Starting to add worse: Effects of learning to multiply on children's addition

Cognikm, 37 (1990) 213-242

KEVIN F. MILLER University of Illinois at Urbana-Champaign DAVID R. PAREDES University of Texas at Austin Received September 18, 1989, final revision accepted June 12, 1990

Abstract Miller, K.F., and Paredes, D.R., 1990. Starting to add worse: Effects of learning to multiply on children’s addition. Cognition, 37: 213-242.

A major stumbling block in acquiring a new skill can be integrating it with old but related knowledge. Learning multiplication is a case in point, because it involves integrating new relations with previously act,luired arithmetic kno wledge (in particular, addition). Two studies explored developmental changes in the relations between single-digit addition and multiplication. In the first study, third-graders, fifth-graders, and adults performed simple addition or multiplication in mixed- and blocked-operations formats. Substantial interfering effects from related knowledge were found at all age levels, but were more pronounced for younger subjects. Thus in the early stages of learning multiplication, one consequence of learning a new operation is interference in performance of an earlier, related, but less recently studied skill. Consideration of error patterns supported the view that the problem of integrating operations is a prominent one even in the early stages of mastering multiplication. Patterns of errors were generally consistent across all age groups, and all groups were much more likely to give a correct multiplication response to an addition problem than the reverse. A second, longitudinal study confirmed this finding, showing evidence for ._

_.

*The research reported here was supported by NSF Grant BNS-8510546 and a Spencer Fellowship to the first author. We would like to thank Leslie Cohen, Colin MacLeod. Michael McCloskey, Herbert Pick, and James Stigler for helpful comments on an earlier draft of this paper, and acknowledge the late Sara Halpern for important contributions to the planning of Study 1. We are also grateful to the students and staff of Cedar Creek Elementary School (Eanes Independent School District) and Odom Elementary School (Austin Independent School District) for their assistance with Studies 1 and 2, respectively. Address reprint requests to: Kevin F. Miller, Department of Psychology, University of Illinois, 603 E. Daniel, Champaign, IL 61820.

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lder children almost always outperform younger children, and the ubiquity of age-related improvements in performance has made findings of nonmonotonic developmental changes a source of recurrent interest for psychologists. The phenomenon of U-shaped developmental curves is perhaps most familiar in the case of children temporarily abandoning correct but irregular syntactic structures in fav of over-generalization of regular Bowerman, 1982; Brown, 1973; bee & Slobin, 1982; Ervin, 1964; 1978), but patterns of non-monotonic development have been rein domains ranging Tom infant imitation (Maratos, 1982) to the intuve physical concepts of older children (Kaiser, McCloskey, & Proffitt, 1986; iloff-Smith, 1984). Interpretations of non-monotonic developments Interest in non-monotonic development has itself shown a non-linear pattern over time. A review of the PsycLIT database shows that 35 articles were published in 1982-1983 involving human subjects and using either nonmonotonic or “U-shaped” in their abstracts (along with at least two edited volumes dealing with the topic: Bever, 1982a; Strauss & Stavy, lY82), while there were only 13 articles fitting this criterion in the four years from 1984 to 1987. One reason for this drop-off in interest in non-monotonic developmental curves has been the argument (Bever, 1982b; BowLerman, 1982; Carey, 1982; Klahr, 1982) that non-monotonic changes reflect the same developmental processes that underlie changes that are monotonic in course.

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Yet even if non-monotonic changes in performance do wx uniquely specify reorganizations of cognitive abilities, they may nonetheless be useful in identifying the points at which children reorganize their knowledge in a given domain. In this view, reorganization of skills need not imply non-monotonic changes in performance, but U-shaped developmental curves do imply some kind of reorganization (e.g., Carey? 1982). A more sweeping rejection of the view that non-monotonic changes are of particular interest to developmentalists was presented by Klahr (1982) and Siegler (1983), who argued that ‘U-shaped curves are due to limitations in the way children’s knowledge is assessed by researchers. Klahr (1982) argued that findings of non-monotonic developmental changes reflect artifacts of the procedures by which competence is being assessed, with children’s performance always changing in the direction of greater competence. Siegler (1983) made a similar point, arguing that “U-shaped curves arise when children adopt a new rule that is more predictive when the total task environment is considered but that is less predictive in a particular portion of the task environment” (p. 269). The position taken by Siegler and Klahr dismisses non-monotonic developmental changes as being of any special significance, and the impact of this argument may account for the diminution in interest in U-shaped growth curves. It is worth noting, however, that the argument assumes that U-shaped development is a consequence of children integrating new and old knowledge within a domain (or “total task environment”). The assumption that children are integrating new and old knowledge in this manner is not a trivial one. The fact that analogous or even identical concepts end abilities differ across different domains of knowledge is a well-established phenomenon of human cognition. Many of what were once. believed to be general developmental changes, such as the ability to distinguish defining from characteristic features (Keil& Batterman, 1984) or the use of active retrieval strategies (Chi, 1985) have been shown to vary across domains of knowledge as a function of children’s understanding of these domains. Furthermore, the domain dependence of knowledge is clearly not a transitory phenomenon of childhood. Descriptions of expertise beginning with deGroot’s (1965) classic study of chess experts have consistently emphasized domain-specific knowledge rather than general strategies or skills as the key feature that distinguishes experts from non-experts. One question that has not been directly addressed in work that demonstrates the domain-specific nature of much of knowledge is how one is to demarcate a domain. In many cases defining a domain presents no real difficulty: experts in chess or calculation have their own ideas of their field. In other areas, however, the relevant domains are less clear. In the case of

ller and DA

Pmedes

65) argued that “addition and mulmathematics, for example, Piaget (19 e implicit in the construction tiplication of classes, re number” (p. 241). Although these of every class, every re mbers, Piaget realized that these relations are implicit in the structure by children. Whether children’s implications are not automatically re ndent, !arge!y modular skills, arithmetic is best understood as a set sses is unclear. Studving the or as an interconnected matri matical skills provides a means developing relations between d skill (or skills), and findings of of characterizing the nature of s are of special significance. NonU-shaped changes in particular pr rtise may provide a way to monotonic changes with develop apply. To the extent that rules or demarcate the domains over which ski skilis invoive different domains pect to find little impact of new knowledge on old skills or be uch interactions, on the other hand, implies that the interac 11within the same functional domain. The development of simple arithmetic (adding and multiplying single-digit numbers) provides a natur ntext for looking for possible effects of new skills on previously learn cesses. To the extent that children’s knowledge about arithmetic is grated into a single interrelated system, one ought to find interference tween related operations at some point in the acquisition of new know1 Existing models of the development of simple arithmetic will be discu with an emphasis on their implications for interrelations between the re skntations of different operations.

