Working memory demands of exact and approximate addition

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Working memory demands ofexact and approximate additionDarren A. Kalaman a & Jo-Anne Lefevre aa Carleton University , Ottawa, Ontario, CanadaPublished online: 22 Jan 2007.

To cite this article: Darren A. Kalaman & Jo-Anne Lefevre (2007) Working memorydemands of exact and approximate addition, European Journal of Cognitive Psychology,19:2, 187-212, DOI: 10.1080/09541440600713445

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Working memory demands of exact and approximate

addition

Darren A. Kalaman and Jo-Anne LeFevre

Carleton University, Ottawa, Ontario, Canada

We compared the working memory requirements of two forms of mental addition:

exact calculation (e.g., 63�/49�/112) and approximation (e.g., 63�/49 is about 110).

In two experiments, participants solved two-digit addition problems (e.g., 63�/49)

alone and in combination with a working memory task (i.e., remembering four

consonants). In Experiment 1, participants chose an answer from two alternatives

(e.g., exact: 112 vs. 122; approximate: 110 vs. 140). In Experiment 2, participants

responded verbally with exact or approximate answers. In both experiments, the

working memory load impaired exact and approximate addition performance, but

exact addition was affected more. Load also impaired performance on problems

with a carry operation in the units (e.g., 28�/59 or 76�/57) more than on problems

without a unit carry (e.g., 24�/53 or 76�/52). These results identify the carry

operation as the source of the working memory demands in multidigit addition.

When people solve arithmetic problems such as 38�/27, they can produce an

exact (i.e., 65) or an approximate solution (i.e., about 70). How are exact

calculation and approximation different? El Yagoubi, Lemaire, and Besson

(2003) argued that people use different strategies to solve exact and

approximate arithmetic problems (see also LeFevre, Greenham, & Waheed,

1993; Siegler & Booth, 2005). In accord with this possibility, researchers who

have used brain imaging or event-related potentials to study exact and

approximate arithmetic have found that brain activation differs during these

two forms of arithmetic (Dehaene, Spelke, Pinel, Stanescu, & Tsivkin, 1999;

Correspondence should be addressed to Jo-Anne LeFevre, Centre for Applied Cognitive

Research, Department of Psychology, Carleton University, Ottawa, Ontarion, Canada K1S 5B6.

E-mail: [email protected]

This research was funded by the Natural Sciences and Engineering Research Council of

Canada through a PGS-A scholarship to DK and a Discovery Grant to JL. We thank the

Mathematical Cognition group at Carleton University and Diana DeStefano for their helpful

comments on earlier versions of this work. DK is also grateful to Katherine Arbuthnott for her

contributions to research that preceded the work reported in this paper. A version of these results

were presented at the annual meeting of the Canadian Society for Brain, Behaviour, and Cognitive

Science, St. John’s, Newfoundland, June 2004.

EUROPEAN JOURNAL OF COGNITIVE PSYCHOLOGY

2007, 19 (2), 187�212

# 2006 Psychology Press, an imprint of the Taylor and Francis Group, an informa business

http://www.psypress.com/ecp DOI: 10.1080/09541440600713445

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El Yagoubi et al., 2003; Stanescu-Cosson et al., 2000). Stanescu-Cosson

et al. (2000) reported that retrieval of small arithmetic facts (e.g., 3�/2, 4�/1)resulted in greater activation of left hemisphere regions, whereas approx-

imation and exact calculation of larger facts (e.g., 9�/6) produced activation

of both left and right parietal regions (see also Dehaene, Piazza, Pinel, &

Cohen, 2003). Given the findings that exact and approximate arithmetic

appear to engage different combinations of mental processes, and that

approximate arithmetic is typically easier than exact arithmetic, we

hypothesised that exact arithmetic would be more demanding of working

memory resources than approximate arithmetic.Research on the cognitive processes involved in exact calculation has

shown that problem complexity for multidigit problems is linked to working

memory demands (reviewed by DeStefano & LeFevre, 2004; LeFevre,

DeStefano, Coleman, & Shanahan, 2005). In particular, more difficult

problems (e.g., those requiring a carry such as 43�/59) involve more working

memory resources than similar no-carry problems (e.g., 43�/52; Furst &

Hitch, 2000; Seitz & Schumann-Hengsteler, 2000, 2002). Working memory

refers to those processes that control, regulate, and maintain specificinformation during the execution of complex cognitive tasks (Ashcraft,

1995; Baddeley, 1996; DeStefano & LeFevre, 2004; LeFevre et al., 2005).

Although there is a variety of working memory models (Miyake & Shah,

1999), the majority of research on working memory and arithmetic uses

Baddeley’s multicomponent model as a framework (Baddeley, 1986, 1996,

2000, 2002; Baddeley & Hitch, 1974). Baddeley’s model consists of four

components: the central executive, the phonological loop, the visual-spatial

sketchpad, and the episodic buffer. Functions of the central executiveinclude the coordination of simultaneous activities, activation of informa-

tion from long-term memory, and inhibition of irrelevant information

(Baddeley, 1996). The phonological loop provides temporary storage and

processing of phonological information. The visual-spatial sketchpad

provides temporary storage and processing of visual and spatial information

(Logie, 1995). The episodic buffer is hypothesised to combine information

from long-term memory with information from the phonological loop and

the visual-spatial sketchpad (Baddeley, 2002). Although all of thesecomponents of working memory may be involved in arithmetic (LeFevre

et al., 2005), in the present research we focused on the role of the central

executive and the phonological loop in exact versus approximate calculation.

To produce the exact answer to a problem such as 38�/27, the standard

(right-to-left) calculation algorithm requires (a) the addition of 8�/7�/15,

(b) retention of the unit part of the answer (5), (c) carrying the 10 portion,

(d) the addition of 10�/30�/20�/60, and (e) assembly of 60 and 5 to produce

an answer. In contrast, LeFevre et al. (1993) found that when people wereasked to provide approximate solutions to multidigit multiplication

188 KALAMAN AND LEFEVRE

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problems, they simplified the numbers to allow for less complex calculations.

For example, simplification might involve rounding 38�/27 to 40�/30�/70(see also Lemaire & Lecacheur, 2002; Lemaire, Lecacheur, & Farioli, 2000).

On this view, exact arithmetic requires more calculations and more storage

of interim results during the solution process than approximate arithmetic.

LeFevre et al. (2005) suggested that as problem complexity increases for

arithmetic problems, so do the working memory demands. On this view,

exact arithmetic may be more difficult than approximate arithmetic because

the former puts a greater demand on the processing and storage capacity of

working memory. The central executive is assumed to control the proceduresinvolved in calculation and to provide cognitive resources to maintain

interim results whereas the interim results are likely to be stored in the

phonological loop (DeStefano & LeFevre, 2004).

