Mental Multiplication Skill - American Psychological Association

CANADIAN JOURNAL OF PSYCHOLOGY, 1985, 39(2), 338-366

Mental Multiplication Skill: Structure, Process, and Acquisition* Jamie I. D. Campbell and David Jeffrey Graham University of Waterloo ABSTRACT Response time (RT) and error data on multiplication problems up to 9 x 9 were collected from children in Grades 2 through 5 and from adults. Even at primitive stages of learning, most errors involved correct products to other problems, and there was a developmental trend where, by Grade five, the specific errors made by children mirrored adult errors. The error patterns indicate that an associative network evolves in which problem operands become linked to specific sets of candidate answers. Retrieval is governed by a process that activates candidates, and accessibility of correct answers is impeded by competing associations: At all skill levels, both problem-error rates and product-error rates (i.e., how often a problem's correct product occurs as an incorrect response to other problems) contributed to predicting correct problem RT in multiple regression analyses. These interference variables always yielded higher correlations than did structural variables (e.g., the numerical size of problem operands), the latter having provided the basis for previous models of arithmetic memory. A network-interference account is proposed that explains the slow course of acquisition and differential problem difficulty in terms of interference by false associations. RESUME Les temps de reponses (RT) et le type d'erreur a des problemes de multiplication allant jusqu'a 9 x 9 sont enregistres chez des enfants de la 2 e a la 5C annee primaire et chez des adultes. Meme aux premiers stadcs de l'apprentissage, la plupart des erreurs sont des reponses correctes a d'autres problemes. II y a aussi un pattern de developpement qui fait qu'en 5C annee les erreurs faites par les enfants sont semblables a celles des adultes. Les patterns d'erreurs montrent qu'il y a etablissement d'un reseau associatif dans lequel les donnees d'un probleme sont reliees a un ensemble specifique de reponses possibles. Le recouvrement est determine par un processus qui active les reponses possibles et l'acces a la reponse correcte est entrave par les associations competitrices: a tous les niveaux d'habilete, le taux d'erreur-probleme et le taux d'erreur-produit (i.e., combien de fois le produit correct d'un probleme devient-il la reponse incorrecte a d'autres problemes) contribuent tous deux a la prediction du RT des problemes corrects dans une analyse a regressions multiples. Ces variables d'interference produisent toujours des correlations plus elevees que les variables de structure (e.g., la grandeur numerique des elements du probleme), ces dernicres etant a la base des modcles anterieurs de memoire mathematique. Les auteurs proposent une explication en terme de reseaud'interference qui rendrait compte de la lenteur de l'acquisition et de la difficulte differentielle des problemes par 1'interference de fausses associations. *This research was supported, in part, by Postgraduate Scholarships to each author from the Natural Sciences and Engineering Research Council of Canada. We are grateful to D.W. Higgins and K. Thompson of the Perth County Board of Education, as well as to Janice Graham for her practical advice and assistance. We also thank Mark Ashcraft, Patrick Brown, Neil Charness, Danica Lavoie, Christie McKenzie, and an anonymous reviewer, for helpful comments on an earlier draft. Reprint requests should be sent to J. Campbell, Department of Psychology, University of Waterloo, Waterloo, Ontario, Canada N2L 3GI. 338

Multiplication skill

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This paper is concerned with the nature and acquisition of basic arithmetic skills, with particular focus on simple multiplication problems like 3 x 4 or 7 x 9. A concern with the basic arithmetic operations is motivated by two main considerations. First, despite some 70 years of experimental research, no firm conclusions can be made regarding how basic arithmetic should be taught to children, or about why the acquisition of simple number facts often presents a serious challenge to children (Baroody, 1984; Wickelgren, 1979, p. 242). Since Thorndyke (1921, 1922) argued for systematic drill as the best method of instruction, there has been controversy over the optimal teaching method. Some have argued that a criterion of computational efficiency de-emphasizes the meaningful understanding of fundamental arithmetic concepts necessary for applying arithmetic skills to everyday problems, or for advancing to more complex or abstract mathematical reasoning (Baroody, 1984; Brownell, 1935). Others (e.g., Resnick & Ford, 1981) have argued that drill is necessary because the number facts need to be available effortlessly to avoid competition with higher level problem-solving processes. The resolution of the optimal training question depends upon a sound theory of basic computational skill. A second reason for studying simple arithmetic is the growing use of arithmetic tasks in cognitive research not concerned specifically with the nature of arithmetic. Mental arithmetic has proven to be a convenient cognitive domain for studying working memory (Hitch, 1978), the relationship between cognitive operations and the symbols that initiate them (Gonzalez & Kolers, 1982), automatic versus effortful processing (Zbrodoff, Logan, & Barber, 1984), hemispheric specialization (Katz, 1980), and skill acquisition (Campbell, 1982). Models of basic arithmetic processes constrain interpretations of arithmetic data collected in any theoretical context and are important for defining appropriate experimental controls. The more thoroughly understood the basic operations are, the more useful arithmetic tasks become as experimental tools. Our focus in this paper is on arithmetic fact knowledge as a memory retrieval phenomenon. Consequently, we do not discuss the learning of the basic operations as mathematical concepts. Although the development of concepts of numerosity, magnitude judgements, and simple counting strategies undoubtedly contribute to the learning of the basic combinations, our principal concern is to identify the nature of the retrieval system that develops. Adults can produce a correct response to most simple multiplication combinations in 1 second or less. What is the nature of this skill, and what is the course of its acquisition? To address these questions we begin with a critical review of recent attempts to understand children's and adults' addition and multiplication performance in terms of structural variables, which imply that differences in difficulty are determined by the numerical magnitude of the problems. Following this, an experiment is reported that compares adult performance on simple multiplication to the performance of children from Grades 2 through 5. The results indicate that current models based on structural variables are inadequate, and we propose a network-interference account of multiplication skill and its acquisition.

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The Problem-Size Effect Many of the systematic studies on children's acquisition of arithmetic skills took place between 1900 and 1940 in the United States. Buswell (1930) catalogued 518 quantitative investigations dated prior to 1929. One of the most extensive efforts was undertaken by Clapp (1924). In 1921 he initiated a massive program of data collection that is astonishing even by the standards of modern computerized laboratories: A total of 10,945 students from Grades 3 to 8 from 49 different schools, from four different cities, generated a total of 3,862,332 answers to combinations. The main results of this effort were tables rank-ordering by difficulty all the simple combinations for addition, subtraction, multiplication, and division. Difficulty was established in terms of the percentage of students knowing each combination at each grade level and by tabulating error frequencies for each problem. Clapp (1924) observed that the simple combinations with larger numbers tended to be more difficult. For example, 6 x 9 = 54 is more difficult to learn than 2 x 6 = 12. This is one of the first reports of the problem-size effect. The effect can be measured in a variety of ways: The simple combinations with larger numbers take more trials to learn, more trials to maintain learning, are more prone to error, and require longer RTs than the smaller combinations (e.g., Norem & Knight, 1930). The effects for retrieval times and errors also hold for adult addition and multiplication (e.g., Ashcraft & Battaglia, 1978; Miller, Perlmutter, & Keating, 1984; Parkman, 1972). The production data reported by Miller et al. characterize the size of the RT effect in adults. They found a range of about 400 msec from the slowest to the fastest problems in both addition and multiplication. The effect of problem-size on RT has given rise to two main classes of models for simple arithmetic skill (but see also Restle (1970) and Aiken & Williams (1973) for a discussion of analog models). One idea has been that the effect reflects the execution of counting procedures. Large-number problems take longer because they require more repetitions of a counting or incrementing operation. There is good evidence that children initially use counting strategies to perform simple addition (e.g., Ashcraft & Fierman, 1982; Groen, 1967; Siegler & Shrager, 1984; Suppes & Groen, 1967), and a number of theorists (e.g., Baroody, 1983,1984; Parkman & Groen, 1971) have suggested that the procedural methods used by children are preserved in adult performance. The problem-size effect in adults might reflect the presence of unconscious, automatic counting procedures. Other researchers have argued for associative-network models of arithmetic memory (Ashcraft, 1982, 1983, 1984; Ashcraft & Battaglia, 1978; Ashcraft & Stazyk, 1981; Miller et al., 1984; Parkman, 1972; Stazyk, Ashcraft, & Hamann, 1982). Encoding a problem into the network causes associated nodes (i.e., answers to problems) to be activated and available for report or other processing if they are activated sufficiently. In these theories, the problem-size effect is supposed to derive largely from the search distance in the network. Small-number problems yield fast RTs because their operands are closely associated with the


