working memory and children's mental addition

JOURNAL OF EXPERIMENTAL CHILD PSYCHOLOGY 67, 21–38 (1997)ARTICLE NO. CH972397

Working Memory and Children’s Mental Addition

John W. Adams

Department of Psychology, University of York, Heslington,York YO1 5DD, United Kingdom

and

Graham J. Hitch

Department of Psychology, University of Lancaster, Bailrigg,Lancaster LA1 4YF, United Kingdom

Two experiments investigated the extent to which children’s mental arithmetic isconstrained by working memory rather than their arithmetical competence. A spanprocedure was used to measure the limit on English- and German-speaking children’sability to add together pairs of multidigit numbers. The children’s ages ranged from7 years 7 months to 11 years 5 months. Spans for mental addition were higher whenthe numbers to be added were visible throughout calculation than when they werenot, consistent with a working memory constraint. Variation in addition span withchildren’s age and with difficulty of the arithmetical operations approximated to alinear function of the speed of adding integers. A similar speed/span relationship haspreviously been observed for counting span, an artificial task designed to load workingmemory by combining separate processing and storage subtasks. We conclude thatthe natural task of mental addition, which combines processing and storage as intrinsiccomponents, reflects working memory in a similar way. Results were remarkablysimilar both between cultures and across age groups, consistent with the notion ofworking memory as a general-purpose resource with dynamics that are indifferent tothe detailed nature of operations. q 1997 Academic Press

In seminal studies, Brown (1958) and Peterson and Peterson (1959) showedthat performance of a short-term memory task is disrupted by a few secondsof mental arithmetic interpolated between presentation and recall of the items.

John Adams was supported by an ESRC research studentship, and part of the work wassupported by a grant from the British–German Academic Research Collaboration Programmeto Graham Hitch and Ruth Schuman-Hengsteler, University of Mainz, Germany. We are gratefulto Ruth Schuman-Hengsteler, Katja Seitz, and Christiane Iwanoff for their help in runningExperiment 2, and to Vicki Culpin, Ansgar Furst, and Janet McLean for helpful discussions.

All correspondence and requests for reprints should be sent to John Adams, PsychologyDepartment, University of York, Heslington, York, YO1 5DD, UK. E-mail: [email protected].

0022-0965/97 $25.00Copyright q 1997 by Academic Press

All rights of reproduction in any form reserved.

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AID JECP 2397 / ad12$$$$61 09-17-97 20:02:04 jecpal AP: JECP

22 ADAMS AND HITCH

Recall gets worse when the arithmetical operations are made harder, indicatingthat both tasks compete for common limited capacity resources (Posner &Rossman, 1965). Other evidence suggests that mental operations and tempo-rary information storage load a general-purpose working memory systemwhich normally combines these functions (Baddeley & Hitch, 1974; Dane-man & Carpenter, 1980). Although much has been learned about the character-istics of the working memory system (see, e.g., Baddeley, 1986), its specificrole in supporting mental calculation is still not well understood. It is knownthat dual task procedures which disrupt working memory impair mental calcu-lation in adults (Ashcraft et al., 1992; Logie, Gilhooly & Wynn, 1994; Le-maire, Abdi & Fayol, 1996) and children (Hitch et al., 1987). Further, errorsof mental calculation in adults can be predicted by assuming rapid decay ofmemory traces of the problem information (Hitch, 1978). There is also someevidence for working memory deficits in children with arithmetical learningdifficulties (see Geary, 1993 for a review; see also Hitch & McAuley, 1991;Siegel & Ryan, 1989). However, the empirical base has been described asslim (Ashcraft, 1995) and the role of working memory in arithmetic has beenquestioned by recent evidence. For example, an adult neuropsychologicalpatient has been described who could perform complex multidigit mentalcalculations despite having an exceptionally low digit span (Butterworth,Cipolotti & Warrington, 1996).

Multidigit mental addition is of particular interest, especially in children,as it is a basic educational skill with many everyday applications. From acognitive developmental perspective, any constraints on mental arithmeticperformance due to working memory are expected to be more severe inchildren than adults (see, e.g., Case, Kurland, & Goldberg, 1982; Hitch &Halliday, 1983). However, an important further consideration is that children’smultidigit mental addition may be constrained by their arithmetical compe-tence. A priori, this would seem most likely for younger children still acquir-ing elementary arithmetical skills and concepts. However, it is not clear ifthere is a progressive changeover from limitations of competence to con-straints imposed by working memory in children’s mental addition, nor whenany such change takes place. The present paper explores these issues.