Recent reviews of working memory and mental arithmetic suggest that

calculation demands central executive resources, even when the problems are

simple single-digit sums such as 3�/4 (DeStefano & LeFevre, 2004; LeFevre

et al., 2005). Further, problem complexity in the form of more calculations

or operations such as carrying or borrowing seem to increase workingmemory demands (Ashcraft & Kirk, 2001; DeStefano & LeFevre, 2004;

Furst & Hitch, 2000; Seitz & Schumann-Hengsteler, 2000, 2002; Seyler,

Kirk, & Ashcraft, 2003). On this view, we hypothesised that problems

involving carrying from the units to the tens (e.g., 37�/48) would demand

more working memory resources (both storage and processing) than similar

problems that did not require unit carries (e.g., 37�/41). In the present

research a dual-task paradigm was used to explore the working memory

requirements of exact and approximate addition. Participants solvedaddition problems such as 37�/56. They also performed a letter memory

task, both alone and in combination with addition problems. For the

memory load, they remembered four consonants (e.g., XPTZ). In the dual-

task condition, they memorised the letters, then performed the arithmetic

task, and then recalled the letters. According to dual-task logic, if letter

memory and arithmetic require the same working memory resources, then

performance on one or both tasks should be worse when they are performed

together, as compared to when they are performed alone.Letter recall was also used by Ashcraft and Kirk (2001, Exp. 1) and by

Seyler et al. (2003, Exp. 3) to assess the working memory demands of

addition and subtraction, respectively. Presumably, letter memory involves

both the phonological loop and the central executive (see also Seitz &

Schumann-Hengsteler, 2000), as does calculation of exact arithmetic

problems. The two tasks combined may therefore compete for working

memory resources. Central executive resources are required to coord-

inate among task demands, execute the rehearsal process, and to maintainthe serial order of the letters (Jones, Farrand, Stuart, & Morris, 1995). The

EXACT AND APPROXIMATE ADDITION 189

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phonological loop is likely used to store interim results. Thus, if letter recall

implicates working memory processes, then we expect that exact calculation

will be more affected than approximation by a letter load. Further, if

working memory demands vary with problem complexity, then we expect to

find that carry problems require more working memory resources than no-

carry problems.

A second objective of this research was to explore patterns of working

memory demands across variations in task requirements. Participants in

Experiment 1 chose between a correct and an incorrect answer (Dehaene

et al., 1999), whereas participants in Experiment 2 produced an answer. The

actual problems were the same in the two experiments, however. The

comparison between experiments allowed us to determine whether relations

between performance and working memory demands varied with specific

task requirements. To the extent that the working memory demands of

arithmetic occur during the calculation stage, no differences would be

expected because calculation should be required in both Experiments 1 and

2. However, if the choice task allows participants to circumvent or reduce the

demands of calculation, or if the simplified response requirements for the

choice task (i.e., press one of two keys vs. formulate a spoken response) then

we might expect a different pattern of working memory demands in

Experiment 1 as compared to Experiment 2.

EXPERIMENT 1

In this experiment, participants performed two different tasks, multidigit

addition and letter memory, alone and in combination. Letter memory was

assumed to involve central executive aspects of working memory and the

phonological loop. We hypothesised that exact arithmetic would be more

demanding of working memory than approximate arithmetic. However,

previous research has shown that problems requiring carries demand more

central executive resources than problems that do not require carries. To the

extent that carry problems are more affected by letter load for both

approximate and exact arithmetic, we hypothesised that working memory

demands would vary with the requirement to carry from the units to the tens.

Method

Participants

Twenty-four adults (16 males and 8 females) were selected for Experiment

1 based on their responses to a brief questionnaire that was distributed to all

Introductory Psychology students at the beginning of the fall academic term.

Students were asked about their skill level in solving basic arithmetic


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problems (e.g., addition, subtraction, multiplication, and division) and how

often they solved arithmetic problems in ways other than retrieving theanswers from memory. For the purpose of the study, participants were

selected if they rated themselves from moderate to high skill in solving basic

arithmetic problems. The selection criteria ensured that participants would

be reasonably comfortable solving two-digit addition problems. Participants

ranged in age from 18 to 32 years, with a median age of 20 years.

Participants received course credit (as partial fulfilment of a course

requirement) in exchange for their participation. Twenty-two of the

participants reported that all of their schooling had been in Canada; tworeported that they were schooled elsewhere.

Apparatus

An IBM Intel Pentium computer was used to present stimuli on a com-puter screen. Participants sat approximately 70 cm in front of the computer

screen. Responses were recorded using an IBM compatible keyboard.

Materials

Addition problems. There were eight sets of 12 two-digit addition

problems. Each set of addition problems had sums in the range from 41

to 159, with one problem from each sum decade (e.g., 25�/17�/42, 20�/37�/

57, 49�/14�/63) in each set. Half of the problems in each set involved a carry

operation in the ones column (e.g., 13�/38) and half did not (e.g., 47�/11).

Half of the problems were presented with the smaller operand first (e.g., 13�/

38) and half were presented with the larger operand first (e.g., 47�/11). Halfof the sums in each set were odd (e.g., 51) and half were even (e.g., 58). Each

problem was created with two correct solutions and two incorrect solutions.

For example, for the problem 13�/38, the correct answer in the exact

condition was 51. The incorrect answer was plus or minus 10 from the exact

answer, so 41 or 61. In the approximate condition, the correct answer was 50.

The incorrect solution was plus or minus 30 from the approximate answer, so

20 or 80 in this example. Approximately half of the incorrect solutions were

greater than the correct and approximate answers. Sets of addition problemswere assigned to working memory conditions, but were counterbalanced

across arithmetic tasks (exact vs. approximate calculation).

Letter sequences. Letter sequences were assigned to the addition

problems (e.g., RDCG). All letter sequences consisted of only consonants,

and each letter in a sequence appeared once. The letter sequences were not

counterbalanced across the addition problem sets. Thus, a particular

addition problem (e.g., 25�/17) was paired with a particular letter sequence(e.g., RDCG).


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Paper-and-pencil tasks. Participants completed two paper-and-pencil

tasks at the conclusion of the experiment. First, participants completed

the addition and the subtraction-multiplication subtests from the French Kit

(French, Ekstrom, & Price, 1963). The number of correct answers on each

page of the test were added together to form a total fluency score. Second,

participants completed the Math Background and Interests Questionnaire

(LeFevre, Smith-Chant, Hiscock, Daley, & Morris, 2003). The questionnaire

asked participants about their background, experience, attitudes, and beliefs

about mathematics. The questionnaire also asked participants whether they

had experienced any head injuries (e.g., concussion) or neurological

disorders (e.g., seizure). Participants in this experiment did not report any

seizures or severe head injuries.