The ability to rapidly and reliably retr’ the sum or product of two singledigit integers is surelv one of the most c of the skills mastered in the eariy school years. Two decades of researc ognitive processes underlying this simple task has shown the vari genuity of the methods that children use in performing simple a rthermore, there is a nearconsensus across models that children ally shift from procedural algorithms for performing simple arithmetic, such as adding by various counting algorithms, to direct retrieval of known a swers (see Baroody & Ginsburg, 1986, for an exception to this view). This search has also left a number of puzzles, perhaps chief among them the ding (e.g., Ashcraft, 1982) that much of the change in the time-course of children’s performance of simple addition occurs in middle elementary school, after the point at which instruction in basic addition has been completed. After reviewing past research on the development of simple arithmetic, we will discuss possible explanations for such late changes in the processes children use to perform simple addition.

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Children’s strategies for performing simple arithmetic There are a number of ways that one might perform simple arithmetic, and interview studies of children’s procedures indicate that young children use most of them. In the )case of addition, one could add two numbers by: (aj directly retrieving their sum, (b) reconstructing the answer based on knowing the answer to a relate&j problem (e.g., adding 6 + 5 by retrieving the sum of 5 + 5 (10) and adding one), (c) counting-on from one of the addends a number of counts corres‘ nding to the other addend (e.g., counting 6 + 5 by saying “7. 8, 9, 10, ll”), or (d) counting-all of the addends. In the case of multiplica In, one can (a) retrieve the answer, (b) reconstruct the answer from a known product (“6 x 4 = 24 SO7 x 4 is 4 more9’), (c) use repeated addition to compute the answer (6 x 4 = 6 + 6 + 6 + 6)9 or (d) count by one of the multiplicands (6 x 4 = “6, 12, 18, 24”). These procedures vary greatly in the amount of cognitive record-keeping involved (Baroody & Ginsburg, 1986; Fuson, 1982) and developmental shifts seem to be in t direction of increasing use of more efficient strategies. Several interview studies of the strategies children use in performing simple addition (that involving single-digit numbers) have shown both the variety of ctrfitpgies children use and developmental shifts in favor of increasing use of I_____ retrieval of answers without overt strategy use and counting-on from the larger addend. Houlihan and Ginsburg (1981) interviewed first-and secondgraders concerning strategies used for both simple and complex addition :crm+:m”PLE&l&ais1&Tg ~w~l.~A~*r~ ;nAptnrndn~tn nr inan_ allu"i-"rrii.6liPP~is"-4s ill-L problems. For single-digit comblllCiLlVllS9 propriate procedures (the latter category limited to first-graders) 5% of tri;s were direct retrieval, 40% were counting from one to the sum, and 44% were counting-on from one addend to the sum. By second grade, 31% of the trials for which a procedure could be determined were direct retrieval, 3% used counting from one, 52% of trials used counting-on, and 14% involved reconstruction from a known sum. Siegler (1987a) reported a similar developmental progression between kindergarten and second grade from an initial reliance on counting-all to an increasing use of retrieval and counting-on procedures, although he emphasized that children continue to use a mixture of strategies throughout this period. Looking at Swedish children, Svenson, Hedenborg, and Lingman (1976) found a similar pattern, with tl-&-graders showing direct retrieval on about a third of trials, and various counting procedures about half the time. Two longitudinal studies of children’s strategy addition during the first three years of elementary school - Carpenter oser (1984) and Svenson and Sjijberg (1983) - both describe a gradual decrez.se in counting-all procedures, with an increase in direct retrieval and counting-on from one of the addends, usually the larger one.

There have been fewer studies rep0

n use in multihave described

direct retrieval of answers. de1 of the relaare consistent with a rational ports of strategy us multiplication, in which chil n use efficient between addition ithms. For addition, this strategies to short-cut more labor-intensive nL.g. For multiplication, means that retrieval is a short-cut to repeate ucts is a short-cut to repeated addition and other back-up retrieving en children don’t know the answer to a particular problem, strategies. strategy” (Siegler, 1988; Siegler & Shrager, they fall back 01, a “b&-u 1984) that permits them to calculate an answer they are unable to retrieve. New arithmetic operations build on previous knowledge (which provides the back-up strategies), but do not seem to affect it, at least in terms of the overt strategies that children report using. Chronometric research on the performance of simple arithmetic operations supports a similar developmental picture of a movement from calculation to direct retrieval of known answers, but suggests (at least for adults) a more direct connection between the representations of addition and multiplication.

~~r~n~rnetric models for

arithmetic

en (1967) described a series of counting-based models that in performing simple arithmetic. These models shared the t children used au internal counter, which could be set to any incremented by ones. Under the further assumptions that the time required to set the counter is independent of the value to which it is set, and that incrementing the counter takes a constant amount of time, algorithms for add distinctive @edictions concerning the relation between the size of and the time required to add them. If, for example, children added by starting one and counting up each of the addends in turn, then reaction time ( should be a function of the sum of the two addends (the number of counts required). On the other hand, if children began counting from the larger of the addends, RT would correspond to the smaller addend (the number of counts required in this, the most efficient counting algorithm). This was termed the min counting model. Croen and Parkman (1972) found that the best predictor of simple addition in first-graders adding numbers summing to ten or less was the smaller of the two addends, suggesting that first-graders use an efficient counting procedure in performing simple addition.

