Procedural vs. direct retrieval strategies in arithmetic: A comparison between additive and multiplicative problem solving

Abstract

The opposition between declarative and procedural knowledge is used to account for the solution by more or less expert and novice arithmeticians of simple additions and multiplications presented either in mixed blocks (Experiments 1 and 3) or unmixed blocks (Experiment 2) in an equation verification task. In the three experiments, presenting the sign (+ vs 2) before the operands had a stronger effect in additions than in multiplications. This priming effect indicates that many participants use a counting procedure for additions that coexists with the declarative knowledge stored in the associative network. In contrast, the small size (and sometimes the absence) of a priming effect for the ''x'' sign, together with the weak effect of size and the frequency of interaction effects, reveals the essentially declarative nature of multiplication solution.

Full text

Procedural vs. direct retrieval strategies in
arithmetic: A comparison between additive and
multiplicative problem solving
Jean-Louis Roussel
UniversiteÂde Bourgogne, Dijon, France
Michel Fayol
Universite
ÂBlaise Pascal, Clermont-Ferrand, France
Pierre Barrouillet
UniversiteÂde Bourgogne, Dijon, France
The opposition between declarative and procedural knowledge is used to account
for the solution by more or less expert and novice arithmeticians of simple addi-
tions and multiplications presented either in mixed blocks (Experiments 1 and 3) or
unmixed blocks (Experiment 2) in an equation verification task. In the three
experiments, presenting the sign (+ vs £) before the operands had a stronger effe ct
in additions than in multiplications. This priming effect indicates that many par-
ticipants use a counting procedure for additions that coexists with the declarative
knowledge stored in the associative network. In contrast, the small size (and
sometimes the absence) of a priming effect for the ‘‘x’’ sign, together with the
weak effect of size and the frequency of interaction effects, reveals the essentially
declarative nature of multiplication solution.
The variety of cognitive arithmetic strategies observed in the solution of simple
multiplication and addition problems can be broken down into two main types:
strategies for the direct retrieval of answers from memory and calculation
algorithms. This distinction is not limited solely to the field of cognitive
arithmetic but instead concerns all human cognitive functioning . It intersects
with Anderson’s (1983, 1993; Anderson & LebieÁre, 1998) distinction between
declarative and procedural memory. We suggest that direct retrievals from
memory take the form of the activation of declarative knowledge, whereas
EUROPEAN JOURNAL OF COGNITIVE PSYCHOLOGY, 2002, 14 (1), 61–104
Requests for reprints should be addresse d to M. Fayol, UniversiteÂBlaise Pascal, L APSCO-
CNRS, 34 Avenue Carnot, 63037 Clermont-Ferrand, France.
Email: Michel.Fayol@srvpsy.univ-bpclermont.f r
#2002 Psychology Press Ltd
http://www.tandf.co.uk/journals/pp/09541446.html DOI:10.1080/09541440042000115

calculation algorithms are based on the implementation of procedures. The
characteristics of these two strategies allow us to (1) integrate all the, sometimes
disparate, facts observed, and (2) generate hypotheses, some of which are new in
the field of cognitive arithmetic.
How do adults solve simple additions and multiplications involving combi-
nations of digits between 1 and 9? One of the most important results of the work
performed in the cognitive arithmetic field over the last 20 years is that both
these operations can be performe d using a variety of strategies (e.g., retrieval,
counting, rules, derived facts, cf., Ashcraft, 1982, 1992; Siegler, 1996; and see
Fayol, 1990 for a review). This diversity would make it possible to account for
variations in reaction time (RT) and error levels as well as for differences in
development. It might also allow us to identify similarities between the types of
processing mobilised for the solution of the two categories of operation. Such
similarities stand in opposition to the initial data reported in the field of cog-
nitive arithmetic (Groen & Parkman, 1972) and coexist uneasily with the history
of the acquisition of these two operations (Reder, 1987). In effect, addition is
acquired early and informally through the use of a counting procedure. Multi-
plication, on the other hand, is learnt formally, primarily through the memori-
sation of associations between pairs of digits and answers.
The purpose of this article is to show that, despite the similarities observed in
the solution of additions and multiplications (LeFevre, Bisanz, et al., 1996;
LeFevre, Sadesky, & Bisanz, 1996), there are fundamental differences in the
processes mobilised by adults when performing these two operations. To put
matters briefly, the solution of addition problems would appear to mobilise both
a procedural component and a declarative component through the activation of
number facts within an associative network, whereas the solution of multi-
plication problems appears to be primarily the result of a declarative component.
This difference appears clearly in a sign priming task (+ vs £). First we shall
describe the data taken from the literature dealing with the use of these two types
of strategy for the solution of simple arithmetical problems. We shall then
present a theoretical account of the two types of representation, inspired by
Anderson’s ACT-R model, and, finally, we shall identify the hypotheses in the
field of cognitive arithmetic that can be derived from our theory.
FACTS AND PROCEDURES IN COGNITIVE
ARITHMETIC
Groen and Parkman (1972) have contrasted two types of process: the direct
retrieval of facts stored in memory, and the use of rules (e.g., counting) to
generate facts. In both first graders and adults, a problem-siz e effect is observed
in the solution of elementary additions with a sum less than 9. This effect could
be simulated using a counting algorithm which starts with the larger of the two
numbers (m) and increments this by 1 ntimes along the numberline. Although
62 ROUSSEL, FAYOL, BAR ROU ILLET

this model was very plausible for children (Baroody, 1987; Carpenter & Moser,
1983; Fuson, 1982), it was not so for adults given the speed at which counting
was performed. Groen and Parkman therefore suggested that the adults did not
calculate, but instead retrieved the number facts associated with pairs of oper-
ands directly from memory (e.g., 3 + 4 ?7) (Ashcraft & Battaglia, 1978;
Ashcraft & Fierman, 1982; Ashcraft & Stazyk, 1981; Siegler & Shrager, 1984;
Svenson & Broquist, 1975). The counting algorithm should then serve as a back-
up strategy in the event of temporary difficulties. Many empirical arguments
support this theory (Compton & Logan, 1991; Fuson , 1982; Geary &
Burlingham-Dubree , 1989; Logan & Klapp, 1991; Siegler & Shrager, 1984).
The memory retrieval procedure has mainly been studied using verification
(e.g., 3 £4 = 9 or 2 + 6 = 8: true or false) or production tasks (3 £4 = ?; 2 +
6 = ?). Most of the time, adults would directly retrieve from memory arithmetic
facts (e.g., 2 £7 = 14; 3 + 2 = 5) stored in interconnected associative networks
and recovered through the mechanism of spreading activation (Anderson, 1993;
Sokol, McCloskey, Cohen, & Aliminosa, 1991). The presentation of the terms of
a problem (2 £7) should induce the activation of the corresponding nodes (2
and 7) and their properties (Campbell, 1994, 1995). Activation would spread
from these nodes to associated nodes, thus resulting in the activation of infor-
mation which influences performanc e in production or verificat ion tasks.
Though it remains unclear whether arithmetic verificati on and production are
mediated by the same memory processes (Campbell & Tarling, 1996; Zbrodoff
& Logan, 1990)—a problem we will discuss later—many empirical facts sup-
port the model of direct retrieval from associative networks: priming effect
(Campbell, 1987a,b, 1991; Meagher & Campbell, 1995); distance effect
(Zbrodoff & Logan, 1990); parity effect (Dehaene, Bossini, & Giraux, 1993;
Hines, 1990; Krueger & Hallford, 1984; Lemaire & Fayol, 1995; Miller &
Gelman, 1983; Shepard, Kilpatric, & Cunningham , 1975; Sudevan & Taylor,
1987); interference effect (Hamann & Ashcraft, 1985; LeFevre, Bisanz, &
Mrkonjic, 1988; Lemaire, Barrett, Fayol, & Abdi, 1994; Lemaire, Fayol, &
Abdi, 1991); and size effect (Ashcraft & Battaglia, 1978; Parkman, 1972;
Siegler, 1988a,b; Zbrodoff, 1995).
Concerning the interference effect, in the case of addition, as for multi-
plication, the proportions of errors and response latencies in verification tasks
increase when the judgement relates to false operations, the answer to which
would be correct for a different operation (e.g., 2 £7 = 9: true or false?)
(Stazyk, Ashcraft, & Hamann, 1982; Winkelman & Schmidt, 1974; Zbrodoff &
Logan, 1986). This type of associative confusion suggests that additive facts are
represented in an interconnected network which includes multiplicative facts.
The size effect corresponds to the fact that people need more time and make
more errors when solving problems with large digits than problems with small
digits, for both additions and multiplications. It is easily explained in terms of
counting processes: the more steps required by the solution procedure, the
PROCEDURES A ND RETRIEVAL IN A RITHM ETIC 63

greater the time required to solve the problem. Theories that invoke direct
retrieval processes have more difficulty explaining this effect. These theories
have recourse to two factors: the strength of the associations and interference
(Zbrodoff, 1995). If we consider the first factor, the difficulty of retrieving
number facts depends on the strength of the association between the operands
and the answer. This strength should increase with the frequency of occurrence
of the association (Hamann & Ashcraft, 1985). The interference effect relates to
the fact that the more one response interferes with other responses, the slower
(increased RT) and more difficult (increased error level) its retrieval will be
(Barrouillet, Fayol, & LathulieÁre, 1997; Campbell & Graham, 1985; Siegler &
Shrager, 1984).
As the problem-size effect and the effect of associative confusions (special
type of interference effect) can be observed in both addition and multiplication
(Lemaire et al., 1991; 1994; Miller & Paredes, 1990; Miller, Perlmutter &
Keating, 1984), they help emphasise the similarity of the processes mobilised by
the two operations (Winkelman & Schmidt, 1974). However, the relative
importance of the impact of associative confusions of multiplications on addi-
tions (3 + 4 = 12: true or false?) and of additions on multiplications (3 £4 = 7:
true or false?) has not been studied in isolation (Lemaire et al., 1991; Winkel-
man & Schmidt, 1974). One difficulty concerns the relative proportion of
intrusions of additive or multiplicative responses in the solution of multi-
plications and additions respectively.
Overall, an examination of the literature might lead us to suppose that we
have moved to the idea that addition and multiplication both possess a pro-
cedural and a declarative component , the second of which is always dominant, at
least in adults. The two operations should therefore be essentially similar. Both
would be primarily solved through the retrieval of number facts from a shared
network, thus explaining cross-operation interference. If retrieval fails, both
operations could then be solved through the implementation of back-up strate-
gies, usually based on counting. Nevertheless, this conception raises a number of
problems.
First, the data relating to cross-operation interference is not totally consistent.
Some researchers report that additive responses frequently interfere with mul-
tiplications (Miller & Paredes, 1990), whereas others claim that such errors are
rare (Campbell & Graham, 1985; Miller et al., 1984; Winkelman & Schmidt,
1974).
Second, the back-up strategy used for additions does not have the same status
as those observed in multiplication solving, i.e., the number series and derived
facts strategies. Counting appears to play a role in connection with addition
which no other back-up strategy plays in the case of multiplication. LeFevre and
Kulak (1994) have reported that not all the adults they studied exhibited an
interference effect, and Little and Widaman (1995) have noted that the min
model approximates better than the retrieval model to the chronometric per-
64 ROUSSEL, FAYOL, BAR ROU ILLET

formances of some adults. As far as we know, adults always retrieved multi-
plicative facts from memory and exhibited interference or priming effects in all
the studies conducte d in this domain.
Finally, children start to count informally and over a prolonged period well
before they start to memorise additive facts (Fayol, 1990). In contrast, even if
multiplication is presented to children as a form of addition in order to help them
grasp its sense, the procedure for performing calculations which might be
derived from it is not used in a systematic, repetitive way which, as is the case
with counting, might permit automation (Lemaire & Siegler, 1995). In contrast,
the direct memorisation of multiplicative facts is the preferred approach and
forms a separate school activity (i.e., learning multiplication tables).
To summarise, the analysis of adult performance in solving additions and
multiplications suggests that addition retains a procedural dimension which
exists in parallel with the declarative dimension. In contrast, the declarative
dimension dominat es in the solution of multiplication problems. Thus simple
addition and multiplication might be distinguished by the strategy used for their
solution: retrieval of the answer from memory or counting procedur e for addi-
tion, more systematic retrieval of numbe r facts for multiplication. This diversity
of strategies correspond s to Anderson’s (1983, 1993; Anderson & LebieÁre,
1998) distinction, proposed in the ACT* and ACT-R models, between
declarative memory and procedura l memory. Declarative and procedural
memories have different properties. The different functions and properties of
these two memories make it possible to derive certain hypotheses which can be
tested directly in the field of cognitive arithmetic.
DECLARATIVE AND PROCEDURAL MEMORIES
Anderson (1983, 1993) conceived of declarative memory in terms of schematic
structures (chunks) that define the category to which the knowledge belongs
(e.g., multiplicative fact number) and contain the units which encode their
contents. Figure 1 presents a graphic representation of the encoding of the
number fact 3£4 = 12. According to the ACT-R theory, the activation Aiof a
chunk (i.e., an item of knowledge ) is the sum of the sources of activation which
it receives from the elements (Wj) already present within the field of attention
(i.e., the values 3 and 4) and depends on the strength of the associations
(parameters Sij) between these elements and the chunk (Ai = §WjSij). The
probability of retrieving an item of knowledge from long-term memory would
be an exponentia l function of its activation and access time a negative expo-
nential function of activation. The greater the activation received, the greater the
probability of retrieval and the shorter the access time (Anderson, Reder, &
LebieÁre, 1996).
An item of declarative knowledge encodes specific contents. It is activated by
the elements which form it: e.g., the chunk which encodes the number fact 3£4
PROCEDURES A ND RETRIEVAL IN A RITHM ETIC 65