Two empirical questions were addressed. The first concerned the effect ofthe way mental addition problems are presented. Previous research showedthat adults made fewer mental calculation errors when problem informationwas made continuously available by providing a written record (Hitch, 1978).This is readily explained by assuming that the written page acts as an externalmemory thereby reducing the load on working memory (see, e.g., Newell &Simon, 1972). To the extent that children’s mental arithmetic performance isconstrained by working memory, we would expect them to show a similarbenefit from having a written record. If on the other hand children’s calcula-tion performance is limited by their arithmetical competence, then reducingthe load on working memory should not help them. To investigate thesepredictions, visual and oral procedures for problem presentation were com-


23WORKING MEMORY AND ARITHMETIC

pared. The visual procedure was such that the addends remained continuouslyavailable for inspection during the calculation. The oral procedure involveda single brief presentation of the problem.

The second question concerned the relationship between the accuracy ofchildren’s mental addition and the speed at which they can execute additionoperations. Previous research revealed that accuracy in a ‘counting span’task, which required counting operations to be combined with temporaryinformation storage, could be predicted from the speed of the operations(Case, Kurland, & Goldberg, 1982; see also Case, 1985, 1995). Thus, childrenaged between 6 and 12 were asked to count the number of targets on eachof a series of cards, while at the same time remembering totals for previouslycounted cards. Targets were spots of a different color from distractors. Thenumber of cards was increased over trials and counting span measured theupper limit on the number of totals that could be recalled. Speed of countingoperations was measured independently by recording the time taken to counttargets with no memory load. The results showed that age differences incounting span and counting time were linearly related, such that older chil-dren’s higher spans could be predicted from their faster counting. In order toinvestigate the basis of this relationship, Case et al. (1982) carried out afurther study in which they taught adults to count using nonsense syllables.This slowed their counting speed to that typical of 6-year-olds. The interestingeffect was that adults’ counting spans were reduced to the level of 6-year-olds,as one would expect if counting span is determined by speed of operations.

Case et al. (1982) interpreted their results in terms of a trade-off model inwhich a limited total processing space is divided between mental operations(counting) and information storage (remembering previous totals). For presentpurposes processing space can be equated with working memory capacity,and detailed differences between the two theoretical concepts can be ignored.Case et al. (1982) assumed that the speed of mental operations reflects theefficiency of processing and that more efficient operations require less work-ing memory capacity, leaving more free for storage. Given the further assump-tion that capacity does not change during development, the linear increase incounting span as a function of counting speed can be explained. Subsequently,Towse and Hitch (1995) proposed an alternative to the capacity trade-offaccount according to which the relationship between span and speed of opera-tions is because faster operations allow less time for information stored inworking memory to be forgotten. In support of this, Towse and Hitch (1975)found that counting span was lower when counting time was lengthened byincreasing the number of targets. This manipulation was assumed to have noeffect on how much working memory capacity was required for countingoperations. On the other hand, when the processing difficulty of discriminatingtargets from distractors was altered, there was no effect on counting spanover and above what would be expected from the change in counting time.Towse and Hitch (1995) therefore argued against a resource trade-off interpre-tation of the relationship between counting speed and counting span. However,


24 ADAMS AND HITCH

they agreed with Case et al. (1982) on the basic point that counting spanreflects limitations due to working memory. Thus, if mental addition is con-strained by working memory in the same way as counting span, we shouldobserve a similar relationship between the accuracy of mental addition andthe speed of addition operations. In order to test this prediction it was firstnecessary to devise a suitable measure of mental addition accuracy.

One such measure is suggested by the operation span procedure used byTurner and Engle (1989) in which adults verified a series of arithmeticalrelationships [e.g. (9/3) 0 2 Å 1] and later recalled the answers stated on theright hand side of the equations. This procedure combines separate subtasksinvolving operations and storage in the same way as counting span and otherworking memory span tasks such as reading span (Daneman & Carpenter,1980) and complex word span (La Pointe & Engle, 1990). Moreover, opera-tion span could readily be adapted so that children were given mental additionsto verify. However, it would remain an artificial task in which the storagesubtask is extrinsic to the mental addition component. What we were moreinterested in was whether the intrinsic storage and processing demands ofmental addition tax working memory in children. The storage demands ofmental addition seemed likely to be considerable and to include keeping trackof partial results and internal counting operations as well as remembering theoriginal numbers to be added. We therefore sought a method for measuringthe limit on the accuracy of children’s mental addition as a function of itsintrinsic processing and storage demands. We achieved this by devising anaddition span procedure in which children were given progressively morecomplex additions, each involving a pair of numbers, to find the limit atwhich errors were made. Complexity was altered by increasing how manydigits there were in the two numbers to be added together.

