Complex Mental Arithmetic: The Contribution of the Number Sense

Abstract

Young adults were asked to solve two-digit addition problems and to say aloud the result of each calculation step to allow the identification of computation strategies. We manipulated the position of the largest addend (e.g., 25 + 48 vs. 48 + 25) to assess whether strategies are modulated by magnitude characteristics. With some strategies, participants demonstrated a clear preference to take the largest addend as the starting point for the calculation. Hence, rather than applying strategies in an inflexible manner, participants evaluated and compared the operands before proceeding to calculation. Further, mathematically skilled participants tended to use those magnitude-based strategies more often than less skilled ones. The findings demonstrate that magnitude information plays a role in complex arithmetic by guiding the process of strategy selection, and possibly more so for mathematically skilled participants.

Full text

Complex Mental Arithmetic: The Contribution of the Number Sense
Julie Nys and Alain Content
Universite´ Libre de Bruxelles
Young adults were asked to solve two-digit addition problems and to say aloud the result of each
calculation step to allow the identification of computation strategies. We manipulated the position of the
largest addend (e.g., 25 ⫹48 vs. 48 ⫹25) to assess whether strategies are modulated by magnitude
characteristics. With some strategies, participants demonstrated a clear preference to take the largest
addend as the starting point for the calculation. Hence, rather than applying strategies in an inflexible
manner, participants evaluated and compared the operands before proceeding to calculation. Further,
mathematically skilled participants tended to use those magnitude-based strategies more often than less
skilled ones. The findings demonstrate that magnitude information plays a role in complex arithmetic by
guiding the process of strategy selection, and possibly more so for mathematically skilled participants.
Keywords: approximate number system, mathematical ability, mental arithmetic, number sense, strategy
There is now a large body of evidence showing that humans as
well as several other animal species possess a specific represen-
tational system for quantity, which provides an approximate code
for abstract numerical magnitude. This approximate number sys-
tem is considered as the foundation of the “number sense”
(Dehaene, 1997, 2001), a cover term accounting for the core
nonverbal abilities allowing to quickly apprehend, estimate, and
roughly manipulate numerosities. How the approximate number
sense relates to arithmetical abilities is however an open question.
The present research aimed at examining the role of the number
sense in complex arithmetic by investigating whether the relative
magnitude of operands influences the way strategies are selected
and applied. Evidence for an influence of the magnitude would be
indicative of a contribution of the number sense, because it en-
compasses magnitude apprehension and comparison.
How do we solve problems such as “48 ⫹25”? Previous studies
have shown that many strategies can be used, and that different
people choose amongst different strategies for different problems
(e.g., Blöte, Klein, & Beishuizen, 2000; Fuson et al., 1997; Green,
Lemaire, & Dufau, 2007; Heirdsfield, 2000; Lemaire & Arnaud,
2008; Lucangeli, Tressoldi, Bendotti, Bonanomi, & Siegel, 2003).
Amongst problem features, carry, problem size and parity have
been shown to influence arithmetical processing (e.g., Fu¨rst &
Hitch, 2000; Imbo, Vandierendonck, & de Rammelaere, 2007;
LeFevre, Sadesky, & Bisanz, 1996; Lemaire & Reder, 1999).
Surprisingly, other features such as the position of the largest
addend, or more generally the magnitude relations across oper-
ands, have hardly been examined despite indications that they
might influence the selection and application of calculation strat-
egies. With single-digit addition problems, young children gener-
ally prefer to add the value of the smallest addend to the other digit
(Groen & Parkman, 1972). This technique, known as the Min
strategy, implies that the magnitude of the addends was appre-
hended precociously during processing and determined the execu-
tion of the incrementation. Interestingly, Butterworth, Zorzi,
Girelli, and Jonckheere (2001) produced evidence for an addend-
comparison stage in adults’ arithmetical facts retrieval (but see
Robert & Campbell, 2008).