Procedure

The computer portion of the session consisted of 24 practice trials and 72

experimental trials. Participants either received the exact addition condition

trials (12 practice trials, 36 experimental trials) followed by the approximate

addition condition trials (12 practice trials, 36 experimental trials) or the

approximate addition condition trials followed by the exact addition

condition trials. Participants completed three tasks for the exact addition

condition and three tasks for the approximate addition condition (each task

consisted of 4 practice trials and 12 experimental trials). The task order

(single-task addition, single-task letter naming, and dual-task) was com-

pletely counterbalanced across participants.

Single-task addition. For the two single-task addition conditions,participants chose either the exact answer from two options (the correct

answer and the wrong answer) or the approximate answer from two options

(the approximate answer and the wrong approximate answer). Participants

were asked to solve all addition problems as quickly and as accurately as

possible. Each trial began with a black asterisk (*) centred on the computer

screen. The asterisk remained on the computer screen for 1000 ms and was

followed by a 1000-ms blank interval. A four consonant letter sequence then

appeared in black font (e.g., RDCG) for 2500 ms. The experimenter told

participants to speak aloud each letter of the sequence but participants were

not required to memorise the letters. The presentation of the letters was

followed by a 1000-ms blank interval. Following the blank interval, an

addition problem presented in black (Courier 18 point) appeared centred on

the computer screen. Each addition problem appeared with two possible

answers in black font below the addition problem. The two possible answers

were positioned with 10 spaces between them such that one possible answer

was more to the left side of the screen and the other possible answer was


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more to the right side of the screen. For exact problems, participants were

told to choose the answer that was correct. For approximate additionproblems, participants were told to choose the answer that was closest (in

size) to the correct answer. Participants indicated their choice by pressing

either the ‘‘1’’ or the ‘‘3’’ key on the numeric keypad with their right hand. If

participants decided that the possible answer to the left side of the screen

was correct, they pressed the ‘‘1’’ key. If participants decided that the

possible answer to the right side of the screen was correct, they pressed the

‘‘3’’ key. The correct response was presented on the right side of the screen

on half of the trials. Participants were asked to respond quickly butaccurately.

The addition problem with the possible answers remained on the screen

until participants pressed a key. The key press was followed by a 1000-ms

blank interval. Following the blank interval, the phrase ‘‘RECALL

LETTERS’’ was centred horizontally on the computer screen in black

font. Below the phrase, the letter sequence that appeared at the beginning of

the trial (e.g., RDCG) was centred on the computer screen in black font. The

experimenter told participants to speak aloud each letter of the sequence.The experimenter typed the letters on the keyboard as they were spoken. A

1000-ms delay occurred after the experimenter typed the last letter of the

sequence. Following the delay, the asterisk appeared, signalling the next trial.

Single-task letter recall. For the two single-task letter recall conditions,

participants memorised and recalled a four consonant letter sequence. The

procedure used here was the same as the procedure used for the single-task

addition condition, except for the following differences. First, whenparticipants saw the letter sequence after the asterisk appeared they were

to say each letter aloud and also to memorise the letter sequence. Second,

when participants saw the addition problem presented with the two possible

answers, the correct answer appeared with an asterisk (e.g., 20�/37�/57* vs.

67). The experimenter told participants to read the addition problem silently

and press the appropriate key (‘‘1’’ or ‘‘3’’) that corresponded to the answer

with the asterisk (e.g., ‘‘1’’). Third, when participants saw the phrase

‘‘RECALL LETTERS’’, the letter sequence did not appear. Instead,participants were required to recall aloud the letters in the order they were

initially presented. Participants were asked to respond quickly but accu-

rately.

Dual-task condition. For the two dual-task conditions, participants both

solved the addition problems (exact or approximate addition) and mem-

orised and recalled a four consonant letter sequence. Thus, participants had

to recall the letter sequence from memory and provide an answer to theaddition problems by pressing either the ‘‘1’’ or the ‘‘3’’ key on the computer


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keyboard. Participants were asked to respond as quickly but accurately as

possible on both tasks.

Paper-and-pencil tasks. Participants were given 2 min per page to

complete the arithmetic fluency test. Two pages contained addition problems

and the other two pages contained a combination of subtraction and

multiplication problems. Participants were told to solve the problems as

quickly and as accurately as possible. Finally, participants completed the

Math Background and Interests Questionnaire. Participants were told to

answer the questions as honestly as possible. Most participants took lessthan 5 min to complete the questionnaire.

Results

The mean score on the arithmetic fluency test was 76.8 (SD�/21.1). Thus,

consistent with the recruiting criteria, participants’ arithmetic skills did not

significantly differ from the expected mean for this population (i.e., 80),t(23)�/�/0.73, p�/.05 (LeFevre et al., 2003). Furthermore, groups of

participants who received different task orders (exact�approximate vs.

approximate�exact) did not differ in fluency scores, t(22)�/�/0.79, p�/.05.

Participants completed a total of 1152 trials where responses to arithmetic

problems were recorded (i.e., 576 single-task arithmetic and 576 dual-task

trials). Only trials for which participants chose the appropriate answer (exact

or approximate answer) were included in the analysis of response times (515

exact trials and 523 approximate trials). Following Seyler et al. (2003; seealso Trbovich & LeFevre, 2003), a combined error score was calculated for

dual-task trials. Trials were scored as errors if participants made an error on

either letter recall or addition (or both). Thus, to receive a correct score on a

dual-task trial, the arithmetic problem had to be solved correctly and all of

the letters had to be recalled correctly in order. For the single-task measure,

consistent with Seyler et al., participants’ error percentages from the two

single-task conditions were averaged. The separate analyses of errors on the

arithmetic task and the letter memory task are described in the Appendix.All of the effects that are reported in the combined analysis (for both this

experiment and Experiment 2) were significant in one or both of the separate

analyses. Thus, the combined analysis presents an integrated picture of the

results. Furthermore, the combined errors in the dual-task condition

represent the joint demands of the arithmetic and letter memory tasks and

thus are a more complete picture of the tradeoffs in performance across the

primary (arithmetic) and secondary (letter memory) tasks.