We investigated the relationship between mental addition span and thespeed of addition operations by varying factors likely to affect the speed ofadding integer pairs. Age was one variable expected to have a large effect,given that there is a qualitative change from early dependence on slow,algorithmic counting procedures through to fast and accurate retrieval ofstored number facts from long-term memory (Ashcraft & Fierman, 1982;Hamann & Ashcraft, 1985; Ashcraft, 1990; see also LeFevre et al., 1996).Speed is also influenced by the problem-size effect (Ashcraft, 1992, 1995),according to which larger digit pairs tend to take longer to add (e.g., 8 / 7takes longer than 3/ 1). Addition span was therefore determined for additionswhich differed in the level of difficulty of their component integer operationsand for children of different ages. The question was whether changes inaddition span with age and difficulty would be systematically related tochanges in integer addition speed with age and difficulty, by analogy with themonotonic relationship between counting speed and counting span reported byCase et al. (1982).

A further aim of the investigation was to assess the generality of conclusionsby repeating the study using samples of children speaking different languages



and undergoing schooling in different cultures. An opportunity arose to repli-cate an initial experiment on British children attending schools in Englandusing a group of German children attending German schools. One reasonsuch a replication is interesting is that the syntax of the German numbersystem differs from English. For example, the ‘‘tens’’ and ‘‘units’’ are re-versed when spoken as in ‘‘vierundachtzig’’ for ‘‘eighty-four.’’ This differ-ence in syntax may be reflected in German speakers using algorithms foraddition in which the order of operations is different. There are also othercultural differences, such as the fact that children start schooling rather laterin Germany and mathematics teachers tend to place far more emphasis onpracticing oral mental arithmetic skills (Bierhoff, 1996). Repeated practiceinvolving basic addition has been cited as the process whereby children formrepresentations of number facts in long-term memory (Siegler & Schrager,1984; LeFevre et al., 1991).

We were interested in whether there would be similarities in patterns ofperformance across cultures, despite numerous differences of the sort notedabove. We argued that if working memory constrains the dynamics of combiningoperations with temporary storage in mental addition, differences in the detailednature of the storage and processing operations, such as those associated withlanguage and culture, should be relatively unimportant. Thus in both English andGerman-speaking children we expected to observe the same form of relationshipbetween addition span and speed of arithmetical operations and similar differ-ences between spans for oral and visual presentation of the problems.

As there were only minor differences in the designs, procedures, and resultsof the two experiments, they are reported together.

EXPERIMENTS 1 AND 2

Method

Subjects. In Experiment 1, participants were 80 children from four yeargroups attending an urban primary school in the Lancaster area. The composi-tion of the groups and their mean ages were as follows: Year 3 (10 male, 10female) 7 years 11 months, SD 3 months; Year 4 (5 male, 15 female) 9 years1 month, SD 3 months; Year 5 (10 male, 10 female) 10 years 1 month, SD3 months; and Year 6 (12 male, 8 female) 10 years 11 months, SD 4 months.Children were selected by their class teachers on the criterion that they couldsolve single digit additions up to and including 9 / 8.

In Experiment 2, participants were 102 children drawn from three yeargroups in two rural primary schools near the German city of Mainz. Grade2 children (16 male, 18 female) had a mean age of 8 years 6 months, SD 4months; Grade 3 (19 male, 15 female) were aged on average 9 years 6 months,SD 4 months; and Grade 4 pupils (14 male, 20 female) 10 years 6 months,SD 4 months. Selection was based on parental consent.

Design. Experiment 1 used an entirely within-subjects design in which allchildren performed both visual and oral addition speed and visual and oral


26 ADAMS AND HITCH

addition span tasks. Each task was performed under each of three conditionsof difficulty of the integer operations used to construct the multidigit additions(see below). Order of presentation of the speed and span tasks was counterbal-anced. Half the subjects received the visually presented tasks first, half theoral. Also, the order in which each condition of difficulty was presented inthe speed task was rotated.

The design of Experiment 2 was identical save that as access to each childwas limited to a maximum of 15 min., presentation mode was varied as abetween-subjects factor. Thus half the children were presented with the speedand span tasks in the oral mode, and half in the visual mode. Children wererandomly assigned to each of these two conditions.