= 12 is therefore only activated by the values 3and 4and not by the values 6and
7. If the additive fact 3 + 4 = 7 is also encoded in a declarative chunk, it would
be activated by the values 3 and 4 in the same way as 3£4 = 12. Thus retrieval
in declarative memory is subject to item-based interferences, particularly in
arithmetic where a very large number of items of knowledg e are activated by a
small number of elements (i.e., the ten digits from 0 to 9, Anderson , 1995, x7).
Procedural knowledge consists of ‘‘production rules [productions] that are if-
then or condition-action pairs. The if, or condition, part specifies the circum-
stance under which the rule will apply. The then, or action, part of the rule
specifies what to do in that circumstance’’ (Anderson, 1993, p. 4, italics added).
Four properties distinguish productions from declarative chunks: They are
modular, abstract, goal-directed and exhibit an asymmetry between condition
and action. The first two of these characteristics are particularly relevant for the
processes mobilised in cognitive arithmetic.
Figure 1. Example of declarative chunk encoding the multiplicative fact 3 £4= 12 in accordance
with ACT-R. Adapted from Anderson , Reder, and LebieÁre (1996).
66 ROUSSEL, FAYOL, BAR ROU ILLET

Unlike declarative knowledge , which is organised in networks through which
activation spreads, each production rule is thought of as a modular piece of
knowledge and represents a well-defined step of cognition. Even complex
productions remain modular. There is no priming effect between productions,
which cannot be activated unless their conditions of activation are met.
Again unlike declarative knowledge, production rules are abstract, general
rules. They contain variables rather than specific values and can be applied in
multiple situations: i.e., no production rule is specific to the addition of the
numbers 3 and 4, for example.
To summarise, declarative chunks are specific to certain items, are activated
by the attention focused on the elements which they encode, and are sensitive to
interference effects. In contrast, procedures are general (they contain variables),
are goal activated, modular, and largely insensitive to interference.
We hypothesise that multiplications are most frequently solved through the
direct retrieval of the answer from memory whereas additions are more often
solved through the use of an algorithm. Thus multiplication problems would
appear to be mostly solved through the activation of declarative knowledge,
whereas additions seem to be more often solved through the mobilisation of a
procedure. It is conceivable that such a procedure , which is able to solve all
simple problems, exists for additions (e.g., incremental counting by 1). In
contrast, the existence of an equivalent computational procedure for multi-
plication (e.g., iterative additions) is implausible. The mobilisation of such a
procedure would rapidly become costly as the size of the operands increases.
Moreover, it would necessarily have to include the additive procedure as a
subcomponent.
The hypothesis that additions are solved by means of a procedure and mul-
tiplications by memory retrieval results in three predictions derived from the
characteristics which are specific to procedures as described by Anderson: their
abstract nature, their goal-directed structure, and their modularity.
If the solution of additions, or at least some of them, mobilises a procedural
component, this procedure should be activated as soon as the subject knows that
the task is to perform an addition, independently of the specific values which are
to be added. As Sohn and Carlson (1998) have recently shown, this goal should
be generated by the simple presentation of the + operator which should then
activate the procedure even in the temporary absence of values that instantiate
the variables. In contrast, the £sign should not have the same effect since the
declarative knowledge that allows subjects to solve multiplications is activated
only by the values that have to be multiplied. Note that Anderso n and LebieÁre
(1998) proposed that the retrieval of arithmetic facts is actually achieved
through the use of a ‘‘retrieval production’’. However, this retrieval production
is different from the algorithmic procedure we hypothesise for solving additions.
It contains only one step and stops as soon as a declarative chunk matches the
content of the current goal. Even in the latest version of ACT-R, version 4.0
PROCEDURES A ND RETRIEVAL IN A RITHM ETIC 67

(Anderson & LebieÁre, 1998), this chunk is still activated by the values to be
multiplied. Thus, this declarative chunk cannot be activated, or even pre-
activated, by the mere presentation of the operator. On the other hand, the +
operator can activate all the productions involved in the algorithmic procedure
and the declarative knowledge this procedure utilises (e.g., the numberline for
the iterative procedure).
Thus, in a verification task (e.g., 6 + 4 = 10? or 7 £5 = 35?), the presentation
of the sign before the operands should have a different effect depending on the
operation to be performed . In the case of additions, solution time should be
shorter when the sign is presented before the operands (negative SOA) than
when it appears at the same time as the operands (SOA 0) (Sohn & Carlson,
1998). In the case of negative SOAs, the + sign would activate the procedure
before the numbers to be added appear, whereas, in the case of SOA 0, the
activation of the procedure by the + sign would occur simultaneously with the
presentation of the values. This would result in shorter verification times in the
negative SOA condition.
If multiplications are solved by means of direct retrieval, the presentation of
the £sign before the operands should not activate any declarative knowledge
since this knowledge is specific to the operands. This knowledge would there-
fore be activated only if the numbers to be multiplied were present in working
memory. The prior presentation of the sign on its own cannot shorten solution
times.
Our model also allows us to make predictions about the effect of the size of
the operands on verification times (size effect) (Barrouillet & Fayol, 1998). The
addition procedure operates step by step by means of the recursive addition of 1,
made possible by the retrieval of the next item in the numberline (e.g., to solve 8
+ 4, start from 8 and advance 4 steps in the numberline: 9–10–11–12, answer
12). Retrieving items and monitoring the number of steps are associated with a
cognitive cost which results in increased error levels as the size of the operand to
be added grows. This model predicts that solution time should increase with the
size of the operand because the increase in the number of steps leads to an
increase in the number of retrievals in the numberline and makes the activity of
monitoring the number of steps already performed more difficult.
In contrast, if multiplications are solved via direct retrieval of the answer
from memory, the size effect should be less prominent. Such an effect exists
(Campbell & Graham, 1985; Stazyk et al., 1982). This might be due to the
greater frequency of small multiplications, accounted for by the ACT-R model.
The more frequently a multiplication occurs, the higher the base level of the
multiplicative fact, the higher the level of activation of the declarative know-
ledge and the shorter the time required for its retrieval. However, the variat ions
in retrieval time due to this phenomenon should not be comparable with the
extra time resulting from the recursive use of a calculation procedure in the case
of additions.
68 ROUSSEL, FAYOL, BAR ROU ILLET

Finally, the modularity of the procedures and the existence of two types of
memory (i.e., declarative and procedural ) lead us to a hypothesis concerning
interference effects. Subjects take longer to evaluate (reject) operations when
two operands and an operator sign are associated with an answer which corre-
sponds to the answer of another operation (e.g., 6 £4 = 10) than they do when
the false answer is not associated with the operand s (e.g., 6 £4 = 25). This extra
time can be explained by the necessary inhibition of an answer that is associated
with the operand s but that does not correspond to the correct operation. It is
likely to be all the greater when subjects use memory retrieval strategies. Since
multiplication problems are solved by the direct retrieval of the answer from
declarative memory, their solution should be particularly sensitive to inter-
ference caused by the activation of declarative knowledge relating to additive
facts.
Let us now suppose that additions tend to be solved more frequently than
multiplications in an algorithmic way through the use of a procedure . Given the
modularity of productions, the operation of this procedure should be relatively
insensitive to the activation of any declarative knowledge which encodes a
multiplicative fact containing the operands for two reasons. First, unlike the
interference generated by two concurrent activations within declarative memory,
the implementation of an additive procedure, on the one hand, and the possible
activation of a multiplicative fact, on the other, involve two different types of
memory (i.e., procedural and declarative) and distinct processes (cf., Anderson ,
1993, x2). This suggests that the activation of knowledge in declarative memory
should have little effect on the progress of any ongoin g procedure when this
knowledge is irrelevant for this procedure. Finally, as the ACT-R model
specifies, the mobilisation and the implementation of a procedure itself claim a
large part of the limited sources of activation. They would therefore reduce the
sources of activation available for the simultaneous activation of declarative
knowledge which is not involved in the procedure. A memory retrieval strategy
does not produce the same phenomeno n since the sources of activation allocated
to the operands cause the simultaneous activation of all the declarative chunks
which contain the elements present in working memory (i.e., the operands to be
processed).
To summarise, if additions are solved more often than multiplications by
means of a procedure and multiplications by the direct retrieval of the answer
from memory, we predict: (1) shorter RTs for verifications of additions when the
sign is presented before the operands, this priming effect being less pronounced
for multiplications; (2) a greater size effect for additions than for multi-
plications; and (3) greater interference effects during the verification of multi-
plications than during that of additions.
These hypotheses were tested in three experiments. In Experiments 1 and 2,
adult subjects who were expert arithmeticians were asked to verify a restricted
number of additions and multiplications presented either in mixed (Experiment
PROCEDURES A ND RETRIEVAL IN A RITHM ETIC 69

1) or unmixed (Experiment 2) blocks. For each type of operation, the operator
sign (+ or £) appeared before, at the same time as, or after the operands and the
answer. Equally, for both types of operati on the operands were either large or
small and the answer to be verified was true or false. False answers were either
interferent (i.e., they corresponde d to the answer of the other operation) or non-
interferent (correct answer §1). Experiment 3 contrasted the performance of
experts and non-exper t subjects and extended the study to a larger range of
problems.
EXPERIMENT 1
The problems in this experiment were presented in mixed blocks: The subjects
were told that two types of operation would be presented (addition and multi-
plication) but they could not predict the type of operation they would be called
on to process before any given trial.
Method
Participants. Twenty-four primary school teachers volunteered to take part
in this experiment. Their average age was 33 years and 7 months (span: 22 years
7 months–53 years 3 months).
Material. Eight pairs of operands were selected and presented with either
an addition or multiplication sign, thus yielding 16 operations in all (the smaller
operand was presented to the left of the sign in half the pairs and to the right of
the sign in the other half). The smaller of the two operands was small (2 or 3) in
half of the pairs and large (4 to 7) in the remainder. This resulted in 4 ‘‘easy’’
problems and 4 ‘‘difficult’’ problems for each type of operation (cf., Appendi x
1). The problems were selected on the basis of the size of the product of the two
operands in the light of earlier studies of the relative difficulty of problems
(Campbell, 1987a,b ; Hamann & Ashcraft, 1985; Siegler, 1988a). In the easy
problems, the mean of the products was 15.8 and the mean of the sums 9. These
means were 44.8 and 13.75 for the difficult problems. The answers proposed for
each operation were either correct or incorrect. The incorrect answers were
either interferent and, in the case of multiplications, corresponded to the sum of
the two operands (e.g., 6 £8 = 14) or, in the case of additions, to their product
(6 + 8 = 48), or were non-interferent and differed from the correct answer by §1
(e.g., 6 + 8 = 13 or 6 £8 = 47). This allowed us to control the average distance
separating the incorrect answers proposed from the correct answers for both
types of problem in order to avoid any split effect (Ashcraft & Battaglia, 1978;
Lemaire & Fayol, 1995; Zbrodoff & Logan, 1986). The ‘true’ problems were
each presented twice in order to balance the number of true and false problems
presented.
70 ROUSSEL, FAYOL, BAR ROU ILLET

