Psychology Science, Volume 47, 2005 (1), p. 74 - 83
The addition of two-digit numbers:
exploring carry versus no-carry problems
MAUD DESCHUYTENEER1, STIJN DE RAMMELAERE & WIM FIAS
Abstract
Participants orally solved two-digit addition problems that were visually presented. The
problems either were carries (e.g., 39 + 48) or no-carries (e.g., 34 + 53). As predicted by the
triple-code model of Dehaene (1992; Dehaene & Cohen, 1995), (a) no-carries were solved
by separately adding the units and the tens so that sums with small tens and large units (e.g.,
32 + 16 = 48) were not solved faster than sums with large tens and small units (e.g., 23 + 61
= 84), even though problem size is much larger for the latter sum. Moreover, (b) carries were
found to be solved with the aid of “semantic elaboration” on the number line: The logarithm
of the correct sum was a more important predictor for carries than for no-carries, even
though both types of problems were matched for problem size. These findings were found
when the numbers were presented under each other (Experiment 1), next to each other (Experiment 2) and far from each other (Experiment 3).
Key words: two-digit addition problems, semantic elaboration, carry and no-carry problems
1
Maud Deschuyteneer, Dept. Experimental Psychology, Ghent University, H. Dunantlaan 2, B-9000 Gent,
Belgium; Phone: +32 9-2646433; Fax: +32 9-2646496; E-mail: [email protected]
The addition of two-digit numbers: exploring carry versus no-carry problems
75
Introduction
Mental arithmetic, an important skill in the everyday life of many adults, has been studied intensively. Most of the research efforts however, focussed on simple arithmetic (i.e., the
so-called arithmetical tables such as 8 + 4 or 7 x 4). For instance, in simple arithmetic studies, differences between production and verification tasks have been explored (e.g., Ashcraft,
Fierman, & Bartolotta, 1984; Dagenbach & McCloskey, 1992), arithmetical strategies have
been investigated (e.g., Lemaire & Fayol, 1995; Lemaire & Siegler, 1995), models of arithmetical fact organization in long-term memory have been proposed (e.g., Ashcraft &
Battaglia, 1978; Butterworth, Zorzi, Girelli, & Jonckheere, 2001; Widaman, Geary, Cormier,
Little, 1989), and the format in which arithmetical facts are stored has been studied (e.g.,
Campbell, 1998, 1999; Dehaene & Cohen, 1995; McCloskey & Macaruso, 1994; Noël, Fias,
& Brysbaert, 1997; Noël & Seron, 1997). Complex arithmetic (e.g., 23 + 48) however, received much less research attention and authors heavily focussed on the involvement of
working memory (e.g., Fürst & Hitch, 2000; Hitch, 1978; Logie, Gilhooly, & Wynn, 1994).
The goal of this paper is to make a contribution to our understanding of complex arithmetic
in general and two-digit addition in particular. Although several number processing models
have been proposed (e.g., Campbell, 1994; Campbell & Clark, 1992; McCloskey, 1992;
McCloskey, Sokol, & Goodman, 1986), to our knowledge only the triple-code model of
Dehaene (1992; Dehaene & Cohen, 1995) includes testable a priori predictions about twodigit addition.
The model states that there are three kinds of number codes in the human brain: a visualArabic, an auditory-verbal and an analog-magnitude number code. Each number-related task
is supposed to require a specific numerical format. For instance, arithmetical fact retrieval
(e.g., 3 x 4 is ?) utilizes the verbal code, whereas the analog-magnitude code is used to compare numbers (e.g., which number is larger: 67 or 72?). As described in a detailed formulation of the triple code model’s way of functioning during various mental operations
(Dehaene, 1992; Dehaene & Cohen, 1995), multi-digit calculations (e.g., 34 + 53 is ?) are
supposed to involve a mental manipulation of a spatial image of the operation in visualArabic notation. The addition of two-digit numbers (e.g., 34 + 53) for instance, involves the
combination of the elementary arithmetical facts using a specific algorithm (i.e., adding the
units and the tens), which most adults have learned at school by placing the numbers under
each other. The numbers are thus assumed to be placed on a visuo-spatial scratchpad in their
visual-Arabic number form. Next, the arithmetical facts of the units and of the tens will be
retrieved (4 + 3 and 3 + 5 in our example). Because arithmetical facts are supposed to be
stored in a verbal code, a translation from the visual-Arabic code to the verbal code will have
to take place. When the arithmetical facts are retrieved, the correct answer will be translated
back from the verbal code to the visual-Arabic code so that the retrieved number can be
placed on the visuo-spatial scratchpad. When both the correct answers of the units and of the
tens are retrieved and placed on the visuo-spatial scratchpad, the resulting number is translated back to the verbal code so that the correct sum of the arithmetical problem can be pronounced.