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correct answer node in the memory structure, whereas larger problems take longer because their answers occupy semantically remote regions of the network. Both of these accounts of the problem-size effect are supported primarily by correlations between RT and numerical quantities in each problem, so-called structural variables. For example, in multiplication, correlations in the neighbourhood of + .80 have been found between RT and the numerical magnitude of the correct product (Miller et al., 1984; Norem, 1928; Stazyk et al., 1982). In both addition and multiplication, reliable correlations have also been found between RT and the minimum or maximum operand, or the sum or squared sum of the operands (e.g., Miller et al., 1984; Stazyk et al., 1982). In procedural models, the values of structural variables estimate the number of steps executed (e.g., number of successive increments in a counting series). In network-distance models, the structural predictors index the network's organization, an organization that is supposed to preserve arithmetic magnitude faithfully. Limits of Structural Variables There are a number of reasons to doubt the assumption that the correlations with structural variables directly reflect the memory processes involved. Tie problems (e.g., 2 + 2 , 6 x 6, etc.) do not obey a general problem-size rule. Ties are often analyzed separately because RTs for ties tend to be faster than for other problems, and the slopes of their correlations with structural variables are lower than for the nontie problems (Groen & Rarkman, 1972; Miller et al., 1984; Rarkman & Groen, 1971; Stazyk e t a l . , 1982). Ties, however, appear not to be the only exceptions. We used the matrix of adult multiplication RTs provided by Miller et al. (1984) and computed an average correct RT for each times-table. As it is generally accepted that problems involving 0 or 1 are solved using procedural rules by both children and adults (i.e., a X 0 = 0, a x 1 = a, see Ashcraft, 1983; Baroody, 1983), we used only the problems ranging from 2 x 2 to 9 x 9, but excluded tie problems. Across this set, each of the eight digits 2 through 9 occurs in 14 different combinations, and we calculated the mean of the median correct RTs (with product naming times subtracted) for the problems involving a 2, a 3, and so on through the 9 timestable. This analysis provides a convenient measure of the problem-size effect by which to assess the predictive power of structural variables: Regardless of which structural variable is used, the mean value of that variable for each times-table predicts increases in RT with successive times-tables, although the predicted shape of the function differs across structural predictors. We need not be concerned with these different functions, however, because the observed pattern is inconsistent with that predicted by any of the structural variables. RTs across the 2-9 times-tables were 2 X : 273.9; 3 X : 298.3; 4 X : 288.6; 5 x : 272.4; 6 x : 318.1; 7 x : 292.4; 8 x : 361.3; and9 X : 390.9. Five times problems were about 26 msec faster than 3 times problems, about 14 msec faster than 4 times problems, and about I msec faster on average than the 2 times-table.

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Further, the 7 times-tabJe was faster than either the 3 times or 6 times problems. Among the eight different times-tables, only the 4, 8, and 9 times problems occupied the rank order RT position predicted by structural variables. Based on the Miller et al. (1984) data, it appears that there are general patterns in RT not captured by the structural variables. Accessibility and Associative Interference To accommodate exceptions to the simple problem-size rule, an accessibility factor that can vary independently of problem-size has been proposed. Miller et al. (1984) distinguish between the location of information in a representation (i.e., network distance which is indexed directly by structural variables) and accessibility of the information. Differences in accessibility are largely thought to reflect variations in practice or familiarity (Miller et al., 1984; Stazyket al., 1982). Miller et al. also suggested, however, that accessibility might be sensitive to the presence of competing confusion responses. Although they did not elaborate this idea, it suggests that, in addition to a correct association, problems may also have false associations with a number of other responses and that false associations interfere with a correct retrieval. Wickelgren (1979, p. 242) claimed that the formation of competing associations is precisely why the arithmetic combinations are difficult to learn. His discussion of the subject provided an introduction to a textbook chapter on interference in paired-associate learning. Given the similarity between drilled learning of arithmetic facts and learning paired-associate lists and the prominent role the concept of interference has played in theories of paired-associate learning, it is surprising that interference has not been explicitly considered as an important factor in modern theories of arithmetic skill. It has been known for many years that similarity among items to be memorized is a factor that promotes interference (e.g., Osgood, 1949). When two lists of paired-associates have items in common, learning the first list can impair learning of the second list (proactive interference), and learning the second list can impair retention of the first list (retroactive interference). Because all problems are constructed out of the same 10 digits, arithmetic fact learning may be especially prone to interference. For example, learning 7 x 9 = 63 as an isolated fact would surely be easy. However, when other problems involving 7 and 9 have been learned previously (e.g., 4 x 9 = 36, 7 x 8 = 56, etc.), the new association must be established in the presence of existing associations with each operand. When 7 x 9 is presented, in addition to a correct association with 63, the problem may also access associations with multiples of 7 and 9 that are false in the context of 7 x 9. There is evidence that such competing associations are sources of confusion during learning and that their impact is preserved in adult performance. Grant Norem, in a 1928 Master's thesis (see also Norem & Knight, 1930), traced learning of the simple multiplication facts by 25 third grade students. He found highly systematic patterns in the errors made by these children: Ninety-one percent of some 5,400 errors he recorded were correct responses to other


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multiplication combinations. Of these, about 70% were correct answers to "closely allied combinations," usually ones having a multiplier in common (i.e., from within the same times-table). Miller et al. (1984), testing six adult subjects, also found that a high percentage of errors (41%) involved answers that were correct to neighbouring problems (e.g., 4 X 8 = 2 4 o r 4 x 8 = 36). It appears that errors frequently result from associations with a problem's operands that are correct in the context of another problem. Assuming that specific errors reflect associated candidate responses that compete for retrieval, is there evidence that such competition has an impact on RT? There are two kinds of evidence that argue that it does. Stazyk et al. (1982, Exp. 3), using a true/false verification task (A x B = C), found a reliable 58-msec interference effect when a false answer was from the same times-table as the presented problem (a confusion product), compared with when the false answer was an unrelated number. The result argues that activating a false associate of a problem slows down correct retrieval. We can think of the confusion effect in verification as externally generated interference (i.e., the interfering product is presented simultaneously with the problem). The fact that confusion products are generated as errors in a recall task, however, suggests that associated responses are activated whenever a problem is encoded. Just as external presentation of confusion products interferes with retrieval of the correct answer, the internal activation of confusion products may interfere similarly. If the frequency of any particular error is taken to indicate the strength of an associative trace linking a problem and that answer, each problem can be thought of as having a set of candidate answers that are activated according to their trace strengths. Siegler and Shrager (1984) have conceptualized errors similarly to define the distribution of associations in children's knowledge of simple addition facts. Time to access a correct item in the structure might vary as a function of the number of different candidates or as a function of the relative strengths of competing candidates activated when a problem is encoded (cf. Anderson, 1981, 1983). If this sort of internal confusion occurs, there should be a positive correlation between problem-error rates and correct RT. Indeed, Norem (1928) found that average correct RT and average number of errors on a multiplication problem were correlated + .49. Similarly, Miller et al. (1984) found a correlation of + .54 for adult multiplication and also found that problem-error rates in simple addition and multiplication contributed independently of problem-size to predicting correct RT. Given the systematic nature of multiplication errors, the fact that problemerror rates are positively correlated with RTs suggests that differential accessibility may be due substantially to an associative interference process. false associations with a problem's operands may not be the only source of interference. While Norem (1928) found that the magnitude of a problem's product was a good predictor of difficulty, he also claimed that a multiplication problem was difficult to learn if its product was susceptible to what he called "promiscuous connection forming" (Norem, 1928, p. 29). More specifically, he found that the responses most often occurring as errors were frequently the correct

