1) The study investigated adults' strategy use for two-digit subtraction problems. Specifically, it examined whether adults use an "addition by subtraction" strategy, where they solve subtraction problems by converting them to addition problems. 2) The study found that adults flexibly switch between direct subtraction and addition by subtraction strategies depending on the relative size of the number being subtracted (the subtrahend). If the subtrahend was smaller than the difference, direct subtraction was used more; if the subtrahend was larger, addition by subtraction was dominant. 3) However, this pattern only held when the distance between the subtrahend and difference was large. When they were close in size, there was no preference for either

1. Adults’ use of subtraction by addition
Greet Peters a,
⁎, Bert De Smedt b
, Joke Torbeyns a,c
, Pol Ghesquière b
, Lieven Verschaffel a
a
Centre for Instructional Psychology and Technology, K.U.Leuven, Belgium
b
Centre for Parenting, Child Welfare, and Disabilities, K.U.Leuven, Belgium
c
GROEP T – Leuven Education College, Belgium
a b s t r a c t
a r t i c l e i n f o
Article history:
Received 17 March 2010
Received in revised form 23 August 2010
Accepted 25 August 2010
Keywords:
Multi-digit subtraction
Strategy use
Subtraction by addition
The present study investigates adults’ use of addition to solve two-digit subtractions. Inspired by research on
single-digit arithmetic, we first examined regression models in which different problem characteristics predicted
participants’ reaction times. Second, we compared performance on two-digit subtractions presented in 2
presentation formats, i.e., the standard subtraction format (81–37=.) and an addition format (37+.=81). Both
methods lead to the conclusion that the participants switched between direct subtraction and subtraction by
addition depending on the relative size of the subtrahend: If the subtrahend was smaller than the difference,
direct subtraction was mainly used; if the subtrahend was larger than the difference, subtraction by addition was
the dominant strategy. However, this performance pattern was only observed when the distance between the
subtrahend and the difference was large; when the subtrahend and the difference were close to each other, there
was no subtrahend-dependent selection of direct subtraction vs. subtraction by addition. These data indicate that
theoretical models of people's strategy choices in subtraction should include the relative size of the subtrahend as
an important factor in the strategy selection process.
© 2010 Elsevier B.V. All rights reserved.
1. Introduction
For a long time, researchers have investigated how people solve
symbolically presented subtraction problems of the type M−S=.
(Barrouillet, Mignon, & Thevenot, 2008; Blöte, Klein, & Beishuizen,
2000; Fuson, 1992; Robinson, 2001; Selter, 2001; Verschaffel, Greer, &
De Corte, 2007; Woods, Resnick, & Groen, 1975). The bulk of this
research relates to single-digit subtraction, whereas only few studies
have looked at people's strategy use on multi-digit subtraction. This
is surprising, given the considerable differences between single-digit
and multi-digit arithmetic in the procedures and underlying arithmetic
concepts being used (Fuson, 1992; Verschaffel et al., 2007). This study
therefore aimed at examining people's strategy use in multi-digit
subtraction, by applying – for the first time – research methods that
have been frequently used in studies on single-digit arithmetic. In the
remainder of this introduction, we review the research on strategy use in
single-digit and multi-digit subtraction, together with the applied
research methods.
Woods et al. (1975) investigated how children solved symbolically
presented single-digit subtractions of the form M−S=. by fitting their
reaction times (RTs) to different linear regression models. They
observed that children solved these problems by counting up or down,
whichever involved the fewest counting steps (i.e., counting down for
9−2 vs. counting up for 9−7). After a period wherein accuracy and RTs
were the main data sources to investigate strategy use (Ashcraft, 1982;
Groen & Poll, 1973), verbal protocols became an increasingly
important research method (Campbell & Xue, 2001; Geary, Frensch,
& Wiley, 1993; LeFevre, DeStefano, Penner-Wilger, & Daley, 2006;
Seyler, Kirk, & Ashcraft, 2003). These verbal protocol data showed that
adults did not rely exclusively on direct retrieval when solving single-
digit subtractions, which was assumed to be the expert way of doing
single-digit arithmetic (e.g., Ashcraft, 1992; Siegler & Shrager, 1984).