However, the triple-code model also assumes that more difficult arithmetical problems
usually are solved with the aid of the analog-magnitude code on the logarithmic number line,
a process called “semantic elaboration”. For instance, the problem 39 + 48 requires a carry
so that participants will be slower in solving it than in solving 34 + 53, even though both
76
M. Deschuyteneer, S. De Rammelaere & .W Fias
problems sum up to the same number. It has been shown that errors are made frequently
during mental arithmetic when a carrying operation is involved (e.g., Fürst & Hitch, 2000).
When confronted with a difficult problem, subjects often resort to strategies other than adding the units and the tens. For instance, the problem may be decomposed into a simpler problem such as 40 + 50 – 3. These semantic elaboration strategies require a good understanding
of the quantities presented in the original problem, and therefore are expected to address the
logarithmic number line.
Following the assumptions of the triple-code model, we outline two predictions that were
tested in our experiments. (A) First, if it is true that easy (no-carry) two-digit problems are
solved by separately adding the units and the tens, then we should find that both the addition
of the units and the addition of the tens have a main effect, without the appearance of an
interaction. Moreover, problems of which the correct sum consists of a large decade and a
small unit should be solved equally fast as problems of which the correct sum consists of a
small decade and a large unit. For instance, the two-digit problems 23 + 61 = 84 and 32 + 16
= 48 should be solved equally fast because they both involve the separate arithmetical fact
retrievals 3 + 1 and 2 + 6. Note that this is a strong prediction because, at the level of the
magnitude of the two-digit number, it seems not to be in agreement with the problem size
effect. This effect is in simple mental arithmetic research the most replicated and most often
reported finding and refers to the fact that arithmetical problems get harder as their numerical size increases (see Ashcraft, 1995; Campbell & Xue, 2001, for recent overviews). (B)
Second, if (difficult) carry problems are solved more frequently than (easy) no-carry problems by means of semantic elaboration, then the logarithm of the correct sum should be more
important for carry than for no-carry problems. This prediction can be tested by conducting
separate individual regression analyses for carries and no-carries, with the logarithm of the
sum as predictor and the latency as the dependent variable. If the logarithm of the correct
sum is more important for carries than for no-carries (as the triple-code of Dehaene predicts),
then slopes should be larger for carries than for no-carries.
Experiment 1
Method and procedure
Twenty first-year psychology students (15 females, mean age: 18.2 years) of Ghent University participated for course requirement and credit. They all solved a set of 252 two-digit
addition problems (84 no-carries, 84 carries and 84 fillers), of which the correct answer was
another two-digit number. Importantly, problem size was matched in carry and no-carry
conditions. Number of problems was more or less equally distributed across the different
problem size. Problems with ties or multiples of ten (in operand or as result) were not included. For the no-carries, there were four types of problems, as a function of small or large
units or decades (see Table 1). The order of the stimuli was randomized for each participant.
The operands were presented under each other in a columnwise Arabic notation in the center
of a computer screen (in white color on a black background). The participants were instructed to solve the problems by pronouncing the answer into a voice-key microphone that
was connected to the gameport of the computer. Latencies were measured with an accuracy
of 1ms. Accuracy and speed of responding were equally stressed.
The addition of two-digit numbers: exploring carry versus no-carry problems
77
Each trial consisted of the following steps. First, a fixation point was presented during
500 ms in the center of the computer screen. Then, the arithmetical problem was presented
and remained on the screen until the participant responded (unless there was still no response
after 10s). When the voice-key was triggered, the problem disappeared and a black screen
was presented. The experimenter typed the answer of the participant into the numerical keypad and struck the enter key (this lasted about 1s per trial). Immediately hereafter, the next
arithmetical problem appeared. Before the start of the experiment, the participants solved 20
practice trials (which were filler items), in order to get familiarized with the apparatus, the
procedure and the stimulus display. There was a short break after each 63 trials.
Results and discussion
159 of the 3360 experimental trials (84 carries and 84 no-carries solved by 20 participants) or 4.7% of the trials were dropped due to coughing or voice-key failure. The correlation between the mean latencies and the number of errors per item was .61 and significant [p
< .05], suggesting no speed-accuracy trade-off. Both speed of responding and proportions of
errors between carries and no-carries differed significantly [2383ms versus 1619ms, t(19) =
8.42, p < .001, and 0.13 versus 0.03, t(19) = 7.20, p < .001].
