Ties Effect Running head: The Ties Effect

A memory-based account of the arithmetic tie-effect

Michele Burigo§, Luisa Girelliº

§

Centre for Thinking and Language, School of Psychology,University of Plymouth, UK ºDepartment of Psychology,University of Milan – Bicocca, Italy

Address for correspondence: Michele Burigo University of Plymouth, School of Psychology B 221 Portland Square, Drake Circus Plymouth, PL4 8AA, UK Phone: +44 (0) 1752-263216 Fax: +44 (0) 1752-233362 Email: [email protected]

Keywords: arithmetical facts; tie effect; memory advantage; mathematical cognition

Word Count: 2976

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Ties Effect Abstract The ties effect is the tendency to solve arithmetic problems formed by repeated operands (e.g. 3x3) faster and more accurately than comparable non-tie problems (e.g., 3x4). Recently, encoding based (Blankenberger, 2001) and retrieval based explanations (Campbell & Gunter, 2002) of this phenomenon were proposed. Two experiments investigate these accounts. Experiment 1 required participants to multiply a visually presented digit (e.g., …x3) against a second one kept in memory (e.g., five). Experiment 2 required participants to classify a visually presented number (e.g., 25) as multiple or not of a target number (e.g., 4). A significant tie effect emerged systematically despite, in both experiments, encoding advantage for tie problems were eliminated. Overall these results favour the access hypothesis according to which frequency of direct retrieval and efficiency of retrieval processes are largely responsible for the tie effect in simple arithmetic (Campbell & Gunter, 2002).

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Ties Effect Introduction

Within mathematical cognition research, it is well known that, not all operations are equally difficult. Differences in problem difficulty (measured by response time and accuracy) are correlated with the magnitude of the result (e.g., 7x8 is more difficult than 3x4), the well-known “problem-size effect” (Ashcraft, 1992, Campbell & Xue, 2001; LeFevre, Sadesky & Bisanz, 1996). This effect concerns almost all simple arithmetic problems but not those composed by repeated operands (i.e., 3x3, 7x7, etc.), where the “problem-size effect” is much smaller (Butterworth, Zorzi, Girelli & Jonckheere, 2001; Groen & Parkman, 1972). The advantage for the repeated operand problems is known as the “tie effect” and indicates the tendency to solve tie problems faster and more accurately than comparable non-tie operations (Campbell, 1999; Miller, Perlmutter & Keating, 1984). This effect concerns additions (Hamann and Ashcraft, 1985; LeFevre, Sadesky & Bisanz, 1996), multiplication, division and subtractions (Campbell, 1997; Campbell & Gunter, 2002). This phenomenon received several explanations. A strategy-based account of the tie effect was proposed originally by Groen and Parkman (1972) and, more recently, by LeFevre and colleagues (LeFevre et al., 1996; LeFevre, Bisanz, Daley, Buffone, Greenham & Sadesky, 1996). They claimed that ties are solved faster because, compared to non-tie problems, are more frequently solved by direct retrieval rather than by using procedural strategies. Alternatively it has been suggested that the tie advantage is grounded on difference in practice: tie problems would occur more frequently than non-tie problems and, thus, receive more practice during learning

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Ties Effect (Siegler, 1998; but see Hamann & Ashcraft,1986). Yet, the fact that a tie effect emerges even with alphanumerical stimuli presented with controlled occurrence made the practice hypothesis rather unlikely (Graham & Campbell, 1992). In contrast, Gallistel and Gelman (1992) proposed that tie problems would benefit from an easier encoding process. Their “bi-directional mapping” theory assumes that problem operands (e.g., 3 x 4) have to be “mapped” to a mental number line. Thus, since the mapping process for ties occurs only once (operands are identical), problem encoding is less error prone and response times are faster. Recent empirical support to the encoding hypothesis comes from a study by Blankenberger (2001). The author reported that addition and multiplication problems presented in mixed format (e.g. three x 5) do not yield any tie effect, in contrast with non-mixed operations (e.g., three x five, 3 x 5). Blankenberger interpreted these results by assuming that the tie effect is mainly due to fast encoding of perceptually identical stimuli. A controlled replication of this study, however, identified differential practice in tie and non-tie problems as potential confounding for eliminating the tie effect (Campbell & Gunter, 2002). Moreover, challenging the encoding hypothesis, Campbell and Gunter (2002) reported a tie effect also in division and subtraction problems (where ties are not represented by identical operands) as well as reduced tie effect as a function of arithmetic skill level and of problem-size. Although these results are difficult to account within the encoding hypothesis, the critical factors that contribute to the advantage for the tie problems in the answer/access stages remain to be established.