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One problem with the min counting model proposed by Suppes and Groen was that it fit adult data at least as well as it did that of children (Parkman & Groen, 1971). Groen an Parkman (1972) suggested that adult data might result from the combination of two competing processes: rapid retrieval (with no consistent relationship between problem size and RT) and reversion to a min counting procedure in cases of retrieval failure. Network retrieval models Later studies of mental addition in adults (Ashcraft & Battaglia, 1978; Geary, Widaman, & Little, 1986; Miller, Perlmutter, & Keating, 1984) provided no srlppt for the idea that college students ever use a slow counting procedure in performing simple arithmetic. Instead, these researchers argued that calculation time reflects the difficulty of accessing a stored answer in a memory network. In the studies cited above, it was argued that the general increase in RT with increasing problem size could reflect the “distance” between entry points and answers in a general table-like representation. Finding that variables such as the square of the sun or the product provided the best fit with adult RT, it was proposed that such a table was stretched in the direction of larger numbers (Ashcraft & Battaglia, 1978), or that RT was a function of the area of a matrix activated by a given pair of numbers (Geary et al., 1986). Miller et al. (1984) presented evidence that simple addition and multiplication are performed in a similar fashion by adults, and argued that the difficulty of procedural approaches to multiplication (e.g., repeated adding) implied that this must involve a retrieval process of some sort. Furthermore, Miller et al. found that the best predictor of RT for a given problem for a given operation (e.g., addition) was the time taken to perform the other operation (multiplication) on the same pair of numbers, suggesting that addition and multiplication in adults involve interrelated networks of stored information. Miller et al. noted that problem size alone gave limited prediction of the time adults took to add or multiply two numbers, and argued that it was necessary to distinguish between a problem’s position in a network (indexed by its size) and its accessibility (as indexed by the number of errors subjects made on the preblsm). Geary et al. (1986) found that the same regression equations with the same parameters could account for the time required for both multiplication and addition in adults. Several important shortcomings exist in network retrieval models that assume that aspects of problem size are incorporated directly into the structure of adults’ representation of arithmetic (Campbell & Graham, 1985; Comet, Seron, Deloche, & Lories, 1988; Siegler, 1938). Ashcraft (1987) argued that the correlation between problem size and RT in adults’ addition is an artifact of the frequency of problem presentation, and developed a simulation ii1

-whk.h strengths of associations between answe

calculate the answer to “3 x 9").Graham (1987) provided developmental evidence for this process by manip ng the order in which multiplication ird-grade subjects. Effects of problem were learned by a set of T were found only for ch who learned problems in the (standard) order beginni ms and working up to larger ones. For all subjects, a significant relation was found between learning order and performance, with problems learned first easier to recall. smm?gy choice models

1988; Siegler & Shrager, 1984) developed the first Siegler (1987a, 198 models for mental arithmetic that explicitly incorporated multiple-solution strategies into a working simulation of children’s arithmetic performance. Each of these models incorporate a variety of procedures for solving arithmetic problems, including direct retrieval and several back-up strategies, including counting for addition and repeated addition in the case of zalultiplication. Children begin with an initial distribution of associations of varying strengths between each problem and particular answers. Answers are retrieved for a lem as a function of the strength of their association to that probthe likelihood that a given answer will be retrieved is a function wn strength of association and of the association between other that problem; problems that showed “peaked” distributions of associations have one answer that is likely to be retrieved, while retrieval for problems with flatter distributions will produce more variable answers. Problems for which a retrieved answer exceeds a confidence criterion are solved by direct retrieval, while other problems are solved by means of a back-up procedure. Doing arithmetic changes the structure of associations between answers and problems, because answers to a problem (whether produced by retrieval. or b$ back-up strategies) are strengthened as associates to that problem. Over time, the distribution of associations is changed as a function of (a) the characteristics of back-up strategies (including the errors that they can produce), and (b) the frequency with which problems are presented. Although Siegler has developed strategy-choice models for both addition and multiplication, the only relation between the two operations in these

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models is that repeated addition serves as the

up strategy for multiplica-

tion in cases where retrieval fails. Si begins with a uniform distribution of and each problem; the addition simulation (1984) does not take into account any effe addition.

multiplication simulation etween all numbers &lo0 ed by Siegler and Shrager ultiplication on children’s

How arithmetic operations are related

.

There is substantial evidence for adults tha representations of addition and multiplication are interrelated. Winke and Schmidt (1974) noted that subjects had difficulty rejecting proble senting the correct product of two addends (such as 3 + 4 = n ta:ik. Miller, Perlmutter, and Keating (1984) found that such cross-operation errors constituted about a quarter of the mistakes that adults made while performing separate blocks of simple addition and multiplication. IXT data from both production and verification tasks also suggest that similar processes are involved in multiplication and addition by adults (Geary, Widaman, & Little, 1986; Miller et al., 1984; Stazyk, & Hamman, 1982). The question of how these operations becom ted and when in development such integration occurs has not been addressed. Understanding at what point children integrate their Tepresentations of addition and multiplication may help to resolve several controversies in the development of arithmetic skill. Baroody (1983,1984; Baroody & Ginsburg, 1986) has argued that it is not parsimonious to assume that children shift from a procedural (algorithmic) representation of addition COa &clarative (retrievnl) process, suggesting instead the gradual development and automatization of calculation procedures coupled with an understanding of mathematical principles (such as commutativity) that underlie arithmetic. If children solve addition problems through counting procedures, conf’usiczc, with the products of the two numbers presented chn~~!/r!X ieiatively rare, and limited to those cases in which children misread the operation sign (Baroody, 1984). Thus evidence of cross-o eration effects, particularly in blocks of trials composed solely of addition problems, suggests that children are using retrieval procedures. Baroody and Ginsburg (1986) have argued that children’s arithmetic knowledge consists of a set of interrelated experiences and relationships rather than a set of independent associations. Evidence of effects of multiplication on children’s addition is consistent with this aspect of Baroody’s model, suggesting that children are integrating their new skill with their previous arithmetic knowledge. It may be that such interference is a late developing process. Hamann and Ashcraft (1985) did not find consistent evidence

222

of a confusion effect in a verification paradig until tenth grade, although as only a small aspect of a much larger study. thi children do represent arithmetic operations as a set of interrelated processes, then managing the relations between these operations should be an important aspect of developing mathematical skill. The first study was designed to collect cross-sectional data on the developing relation between addition and multiplication, by looking at children’s performance of simple (single-digit) addition and multiplication in two different contexts. In one context (blocked presentation), children saw sets of single-operation problems (all multiplication or all addition). In the other context (mixed presentation) children saw mixed sets of both o ggested that a transition counting to primary use (as shown by I&T data) occurs between the third and fourth s is also the period in which children are actively engaged in mastering multiplication. Therefore we looked at children in this period, to look for changes in the interrelation between the performance of addition and multiplication. Two Questions of particular interest are: (1) at what level of skill do cross-operation errors become a significant source of children’s mistakes in performing simple arithmetic; and (2) to what extent is there evidence that learning a new procedure (multiplication) affects the performance of a previously mastered skill (addition)?