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products to the problems requiring the most learning trials. Norem's explanation for this relationship was that children establish false associations during learning that weaken or interfere with the formation of correct associations. The observation that product-error rate (i.e., how often a problem's correct answer occurs as an incorrect response to other problems) predicts the number of learning trials suggests that competing associations with a problem's product may be a source of interference additional to competing associations with a problem's operands. Interference and the Problem-Size Effect A consideration of interference processes operating in arithmetic memory leads to a different interpretation of the problem-size effect. In network-distance and procedural models of arithmetic memory, the problem-size effect is a consequence of numerical magnitude. In the network-distance case, the effect occurs because the organization of the hypothetical network structure that is acquired preserves the relative numerical magnitude of the stored answers. In procedural models, the reconstructive arithmetic processes that develop are controlled directly by the magnitude of the problem's operands. A network interference account, however, suggests that correlations with structural variables may reflect less the fact that problem difficulty is a consequence of magnitude than that it is coincident with magnitude as a result of the conditions of initial learning. Specifically, it has long been held that frequency of occurrence determines the strength of associations (e.g., Anderson, 1983; Thorndyke, 1922, 1933), where strength controls the probability and speed of a correct retrieval. For the four basic arithmetic operations, Clapp (1924) found respectable correlations (averaging about — .40) between a problem's difficulty ranking and its frequency of occurrence in textbooks. A strong correct association would be less susceptible to interference than a weak correct association. Because small-number problems are encountered more often than large-number problems, the predictive power of structural variables may derive, in part, from a correlation with frequency (Stazyk et al., 1982). Another factor that may contribute to the predictive power of structural variables is the order in which the arithmetic combinations are learned. Typically, within operations, the small-number combinations are encountered first and the large-number problems are introduced later. Initially, there would be only a few candidate responses in the child's arithmetic lexicon or arithmecon, as it might be called. As each combination is added to the structure, all prior existing responses constitute potential sources of interference. Indeed, Norem (1928) found a correlation of + .51 between difficulty rankings and the order in which the multiplication combinations were learned. Taken together in the context of an associative interference account of accessibility, the effects of frequency of occurrence and order of acquisition may be sufficient to explain the problem-size effect. To investigate the possibility that interference processes play a major role in arithmetic performance and its development, simple multiplication data from


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public school children in Grades 2 through 5 and from a large adult sample were collected. We argue from error analyses that children acquire an associativenetwork structure linking problems to candidate answers. In contrast to theories that attribute differential problem difficulty directly to numerical magnitude, we argue that difficulty is determined by factors affecting the patterns and strengths of associations in the developing network. Based on the relationship between error rates and RT, we conclude that problem difficulty is determined largely by associative interference. METHOD The Grade School Sample Data were collected from children in Grades 2 to 5 at a public school in south-western Ontario. Eighty-six children (41 males) ranging in age from 7 to 12 were tested. There were 10 children in the Grade 2 class, 23 in Grade 3,21 in Grade 4, and 26 in Grade 5. It was generally true for this sample that problems had been taught in the context of times-tables. However, the children were first taught the set of problems with products less than 25. Within this set, the 2 times-table was taught first, followed by the 3, 4, and 5 times-tables. During the initial introduction to a timestable, problems were practiced in order of increasing magnitude. New problems were then embedded in written practice sheets along with the problems introduced earlier. The 0 and 1 times-tables were learned after the "below 25" problems. Within a times-table, problems were learned in both operand orders (e.g. ,5 x 8 and 8 x 5) and practice sheets included both orders. The Grade 3,4, and 5 classes were tested in March of 1984. The Grade 3 class was tested again 7 weeks later, and again 5 weeks after the second testing session. Grade 2 was tested in June 1984 after 1 month of training on problems up to and including the 5-times problems. The Problem-Set For Grades 3 through 5, the simple multiplication combinations from 0 x 0 to 9 x 9 were testedin2blocksof55trials.Thel0tieproblems(e.g.,2 x 2,3 x 3, etc.) were tested once in each block. For the 45 nontie problems, operand order was determined randomly in the first block and reversed in the second block. Problem order was random for each subject, but was constrained to avoid repetition of operands or correct answers on consecutive trials. The Grade 2 class was tested only on problems up to 6 x 6, omitting the other problems with both operands greater than or equal to six. Apparatus and Procedure Students were tested individually in a quiet office using a PET 2001 microcomputer connected to a black and white CRT screen. The problems were presented horizontally at the centre of the screen in a five character field: the two multipliers separated by an uppercase X with adjacent blank characters. The Commodore Business Machine standard character set (4mm high x 3mm wide) was used. Subjects sat facing the screen at a distance of about 50 cm and were instructed to respond as quickly and accurately as possible. The experimenter initiated each block of trials. On each trial a flashing fixation dot appeared at the centre of the screen. The problem appeared on what would have been the third flash with the X appearing at fixation, and the problem remained on the screen until the subject responded. Response times were measured using a hardware clock accurate to ± 17 msec. Timing began when the problem appeared, and the experimenter stopped the timer by pressing a button when a response was given. Manual timing was used instead of a voice activated relay to avoid trials being spoiled by extraneous vocalizations. Upon the response, the experimenter immediately entered the given answer at the computer keyboard. Providing this was accomplished within 3 seconds, a constant 5-sec interval separated the end of one trial from the beginning of the next.

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Regardless of the duration of response entry, however, the subjects always received the two warning flashes before a problem appeared. Blocks were separated by a minimum of 15 seconds. Correct feedback was provided on error trials. The set of 110 trials was tested in a period ranging from 12 to 20 minutes.

The Adult Sample The adult sample of 60 subjects (25 females) were the subjects from four different experiments conducted by the first author. All subjects were graduate or undergraduate students from the University of Waterloo, who participated as unpaid volunteers or as part of a course requirement. Age ranged from 19 to 31 years. Eight of the subjects had performed multiplication in a verification task prior to the generation trials included here, and another 44 had performed a prior number naming task that included naming the answers to the multiplication combinations. Of the latter subjects, eight were tested on simple mental division problems prior to the multiplication task. The procedure for the multiplication trials was identical for all adult subjects, but differed from that for the children in that the adult subjects were tested only on the problems ranging from 2 x 2 to 9 x 9. Ignoring operand order (i.e., treating, for example, 5 x 8 and 8 x 5 as the same problem), there are 36 problems in the range from 2 x 2 to 9 x 9. There are 64 combinations when the operand order distinction is preserved. The adults were not tested on the problems involving 0 or 1 which, it is generally believed, are solved by rules (i.e., a x 1 = a, a x 0 = 0, see Ashcraft, 1983; Baroody, 1983, 1984). We do not report analyses of the children's performance on these items here. The adults were tested individually using the same apparatus and stimulus display arrangement as was used for the children. However, response times were determined using a software clock accurate to ± 1 msec, that was controlled by a voice activated relay. Timing began with the onset of the stimulus and was terminated by the subject's verbal response. Each problem was tested once in each of 4 randomized blocks of 36 trials, with order constrained to avoid repetition of operands or correct products on consecutive trials. For each subject, 14 of the 28 nontie problems were randomly assigned to have the minimum multiplier appear on the left (min-left) in the first block. Operand order for each problem then alternated across blocks so that each nontie problem was tested twice with each order. After an error trial, subjects were instructed to provide the correct answer.