Materials. The speed task consisted of sets of ten additions of pairs of integers.The difficulty of the operations in each set was designated as either Easy (e.g.,2 / 3), Hard (e.g., 4 / 5) or Carry (e.g., 6 / 7). The differentiation betweenEasy and Hard pairs was based upon RT data reported by Groen and Parkman(1972). Of the 35 integer pairs summing to less than 10 (obtained by excluding1 / 1 and pairs involving zero), 19 were classified as Easy (RT õ 3.2 s) and16 as Hard (RT ú 3.2 s). The Carry additions were all the 32 integer pairs(excluding ties) with sums between 11 and 17 inclusive. Two sets of additionsfor each type of operational difficulty (Easy, Hard, and Carry) were constructedby random sampling from these item pools. One set was assigned to the oralcondition, the other to the visual condition. For visual presentation, the additionswere written onto pastel colored cards (size 105 1 70 mm) in black nylonmarker pen 45-mm high in a standard horizontal format (e.g., 7 / 8 Å).

Materials for measuring addition span were three lists of additions of in-creasing complexity (as reflected by the number of digits and the number ofinteger additions, e.g., 8 / 1, 21 / 7, 122 / 3, 31 / 26, 231 / 16, etc.).There was one list for each of the three types of difficulty of the additionoperations (Easy, Hard, and Carry). Each list contained two additions at eachlevel of complexity. Problems for the Easy Operations and Hard Operationslists were constructed by randomly selecting integer pairs from the relevantitem pools, with any unpaired integers selected at random from the integers1 and 2 (see Table 1). Problems in the Carry lists contained one randomlychosen integer pair from the Carry item pool per problem, with all otherinteger pairs taken from the Easy item pool and any unpaired integers asabove. (Pilot work indicated floor effects if all the pairs were taken from theCarry pool.) The position of the carry was varied randomly so that subjectscould not anticipate in advance where it would occur.

Two sets of addition span materials were constructed for use in the visualand oral presentation conditions, respectively. Problems for visual presenta-tion were written onto pastel coloured cards (size 210 1 70 mm). Numeralswere written in black nylon marker pen 45-mm high in the standard horizontalformat (e.g., 723 / 813 Å).

In Experiment 2 the materials were identical except that the digit 7 waswritten in the continental style (i.e., with a horizontal cross line).



TABLE 1Incrementation of Sum Complexity for the Three Levels of Operational Difficulty

Sum complexity Easy Hard Carry

Level 1 8 / 1 3 / 5 5 / 9Level 2 21 / 7 22 / 6 23 / 9Level 3 122 / 3 123 / 4 127 / 4Level 4 31 / 26 76 / 23 51 / 66Level 5 231 / 16 233 / 45 296 / 21Level 6 611 / 136 574 / 422 628 / 931Level 7 2412 / 123 2242 / 527 2834 / 624Level 8 4813 / 1152 3623 / 5356 1512 / 5842

Procedure. In Experiment 1, children first completed the six speed tasks(2 presentation modalities 1 3 operational difficulties) under instructions tosay the answers as fast as possible without mistakes. The ten integer additionsin each set were presented in turn, at a rate determined by the child’s answers.In the oral presentation condition the male experimenter read aloud eachaddition (e.g., ‘‘six add three’’). In the visual presentation condition eachcard was presented and remained visible until the answer was given. Thetotal time to answer all ten sums from the start to the end of the task wasdetermined using a stopwatch. All answers were recorded.

The addition span task involved presenting problems of increasingly com-plex additions to find the limit beyond which accurate performance brokedown. In the oral condition, problems were read aloud by the experimenterusing natural expression (e.g., ‘‘one hundred and twenty-one add seven’’).In the visual condition, problems were presented on cards, each card remainingin view until the subject had given an answer. Half the subjects were presentedwith the oral task first, half the visual. For each modality of presentation thethree types of problem, differing in operational difficulty, were examinedconcurrently. This was to avoid children anticipating the precise type ofproblem in advance and adopting atypical strategies. Thus, at each level ofcomplexity, an Easy Operations problem was followed by a Hard Operationsproblem and then a Carry Operation problem. For example, a child mightfirst be presented with 8 / 1 Å ?, then 3 / 5 Å ?, and then 5 / 9 Å ? Ifthe child correctly answered all three problems, he or she would progress tothe next level of complexity, starting with an Easy Operations problem (e.g.,21 / 7 Å ?; see Table 1). If a problem was incorrectly answered, a secondproblem with the same structure was given. A correct response on the secondattempt allowed the child to progress to the next level of complexity forproblems with that type of operation. However, a second error terminated thespan procedure for problems of that type. The entire procedure was terminatedwhen the subject had made two consecutive errors for each of the three typesof problem. A span was determined for each type of problem by awarding


28 ADAMS AND HITCH

TABLE 2Mean Addition Span Scores (Standard Deviations), Oral and Visual Presentation—