Subjects

A total of 94 subjects (33 third-graders, 32 fifth-graders, and 29 adults) took part in two 20-minute sessions. Children came from the third and fifth grades of a public school serving a middle- to upper-class suburban nity in the southwestern United States. Data from the Texas E cy sho-a that 97% of the third-graders in the school used in this study d the mathematics section of the Texas Educational Assessment of Skills ) test, a measure of basic academic competence 11thir in Texas schools. This compares to a state-wide average passing rate of $6% for the 1985-1986 school year, when this study was conducted. Third-grade subjects (15 males, 18 females) ranged from 8.4 ge. Fifth-grade subjects (19 males, 13 females) ears old. Adult subjects (13 males, 16

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females) were introductory psychology students who participated for course credit. Stimuli

Stimuli consisted of single-digit addition and multiplication problems of the form “a -t b” and “a X b” arranged in blocks of pure addition, pure multiplication, and mixed addition and multiplication. Pure blocks consisted of 55 problems each: 45 unique pairs of 10 digits (ignoring order) plus the ten pairs 0f ties formed by pairing each digit with itself. Blocks of addition and multiplication were divi_ded approximately in half and recombined to form complementary sets of mixed blocks of 55 problems each: that is, any set of two blocks of mixed operations inclu ded all unique combinations of single-digit problems and tie problems for both addition and multip!ication. Problems were sorted randomly within blocks, with the constraint that no two consecutive problems produced the same correct answer nor could the same number occupy the same position (left or right addend/multiplicand) across consecutive problems. In the case of mixed operations, the same operation was not allowed to appear more than three times in a row. A total of 54 such random orderings of problems were gererated, and were randomly assigned to subjects. Apparatus

Stimuli were presented on an IBM-PC AT microcomputer equipped a microphone connected to a voice-operated relay. An internal timing gram using the computer’s hardware timer and a voice communications gram provided millisecond timing between the presentation of a problem a spoken response.

with proproand

Procedure

Adult subjects were tested in a small, sound-proofed experimental room. School-aged subjects were tested individually in a mobile laboratory van parked outside the subjects’ school during the last quarter of the school year. The van contained two subject rooms which were sound-proofed to permit two subjects simultaneously to be run at d time. Subjects were taken from their classrooms and tested individually in front of the computer monitor and microphone. The experimental task was explained and instructions were presented on the computer screen and read to subjects. Subjects were shown 5 demonstration trials, with pauses between trials to answer any questions they had and, if necessary, to remind them of the need to respond quickly. The experimenter remained seated out of the subject’s field of vision during the

ent, and recor ed any en-cm

rices in which the microphone

t period separating the gle-operation) blocks

blocks sessions was counterbalanced across subjects. Testing sessions lasted from 15 to 30 minutes and time between sessions ranged from 2 to 9 days (M

verall error rate in the set of 20,680 responses was 8.13%‘) which varied from 15.99% (addition problems presented in the mixed-block condition for third-graders) to 3.17% (addition problems presented in the pure-block condition for adults). There were a total of 398 (1.93%) microphone errors, in icrophone failed to react to a subject’s response (this also includes subjects coughed, or made extraneous remarks that were registered as a reponse). Table 1 presents the substantive errors: those in which subjects produced a number other than the correct response to the problem Pr nted. particular interest are the cross-operation confusion errors, which consistently accounted for roughly twice the proportion of errors in blocks of pure addition as in blocks of pure multiplication. When making errors, adults as well as children were about twice as likely to produce multiplication answers to addition problems than to offer addition answers to multiplication problems. When addition and multiplication were mixed within blocks, incidents of cross-operations rose both in addition and multiplication; however, all age groups continued to produce more multiplication response to addition problems than vice versa. This asymmetry of cross-operation errors suggests that retrieval procedures for doing simple multiplication begin to compete with addition strategies as early as the tnird grade and that some effects of that competition persist to adulthood.

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Table 1.

225

Error patterns: Experiment I Breakdown of errors by operationa Pure addition Thirdgraders Overall error rate 7.50 % of total errors: Cross-operation 58.59 Off by factor of l 46.88 Off by factor of 2 10.94 Zero in factor 7.811 One in factor 17.97 Other 8.95 __ _______ __-____-_-____

Overall error rate % of total errors: Cross-operation Off by factor of 1 Off by factor of 2 Zero in factor One in factor Other -____ -~~-

Pure multiplication

Fifthgraders

Adults

Thirdgraders

Fifthgraders

Adults

4.08

3.17

6.67

3.49

5.72

42.03 59.42 15.94 0.08 21.74 5.80

25.00 62.50 20.38 0.00 6.25 6.25

28.95 42.86 12.28 14.04 9.65 37.72 ___.-

22.03 71.19 16.95 16.95 5.08 18.64 -____

9.88 49.38 27.16 7.41 1.23 32.10

Mixed addition - ------.-

Mixed multiphcation -_____-

Thirdgraders

Fifthgraders

Adults

Thirdgraders

Fifthgraders

Adults

15.99

11.15

5.18

9.55

7.54

5.48

81.32 28.57 5.13 15.02 19.41 7.33

77.78 30.69 8.47 17.46 18.52 6.88

48.65 39.19 29.27 6.76 17.57 10.81

59.88 46.91 11.11 16.67 26.54 17.9 _-____

55.56 45.24 3.35 15.87 16.67 19.84

26.32 47.37 25.09 9.21 6.58 28.95

_

-

aBased on all wrong answers given by subjects (i.e., ignoring microphone errors). Because error types may overlap, sums of error rates can exceed 100%. Error types are defined as follows: Cross-operation: Giving the correct product to an addition problem, or the correct sum to a multiplication problem (e.g., saying “12” to “3 + 4”). Off by factor of I: Giving an answer correct for one of the arguments +l (e.g., saying “24” to “3 X 7”). Off by factor of 2: Giving an answer correct for one of the arguments +2 (e.g., saying “15” to “3 x 7”). Zero in factor: Problems with a “0” as an addend/multiplicand. One in factor: Problems with a “1” as an addend/multiplicand. Other: All other errors.