Adults sometimes used procedural strategies, such as counting or
decomposing the problem into smaller facts, particularly on the so-
called large single-digit subtractions, i.e., with a minuend N10. Most
interestingly within the context of the present study is the observation
that adults sometimes used addition-based strategies to solve single-
digit subtractions (e.g., solving 9−7 by means of the known addition
fact 7+2=9, or solving 13−9 by counting or adding up from 9 to 13:
9+1=10 and 10+3=13, so the answer is 1+3=4) (Campbell &
Xue, 2001; Geary et al., 1993; LeFevre et al., 2006). However, these
addition-based strategies are mostly executed in a very fast and quasi-
automated way, which makes them difficult to investigate by means of
verbal protocols (Kirk & Ashcraft, 2001).
Campbell (2008) recently introduced another method to investigate
whether people use addition to solve single-digit subtractions. Based on
a design originally developed for multiplication and division by Mauro,
LeFevre, and Morris (2003), he presented single-digit subtractions in
subtraction (e.g., 12−9=.) and addition format (e.g., 9+.=12). He
Acta Psychologica 135 (2010) 323–329
⁎ Corresponding author. Andreas Vesaliusstraat 2, box 3770, 3000 Leuven, Belgium.
Tel.: +32 16 32 62 47; fax: +32 16 32 62 74.
E-mail address: Greet.Peters@ped.kuleuven.be (G. Peters).
0001-6918/$ – see front matter © 2010 Elsevier B.V. All rights reserved.
doi:10.1016/j.actpsy.2010.08.007
Contents lists available at ScienceDirect
Acta Psychologica
journal homepage: www.elsevier.com/ locate/actpsy

2. reasoned that if subtractions are solved by addition, the problems
presented in addition format should be solved faster than those in
subtraction format. This should be so because a subtraction presented in
addition format does not require the mental re-representation of the
subtraction into an addition,whereassuch a re-representation is needed
when the problem is presented in subtraction format. In other words,
the addition format might be a better cue for retrieving the addition fact
than the subtraction format and/or it might be a facilitator for executing
an addition-based procedure. Campbell successfully demonstrated that
large single-digit subtractions were solved significantly faster in
addition than in subtraction format, which indicated that adults used
addition to solve large single-digit subtractions. No such format
differences were observed on small single-digit subtractions.
Peters, De Smedt, Torbeyns, Ghesquière, and Verschaffel (2010)
recently refined Campbell's findings, by showing that this use of
subtraction by addition in large single-digit subtractions depends on
the relative size of the subtrahend. They suggested that adults solve
large single-digit subtractions by switching between subtraction by
addition and direct subtraction, taking into account which strategy is
most efficient. This mirrors the above-mentioned finding of Woods et
al. (1975) about children's flexible switches between counting up or
down on small single-digit subtractions.
Can these findings on strategy use in single-digit subtraction be
generalised to multi-digit subtraction? Within the mathematics
education literature, several authors have reported the use of
addition-based strategies on multi-digit subtractions (e.g., 71−57
by 57+10=67, 67+4=71; so the answer is 10+4=14), thereby
using different terms to denote this strategy, such as the forward
strategy (Brissiaud, 1994), solving subtractions by means of addition
(Beishuizen, 1997), the short jump strategy (Blöte et al., 2000), the
adding up strategy (Selter, 2001), or indirect addition (Torbeyns,
Ghesquière, & Verschaffel, 2009). In this article, we will call this
strategy subtraction by addition, whereas direct subtraction will be
used to refer to the common subtraction strategy where the
subtrahend is directly subtracted from the minuend.
Torbeyns et al. (2009) investigated adults’ use of subtraction by
addition on three-digit subtractions with the choice/no-choice para-
digm andsystematically manipulated therelative sizeof thesubtrahend.
Accuracy and speed data from the two no-choice conditions (wherein
participants had to solve all items first by means of direct subtraction
and afterwards via subtraction by addition, or vice versa) indicated that
subtraction by addition was a highly efficient strategy for three-digit
subtraction, irrespective of the size of the subtrahend. The verbal
protocol data from the choice condition indicated that this subtraction
by addition strategy was mainly selected on problems with a relatively
large subtrahend(e.g., 812−783),whichsuggests that participants took
into account the size of the subtrahend when selecting their strategy.
The use of subtraction by addition in multi-digit subtraction has
also been explored in children, again with verbal protocols. Different
from adults, children appeared to hardly apply this strategy
spontaneously, not even when they were confronted with problems
with a very large subtrahend as in 81−79 (Selter, 2001; Torbeyns, De
Smedt, Ghesquière, & Verschaffel, 2009b), when they were explicitly
stimulated to generate a strategy other than direct subtraction
(Torbeyns, De Smedt, Ghesquière, & Verschaffel, 2009a), or after
they received explicit training in subtraction by addition (De Smedt,
Torbeyns, Stassens, Ghesquière, & Verschaffel, 2010).