Experiments 1 and 2

Oral span Visual span

Group Easy Hard Carry Easy Hard Carry

Experiment 1 (UK)Yr.3 (age 8) 3.5 (1.2) 3.2 (1.0) 2.7 (1.0) 4.2 (2.1) 4.0 (2.0) 3.1 (1.4)Yr.4 (age 9) 4.2 (1.0) 3.4 (1.1) 3.3 (0.7) 6.0 (1.6) 5.1 (1.7) 4.1 (1.3)Yr.5 (age 10) 4.7 (0.8) 3.8 (0.7) 3.4 (0.9) 6.8 (1.7) 6.2 (2.1) 4.4 (2.0)Yr.6 (age 11) 5.1 (0.7) 4.1 (0.9) 3.6 (0.9) 7.4 (1.4) 7.0 (1.7) 5.7 (1.9)

Experiment 2 (Germany)Gr.2 (age 8) 4.3 (1.3) 3.4 (1.2) 3.4 (1.1) 5.4 (1.9) 5.1 (2.1) 4.2 (2.0)Gr.3 (age 9) 4.7 (0.8) 4.1 (0.9) 3.5 (1.1) 7.2 (1.6) 6.8 (1.8) 5.7 (1.9)Gr.4 (age 10) 5.4 (0.9) 4.7 (1.1) 4.2 (1.0) 7.9 (0.5) 8.0 (0.0) 7.1 (1.8)

one point for each level of complexity at which the subject correctly answeredat least one problem (e.g., correct solution of Easy Operations problems upto and including the level typified by 141 / 15 Å was given a span score of5; see Table 1).

Experiment 2 used the same procedure except that each child performedthe speed and span tasks in only one modality, and the experimenter whopresented the oral problems was a female native German speaker.

Results

Span tasks. In both experiments, mean addition spans were higher for visualthan oral presentation, they decreased with increasing operational difficulty,and they increased with age (see Table 2). Separate three-way (age 1 opera-tional difficulty 1 presentation mode) analyses of variance on the span datafrom each experiment revealed similar results (see Table 3). Thus, all themain effects were significant. There was an interaction between age and modeof presentation, reflecting a larger benefit from visual presentation for olderchildren (see Figs. 1 and 2). There was also a significant interaction betweenmode of presentation and operational difficulty, corresponding to a largeradvantage for visual presentation for problems with lower operational diffi-culty. The remaining interactions were nonsignificant.

Planned comparisons were used to examine the effect of presentation mo-dality in greater detail. In both experiments these showed that spans weresignificantly higher (p õ .05) in the visual modality at all three levels ofoperational difficulty and for all age groups except the youngest children forCarry problems. Although the youngest children did record marginally higherspans for the carry problems in the visual task, the differences were nonsig-nificant (Experiment 1: F(1,76) Å 1.54; Experiment 2: F(1,96) Å 2.1).

Speed tasks. Mean integer addition times are shown in Table 4 (note that



TABLE 3Analyses of Variances for Addition Span Data—Experiments 1 and 2

Source of variance Experiment 1 Experiment 2

Age (A) F(3, 76) Å 13.6*** F(2, 96) Å 22.8***Operational difficulty (OD) F(2, 152) Å 94.1*** F(2, 192) Å 37.8***Presentation modality (PM) F(1, 76) Å 163.*** F(1, 96) Å 87.6***A 1 PM F(3, 76) Å 9.19*** F(2, 96) Å 4.87**OD 1 PM F(2, 152) Å 12.5*** F(2, 192) Å 3.68*A 1 OD F(6, 152) Å 1.91 F(4, 192) Å 0.62A 1 OD 1 PM F(6, 152) Å 1.16 F(4, 192) Å 0.49

* p õ .05.** p õ .01.

*** p õ .0001.

trials containing errors were not excluded). In both experiments, times de-creased with age, increased with increasing operational difficulty, and werefaster for visual than for oral presentation. Separate three-way (age 1 opera-tional difficulty 1 presentation mode) analyses of variance were carried outon the speed data from each experiment (see Table 5). The results were thesame in all respects save that the interaction between operational difficulty andpresentation modality was significant in Experiment 1 but not in Experiment 2.The discrepancy is merely noted here as the aims of the study do not relateto this interaction in any obvious way.

FIG. 1. Graph of group mean addition spans by age, operational difficulty, and presentationmode (Experiment 1).


30 ADAMS AND HITCH

FIG. 2. Graph of group mean addition spans by age, operational difficulty, and presentationmode (Experiment 2).