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Ties Effect The present study contributes to this debate providing additional evidence against an encoding-based interpretation of the tie effect (Experiment 1), and pointing to the memory distinctiveness of tie problems among the main responsible of their speed-accuracy advantage (Experiment 2). Experiment 1 controls for encoding facilitation by requiring subjects to multiply a visual number against a memorised operand. Experiment 2 entails a true/false classification of numbers as multiple or not of a given operand. If the tie-effect is access/memory based, tie-answers (e.g., 25, 36, …etc) should equally benefit from a speeded retrieval.

Experiment 1 Method Participants Twenty undergraduates (age ranged from 19 to 30 years) from the University of Milano-Bicocca participated as volunteers in the experiment. All participants were Italian native speakers and had normal or correct to normal vision. All of them were naïve to the experimental hypothesis.

Design and materials The experiment included two tasks: a “memory task” and a “production task”. In the memory task participants were required, first, to memorise a number and then, to multiply it for the number shown afterwards on the screen. In the production task,

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Ties Effect participants were required to multiply the two operands presented simultaneously on the screen according to the standard methodology. In the production task subjects were presented with two experimental blocks, each block included 64 multiplications (from 2x2 to 9x9) presented twice, for a total of 256 trials. Within each block, problems were presented in pseudorandom order with the criterium that consequent trials differed in both operands (e.g., 3x4, 5x6). In the memory task subjects were first presented with a target single digit (e.g., 5), followed by a sequence of 16 numbers, including all numbers from 2 to 9 presented twice in randomised order. Subjects were required to multiply each number in the sequence by the first one. They were presented with eight sequences, one for each number from 2 to 9 in a single experimental block. Order of the target numbers was randomised within each of two experimental blocks for a total of 256 trials. Presentation order of the two tasks was counterbalanced between subjects. Each task started with a warm – up (32 items). Instructions emphasised both speed and accuracy. Oral responses were both taped and manually recorded.

Procedure Participants were tested individually in a quite and slightly dark room. They sat about 50 cm far from the screen. The stimuli, black on white background, were displayed in the centre of the screen and were 0.24º high and 0.08º width. The sequence started with a 500 ms fixation point followed by a target number (memory task) or a problem (production task). The target remained on the screen until subjects

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Ties Effect produced the vocal response. The ISI was 1500 ms; the experiment lasted about 35 minutes.

Apparatus The stimuli were shown by an IBM-compatible PC connected with a 17’’color monitor. A Sony microphone was interfaced with an audio board I/O SoundBlaster trigged by vocal responses. RTs were recorder by software program SuperLab Pro ver. 2.0 by Cedrus, accurate to the millisecond.

Results Product task RT results Correct answers only were submitted to statistical analysis. In order to eliminate outliers a filter of 3 standard deviation was adopted. This procedure eliminated 84 responses (1.74 %). Filtered data were entered in a repeated measure ANOVA with tie (ties vs. non-ties) and problem-size (small - problems with both operands smaller than or equal to 5 - vs. large - problems with both operands larger than 5) as within-subject factors. Following Campbell and Gunter (2002), excluding problems like 2x7 or 9x3 we ensured that tie and non-tie problems had operands of comparable magnitude. Both main effects were significant: tie effect [F(1,19) = 31.445, MSE = 70918.57, p < .0001] and problem-size effect [F(1,19) = 49.67, MSE = 70434.77, p < .0001]. Tie problems were solved faster (879 ms) than non-tie problems (1214 ms). The same advantage was found for small problems (837 ms) compared to large ones

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Ties Effect (1256 ms). Moreover, the interaction was significant [F(1,19) = 26.3, MSE = 62755.57, p < .0001] (Figure 1). Post-hoc tests (Scheffé) pointed out that the problemsize effect holds true for the non-tie problems only (1569 ms vs. 859 ms, p < .0001). Similarly, the tie advantage emerged in large problems only (1569 ms vs. 942 ms, p < .0001).

Error analysis Overall, 290 (5.7%) errors were recorded: 108 were device errors (2.1%) and 182 were production errors (3.55%). The analysis of the production errors revealed a strong tie effect with an higher number of error for non-tie problems (177; 3.95%) than for tie problems (5; 0.8%) (t(19) = 6.046, p < .0001). Moreover, 33 (18.6%) out of the 177 errors for non-tie problems corresponded to tie answers (e.g., 8x9=81) while none of the few errors for ties were tie answers.

Memory task RT results Correct answers only were submitted to statistical analysis. Responses were sent to a filtering process (set to 3 SDs) that eliminated 91 responses (1.88 %). Filtered data were entered in a repeated measure ANOVA with tie (ties vs. non-ties) and problemsize (small vs. large). Similarly to production task, small trials were defined as those in which both the memorised and the visually presented operands were smaller than 5, large trials were defined as those in which both operands were larger than 5.