iller and D. R. Paredes

Response

time

results

ade (3: third, fifth, and college) X Sex (2: male, female) X Operation ~ltiplicati~n) x Context (2: pure VS.mixed blocks) MANOVA and context as within-subjects factors) was calculated, using et’s median IXTs for all correct responses within each Operation X ontext condition ;ds the dependent measure. of grade, F(2,86) = 54.21, p < .oOl, and operation, F(l,86) = 8.77, p c ntext, F(2,86) = 17.19, p e .OOl, and = 5.89, p < .005, with a marginal Grade X = 2.95, .05 < p e .06. The effect of speed between all adjacent age groups, ntly faster in the pure blocks than in the and each age group tests for both post-hoc comparisons). The mixed blocks (using effects of context and operation differed by grade, as Figure 1 suggests, and y..~xts \vere assessed by a the relations between operatiora in the differen: CW+ series of tests of simple effects, with I% qferroni adjustment of cc-levels to reflect the number of comparisons made (Keppel, 1982; Stevens, 1986). For third-graders, addition was significantly slower than multiplication in the pure block context, F( 1,30) = 7.81, p < .OS.while there was no significant effect of arithmetic operation in the mixed block context, F(1,30) = 1.86, n.s. For fifth-graders, there was no significant effect of operation in the pure block context, F( 1,29) = .038, n.s., but multiplication was faster than addition in the mixed block context, F(1,29) = 5.30, p e .05. Finally, adults teresting pattern, with addition faster than multiplication in the xt, F(1,27) = 13.20, p e .05, and slower than multiplication ock context? F(l,27) = $.76, p e .05. udy 1 are consistent with the view that learning to multiply has an early and substantial impact on children’s addition. Interference between addition and multiplication (as shown by errors in addition) occurs at an early stage of learning to multiply, and third-graders were faster at performing multiplication than addition, a pattern the reverse of that observed in adults. Even for adults, performing simple addition and multiplication is significantly affected by having to do so in a context that involves the other operation, and cross-operation errors are a common source of mistakes for adults as well as younger children. ese results show that the fact that third-graders can multiply is certainly relevant to their performance of addition, but they certainly do not show that children’s addition becomes worse as they learn to multiply. Furthermore, it may be that any disruptive effects of multiplication upon children’s addition are limited to lower-ability children who do not understand the conceptual

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relations between these operations. Eampert (1986) has suggested that a variety of “bugs” in children’s multiplication are due to their failure to connect ious kinds of conceptual knowledge (termed intuitive, principled, and condge) with their computational procedures. Perhaps interference tion and multiplication is limited to children who lack a general understanding of mathematical concepts, resorting instead to rote and mechanical computational techniques. A second study followed a set of second-, third-, and fourth-graders over Figure 1.

Developmental changes in time required for simple (single-digit) addition and multiplication when performed in pure (single operation) blocks, or mixed operation blocks. Data plotted are the average for each grade of each subject’s median reaction times for correct answers. Pure operation blocks are plotted with solid lines, mixed operation blocks with dashed lines. Thirdgraders are significantly slower at simple addition than multiplication in pure blocks, an effect that reverses for older subjects. 6

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ller and D.R. Paredes

oking at longitudinal changes in performance of simple ication. To the extent that children are developing an presentation ‘of arithmetic knowledge, learning to multiply a reorganization of children’s addition, with a temporary deation is a necessary part of learning to uld be observable in subjects from both advanced and regular math classes.

ee

tal of 93 children from the second, third and fourth grades of a middlean school in the southwestern United States participated in all rounds udy (a total of IQ9 children took part in at least one session). As a ) this sample was somewhat less mathematically skilled than subjects in t study; 81% of third-graders in this school passed the TEAMS test, : ed with a state-wide average passing rate of 91% during the 1987-1988 ear, when this study was run. Of the 93 students who took part in two second-graders were dropped from analysis, one because of rors in addition (35% errors vs. 4.3% for the rest of the second another because her final testing session was interrupted when to go to the bathroom. A total of 91 students qontiibuted data al analyses: 30 second-graders (16 females, 14 males), 25 third-graales, 12 males), and 36 fourth-graders i.24 females, 12 males). At the beginning of testing, second-graders ranged from 7.0 to 8.5 (M = 7.1) third-graders were from 7.9 to 10.1 (M = 9.0) years old; and s ranged from 9.1 to 11.1 (M = 9.8) years old. nts in the second and third grades attended two levels of math classes: classes and high (advanced) classes. Fourth-graders attended four levels of math classes. The four levels of the fourth grade were collapsed into two categories (regular and high classes) for comparison with the lower grades= Students in the advanced math classes proceeded through the regular curriculum at an accelerated pace in addition to covering more advanced gh math second-graders were introduced to multiplication, for inthe first third of the school ye hile their peers in regular *math s were still mastering addition. ta from administration of the atics section of the Iowa Test o asic Skills in the spring showed

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229

there were substantial achievement differences between these groups: second-grade regular math class students had a mean grade equivalent of grade 2.97 (range 2.4-3.3), while the igh math class had a mean of 4.36 (range 3.4-5.4). Third-grade regular math students had a mean of 3.64 (range 2.35.3), while the high math class had a mean of 4.73 (range 4.1-5.2). For fourth-graders, the regular math class had a mean of 4.32 (range 3.3-5.1) and the high math class had a mean of 6.27 (range 5.0-8.5). Stimuli

Stimuli consisted of single-digit addition and multiplication problems which were presented in separate single-operation blocks. Stimulus problems were generated from the complete set of all (55) combinations of the numbers O-9 x O-9, ignoring order and including ties (such as 5 + 5). Ten sets of stimuli were produced by randomly varying which number appeared first within a pair. For addition, a second block of 55 stimuli was produced for each set by reversing the order of addends; thus if “3 + 5” appeared in the first block, then “5 + 3” appeared in the second block. Order of problems within a block was randomly determined subject to the constraint that no two consecutive problems had the same answer or had an addend in the same position in the prcbiem. For multiplication as for addition, ten sets of stimuli were generated, each containing all 55 combinations of single-digit integers, with random ordering of multiplicands and the same constraints on consecutive answers or multiplicands as with the addition stimulus. Because this study was primarily focused on addition, subjects saw only one block of 55 multiplication problems. Apparatus

The apparatus used in Study 2 was identical to that used in Study 1. Procedure

Subjects were tested in the same mobile laboratory using the same: computers used in Study 1. The procedure generally corresponded to that used in Study 1, except that all subjects were first presented with a pure addition task: two blocks of 55 addition problems each, with a short rest between blocks. Each block began with 6 practice trials, the data from which were not analyzed. Following completion of the addition task, subjects who chose to do so continued to take part in the multiplication task. Because not ail subjects had been taught multiplication, the instructions stressed the optional nature of this task. Subjects were then shown 10 practice multiplication problems. Subjects in the second and third grades who scored above @Go/d correct on the practice

iller and

. R. Paredes

trials were given the option of continuing or itting at that point. All fourthose second- and third-graders who chose to continue were then of 61 multiplication problems (including 6 practice problems that were not analyzed). Testing sessions lasted from 15 to 35 minut cts were tested in three sessi ns in December, February, and subjects and equivalent 01 year. The same procedure with the sa ghty days separated the s was repeated for each round of testing beginning of the first round of testing from the beginning of the second round, and the third round began 85 days after the second round. The final round of testing concluded within two weeks of the end of the school term. Resuits and discussion Addition resdts