We remind that the available studies on subtraction by addition in
multi-digit arithmetic relied mainly on verbal protocol data. As for
research in single-digit arithmetic, it can be argued that verbal protocols
may be less suited to identify the subtraction by addition strategy in
multi-digit subtraction because this strategy may be executed very fast
and quasi-automatic. For example, when solving a problem such as
81−79, people may be unaware of the calculation steps they executed,
or they may have difficulties in articulatingprecisely how they foundthe
answer. This may have resulted in an underestimation of the number of
subtraction by addition strategies in the available studies on multi-digit
subtraction. In this respect, we point out that both the data of Torbeyns
et al. (2009) and of De Smedt et al. (2010) suggest that participants
sometimes used subtraction by addition while reporting the direct
subtraction strategy. If participants only used direct subtraction, an
increase in RTs should have been observed from items with small
subtrahends over items with medium-sized subtrahends to items with
large subtrahends because subtracting a larger subtrahend from a given
minuend requires more and/or larger calculation steps. However, both
studies showed that not only problems with small but also with large
subtrahends were solved faster than problems with medium-sized
subtrahends. This indicates that participants did not always use direct
subtraction,and it stronglysuggests that the verbal protocol data did not
always capture the actually applied strategy. Therefore, it justifies the
application of other methods of inferring participants’ strategy use that
have been successfully used in single-digit arithmetic, namely the
regression-based approach (Groen & Poll, 1973; Woods et al., 1975) and
the manipulation of presentation formats (Campbell, 2008; Peters et al.,
2010).
2. The present study
We used a stepwise regression analysis and the manipulation of
presentation formats to investigate the use of subtraction by addition
in multi-digit subtraction. We presented adult participants with two-
digit subtractions1
in the standard subtraction format (83−4=.) and
in its corresponding addition format (4+.=83), manipulating the
relative size of the subtrahend.
Following Groen and Poll (1973) and Woods et al. (1975), we first
calculated a stepwise regression model in which participants' RTs
on two-digit subtractions were predicted by the presentation format
(M−S=. or S+.=M), three variables referring to strategy use, and
their respective interactions. The variables referring to strategy use
were the to-be-determined difference (D), the known subtrahend (S),
and the minimum of difference and subtrahend (min[D, S]), which
reflect the use of subtraction by addition, direct subtraction, and
switching between both strategies depending on the size of the
subtrahend, respectively. In view of Peters et al. (2010) findings in
single-digit subtraction, we expected that a model including the min
(D, S) predictor as the variable that explains most of the variability in
RTs would provide the best fit, suggesting the mixed use of subtraction
by addition and direct subtraction in both presentation formats.
We also compared participants’ performance on different problem
types in the two presentation formats. We predicted an interaction
between thepresentationformat(M−S=. vs. S+.=M), themagnitude
of S compared to D (SND vs. SbD), and the numerical distance between
S and D (large vs. small). For large-distance problems, we predicted that
subtraction by addition would be used to solve SND problems and that
the direct subtraction strategy would be used to solve SbD problems.
The SND subtractions presented in addition format are thus expected
to be solved faster than the corresponding problems in subtraction
format, whereas SbD subtractions are expected to be solved faster in
subtraction than in addition format, this because the addition format
may facilitate the execution of the subtraction by addition strategy
more than the subtraction format (and vice versa for direct subtraction).
For small-distance problems, we predicted no interaction between
the magnitude of S compared to D and presentation format because
the computational advantage of one of the two strategies is less clear.
1
‘Two-digit subtraction’ has different meanings in the research literature. According
to some authors, it refers only to the subtraction of a two-digit number from a two-
digit number yielding a two-digit result (e.g., 75−36=.). However, it can also be used
to refer to problems with a single-digit subtrahend subtracted from a two-digit
minuend resulting in a two-digit difference (e.g., 71−2=.) or to refer to problems
with a two-digit subtrahend subtracted from a two-digit minuend resulting in a
single-digit difference (e.g., 71−69=.), as we did in the current manuscript (see also
Brissiaud, 1994; Van Mulken, 1992).
324 G. Peters et al. / Acta Psychologica 135 (2010) 323–329

3. 3. Method
3.1. Participants
Participants were 25 university students (5 men, 20 women;
Mage =26 years, SDage =7 years) from the Department of Educational
Sciences of the Katholieke Universiteit Leuven. All of them had at least
a bachelor degree. They had normal or corrected-to-normal vision and
none of them reported learning difficulties.