Procedure. The problems were presented left-to-right (e.g., 6 £8 = 14) in
the centre of a computer screen (IBM PS2). The screen was switched to 40
characters per line mode (each character 0.7 cm high and 0.5 cm wide). A signal
(a line of five as at the centre of the screen), which lasted for 1 s, preceded the
presentation of each problem (consisting of two operands , the operator sign + or
£, the = sign, and the answer). Eight Stimulus Onset Asynchronie s (SOA)
determined the delay between the presentation of the operands, the answer, and
the = sign, on the one hand, and that of the operator sign (+ or £) on the other.
The operator sign appeared either before (negative SOAs of ¡300, ¡150, and
¡70 ms), at the same time as (zero SOA) or after (positive SOAs of +150, +240,
+300, and +500 ms) the remainder of the equation. As this experiment was
initially designed to compare performance of children and adults, a large number
of SOAs were selected in order to track the temporal course of the activation
process. The data concerning children will not be reported here. The first three
positive SOA values were chosen in the light of earlier research into the
temporal progression of the cognitive processe s involved in cognitive arithmetic
(Campbell, 1987a,b, 1991; LeFevre et al., 1988, LeFevre, Kulak, & Bisanz,
1991; Lemaire & Fayol, 1995; Zbrodoff & Logan, 1990). In total, 8 pairs of
operands £2 operations £4 answers (2 true and 2 false, interferent and non-
interferent) £8 SOAs yielded 512 trials.
The subjects had to respond ‘‘true’’ or ‘‘false’’ as quickly as possible by
pressing one of two keys situated at the right- and left-hand sides of the com-
puter keyboard. For half the subjects, the ‘‘true’’ key was located at the left-
hand side of the keyboard; it was situated at the right-hand side for the other
half. The measured reaction time was the time which elapsed between the
display of the final item on the screen (i.e., the operands and the answer in the
case of negative SOAs and the operator sign in the case of positive SOAs) and
the production of the response on the part of the subject. The 512 problems were
presented in random order in blocks of 128 problems which were separated by a
pause. The experimental items were preceded by 20 training trials, which were
not considered in the results.
Results
We analysed in turn the reaction times of the verification of true problems, of
the rejection of false problems, and the rates of errors. As we will show, our
main predictions were confirmed. The presentation of the + operator before the
operands resulted in a priming effect and reduced reaction times for additions
compared to the SOA 0 condition where the anticipated presentation of the £
sign did not have any significant effect. Additions generated a larger size effect
than multiplications, which were verified faster and more accurately than
additions. Finally, the verification of multiplications was more prone to inter-
ference than that of additions.
PROCEDURES A ND RETRIEVAL IN A RITHM ETIC 71

Reaction times for the verification of true problems. Table 1 presents the
mean of the median RTs of correct responses as a function of the type of
operation, the size of the operands, and the SOA value for the verification of true
problems. We performed a 2 (operations: addition vs multiplication) £2 (size of
operands: small vs large) £8 (SOA: ¡300 ms, ¡150 ms, ¡70 ms, 0 ms, +150
ms, +240 ms, +300 ms, and +500 ms) ANOVA with repeated measures on all
the factors. All the main effects and all the interactions were significant. The
verification of additions (M= 865 ms) was slower than that of multiplications
(831 ms), F(1, 23) = 5.33, p<.05, MSe = 41371, and large operands resulted in
slower verifications (879 ms) than small operands (817 ms), F(1, 23) = 21.98, p
<.001, MSe = 33289, while RTs varied as a function of SOA (917 ms, 889 ms,
936 ms, 944 ms, 835 ms, 786 ms, 752 ms, and 728 ms for the SOAs ¡300,
¡150, ¡70, 0, +150, +240, +300, and +500 ms respectively), F(7, 161) = 97.66,
p<.001, MSe = 7187.
Errors on true problems were somewhat rare (2.68%). An ANOVA with the
same design as for RTs was performed on the rate of errors. There was only an
effect of operation: additions elicited more errors (3.42%) than multiplications
(1.93%), F(1, 23) = 4.33, p<.05, MSe = 98.17. This effect suggests that the
faster verification of multiplications was not due to a speed/accuracy trade-off.
Our initial hypothesi s predict ed that presenting the sign before the operands
and the answer (negative SOA) would result in shorter solution times (compared
with simultaneous presentation) for additions only, whatever the size of the
operands. We performed a priori comparisons of mean RTs for each negative
SOA value and the mean RTs for SOA 0 for both types of operation and both
TABLE 1
Mean of median RTs in ms (and errors rate in perc ent) for the verification of true
problems as a functio n of operand type, operand size, and S OA value in Experiment 1
(mixed blocks)
SOA value
Operation 7300 7150 770 0 +150 +240 +300 +500
Additions
Small 843 841 897 916 825 709 677 738
(errors) (2.6) (1.0) (3.1) (1.6) (3.7) (3.1) (1.6) (6.3)
Large 986 947 1048 1041 854 888 837 796
(errors) (6.8) (5.2) (3.7) (4.7) (3.1) (1.0) (3.7) (3.7)
Multiplications
Small 885 885 888 900 825 769 768 711
(errors) (1.0) (4.7) (2.1) (3.1) (0.6) (1.0) (2.6) (3.7)
Large 952 883 910 919 834 778 725 666
(errors) (2.1) (0.0) (1.0) (1.1) (2.1) (1.6) (1.6) (2.6)
72 ROUSSEL, FAYOL, BAR ROU ILLET

sizes of operand (Table 2). Whatever the size of the operands, the verification of
additions at the SOAs ¡300 (843 ms for small operands and 986 ms for large
operands) and ¡150 (841 ms and 947 ms respectively) was significantly faster
than for SOA 0 (916 ms and 1041 ms respectively). In contrast, the early
presentation of the multiplication sign did not cause significantly shorter solu-
tion times. The verification of additions (all operand sizes taken together) was
facilitated more greatly than that of multiplications both at an SOA of ¡300
(reduction of 63 ms for additions compared to an increase of 9 ms for multi-
plications, F(1, 23) = 11.95, p<.01, MSe = 5325) and of ¡150 (reduction of 84
ms for additions compared to a reduction of 26 ms for multiplications, F(1, 23) =
6.66, p<.02, MSe = 6104).
In contrast, no effect was observed at SOA ¡70 ms. This suggests that the
activation of the additive procedure by means of the + sign requires longer than
70 ms or, alternatively, that the level of activation attained at 70 ms is too low to
result in a noteworthy reduction in verification time.
The fact that reduced times were observed for the verification of additions
only at the SOAs ¡300 ms and ¡150 ms conforms to our hypothesis that
calculation is based on the use of a procedure which is triggered, in part, by the
presence of the addition sign. Even though the size of the operands had a strong
effect on verification time for additions, the same reductions in time were
observed for the negative SOAs whatever the size of the operands (F<1 at SOA
¡300 and at SOA ¡150).
TABLE 2
F(and MSe) values for comparison s betwee n negative
SOAs and SOA 0 for ea ch negativ e SOA value as a
function of operand type and operand size in
Experiment 1 (mixed blocks)
SOA value
7300 7150 770
Additions
Small 13.89** 29.77** 1.03
(4632) (2241) (4378)
Large 4.37* 9.45** <1
(8063) (11036) (7573)
Multiplications
Small <1<1<1
(3570) (4573) (5372)
Large 1.50 2.63 <1
(8704) (6075) (9943)
** p<.01, * p<.05.
PROCEDURES A ND RETRIEVAL IN A RITHM ETIC 73

The fact that the time reductions are independen t of the size of the operands
while this size had a strong effect on verification time conforms to the
hypothesis that a procedure underlies the solution of additions. The preliminary
presentation of the + sign alone would appear to activate the procedure in
memory. The procedure would then be ready to function as soon as the operand s
appear. The resulting time reduction is independent of the size of the operands,
which are involved in the procedure as variables only. Once these are available,
the procedure follows its course which depends on the size of the operands (the
larger these are, the greater the number of steps to be performed) .
Nothing of the sort was observed when the £sign was presented before the
operands. This conforms to the hypothesis that the declarative knowledge that
stores the multiplicative facts is retrieved from memory and that this retrieval
can only be activated by the presentation of the operands themselves.
The effects observed for positive SOA values (i.e., presentation of the sign
after the operands and the answer) confirme d the data reported in the literature.
As of the SOA + 150 ms, we observed a significant time reduction for both
additions, F(1, 23) = 60.14, p<.001, MSe = 7699, and multiplications, F(1, 23)
= 25.85, p<.001, MSe = 5936, irrespective of the size of the operands. These
time reductions tended to increase with SOA (cf., Table 1). Taken overall, the
RTs for the positive SOAs were smaller than the RTs measured for SOA 0,
F(1, 23) = 341.49, p<.001, MSe = 6434, and this effect did not interact either
with the type of operation, F(1 , 23) = 3.04, p= .09, MSe = 9074, or with the size
of the operands, F(1, 23) = 3.44 , p= .08, MSe = 4236.
The facilitation observed with positive SOAs for both additions and multi-
plications confirms the validity of the paradigm used and argues in favour of the
most important fact obtained by manipulating the SOAs: when the operands are
not presented before, but after, the sign, the reductions in verification time which
result from these asynchronies in presentation are observed in connection with
additions only whereas multiplications are unaffected. The only explanation for
such a phenomenon is that the two types of operation are solved using different
procedures.
Our second hypothesi s predict ed that operand size should have a greater
effect on additions than on multiplications. In fact, the operand size interacted
with the type of operation, F(1, 23) = 16.44, p<.001, MSe = 38356. Large
additions (925 ms) were verified more slowly than small ones (806 ms), F(1, 23)
= 27.60, p<.001, MSe = 49299, which was not the case for multiplications (833
ms and 829 ms respectively, F<1). This Operand size £Operation interaction
was significant (p<.05) for all SOA values with the exception of SOA ¡300
(for which the verification of large multiplications took 67 ms longer than that of
small ones) and SOA +150 at which size effects were particularly diminished
(see Table 1). These variations resulted in an Operand size £Operation £SOA
interaction, F(7, 161) = 3.06, p<.01, MSe = 6133, for which we have no
explanation.
74 ROUSSEL, FAYOL, BAR ROU ILLET

Independently of the various effects linked to the position at which the
operator sign was presented, the size effect conformed to our predictions at SOA
0 at which the sign was presented at the same time as the operands. In this
condition, the large additions were verified more slowly than the small ones
(1041 ms vs 916 ms), F(1, 23 ) = 19.98, p<.001, MSe = 9338, while the size
effect for multiplications was not significant (919 ms vs 900 ms respectively),
F<1. The Operand size £Operation interaction was significant for this SOA,
F(1, 23) = 8.97, p<.01, MSe = 7418. These results are in total conformity with
the hypothesis that additions are solved by means of a procedure, whereas
multiplications are solved by the direct retrieval of knowledg e from declarative
memory. The absence of a size effect for multiplications, though frequently
reported in the literature (Campbell & Graham, 1985; Parkman, 1972; Stazyk et
al., 1982) might be due either to the considerable expertise of the participants,
which makes strategies other than direct memory retrieval unlikely, or to the
small number of problems presented (see later and Experiment 3).
Our third hypothesis related to interference effects caused by the presenta-
tion, for any given operation, of the answers of the other operation. This requires
an analysis of the median RTs for the rejection of incorrect problems.
Reaction time for the rejection of false problems. We performe d a 2
(operations: addition vs multiplication) £2 (operand size: small vs large) £2
(type of answer: interferent vs non-interferent) £8 (SOA: ¡300 ms, ¡150 ms,
¡70 ms, 0 ms, +150 ms, +240 ms, +300 ms, and +500 ms) ANOVA with
repeated measures for all the factors on the median RTs for the rejection of false
problems. As for the true problems, subjects took longer to reject false additions
(969 ms) than false multiplications (924 ms), F(1, 23) = 18.06 , p<.001, MSe =
41769. They also took longer to reject large operations (965 ms) than small ones
(929 ms), F(1, 23) = 17.63, p<.001, MSe = 28700, and rejection time varied
with SOA (cf., Table 3).
Rejection times for interferent problems (962 ms) did not differ significantly
from the rejection times for non-interferent problems (932 ms), F(1, 23) = 4.02,
p= .06, MSe = 91783. However, the type of answer interacted weakly with the
SOA, F(7, 161) = 2.10, p= .05, MSe = 12901. In contrast, and as predicted by
our hypothesis, the Type of answer £Operation interaction was significant,
F(1, 23) = 24.23, p<.001, MSe = 28879. Subjects took longer to reject false
interferent multiplications (961 ms) than non-interferent multiplications
(888 ms), F(1, 23 ) = 31.70, p<.001, MSe = 32965. This effect was not observed
for additions (963 and 975 ms respectively, F<1). The Type of answer £
Operation interaction interacted neither with the SOAs nor with the operand size
(Fs<1).
Once again, the interaction between the type of answer and type of operation
for verification was in conformit y with our hypotheses and replicated a result of
Zbrodoff and Logan (1986). The strategy of retrieving number facts directly from
PROCEDURES A ND RETRIEVAL IN A RITHM ETIC 75