RESULTS AND DISCUSSION The focus of this paper is on differential problem difficulty; consequently, all analyses were based on statistics for each multiplication problem. Initial analyses indicated that it would be appropriate to treat combinations with the same operands (e.g., 3 x 4 and 4 x 3) as one problem when dealing with the children's data, so all statistics concerning their performance on nontie cases were computed collapsing over operand order. This was not the case for the adults, however, who were 18 msec faster, t(26) = 2.39, p < .05, and 21% less likely to commit an error, t(26) = 4.40, p < .001, when the left-most number was the larger.' Except where otherwise indicated, the operand order distinction was preserved for adult problem statistics. 1 For the adults, errors were promoted when the position of a digit in a problem (left or right) mapped on to the position of the same digit inafalse product. For example,9 x 6 = 63 occurred only twice, while 6 x 9 = 63 occurred ten times as an error (see Appendix A). Such positional digit correspondence occurs more often for min-left than for max-left combinations and appears to account entirely for the higher min-left error rate.


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TABLE 1 Response Time (in seconds) and Error Data Group Variables

2

3A

3B

Mean RT SD % Errors # Errors

— — — —

3.83 1.92 29.65 491

2.91 1.05 20.65 342


7.48 3.08 32.59 176

3.21 1.28 15.29 190

2.54 0.75 10.06 125


— — — —

5.71 2.36 72.71 301

4.02 1.11 52.41 217

3C

4

5

3.63 1.38 25.31 492

1.87 0.42 16.77 314

0.83 0.09 7.65 661

3.48 1.35 16.25 237

1.80 0.42 12.53 176

0.80 0.08 5.63 365

4.10 1.44 52.47 255

2.06 0.39 29.48 138

0.87 0.10 13.70 296

Adults

All Problems 2.78 0.86 18.90 313 Set 1 Problems 2.50 0.63 11.75 146 Set 2 Problems 3.63 0.95 40.33 167

Note. The Grade 2 data involve only the problems in Set 1. Grade 3 was tested in three sessions, A, B,and C. Grades 4 and 5 were tested at the same time as Grade 3, Session A.

For each group a mean RT for each problem was computed by pooling all correct RTs for a problem after applying an outlier rule. Outliers were defined by computing each subject's grand mean and standard deviation using all correct RTs for the problems ranging from 2 x 2 to 9 x 9 (2 x 2 to 6 x 6 for Grade 2) and discarding any RT more than 2.5 times the standard deviation from the mean. Over all the children's sessions, there were 0.21% spoiled trials, and an average of 4.48% (SD = 0.32) of correct unspoiled RTs were discarded as outliers. In the adult data, 3.3% of trials were spoiled by failures of the voice key, and 3.02% of correct RTs were discarded as outliers. Development of the Problem-Size Effect Table 1 shows the mean of the mean problem RTs, the standard deviation, and error rate data for each group for all problems and with the problem set divided into two subsets. Set 1 includes all the problems with minimum multipliers ranging from 2 to 5, plus 6 X 6. In Set 1, there are five tie problems (e.g. , 2 x 2 , 3 x 3, etc.) and 22 nontie problems (44 nontie combinations when operand order is preserved). Set 2 was composed of the remaining problems ranging from 6 x 7 to 9 x 9 which have 6 and 7 as the smallest minimum and maximum operands. Grade 2 was not tested on the Set 2 problems. There are three tie problems and six nontie problems (12 nontie combinations) in Set 2. The Set 1 and Set 2 breakdown


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8.5 -I

7.5• 3A

RT (SECS) 4-

3-

2-

9OO

ADULTS

RT (MSEC) 8OO

2x

3x

4x

Sx

TIMES

6x

7x

8x 9x

TABLE

Figure 1. RT in seconds (msec for the adults) as a function of times-table.

is shown in Appendix B, which includes for each problem for each group: mean correct RT, number of problem-errors, and number of product-errors (i.e., how often a problem's correct answer appeared as an error response to other problems). It is clear that mastery of the basic multiplication facts is not a trivial task. Under moderate speed pressure, after 4 years of experience (i.e., Grade 5), the children were still making errors on 12.5% of the trials involving the Set 1 problems. Performance on Set 2 problems lagged far behind that of the Set 1 problems. Error rates for Grades 3,4, and 5 averaged 36.9% higher for Set 2 than Set 1. Adults also made more errors on the large-number problems (13.7% vs. 5.6%). Similarly, correct RTs were consistently longer for Set 2 than Set 1. The differences between performance on Set 1 and Set 2 reflect the basic problem-size rule that large-number combinations tend to be more difficult than small-number combinations. This trend is captured in Figure 1 which shows


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correct RT in seconds (msec for the adults) as a function of times-table for each group. For the problems ranging from 2 X 2 to 9 x 9, collapsing over operand order yields 36 problems, and each of the numbers 2 through 9 occurs in eight different problems. Each point in the figure represents the mean of the mean RTs for the eight problems. As is readily seen, the general problem-size rule holds for all groups, with a definite trend for RTs to become faster and for the functions to flatten out with increasing skill levels. It is also clear, however, that there is not a simple linear or curvilinear increase in RT with increasing numerical magnitude, as theories based on correlations with structural variables would predict. Recall that structural variables predict a systematic increase in RT across successive times-tables. For every group, problems in the 5 times-table were faster than their average numerical size would predict. Similarly, 9-times problems tended to be faster than the 8-times problems in the children's data, and RTs for the 8- and 9-times problems were about the same for the adults. These results also contradict the general problem-size rule. Our claim will be that structural variables do not directly reflect the structure or processes of arithmetic memory but, instead, index factors that affect the formation of associative traces linking problems to candidate answers. Consistency in the Relative Difficulty of Problems Although the pattern of RTs across times-tables deviates from the pattern predicted by structural variables, it is clear, nevertheless, that problems vary in their difficulty to a marked degree. It is of interest to determine how consistent performance across problems is among adults, and at what grade level the children's data begins to mirror the adult pattern. Correlations of the adult data with each of the grade school groups were calculated for problem-error rates, product-error rates, and RTs. For these comparisons the adult data were collapsed over operand order to match the children's data. Adult performance on each problem was correlated with the children's data for each problem using error rates and mean correct RTs for 36 problems (27 for Grade 2). As discussed below, errors frequently involved answers to other multiplication problems. The frequency of each product occurring as an error response served as the data for the product-error correlations. There are 31 different products for the problems ranging from 2 x 2 to 9 x 9, and 23 products in the Grade 2 set which ranged up to 6 x 6. Thirty of the adult subjects were randomly selected and their data correlated with the results from the remaining 30 subjects. The split-half correlations indicated that the adults were highly consistent in the errors they made, both with respect to problem-error rates, r (34) = .82, and product-error rates, r (29) = .71. The split-half correlation for the adult RTs was r(34) = .89. There was a clear developmental trend for the children's error patterns to converge on the adult pattern. The problem-by-problem correlation of the adult error frequencies with the Grade 2 error frequencies was .57, for Grade 3 in Session A it was .62, for Grade 4 it was .74, and for Grade 5 it was .84. By Grade 5, differential problem difficulty, as measured by problem-error rates, was

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J.I.D. Campbell & D.J. Graham TABLE 2 Percent of Errors that were Answers to Other Problems or Miscellaneous Group

Type"