Combined errors and correct response times (arithmetic only) wereanalysed in separate 2 (order: exact-approximate vs. approximate-exact)�/2


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(task: exact, approximate)�/2 (complexity: no carry, carry)�/2 (load: single,

dual) ANOVAs, with repeated measures on the last three factors. The resultsfor the two dependent variables are discussed together, with an emphasis on

the combined errors because load effects (single and dual task differences)

occurred mainly in errors. Participants appeared to preserve arithmetic

response times across the load conditions. Note that, for both experiments,

error bars shown in all figures are the 95% confidence intervals based on the

mean-square error values for the displayed interaction (Masson & Loftus,

2003). Reported effects were significant at pB/.05, unless otherwise

indicated. In all figures, the interactions are shown for both dependentmeasures, even when the interaction is only significant for one of those

measures (discrepancies between measures are noted).

Participants solved approximate addition problems more quickly than

they solved exact addition problems (2123 vs. 2970 ms), F(1, 22)�/47.64,

MSE�/723,148, and they made fewer combined errors on approximate than

on exact problems (12% vs. 17%), F(1, 22)�/15.28, MSE�/85, pB/.05. Thus,

approximate addition was easier than exact addition. These findings are

consistent with the assumption that approximation engages somewhatdifferent processes than exact calculation.

Participants solved no-carry problems more quickly than they solved

carry problems (2170 vs. 2923), F(1, 22)�/26.76, MSE�/1,017,134, and

made fewer errors on no-carry than on carry problems (10% vs. 19%), F(1,

22)�/20.61, MSE�/162. These findings are consistent with other research on

multidigit exact addition (e.g., Furst & Hitch, 2000; Heathcote, 1994, Exp. 1;

Seitz & Schumann-Hengsteler, 2002) and support the assumption that the

choice task engages similar processes as the production tasks that haveusually been reported.

There were also significant interactions between task and complexity for

both latencies, F(1, 22)�/28.70, MSE�/266,104, CI�/218 ms, and for

combined errors, F(1, 22)�/4.603, MSE�/126, CI�/5%. On exact problems

(Figure 1, left panel), participants were slower and made more errors on

carry than on no-carry problems. For approximate arithmetic, in contrast,

the difference between no-carry and carry problems was not significant for

either latencies or combined errors. Thus, an important aspect of problemcomplexity for exact arithmetic, the presence of a carry in the units digits, is

a less important aspect of approximate arithmetic. These results support the

contention of El Yagoubi et al. (2003; LeFevre et al., 1993) that participants

use different strategies to solve exact and approximate arithmetic.

In support of the assumption that the letter recall and the arithmetic tasks

engaged some of the same working memory processes, participants made

more errors in the dual- than in the single-task condition (22% vs. 8%), F(1,

22)�/20.28, MSE�/456. Response latencies on the arithmetic problemsshowed the opposite pattern, however, in that participants solved addition


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problems more quickly in the dual-task than the single-task condition (M�/

2473 vs. 2620 ms), F(1, 22)�/5.22, MSE�/201,164. The effect for latencies

was qualified by the interaction of arithmetic task and load, F(1, 22)�/6.06,

MSE�/185,654, CI�/182 ms. [Note that the interaction for combined errorswas not significant, F(1, 22)�/1.90, MSE�/149, p�/.182]. As shown in

Figure 2 (left panel), participants were faster but made more errors in the

dual-task condition than in the single-task condition for approximate

arithmetic, suggesting that there was a speed�accuracy tradeoff in perfor-

mance between the two tasks. In contrast, although latencies on exact

problems did not vary with load, participants made substantially more

combined errors in the dual- than in the single-task condition. These results

provide support for the hypothesis that exact arithmetic requires more

working memory resources than approximate arithmetic.The interaction between complexity and load for combined errors is

shown in Figure 3 (left panel), F(1, 22)�/8.76, MSE�/169, CI�/5%. (Note

that the interaction is not significant for latencies.) Participants made

substantially more errors in the dual- than in the single-task condition, but

this difference was greater for carry than for no-carry problems. These

results are consistent with previous research on exact arithmetic (e.g., Furst

& Hitch, 2000) in which carry problems were more affected by a central

executive load than no-carry problems.

Because the effect of problem complexity was significant only for exactarithmetic (Figure 1), and the memory load only affected carry problems

Figure 1. Interactions between arithmetic task (exact vs. approximate) and problem complexity

(no-carry vs. carry) for response latencies on arithmetic problems (in ms) and combined errors

(percentage). Whiskers represent the 95% confidence intervals for the illustrated two-way

interactions.


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(Figure 3), we might expect to find a three-way interaction of task,

complexity, and load. This interaction was not significant, however, for

either latencies, FB/1, or combined errors, F(1, 22)�/2.56, MSE�/144, p�/

.124. Inspection of the combined error means for this interaction (see Table

1), indicated that carry problems in the exact condition were most affected

by the memory load. In general, these results support the view that the

complexity of an arithmetic problem is an important determinant of

working memory demands for exact arithmetic (LeFevre et al., 2005).

However, they also suggest that the processes required to solve carry

problems on approximate arithmetic problems may be somewhat more

demanding of working memory than those used to solve no-carry problems.

In this experiment, at least one and sometimes both of the operands in no-

carry problems were closer to the lower decade unit and thus could be

truncated or rounded down. Rounding down is a less demanding procedure

than rounding up (Lemaire & Lecacheur, 2002). Thus, carry versus no-carry

problems produce greater working memory demands for both exact and

approximate arithmetic, but for different reasons.These patterns are further clarified by examining some effects of order.

The main effects of task order (exact first, approximate second vs. exact

second, approximate first) were not significant, Fs(1, 22)�/1.17 and 0.22 for

latencies and combined errors, respectively. However, in the analysis of

Figure 2. Interactions between arithmetic task (exact vs. approximate) and working memory load

(single vs. dual) for response latencies on arithmetic problems (in ms) and combined errors

(percentage). Whiskers represent the 95% confidence intervals for the illustrated two-way

interactions.


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combined errors, order interacted with task and complexity, F(1, 22)�/9.02,

MSE�/126, and with Task�/Complexity�/Load, F(1, 22)�/4.00, MSE�/

145, CI�/5%, p�/.058 (means are shown in Table 2). Examination of the

means for these interactions suggests that the interaction of task and

complexity (i.e., the larger difference between carry and no-carry problems

for exact than approximate arithmetic) was stronger in the first block of trials

than in the second block and that the Task�/Load interaction (i.e., the larger

difference between single and dual task for exact than for approximate

addition) was also greater in the first than in the second block of trials. Thus,

TABLE 1Mean percentage of combined errors for each experiment: Interaction of task, load,

and complexity

Single Dual

Experiment Task No-carry Carry No-carry Carry

1 Exact 7.3 10.8 15.3 35.4

Approximate 5.2 13.9 7.2 21.5

2 Exact 24.6 28.8 43.8 66.0

Approximate 31.9 26.0 36.8 44.4

Figure 3. Interactions between problem complexity (no-carry vs. carry) and working memory load

(single vs. dual) for response latencies on arithmetic problems (in ms) and combined errors (per-

centage). Whiskers represent the 95% confidence intervals for the illustrated two-way interactions.