3.2. Materials and procedure
Participants were asked to mentally solve 32 subtraction problems,
which had a two-digit minuend larger than 30 and required borrowing.
Problems were presented in the horizontal subtraction format2
(52−4=.) and in their corresponding addition format (4+.=52),
yielding a total set of 64 trials. They were divided into four problem types
based on the combination of the magnitude of S compared to D (SND or
SbD) and the numerical distancebetweenS and D (small or large). Small-
distance problems were defined by S and D differing by less than 10,
whereas in the large-distance problems S and D differed by at
least 10 and either S or D was a one-digit number. This resulted
in the following problem types: (a)large-distance SbD problems,
with subtrahends smaller than 10 (e.g., 83−4=. and 8+.=34);
(b)large-distance SND problems, with differences smaller than 10
(e.g., 77−68=. and 37+.=42); (c)small-distance SbD problems
(e.g., 92−44=. and 36+.=75); and (d)small-distance SND
problems (e.g., 32−17=. and 29+.=53) (see Appendix).
Threepracticeproblemsperpresentationformatwereadministeredto
familiarise participants with the task administration. The experimental
items were presented into six blocks, assigned in a pseudo-random
manner with the following constraints: (a) items from the same problem
typedid notoccuronmorethan twoconsecutive trials; (b)itemsthatonly
differed in presentation format (e.g., 31−28=. and 28+.=31) and
complementary items (e.g., 31−28=. and 31−3=.) were not
presented in the same block. Task instructions stressed both accuracy
and speed.
Each trial started with a fixation asterisk that appeared for 1000 ms
in the centre of a DELL personal computer screen. Next, the item was
presented horizontally in the middle of the screen. The characters were
2×2 cm large, black, separated by adjacent spaces and presented
against a white background. Time started to run when the item
appeared, and ended when the verbal response triggered the sound-
activated voice key. The item remained on the screen until the voice key
was triggered, with a maximum of 30 s. Participants were instructed to
articulate only the final answer and not the intermediate solution steps.
A computer connected to the voice key registered RT with millisecond
precision. The experimenter then entered the answer via the keyboard.
After that, the next trial was initiated. No feedback was given. Trials with
incorrect voice key registrations were presented again at the end of each
block, to avoid losing too many data3
.
We instructed participants on how to answer during the practice
trials, asking them to say the numerical answer to the problem as soon
as they knew the answer and without any preceding words or
phrases, such as “that makes…” or “the outcome is”. Furthermore, we
started the experiment by asking participants to name a given set of
numbers in order to make sure that the voice key reliably detected the
different initial phonemes of potential answers. Participants com-
pleted the task in a quiet room. They solved the task individually on a
computer in the presence of an experimenter.
4. Results
A total of 1600 items was solved. Seventeen items were excluded
because of voice key errors (1.06%). For RT analysis, 131 items were
deleted because of incorrect answers (8.49%) and 39 items (2.44%)
were additionally discarded as outliers (i.e., more than 2 standard
deviations from a participant's cell mean for the correct items per
problem type).
Results are presented in two parts. The first part involves the
evaluation of the regression models that predict participants’ RTs. In
the second part, we compare performance on both presentation
formats.
4.1. Regression analysis
Descriptive analyses showed that, for both presentation formats,
problems with a large numerical distance between S and D were
solved considerably faster than those with a small numerical distance
(Mlarge =2362 ms; SD=1208 ms vs. Msmall =4978 ms; SD=2379 ms
for the subtraction format; and Mlarge =2424 ms; SD=1196 ms vs.
Msmall =4774 ms; SD=2567 ms for the addition format). This was
observed in SbD problems as well as in SND problems (Table 1). These
descriptive data suggest that participants did not use the same
strategy on all problem types of the same presentation format, but
potentially switched between subtraction by addition and direct
subtraction depending on the relative size of the subtrahend.
To test this strategy switch hypothesis more formally, we carried
out a stepwise multiple regression analysis based on the work of
Groen and Poll (1973) and Woods et al. (1975), using the stepwise
selection option for regression analysis in the statistical program SAS
(version 9.1), with the criterion for inclusion of a parameter in the
model at p=0.05. We predicted the RTs of the 64 administered
problems by presentation format (coded as subtraction format=1
and addition format=−1). Additional predictors were the size the
subtrahend, the size of the difference, the minimum of D and S, and
the interactions between these predictors and presentation format.