Because magnitude characteristics have an impact on the strat-
egies used to solve simple arithmetic problems in children, and
possibly on the organisation of the arithmetical facts network in
adults as suggested by Butterworth et al. (2001), they might also
intervene in complex arithmetic. Indeed, Trbovich and LeFevre
(2003) manipulated operand order and found that, with horizontal
presentation, participants were faster to solve two-digit ⫹one-
digit problems (e.g., 52 ⫹3) than the reverse (3 ⫹52). Although
the study did not examine calculation strategies, this finding sug-
gests that the relative magnitude of operands may influence pro-
cessing. Here, we wondered whether strategies are influenced by
magnitude in adults. In a pilot study with complex additions,
participants reported preferring to start the calculation from the
largest of the two operands rather than systematically from the left
(or right) one. Therefore, in the present experiment, we varied the
position of the largest addend in a controlled way by presenting the
problems twice, as a⫹band b⫹a, to test whether adults evaluate
magnitude and use the largest operand as calculation anchor,
whatever its position.
Strategies were identified by requiring participants to verbalise
online the result of each calculation step. For instance, when
solving the “48 ⫹25” problem, one participant would successively
produce 68, then 73. Another might say 50, 75, 73, or 60, 13, 73.
We assumed that the Intermediate Results Verbalization (hence-
forth, IRV) technique would explicitly reveal mental calculation
Julie Nys and Alain Content, Laboratoire Cognition, Langage & De´ve-
loppement (LCLD), Universite´ Libre de Bruxelles.
The present work was partially supported by Concerted Research Action
(ARC - 06/11 - 342) funded by the Direction ge´ne´rale de l’Enseignement
non obligatoire et de la Recherche scientifique-Communaute´ franc¸aise de
Belgique. We thank Jacqueline Leybaert, Wim Gevers, and the reviewers
for their helpful comments.
Correspondence concerning this article should be addressed to Julie
Nys, Laboratoire Cognition, Langage et De´veloppement (LCLD), Univer-
site´ Libre de Bruxelles, Av. Roosevelt, 50/CP 191, B - 1050 Brussels,
Belgium. E-mail: julienys@ulb.ac.be
Canadian Journal of Experimental Psychology © 2010 Canadian Psychological Association
2010, Vol. 64, No. 3, 215–220 1196-1961/10/$12.00 DOI: 10.1037/a0020767
215

steps without imposing a strong cognitive overload that would
modify the course of processing (Kirk & Ashcraft, 2001; Smith-
Chant & LeFevre, 2003), while at the same time avoiding the risk
of post hoc reelaboration (Ericsson & Simon, 1980). Latencies to
the first intermediate result, as an index of the time required for
strategy choice and initialization, and IRV sequences were re-
corded. Participants also received a standard mathematical test to
evaluate the relation between mathematical ability and strategic
preferences.
Method
Participants
Fifty undergraduate French-speaking students at the Universite´
libre de Bruxelles (mean age ⫽20.96 years, 25 men) took part. All
had normal or corrected vision. Most of them had received their
education in the French Community of Belgium or in France.
Stimuli
Forty-two pairs of addition problems were used. All contained
two-digit operands and required a carry. Addends were sampled
across all tens (excluding the tens themselves) to avoid biases in the
response range. In each pair, the position of the largest addend was
manipulated by reversing the order of the two addends, so that the
largest was located to the right or left side of the sign (see Table 1).
Unit-decade compatibility effects have been reported in Arabic num-
ber comparison studies (e.g., Nuerk, Weger, & Willmes, 2001), such
that interference was observed when the comparison of the tens and of
the units did not match (for instance, 64 vs. 37 slower than 67 vs. 34).
Here, we only used compatible addends, so that the unit of the largest
addend was always larger than the unit of the smallest addend.
Moreover, we modulated the salience of the position manipulation, by
varying the numerical distance between the addends. For small dis-
tance problems, the difference between addends was smaller than 40;
for large distance problems, it was larger than 40. Distance was partly
confounded with problem size, which ranged between 30 and 140 and
from 70 to 150, respectively (r⫽.58).
The set of problems was divided into three blocks, in such a way
that each block comprised equal numbers of the four kinds of
problems. Paired problems appeared in different blocks.
Procedure
Testing occurred individually in a quiet room and required about 40
min. The word-problems subtest of the WAIS (“Arithmetic subtest,”
Wechsler, 1989) was administered first, as a measure of mathematical
ability. Then, participants were asked to mentally solve the experi-
mental addition problems. They were asked to say the result of each
calculation step as soon as possible. One training block (14 trials) was
then passed, followed by three blocks of 28 trials presented in a fixed
pseudorandom order. Block order was varied across participants and
short breaks were allowed between blocks.