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participants were influenced by task order such that they solved exact and

approximate addition more similarly in the second than in the first block of

trials. Despite these order effects, exact arithmetic problems always showed asubstantial carry effect in the dual-task conditions whereas approximate

arithmetic only showed the effect when it was presented first. Similarly, there

were always substantial dual-task decrements on carry problems, but on no-

carry problems, the differences between single and dual-task performance

was smaller and not significant in some cases. Hence, the expected three-way

interaction of task, complexity, and load is more evident in the second block

of trials, indicating that participants may have refined their strategies with

practice. These carry-over effects do not appear to compromise the mainconclusions, although they may indicate that familiarity with the letter recall

task, as well as practice with the arithmetic problems, influences performance.

Discussion

We hypothesised that (1) exact calculation typically requires more processing

steps and therefore has greater central executive demands than approxima-

tion and (2) problems requiring carries would be more demanding of

working memory than problems that did not require carries. Both these

hypotheses were supported, in that participants showed greater dual-task

decrements in exact than in approximate arithmetic and greater dual-task

decrements on carry than on no-carry problems. Decrements occurred

primarily in errors; participants appeared to maintain arithmetic latencies

TABLE 2Mean percentage of combined errors in each experiment: Interactions of task order,

task, complexity, and load

No-carry Carry

Order Task Single Dual Single Dual

Experiment 1

E-A Exact 7.6 8.3 9.7 37.5

Approximate 6.2 18.0 4.2 18.0

A-E Exact 6.9 22.2 11.8 33.3

Approximate 4.2 9.7 10.4 25.0

Experiment 2

E-A Exact 25.0 40.3 25.0 68.1

Approximate 29.2 41.7 28.5 40.3

A-E Exact 24.3 47.2 32.6 63.9

Approximate 34.7 31.9 23.6 48.6

E-A indicates exact arithmetic first, approximate second; A-E indicates approximate arithmetic

first, exact second.


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across single- and dual-task conditions. Decrements related to carrying have

been connected to central executive requirements of working memory (Furst

& Hitch, 2000; Seitz & Schumann-Hengsteler, 2000, 2002) but they may also

implicate storage limitations in the phonological loop (Logie et al., 1994).

Thus, our results suggest that exact arithmetic is more demanding of

working memory than approximate arithmetic, and point to differences in

the complexity of the strategies that participants adopt for these tasks

as potentially important for understanding these differential demands

(Duverne, Lemaire, & Michel, 2003; El Yagoubi et al., 2003).

One limitation of the methodology used in Experiment 1 is that the

response demands of the exact versus approximate task were confounded

with a magnitude comparison task. In arithmetic verification tasks, people

reject false answers that are far away from the correct answer (e.g., 6�/8�/

72) more quickly and accurately than they reject false answers that are close

to the correct answer (e.g., 6�/8�/49; Ashcraft & Battaglia, 1978; Campbell,

1987). Thus in the present research, selecting the answer may have been more

difficult for exact arithmetic (where the difference between the correct and

false answer was always 10) than for the approximate task (where the

difference between the approximate and false answer was always 30). In both

cases, participants could have calculated an answer and then chosen the

better alternative from the two possibilities. Exact solutions may require a

more complete calculation than approximate solutions, but the answer

choice process would also have been more difficult. On this view, the

increased working memory demands for exact arithmetic as compared to

approximate arithmetic may have reflected differences across tasks in a

decisional stage of processing and not in the actual calculation process. To

assess this explanation for the differences between the two tasks, we used a

production paradigm in Experiment 2. Exact calculation versus approxima-

tion was manipulated with instructions, rather than with answer choices.

EXPERIMENT 2

As in Experiment 1, participants performed multidigit addition and letter

memory both alone and in combination. In contrast to Experiment 1,

however, participants were required to produce either exact or approximate

answers to the arithmetic problems. If the differences between the tasks were

related to the requirement to choose between two potential answers, we

expected smaller differences between tasks in this experiment than in

Experiment 1. However, if the differential working memory demands of

exact and approximate arithmetic are related to the complexity of the

calculations involved in these tasks, then we expect to find a very similar

pattern of results to those in Experiment 1.


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Method

Participants

Twenty-four students in Introductory Psychology (14 males and 10

females) participated in the experiment. No selection criteria were used.

None of the participants had participated in Experiment 1. Participants

ranged in age from 18 to 40 years, with a median age of 21.5 years.

Participants received course credit (as partial fulfilment of a course

requirement) in exchange for their participation. Participants did not report

having had any seizures or severe head injuries. Nine of the participants

reported that all of their schooling had been in Canada and fifteen had

received some of their schooling elsewhere. In comparison to Experiment 1,

therefore, these participants came from a wide variety of language and

educational backgrounds. Nevertheless, arithmetic performance was very

similar across experiments.

Apparatus

The same apparatus was used in this experiment as in Experiment 1. In

addition, participants wore a headset with an attached microphone. The

microphone was connected to a Psychology Software Tools Serial Response

Box, with a timing accuracy of 1.25 ms.

Materials, design, and procedure

The materials, design, and procedure were identical to Experiment 1,

except that participants provided answers to arithmetic problems by

speaking into the microphone. For exact addition problems, the experi-

menter told participants to speak the correct answer into the microphone

(e.g., say ‘‘57’’ for the problem 20�/37). For approximate addition problems,

the experimenter told participants to speak an answer to the nearest decade

of the exact answer (e.g., say ‘‘60’’ for the problem 20�/37). Immediately

after participants spoke the answer into the microphone, a blank screen

appeared, and the experimenter typed the answer on the keyboard. After the

experimenter typed the answer, there was a 1000-ms blank interval, and then

the message ‘‘RECALL LETTERS’’ appeared on the screen.

For the single-task letter naming condition, participants read the addition

problem and the answer aloud before proceeding to the ‘‘RECALL

LETTERS’’ screen (e.g., for the problem 20�/37�/57, they were instructed

to say ‘‘twenty plus thirty-seven equals fifty-seven’’). After the participants

read the addition problem and the answer aloud, the experimenter pressed a

key to trigger the ‘‘RECALL LETTERS’’ screen. This screen was preceded

by a 1000-ms blank interval.