If participants only used direct subtraction, RTs should be best
predicted by the size of the subtrahend because it takes longer to
subtract 84 from a given number than to subtract 3 from that number.
If participants only used subtraction by addition, RTs should be best
predicted by the size of the difference because it takes longer to
determine how much needs to be added to get at a given number
when the difference between both numbers is large (“How much
needs to be added to 3 to have 91?”) than when it is small (“How
much needs to be added to 88 to have 91?”). If participants switched
2
The horizontal format is the standard format for doing subtraction in Flanders,
Belgium, as in many other European countries (e.g., The Netherlands, Germany).
Flemish children learn to subtract in this way in the first two grades of elementary
school before they are introduced to the ‘vertical’ columnar algorithm, which they only
learn after mastering subtractions in the horizontal format in the number domain up
to 100. This differs from North America, where the vertical format is the dominant one.
3
About 5% of all administered trials were repeated at the end of a block (125/2400).
The distribution of incorrect voice key registrations did not differ across problem
types, χ²(3)=2.93, p=0.40. We repeated the analyses without the trials with
incorrect voice key registrations and the results remained generally the same,
indicating no main effect of format [F(1,24)=1.13, p=0.30], a significant format×-
magnitude interaction [F(1,96)=14.02, pb0.01], and a format × magnitude×distance
interaction [F(1,96)=3.92, p=0.05]. Since repeating problems did not affect our
findings, we did not to exclude these trials from the results.
Table 1
Mean reaction times of the 4 problem types in both presentation formats.
S b D S N D
Presentation
format
Large
distance
Small
distance
Large
distance
Small
distance
Subtraction format 2201 ms 4752 ms 2530 ms 5216 ms
Addition format 2735 ms 4612 ms 2120 ms 4942 ms
325
G. Peters et al. / Acta Psychologica 135 (2010) 323–329

4. between both strategies depending on the most efficient strategy, RTs
should be best predicted by the minimum of subtrahend and
difference: Short RTs are then expected for problems with a small
subtrahend (91−3=.) because they are easily solved by direct
subtraction, and for problems with a small difference (91−88=.)
because they can be easily solved by subtraction by addition. If
participants use different strategies on the two presentation formats,
there should be an interaction between presentation format and (one
of) the other predictors. If their strategy use was similar across both
formats, no such interactions are expected.
The results of the stepwise multiple regression analysis revealed
that the model containing only the minimum of S and D as predictor
provided the best fit to the data (Table 2), F(1,62)=222.65, pb0.01,
R²=0.78. This is in line with our hypothesis that, for both presentation
formats, participants switch between subtraction by addition and
direct subtraction depending on the relative size of the subtrahend.
This regression analysis aggregated performance over the various
participants and may mask performance patterns at the individual level.
We therefore also analysed the RTs for each participant individually. All
but two participants solved all types of large-distance problems faster
than the corresponding small-distance problems (Subjects 22 and 23, see
Fig. 1). We repeated the abovementioned stepwise regression analysis
for each participant individually as well. The model with min(D, S) as the
only predictor provided the best fit of the RTs for all 25 participants,
indicating at the individual level that all participants switched between
direct subtraction and subtraction by addition.
4.2. The effect of presentation format
The mean RT and accuracy per problem type and presentation
format are depicted in Fig. 2. We performed a 2×2 × 2 repeated
measures ANOVA on the RT and accuracy data with magnitude (SbD
vs. SND), numerical distance (small vs. large) and presentation format
(subtraction vs. addition) as within-subject factors. Tukey-Kramer
adjustments were used for post hoc comparisons.
4.2.1. Reaction time
There was a main effect of numerical distance, F(1,24)=122.23,
pb0.01, with large-distance problems being solved 2512 ms faster
than small-distance problems. There were no main effects of format,
F(1,24)=1.03, p=0.32, or magnitude, F(1,24)=2.98, p=0.10. The
magnitude×distance interaction was significant, F(1,96)=11.46,
pb0.01: Small-distance SbD problems were solved 484 ms faster than
small-distance SND problems, t(96)=-3.47, pb0.01, whereas large-
distance S N D problems were solved faster than large-distance S b D
problems, but this latter difference was not significant, t(96)=0.87,
p=0.44. More importantly, a significant format × magnitude interac-
tion was found, F(1,96)=9.29, pb0.01. S N D problems were solved
faster in addition than in subtraction format, t(96)=2.87, pb0.01, while
for the S b D problems the subtraction format tended to elicit shorter
RTs, but this latter effect was not significant, t(96)=-1.44, p=0.15.
