0.11
0.51
0.10
Note. Accuracy is expressed in proportion correct; L+s=Problems with a large first and a small second addend, like
8+3=.; s+L=Problems with a small first and a large second addend, like 3+8=.; Tie=Tie sum, like 6+6=.; Almosttie=Almost-tie sum, like 6+7=.
Adaptiveness of strategy choices in the choice condition
The differences in the frequency of strategy use between the four problem types in the
choice condition are a first source of evidence for the adaptive nature of children’s strategy
choices in the latter condition. The children solved tie sums, which were answered more
accurately than non-tie sums in the retrieval condition, in the choice condition more frequently
with retrieval than the other problem types; they answered the former also less often with a
decomposition strategy than the latter. This indicates that the children adapted their strategy
choices to (objective) problem-specific characteristics.
Starting from Siegler’s definition of an adaptive strategy choice as choosing the strategy
that leads the individual fastest to an accurate answer to the problem, we further intended to
describe the adaptiveness of strategy choices in the choice condition on the basis of the
accuracy and the speed of the adding up to 10 and retrieval strategy in the respective nochoice conditions. However, as explained in Footnote 4, we could not gather reliable data
about the speed of strategy execution in the no-choice conditions. Consequently, we had to
rely on the accuracy data only.
We first estimated the adaptiveness of strategy choices in the choice condition at the group
level. In line with the analytic approach of Siegler and Lemaire (1997; Lemaire & Lecacheur,
2001a,b, 2002; see also Geary & Burlingham-Dubree, 1989), we calculated for each problem the
correlation between, on the one hand, the relative frequency with which the strong, the moderate
and the weak pupils applied retrieval, decomposition, and counting strategies in the choice
condition, and, on the other hand, the accuracy with which they answered this problem in the
no-choice conditions. Based upon the hypothesis that all three ability groups would at least to
some extent be adaptive in their strategy choices, we expected the following correlations:
1) A positive correlation between the accuracy rate for a particular problem in the
retrieval condition and the relative frequency of retrieval for that problem in the choice
condition,
2) A negative correlation between the accuracy rate for a particular problem in the
retrieval condition and the relative frequency of decomposition for that problem in the
choice condition,
3) A negative correlation between the accuracy rate for a particular problem in the
retrieval condition and the relative frequency of counting for that problem in the
choice condition,
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4) A positive correlation between the accuracy rate for a particular problem in the adding
up to 10 condition and the relative frequency of retrieval or decomposition for that
problem in the choice condition,
5) A negative correlation between the accuracy rate for a particular problem in the adding
up to 10 condition and the relative frequency of counting for that problem in the choice
condition.
The first correlation analysis revealed that the children, irrespective of their general
mathematical ability, adapted their strategy choices to the accuracy with which they knew a
problem by heart, as evidenced by their scores in the retrieval condition. We observed a
positive correlation between the accuracy of problem solving in the retrieval condition and the
frequency of retrieval in the choice condition (hypothesis 1). The more accurately the strong,
the moderate, and the weak children knew a problem by heart, the more frequently they used
retrieval to solve this problem in the choice condition (rstrong=0.73, p=.0002; rmoderate=0.88,
p<.0001; rweak=0.75, p=.0001).
As can be expected on the basis of the results of the first correlation analysis, we found a
negative correlation between the accuracy of retrieval in the retrieval condition and the
frequency of decomposition and counting in the choice condition (hypothesis 2 and 3). The
more accurately the strong, the moderate, and the weak children knew a problem by heart, the
less frequently they solved this problem by means of decomposition in the choice condition
(rstrong=-0.73, p=.0003; rmoderate=-0.86, p<.0001; rweak=-0.72, p=.0003); the more accurately
the children with moderate and weak mathematical abilities answered a problem in the
retrieval condition, the less frequently they counted the answer to this problem in the choice
condition (rstrong=-0.04, p=.8551; rmoderate=-0.61, p=.0040; rweak=-0.51, p=.0208).