Three measures o addition facility were analyzed: accuracy of responses, proportion of cass-operation errors for incorrect responses, and response times for correct answers. Each of these measures were applied to compare second, third, and fourth grades (and high and regular Imath classes -within the grades) at each of the three rounds of testing, and to compare performances of each grade and group across tilme. Accuracy (addition)

Overall, the children were highly accurate at simple addition: with the three grades scoring from 94% to 97% accuracy at each testing session. The ers were at the top end of this narrow range at each session. Three hundred and twenty of the total 10,010 aadition responses (3.2%) were lost due to microphone errors, in which the microphone failed to react to a subject’s response. Cross-operation errors (addition)

At the first testing session, cross-operations errors (responding with a multiplication answer to an addition problem), accounted for 48.21% of all addition errors made by fourth-graders, 14.52% of addition errors made by third-graders, and 4.00% of such errors for second-graders. By the second round, the proportion of cross-operation errors had declined for fourth-graders (25.71%), third-graders (11.88%), and second-graders (2.33%). By the third round, fourth-graders were near the starting point for third-graders in proportion of cross-operation errors, with such mistakes accounting for 16.50% of their addition errors. Third-graders continued to make more errors (see above) and a greater proportion of those errors (29.53%) were cross-operations. Finally, second-graders, who were beginning to learn multiplication,

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Table 2.

231

Error patterns: Experiment 2 __~

Breakdown of errors by operationa _ _ _.- _- - _-.-__ - -_ - __- -

--

Addition Second grade _-._-- -- ---

December

5.50

3.50

4.00 54.40 15.20 0.80 8.08 110.40 19.20

2.33 65.70 13.37 0.00 6.48 2.91 17.44 -.

9.82 66.67 11.61 0.00 9.80 4.46 15.18

No second-grader completed all multiplication rounds

May

December

February

5.60

25.20

18.40

9.70

29.53 14.52 al.88 57.43 66.13 54.38 12.90 10.89 17.45 2.42 0.00 13.42 14.09 14.52 12.87 8.06 15.84 1.34 10.07 9.68 10.89 ___ ___ ..---- - - - ~--

l.50 10.14 3.38 0.68 2.03 77.70 8.11

5.90 12.04 5.56 O.fK! 0.93 76.85 6.48

15.80 18.97 5.17 5.1? 6.90 56.9s 12.07

February

Third grade Overall error rate % of total errors: Off by factor of l Off by factor of 2 Zero in factor One in factor No answer Other ____I__-~ --. Fourth grade _._-- -- -~ Overall error rate % of total errors: Cross-operation Off by factor of 1 Off by factor oE2 Zero in factor One in factor No answer Other --

May

3.90

Overall error rate % of total errors: Cross-operation Off by factor of 1 Off by factor of 2 Zero in factor One in factor No answer Other

C~~~5-0pt2Gh%i

February

Multiplication

3.80

December

February

-- ---

May

December _

February _.___-_---_

May

May __ -_~-

2.90

2.70

2.70

7.00

6.40

5.20

48.21 69.64 11.61 30.36 30.36 0.89 2.68

25.71 77.14 6.67 1.90 12.38 0.00 3.81 ._

16.50 75.73 13.59 0.97 17.48 II.94 7.77

9.iB

29.01 9.92 3.82 1.53 38.17 20.61

17.89 39.02 10.5? 10.57 I.63 23.58 26.83

28.00 38.00 13.00 9.00 7.00 10.00 30.00

aBased on all wrong answers given by subjects (i.e., ignoring microphone errors). See Table 1 for definitions of error types.

ilkr and D. R. hredes

increased their proportion of cross-operation errors to 9.82% of total errors t should be noted th in this study addition problems were given istinct from mixed) ocks only, and all subjects were given the lems before multiplication, hence any bias in the testing situave tended to promote cross-operation errors in multiplication rather than addition. he predominance of cr -operation errors in the addition sessions of this udy presents evidence concluding that the acquisition of multi hcation has a disruptive effect upon children’s addition in the latter part of the third and the beginning of the fourth grade of elementary school. were calculated to represent typical response e 2 shows changes over the period of the study y by grade and math class. A Grade (3: in addition RT, plotted sep second, third, and fourth) x class 22: high and regular) x Sex (2: male, female) x Testing round (3: December, February, and May) MANOVA, with round of testing as a -within-subjects factor, was calculated, using each subject’s median RT for correct responses during a given round as the depeneasure. Significant effects for comparisons over time were found for < .05, Grade x Testing round, F(4,156) = 2.71, ath class x Testing round, F(4,156) = 2.30, = < .05. The grade effect resulted from fourth-graders calculating signifithan the earlier grades. Third-graders were marginally slower -graders, but this difference was not significant. on of Figure 2 suggests that changes in adding speed over time were very different across the different grades and math classes. Changes over time were assessed by polynomial trend analyses (Bock, 1979) within each Grade x Math class group. Given three equally spaced hessions, a significarlt linear trend reflects a significant difference between the first and third sessions, while a significant quadratic trend corresponds to some significant violation from this linearity for the second testing session. For second-graders, high and low math classes show different patterns of change-over time. igh math children show a significant increase in the time required to perform addition over the course of the year, F(lJ5) = 8.99, p c .05, for the linear component, losing the speed advantage they showed over the regular math group by the end of the year. Regular math third-graders show an apparent decrease in addition time, although this was not significant, F(l,ll) = 0.52, n.s., for the linear component. Third graders showed a different pattern of changes over time as a function ol math ability. The regular math class subjects showed a significant increase

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Figure 2.

233

Changes in speed of simpIe (single-digit addends) addition by grade and math class. Data plotted are the average for each grade, math class (regular vs. advanced math class) and testing round of each subject’s median reaction times for correct answers. Each subject took part in three sessions during the course of the school year. Solid lines connect data points based on the same subjects; dashed lines connect corresponding #mathclasses across grades.