Finally, the expected magnitude × distance × format interaction was
significant, F(1,96)=7.02, pb0.01. Post hoc t-tests showed that only for
large-distance subtractions, S b D problems were solved significantly
faster in subtraction than in addition format, t(96)=-3.09, pb0.01, and
S N D problems were solved faster in addition than in subtraction format,
t(96)=2.60, p=0.01. When thedistancebetween S and D wassmall, no
format effect was found.
4.2.2. Accuracy
Accuracy levels were high for all problem types. There was only a
main effect of numerical distance, F(1,24)=23.63, pb0.01, with
large-distance problems (M=0.96) being solved more accurately
than small-distance problems (M=0.87). No significant magnitude ×
distance × format interaction was found, F(1,96)=0.53, p=0.47.
Table 2
Outcome of the stepwise regression analysis in which all 7 predictors were
incorporated.
Predictor Regression weight t Changes
in R²
p
Raw Standardized
min(S, D) 98.65 0.92 14.46 0.72 b0.01
Format* size Subtrahend S 1.79 0.05 0.35 0.73
Format* min(S, D) -3.68 -0.05 -0.54 0.59
Size of difference -5.23 -0.08 -1.03 0.31
Format -317.51 -0.21 -1.16 0.25
Size of subtrahend -8.69 -0.13 -1.71 0.09
Format*size difference D 9.50 0.24 1.87 0.07
The predictors given in italics were not considered in the final model.
Subj.22
Subj.23
0
1000
2000
3000
4000
5000
6000
7000
8000
9000
10000
large
distance
small
distance
small
distance
large
distance
large
distance
small
distance
small
distance
large
distance
S < D S > D S < D S > D
Problem Type
RT
(ms)
Fig. 1. Mean RTs per participant per problem type for the problems presented in addition format (left panel) and subtraction format (right panel). S = subtrahend; D = difference.
326 G. Peters et al. / Acta Psychologica 135 (2010) 323–329

5. 5. Discussion
The present study examined adults’ use of the subtraction by
addition strategy in two-digit subtraction by means of a stepwise
regression analysis and by contrasting performance on problems
presented in subtraction and addition format. Both methods con-
verged to the conclusion that adults solve two-digit subtractions by
switching between subtraction by addition and direct subtraction
depending on the relative size of the subtrahend.
These findings fit with previous research on adults’ use of
subtraction by addition in multi-digit subtraction (e.g., Torbeyns et
al., 2009), wherein verbal reports were the major data source. Because
the validity of such protocols has been questioned (Kirk & Ashcraft,
2001), we started from other methods that have been successfully
applied in research on single-digit subtraction (Campbell, 2008;
Woods et al., 1975) but not in research on multi-digit problems.
Based on Groen and Poll (1973) and Woods et al. (1975), we first
carried out a stepwise multiple regression analysis that predicted RTs by
the problem characteristics: presentation format, size of the difference,
size of the subtrahend, the minimum of difference and subtrahend, and
their respective interactions. The model involving the minimum of
difference and subtrahend, referring to a switch between direct
subtraction and subtraction by addition depending on the most efficient
strategy, emerged as the best predictor of the RT data. The absence of an
interaction between presentation format and the other predictors
indicates that this is thesame for bothpresentation formats. This pattern
was replicated at the individual level, as all participants’ RTs were best
predicted by that model. This all indicates that participants did not apply
the same strategy on all problem types, but switched between direct
subtraction and subtraction by addition. Our findings extend those
obtained by Groen and Poll (1973) and Woods et al. (1975) by showing
that their observations also apply to numbers beyond 10.
We also compared performance on problems presented in
standard subtraction format and in its corresponding addition format.
Our findings revealed that there was an interaction between the
relative size of the subtrahend and the presentation format: When the
numerical distance between the subtrahend and difference was large,
S N D problems were solved faster in addition than in subtraction
format, while S b D problems were solved faster in subtraction than in
addition format. When the numerical distance was small, there were
no format effects. These findings echo those of Peters et al. (2010) for
single-digit subtraction and further indicate that adults switch
between subtraction by addition and direct subtraction to solve
multi-digit subtractions depending on the relative size of subtrahend.