As expected, errors in adding pairs of integers were infrequent (see Table6). However, they provided useful confirmation of the effectiveness of themanipulation of operational difficulty, as errors typically increased going fromEasy, to Hard, to Carry pairs. Errors also showed the expected decrease withage, and the oral task produced slightly more errors than the visual task.

Speed/span correlations. For each experiment, individual differences wereexamined by correlating children’s mean times for integer additions with theiraddition spans. This was done separately for each level of operational diffi-

TABLE 4Mean Calculation Times (s) and (Standard Deviations) for the Addition Speed Task,

Oral and Visual Presentation—Experiments 1 and 2

Oral speed Visual speed

Group Easy Hard Carry Easy Hard Carry

Experiment 1 (UK)Yr.3 (age 8) 34.5 (13.6) 53.4 (23.3) 80.0 (35.6) 27.0 (10.1) 45.0 (20.5) 70.4 (39.4)Yr.4 (age 9) 29.1 (9.1) 43.7 (13.2) 73.5 (23.1) 24.0 (7.3) 36.2 (10.5) 57.3 (15.6)Yr.5 (age 10) 23.2 (4.2) 35.8 (8.2) 56.4 (12.5) 20.3 (3.0) 30.8 (6.1) 45.2 (10.7)Yr.6 (age 11) 24.7 (8.8) 32.2 (10.3) 49.6 (16.8) 19.3 (3.4) 26.8 (7.9) 41.3 (16.3)

Experiment 2 (Germany)Gr.2 (age 8) 32.0 (5.6) 44.7 (14.6) 81.2 (35.8) 26.6 (8.2) 38.0 (10.2) 70.4 (23.7)Gr.3 (age 9) 24.5 (2.2) 31.9 (5.6) 51.3 (12.9) 17.9 (3.6) 23.1 (6.4) 38.2 (13.6)Gr.4 (age 10) 21.3 (2.1) 25.1 (6.3) 33.9 (7.4) 16.8 (2.9) 18.9 (3.4) 27.2 (5.6)



TABLE 5Analyses of Variance for Addition Speed Data—Experiments 1 and 2

Source of variance Experiment 1 Experiment 2

Age (A) F(3, 76) Å 8.00*** F(2, 96) Å 60.4***Operational difficulty (OD) F(2, 152) Å 267.*** F(2, 192) Å 205.***Presentation modality (PM) F(1, 76) Å 124.*** F(1, 96) Å 16.1***A 1 PM F(3, 76) Å 1.37 F(2, 96) Å 0.32OD 1 PM F(2, 152) Å 16.9*** F(2, 192) Å 1.46A 1 OD F(6, 152) Å 5.22*** F(4, 192) Å 29.5***A 1 OD 1 PM F(6, 152) Å 2.07 F(4, 192) Å 0.13

*** p õ .0001.

culty and for the oral and visual presentation conditions, both within agegroups and for all ages combined (see Table 7). The correlations were predom-inantly negative, indicating that in general, faster operations were associatedwith higher spans. More specifically, 14 of the 24 within age-group correla-tions for Experiment 1 were significant at the 5% level. However, only 1 ofthe 18 corresponding correlations was significant in Experiment 2. In theanalyses of data from all age groups combined, all 6 correlations were signifi-cant in both experiments. Eleven of these 12 correlations remained significantwhen age was partialed out.

The relationship between addition speed and addition span across age groupsand operational difficulties was examined using regression analyses. Figures 3and 4 illustrate mean mental addition span as a function of mean speed ofaddition for oral and visual presentation in the two experiments. The dataindicate relatively good fits to linear functions in all cases. Thus, regressionanalyses on the data from Experiment 1 showed that oral span Å 5.13 0 0.03C

TABLE 6Speed Task Errors (%) for Each Age Group by Operational Difficulty

and Presentation Modality

Oral Visual

Easy Hard Carry Easy Hard Carry

Experiment 1 (UK)Year 3 4.0 6.0 21.0 3.0 6.0 14.5Year 4 2.5 5.5 16.5 1.0 6.0 17.5Year 5 0.5 5.5 14.5 0.0 1.0 7.5Year 6 1.5 2.0 6.5 0.5 3.5 5.0

Experiment 2 (German)Grade 2 1.2 2.4 12.9 4.1 1.8 11.2Grade 3 1.2 1.8 8.2 0 0 8.2Grade 4 0.6 2.4 8.2 0 0 3.5


32 ADAMS AND HITCH

TABLE 7Correlations of Addition Speed and Addition Span for Each Age Group

and Combined—Experiments 1 and 2

Oral Year 3 Year 4 Year 5 Year 6 All ages APO

Experiment 1 (UK)Easy 00.59* 00.46* 00.58* 00.11 00.56*** 00.44***Hard 00.49* 00.60* 00.32 00.40 00.54*** 00.45***Carry 00.53* 00.23 00.33 00.22 00.45*** 00.34**