2ndgrade

my

-F-=Y 3rd grade

I

IkambaFcb~ 4th

I

I

tiy grade

Gmk and Month in School Year

in calculating time, F( 1,12) = 6.37, p c .05, for the linear component, mirrorigh math thirding the slowing shown by the high math second-graders. graders showed a marginal decrease in calculating speed during the study, F(l,9) = 0.44, n.s., for the linear component. Fourth-graders showed an overall pattern of faster addition over the course of the study. For the regular math group this trend WASmarginally significant, qI,9) = 4.70, .05 < p < .06, while there was a significant decrease in calculation time for the high math group, F(l,l2) = 10.49, p c .Ol, for the linear component. A similar pattern is found if one looks at changes in individual children and simply counts the number of children whose addition time is slower or faster across rounds, as presented in Table 3. Looking at second-graders, 71% of the high math class were slower 3n May than in December, while 69% of the regular math class were faster. For third-graders, 71% of the regular

Table 3.

Developmental trends in respome times for simple addition: Experiment 2 Grade

Math class

December-February _ _. __ -. _

February-May -- ___- ___---___

_.__- __ _ _~ ._.

Slowera

Faster

Slower

Faster

Slower

Faster

Regular High - -__.-.

30.77 70.59

69.23 29.41

38.46 “6.47

61.54 23.53

30.77 70.59

69.23 29.41

3 3

Regular High

78.57 54.55

21.43 45.45

64.29 36.36

35.71 63.64

71.43 45.45

28.57 54.55

4 4

Regular High

27.27 24.

72.73 76.

36.36 40.00

63.64 60.00

9.09 24.00

90.91 76.00

2 2

_

“Dependent measure is percentage of children whose median addition time was slower/faster over the time period indicated.

math children were s ay than in December, while 55% of the high 1% of the regular math and 76% s *werefaster. For fourth-grade y than in December. Thus the gh math class were faster in phenomenon of decreasing addition speed over time is not due to extreme changes on the part of a few children: most subjects within a group whose time increases showed this pattern. contrast to the general tendency for skills to improve of subjects showed a significant increase over a 6-mont quired to perform simple addition. Both high math ath third-graders were wer to perform the same set of probhan they had been in cember. The fact that these were the groups that showed this effect suggests an explanation for this effect. Consultations with the students’ teachers and with the mathematics curriculum used lls, 1985) indicated that the slowing roughly corresponded with the time that children were actively engaged in learning multiplication. ltiplication

Because subjects were given the option of discontinuing the task after the are only available for addition component, co plete data on multiplication about half the subjects (46 out of 90). No second-grader completed all sessions, 11 third-graders did (4 in the regular math class, and 7 in the high math class), and 35 fourth-graders (10 in the regular math class, 25 in the high math c!ass). As would be expected, th_e 1ess able students (as measured by ITBS scores) tended not to take part in the multiplication task. Looking at the

A

r -‘g

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235

ITBS math scores, the regular math third-graders who took part in the multiplication task had a mean grade equivalent of 4.15 (range 3.8-4.5) compared to 3.64 for the entire group of subjects, while the high math third-graders in the multiplication task had a mean of 4.85 (range 4.1-5.2) compared with 4.73 for the entire group. The single fourth grader from the regular math c!ass who did not take part in the multiplication task had the lowest ITBS score in the group, resulting in a mean of 4.43 (range 3.6-5.1) compared with 4.32 for the group as a whole. This self-selection means that grade and math class differences need to be interpreted cautiously, but we can compare changes in multiplication and addition across time for this subsample of children. Accuracy (multiplication) The 11 third-graders who completed all three rounds of multiplication were 75% accurate in their responses in December and improved their accuracy to 90% correct by the end of the school year. Fourth-graders began with 93% accuracy in multiplication and ended the year correct on 95% of problems. Cross-operation errors (multiplication) Cross-operation errors of the kind “4 x 3 = 7” accounted for 2.7%‘) 6.5%‘) and 17.2% of total errors for third-graders in rounds one, two, and three, fo_8 tiea e:-rors for fourth-graders at respectively. The percentages of cross-o;:v+T ‘I. each round amounted to 9.2%) 17.9%, and 28% a Cne student in each grade accounted for 19% of that grade’s cross-opera:;tin errors. Dropping these outliers from analysis, the percentage cf cross-operation errors becomes 1.5%, 5.9%, 15.8% for third-graders and 7.6%‘) 13.6%, 26.8% for fourthgraders (see Table 2). Response times (multiplication) A Grade (3: second, third, and fourth) x Math class (2: high and regular) X Sex (2: male, female) X Testing round (3: December, February, and May) X Operation (2: addition, multiplication) MANOVA, with round of testing and operation as within-subjects factors, was calculated, using each subject’s median RT for correct responses for each operation during a given round as the dependent measure. Significant effects for comparisons over time were found for operation, F( 1,38) = 12.95, p < .Ol, Grade x Operation, F(1,38) = 4.44, p < .05, Operation x Testing round, F(2,37) = 4.17, p < .05, and Grad.2 x Operation x Testing round, F(2,37) = 8.75, p c .Ol. The overall operation effect was due to multiplication being slower than addition overall. Figure 3 shows the relation between addition and multiplication times for those subjects who completed all rounds of both operations.

iller and D. R. Paredes

Figure 3.

Changes in speed of simple (single-digit) addition and multiplication by grade, based on subjects who took part in all three rounds of both tasks. Data plotted are the average for each grade, operation and testing round of earl* subject S median reaction times for correct answers. Each subject took part in three sessions during the course of the school year. Solid lines connect data points based on the same subjects; dashed lines connect corresponding operations across grades. Changes in Speed of Simple Addition & Muitip!ication by Grade --

2400 T

4lb

grade

Grade & Month ia !kbaol Year

Third-graders showed an overall operation effect, F( 1,8) = 41.61, p C -01, with faster times for addition than multiplication. There was also a significant Operation x Time interaction, F(2,7) = 8.27, p < .05, due to a significant decrease in multiplication speed over time, F&8) = 11.2, p = .Ol for the linear trend, and a non-si nificant increase in addition speed during the same ., for the linear trend. a somewhat different pattern. The operation effect was only of marginal significance, F(l,30) = 3.90, .05 c p < .lO. There was an overall time effect, F(2,29) = 4.10, p C .05, but the interaction between operation and time was not significant at fourth grade, F(2,29) = .84, (n.s.), indicating that the two operations showed a similar pattern of change over time. ddition became significantly faster over the year, F(1,30) = 12.78, p < .@I for the linear trend, while multiplication showed a marginally significant drop in time, F(1,30) = 3.50, .05 < p c .lO for the linear trend.