Presentation of the stimuli and timing were controlled by
PsyScope running on an Apple Macintosh computer. Stimuli were
displayed at the centre of the screen using Arial 120 and partici-
pants were seated at a comfortable viewing distance (⬃50 cm).
Each trial started with a central fixation point (a “⫻” sign) during
600 ms. Then, the problem appeared in a horizontal configuration,
and remained on the screen until the end of the trial. The experi-
menter recorded IRV sequences as well as final responses. Re-
sponse latencies from the appearance of the stimulus to the first
intermediate result produced were collected through a microphone
and voice key. When ready, the experimenter pressed the space
key to initiate the next trial after an 800 ms blank interval.
Results
An ␣level of .05 was used for all statistical tests. Overall, partic-
ipants made very few errors when solving the problems (M⫽7.1%,
SD ⫽7.4). Error rates were entered in a within-subject ANOVA with
Position (left vs. right largest addend) and Distance (small vs. large).
The analysis revealed no significant effect (Fs⬍1).
1
Regarding calculation processes, we first analysed the latencies
of first verbalizations. All latencies above 500 ms, except voice-
key malfunctions, were taken into account (8% discarded), what-
ever the final response. The ANOVA revealed a significant effect
of Position, F(1, 49) ⫽5.90, MSE ⫽317,873, p⫽.019, ␩
2
⫽.11.
Indeed, latencies were shorter when the largest addend was located
on the left side of the operation sign (2,871 vs. 3,065 ms). No other
significant effect was found (Fs⬍1).
Based on previous studies (Blöte et al., 2000; Heirdsfield,
2000), we classified the IRV sequences into four categories
(Aggregation, Rounding, Tens & Units, and Pen & Paper). Virtu-
ally all responses (97.7%) could be categorized according to that
scheme. The four categories generally entailed distinct IRV se-
quences (see Table 2), so there was little ambiguity in the classi-
fication.
2
Tens & Units was the most used strategy in terms of
overall frequency (34.9, 24.9, 23.7, and 14.3%, respectively, for
Tens & Units, Aggregation, Rounding, and Pen & Paper), as well
as in terms of number of participants (28, 19, 20, and 11 partici-
pants used the respective strategies at least once). About 60% of
the participants used only one strategy (11, 6, 6, and 6, respec-
tively), 30% used two, with virtually all possible combinations,
and few showed evidence of using more than two strategies.
To assess whether problem characteristics influence strategy
selection, we conducted ANOVAs on the percentage of use of each
strategy, with Position and Distance as within-subject factors. The
only significant effect was that Rounding was employed more
frequently when the distance was large (26.0%) than when it was
small (21.3%), F(1, 49) ⫽11.52, MSE ⫽95.9, p⫽.001, ␩
2
⫽.19.
1
This analysis and the following ones were also run with Gender as an
additional factor. No significant effect of Gender was observed, and the
inclusion of Gender did not change any of the conclusions.
2
Interrater agreement on a sample of 1,680 items (20 participants) was
96.8%.
Table 1
Parameters of the Problems
Distance
Position of the largest addend
Left Right
Small 58 ⫹37 37 ⫹58
Large 78 ⫹16 16 ⫹78
216 NYS AND CONTENT

To assess whether problem characteristics influence strategy
application, Aggregation and Rounding were further differentiated
as different IRV sequences occur depending on the anchor addend,
the operand to which the strategy is applied. In Left-anchored
Aggregation, the left addend is used without decomposition (e.g.,
48 ⫹25 would produce the sequence “68, 73”); whereas the
opposite happens in Right-anchored Aggregation (i.e., starting
from 25: “65, 73”). Similarly, Rounding was split into two vari-
ants, which correspond, respectively, to the rounding up of the left
(“50, 75, 73”) or right addend (“30, 78, 73”). Overall, left-
anchored strategies were used more frequently than right-anchored
strategies, suggestive of a default left-to-right preference (35.3 vs.
14.6% for Left- and Right-anchored Aggregation; 25.3 vs. 22.0%
for Left- and Right-anchored Rounding).