VisualEasy 00.70* 00.56* 00.37 00.75* 00.67*** 00.58***Hard 00.59* 00.54* 00.06 00.35 00.56*** 00.40**Carry 00.67* 00.36 00.52* 00.60* 00.57*** 00.45***

Oral Grade 2 Grade 3 Grade 4 All ages APO

Experiment 2 (Germany)Easy 0.13 00.44 00.53* 00.44** 00.33*Hard 00.48 00.44 00.33 00.57*** 00.46**Carry 0.23 00.22 00.34 00.33* 00.20

VisualEasy 00.31 00.37 0.03 00.56*** 00.42**Hard 00.38 00.05 0.00 00.58*** 00.39**Carry 00.19 00.35 0.03 00.51*** 00.33*

APO, Age partialed out.* p õ .05.

** p õ .001.*** p õ .0001.

(where C is calculation time and r Å 0.85), and that visual span Å 8.05 00.07C (r Å 0.83). Corresponding analyses on the data from Experiment 2showed that oral span Å 5.32 0 0.03C (r Å 0.81) and that visual span Å 8.460 0.07C (r Å 0.86). It is noteworthy that the parameters of the best-fittingfunctions appear to be consistent across experiments, while the parameters forthe oral and visual tasks appear to be systematically different.

Discussion

The main aim of the study was to address the role of working memory inlimiting children’s oral mental addition performance. Two aspects of theresults bear centrally on this issue. One is the prediction that visual mentaladdition spans should be higher than oral spans, as the visual display providesan external record of the addends to supplement working memory (Hitch,1978). The other is the prediction of a relationship between mental additionspan and integer addition speed of the type shown previously for the countingspan task (Case et al., 1982).

The results of both experiments agreed in showing that children’s visualaddition spans were consistently higher than their oral spans across all three



FIG. 3. Correlation of group mean addition speed and span scores for all three levels ofoperational difficulty (oral and visual presentation, Experiment 1).

levels of operational difficulty and all age groups. Thus, providing an externalrecord of the addends is helpful, suggesting that remembering the addendsrather than arithmetical competence constrains children’s mental addition spansunder oral presentation. It could be argued that the effects of presentationcondition reflect the modality rather than duration of presentation. However,immediate memory for verbal materials is similar for auditory and visual presen-tation, with the main exception that auditory presentation gives rise to betterrecall of the most recent item or two (Murdock, 1967). It is therefore difficultto see how modality per se could explain the span differences reported here.

The interaction effects involving presentation modality of the addition spantask provide further evidence for the role of working memory. One of theseis that older children benefited more from visual presentation than youngerchildren. This can be explained by noting that older children were able to dooral sums involving longer addends. Therefore, at the limit of complexity setby oral span, visual presentation provided older children with more informa-tion to supplement working memory than younger children. A related findingis that the benefit of visual presentation was greater for calculations involvingeasier integer operations. This can be explained in a similar way by notingthat oral spans were higher for problems involving easier operations, so once


34 ADAMS AND HITCH

FIG. 4. Correlation of group mean addition speed and span scores for all three levels ofoperational difficulty (oral and visual presentation, Experiment 2).

again, at the limit set by oral span, visual presentation supplemented workingmemory with a greater amount of information.

More generally, the results indicate boundary conditions beyond whichcompetence and not working memory may be the limiting factor. Thus, whenthe youngest children were given additions involving carrying, their spanswere low and the small benefit from visual presentation was not significant.This suggests that some of the youngest children’s performance on theseproblems may have been constrained by a lack of competence with the carryoperation. The accuracy of these children’s performance on the speed tasksuggests they had some problems in adding pairs of digits that generate acarry (see Table 6). Despite this it seems that for most school-aged children,oral mental addition is limited by working memory and not by competence.However, competence limitations would clearly be expected to be more perva-sive with less practiced mental arithmetic skills than addition.

The second major prediction concerned the relationship between mentaladdition span and the speed of adding integer pairs. The results fulfilled theexpectation that the speed of addition would be sensitive to children’s ageand the difficulty of the operations, consistent with previous research on agedifferences (e.g., Ashcraft & Fierman, 1982; Kaye et al., 1986; Geary et al.,



1991) and the problem-size effect (Ashcraft, 1992, 1995). As regards thespeed/span relationship, children who were fast at performing integer addi-tions tended to have higher addition spans, and these differences remainedimportant even when age was partialed out. However, this correlation doesnot constrain interpretation greatly, given that the speed and span tasks involvesimilar processes. The critical result is the observation of an approximatelylinear relationship between span and speed across age-groups and differencesin difficulty of the calculation operations (see Figs. 3 and 4). This is the sametype of relationship that Case et al. (1982) found for counting span, and weinterpret this as suggesting that both types of span reflect a similar underlyingconstraint imposed by working memory.