long-term memory for the verification of multiplications exposes subjects to
interference phenomena inherent in the structure of declarative memory in which
the activations resulting from attentional centrations on the operands propagate.
Since these networks contain both multiplicative and additive facts, the pre-
sentation of the operands activates the answers of the two operations as well as
information concerning the type (multiplicative or additive) of the number fact
activated. When a multiplication is presented with the sum (rather than the
product) of the two operands as the answer, the information contained in the
chunk representing the additive declarative knowledge activated by the operands
corresponds to the current contents of working memory, which contains the
operands and the answer for evaluation. The inhibition of this knowledge in order
to produce a correct response (i.e., rejection) would explain the extra time
observed in the case of interferent multiplications. In contrast, the modular nature
of the additi ve procedure, which we consider to be involved in the verification of
additions, would prevent the risk of interference but would result in longer
rejection times for additions (969 ms) than for multiplications (924 ms).
TABLE 3
Mean of median RTs in m s (and error rates in perce nt) for the rejection of false
problems as a function of operan d type, operand size, and type of incorrect answer
presented (Interferent vs Non-interferent ) and SOA valu e in Experime nt 1
(mixed blocks)
SOA value
Operation 7300 7150 770 0 +150 +240 +300 +500
Additions
Interferent
Small 981 1023 1031 1105 932 855 826 845
Large 997 1056 1040 1078 977 896 927 838
Mean error rate (2.1) (3.7) (3.7) (3.7) (2.1) (2.1) (2.6) (5.7)
Non-interferent
Small 987 967 981 1066 973 886 912 808
Large 1011 1048 1100 1124 962 964 911 894
Mean error rate (5.2) (4.7) (5.2) (4.2) (4.7) (1.6) (4.2) (4.2)
Multiplications
Interferent
Small 970 1041 1101 1064 969 827 815 794
Large 995 1031 1075 1080 990 917 884 830
Mean error rate (4.2) (2.1) (3.7) (6.3) (4.7) (2.6) (2.1) (6.3)
Non-interferent
Small 880 929 978 1014 842 750 770 790
Large 895 951 1014 1031 868 904 805 780
Mean error rate (3.1) (1.6) (1.0) (2.6) (3.1) (1.0) (4.2) (2.1)
76 ROUSSEL, FAYOL, BAR ROU ILLET

This point is confirmed by the analysis of the number of errors made during
the evaluation of false problems.
Errors during the evaluation of false problems. The participants made few
errors during the evaluation of false problems (3.5%). We performed an
ANOVA on the rate of errors made on false problems. This was identical to the
one performed for the RTs. Only two effects were significant. We observed a
weak SOA effect, F(7, 161) = 2.10 , p= .05, MSe = 61.65 and, more importantly,
a Type of answer £Operation interaction, F(1, 23) = 6.69, p<.02, MSe =
102.31. For the additions, the mean rate of errors made in the interferent
problems (3.19%) was less than that for the non-interferent problems (4.23%)
whereas the reverse was observed in the case of multiplications: false interferent
multiplications resulted in more errors (3.97%) than non-interferent multi-
plications (2.34%).
It should be noted that this interaction could not be due to a split effect
(an incorrect response is all the more difficult to reject, and therefore a more
potent source of error, the closer it is to the correct response). In effect, for
both types of operation, the false responses which were furthest from the cor-
rect response corresponded to interferent answers (answer of the other oper-
ation), whereas the non-interferent responses were the closest (§1) to the
correct response. A split effect does not account for the Type of answer £
Operation interaction.
However, this interaction can be explained if, as we suggest, subjects solve
multiplication problems by means of a direct memory retrieval strategy while
solving additions on the basis of a procedure. For additions, errors were
observed most frequently in the non-interferent condition (i.e., correct response
§1) and, for multiplications, in the interferent condition (i.e., answer associated
with the correspondin g addition). As we have already explained, a strategy of
directly retrieving the answer from memory in the case of multiplications
exposes subjects to interference effects and the possibility of error when the
incorrect answer corresponds to the answer of another operation (i.e., the
addition). In contrast, this strategy reduces the likelihood of error when the
incorrect answer is close to the correct answer (§1). In this case, no declarative
knowledge links the operands to the answer for evaluation and there is only a
low probability that the equation will be judged to be correct.
In contrast, if additions are solved through the use of a step-by-step count-
ing strategy, non-interferent answers (i.e., differing from the correct answer
by §1) represent a greater potential source of error than the answer for the
concurrent operation (i.e., the multiplication). In effect, this error is all the
more likely to occur, the lower the number of steps required to achieve the
incorrect answer. This means non-interferent answers induce more errors than
interferent answers.
PROCEDURES A ND RETRIEVAL IN A RITHM ETIC 77

Discussion
The three predictions which result from the hypothesi s that a procedural com-
ponent is mobilised during the solution of addition problems whereas multi-
plication problems are solved through the direct retrieval of the answer from
memory have been verified. First, presenting the sign before the operands
(negative SOAs) resulted in a larger reduction in verification times for the
additions than for the multiplications. As predicted by the theory, the + sign had
the effect of priming the additive procedure independently of the size of the
operands subsequently presented, whereas the simple presentation of the £sign
did not result in any significant priming effect for any of the multiplicative
knowledge stored in declarative memory. This reduction in verification time
resulting from the presentation of the addition sign conforms to the ACT-R
model, which stipulates that the mobilisation of a procedure requires the
matching of a set of information contained in working memory with the
elements that constitute the conditions of the procedure. The time necessary to
fire a procedure might be equal to the sum of the times required to match the
clauses in the condition of the production (Anderson, 1993, p. 48). Unlike in the
ACT* model, the conditions for the application of the procedure would be
matched in series. Thus the time reduction caused by the early presentation of
the sign, when compared to simultaneous presentation, would be due to the fact
that the ‘‘goal’’ component of the procedure has already been matched when the
numbers to be added are displayed. The absence of any time reduction at SOA
¡70 ms suggests that this matching takes between 70 and 150 ms.
Second, the operand size effect was greater for additions than for multi-
plications. The time taken by the step-by-step procedur e to reach its goal
depends on the size of the operands whereas the time required to access the
declarative knowledge in which number facts are stored is, a priori, independent
of the size of the numbers. In this latter case, only a retrieval frequency effect
may occur, with some interference between stored knowledge (Zbrodoff, 1995).
Finally, the interference effects were more marked for multiplications than for
additions. The retrieval of multiplicative facts from declarative memory is
sensitive to the concurrent activation of additive facts within the same memory
register, whereas the mobilisation of procedural knowledge for additions pre-
vents, at least in part, the effects of any retrieval of information concerning
multiplications from declarative memory.
Our theory is able to account in an economical way for all these phenomena.
The solution of additions would mobilise a procedural component, even in
expert subjects such as the participants in our experiment. In contrast, the same
participants made very considerable use of a direct memory retrieval strategy
when performing multiplications. The result was a weak size effect and a strong
interference effect. This interference effect appears to be caused by the presence
in declarative memory of additive facts produced by the recurring imple-
78 ROUSSEL, FAYOL, BAR ROU ILLET

mentation of the additive procedure from a very young age. Thus, expert sub-
jects should possess both procedura l and declarative knowledge bearing on
addition, whereas, to a very great extent, the solution of multiplication problems
is based on declarative knowledge. In fact, as mentioned earlier, the use of a
procedure for the calculation of even elementary multiplicative facts would
result in long and costly solution times as the data reported by LeFevre, Bisanz,
et al. indicate (1996, Table 2).
Everything therefore seems to indicate that expert adult subjects possess a
variety of additive strategies that allow them to mobilise declarative or pro-
cedural knowledge. Thus, the main problem is not to determine whether the
knowledge involve d in addition is procedura l or declarative in nature. The
priming effects caused by the presence of the + sign and the operand size effects
seem to indicate the use of a procedure, while, for their part, the interference
effects in multiplication suggest the existence of declarative knowledge. Indeed,
additions took longer to verify than multiplications. It is therefore unlikely that
interference during multiplication is due to the answer being simultaneously
accessed by the additive procedure which is slower than the memory retrieval
process. That is why much of the literature suggests that subjects possess
declarative knowledge concerning addition and that this is activated by the
presentation of the operands.
In contrast, the problem is to understand why the procedural component is
mobilised in the solution of simple additions, even in expert adults who possess
declarative knowledge concerning additive facts. Three reasons can be
advanced. The first is that children construct the addition procedure at a very
young age (before any academic input occurs) and that it is one of their first
number skills. The second is that when two types of strategy, which are
applicable to the same class of problem, coexist, the choice of a given strategy is
determined by factors more complex than, for example, the relative speed of the
strategies or the existence of facts stored in memory which would result in the
systematic selection of a retrieval strategy.
The third reason is that recourse to a counting procedure to solve addition
problems and memory retrieval to solve multiplications was the strategy adopted
by participants faced with a specific situation in which the problems were
presented in mixed blocks. It should be remembered that when the signal
announcing the appearance of a new problem appeared, the participants were
unable to predict the nature of the operation. It is therefore possible that the
participants adapted to the requirements of the task (rapid, correct responses) by
mobilising a different process for each type of possible operation. This might
have the advantage of reducing interference. If this is indeed the case, the results
of Experiment 1 suggest that a procedura l component for solving additions
persists in expert adults and that this can still be mobilised, but only for very
specific, non-ecological conditions. This point, which can easily be tested, was
the object of Experiment 2. It is also possible that the use of an algorithmic
PROCEDURES A ND RETRIEVAL IN A RITHM ETIC 79

strategy by experts is due to the fact that teachers spend a substantial amount of
time practising such procedure s in their teaching. This latter point was the object
of Experiment 3, which compared teachers with university students.
EXPERIMENT 2
The aim of this experiment was to verify that the recourse to a procedural
component for the solution of addition problems was not linked to an adaptive
strategy implemented in order to permit subjects to provide optimum responses
within the constraints specific to the presentation of operations in mixed blocks.
The same problems as were used in Experiment 1 were presented to a new group
of expert adults. However, in this new experiment the problems were presented
in two blocks in two separate sessions, one devoted to the solution of additions
and the other to that of multiplications. At the start of each session, the parti-
cipants were told which type of operation (addition or multiplication) would be
presented to them.
If the mobilisation of the procedural component in the solution of additions
was linked to constraints caused by the mixed-block presentation, the removal of
these constraints should lead to additions being solved more frequentl y through
the retrieval of the answers from memory. In this case, similar result patterns
should be obtained for both addition and multiplication since the solution of the
two operations would involve the same types of process. The effect of the
preliminary presentation of the sign on additions (negative SOAs) should dis-
appear and the size and interference effects should be comparable for both
operations. In contrast, if mobilisation of the procedural component is inherent
in the solution of additions independently of the problem-solving situation, the
results of Experiment 2 should be comparabl e to those of Experiment 1: pre-
liminary presentation of the sign should result in reduced verification times for
additions only, the operand size effect should be greater for additions than for
multiplications and interference effects should be more pronounced for multi-
plications.
Method
Participants. Thirty-one primary school teachers volunteered to take part in
the experiment. Their mean age was 39 years and 6 months (range: 22 years 8
months to 58 years 10 months). None of them had taken part in the preceding
experiment.
Material and procedure. The material was identical to that used in
Experiment 1. The only difference lay in the way the problems were presented.
This was performed in two sessions separated by a period of one week. The
additions were presented during the first session and the multiplications during
the second for half the participants and this order was reversed for the others.
80 ROUSSEL, FAYOL, BAR ROU ILLET