2b

Table Misc

— —

3A

3B

3C

4

5

Adults

All Problems

Table Misc Table Misc

78.4% 21.6 — —

63.9% 36.1

68.7% 31.3

83.2 16.8

80.8 19.2

51.8 48.2

61.7 38.3

77.6% 22.4

69.1% 30.9

82.8% 17.2

92.6% 7.4

Set 1 Problems 81.5 18.5

79.3 20.7

86.4 13.6

94.5 5.5

Set 2 Problems 74.3 25.7

59.6 40.4

78.3 21.7

90.2 9.8

"Table = Errors that were answers to other problems. Misc = Errors that were not answers to other problems. b Grade 2 was not tested on Set 2 problems.

substantially the same as in the adults, although, of course, the adults were faster overall and less prone to error. The correlations for product-error rates were .41, .55, .69, and .73 for Grades 2, 3 (Session A), 4, and 5, respectively. Thus, not only was there a developmental convergence towards the adults for relative problem-error rates, but also for the particular responses that occur as errors. Similarly, the correlations involving RT indicate that by Grade 3 (Session A) the pattern of mean correct RT across problems already mapped closely on to the adult pattern yielding a correlation of .87 across the 36 problems. Taken together, these results indicate that a problem's difficulty is established very early in the learning process and is preserved in adulthood. To understand why problems differ in difficulty, we need to consider several aspects of the memory system involved: the internal organization of the memory structure, how the structure evolves, and what kind of processes operate to accomplish retrieval. Growth of the Associative Candidate Structure The specific errors that subjects make in simple multiplication are an important source of evidence about the cognitive structures that control retrieval. In particular, the errors that occur can be taken to reflect associative links between problems and candidate answers (Siegler & Shrager, 1984). Table 2 presents error rate data for each group for problem Sets 1 and 2 with errors classified into two categories. Table-errors are incorrect responses that involve answers that are correct to another simple multiplication problem (e.g., 4 x 8 = 24), and miscellaneous errors refer to any other incorrect response (e.g., 4 X 8 = 23). Consider first the errors that occurred on the Set 1 problems. On average over groups, 97% of responses to these problems (including errors) involved numbers in the range from 4 to 45. If we take this as the range of potential responses, there


351

TABLE 3 Percent of Errors that were Table-Related, Table-Unrelated, or Miscellaneous Group Type

2a

3A

3B

3C

4

5

Adults

Table-Related

46.0%

43.1%

47.7%

56.2%

47.6%

68.5%

79.1%

Table-Unrelated

32.4

20.8

21.0

21.4

21.5

14.3

13.5

Miscellaneous

21.6

36.1

31.3

22.4

30.9

17.2

7.4

a

Grade 2 was not tested on Set 2 problems.

are 42 integer values in the response set. For the 27 Set 1 problems, there are 1107 (27 x 41) possible different Set 1 errors. Among these, 594 (53.7%) would involve table-errors and 513 (46.3%) would involve miscellaneous errors. For no group, however, did the rate of miscellaneous errors approach this value. Even for the Grade 2 subjects, with approximately 1 month of experience on the Set 1 problems, only 21.6% of errors were miscellaneous while 78.4% involved answers to other multiplication problems. For Grade 3 (Session A), table-errors accounted for 83.2% of incorrect responses; for Grades 4 and 5, the percentages were 79.3 and 86.4 respectively. At adult skill levels, 94.5% of Set 1 errors involved table-errors. We can observe the same pattern on the Set 2 problems beginning, however, at a more primitive stage of learning. For Grade 3 in Session A, 72.7% of Set 2 trials involved an error. Of these, 51.8% were table-errors and 48.2% were miscellaneous errors. At this early stage of learning, children frequently guess nonproducts to provide an answer for an unlearned problem. For Grades 4 and 5 and the adults, the incorrect Set 2 responses that were table-errors were 59.6%, 78.3%, and 90.2%, respectively. As skill progresses, the set of candidate responses becomes increasingly constrained to the answers familiar as correct responses to some multiplication problem. This indicates that learning of multiplication facts proceeds by the development of an associative candidate structure. The error data presented in Table 3 are relevant to identifying the specific organization of the retrieval structure and the process by which it is acquired. Errors are classified into three categories: table-related errors which are tableerrors that are also a multiple of one of the problem's operands (e.g., 4 X 8 = 36); table-unrelated errors that involve answers to simple multiplication problems that are not multiples of the problem's operands (e.g., 4 x 8 = 27); and, again, miscellaneous errors that are neither type of table-error (e.g., 4 x 8 = 34). Embedded within the developmental trend for errors to involve table answers, is a trend for errors to be multiplicatively related to one or both of the operands in a problem. For Grade 3 (Session A), 43.2% of errors were table-related, 47.6% in the Grade 4 errors, 68.5% for Grade 5, and 79.1% of adult errors were multiplicatively related.

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This phenomenon addresses the issue of whether the formation of the retrieval structure is problem driven or operand driven. In other words, should, for example, 3 x 4 be considered a gestalt that forms a single association with 12, or do both 3 and 4 each have a link to 12? The developmental trend for errors to become table-related supports the idea that an associative structure evolves with links between operands and answers, as opposed to problems and answers, and that retrieval is operand driven. The contiguous occurrence of problems and correct answers during initial learning and on subsequent correct retrievals establishes links between each operand in a problem and the correct answer. One can think of a network structure in which all problems that share an operand in common (i.e., all problems within each times-table) are linked predominantly to products from the corresponding times-tables. When a problem is encoded, a set of associated candidates becomes activated, and an error reflects the selection of a false candidate. The correct candidate would receive double activation because it is associated with both operands (Stazyk et al., 1982); but, as discussed below, this cannot be the only basis for discriminating the correct answer. Additional evidence for operand driven activation is that the products that are correct for two problems (i.e., 12, 16, 18, 24, 36) are very common as error responses. For the adults, 30.6% of table-errors involved one of these five answers and for the children the percentage ranged from 21 to 29. This would be expected if retrieval were operand driven because these products would be directly associated with more problems than are the other products. Further, when both of a problem's operands are multiplicatively related to one of these answers, that answer is typically the most common error (e.g., 4 x 8 = 24,3 x 9 = 18). Indeed, for 8 x 4 under instructions for speed, the adults responded with 24 on nearly 17% of trials. Taken together, these results argue that associations develop between operands and candidate answers and that retrieval is governed largely by a process of spreading activation that is operand driven. Retrieval cannot be completely operand driven, however. If it were, it would be impossible, for example, to distinguish 32 from 24 when 8 x 4 (or 4 x 8) was the source of activation, because both operands have correct associations with each product (4 x 6 = 24, 3 x 8 = 24,4 x 8 = 32). Since, in such cases (e.g., 3 x 9 = 18, 4 X 8 = 24, 6 x 9 = 36, etc.), the correct response has a higher probability of retrieval than the error response, it appears that activation is both problem driven and operand driven. There is other evidence that retrieval is at least partially driven by the problem as a gestalt. With increasing skill there was a tendency for errors to be numerically closer to the correct answer. We computed the proportion of error answers for each group that were correct answers to neighbouring problems, ones with one operand the same and the other differing by ± 1. For Grades 3 (Session A), 4, 5, and adults, the percentages were 23.6, 32.5, 50.9, and 48.4 respectively. This suggests that, in addition to operandproduct links, pairs of operands may become associated with an abstract magnitude representation that constrains the numerical range of plausible candidates. This is consistent with the finding that in a true/false verification task (i.e.,.A x B = COT A + B = C), time to disconfirm a false answer is negatively