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After each arithmetic condition, the experimenter asked participants to

describe how they solved the addition problems in that block. Theexperimenter showed the participant a problem from the block of experi-

mental trials. Then he asked the participant, ‘‘How, in general, did you solve

this problem?’’ Answers were transcribed and coded later.

Results

Arithmetic fluency

The mean score on the arithmetic fluency test was 77.5 (SD�/26.9).

Participants’ fluency scores did not significantly differ from the expectedmean for fluency (i.e., 80) in this population, t(23)�/�/0.46, p�/.05, and were

not different from those of participants in Experiment 1 (i.e., M�/76.8).

Fluency scores did not differ between order groups, t(22)�/0.30, p�/.05.

Strategy descriptions

For exact addition, participants reported two general categories of

strategies. Seventeen participants reported using the standard right-to-left

addition algorithm: They added the ones digits first and then added the tens

digits (e.g., for the problem 34�/19, they added 4�/9�/13, then 30�/10�/

10�/50, and 50�/3�/53; interim results are shown in italics). Seven

participants reported adding the tens digits first followed by adding theones digits (e.g., 30�/10�/40, 4�/9�/13, 40�/13�/53). For approximate

addition, three strategies were reported. Sixteen participants reported adding

the tens digits first followed by performing a rounding operation on the ones

digits and adjusting the answer if the sum of the ones digits was greater than

5 (e.g., 30�/10�/40, 4�/9�/5 therefore 40�/10�/50). Six participants

reported rounding both operands and then adding them (e.g., 30�/20�/

50). Two participants reported adding the ones column first followed by

adding the tens column (e.g., 4�/9�/5, 10�/30�/10 answer 50). Thus, thedifferences between exact and approximate strategies were that (a) fewer

interim results needed to be maintained during subsequent calculations, and

(b) fewer calculations were required. Note that most participants (22 of 24)

added the tens digits first in the approximate condition, whereas they were

more likely to add the ones digits first for exact arithmetic. The strategy

reports are consistent with the findings from Experiment 1 that exact

arithmetic requires more steps than approximate arithmetic.

Experimental task performance

Participants completed a total of 1152 trials where responses to arithmetic

problems were recorded (i.e., 576 of each single- and dual-task trial). Only

correct trials were included in the analysis of response times (482 exact


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addition trials and 526 approximate addition trials). For exact addition

problems, a trial was scored as correct if participants provided the exactanswer to the addition problems (e.g., 32�/12�/44). For approximate

addition problems, a trial was scored as correct if participants provided an

approximate answer within 10 of the exact answer (e.g., for 32�/12�/42,

acceptable approximations were 40 and 50). Combined error scores were

calculated as in Experiment 1. Combined errors and correct response times

were analysed in separate 2 (order: exact-approximate vs. approximate-

exact)�/2 (task: exact, approximate)�/2 (complexity: no carry, carry)�/2

(load: single, dual) ANOVAs, with repeated measures on the last threefactors. The results for the two dependent variables are discussed together,

with an emphasis on the combined errors. As for Experiment 1, an analysis

of each error score separately is given in the Appendix. All of the significant

effects described for the combined errors were significant either in both error

measures or in one measure. Note that, for both experiments, error bars

shown in all figures are the 95% confidence intervals based on the MSE

values for the illustrated interaction (Masson & Loftus, 2003). Reported

effects were significant at pB/.05, unless otherwise indicated.As in Experiment 1, participants solved approximate addition problems

more quickly than exact addition problems (2075 vs. 2868 ms), F(1, 22)�/

24.34, MSE�/1,240,433, and made fewer errors on approximate than exact

problems (35% vs. 40%), F(1, 22)�/7.03, MSE�/245. Note that the overall

combined error rates were greater in Experiment 2 than in Experiment 1

even though the response times and patterns of performance in exact versus

approximate addition were very similar across experiments. The increase in

combined error rates in Experiment 2 was largely due to a generally higherrate of letter recall errors (44% vs. 10% in Experiment 1). Overall rates of

arithmetic errors were similar across experiments (13% vs. 10% in Experi-

ments 2 and 1, respectively), as were response latencies (as illustrated in the

figures). At least three factors may have contributed to these differences: (1)

Participants found it more difficult to share the letter memory task with the

spoken response because both involved internal phonological codes, (2) the

requirement to verbally produce the answer interfered with maintenance of

the memory load in the phonological loop, and (3) the varied languagebackgrounds of the participants in this experiment may have made the letter

recall task more difficult. Of most interest are the patterns of performance,

as described below. The patterns were very similar to those found in

Experiment 1, but more evidence for tradeoffs between the arithmetic and

letter memory tasks was also evident.

As in Experiment 1, participants solved no-carry problems more quickly

than carry problems (2064 ms vs. 2879 ms), F(1, 22)�/49.91, MSE�/

638,045, and made fewer combined errors on trials with no-carry thancarry problems (34% vs. 41%), F(1, 22)�/14.68, MSE�/162. Further, task


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and complexity interacted, for latencies, F(1, 22)�/23.06, MSE�/630,526,

CI�/336 ms, pB/.05, and for combined errors, F(1, 22)�/13.53, MSE�/135,CI�/5%. As shown in Figure 1 (right panel), latencies and errors were not

different for exact and approximate problems on no-carry problems. On

carry problems, however, participants were significantly slower and more

error-prone in the exact than in the approximate condition. These patterns

are strikingly similar to those found in Experiment 1 (left panel), supporting

the assumption that the task requirements were similar across experiments.

Furthermore, there was no effect of complexity for approximate arithmetic.

As shown in the figure, exact arithmetic was more difficult than approximatearithmetic only when a carry operation was required.

There were no main effects of load in this experiment for latencies. As in

Experiment 1, participants appeared to maintain or emphasise latencies on

the arithmetic problems relative to accuracy on the letter memory task.

Replicating Experiment 1, participants made more combined errors in the

dual-task than the single-task conditions (48% vs. 28%), F(1, 22)�/44.00,

MSE�/431. The interactions between task and load that were found in

Experiment 1 were also significant in this experiment, for latencies, F(1,22)�/5.05, MSE�/201,642, CI�/190 ms, and for combined errors, F(1,

22)�/7.36, MSE�/444, CI�/9%. As shown in Figure 2 (right panel),

latencies tended to increase with load for exact arithmetic and decrease

with load for approximate arithmetic, although neither change was

significant. Furthermore, the decrease in solution latencies on approximate

problems was accompanied by a corresponding increase in combined errors.