Results
Reaction times. Table 4 gives the mean of the median RT’s for correct
responses as a function of the type of operation, the operand size, and the SOA
value for the verification of true problems. We performed a 2 (operations:
addition vs multiplication) £2 (operand size: small vs large) £8 (SOAs)
ANOVA with repeated measures on all the factors. As previously, all the main
effects and all the interactions were significant with the exception of the Size £
SOA interaction. Subjects took longer to verify additions (M= 881 ms) than
multiplications (784 ms), F(1, 30) = 11.74, p<.01; MSe = 198211. Large
operands resulted in slower verifications (869 ms) than small operands (796 ms),
F(1, 30) = 26.77, p<.001, MSe = 49246 and the RTs varied as a function of the
SOAs, F(7, 210) = 31.53, p<.001, MSe = 10582.
Placing the sign before the operands produced the same effect as had been
observed for the mixed block presentation. For additions, the RT’s at SOA ¡300
(913 ms) and ¡150 (909 ms) were shorter than at SOA 0 (951 ms), F(1, 30) =
13.33, p<.001, MSe = 8385 and F(1, 30) = 22.98, p<.001, MSe = 5445
respectively, whereas there was no significant time reduction at SOA ¡70
(951 ms), F(1, 30) = 1.44, p= .24, MSe = 10134. No effect of this type was
observed when the multiplication sign was presented before the operands
(821 ms, 820 ms, 799 ms, and 807 ms for SOAs ¡300, ¡150, ¡70, and 0
respectively). In the case of additions, a reduction in verification time was—as
previously—observed for both operand sizes with the exception of small addi-
tions at SOA ¡150 where the time reduction showed only a trend (Table 5).
TABLE 4
Mean of media n RTs in ms (an d error rates in percent) for the verificatio n of true
problems as a functio n of operand type, operan d size, and SOA value in Experiment 2
(unmixed blocks)
SOA value
Operation 7300 7150 770 0 +150 +240 +300 +500
Additions
Small 851 836 874 873 792 775 728 674
(errors) (0.4) (1.2) (2.4) (1.6) (0.8) (0.4) (0.8) (0.4)
Large 975 983 1028 1072 960 941 945 792
(errors) (3.6) (3.2) (3.6) (3.6) (2.8) (1.6) (2.0) (4.0)
Multiplications
Small 804 841 803 817 772 794 788 720
(errors) (4.8) (4.8) (2.4) (4.0) (3.6) (1.2) (0.8) (1.6)
Large 837 799 795 797 798 744 670 770
(errors) (2.4) (1.6) (1.2) (1.6) (0.8) (0.0) (1.2) (2.8)
PROCEDURES A ND RETRIEVAL IN A RITHM ETIC 81

Similarly, the size effects were in line with our observations in Experiment 1.
Operand size interacted with operation type, F(1, 30) = 58.14, p<.001, MSe =
33687. Small additions (800 ms) were verified more quickly than large ones
(962 ms), F(1, 30) = 63.45, p<.001, MSe = 51145, while operand size had no
effect on RTs for the verification of multiplications (792 ms for small operand s
vs 776 for large operands), F(1, 30) <1, p= .33, MSe = 31787. The effect of
operand size on the RTs for the verification of additions was significant at all the
SOAs (p<.01). At SOA 0, the size effect for additions was significant (873 ms
for small operands vs 1072 for large ones), F(1, 30) = 30.38 , p<.001, MSe =
20270, whereas at the same SOA, small multiplications were verified slightly
more slowly (817 ms) than large ones (797 ms).
Finally, the interference effects in the verification of false responses were
identical to those observed in Experiment 1. The type of answer (interferent vs
non-interferent) interacted with the operation type, F(1, 30) = 16.69, p<.001,
MSe = 29765. The multiplications for which the false answer was interferent
(i.e., sum of the two operands) were verified more slowly than those with non-
interferent answers (912 ms vs 886 ms), F(1, 30) = 5. 32, p<.03, MSe = 30937,
whereas the opposite effect was observed for additions (interferent: 983 ms;
non-interferent: 1020 ms), F(1, 30) = 5.92, p<.02, MSe = 58993. This latter
effect could be a split effect. Since the operations were presented in unmixed
blocks, the false interferent answers (i.e., the product of the two operands) would
appear to be implausible for additions because of their size, especially in the
case of large operands . In fact, this effect was observed for large operands
TABLE 5
F(and MSe) values for com parisons betwee n negativ e
SOAs and SOA 0 for each negative SOA value as a
function of operand type and operand s ize in
Experiment 2 (unmixe d blocks )
SOA value
7300 7150 770
Additions
Small 1.47 3.82 (p<.06) <1
(5390) (5720) (6571)
Large 10.54** 10.58** 2.02
(13982) (11752) (15067)
Multiplications
Small 1.69 2.02 <1
(5168) (5050) (4948)
Large 2.46 <1<1
(10083) (4276) (9155)
** p<.01.
82 ROUSSEL, FAYOL, BAR ROU ILLET

(interferent: 986 ms; non-interferent : 1066 ms) but not for small ones (inter-
ferent: 981 ms, non-interferent: 975 ms). At SOA 0, interferent multiplications
were verified more slowly (995 ms) than non-interferent multiplications
(958 ms), F(1, 30) = 3.97, p<.06, MSe = 10547, whereas the opposite effect was
observed for additions (1141 ms vs 1077 ms).
Participants made very few errors in the evaluation of true problems (2.12%).
The SOA effect was significant, F(7 , 210) = 2.71, p<.01, MSe = 25.38. This
effect was mainly due to lower frequencies of errors on the SOAs +240 and
+300 than on the others (Table 4). The Operation £Size interaction was also
significant, F(1, 30) = 12.61 , p<.01, MSe = 61.19: The small additions elicited
less errors (1.01%) than the large additions (3.07), whereas the reverse was true
for the multiplications (2.92 and 1.46%for small and large operands respec-
tively). No other effect was significant.
As far as false problems are concerned, additions (3.00%) elicited more errors
than multiplications (2.02%), F(1, 30) = 8.98, p<.01, MSe = 53.36. The
observed SOA effect was the same as for true problems, F(7, 210) = 3.13,
p<.01, MSe = 41.22.
To summarise, the effects observe d in Experiment 1 were replicated, thus
reinforcing the hypothesis that different processes are mobilised to solve addi-
tions and multiplications. Whether the problems were presented in mixed or
unmixed blocks, the predictions resulting from our model were borne out. The
mobilisation of a procedura l componen t in the solving of additions was con-
firmed by the reduction in verification times caused by the early presentation of
the sign, a strong size effect, and weak interference effects. In contrast, the use
of a memory retrieval strategy for multiplications was demonstrated by inter-
ference effects, a weak size effect and subjects’ lack of sensitivity to the pre-
liminary presentation of the sign.
Discussion
The results of this experiment show that the main phenomen a observe d in
Experiment 1 were not due to the way in which the problems were presented
(i.e., mixed blocks). Nevertheless, these results still suffer from three limitations
relating to (1) the small numbe r of different problems presented (only eight
problems) and their repeated presentation, (2) the characteristics of the studied
population, and (3) the nature of the non-interferent results presented for the
false problems.
As far as the first point is concerned, it is possible that the results observed in
the earlier experiments were due to the repeated use of a small set of operands.
This might have led the participants to the rapid identification of the subset of
operand pairs to be processed, encouraging them to adopt ad hoc problem-
solving strategies (e.g., retrieval for multiplications and algorithmic strategy for
additions). In effect, the repeated presentation of the same pair of operands
PROCEDURES A ND RETRIEVAL IN A RITHM ETIC 83

cannot fail to reinforce the activation level of the (multiplicative or additive)
number facts associated with it. This could lead to the permanent maintenance of
answers in working memory (Conway & Engle, 1994). The risks of interference
which result from this might lead to the adoption of back-up strategies where
these exist (i.e., for addition). This might explain why multiplications were
solved more quickly than additions, whereas the opposite is usually observed—
at least for small operands. Moreover, a high level of activation of the multi-
plicative facts might accelerate their retrieval and mask the size effect which is
habitually observed in connection with this operation.
As far as the selected population is concerned, it is possible that school
teachers consider a high number of errors in such simple addition problems to be
unbefitting to their professional status and consequently adopt a strategy which
favours precision at the expense of speed. In a study of individual differences,
Siegler (1988a) observed that among high-performanc e pupils there was a group
of so-called ‘‘perfectionist’’ pupils who placed the emphasis on precision in the
solving of addition and subtraction problems. These subjects made far less
frequent use of a memory retrieval strategy than the others when solving
problems. It is therefore possible that the experts in Experiments 1 and 2 adopted
a ‘‘perfectionist’’ strategy and solved the majority of additions by means of
back-up strategies. This would explain why the multiplications were solved
more quickly than the additions.
Finally, the non-interferent answers were always close to the correct answer.
Their interferent character was therefore confounded with the split. It is possible
that the multiplicative answer interferes with the rejection of false additions
when the split is controlled. For example, in the case of the problem 4 + 9, the
answer 36 was rejected more quickly than 14, but was sometimes also rejected
more slowly than 34. At the same time, the non-interferent answer for the false
multiplications did not respect the parity of the correct answer and this might
have led to their rapid rejection (Lemaire & Fayol, 1995). The purpose of
Experiment 3 was to surmount these three difficulties.
EXPERIMENT 3
This experiment compared the performances of expert (school teachers) and
‘‘non-expert’’ (undergraduat e psychology students) subjects in the verification
of 28 additions and 28 multiplications instead of the 8 problems used in the
previous experiments. Only three SOAs were used (¡150, 0, and +150 ms) in
order to reduce the number of presentations of each pair of operands. SOA ¡150
is the one at which the priming effect of the + sign had been observed to be the
most regular (Experiments 1 and 2). A positive SOA (+150 ms) was retained in
order to ensure that any difference between the operations at SOA ¡150 was due
solely to the preliminary presentation of the sign (+ vs £) and not to any
asynchrony in the presentation of the information which might affect addition
84 ROUSSEL, FAYOL, BAR ROU ILLET

more strongly than multiplication. The time gain due to the positive SOA should
be identical for the two types of operation. In effect, it results from the fact that
the processing made possible by the presentation of the operands and the result
has already started when the timer is started by the appearance of the sign. The
time gain relative to SOA 0 should be identical whatever the presented operation
and the strategy used.
In the case of the false problems, the non-interferent answer which were
distanced from the correct answer (e.g., 4 + 9 = 34) were added to the ‘‘close’’
interferent and non-interferent answers used earlier. For both subject groups, the
predictions were the same as before.
Method
Participants. Twenty-eight students at the UniversiteÂde Bourgogne (Mean
age: 20 years 2 months) and 36 school teachers (mean age: 40 years 7 months)
volunteered to take part in this experiment.
Material and procedure. The additions and multiplications were presented
in mixed block using the same procedure as in Experiment 1 with the exception
of the presented operand pairs and the SOAs (three SOAs were used: ¡150, 0,
and +150 ms). All the single-figure operand pairs were used with the exception
of ties and pairs containing 0 or 1 (56 pairs, see Appendix 2). The 28 pairs with a
sum smaller than or equal to 10 and a product smaller than or equal to 24 were
considered to be composed of small operands, and the others of large operands.
The answer displayed on the screen was either correct or incorrect. In the latter
case, 3 types of answer were presented: interferent, close non-interferent (sum §
1 for additions, product §2 for multiplications), or remote non-interferent (sum
§1 for multiplications, product §2 for additions). The remote non-interferent
answers were introduced to prevent the interferent or non-interferent nature of
the proposed answer from being confounded with the distance to the correct
answer. The close non-interferent multiplication answers were obtained by
adding or subtracting 2 to or from the correct answer in order to control for a
possible parity effect.
In order to simplify the participants’ task, the 56 pairs of operands were split
into two blocks of 28 pairs each (14 small and 14 large operands). The dis-
tribution was symmetrical (i.e., the pair 2–4 belonged to one block and the pair
4–2 to the other). Half of the participants saw one block in additive and one
block in multiplicative mode, while the other half saw the blocks in the opposite
modes. Each participant saw each problem twice with a correct answer, once
with an interferent answer and once with a non-interferen t answer (for each SOA
value). Each subject saw half of the pairs with a remote non-interferent answer
and the other half with a close non-interferen t answer. The permutation we
employed allowed us to present all the pairs with both operations and both types
PROCEDURES A ND RETRIEVAL IN A RITHM ETIC 85