353

related to the magnitude of the numerical difference between the false answer presented and the correct answer (e.g., Ashcraft & Stazyk, 1981; Stazyk et al., 1982). Another possible explanation for the preponderance of neighbour errors is that when the learning sequence is structured by times-tables, groups of neighbouring problems and their products are introduced at the same time. Since the weakest associations would exist among new problem-product pairs, one could expect that confusions among these items would become common sources of errors. To summarize, the data from both children and adults indicate that most multiplication errors reflect the activation of false candidates in an associative network structure. Associations that are correct in the context of one problem (e.g.,4 x 6 = 24,3 x 8 = 24), promote errors in the context of another problem ( 4 x 8 = 24). Although most errors appear to reflect specific operand-product associations, the fact that table-unrelated errors occur with some frequency (see Table 3) suggests that there are other factors that contribute to associative confusions. Many of the table-unrelated errors involve digit-overlap between the correct answer and the error answer (e.g., 6 x 7 = 32,6 x 9 = 56,7 X 8 = 54, see Appendix A), indicating that similarity among products is also a dimension of association. We now develop the idea that differential problem difficulty largely reflects differences in the amount of interference caused by competing associations. Interference and the Problem-Size Effect As discussed earlier, one plausible explanation of why small-number problems are learned better than large-number problems is that the former are tested more frequently (Miller et al., 1984; Stazyk et al., 1982). This occurs simply because small-number problems are usually encountered first in the learning sequence and continue to receive maintenance practice as later problems are introduced. Moreover, the small-number problems are more common in everyday computation than are the larger problems. The facilitating effects of practice trials on accuracy and RT is one of the most robust phenomena in cognitive psychology (cf. Newell & Rosenbloom, 1981). Small-number problems may benefit in another way, relative to the other problems, as a consequence of entering first in the learning sequence. It may be that learning later groups of problems is impaired by the retrieval structure existing for problems learned earlier. The Grade 3 error data provide evidence for such a proactive effect, where answers already familiar as candidates (i.e., Set 1 products) interfere with learning the correct answers for new problems (i.e., Set 2 problems). In Session A, there was a 72.2% error rate on Set 2 problems, and 48.2% of the errors were miscellaneous guesses. Of errors that were table answers, 37.8% (59 of 156) involved answers to Set 1 problems and the remainder were Set 2 products. In Sessions B and C the percentages were 26.1 (35 of 134) and 20.9 (26 of 124). These percentages cannot be attributed to miscellaneous guessing and substantially reflect bona fide intrusions of Set 1 products: The answers to Set 1 problems span the numerical range from 4 to 45. Numbers less

354


than 20, however, were not considered to be plausible candidate answers to Set 2 problems. Over the three Grade 3 sessions, 98.2% of Set 2 error responses in the Set 1 range of answers involved numbers from 20 to 45. Twelve of the 26 numbers in this range are correct answers to some Set 1 problem. Therefore, a random selection of responses in the 20 to 45 range would yield Set 1 products 46.1% of the time. Across the three sessions, 79.7%, 62.5%, and 74.3% of Set 2 errors falling in this range involved numbers that are correct responses to Set 1 problems. On average, this is 26.1% greater than chance guessing would predict. When the new set of problems is first being learned, answers to problems encountered earlier are the source of a substantial percentage of the specific error responses. Our claim is that errors reflect associative links between problems and false candidates. Thus, associations formed between new problems and old answers represent a form of proactive interference. Given that problems are learned, generally, in order of increasing numerical magnitude, a build-up of proactive interference could be substantially responsible for the problem-size effect. As each set of problems is encountered, the pool of associations with each operand is increased. Thus, problems introduced later in the sequence must be learned in the context of many competing associations, whereas problems learned early are relatively free from competition. Putting the effects of competition together with relatively lower frequencies of occurrence, the sum effect would be that problems encountered later in the learning sequence, typically large-number problems, would tend to have weaker correct associations and more or stronger competing associations. If this is correct, the power of structural variables to predict RT derives largely from coincidental correlations with frequency of occurrence and with the order in which the problems are learned. In other words, the structural variables may be indirect indices of the extent to which interference is having an impact on any particular problem. If this is so, more direct measures of interference should provide better predictions of RT than the structural variables. Errors, RT, and Interference Given that errors reflect the organization of an associative candidate structure, the impact of interference on correct RT can be assessed by entering the error proneness of a problem, and the error proneness of the problem's product into a multiple regression predicting RT. Our claim is that the positive relationship between problem-error rates and RT observed by Norem (1928) and Miller et al. (1984) reflects associative competition from false candidates activated when a problem is encoded. This assumes that problem-error rates index the cumulative strengths of false associations with a problem. On trials where a false candidate is not retrieved, activation of competing candidates, nevertheless, contributes interfering noise in the retrieval process. If this is correct, there should be a strong positive correlation between problem-error rates and correct RT. Miller et al. (1984) found a rather modest correlation between problem-error rates and RT in multiplication (.537). Their subjects were tested on both addition problems and a


355

numerical comparison task, however, and 24% of the errors on multiplication trials were attributed to performing the wrong operation. When such errors occur, the multiplication retrieval structure might not be accessed and a high proportion of cross-operation errors could easily conceal a more intimate relationship between correct RT and within-operation retrieval errors. Norem's (1928) observation that product-error rate (i.e., how often a problem's correct product occurs as an error response to other problems) predicts learning difficulty of the problem suggests that accessibility of a product as a correct response is interfered with by competing associations with that product. This idea predicts that product-error rates should also be positively correlated with correct retrieval time for products. Thus, each problem can be thought of as having two basic error aspects that index interfering associations in the retrieval structure: problem-error rates and product-error rates. A correlation between problem-error rates and correct problem RT is open to criticism that we are correlating two dependent measures, both of which measure the -same thing: problem difficulty. The observation that most errors are multiplicatively related to problems' operands, however, lends considerable plausibility to an interference interpretation of the relationship. Moreover, while product-error rate is also a dependent measure, a positive correlation with correct RT is not a matter of correlating difficulty with difficulty for the same problem. A particular problem's product-error rate is determined by performance on other problems. Given that errors reflect associative links between problems and answers in a network structure, a product-error/RT correlation indicates that false associations with a product have an impact on the correct retrieval of that product. We have already shown that, with increasing skill levels, the pattern of specific errors made increasingly approximates the adult pattern, but that even at the earliest grades there are impressive similarities in the RT and error patterns. These similarities, we believe, reflect the fact that the same associative interference process is operating in children and adults; indeed, that adult performance is a direct reflection of the structure's initial development. If this is true, we should find a strong relationship between correct RT and problem-error and product-error rates in both the children's and adult's data. To investigate this, a series of multiple regressions predicting correct RT were carried out. For the children's data, the RT and error statistics for each problem were computed collapsing over operand order since there was no evidence that the order of operands affected their performance. Thus, except for Grade 2, the children's analyses were based on 36 problems, including 28 nontie problems and eight ties. The Grade 2 analysis included 22 nontie and five tie problems. Since significant operand order effects were observed in the adult data, all 56 nontie combinations were included in addition to the eight tie problems. The set of predictors included the standard structural variables (the minimum and maximum operand, the sum of the operands, the sum squared, and the correct product). Given the significance of problem-errors and product-errors to our theoretical account, however, the error variables were forced to enter the

356

J.I.D. Campbell & D.J. Graham TABLE 4 Correlation Matrices for the Dependent and Structural Variables Used in the Multiple Regression Analyses Group 3A (N = 36)

2(N = 27]1 RT PBE PDE PROBERR PRODERR M1N MAX SUM SUM SQR PRODUCT TIE/NON

.77 .33 .52 .59 .29 .63 .61 .40 .60 .75 .45 .76 .76 .37 .74 .77 .34 .76 - . 2 4 -- . 1 6 - . 1 4 4(N = 36;1

.69 .38 .59 .64 .24 .57 .56 .70 .47 .68 .39 .68 .35 - . 4 5 --.09

.33 .76 .27 .61 .54 .79 .46 .80 .38 .81 .34 .06 - . 1 7

5 (N = 36:1

RT PBE PDE PROBERR PRODERR MIN MAX SUM SUM SQR PRODUCT TIE/NON

RT PBE PDE .75 .49 .53 .69 .70 .68 .66 -.36

.25 .44 .40 .33 .30 .19

RT PBE

PDE

.76 .73 .55 .27 .51 .18 .62 .61 .48 .64 .51 .38 .43 .61 .32 .38 .59 .28 -.41 -.11 -.38