However, over and above this default preference, participants
who used these two strategies tended to start quite systematically
from the largest operand. As shown in Figure 1, Left-anchored
Aggregation was used more when the largest addend was on the
left (23.5%) than when it was on the right (11.8%), and the
opposite trend was observed for Right-anchored Aggregation (1.3
vs. 13.3%). Similarly for Rounding (see Figure 2), the left-
anchored strategy prevailed when the largest addend was on the
left (24.1%) relative to the right (1.2%) and the mirror pattern
came out for Right-anchored Rounding (0.7 vs. 21.3%, respec-
tively). As a result, 74% of Aggregation IRVs and 96% of Round-
ing IRVs were based on the largest addend, and both rates were
significantly higher than the value expected if participants started
from either addend at random, t(18) ⫽2.45; p⫽.025; t(19) ⫽
45.60; p⬍.001 for Aggregation and Rounding, respectively. The
magnitude-based preferences did not vary significantly according
to distance, both Fs⬃1.
Because unit values were systematically larger in the largest
addend, one might wonder whether the magnitude-based prefer-
ences described above rely on the comparison of the addends
values themselves or of their respective units magnitudes. To
disentangle the two factors, we ran multiple regression analyses on
the percentage of use of Aggregation and Rounding across prob-
lems, using the Addends Difference and the Units Difference as
predictors. The Addends Difference (left addend value minus right
addend value) accounts for the distance between operands and for
the position of the largest addend (positive difference when the left
operand is the largest, negative otherwise), and similarly for the
Units Difference. Hence, if the trend to start from the largest
operand occurs because of a comparison of the unit values rather
than of the addends values, Units Difference should come out
rather than Addends Difference. Furthermore, as the distance ma-
nipulation was partially confounded with problem size, the Ad-
dends Sum was entered as an additional predictor to capture
problem magnitude. Addends Difference, Units Difference, and
Addends Sum were entered simultaneously in the analyses.
Each model explained about 75% of the variance, with inde-
pendent contributions of both Units Difference and Addends Dif-
ference (see Table 3). The ␤coefficients were positive for Left-
0
5
10
15
20
25
30
thgiRtfeL
Position of the largest addend
Left Rounding
Right Rounding
Figure 2. Rate of use (%) of Left and Right-anchored Rounding as a
function of the position of the largest addend. Vertical lines depict SEMs.
Table 2
Description of Strategies and Examples of IRV Sequences for
the “48 ⫹25” Problem
Strategy Description
IRV
sequence
Aggregation One addend, undecomposed, is added with
the tens of the other
68
Then the units are handled 73
Rounding One addend is rounded up 50
The result is added with the other addend 75
The surplus value is then corrected 73
Tens & Units Units and tens are processed separately;
tens are added
60
Units are added 13
Tens and units results are added 73
Pen & Paper Units are added, unit digit of the response
is reported
3
Tens are combined, taking carry into
account if necessary
7
Final response is put together 73
Note. The “Tens & Units” category includes IRV sequences in which
tens are processed before units as well as the reverse (13, 60, 73). The label
“Pen & Paper” refers specifically here to the mental application of the
standard written algorithm.
0
5
10
15
20
25
30
thgiRtfeL
Position of the largest addend
Left Aggregation
Right Aggregation
Figure 1. Rate of use (%) of Left and Right-anchored Aggregation as a
function of the position of the largest addend. Vertical lines depict SEMs.
217
NUMBER SENSE IN COMPLEX MENTAL ARITHMETIC

anchored strategies, indicating that their use augmented with both
larger left addends and larger left units. Conversely, the ␤were
negative for Right-anchored strategies, suggesting that Right-
anchored Aggregation and Rounding were increasingly used with
larger right addend and unit values. In sum, the regressions con-
firm that Aggregation and Rounding are magnitude-based strate-
gies in that they are sensitive to the relative magnitude of both the
addends and the units.