The children with moderate mathematical abilities took into account the accuracy of the
adding up to 10 strategy too (hypotheses 4 and 5): The more accurately the moderate pupils
could solve a problem with the latter strategy, as evidenced by their scores in the adding up to
10 condition, the more often they chose the retrieval or a decomposition strategy (r=0.51,
p=.0204), and the less frequently they applied counting (r=-0.51, p=.0204) to solve the
problem in the choice condition.
We found no group differences in the strength of the correlation between the accuracy
rate in the no-choice conditions and the frequency of strategy use in the choice condition.
Next – and complementary to the group level analyses described above – we estimated the
adaptive nature of strategy choices in the choice condition at the individual level. Therefore, we
compared for each child and for each problem the accuracy of strategy execution in the adding up
to 10 and the retrieval condition with the particular strategy this child had chosen to solve this
specific problem in the choice condition. We formulated the following general criterion to
determine the adaptive nature of strategy choices in the choice condition: A child makes an
adaptive strategy choice if and only if he or she solves a problem in the choice condition with a
strategy that belongs to at least the same level of strategy use as the highest level available to this
child, as indicated by his or her task performance on the same problem in the adding up to 10 and
the retrieval condition. We concretised this general adaptiveness criterion as follows: A child
makes an adaptive strategy choice in the choice condition if he or she:
1) prefers retrieval to solve a problem he or she can answer accurately with this strategy
in the retrieval condition,
2) prefers the adding up to 10 strategy to solve a problem he or she can not answer
accurately with retrieval in the retrieval condition, but with the adding up to 10
strategy in the adding up to 10 condition; if the child prefers another decomposition
strategy (like, for instance, the tie strategy) or retrieval to answer such a problem in the
choice condition, this choice is also scored as an adaptive one, provided it resulted in a
correct answer,
3) prefers a counting strategy to solve a problem he or she can not answer accurately with
retrieval nor the adding up to 10 strategy in the respective no-choice conditions; if the
STRATEGIC COMPETENCE: SIMPLE ADDITION
285
child prefers a decomposition strategy or retrieval to answer such a problem in the
choice condition, this choice is scored as an adaptive one, provided it led to an
accurate answer.
The results of this data analysis at the individual level reveal clear differences in the
adaptiveness of strategy choices (F(2,1428)=7.06, p=.0009). Children with strong
mathematical abilities made more adaptive strategy choices than children with moderate
mathematical abilities (M=89.20 and M=80.69, respectively), who, at their turn, chose more
adaptively between the different types of strategies than children with weak mathematical
abilities (M=70.20). The latter children frequently used a (more primitive) counting strategy to
solve problems they already could solve accurately with the (more advanced) adding up to 10
strategy. Furthermore, we observed differences in the adaptiveness of individual strategy
choices between the different problems (F(18,1428)=1.71, p=.0324): The children answered
9+5 least adaptively (M=67.21), and 5+6 most adaptively (M=90.97).
In sum, the results of the group level analyses as well as the results of the analyses at the
individual level indicate that the children were generally quite adaptive in their strategy
choices in the choice condition. The group level analyses showed that children fitted their
strategy choices to the accuracy of the adding up to 10 strategy and, even more, of the
retrieval strategy, and revealed no group differences in the adaptiveness of strategy choices.
Next, the analyses at the individual level demonstrated that children made adaptive strategy
choices, and revealed that children with stronger mathematical abilities chose more adaptively
between the retrieval, decomposition, and counting strategies than children with weaker
mathematical abilities.
Conclusions and discussion
This study aimed at describing the repertoire, frequency, efficiency, and adaptiveness of
the strategies second-graders apply to solve simple additions with the bridge over 10, using
the choice/no-choice method. On the one hand, our study replicates the findings of previous
research on strategy use in the domain of simple addition up to 20, in which empirical
evidence was found for the validity of Siegler’s model of developmental changes in the
repertoire and frequency of strategies. On the other hand, the results of our study deepen our
insight into the efficiency of strategy execution and the adaptiveness of young children’s
strategy choices, and document the value of the choice/no-choice method to determine
strategy efficiency characteristics and, especially, the adaptive nature of strategy choices.