3B (N = 36) RT PBE PDE .81 .57 .42 .52 .61 .32 .63 .58 .49 .66 .63 .52 .63 .61 .48 .61 .62 .43 -.32 -.03 -.24

3C (N = 36) RT PBE

PDE

.80 .77 .63 .41 .36 .58 .63 .44 .55 .60 .59 .52 .55 .47 .56 .52 .57 .43 - . 3 7 - . 0 2 --.22

Adults (N = 64) RT

PBE PDE

.85 .45 .31 .45 .19 .46 .72 .46 .63 .68 .63 .38 .64 .62 .31 .60 .26 .59 -.40 -.28 -.21

Note. RT = Response Time. PROBERR (PBE) = Problem errors. N = Number of problems in regression. PRODERR (PDE) = Product errors. MIN = Minimum operand. MAX = Maximum operand. SUM = Sum of the operands. SUM SQR = SUM squared. PRODUCT = Problem's correct product. TIE/NON = Ties coded as 1, nonties coded as 0. Critical values for two-tailed tests at p < .05: r(25) = .38, r(34) = .33, r(62) = .24.

regressions first in separate steps. The 5% (on average over groups) of errors that apparently involved adding operands or naming an operand were not included in the error tallies, since, as pointed oflt above, such errors probably do not involve accessing the multiplication retrieval structure. Recall that the adult subjects received each nontie problem twice in each operand order and each tie problem four times. To correct for the difference in frequency of testing when operand order is preserved, the adult problem-error totals for tie problems were halved before entering the regression. Typically, tie and nontie problems receive separate analyses because tie problems do not appear to lie on the same regression line as the nonties when structural variables are used as predictors. We wanted to determine whether the distinction between tie and nontie cases would still hold when the variance associated with the interference variables was accounted for. A TIE/NONTIE

357

Multiplication skill TABLE 5 Multiple Regression Analyses of Problem Response Time

Group

Variables Entered

Final Beta Weight

Cumulative R Squared

F for Final Equation (df)

.393 .269 .376

.593 .676 .732

20.98 (3, 23)

PROBERR PRODERR TIE/NONTIE

.711 .194 -.370

.567 .632 .764

34.46 (3, 32)


.715 .216 -.249

.661 .727 .785

38.98 (3, 32)


.596 .328 -.304

.754 .838

54.99 (3, 32)


.464 .353 -.343

.392 .534 .653

20.09 (3, 32)


.543 .343 -.223

.573 .712 .753

32.54 (3, 32)

PROBERR PRODERR TIE/NONTIE SUM

.600 .126 -.205 .250

.728 .766 .786 .819

66.55 (4, 59)

PROBERR PRODERR PRODUCT 3A

3B

3C

Adults

.663

Note. PROBERR, PRODERR, and TIE/NONTIE were forced into the equation in that order, and remained in the equation only if they were significant atp < .05. Structural variables were then allowed to enter by a forward inclusion procedure, subject to the same criterion. Tests of the F-ratio at the final step were all significant at p < .001.

vector coding this distinction (tie = 1, nontie = 0) was entered third after the error variables in each analysis. The five structural variables were then allowed to enter by a forward inclusion procedure. Only those variables that contributed significantly (p < .05) were retained in the final equation. A simple-correlation matrix for each group for the dependent and structural variables in the regressions appears in Table 4, and the outcome of each analysis is presented in Table 5. For the adult data, the correlation with problem-errors (PRBERRS) yielded the highest simple correlation with RT of any of the variables tested, accounting for 72.8% of the variance in RT across the 64 combinations. This was 20% more than the best predicting structural variable (MAX). The frequency of a problem's correct product occurring as an error response to other problems (PRDERRS) accounted for a significant proportion of the residual, and, together, the error

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J.ID. Campbell & D.J. Graham

variables predicted 76.6% of RT variance. The correlations between RT and the error variables is evidence that adult performance is controlled by interference among associated candidate facts represented in a network structure. When the contribution of the interference variables was factored out, RT differences between tie and nontie problems were significant. TIE/NONTIE captured an additional 2.0% of the variance with beta-to-enter equal to — .149. The negative coefficient indicates that ties tended to be faster than nonties. That ties are faster is consistent with a theory where associations form between operands and products, and retrieval is slowed by false associations. Ties, having only one number, will activate fewer interfering false candidates. The difficulty for the present account is that ties were faster than their error rates would predict, given that error rates are a general index of interference. The independent contribution of TIE/NONTIE, however, does not mean that tie problems are immune to the interference effects implicated by the overall correlation with the error variables. A separate regression analysis for the eight tie problems revealed that the only significant predictor of RT from among all of those included in the main analysis was PRBERRS, r(6) = .812, p < .02. Further, the slopes of the simple regressions relating problem-error rates to correct RT were essentially the same for ties (7.08 msec per error) and nonties (7.75 msec per error). This suggests that the impact of interference as indexed by problem-error rates is the same for ties and nonties, and that the intercept difference is associated with a different process. It could be that operands are encoded successively, and when the second operand is the same as the first, it can be encoded faster. Such a repetition effect would not affect the slope for ties, but could produce a small uniform RT advantage for the tie problems. Another possibility is that our design provided for more practice on ties than on the nontie problems. Each time the nonties were tested with one operand order, each tie problem was also tested. It may be that there is not complete transfer of practice across operand order for the nonties; in this case, the ties can be considered to have received more practice. A structural variable, the SUM of the operands, entered after the forced predictors and accounted for an additional 3.2% of RT variance. The multiple R for the final equation was .904, with all four predictors contributing beta weights reliably different from zero. The regressions for the children's data yielded essentially the same pattern of results. For every group, problem-error rate was the predictor yielding the highest correlation with RT, on average accounting for 56.8% (SD = 9.4) of the variance in correct RT across the 36 problems (27 problems for Grade 2). Further, in every analysis the error proneness of a problem's product contributed independently of problem-errors to RT. Together, the error variables predicted, on average, 67.2% of RT variance across all the children's sessions. With the exception of the Grade 2 data, TIE/NONTIE accounted for a significant proportion of the residual in each group's analysis. In each case, ties tended to be faster than nonties. Separate analyses of these children's performance on the eight tie problems indicated that the predictor yielding the highest


359

correlation with correct RT was always PRBERRS or PRDERRS, although there was no reliable predictor of tie RTs for Grade 4. Among the children's analyses, only in the Grade 2 regression (with the TIE/NONTIE vector excluded) did a structural variable (PRODUCT) enter after variance associated with the interference variables was removed. These results support the conclusion that the multiplication performance of both children and adults involves fact retrieval from an associative network. Both the error proneness of the problem and the product contribute to predicting correct retrieval time, a result that indicates that competing associations with both the problem and the product interfere with a correct retrieval. The contribution of PRODUCT in the Grade 2 regression may indicate that these subjects sometimes resorted to counting strategies. The significance of the SUM of the operands entering the regression for the adults, however, is not simple to interpret. Miller et al. (1984) have suggested that correlations with structural variables reflect the location of information in a network structure; however, in their account this distance factor is supposed to be the major variable controlling correct RT. Here, when the contribution of the interference variables was removed, the structural variables could account for only about 3% of additional RT variance in the adult data. The regression analyses indicate that, for both children's and adults' performance, factors derived to index network interference best predict variability in RT across problems. Summary and Conclusions Our claim in this paper is that the acquisition of simple multiplication skill is well described as a process of associative bonding between problems and candidate answers. Performance is determined by the relative strengths of correct and competing associations represented in a network structure that is searched by a process that activates associated products. The activation of competing products and competing associations with the correct product both interfere with a correct retrieval. By this account, the problem-size effect is not a consequence of the distance searched through the network, or of the execution of reconstructive arithmetic procedures. Instead, it is largely the result of large-number problems being tested less often and occurring later in the learning sequence, factors that result in relatively weak correct associations and more, or stronger, competing associations. Although we suspect that the confounding of order and numerical magnitude can substantially explain the predictive power of structural variables, there is variable difficulty that is intrinsic to the problem-set. For example, 4 x 8 (or 8 X 4) is one of the most difficult problems even though it is usually encountered in the middle of the sequence. Its difficulty is probably due to a strong association with 24, promoted by the overall similarity of 3 x 8 = 24 and 4 x 8 = 32, and because both 4 and 8 have correct multiplicative associations established with 24. Conversely, 8 x 9 (or 9 x 8) is a relatively easy problem (for Grade 3 Session C, Grades 4 and 5, and adults its averaged ranked RT was below the 75th percentile) even though it is likely one of the last problems to be learned. This probably