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Ties Effect The ANOVA revealed a significant main effect of tie [F(1,19) = 35.691, MSE = 39360.48, p < .0001]: Tie problems were solved faster (908 ms) than non-tie problems (1154 ms). Moreover, large problems yielded slower RTs than small problems [1314 ms vs. 888 ms; F(1,19) = 64.48, MSE = 33374.44, p <.0001]. The interaction tie x problem-size was also significant [F(1,19) = 34.922, MSE = 26811.1, p < .0001]. Post-hoc tests (Scheffé) pointed out that the problem-size effect holds true for the non-tie problems only (1445 ms vs. 899 ms, p < .0001). Similarly, the tie advantage emerged in large problems only (1445 ms vs. 964 ms, p < .0001) (Figure 1). The tie effect was further tested by maximising the similarity between ties and non-ties. To this purpose the RT for each tie problem was compared to the mean RT for the preceding (e.g., 4x4 vs. 4x3-3x4) and the following problems (e.g., 4x4 vs. 4x55x4) in the table. All comparisons between ties and neighbour problems were significant (all, p < .05) but the tie “5x5”. A final analysis compared complement problems (e.g. {…x 3} vs. {3 x…}) revealing a null effect of the operand order (1129 ms vs. 1122 ms, t(27)=.5214, n.s.).

Error Analysis Overall, 286 (5.6%) errors were recorded: 125 were device errors (2.4%) and 161 were production errors (3.2%). The analysis of the production errors revealed a strong tie effect with an higher number of error for non-tie problems (152; 3.4%) than for tie problems (9; 1.4%) (t(19) = 5.88, p < .0001). Moreover, 36 (23.7%) out of the

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Ties Effect 152 errors for non-tie problems corresponded to tie answers (e.g., 8x9=81) while none of the few errors for ties were tie answers.

(Please, insert figure 1 about here)

In summary, in the production task we replicated the standard tie effect and its interaction with the problem-size; in fact, as reported by Campbell (2002), the tie effect holds true for large problems only. On the other hand, the memory task required subjects to multiply a visual number against a memorised operand. In this way, encoding of the two operands took place at different time, thus minimising, following Blankenberger (2001), the facilitation for ties. Nonetheless, tie problems yielded faster (245 ms and 206 ms for production and memory tasks respectively) and more accurate responses than other problems. However, one may suggest that encoding of a visual number matching a memorised one benefits from facilitated recognition, making ties still easier than non-ties. Experiment 2 aimed to overcome this possible confound by requiring participants to focus on multiplication results only.

Experiment 2

Following the access/memory interpretation of the tie-effect, we reasoned that tie answers (e.g., 25, 36, etc) should be easier to recognise as belonging to their tables than comparable non-tie answers (e.g., 24, 35, etc). On this basis, participants were presented with a true/false classification of table results.

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Ties Effect

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Method Participants Twenty students (age ranged from 20 to 32 years) from the University of Milano-Bicocca volunteered in the experiment. All participants were Italian native speakers with normal or corrected to normal vision. None of them participated in Experiment 1 and were aware of the experimental hypothesis.

Design and Materials The experimental task required participants to classify a visually presented number as belonging to a “target table” or not; for example, given the table of 5 as the target one, subjects had to decide whether the number 36 was included in the table or not. Within each “target table” from 2 to 9 a randomised sequence of 64 numbers was presented: thirty-two numbers required a “true” answer (the eight multiples repeated four times), and thirty-two required a “false” answer. The false trials included 16 “table” numbers (i.e. numbers that belong to tables different from the target one) and 16 “non table” numbers (i.e. numbers out of any table) matched for magnitude to the correct multiples. Overall, the experiment included 8 blocks (for a total of 512 stimuli), one for each “target table”, presented in random order to each participant. The experiment started with a warm-up including 16 numbers with the 10 table as the “target”. Position of true/false answer buttons was counterbalanced between subjects. Instructions emphasised both speed and accuracy.

Ties Effect

Procedure Participants were tested individually in a quite and slightly dark room. They sat about 50 cm far from the screen. Before each block the experimenter specified to the participant the “target table”. The stimuli, black on white background, were displayed in the centre of the screen with viewing angles of 0.7º horizontally and 0.12º vertically . The sequence started with a 300 ms fixation point followed by the number to be classified. The target remained on the screen until subjects pressed the answer key. The ISI was 500 ms. The experiment lasted about 30 minutes.

Apparatus The same apparatus as Experiment 1 was used.