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237

The development of mental multiplication showed a different developmental pattern than did simple addition, with a continuous drop in the calculation time over the period studied. One important difference between the results of Studies 1 and 2 is evident in comparing Figures 1 and 3. No group of subjects in Study 2 showed the pattern shown by the third-graders in Study 1, of faster multiplication than addition. Subjects from the corresponding point in the school year (third-graders in Round 3) were faster to add and slower to multiply than their peers in Study 1. The same pattern was found in the error data, with third-graders in Study 2 making more errors than their peers in Study 1 on the pure multiplication blocks, and fewer errors on the pure addition blocks. Children’s relative skill at addition and multiplication is almost surely a function of the effort placed in developing this skill. As noted above, the two schools used in Studies 1 and 2 differed substantially in the overall affluence of the subjects’ families and available data suggest there are also differences in achievement levels of these two samples. What does appear to be constant across both settings, however, is that the process of learning to multiply has temporary deleterious consequences on children’s ability to perform a previously mastered arithmetic operation.

aken together, the data depict two lines of converging evidence for interference effects flowing from multiplication to addition. Error patterns Several researchers have noted the presence of confusion errors crossing between addition and multiplication and have taken them as evidence of use of memory-retrieval processes in doing simple arithmetic (Ashcraft, 1987; Stazyk et al, 1982; Winkelman and Schmidt, 1974). Miller et al. (1984) commented on the asymmetry of such errors among their adult subjects: 35% of all addition errors and 24% of multiplication errors were cross-operation confusions, revealing a greater tendency to reduce multiplication answers to addition problems than vice versa. The present studies replicate those findings: every age group under every condition of testing made proportionally more cross-operation confusions in addition than in multiplication.

Response times

interaction over time between rates of addition and third-graders being faster at multiplication ana adults ks. being faster at addition when the problems were presented in ers Study 2 showed that advanced second-graders and regular-class nse time to addition problems. own during the school year in r are capable of doing so with a at children have learned to add high degree of accuracy before leakming to multiply, the question arises as to why multiplication should become faster than addition for some period of time for at least some children (Study 1) and why speed of addition should decrease concurrently with the acquisition of multiplication skills (Study 2). One answer to these questions is suggested by a third-grader in the second study. She remarked that she greatly preferred doing multiplication rather than addition, ‘*because with multiplication”, she explained, “you just know it and you don’t have to work it out”. A straightforward mechanism by which multiplication uld lead to increasing retrieval for addition is consistent with these results a this anecdote. The processes for solving addition and multi& plication problems and the knowledge structures that result from these processes are interrelated. Counting procedures for doing addition are easily learned and generally accurate, and there is little incentive for adopting more efficient retrieval strategies. Si ar procedures for multiplication, however, are difficult and easily subject error. They involve counting by arbitrary or repeatedly adding series of numbers. The difficulty of such strategies rforming simple multiplicatio&J may result in an early emphasis on retrieval over reconstructive processes for multiplication. Because retrieval is a fast procedure, this in turn may have the consequence of making products more accessible than sums for some problems during the period in which children are learning to multiply. Early retrieval of products to multiplication problems may provide both a model zlnd a motivation for children to memorize the sums of addition problem s, in order to avoid producing the cross-operation errors for addition problems that were so prominent in the mixed-operation blocks of Study 1. Once the pattern of interference from multiplication to addition has been established, however, it appears that while the extent of interference may be tempered by practice, the asymmetry of the direction of effects persists through adulthood. Implicationsfor defining domain? As noted above, research on expersise has generally found that expert skill is constrained to rather small domains of knowledge (e.g., Chi, 1985; de-

Adding worse

239

Groat, 1965), although Ericsson and Polson (1988) present a counter-examof what constitutes a domain has generally been by focusing on clearly defined domains, including as chess (deGroot, 1965; Simon & Chase, 1973), go (Eisenstadt 1975; Reitman, 1976), bridge &harness, 1979; Engle $c Bukstel, 1978), and baseball (Chiesi, Spilich & Voss, 1979), and specialized skills such as reading electronic circuit diagrams (Egan & Schwartz, 1979) or writing computer pTograms (Adelson, 1981; McKeithen, Reitman, Rueter, & Hirtle, 1981; S4oway, Adelson, & Ehrlich, 1988). In one study explicitly looking at relations between different strategic games using the same board, Eisenstadt and Kareev (1975) found substantial interference between the two games of go and gomoku. In a study more directly related to the current research, Staszewski (1988) studied two subjects who were given extensive practice (175. 300 hours) over several years mentally multiplying numbers (up to 2 x 5 digit multiplicands). Qf particular interest is the report for one subject of a temporary increase in RT as he moved to a strategy emphasizing recognizing familiar patterns in multiplying large numbers. The generality of the phenomenon described here is not clear, although there is no reason to think that it is limited to acquisition of simple arithmetic. Anecdotal evidence suggests that this may indeed be a more general phenomenon; a clear example is presented in the cellist YO-YO Ma’s characterization of his own development (Blume, 1989): I’ve never been able to learn in smooth transitions. I’ve found that to go from one level to another you often have to destroy some knowledge in order to gain other knowledge. You realize after a while that you’re not happy with what you’re doing; you’re at a standstill. Then, from the various ideas you’ve collected, you find one little piece of information that helps. It brings about a chemical reaction, and - ZOOM! - things finally fall into place . . Then you’re at the next level, awaiting further answers. (p. 46) l

Within a domain of knowledge, learning new skills may often require one to rethink or reorganize previous knowledge, leading in turn to a temporary disruption of performance. Finding that such disruptions occur as one acquires new knowledge suggests that new knowledge is being incorporated into a structure containing the previous skill, This may provide an empirical method of using developmental data to define the limits of a domain.

A?&s:t, J., & Wells, D. (1985). Mathematicstoday: Teacher’s edition level 3. New ‘il~fk: HG’LXUL Jovanovich.

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Campbell, J.I.D. (1987a). Network interference and mental multiplication. Journal of Experimental Psychology: Learning, Memory, and Cognition, 13, 109-123.

Campbell. J.I.D. (1987b). Production, verification, and priming of multiplication facts. Memory and Cognition, 15. 349-364.

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Carpenter. T.P.. & Moser, J.M. (1984). The acquisition of addition and subtraction concepts in grades one through three. Journal for Research in Mathematics Education, 15, 179-202. Chamess, N. (1979). Components of skill in bridge. Canadian Journal of Psychology, 33, l-50. Chi, iW’I’A-k (1985). Interactive roles of knowledge and Atrategies in the development of organized sorting and recall. In S.J. Chipman, J.W. Segal, & R. Glaser (Eds.), Thinking and learning skills. Vol 2: Research and open questions (pp. 457483). Hillsdale, NJ: Erlbaum.

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