of non-interferent answer. In total, each subject verified 672 problems (56 pairs
£4 answers £3 SOAs, see Appendi x 2) presented in a random order.
The order of the events on the screen, the respons e modes and the instructions
given to the participants were the same as in the preceding experiments. The
experiment was conducted in four phases of 168 trials each, separated by brief
pauses for rest, and lasted approximately 50 minutes.
Results
As we shall see, the results of this experiment confirmed our hypotheses. The
priming effect due to the anticipated presentation of the operator was significant
for the additions only and did not interact with the groups (experts vs students).
The size effect remained larger for additions than multiplications in both groups,
experts exhibiting a smaller size effect than students for both operations. Finally,
the interference effect was less clear than in the previous experiments but did not
disconfirm our predictions. First, we analysed the reaction times for the true
problems, then the reaction times for the false problems, and finally the error
rates in false problems.
True problems. The median time for correct responses to true problem s was
calculated for each subject and each experimental condition (Table 6). A 2
(groups: expert vs student) £2 (operations: addition vs multiplication) £2
(size: small vs large operands) £3 (SOAs: ¡150 ms, 0 ms, +150 ms) ANOVA
TABLE 6
Mean of median RTs in ms (and error rates in percent) for the verification
of true problem s as a function of groups, operand type , operand s ize, and
SOA value in Experime nt 3
SOA values
Experts Students
Operation 7150 0 +150 7150 0 +150
Additions
Small 990 1042 1011 945 1037 917
(errors) (2.9) (3.1) (3.8) (3.6) (3.7) (3.1)
Large 1173 1199 1147 1212 1291 1213
(errors) (6.0) (5.1) (5.5) (7.3) (7.0) (9.1)
Multiplications
Small 1008 1053 1003 975 1000 942
(errors) (3.0) (3.3) (3.2) (2.9) (5.4) (4.0)
Large 1130 1133 1066 1113 1180 1059
(errors) (2.7) (3.2) (3.5) (4.5) (6.8) (6.1)
86 ROUSSEL, FAYOL, BAR ROU ILLET

with operation, size, and SOA as within-subject factors was performed on these
median times.
As had been observed in the preceding experiments, the multiplications were
verified more quickly (1055 ms) than the additions (1098 ms), F(1, 62) = 13.60,
p<.001, MSe = 25494. This effect did not interact with group, F(1, 62) = 1.65,
p= .20, MSe = 25494. This result allows us to discard the hypothesi s that
multiplications were solved more quickly because of the small number of
problems presented in the earlier experiments. Moreover, it is compatible with
the hypothesis that additions are more frequentl y solved through the application
of algorithmic strategies than multiplications. Furthermore, the absence of an
interaction with the group suggests that this effect does not depend on a strategy
which is specific to experts in the verification of additions.
Our first hypothesis predicted that the priming effect caused by the pre-
liminary appearance of the sign (SOA ¡150 vs SOA 0) should be greater for
additions than for multiplications. The main effect of the SOA was significant:
The operations were verified more slowly at SOA 0 (1117 ms) than at the SOAs
¡150 (1068 ms) and +150 (1045 ms), F(2, 124) = 43.29, p<.001, MSe = 7869,
and the a priori comparison between the SOAs ¡150 ms and 0 ms was also
significant, F(1, 62) = 43.31, p<.001, MSe = 6907. In conformity with our
hypothesis, the priming effect was greater for additions (1080 and 1142 ms at
the SOAs ¡150 and 0 respectively, difference: 62 ms) than for multiplications
(1057 and 1092 ms respectively, difference: 35 ms); this interaction was sig-
nificant, F(1, 62) = 5.14, p<.05, MSe = 4472. Neither the Group £SOA (¡150
ms vs 0 ms) £Operation interaction, nor the Operand size £SOA (¡150 ms vs
0 ms) £Operation interaction were significant, F(1, 62) = 1.07, p= .30, MSe =
4472, and F<1 respectively. The priming effect when the operands and the
result were presented before the sign (SOA +150 vs SOA 0) did not differ as a
function of the operation (70 and 74 ms for addition and multiplication
respectively), F(1, 62) <1, MSe = 6555.
These results suggest that the fact that the priming effect due to the addition
sign is greater than that resulting from the multiplication sign which had been
observed in the earlier experiments was due neither to the small numbe r of
problems used in Experiments 1 and 2 nor to a peculiarity specific to the expert
participants in terms of the implemented strategies.
Our second hypothesis predicted that the size effect would be greater for
additions than for multiplications. In general, the problems containing large
operands were verified more slowly (1160 ms) than those containing small
operands (994 ms), F(1, 62) = 116.00, p<.001, MSe = 44916. This size effect
was greater for the students (969 vs 1178 ms for small and large operands
respectively) than for the experts (1018 vs 1142 ms), F(1, 62) = 7.53 , p<.01,
MSe = 44916. As predicted by the hypothesis, the size effect was greater for
additions (990 and 1206 ms for the small and large operands respectively) than
for multiplications (997 and 1114 ms respectively), F(1, 62) = 23.85, p<.001,
PROCEDURES A ND RETRIEVAL IN A RITHM ETIC 87

MSe = 19209. The Group £Operation £Operand size interaction was not
significant, F(1, 62) = 2.02, p= .16, MSe = 19209. This result is consistent with
the preceding and suggests that the experts do not differ from the students in the
strategies they mobilise.
The number of errors made was subjected to an ANOVA identical to that
performed for the RTs. The level of errors in the verification of the true prob-
lems was low (4.51%). As might have been expected, the students committed
more errors (5.27%) than the experts (3.75%), F(1, 62) = 5.57, p<.05, MSe =
6.20. Although verified more slowly, the additions gave rise to more errors (5%)
than the multiplications (4.03%), F(1, 62) = 6.36, p<.02, MSe = 2.15. The
problems containing large operands resulted in more errors (5.54%) than those
containing small operands (3.48%), F(1, 62) = 29.22, p<.001, MSe = 2.16. This
effect of operand size interacted with group, F(1, 62) = 6.27, p<.02, MSe =
2.16, and with the type of operation, F(1, 62) = 10.78, p<.01, MSe = 2.08. As
was the case for the times, the effect of operand size on the error levels was
greater for the students (3.77%vs 6.78%for small and large operands respec-
tively) than for the experts (3.20 and 4.30%respectively), and greater for the
additions (3.35 and 6.64%) than for the multiplications (3.61 and 4.44%). No
other effect was significant.
False problems. It should be remembered that, unlike in the preceding
experiments, two types of non-interferent false answers were presented: non-
interferent answers close to the correct answer (CNI); and remote from the
correct result (RNI). A 2 (Group: expert vs student) £2 (Operation: addition vs
multiplication) £2 (Operand size: small vs large) £3 (type of answer:
interferent, CNI, RNI) £3 (SOAs: ¡150, 0, +150 ms) ANOVA was performed
on the median rejection times for false problems in each of the 36 experimental
conditions seen by each subject (Table 7).
Apart from an SOA effect which did not interact with any other effect, the
false multiplications were rejected more rapidly (1104 ms) than the additions
(1162 ms), F(1, 62) = 33.14, p<.001, MSe = 58125, and the small operands
were rejected more quickly (1105 ms) than the large ones (1162 ms), F(1, 62) =
48.10, p<.001, MSe = 38927. The CNI answers were rejected more slowly
(1218 ms) than the RNI (1076 ms) and interferent (1107 ms) answers, F(2, 124)
= 49.91, p<.001, MSe = 84340. In conformity with the hypothesis that distinct
types of solution are used for the two types of operation, this effect was sig-
nificantly more pronounce d for the additions (CNI: 1292 ms; RNI: 1075 ms; Int:
1121 ms) than for the multiplications (CNI: 1144 ms; RNI: 1076 ms; Int:
1093 ms), F(2 , 124) = 21.65, p<.001, MSe = 54816. The Operation £Operand
size £Type of answer interaction was significant, F(2, 124) = 16.30, p<.001,
MSe = 30545.
In the case of the additions, the large operands greatly accentuated the dif-
ference observed in the response times for the CNI answers, whereas the RTs for
88 ROUSSEL, FAYOL, BAR ROU ILLET

both RNI and interferent answers remained unaffected by operand size (1180,
1081, and 1131 ms respectively for the small operands, and 1403, 1070, and
1110 ms for the large operands). This result was compatible with what we had
already observed in the preceding experiments. The CNI answers took longer to
reject, probably because the algorithmic solution of additions requires partici-
pants to complete the calculation before being able to establish that they are
incorrect. The CNI rejection times were therefore extremely sensitive to the size
of the operands which constrained the period required for calculation. The
interferent answers (i.e., the product of the two operands) and RNI answers were
remote from the correct result and probably very quickly rejected as implausible.
As far as the multiplications are concerned, the effect of operand size on the
difference between the CNI and RNI and interferent answers was less pro-
nounced than for the additions (for the small operands: 1098, 1063, and 1076 ms
for CNI, RNI, and interferent respectively, and 1190, 1090, and 1110 ms for the
large operands). The slower rejection of the CNI answers (1194 ms) than the
interferent answers (1093 ms) did not replicate the results of the earlier
experiments. However, this was mainly due to the students’ results. As pre-
viously observed, the experts rejected the interferent answers more slowly (1143
ms) than the other answers (1120 ms for CNI and RNI). However, this differ-
ence was not significant (p= .12). In contrast, as for the additions, the students
rejected the CNI answers more slowly than the other answers, and this pheno-
TABLE 7
Mean of media n RTs in ms (and error rates in per cent) for the rejec tion of
false problem s as a functio n of operan d type, oper and size, an d type of
incorrect answer presented in Experimen t 3 (mixed blocks)
Type of answer
Experts Students
Operation Int CNI RNI I nt CNI R NI
Additions
Small 1177 1180 1102 1085 1180 1059
(errors) (5.6) (3.8) (1.1) (6.9) (12.1) (0)
Large 1184 1333 1116 1036 1473 1023
(errors) (2.3) (9.8) (1.2) (2.2) (28.1) (0.9)
Multiplications
Small 1112 1104 1089 1040 1091 1036
(errors) (2.8) (3.7) (1.2) (4.7) (6.1) (1.7)
Large 1173 1135 1150 1047 1245 1029
(errors) (1.8) (3.8) (1.9) (1.6) (12.9) (1.5)
Int: Interferent answer ; CNI: Clos e Non-Interferent ; R NI: Remot e Non-
Interferent.
PROCEDURES A ND RETRIEVAL IN A RITHM ETIC 89

menon became more pronounce d the larger the operands were (for small
operands: 1091, 1036, and 1040 ms for CNI, RNI and interferent answers
respectively, 1245, 1029, and 1047 ms for large operands).
Thus, the results obtained for the students revealed a massive split effect for
the two types of operation, something observed in the expert participants in the
case of additions only. Thus the students were quicker than the experts to reject
answers which were remote from the correct result (1037 vs 1114 ms for the
students and experts respectively for the RNI answers; 1052 vs 1162 ms for
interferent answers), whereas they were slower to reject close results (1247 vs
1188 ms for the CNI answers). This interaction was significant, F(2, 124) =
18.05, p<.001, MSe = 84340.
These results differ in part from those of the preceding experiments. As
observed earlier, the experts took longer to reject the interferent answers than the
CNI answers (used previously) in the case of multiplications (1143 and 1120 ms
respectively), whereas the opposite was observed for additions (1181 and 1257
ms). This interaction was significant, F(1, 35) = 9.04, p<.01, MSe = 59227.
However, in the case of multiplications, the students rejected the interferent
answers more quickly (1044 ms) than the CNI answers (1168 ms). This suggests
that the presentation of the operands in the multiplications resulted in only a low
level of activation of the network of additive facts in the case of the students.
Moreover, although, as we expected, the interferent answers for additions
result in faster rejections than the CNI answers in all the participants, only the
experts proved to be even faster in rejecting the RNI answers (Int : 1189 ms,
RNI: 1109 ms, F(1, 35) = 30.88, p<.001). Thus, with a constant split effect
(both the interferent and RNI answers were remote from the correct answer), the
answer of the multiplication (e.g., 6 + 4 = 24) took longer to reject than a remote
answer which was not linked to the operands (e.g., 6 + 4 = 26). This suggests
that the answer of the multiplication exerts an interference effect in the veri-
fication of false additive problems which runs counter to our predictions.
However, in the case of additions, this interference effect was not sufficient to
cause interferent answers to be rejected more slowly than CNI answers.
The number of errors made for the false problems (4.90%overall) was
subjected to an ANOVA with the same design as the previous one. The results
confirmed those obtained for the response times. All the main effects with the
exception of the SOA effects were significant and compatible with those
observed for the response times: the experts made fewer errors (3.25%) than the
students (6.56%), F(1, 62) = 19.51, p<.001, MSe = 318.04, the additions caused
more errors (6.16%) than the multiplications (3.64%), F(1, 62) = 33.80, p<.001,
MSe = 106.09, the large operands more errors (5.66%) than the small ones
(4.14%), F(1, 62) = 11.66, p<.01, MSe = 113.18, and the CNI answers more
errors (10.04%) than the RNI answers (1.17%) and interferent answers (3.49%),
F(2, 124) = 80.52 , p<.001, MSe = 198.83. This suggests that none of the effects
observed for the times were due to a speed/accuracy trade-off.
90 ROUSSEL, FAYOL, BAR ROU ILLET