Finally, to assess whether strategy selection was related to
mathematical skill, we examined the correlations between the
percentage of use of each strategy and the raw score obtained in
the WAIS subtest (Raw score: M⫽13.4; SD ⫽4.7; Standardised
score: M⫽9.7; SD ⫽2.6). No correlation was found for the Tens
& Units strategy (r⫽.009, p⫽.95). The Pen & Paper strategy
was negatively correlated with the WAIS subtest (r⫽⫺.45, p⫽
.001), and there was a nonsignificant trend toward a positive
correlation for Aggregation (r⫽.26, p⫽.07), as well as for
Rounding (r⫽.11, p⫽.45). When the two magnitude-based
strategies were considered together, a positive correlation (r⫽.31,
p⫽.03) was observed, suggesting that mathematically skilled
participants were inclined to use magnitude-based strategies more
often.
Discussion
Does the number sense contribute to complex mental arith-
metic? Our findings lead to a positive answer by demonstrating
that participants use magnitude information to guide and adapt the
application of their calculation strategies. The position of the
largest addend was one critical determinant of the way Aggrega-
tion and Rounding were executed, so that participants choose to
start calculation from the largest addend most of the time. Inter-
estingly, this finding is in line with recent observations on complex
subtraction problems. Torbeyns, Ghesquie`re, and Verschaffel
(2009) manipulated the numerical features of subtractions and
reported that when the subtrahend was large (e.g., 71–59), partic-
ipants used a strategy of indirect addition (59 ⫹10 ⫽69, 69 ⫹
2⫽71, so the response is 10 ⫹2⫽12) more often than when it
was small (e.g., 71–29). Both observations suggest that calculators
evoke the magnitude of the operands, compare them, and use the
result of this comparison to organise the computation.
By contrast, in our study, no evidence indicative of an influence
of magnitude properties was found for the Pen & Paper and Tens
& Units strategies, and one might think that they are more depen-
dent on the formal structure of the Arabic notation than on the
magnitude characteristics of the addends. Interestingly, the corre-
lations with the WAIS subtest revealed that mathematically skilled
participants tend to use Pen & Paper less and either magnitude-
based strategy more than less skilled ones. While the correlations
could be mediated by cultural, educational, as well as instructional
differences (see, e.g., Imbo & LeFevre, 2009), they nevertheless
support the distinction between magnitude-based and non
magnitude-based calculation strategies.
Why would one want to evaluate and compare operands before
doing the computation? Is adding 25 to 48 easier than adding 48 to
25? Every colleague with whom we discussed the present findings
shared the intuition that it is. One possible explanation is that starting
from the largest addend constitutes a remnant of the Min strategy
(Groen & Parkman, 1972). Further, if one assumes, as for the ele-
mentary facts (Butterworth et al., 2001), that sums of tens (e.g., 40 ⫹
20) are stored in long-term memory following a MAX ⫹MIN organi-
sation, the anchoring of Aggregation and Rounding on the largest
addend would suit memory retrieval constraints.
An alternative hypothesis is that starting from the largest addend
enables the excitation of a region of the analogical number repre-
sentation system that is closer to the region of the sum (i.e., 73 is
closer to 48 than to 25; for a similar hypothesis, see Restle, 1970).
This might facilitate the activation of the numerosity detectors
corresponding to the response. The envisaged process is similar to
one mechanism operating in number priming experiments
(Koechlin, Naccache, Block, & Dehaene, 1999; Reynvoet, Brys-
baert, & Fias, 2002), in which a prime facilitates the naming or
numerical categorization of a target, and more so when the prime
and the target are numerically close to each other. Further exper-
iments would be required to determine whether starting from the
largest operand is indeed more efficient than starting from the
smallest, especially when the former is close to the sum.
In conclusion, we believe that the present findings have impli-
cations for the understanding of both arithmetic processing and
individual differences in numerical cognition. Regarding process-
ing, they corroborate the notion of a tight connection between
magnitude representations and exact calculation mechanisms.