This pattern suggests no overall load effects on approximate problems, but

instead that participants traded off letter responses for speed on arithmeticsolutions. In contrast, as in Experiment 1, the load task could not be easily

shared with exact arithmetic, resulting in a significant increase in combined

errors in the dual-task condition. These results support our hypothesis that

the working memory demands of these arithmetic problems are related to

processing complexity.

As in Experiment 1, the interaction of complexity and load was not

significant for latencies, FB/1, but was significant for combined errors, F(1,

22)�/11.35, MSE�/264, CI�/7%. As shown in Figure 3, participants mademore errors on carry than on no-carry problems in the dual-task condition,

whereas error rates were not significantly different across complexity in the

single-task condition. Note, however, that error rates increased with load for

both no-carry and carry problems. These findings suggest that the working

memory demands of carry problems are greater than those for no-carry

problems but both are affected by the working memory demands of the

combined tasks. Finally, despite the significant two-way interactions

between task and load and carry and load, the three-way interaction wasnot significant. Thus, even for approximate arithmetic, carry problems were


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somewhat more demanding of working memory than no-carry problems.

The means for the three-way interaction are shown in Table 1. As in

Experiment 1, the largest percentage of combined errors was for carry

problems in the exact arithmetic task.

Finally, although there were no significant overall effects of order for

either latencies or combined errors, FsB/1, there were interactions of

complexity, load, and order for latencies, F(1, 22)�/5.083, MSE�/836,022,

and a four-way interaction of order with task, complexity, and load for

combined errors, F(1, 22)�/8.46, MSE�/203, CI�/6.0% (as shown in Table

2). As in Experiment 1, scrutiny of these interactions suggests that Load�/

Complexity interactions (i.e., smaller effects of load for no-carry than for

carry problems) were moderated by task order. When approximate

arithmetic was the second task that participants performed, load effects

were similar on carry and no-carry problems but when approximate addition

came first, load effects were greater on carry than on no-carry problems.

Load effects were also moderated in exact problems when they were solved

second. Nevertheless, participants always made more combined errors in the

dual- than in the single-task on carry problems and for most of the

comparisons for no-carry problems. As shown in the table, combined error

percentages were always largest in the dual-task for carry problems in the

exact condition (68% and 64% when exact arithmetic was first vs. second,

respectively). Thus, despite some carryover effects related to practice with

the tasks, these findings suggest that participants’ performance was most

disrupted in the dual-task condition when they were attempting to answer

more difficult problems, that is, those that required carrying or exact

solutions. Again, we suggest that these order effects are consistent with the

possibility that differences in strategy use across tasks developed with

practice.

Comparison across experiments

Although the results shown in Figures 1�3 suggest that the patterns of

results were very similar across experiments, there was a greater frequency of

letter recall errors in Experiment 2. To assess more directly whether this

overall difference affected the patterns of results, we pooled the data from

the two experiments and reanalysed combined errors. As expected, the

participants in Experiment 1 made significantly more combined errors than

participants in Experiment 2 (38% vs. 15%), F(1, 44)�/48.31, MSE�/1071.

However, there were no significant interactions between experiment and any

of the other variables. Furthermore, main effects of task, load, and

complexity were significant, Fs(1, 44)�/20, psB/.001, as were the reported

two-way interactions of Task�/Complexity, Task�/Load, and Load�/

Complexity, Fs(1, 44)�/9, psB/.01. Finally, the four-way interaction of these


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variables with order was significant, as in the separate analyses, F(1, 44)�/

12.34, MSE�/174, CI�/4%. Thus, there was no evidence that the elevatedfrequency of letter recall errors in Experiment 2 changed the pattern of

results. Response mode (choice vs. production) did not interact with the

working memory demands of exact and approximate arithmetic.

Discussion

The results of this experiment were very similar to those of Experiment 1.

Participants solved exact arithmetic problems more slowly and less

accurately than approximate problems, exact arithmetic was more affected

by the working memory load than approximate arithmetic, and carry

problems were more affected by load than no-carry problems. These results

support the view that the working memory demands of exact versusapproximate arithmetic are related to the relative calculational complexity

of the two tasks. They also indicate that carry problems are more demanding

of working memory for both exact and approximate solutions.

GENERAL DISCUSSION

The purpose of the two experiments described in this paper was to examine

potential differences in working memory demands between exact calculation

and approximation for problems such as 37�/29 or 56�/32. Researchers have

suggested that the main difference between exact calculation and approx-

imation for multidigit problems is that exact calculation involves more

computations and greater requirements for maintenance of intermediatesums than approximation (Duverne et al., 2003; LeFevre et al., 1993;

Lemaire et al., 2000). Strategy reports in Experiment 2 supported this view

in that both exact and approximate arithmetic required calculations, but that

participants reported simplifying the problems in the approximate condition

and thereby reduced both the number and complexity of calculations that

were required. The patterns of performance across single- and dual-task

conditions supported the hypothesis that these strategy differences were

linked to working memory demands. Interactions between problem com-plexity and memory load indicated that carry problems were affected more

than no-carry problems by the working memory load and exact arithmetic

was affected more than approximate arithmetic. The present results support

the view that exact solutions require working memory resources, and extend

this conclusion to approximate arithmetic, but suggest that the processes

involved in approximating solutions to carry problems are generally less

demanding than those involved in the execution of a fully fledged exact

algorithm.


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The present research contributes to the limited literature on approximate

arithmetic (Dehaene et al., 1999; Dowker, 2003; Duverne et al., 2003; El

Yagoubi et al., 2003; LeFevre et al., 1993; Lemaire & Lecacheur, 2003;

Siegler & Booth, 2005; Stanescu-Cosson et al., 2000) and supports the view

that participants adopt different strategies on approximate than on exact

calculations. The current findings also provide new information about the

working memory demands of multidigit addition, and suggest that the

greatest portion of these demands for such problems occur as calculation

processes become more involved and complex (LeFevre et al., 2005). Finally,

the similarity in patterns of results across experiments suggests that

variations in working memory demands that are associated with calcula-

tional complexity do not interact with specific task requirements. Patterns of

interference across exact and approximate arithmetic and carry versus no-

carry problems were similar regardless of whether participants chose the

better answer from two alternatives or produced an answer. Researchers have

rarely contrasted arithmetic tasks with different response requirements

within the working memory paradigm (cf. De Rammelaere, Stuyven, &

Vandierendonck, 1999, 2001).