Discussion
To summarise, with the exception of the interference effects, the results of this
experiment confirmed those of the preceding experiments. The priming effect
caused by the preliminary presentation of the sign and the size effect were
greater for additions than for multiplications. These predictions, which result
from the hypothesis that additions are more frequentl y solved by the use of a
procedure than multiplications are, were generalised to a population of students
who were confronted with a large set of problems. Thus the main phenomena
observed in Experiments 1 and 2 were not due to any specific characteristic of
the selected population (experts), or to the small number of problems used or
their frequent reappearance during the experiment.
In this experiment, the size effect which is habitually observed for both types
of operation was observe d for the multiplications. This suggests that the earlier
absence of a size effect was due to the small number of problems presented.
However, in conformity with our hypothesis, this size effect was weaker for
multiplication than for addition. The small multiplications were verified as
quickly (997 ms) as the small additions (990 ms), whereas the large additions took
longer (1206 ms) than the large multiplications (1114 ms). The multiplications
were thus verified more quickly than the additions. This last fact, which is at odds
with the literature, might be related to the specific characteristics of the French
educational system in which the memorisation of multiplication tables is one of
the aims of arithmetic teaching. Starting in third grade, the systematic learning of
these tables continues until they are known by heart, normally up to and including
fifth grade. Addition tables are not taught in the same way. In fact, the multi-
plications resulted in fewer errors (4.03%) than the additions (5%), and the size
effect was smaller for the multiplications (3.61 and 4.44%for the small and large
operands respectively) than for the additions (3.35 and 6.64%).
Only the false problems led to results which differed from those obtained in
the earlier experiments. In effect, a split effect was observed not only for
addition, but also for multiplication in the student participants. It is possible that
the presentation of the RNI answers induced these subjects to adopt a strategy
involving the evaluation of the plausibility of the answers. Indeed, the RNI
answers were in no way related to the operands (e.g., 6 £4 = 11 or 6 + 4 = 26).
The presentation of this type of answer may make a strategy of evaluating the
plausibility of the answer before applying strategies for the solving of the
operation particularly adaptable. In fact, the students were quicker than the
experts to reject remote false answers. In contrast, the use of this strategy led to
an increase in time in cases where the answer was plausible (i.e., close to the
correct answer): the students were slower than the experts in rejecting the CNI
answers. The use of a strategy of evaluating the plausibility of the answers was
confirmed by the error levels. The students made a large number of errors on the
CNI answers (20.1%for the additions, 9.5%for the multiplications).
PROCEDURES A ND RETRIEVAL IN A RITHM ETIC 91

Even though, in the case of the additions, the CNI answers were rejected
more slowly than the interferent and RNI answer by both groups of subjects, an
interference effect was observed in the expert group: the interferent answers
were rejected more slowly than the RNI answers. Two comments can be made
concerning this phenomenon. First, this effect was observed only in the experts,
who were also the only participants to reject the interferent answers more slowly
in the case of the multiplications. It would thus appear that the activation of
irrelevant responses through the simple presentation of the operands depends on
the level of expertise. Second, the interference effect on additions in the expert
subjects was observed only when we compared the RTs for the rejection of the
answers remote from the correct result (i.e., interferent and RNI). In other words,
the interference effect was observed for additions only whe n the split was
constant, which was not the case for the multiplications for which the interferent
answers took the longest time to be rejected by experts.
It is possible that the retrieval of the answer of the multiplication from
memory produces an interference effect only when the answer of the addition is
rejected on the basis of its implausibility (e.g., 6 + 4 = 26). In this case, the
calculation of the addition would not be triggered. In effect, it can be useful to
check by means of calculation that 9 + 7 is not equal to 17. In contrast, such a
calculation seems less necessary in order to verify that 9 + 7 is not equal to 63 or
65. We hypothesised a weak interference effect for additions because the pro-
gress of a procedure should be only slightly disrupted by the activation of
knowledge in a declarative network. Indeed, CNI answers were far longer to
reject than interferent answers. However, it appears that if the implementation of
the additive procedure is supplanted by a rapid evaluation of the (im)plausibility
of the result, then the decision to reject the answer is slowed down by the
activation of the product of the two operands in LTM.
This suggests that participants possess additive and multiplicative networks
which are activated by the presentation of the operands and that the appearance
of interference effects depends, among other things, on the level of expertise and
the strategies used, some of which are more sensitive than others. It nevertheless
remains the case that the only situation in which the interference effect sup-
planted the split effect was the evaluation of false multiplications by experts.
This is in line with our hypotheses.
GENERAL DISCUSSION
Overall, the results of the three experiments presented here confirm our main
hypothesis that simple multiplication and addition problems mobilise different
strategies. The former appear to be solved by means of direct retrieval of the
answer from memory, while the latter are solved through the use of procedur e
and declarative knowledge concerning additive facts. The priming effect
resulting from the preliminary presentation of the sign and the size effect were
92 ROUSSEL, FAYOL, BAR ROU ILLET

stronger for additions than for multiplications. The presentation of interferent
results affected the two types of operation in different ways. Taken as a whole,
these results would not appear to be very compatible with the hypothesis that the
two types of operation are solved by the same process, namely the direct
retrieval of the results from memory.
Alternative hypothesis
As far as the priming effect due to the + sign is concerned, the hypothesis of a
retrieval strategy which is shared by the two operations could be retained if we
suppose that the additive and multiplicative networks differ in their accessibility
and the strength of their associations. In this event, the additive network would
contain the stronger associations, which is highly plausible given that it is
constructed earlier. The greater time gain resulting from the preliminary pre-
sentation of the + sign would then result from the higher level of preactivation of
the additive network and not from the instantiation of a procedure. This
hypothesis would also make it possible to explain why additive facts interfere
more strongly in the rejection of false multiplications than do multiplicative
facts in the rejection of additions. Though plausible, this hypothesi s squares
uneasily with the fact that the additions were solved more slowly than the
multiplications in the three experiments. In effect, all the current associative
network models predict that retrieval will be faster the stronger the associations
are. According to this hypothesis, additions should therefore be solved more
quickly than multiplications. However, this is not what was observed.
Similarly, the fact that the size effect was greater for the additions than for the
multiplications is not incompatible with the hypothesis that a retrieval process is
used for both operations. Zbrodoff (1995) has suggested that, given a high level
of practice, the size effect can be explained by the conjunction of two factors:
strength of associations and similarity-based interference. The strength of
associations depends on the frequency, resulting presumably in stronger asso-
ciations for small than large problems. In contrast, the potential for similarity-
based interference is equal for small and large problems. Thus, if we suppose, as
Zbrodoff (1995) did, that there are differences in strength, stronger problem s
(i.e., small) will interfere more with weaker problems (i.e., large) than vice
versa, hence the size effect. Now, we might suppose that differences in strength
are larger for additions than multiplications, resulting in a stronger size-effect
for the former. For example, small additions and multiplications could have
equal strength (their verification times were approximately equal) whereas large
multiplications could be stronger than large additions. This difference in the
strengths of the associations would also explain why the large multiplications
were verified more quickly than the large additions. Unfortunately, this
hypothesis is incompatible with a higher level of preactivation resulting from the
preliminary presentation of the sign since it would lead us to suppose that,
PROCEDURES A ND RETRIEVAL IN A RITHM ETIC 93

overall, the strength of the associations is less in the additive than in the
multiplicative network.
Thus, even though in isolation the observed facts can be explained in terms of
the hypothesis that both additions and multiplications are solved by means of
memory retrieval, it does not appear to be possible to make this hypothesis
compatible with the entire set of observed phenomena. On the contrary, these
phenomena are compatible with the hypothesis that additions are frequently
solved by means of a procedura l mechanism wherea s multiplications are mainly
solved by a direct memory retrieval process.
At the end of Experiment 1, we suggested three reasons which might explain
why adults use a procedure to solve addition problems: (1) the fact that the
additive procedure is constructed at an early age and that it precedes the creation
of a network of number facts, (2) the existence of strategy selection processes
that involve factors other than the simple relative speed of the available
strategies, and (3) adaptation to a specific situation (mixed block presentation),
which causes subjects to favour a reliable, robust strategy (counting) to a faster,
more habitual strategy (retrieval). Experiment 2 allowed us to discard this third
possibility. Experiment 3 provided evidence that the main effects reported in
Experiments 1 and 2 were due neither to expertise nor to a particular sample of
problems. The remainder of this discussion will therefore concentrate on points
(1) and (2).
The construction of additive procedures and their
early appearance
Counting provides the foundation for the development of basic arithmetic skills
(Kaye, 1986; Resnick, 1983). As of the age of 3–4 years, when determining the
cardinality of a set or solving additions or subtractions, children count one step
at a time up or down through the numberline and match the number names to the
objects they are counting, even before receiving any forma l tuition (Baroody &
Ginsburg, 1986; Fuson, 1988; Fuson, Richards, & Briars, 1982; Groen &
Resnick, 1977; Hatano, 1982; Siegler & Jenkins, 1989). Thus, early interact ions
with numbers are based on counting and result in the development and use of
numberline associations.
These associations between successive items in the verbal sequence of
numbers appear at a very early age and are very frequently used, a fact which
explains the occurrence of the errors reported by Siegler (Siegler & Jenkins,
1989; Siegler & Shrager, 1984): children aged 4–5 years who are asked to solve
the problem 3 + 4 sometimes respond 5 which is the next item in the numberline.
Such errors again occur at 8–10 years (grades 3 and 4) in a number matching
task. When, following the presentation of a pair of digits (e.g., 3 and 4), they are
asked to decide whether a target item was present (e.g., 3, 7, 5, or 9), the
children take longer to reject the target item that immediately follows the second
94 ROUSSEL, FAYOL, BAR ROU ILLET

of the presented digits (i.e., 5; LeFevre et al., 1991). Despite the fact that in adult
subjects, this slowing effect is observed most frequently when the item to be
judged corresponds to the sum of the presented digits (LeFevre et al., 1988), the
following-digit effect is still observed, whatever the subject’s level of ability
(LeFevre et al., 1991; LeFevre & Kulak, 1994).
These facts indicate that the addition procedure which consists of counting
one at a time is one of the first numerical skills learned by children. Its early
appearance and frequent use probably mean that it is a reliable and rapid
strategy. The results obtained by Barrouillet and Fayol (1998) indicate that
second graders still make very great use of it even when faced with simple
additions (from +1 to +4). It is very possible that this procedure attains a high
degree of efficiency and speed of implementation, allowing it to compete as a
strategy with direct memory retrieval (Baroody, 1994; Little & Widaman, 1995).
Moreover, its field of application means that it can never be totally replaced by a
memory retrieval strategy. In effect, a problem such as 198 + 3 is likely to be
solved faster by counting on three in the numberline than by retrieving the
number fact 8 + 3 = 11 and adding this result to 190.
The problem of strategy selection
The problems relating to strategy selection have given rise to a number of
hypotheses. One initial hypothesis is to assume that subjects always start by
trying to retrieve the answer from memory (distributions of associations
model; Siegler, 1987). An algorithmic strategy would not be mobilised unless
retrieval fails, either because none of the memorised answers meets the sub-
jects’ confidence criterion or because the search for the answer is excessively
long (Siegler, 1987). A second hypothesis is to suppose that the two strategies
are in competition and that the fastest wins (race model; Logan, 1988). A
third hypothesis has recently been proposed by Siegler and Shipley (1995) in
their Adaptive Strategy Choice Model (ASCM). According to these authors,
subjects select their strategy (retrieval vs different back-up strategies) on the
basis of information concerning each of the strategies (speed of execution,
precision in the past, mean estimates of these parameters for all the problems
of a given type, etc.). This information determines the strength of the associa-
tion between the problem to be processed and each of the available strategies.
The strategy with the greatest strength will then be chosen. However, accord-
ing to the authors, the procedural solution of additions should lead to the per-
fect memorisation of the associations between problems and answers and to
the systematic retrieval of additive facts (in 99%of cases in the computer
simulation of the model). In effect, the retrieval strategy should almost always
be the fastest and should normally yield precise answers when used (1995,
p. 61). Thus Siegler and Shipley’s model leads to the same predictions as the
race models.
PROCEDURES A ND RETRIEVAL IN A RITHM ETIC 95