In line with the results of previous research, the children who participated in our study
used a rich variety of retrieval, decomposition, and counting strategies to solve simple
additions with the bridge over 10. Notwithstanding the strong instructional focus on the
mastery of the strategy adding up to 10, most children already were able to retrieve the answer
to some problems, while applying a diversity of counting strategies on others. With respect to
the decomposition strategies, the children employed the (well-trained) adding up to 10 strategy,
but also, although to a much lesser extent, the (untrained) tie strategy, reversal strategy, and
“one less than 10” strategy. We observed no differences in the repertoire of strategy use
between children with different mathematical abilities (first dimension).
With respect to the second dimension, the children did not use the different types of
strategies equally frequently. They obviously preferred the adding up to 10 strategy to solve
the problems. But they also used the retrieval strategy rather frequently, and applied the latter
strategy more often than counting strategies. The frequency of strategy use differed clearly in
function of the children’s general mathematical ability: Children with stronger mathematical
abilities solved more problems with retrieval or a decomposition strategy, and solved less
problems by means of counting, than children with weaker mathematical abilities.
The children did not execute all strategies equally efficiently (third dimension). Retrieval
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J. TORBEYNS, L. VERSCHAFFEL, & P. GHESQUIÈRE
was executed fastest – and also very accurately – in the choice condition, and decomposition
strategies were faster and more accurately than counting strategies in the latter condition. The
efficiency of strategy execution in the choice condition differed between children of distinct
mathematical ability. Children of the weakest ability level made more retrieval and
decomposition errors than children with moderate and strong mathematical abilities, and
children with moderate mathematical abilities made more retrieval errors than children of the
strongest ability level. However, as stressed before, these data about the accuracy and the
speed of strategy execution in the choice condition need to be interpreted cautiously. In
particular, the data about the accuracy of the adding up to 10 and the retrieval strategy,
obtained in the respective no-choice conditions, demonstrated that the accuracy of the retrieval
strategy was probably overestimated in the choice condition due to its frequent use on tie
sums, which are – and also were in our study, as evidenced by the data in the retrieval
condition – memorised earlier than non-tie sums (Ashcraft, 1987; Baroody, 1987; Fuson,
1992). Moreover, the accuracy data from the no-choice conditions revealed – in contrast with
those obtained in the choice condition – no differences in retrieval accuracy between children
with strong and moderate mathematical abilities, but clear differences in decomposition
accuracy between these groups of children.
Finally, with respect to the fourth dimension, our results reveal that after one year of
elementary mathematics education, Flemish second-graders are quite able to choose adaptively
between retrieval, decomposition, and counting strategies while solving simple additions with the
bridge over 10. First of all, the differences in the frequency of strategy use between the four
problem types in the choice condition show that the children adapted their strategy choices to
problem-specific characteristics. The children preferred to solve the tie sums, which, as
mentioned before, are and were memorised earlier than non-tie sums, by means of retrieval.
Second, the positive correlations between the accuracy with which the three ability levels
applied retrieval in the retrieval condition and the frequency of retrieval, decomposition, and
counting in the choice condition, illustrate the basically adaptive nature of the strategy choices
too, even of those from the weakest ability level. Thirdly, the fine-grained, individualised
comparison of each child’s strategy choices in the choice condition with the accuracy of
problem solving in the adding up to 10 and the retrieval condition, revealed that the children
made more adaptive strategy choices than non-adaptive ones (in the choice condition), and
that children with stronger mathematical abilities chose more adaptively between the retrieval,
decomposition, and counting strategies than children with weaker mathematical abilities.
However, the latter results contradict the results of the group level analyses. These
contradictory results are probably due to differences in the concretising of the adaptiveness
criterion in both analyses. The individual analyses took into account the accuracy of problem
solving in the no-choice conditions, as well as the accuracy of problem solving in the choice
condition (see part 2 and 3 of the concretising of the criterion), whereas the group level
analyses did not take into account the accuracy of problem solving in the choice condition.
The latter resulted in differences between both analyses in the scoring of a choice as an
adaptive or a non-adaptive one, especially in the group of weak children, which decreased the
group differences in adaptiveness at the group level, and increased these differences at the
individual level (for more details, see Torbeyns, Verschaffel, & Ghesquière, 2001a).