Prediction A. We analyzed the data of the no-carries in a 2 (Size of the tens: small versus
large) by 2 (Size of the units: small versus large) within-subjects ANOVA (see Table 1). For
the latencies, the main effects of tens and units were significant [respectively F(1,19) =
31.06, MSe=28892.18, p < .001, and F(1,19) = 9.94, MSe=64046.43, p < .01], but their interaction was not, F(1,19) = 1.90, MSe=31909.72. The difference between SL and LS problems was not significant, F < 1, MSe=18953.49, despite the fact that the mean problem sizes
of these categories were 42.6 and 78.7 respectively. In the error analyses, only the main
effect of the tens was significant, F(1,19) = 5.02, MSe=0.002397, p < .05.
Table 1:
Mean latencies (in ms) and mean proportions of errors of the four categories of the no-carries in
Experiment 1, 2, and 3. Tens and units are “small” if the sum is smaller than 5 and “large” if the
sum is larger than 5 (cfr. Geary, Bow-Thomas, & Yao, 1992).
Experiment 1
RT
ERR
Experiment 2
RT
ERR
Experiment 3
RT
ERR
Tens
Units
Small
Small
Large
1419
1543
.02
.02
1756
1924
.07
.02
2129
2495
.05
.07
Large
Small
Large
1576
1810
.03
.05
1900
2180
.04
.05
2395
2645
.03
.05
M. Deschuyteneer, S. De Rammelaere & .W Fias
78
Prediction B. Following the suggestions of Lorch and Myers (1990), separate individual
regression analyses for carries and no-carries were run with logarithm of the sum as the
independent variable and latency as the dependent variable. Thus, for each participant a
slope for the no-carries and a slope for the carries were obtained. If carry problems are more
frequently solved by means of semantic elaboration than no-carry problems, then we expect
that the logarithm of the sum will be more important for carries than for no-carries (i.e.,
larger slope for carries). The logarithm of the sum was a significant predictor both for carries
[t(19) = 3.45, p < .05] and no-carries [t(19) = 4.85, p < .05], but it was more important for
carries than for no-carries, 1031 versus 440, t(19) = 2.26, p < .05. This difference remained,
even when the slopes were scaled as a function of processing time, which differed considerably between the two conditions [t(19) = -1.76, p < .05].
In sum, both predictions derived from the triple-code model were confirmed. However,
the two-digit numbers were presented under each other, so that it is possible that we stimulated the strategy that Dehaene assumes to be dominant. Therefore, in Experiment 2 we
wanted to investigate whether the predictions are still confirmed when the numbers are presented next to each other.
Figure 1:
Logarithmic fit for the mean latencies (in ms) of the mean magnitude of the correct sum for the
no-carry and carry problems.
3000
latencies (in ms)
2750
2500
NO-CARRY
2250
CARRY
2000
Logarithmic
1750
Logarithmic
1500
1250
1000
37
45
55
64
74
m ean m agnitude
85
95
The addition of two-digit numbers: exploring carry versus no-carry problems
79
Experiment 2
Method and procedure
Twenty subjects from the same pool as in Experiment 1 participated. There were 18 females and the mean age was 18.8 years. None of the subjects had participated in Experiment
1. All procedural details were the same as in Experiment 1, except that the two-digit numbers
were presented next to each other, separated by a plus sign and a blank on each side of the
plus sign.
Results and discussion
140 or 4.2% of the trials were dropped due to coughing or voice-key failure. The correlation between the mean latencies and the number of errors per item was .44 and significant [p
< .05], suggesting no speed-accuracy trade-off. Both speed of responding and percentages of
errors between carries and no-carries differed significantly [3332ms versus 1979ms, t(19) =
9.39, p<.001, and 0.09 versus 0.04, t(19) = 4.86, p<.001].
Prediction A. The same data-analytic techniques as in Experiment 1 were used. For the
latencies, the main effects of tens and units were significant [respectively F(1,19) = 20.05,
MSe=39913.4, p < .001 and F(1,19) = 8.94, MSe=112079.8, p < .01], but their interaction
was not, F(1,19) = 1.22, MSe=52332.6. The difference between SL and LS problems was not
significant, F < 1, MSe=30362.47. In the error analyses, nothing was significant.
latencies (in ms)
Figure 2:
Logarithmic fit for the mean latencies (in ms) of the mean magnitude of the correct sum for the
no-carry and carry problems.
4000
3750
3500
3250
3000
2750
2500
2250
2000
1750
1500
NO-CARRY
CARRY
Logarithmic
Logarithmic
37
45
55
64
74
mean magnitude
85
95
80
M. Deschuyteneer, S. De Rammelaere & .W Fias
Prediction B. The individual regression analyses showed that the logarithm of the sum
was a significant predictor both for carries [t(19) = 4.18, p < .001] and no-carries [t(19) =
2.40, p < .05], but as in Experiment 1 it was more important for carries than for no-carries,
869 versus 326, t(19) = 2.19, p < .05. Again, the difference cannot be attributed to the general difference in processing time between the two conditions. The difference was preserved
after scaling as a function of RT [t(19) = -1.45, p < .1].