The ASCM approach is akin to the last version of the ACT-R model
(Anderson & LebieÁre, 1998) that assumes that the select ion between strategies
(namely productions) results from the resolution of a conflict between all the
productions that match the current goal. This conflict resolution depends on the
expected gain of each production. Expected gain is a function of both the
expected probability that the goal will be achieved and the cost to achieve this
goal if a given production rule is selected. This cost is in fact an estimation of
the time to achieve the goal. Actually, algorithmic strategies for additions would
be selected only when the strategy of retrieval of a declarative chunk has failed
because the former have higher cost (i.e., are slower) and then are lower valued
in the conflict resolution (Anderso n & LebieÁre, 1998, p. 102). Indeed, the
ACT-R 4.0 lifetime simulation on additive problem solving resulted in a
ubiquitous use of retrieval: ACT-R 4.0 solved only 0.1%of the problem s by
computation.
Thus, all the models presented here predict that adults will almost system-
atically select direct memory retrieval strategy for additions. Indeed, they all
assume that when an item of knowledge which encodes the answer of a problem
is accessible, this knowledge is systematically retrieved, either because retrieval
is tried first (Siegler & Shrager, 1984), or faster than the use of a procedure
(Anderson & LebieÁre, 1998; Logan, 1988), or because the association between
the problem and the answer is stronger than that between the problem and the
procedure (Siegler & Shipley, 1995). Though such a consensus is quite
impressive, there are at least two remaining unanswered questions.
First, the available simulations of adult behaviour do not fit data very well.
Both ASCM and ACT-R 4.0 simulations lead to more than 99%of problems
being solved through retrieval in adults. However, LeFevre, Sadesky, and Bisanz
(1996) found that adults solved about 30%of additions by non-retrieval pro-
cedures, especially when large operands are involved (i.e., problems with sums
greater than 10). In this latter case, about half of the problems are solved using
algorithmic strategies. Of course, this discrepancy casts doubt on the general
validity of the models these simulations implement, but it raises a more thorny
question. Though being inaccurate in predicting the type of strategy adults use
when solving large problems, they none the less simulate the size problem effect.
For example, in ACT-R 4.0, the size effect results from the relative frequency of
occurrence of the addends, the smallest being more frequently encountered than
the largest. Thus, if these simulations account for the size effect by implementing
strategies that do not correspond to the strategies used by adults, we can also
suppose that their account of the size effect is questionable.
Second, all the models we previously reviewed explain the prevalence of
retrieval by its assumed superiority in speed. Are we so confident that algo-
rithms are slower than retrieval? It is generally assumed that small problems are
solved by retrieval. Thus, we can assume that algorithmic procedure s are slower
than retrieval when small operands are to be processed. On the other hand, it is
96 ROUSSEL, FAYOL, BAR ROU ILLET

generally assumed that algorithmic procedures are used, if so it be that they are,
more often for large than for small problems. Thus, we are faced with a kind of
paradox because algorithmic strategies are all the slower the larger the operands
they process. As a consequence, it should be assumed that algorithmic strategies
supersede fact retrieval just when their use is the slowest, that is for large
operands. Of course, it could be argued that procedures are used because
retrieval failed, and that retrieval fails more often for large problems because
they result in weak operand–answer associations (large operands are less fre-
quent than small operands and the algorithmic procedures that lead to memor-
isation of operand–answer associations fail more often for large operands).
However, the available models and simulations do not account for this pheno-
menon, at least in adults, because they postulate (1) that algorithmic strategies
are necessarily slower than retrieval, and (2) that retrieval is a highly efficient
strategy even for large problems. We could also suppose that learning results in
algorithms that are extremely efficient and can compete with retrieval whatever
the size of the operands (Baroody, 1983).
Concluding comments
We have seen that our results were in line with our model that suggest s that
multiplications are solved by means of direct retrieval of the answer from
memory, whereas additions are more often solved by algorithmic procedures.
However, two facts lead us to temper our conclusions. First, some of our results
are at odds with previous observations in cognitive arithmetic. Second, the fact
that the negative SOA advantage for the additions was also observed in small
problems can be considered surprising. Indeed, we assume that the anticipated
presentation of the operator primes the algorithmic procedure but it is usually
assumed that these problems are solved by retrieval.
As far as the first point is concerned, we found that multiplications took
shorter to verify and elicited a lower size effect than additions, whereas reversed
effects are usually reported in the literature. It should be noted that our results
were obtained through a verification task, and it is possible that they are due to
peculiarities linked to this paradigm. Indeed, contrary to Ashcraft’s (1982) claim
that additions are verified through a sequential process of retrieval and com-
parison with the proposed answer, it has been argued that verificati on of mul-
tiplications does not involve a retrieval strategy but a familiarity-based strategy
(Campbell & Tarling, 1996). This familiarity-based strategy would result in a
reduced size effect for multiplications but not for additions (Campbell, 1987b).
Thus, the difference we observed in size effect between these operations can be
due to ad hoc strategies the participants use in verification tasks. In the same
way, different priming effects of the proposed answer on multiplications and
additions could explain why multiplications were more quickly verified than
additions in our experiments.
PROCEDURES A ND RETRIEVAL IN A RITHM ETIC 97

However, some of the characteristics of our results can not be explained by
the paradigm we used. Indeed, Zbrodoff and Logan (1986) observed that mul-
tiplications took longer to verify than additions in a verification task. This
discrepancy in results could be explained by differences in arithmetic back-
ground knowledge between populations. Even when we used a large range of
operations, Frenc h adult experts, university students (Experiment 3), and fifth
graders (Roussel, 2000) all exhibited the same pattern of reaction times with
multiplications consistently faster than additions. We can just note that multi-
plications are easier to verify than additions for French people, a fact probably
due to the way they learn the two operations. These kinds of differences between
populations have already been observed. Geary (1996) found that Chinese
children and adults were faster and less sensitive to the size effect than their
North-American peers in additive problem solving. For example, Chinese third
graders solved small additions more than 1 second faster than their North-
American peers (mean reaction times of 891 and 1893 ms respectively; 847 and
2278 ms for the large problems)! Thus, performances in cognitive arithmetic
depend strongly on the frequenc y of exposure to arithmetic in school.
Concerning the second point, we predicted that the procedure involved in
addition problem solving should be primed by the anticipated presentation of the
operator. As a consequence, the negative SOA advantage should be stronger for
problems frequently solved by computational means (i.e., the large problems).
However, we also observed such an effect for small additions (p<.06) that are
probably solved by retrieval. Indeed, the reaction times for small additions and
multiplications were approximately equivalent and we hypothesised that the
multiplications were solved by retrieval. Three explanations can be put forward
to account for this surprising result. First, we could suppose that the anticipated
presentation of the operator activates the procedure even for small additions
because the operands are not yet available, whereas a retrieval strategy is used
for these additions at SOA 0. In this case, it is possible that a preactivated
procedure is faster than the retrieval process. However, there is no way to test
directly this kind of hypothesis.
Second, the negative SOA advantage for small additions can result from the
sporadic use of a procedur e to solve some small problems, even by a minority of
participants. We tested this point in Experiment 2. Assuming that the procedure
is slower than the direct retrieval of results from memory, the participants who
exhibited the highest RTs for the verification of small additions at SOA 0 should
be the ones who make most frequent use of a procedural strategy. In contrast, the
fastest participants would tend to retrieve the results directly from memory.
According to our hypothesis, the participants who make most frequent use of the
procedure when verifying additions (and who are therefore probably the slowest
subjects) should exhibit greater reductions in verification time at negative SOAs.
The fastest participants, who we assume rarely (or never) use the procedure,
should draw very little benefit from the preliminary presentation of the sign
98 ROUSSEL, FAYOL, BAR ROU ILLET

because this cannot accelerate memory retrieval. Thus, we distinguished four
groups of participants from their mean RTs on small additions at SOA 0. The
three upper quartiles (mean RTs on additions at SOA 0 of 715, 806, and 907 ms
respectively) did not exhibit any significant negative SOA advantage. However,
the participants in the lower quartile who were the slowest in verifying small
additions (mean = 1057 ms) benefited from the anticipated presentation of the +
sign. Their overall mean RT at the negative SOAs (¡300 and ¡150 ms) was
lower than at SOA 0 (977 and 1057 ms respectively) but the difference was not
significant, F(1, 7) = 3.19, p= .12, MSe = 7936. Thus, it is possible that the
negative SOA advantage for small additions results from some rare participants
who used algorithmic strategies for a restricted set of small additions. In
agreement with this, LeFevre, Sadesky, and Bisanz (1996) observed that adults
solved more than 10%of the small additions through non-retrieval strategies
(mainly counting) .
Finally, it is possible that the differential effect of the anticipat ed presentation
of the operator is peculiar to the verification task, and further studies using a
production task are needed. However, even if it turned out that this effect is
peculiar to verification processes, it still needs to be accounted for by cognitive
arithmetic models. As Campbell (1987b) pointed out, verification is not the
villain and productio n the champion of cognitive arithmetic research. The
additive operator effect we observed is probably indicative of the processes
involved in arithmetic problem solving.
Manuscript received September 1999
Revised manuscript received July 2000
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APPENDICES
1. List of presented problems and answers in
Experiments 1 and 2
Answers for additio n Answers for multiplication
Operands True FNI FI True FNI FI
2 7 9 10 14 14 13 9
3 6 9 8 18 18 19 9
5 3 8 7 15 15 14 8
8 2 10 11 16 16 17 10
4 9 13 14 36 36 37 13
6 8 14 13 48 48 47 14
9 7 16 17 63 63 64 16
8 4 12 11 32 32 31 12
FNI: false non-interferent; F I: false interferent.
PROCEDURES A ND RETR IEVAL IN A RITHM ETIC 103

2. List of presented problems and answers in
Experiment 3
Answers for addition Answers for multiplication
Pairs of operands True Int CNI RNI True Int CNI RNI
Small
2 3 5 6 4 8 6 5 8 4
2 4 6 8 7 10 8 6 10 7
2 5 7 10 6 8 10 7 8 6
2 6 8 12 9 14 12 8 14 9
2 7 9 14 8 12 14 9 12 8
2 8 10 16 11 18 16 10 18 11
2 9 11 18 10 16 18 11 16 10
3 4 7 12 8 14 12 7 14 8
3 5 8 15 7 13 15 8 13 7
3 6 9 18 10 20 18 9 20 10
3 7 10 21 9 19 21 10 19 9
3 8 11 24 12 26 24 11 26 12
4 5 9 20 8 18 20 9 18 8
4 6 10 24 11 26 24 10 26 11
Large
3 9 12 27 11 25 27 12 25 11
4 7 11 28 12 30 28 11 30 12
4 8 12 32 11 30 32 12 30 11
4 9 13 36 14 38 36 13 38 14
5 6 11 30 10 28 30 11 28 10
5 7 12 35 13 37 35 12 37 13
5 8 13 40 12 38 40 13 38 12
5 9 14 45 15 47 45 14 47 15
6 7 13 42 12 40 42 13 40 12
6 8 14 48 15 50 48 14 50 15
6 9 15 54 14 52 54 15 52 14
7 8 15 56 16 58 56 15 58 16
7 9 16 63 15 61 63 16 61 15
8 9 17 72 18 74 72 17 74 18
Int: Interferent answer; CNI: Close Non-Interferent; RNI: Remote Non-Interferent.
104 ROUSSEL, FAY OL, BA RROU ILLET