From a methodological viewpoint, our study documents the value of the choice/no-choice
method to gain deeper insight in the efficiency and the adaptive nature of young children’s
strategy use. Above all, this method allowed us to gather unbiased information about the
accuracy with which each child executed the strategy adding up to 10 and retrieval when
solving simple additions with the bridge over 10, which helped us to estimate the adaptiveness
of each child’s individual strategy choices in the choice condition accurately too. However, as
a consequence of two practical problems, we were not able to obtain unbiased information
about the efficiency of all strategies that the participating children used, which made it
impossible for us to get a complete picture of the adaptiveness of each individual’s strategy
choices. First of all, the children used a very rich diversity of decomposition and counting
strategies to solve the problems in the choice condition. Methodological and practical
STRATEGIC COMPETENCE: SIMPLE ADDITION
287
considerations, such as, the methodological (im)possibilities to guarantee that the children use
the strategy they are supposed to be using in each of the required no-choice conditions, and the
large amount of time it would have taken to test each child in each of these conditions,
prevented us to design a no-choice condition for each of the observed strategies (for a more
extensive discussion of this topic, see Torbeyns, Verschaffel, & Ghesquière, 2001b).
Secondly, we were not able to gather reliable information about the speed of the adding up to
10 and the retrieval strategy to solve simple additions with the bridge over 10. As mentioned
earlier, the participating children systematically answered slower in the adding up to 10
condition than in the choice condition, even on those problems they had solved freely by
means of that same adding up to 10 strategy in the choice condition. The time limit in the
retrieval condition made it impossible to gather reliable data about the speed of the retrieval
strategy. This made it problematic to apply the information about the speed of strategy
execution in our operationalisation of the adaptiveness criterion in the group level analyses as
well as in the analyses at the individual level. Taking into account the theoretical strengths as
well as the practical limits of the choice/no-choice method, we think it is a challenge for future
research to try to reconcile Siegler’s methodological ideal with the practical constraints of a
research setting, in order to further improve our understanding of the efficiency of strategy use
and, ever more, the adaptive nature of children’s and adults’ strategy choices in solving
mathematical tasks in particular, and in reasoning and problem solving in general.
Notes
1
As argued by Siegler and Lemaire (1997) and discussed below, the information about the efficiency of strategies
generated by the “choice” method are probably biased by selection effects.
2
Taking into account the results of our pilot-study and the literature concerning the topic, we reduced the maximum
solution time in the retrieval condition to two seconds. Reducing the maximum solution time up to two and a half
seconds, allowed the second- and third-graders who participated in our pilot-study, to use a (fast executed)
decomposition or counting strategy on the problems they did not know by heart. Although a further reduction of the
solution time to two seconds can neither guarantee the use of retrieval, the research literature (Baroody, 1999;
Siegler, 1996) does not support the reduction of the solution time to less than two seconds.
3
In this article, we focus on the results concerning the level of strategy use (retrieval, decomposition, counting). We
refer the interested reader to another publication (Torbeyns, Verschaffel, & Ghesquière, 2001a), in which these
results and the results about the cleverness of strategy use are discussed in more detail.
4
Methodological and practical constraints prevented us to design a no-choice condition for each of the distinguished
retrieval, decomposition, and counting strategies. Taking into account the aims, the content, and the organization of
elementary mathematics education in Flanders (Feys, 1995), we asked the children to use the adding up to 10 strategy
on all problems in a first no-choice condition, and the retrieval strategy in a second one. The data we obtained in the
adding up to 10 and the retrieval condition are limited to the accuracy of the respective strategies. We were not able to
gather reliable data about the speed of the adding up to 10 and retrieval strategy. First, children solved problems slower
in the adding up to 10 condition than in the choice condition, even those problems they had answered spontaneously
with this same adding up to 10 strategy in the choice condition. This is probably due to the fact that most children had
already reached a stage wherein they could execute the adding up to 10 strategy in a (quasi-)automatised way (i.e.,
very fast and without conscious control of all steps involved) in the choice condition, whereas both the instruction and
the presentation format of the problems in the adding up to 10 condition forced them to execute this strategy in a more
articulated and stepwise – and therefore necessarily slower – way. Second, children were forced to answer all
problems within two seconds in the retrieval condition. Whenever a child was not able to answer a problem within this
time limit, the computer registered his or her answer automatically as “out of time”.