360


reflects the fact that its answer (72) is the largest of the nontie answers, and so is not embedded numerically among other candidate answers. Further, it is the only simple multiplication product in the seventies. Any factor that contributes to an answer's discriminability from other answers probably contributes to ease of learning and retrieval. While network interference can substantially explain multiplication performance, this does not preclude a role for the kind of procedural or rule-based knowledge described by Baroody (1983, 1984). For example, the fact that the 5-times problems as a group are easier than many of the other times-tables can be attributed to a rule that states that if a problem contains a 5 then the answer ends in 5 or 0. That such a rule is functional in retrieval is suggested because most of the errors on 5-times problems are multiples of 5, as opposed to multiples of the other operands. The application of this rule during learning may reduce the candidate set for 5-times problems by rendering many of the products associated with the other operand implausible. In general, any rule that constrains the candidate set for a problem should facilitate learning. The associative interference account of multiplication performance proposed here argues that learning of the multiplication facts can be facilitated by minimizing the formation of false associations. This may be accomplished by the following: (a) by assuring complete learning of problems encountered early in the sequence to minimize proactive interference with problems introduced later; (b) by structuring the order of acquisition so that problems prone to specific confusions (e.g., 3 x 9 = 18,4 x 8 = 24,6 x 9 = 36, etc.) are learned prior to learning the competing problems; (c) by establishing strong correct associations for products that are intrinsically prone to forming false associations (e.g., products that are correct for two problems); and (d) by providing rules or mnemonics during learning that constrain the set of plausible candidates for specific problems. We are currently conducting research to determine the effectiveness of these manipulations. The fact that the stimulus domain is relatively small and that training of the number facts is intensive encourages the view that basic arithmetic cognition involves stable, isolated, and relatively simple associative structures. Using simple arithmetic as a paradigm case, it may be possible to define precise mechanisms of interference that have general implications for semantic memory and its development.


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363

APPENDICES

APPENDIX A Adult Multiplication Error Matrix (W = 60) with Mean Correct Response Time in Msec (RT), Standard Deviation (SD), and Totals for Table Errors (TE), Miscellaneous Errors (ME), and Product-errors (PE) PRODUCT PROBLEM

4

2x2

c

2 x 3

3x2 2x4

1

6

c c 1

4 x 2 2X5

9

10 12 14 15 16 18 20 21 24 25 27 28 30 32 35

1

c c

c c

5x2 2x6 6x2 2x7 7x2 2x8 8x2 2x9 9x2

3 1 \

1 1 2 1

1

2 2 1 c c 1 1

1 c

c

2

1

c c 1

c c

2

1

6 x 3

2 c

1

3 x 8 8 x 3 3 x 9

2 1 2

9x3 1 1

3

2 1 1

1 1

1

1

2

3x7 7x3

4 x 4 4 x 5 5 x 4

1 1 8 3 c c

c

1

RT

SD

TE

ME

PROD

667 746 729

136

0 0 1 5 3 4

4 0 0 0 0

2 5 2 5

0 0 0 0

4 6 6 8 8 10 10 12

5 10 6 0 2 5 1 0 2 0 6 8 4

2

750

2 5

36 40

1

2 c c 1 2 1 2

1

3 x 3

3 x 4 4x3 3x5 5x3 3x6

8

c

10 5 8 13 2

2 c c

2

1

c c 1 1 1

7 4

1 4 1 1 3 c c

1

1 1 2

735 724 703 796 734 761 763 805 816 741 762 687 736 717 722 740 898 842 743 759 953 878 893 881 764 761 729

202 156 174 164 134 161 195 218 182 202 236 276 167 205 141 184 163 157 !59 353 259 177 170 324 300 278 286 252 207 196

2 14 10 19 20 9 4 2

2

1 0 0 0 0 0 0 0 0 0 0 1 0 0 0

2 1 0 0 0

PE

5 10 4

5 25

12 14 14 16 16 18 18 9 12 12 15 15 18 18 21 21 24 24 27 27 16 20 20

6 10 55 8

2

20

53 20

4

I

I a o

APPENDIX A (Continued) PRODUCT PROBLEM < 20 20 21 24 25 27 28 30 32 35 36 40 42 45 48 49 54 56 63 4X6 6X4 4X7 7X4 4X8 8X4 4X9 9X4 5 x 5 5X6

2

c c :

3

2

1

3 6 15

20

2 1

1 1

1 1

1 1

c c

1

3 2

c

5

c

1

c c

1

6x5

1 2

5X7 7 x 5 5X8

1 1

1

1

8x5

2

5X9

2

9x5 6X6

2

1

6x7 7X6 6X8

1 1

1

6 4

3 3 5 c c 2 2 3 2

1 1

7X7

2

;

1

1

9x7 8X8

2

8x9

c £

c 2 5

3 11

1 3 3

1

e &

4 2 3 4

1 1

c c 1 c c 11 4 1 3 6 6 4

1 2

2

5 1 1

1 1 2 2 1

c c

4

2

2

c c

10 2

4

c 2

1

7 4 1

6 3 4

c 12 8 1 4 1

5 9 10

1 1 1 2 2

3

c

1

9

1 1

1

3 1

1

1 2

TE

ME

PROD

PE

5 1 12 10 29 28 11 11 5 6 7 11 4 9 8 17 7 7 14 15 14 17 32 31 10 23 17 31 24 16 14

0 1

24 24 28 28 32 32 36 36 25

53

759

9X8

9x9

1 1

1

2

SD 257 287 355 342 310 387 354 291 172 297 206 266 252 314 281 277 223 153 270 320 345 375 400 331 163 341 292 294 359 262 317 303 172

973 975 924 705 827

2

1

7x8 8x7 7X9

RT 835 826 905 892

952 2

c c

8x6 9x6

2

2

1

6X9

1 1 4

1

2 5

c 4 1

2

3

64 72 81

2

1 1

1 1

3 2 2

1 c

5

c c 2 4

c c

7 J 3 c

877 835 908 832 913 833 711 831 919 916 912 1017 1016 720 892 863 959 982 834 916 909 748

6 3

0 0 2 1 3

0 0 0 0 0 0

0 0 0 1 1 2 1 1

0 4 3 1 3 3 1 3 3 3 1 0

30 30 35 35 40 40 45 45 36 42 42 48 48 54 54 49 56

13

22 44 3 11 20 18 19 44 45 23 31 14

58

56 63 63 64 72 72 81

24 12 15 13

Note. Each of 60 adult subjects was tested twice on each nontie combination and four times on each tie problem. Average number of correct RTs per mean: nontie = 100.3, tie = 227. Matrix entries are the total number of times a particular table-answer (product) occurred as an error to a given problem, c = correct product.