In sum, as in Experiment 1, both predictions derived from the triple-code model were
confirmed. However, one could still argue that these findings were obtained because subjects
could see both numbers simultaneously, which could elicit the calculation strategies as predicted by the triple-code model. Therefore, in Experiment 3, we wanted to put our predictions to a last test by presenting the first two-digit number on the leftmost side of the screen,
and the second number on the rightmost side of the screen.
Experiment 3
Method and procedure
Twenty subjects from the same pool as in Experiment 1 and 2 participated. There were
18 females and the mean age was 20.1 years. None of the subjects had participated in the
previous experiments. All procedural details were the same as before, except that the first
and the second operand were presented at the left- and rightmost side of the screen, respectively.
Results and discussion
111 or 3.3% of the trials were dropped due to coughing or voice-key failure. The correlation between the mean latencies and the number of errors per item was .43 and significant [p
< .05], suggesting no speed-accuracy trade-off. Both speed of responding and percentages of
errors between carries and no-carries differed significantly [3645ms versus 2485ms,
t(19)=12.13, p<.001, and 0.09 versus 0.05, t(19)=3.31, p<.01].
Prediction A. The same data-analytic techniques as before were used. For the latencies,
the main effects of tens and units were significant [respectively F(1,19) = 11.77,
MSe=73828.81, p < .01 and F(1,19) = 20.52, MSe=92305.63, p < .001], but their interaction
was not, F(1,19) = 1.38, MSe=48508.68. The difference between SL and LS problems was
not significant, F(1,19) = 3.05, MSe=32314.56. In the error analyses, nothing was significant.
Prediction B. The individual regression analyses revealed that the logarithm of the correct sum was a significant predictor for carries [t(19) = 3.75, p < .01], but not for no-carries
[t(19)<1]. More importantly, the logarithm of the sum was more important for carries than
for no-carries, 933 versus 150, t(19)=2.66, p < .05. Again, the difference survived RT scaling [t(19) = -2.50, p < .05].
In sum, as in the previous two experiments, both predictions were confirmed.
The addition of two-digit numbers: exploring carry versus no-carry problems
81
Figure 3:
Logarithmic fit for the mean latencies (in ms) of the mean magnitude of the correct sum for the
no-carry and carry problems.
4100
latencies (in ms)
3800
3500
NO-CARRY
3200
CARRY
2900
Logarithmic
Logarithmic
2600
2300
2000
37
45
55
64
74
85
95
m ean m agnitude
General discussion
The goal of the present study was to investigate the assumptions of the triple-code model
(Dehaene, 1992; Dehaene & Cohen, 1995) on multi-digit operations. Easy no-carry two-digit
addition problems are assumed to be solved by separately adding the units and the tens,
predicting main effects of the units and of the tens without an interaction. Moreover, sums
with a large digit on the tens and a small digit on the units should be solved equally fast as
sums with a small digit on the tens and a large digit on the units, despite the fact that the size
of the former sums is much larger than the size of the latter sums (e.g., 23 + 61 = 84 and 32
+ 16 = 48). Moreover, difficult carry problems are assumed to be solved with the aid of
“semantic elaboration”, which involves the logarithmic number line. If this assumption is
correct, then the logarithm of the correct sum should be more important for carry than for
no-carry problems. All predictions were confirmed when the operands were presented under
each other (Experiment 1), next to each other (Experiment 2) and far from each other (Experiment 3).
The main effects of the units and the tens that we found for the no-carries are consistent
with the problem size effect in simple mental arithmetic (e.g., Ashcraft, 1995; Campbell &
Xue, 2001). As mentioned in the introduction, this effect refers to the finding that arithmetical facts get harder as their numerical size increases. This is exactly what we found for the
separate fact retrievals of the units and the tens.
Note also that our finding that the logarithm of the correct sum is more important for carries than for no-carries is even more striking if one knows that also the correct sum of the no-
82
M. Deschuyteneer, S. De Rammelaere & .W Fias
carries had to be pronounced as one whole number. Several studies have shown that the
activation of the magnitude of two-digit numbers is fast and automatic, even if this magnitude is irrelevant to the task at hand (e.g., Brysbaert, 1995; Dehaene, Dupoux, & Mehler,
1990). In our experiments, the correct sum of the no-carries had to be pronounced and thus
was task-relevant, increasing the chance that the corresponding magnitude was activated
(which was indeed obtained in the first two experiments). Yet, the logarithm of the correct
sum was still found to be more important for carries than for no-carries. The additional impact of number magnitude during the solution of carry problems strongly suggests that the
number line was involved in the calculation process, confirming the predictions of the triplecode model.
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