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Dans cet article, nous examinons différentes caractéristiques des
stratégies cognitives utilisées par de jeunes enfants afin de résoudre
des problèmes d’addition initiale. Les stratégies sont caractérisées et
étudiées d’après le cadre conceptuel et méthodologique proposé par
Siegler (Lemaire & Siegler, 1995; Siegler & Lemaire, 1997). Il s’agit
notamment de la variabilité, la fréquence et l’efficacité des stratégies
utilisées ainsi que de la faculté d’adaptation des enfants dans la
sélection de stratégie. Soixante-dix-sept élèves en deuxième année de
l’école primaire ont calculé une série de 25 problèmes d’addition
initiale dans 3 conditions différentes. Dans la première condition, les
élèves pouvaient utiliser une stratégie à leur choix. Dans la deuxième et
la troisième condition, les mêmes enfants étaient obligés à résoudre
toutes les additions soit par la stratégie de décomposition jusqu’à 10
soit par la stratégie de mémorisation. Les résultats montrent que des
élèves de deuxième année primaire se servent de plusieurs stratégies
(la stratégie de mémorisation, décomposer, compter) selon le type de
problèmes d’addition initiale. De plus, ces stratégies diffèrent selon
leur fréquence et leur efficacité. Quant à la compétence mathématique
des enfants, nous pouvons conclure que celle-ci n’influence pas le
répertoire des stratégies utilisées, mais qu’elle a toutefois un effet
substantiel sur la fréquence, l’efficacité et la faculté d’adaptation dans
leur choix stratégique. Finalement, notre étude illustre la valeur du
cadre méthodologique de Siegler qui permet d’étudier la faculté
d’adaptation de jeunes enfants dans leurs choix stratégiques en
mathématiques.
Key words: Methodology, Simple addition, Strategic change.
290
J. TORBEYNS, L. VERSCHAFFEL, & P. GHESQUIÈRE
Received: January 2002
Revision received: May 2002
Joke Torbeyns. Research Assistant of the Fund for Scientific Research – Flanders, Centre for
Instructional Psychology and Technology, University of Leuven, Vesaliusstraat 2, B-3000 Leuven,
Belgium, E-mail: [email protected]
Current theme of research:
Strategic aspects of elementary mathematics learning.
Most relevant publications in the field of Psychology of Education:
Torbeyns, J., Verschaffel, L., & Ghesquière, P. (2001). Strategieontwikkeling en strategiekeuze bij cognitieve taken.
Een kritische analyse van Sieglers theorie van “strategic change” [Choice and development of cognitive strategies.
A critical analysis of Siegler’s theory of “strategic change”]. Pedagogisch Tiidschrift, 26, 113-142.
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domain of simple addition, using the “choice/no-choice” method). Pedagogische Studien, 79, 89-102.
Torbeyns, J., Van den Noortgate, W., Ghesquiere, P., Verschaffel, L., Van de Rijt, B.A.M., & Van Luit, J.E.H. (in
press). Development of Early Numeracy in 5- to 7-Year-Old Children. A Comparison between Flanders and The
Netherlands. Educational Research and Evaluation. An International Journal on Theory and Practice.
Lieven Verschaffel. Centre for Instructional Psychology and Technology, University of Leuven,
Vesaliusstraat 2, B-3000 Leuven, Belgium, E-mail: [email protected]
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Mathematics learning and teaching.
Most relevant publications in the field of Psychology of Education:
Verschaffel, L., Greer, B., & De Corte, E. (2000). Making sense of word problems (XVII, 203 pp.). Lisse, The
Netherlands: Swets & Zeitlinger.
Verschaffel, L., De Corte, E., Lamote, C., & Dhert, N. (1998). Development of a flexible strategy for the estimation of
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Pol Ghesquière. Section of Orthopedagogics, University of Leuven, Vesaliusstraat 2, B-3000 Leuven,
Belgium, E-mail: [email protected]
Current theme of research:
Specific learning disabilities (dyslexia, dyscalculia); Special education.
Most relevant publications in the field of Psychology of Education:
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