The observation that small-distance problems (e.g., 75−36) were
solved considerably slower and less accurately than large-distance
problems (e.g., 71−2 or 71−69) merits further comment. A first
possible explanation for this finding is that the solution process of the
former problems is less efficient, regardless whether direct subtraction
or subtraction by addition is used. Another possible explanation might
be that the process of strategy selection is more difficult in cases where
the subtrahend and the difference are close to each other. Indeed, it
might be that the strategy selection process on such small-distance
problems takes more time and/or may lead to more errors than on large-
distance problems because it is less clear whether direct subtraction or
subtraction by addition will be the most efficient (Siegler & Shrager,
1984). Further research focusing on both strategy selection and strategy
execution in multi-digit subtraction is needed to address this issue.
At a more general level, the above findings are in accordance with
Siegler's SCADS* model (Siegler & Araya, 2005; see also Verschaffel,
Luwel, Torbeyns, & Van Dooren, 2009), which postulates that when
confronted with cognitive tasks people make adaptive strategy
choices and take into account knowledge about the efficiency of a
particular strategy for a particular problem type. Indeed, the current
findings reveal that adults rely on two number features involved in
the task when choosing between the direct subtraction strategy and
subtraction by addition to solve multi-digit subtractions, namely the
magnitude of the subtrahend and the numerical distance between
subtrahend and difference. These task features seem to involve a
comparison between the numbers in the problem. This comparison
0
10
20
30
40
50
60
70
80
90
100
Subtraction
Format
Addition
Format
Subtraction
Format
Addition
Format
Subtraction
Format
Addition
Format
Subtraction
Format
Addition
Format
large distance small distance small distance large distance
D
>
S
D
<
S
Error
Rate
(%)
0
1000
2000
3000
4000
5000
6000
Reaction
Time
(ms)
Fig. 2. Presentation format × problem type interactions for accuracy (in error rates; bars represented on the left y-axis) and RT (lines depicted on the right y-axis). Error bars depict 1
SE of the mean. S = subtrahend; D = difference.
327
G. Peters et al. / Acta Psychologica 135 (2010) 323–329

6. stage probably occurs during the orienting or planning phase of the
solution process and relies on fast (quasi-)automatic processes of
estimation and number sense rather than precise calculations of
differences between the numbers. Several studies have highlighted
the importance of magnitude comparison skills (De Smedt, Verschaf-
fel, & Ghesquière, 2009; Holloway, & Ansari, 2009) and number sense
(Gilmore, McCarthy, & Spelke, 2007) for successful mathematical
development. Future research should therefore investigate more
systematically the role of this comparison process in the execution of
strategies in multi-digit subtraction.
It should be noted that multi-digit subtractions can also be solved by
the so-called indirect subtraction strategy, in which people determine
how much needs to be subtracted from the minuend to get to the
subtrahend (e.g., 75−43 by 75−30=45 and 45–2=43; so the
answer is 30+2=32) (De Corte & Verschaffel, 1987). This indirect
subtraction strategy may be particularly efficient on large-distance SND
problems (71−69). However, previous studies on people's strategy use
in subtraction revealed that participants use this strategy only very
rarely or not even at all (Beishuizen,VanPutten, & VanMulken, 1997;De
Smedt et al., 2010; Torbeyns et al., 2009b; Van Lieshout, 1997).
Moreover, our data of the format manipulation indicate that it is very
unlikely that participants applied this strategy. If they had used indirect
subtraction regularly, we should have observed a smaller speed
difference between the two formats; or we should have found at least
a difference in favor of the subtraction format for the large-distance SND
problems because the S+.=M format is not facilitating indirect
subtraction. This was not the case.
After having discussed the main research findings and their theoretical
implications, we now address some limitations of the present study and
some suggestions for further research. A first limitation deals with the
specific set of problem types administered in this study. All the large-
distance problems involved a single-digit number (either the subtrahend
as in 71−2=. or the difference as in 71−69=.) because we wanted to
test the hypothesised format effects for problems where the numerical
distance between subtrahend and difference was as large as possible. If
our participants did not use subtraction by addition on these extreme
large-difference problems, they probably would not use it for problems
with less extreme differences between subtrahend and difference, for
which the subtraction by addition strategy is – compared to the direct
subtraction strategy – clearly less efficient. As a consequence, trials having
a relatively small subtrahend or difference compared to the minuend, i.e.,
a single-digit subtrahend or difference, were eligible to be included in the
stimulus set. However, this limits the generalisability of our findings to
subtraction in the number domain 20-100 with single-digit subtrahends
or differences. It remains to be determined whether our findings can be
applied to the entire domain 20–100 an issue that should be investigated
in future studies. These studies should include large-distance problems in
which the minuend, subtrahend and difference are two-digit numbers
(e.g., 71−15=.) (see Klein et al. (2010) for such a design in their study
about two-digit addition). Furthermore, we cannot exclude that the
exclusive use of these single-digits in the large-distance problems of the
present study might have biased the results, since it may have led to an
increased use of fact retrieval rather than procedural strategies.