Even though it would seem premature to propose a processing
model, the data offer several constraints that should be taken into
account. Left-anchored strategies were used more frequently over-
all than right-anchored strategies, suggestive of a natural left-to-
right preference. The default left-to-right procedure can however
be modulated by the evaluation of operands and units values,
possibly leading to a reorganization of the calculation. Indeed the
Table 3
Multiple Regression Analyses Relating Problem Features to Aggregation and Rounding
Left-anchored
Aggregation
Right-anchored
Aggregation
Left-anchored
Rounding
Right-anchored
Rounding
Predictors ␤R
2
␤R
2
␤R
2
␤R
2
.70
ⴱⴱ
.74
ⴱⴱ
.79
ⴱⴱ
.78
ⴱⴱ
Addends difference .37
ⴱⴱ
⫺.41
ⴱⴱ
.40
ⴱⴱ
⫺.49
ⴱⴱ
Units difference .51
ⴱⴱ
⫺.51
ⴱⴱ
.55
ⴱⴱ
⫺.44
ⴱⴱ
Addends sum ⫺.09 ⫺.01 .004 .06
ⴱⴱ
p⬍.005.
218 NYS AND CONTENT

longer latencies for the first intermediate result when the largest
operand appeared on the right suggest such a reorganization (see
also Trbovich & LeFevre, 2003, for a similar finding). Exactly
how the comparison of operands and of units influence processing
requires further research, but the observation that both come into
play is reminiscent of the current debate on the holistic or com-
ponential nature of mental magnitude (e.g., Nuerk et al., 2001). In
summary, the present findings invite to consider more systemati-
cally the influence of magnitude characteristics in behavioural as
well as in neuroimaging investigations of calculation processes.
Regarding individual differences, it is worth noticing that not all
participants used either Aggregation or Rounding. The trend for
mathematically skilled participants to use magnitude-based strat-
egies more often than less skilled ones is compatible with Gallistel
and Gelman’s assumption that “the acquisition and performance of
verbal arithmetic is mediated by the preverbal system for repre-
sented numerosity and doing arithmetic computation” (Gallistel &
Gelman, 1992, p. 67). One may further wonder whether the stra-
tegic preferences are related to individual differences in the num-
ber sense, given Halberda, Mazzocco, and Feigenson’s (2008)
recent data showing a link between the acuity of the approximate
number system and mathematical ability. Studies bearing on the
role of magnitude in calculation strategies such as the present one
contribute to clarify ways in which the approximate number system
could impact on learning and performing arithmetic. For instance,
children prone to access and use magnitude information would select
adaptive incrementation strategies that might enhance the structure of
their arithmetic facts network and the strength of the stored associa-
tions. This view opens further perspectives for the analysis of the
causal mechanisms through which a core number system deficit might
explain certain dyscalculias (Butterworth, 1999; Mussolin, Mejias, &
Noe¨l, 2010; Price, Holloway, Ra¨sa¨nen, Vesterinen, & Ansari, 2007;
Wilson & Dehaene, 2007) and leads to promote educational and
remediation programs that incorporate training on quantity represen-
tations (e.g., Wilson, Dehaene, Pinel, Revkin, Cohen, & Cohen, 2006;
Wilson, Revkin, Cohen, Cohen, & Dehaene, 2006).
Re´sume´
Des jeunes adultes devaient re´soudre des proble`mes d’addition a`
deux chiffres et de´crire a` haute voix toutes les e´tapes de leurs
calculs afin d’en identifier les strate´gies. Nous avons manipule´la
position de l’ope´rande le plus grand (par ex., 25_48 vs 48_25) afin
de tester si les strate´gies sont module´es par des caracte´ristiques de
magnitude. Avec certaines strate´gies, les participants ont de´-
montre´ une nette pre´fe´rence pour prendre le plus grand
ope´rande comme point de de´part du calcul. Ainsi, plutoˆt que
d’appliquer les strate´gies de fac¸on inflexible, les participants
ont e´value´ et compare´ les ope´randes avant de proce´der au
calcul. De plus, les participants habiles en mathe´matiques ont
montre´ une plus grande tendance a` utiliser ces strate´gies base´es
sur la magnitude. Les re´sultats de´montrent que l’information de
magnitude joue un roˆle dans l’arithme´tique complexe en
guidant le processus de se´lection de strate´gie, et ce, possible-
ment de fac¸on plus marque´e pour les participants habiles en
mathe´matiques.
Mots-cle´s : syste`me de nombres approximatifs, habilete´ math-
e´matique, arithme´tique mentale, sens du nombre, strate´gie
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Received December 29, 2009
Accepted May 17, 2010 䡲
220 NYS AND CONTENT