In research on mental arithmetic, participants always solve problems

that require carries more slowly and less accurately than problems that do

not require carries. In the present research, however, we identified a

situation in which the demands of carry problems were moderated: Carry

problems were solved almost as quickly and accurately as no-carry

problems. On approximate problems, participants simplified the operands

before they calculated and, consequently, carry and no-carry problems

showed similar solution times and accuracy. In contrast, for exact solutions

to the same problems, participants showed a typical effect such that carry

problems were much more difficult than no-carry problems. Despite the

similar performance on carry and no-carry problems for approximate

arithmetic, however, the analysis of combined errors indicated that carry

problems were more demanding of working memory than no-carry

problems even on approximate arithmetic. As shown by Lemaire and

Lecacheur (2002; Lemaire et al., 2000), rounding down is a more

demanding procedure than rounding up, presumably because an increment

must be made (e.g., rounding 42 to 40 vs. rounding 47 to 50). Rounding up

may increase the interim storage requirements of the problem. On carry

problems, one or both of the operands were closer to the larger decade.

Thus, although both exact and approximate arithmetic problems required

more working memory on carry than on no-carry problems, the reasons

were probably different. In conclusion, the present results provide direct

support for the view that the processes involved in carrying are an

important source of working memory demands for multidigit arithmetic


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problems but show that the type of arithmetic task (exact vs. approximate)

influences these demands.

Original manuscript received August 2005

Manuscript accepted February 2006

First published online 5 July 2006

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APPENDIX

Analyses of errors in Experiment 1

Percentage errors on letter recall and arithmetic were analysed in separate

2 (order: exact-approximate vs. approximate-exact)�/2 (task: exact, ap-

proximate)�/2 (complexity: no carry, carry)�/2 (load: single, dual)

ANOVAs, with repeated measures on the last three factors.

Arithmetic

In the arithmetic task, the only significant effects were for complexity and

complexity by order. Participants made more errors on carry than on no-

carry problems (13% vs. 7%), F(1, 22)�/13.22, MSE�/142, but this effect

varied with order. Participants who solved exact problems in the first block

and approximate problems in the second block showed the expected pattern

of more errors on carry than on no-carry problems (18% vs. 7%), whereas

participants who solved problem blocks in the opposite order did not show a

difference between carry and no-carry problems (8% vs. 7%), F(1, 22)�/6.90,MSE�/142. No other main effects or interactions approached significance,

suggesting that participants preserved arithmetic accuracy across the single-

and dual-task trials for arithmetic and further, when participants selected an

answer, errors did not vary across the exact and approximate tasks.

Letter memory

In contrast, letter errors showed effects of the working memory load.Participants made fewer errors on the letters in the single- than in the dual-

task condition (6% vs. 14%), F(1, 22)�/7.77, MSE�/411, indicating that

arithmetic and letter recall mutually interfered even though there was no

load effect on arithmetic performance. Letter memory also varied with

arithmetic task, in that participants made letter recall errors on exact than

on the approximate trials (13% vs. 6%), F(1, 22)�/11.51, MSE�/154.

Participants also made more errors on carry problems than on no-carry

problems (12% vs. 7%), F(1, 22)�/9.98, MSE�/123, and complexityinteracted with load, F(1, 22)�/5.67, MSE�/214. In the single-task

condition (no arithmetic required), errors did not vary with complexity

(6% in both conditions). In the dual-task condition, however, participants

made fewer errors when they were solving no-carry than carry problems (7%

vs. 19%). The interaction of arithmetic task, complexity, and order was

significant, F(1, 22)�/7.27, MSE�/88. Participants made more errors on

carry than on no-carry problems in all combinations of arithmetic task and

order conditions, but the most dramatic difference was for the exact


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condition when it was solved first (7% vs. 21%). When exact arithmetic was

solved second (10% vs. 11%), and when approximate arithmetic was either

first (3% vs. 7%) or second (7% vs. 9%), the differences between carry and

no-carry problems were modest. This pattern suggests that participants who

solved approximate problems first developed more effective solution

strategies by the second block. The four-way interaction for this analysis

approached significance, F(1, 22)�/4.04. MSE�/104, p�/.057. The pattern is

the same as that described in the combined error analysis.

Based on these initial analyses of errors separately in the two tasks, we

decided to proceed with a combined error score. Because complexity effects

occurred in the errors for both tasks, and the other effects of interest showed

up in the letter task, the results suggest that the arithmetic task performance

was emphasised or preserved (presumably because it was relatively easy) and

dual-task tradeoffs were evident in the secondary (letter) task. As described

below, the patterns of errors were more variable for both arithmetic and

letter recall in Experiment 2.

Analyses of errors in Experiment 2

Arithmetic

For the arithmetic task, participants made more errors on exact than on

approximate trials (16% vs. 9%), F(1, 22)�/10.12, MSE�/277, in line with

other research in which error effects are more pronounced for production

than for verification (Campbell, 1987). As in Experiment 1, participants

made more errors on carry than on no-carry problems (9% vs. 16%), F(1,

22)�/8.29, MSE�/252. The interaction of arithmetic task and complexity

approached significance, F(1, 22)�/3.97, MSE�/176, p�/.056, such that

the difference between carry and no-carry problems tended to be larger for

exact (11% vs. 21%) than for approximate problems (7% vs. 10%). The

interaction of arithmetic task and load was significant, F(1, 22)�/5.78,

MSE�/144, such that participants made fewer errors in the single- than in

the dual-task condition when the arithmetic task required exact answers

(12% vs. 20%), whereas they showed no effect of load in the approximate

arithmetic condition (9% vs. 8%). In combination with the patterns in

Experiment 1, it appears that only when participants are required to

produce exact answers (rather than recognise them) is there an effect of

working memory load on arithmetic performance as measured by errors.

As in Experiment 1, however, there were additional patterns in the letter

recall data suggesting that participants traded off performance between the

two tasks.


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Letter memory

For letter recall, there were significant interactions between arithmetic

task and complexity, F(1, 22)�/9.54, MSE�/208, between arithmetic taskand load, F(1, 22)�/6.87, MSE�/684, as well as between complexity and

load, F(1, 22)�/10.82, MSE�/375. Finally, the four-way interaction was

significant, F(1, 22)�/6.32, MSE�/280. These patterns mirror those

described in the combined error analysis. Furthermore, the two-way

interactions in each of the two tasks showed the same patterns as described

for the combined error analyses. Thus, the combined error analysis was

chosen as the most complete reflection of overall performance in this dual-

task situation. In this memory load paradigm where both of the tasks(arithmetic and letter recall) are quite difficult, the combined error analysis

provides a more complete picture of the patterns of performance (Seyler et

al., 2003; Trbovich & LeFevre, 2003).


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Working memory demands of exact and approximate addition

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