It is interesting that the way the mental addition task was presented affectedthe parameters of the speed/span relationship and not its form. This suggeststhat performance is still limited by working memory even when, as withvisual presentation, the numbers to be added do not have to be stored inworking memory. At first sight this might seem anomalous. However, it isclear that making the numbers to be added permanently visible does notobviate the need to keep track of partial results or to combine memory forpartial results with mental operations. Thus, we would argue that mentalarithmetic with oral and written numbers are both tasks that load workingmemory, albeit that the load is lower when the numbers are written.

It is remarkable that the parameters for the span/speed relationship wereso similar to those for the German and British samples. Informal observationsrevealed that over half the British children showed evidence of counting suchas using fingers, nodding, or lip movements. As expected, such behaviorswere more common in the younger children, but several older children alsoshowed them. In contrast, the majority of German children, including theyoungest group, appeared to rely on retrieval, or at any rate, strategies thatdid not involve overt signs of counting. We therefore conclude that the rela-tionship between addition speed and addition span is relatively insensitive tocultural variation in the nature of addition operations. As noted above, apossible reason for the difference in addition behavior is the tendency forGerman children to receive more practice in mental arithmetic (Bierhoff,1996). However, no reliable information was available concerning the educa-tional training that children in the present study had received.

It is also remarkable that the plots of span against speed indicate a smoothlinear function spanning a period of education and development in whichthere are marked qualitative changes in the way arithmetical operations areperformed (Ashcraft & Fierman, 1982; Hamann & Ashcraft, 1985; Ashcraft,1990). We conclude from this that the relationship between addition speedand addition span is relatively insensitive to developmental variation in thenature of addition operations. More generally, a lack of sensitivity to thespecific form of operations (both developmentally and cross-culturally) isprecisely what would be expected on the assumption that working memoryis a central, general-purpose resource.


36 ADAMS AND HITCH

It is informative to compare addition span with tasks that are standardlyused to measure working memory, such as counting span, operation span,reading span, and complex word span (see above). As stated earlier, additionspan differs in being a natural skill with intrinsic storage and processingcomponents that are difficult to separate out. In contrast, separate storage andprocessing components are explicitly built in to the standard tasks. This makesthe standard tasks easier to analyze. Moreover, in the absence of a detailedmodel of multidigit addition it is difficult to interpret addition span in termsof parameters of working memory. However, addition span has an advantagein being a natural task with high intrinsic educational and practical importance.The present evidence that addition span reflects working memory dynamicsin the same way as the standard but more artificial task of counting span,suggests that it may be a useful tool in future investigations.

To conclude, the present results extend previous evidence concerning theinvolvement of working memory in arithmetic in both adults (e.g., Hitch,1978; Logie et al., 1994; Posner & Rossman, 1965) and children (Hitch etal., 1987). They are inconsistent with the claim that performance in reasoningtasks is independent of (verbatim) memory for the problem information(Brainerd & Reyna, 1993). However, Brainerd and Reyna (1992) acknowl-edged the possibility of exceptions to such independence and noted mentalarithmetic as a likely example (see also Brainerd & Reyna, 1988). The presentresults also seem at odds with the observation of preserved calculation abilityin a neuropsychological patient with a marked working memory impairment(Butterworth et al., 1996). However, the patient had been a pharmacist andmay have been relatively expert at arithmetic prior to his illness. In theabsence of an indication of his premorbid arithmetical ability, it is at the veryleast unclear whether his mental calculation was unimpaired.

Thus, to summarize, the studies reported here were designed to evaluatethe role of working memory in schoolchildren’s arithmetic. It was found thatwhen working memory load was reduced by making the addends in a mentaladdition task permanently available, performance improved. It was also foundthat there was a systematic relationship between mental addition span andthe speed of integer addition. These findings generalized over cross-culturaldifferences between British and German children. Taken together the resultssuggest that working memory is an important general-purpose resource sup-porting schoolchildren’s mental arithmetic performance. The evidence thatmental addition reflects the dynamics of working memory in the same wayas the artificial task of counting span is important in generalizing previousfindings to a natural task with direct practical applications.

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RECEIVED: May 28, 1996; REVISED: June 26, 1997


working memory and children's mental addition

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