A second limitation of our problem set is that we only used carry
problems. Although the effect of the carry operation has been studied for
addition (e.g., Klein et al., 2010), it has not been systematically addressed
in subtraction. It seems plausible that people would use more direct
subtraction for problems without borrowing, such as 84−81 (for
example, transposing the problem into 4−1=3 by subtracting 80 from
bothnumbersintheproblem),butthispossibilityneedstobeinvestigated
in future studies.
A third limitation of the study is the number of presented problems.
Due to time restrictions, we only could present students with 64 sub-
traction problems. To obtain data that would provide the most optimal
comparison of RTs and accuracies on the different types of problems,
we decided to select only 16 problems and present them in the four
different presentation formats (e.g., 31−28=., 31−3=., 28+.=31
and 3+.=31). Since each problem was presented four times, we cannot
exclude the possibility that some participants memorised the answer
to particular problems, which in turn might have affected their solutions
on subsequent items. By assigning the problems in a pseudo-random
manner to six blocks, we tried to reduce these possible memory effects
as much as possible.
Fourth, the present study applied two methods to investigate
participants’ strategic behaviour without relying on verbal protocols for
the first time in the domain of multi-digit arithmetic. As argued before,
verbal protocols may be of questionable validity when quick and quasi-
automatic processes are examined (Kirk & Ashcraft, 2001). This
particularly applies to the subtractions with a very small or very large
difference between the two numbers. Participants may be unable to
accurately report their strategy use on these problems because (a) they
are not aware of how they calculate the answer, (b) they do not find the
appropriate words to articulate their strategy use, or (c) they may even
hide the use of a particular strategy because they think that the
experimenter considers it as an inappropriate or even unacceptable
method. These issues disappear when participants are not asked to report
on their solution process. We used two different methods, which both
converged to the same conclusion and thus enhanced the reliability of the
obtained results. Future research should, however, include other non-
verbal methods, such as eye-movement data. Such data could help to
further triangulate the data about people's strategy use. Eye-movement
data might also provide a unique way to investigate peoples’ orienting or
planning processes that precede the strategy execution phase.
Fifth, the current findings only involved adults. Previous work in
children has shown that they rarely use subtraction by addition
spontaneously in symbolically presented two-digit subtractions (De
Smedt et al., 2010; Dowker, 2009; Torbeyns et al., 2009a,b). Because
these findings are mainly based on verbal protocol methods, future
research should replicate the current study with children to find out
whether the previous findings indicating the scarcity of subtraction by
addition in children remain when non-verbal methods to infer strategy
use are used. Developmental studies are needed to examine the origin
and development of the subtraction by addition strategy, particularly
the point at which children start using this strategy. The task, subject,
and context characteristics that influence this change in children's
strategy choice process should be the focus of investigation as well.
Appendix
Overview of the 64 subtraction problems, categorised by numer-
ical distance between subtrahend(S) and difference(D), magnitude of
the subtrahend, and presentation format.
SbD problems S N D problems
Numerical
distance
Subtraction
format
Addition
format
Subtraction
format
Addition
format
Large distance 31−3=. 3+.=31 31−28=. 28+.=31
34−8=. 8+.=34 34−26=. 26+.=34
42−5=. 5+.=42 42−37=. 37+.=42
52−4=. 4+.=52 52−48=. 48+.=52
71−2=. 2+.=71 71−69=. 69+.=71
77−9=. 9+.=77 77−68=. 68+.=77
83−4=. 4+.=83 83−79=. 79+.=83
93−5=. 5+.=93 93−88=. 88+.=93
Small distance 32−15=. 15+.=32 32−17=. 17+.=32
43−18=. 18+.=43 43−25=. 25+.=43
51−25=. 25+.=51 51−26=. 26+.=51
53−24=. 24+.=53 53−29=. 29+.=53
75−36=. 36+.=75 75−39=. 39+.=75
81−37=. 37+.=81 81−44=. 44+.=81
84−38=. 38+.=84 84−46=. 46+.=84
92−44=. 44+.=92 92−48=. 48+.=92
328 G. Peters et al. / Acta Psychologica 135 (2010) 323–329

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