Results RT results Correct answers only were submitted to statistical analysis. Responses were sent to a filtering process (set to 3 SDs) to eliminate outliers. This procedure eliminated 203 responses (2.41 %). Within this data set the first analysis focused on “true” trials only. As expected, tie results were overall answered faster (705 ms) than non-tie results (781 ms) [t(19) = 6.088, p < .00001]. This advantage was nearly replicated when a stringent analysis, minimising the magnitude difference between ties and non-ties, compared directly each tie trial to the closest non-tie trials (e.g. 16 (4x4) vs. 12 (4x3) and 20 (4x5)). Although this item analysis failed to reach significance (t(7)= 1,986, p=.08),

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Ties Effect inspection of the single item revealed an unique odd effect referred to the 2 table (2x2, 573 ms vs. 2x3, 527 ms). Once this item was removed from the analysis, the tie advantage reach significance (t(6)= 2.435, p=.05) (Figure 2). A further analysis focused on the correct “false” trials, revealing, that non-table and table results yielded similar response times (t= -.400, n.s.). However, within table results, tie answers (16 and 36 were excluded from this analysis being multiple of both tie and non-tie problems) were rejected faster than non ties (770 ms vs. 806 ms, t(19) = -2.49; p < .05).

(Please, insert figure 2 about here)

Error analysis Overall, 790 (7.7%) errors were recorded; 372 were “false negative” response (3.6%) and 418 were “false positive” (4.1%). Interestingly, within these latter, table results (N=291, 37%) yielded significantly more errors than non table results (N=127, 16%) (t=-7.937, p<.001). Finally, false negative responses were significantly more frequent for non-tie (7.6%) than for tie answers (4.8%) (χ2= 5,88, p<.05) (Table 1).

(Insert Table 1 about here)

In summary, the outcome of this experiment suggest that tie results were processed faster and were less error prone than non-tie results. Thus, both RTs and accuracy data indicate that tie advantage applies not only to standard retrieval process

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(e.g., 3x3= ?), but also to indirect access to problem knowledge (e.g., “Does 9 belong to the 3 table or not?”).

Discussion

Aim of the presented experiments was to test different accounts of the tie effect in simple multiplication. In particular, Experiment 1 challenges the encoding hypothesis demonstrating that tie problems are faster and more accurate than non-tie problems even when encoding is limited to a single operand. In the memory task, however, we may not exclude that faster encoding of a visual number corresponding to the memorised one contributed to the advantage of ties compared to non-ties. In fact, this specific experimental condition may be considered a special case of repetition priming, where the identity between prime, i.e., the memorised digit, and target, i.e., the visually present digit, improved target processing (Koechlin, Naccache, Block & Dehaene, 1999; Reynvoet & Brysbaert, 2004). Experiment 2 was designed to test the hypothesis that tie advantage is mainly based on difference in accessibility and in particular to memory distinctiveness of tie problems. It has been suggested that, as far as multiplication is concerned, the tie effect reflects mainly better memory for the repeated operand problems compared to other problems (Campbell & Gunter, 2002). Several factors may be responsible for more efficient retrieval; among them, higher frequency of ties during learning, more practice associated to the use of ties as “anchors” in procedural strategies (e.g., 6x7= (6x6) + 7 = 36+7=42), and greater distinctiveness determined by their reduced source of associative interference. In fact, assuming that arithmetic facts representation is mainly based on meaningful association between similar problems (e.g., 4x6 being variably

Ties Effect

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associated to all “4 times” and “6 times” problems; cf. the frequency of operand errors in skilled performance Campbell & Graham, 1985), repeated operand multiplications may be only associated to a single table problems (e.g., 4x4 being associated to all “4 times” problems only). We reasoned that this structural advantage may contribute to greater memory distinctiveness not only in production tasks, but also in a mere classification of problem answers. Similarly to standard verification of simple arithmetic, classification of problem answers may be mainly solved through familiarity-based judgment (Lemaire & Fayol, 1995). This latter is certainly more efficient for easier trials; more difficult trials possibly require verification processes based on fact retrieval or calculation strategies (see Footnote 1) (e.g., 56
56:8 = ? or 8x7= ?). The results of Experiment 2 support our hypothesis, demonstrating that tie answers were faster to be recognised and rarely incorrectly rejected as multiple of a target operand than comparable non-tie answers. Clearly, encoding processes may hardly be responsible for these results. Overall, we argue that this evidence add to the existing literature favouring a memory based account of tie effect in simple arithmetic (Campbell & Gunter, 2002).

Ties Effect References Ashcraft, M. H. (1992). Cognitive arithmetic; a review of data and theory. Cognition, 44, 75 – 106. Blankenberger, S. (2001). The arithmetic tie effect is mainly encoding-based. Cognition, 82, B15 - B24. Butterworth, B., Zorzi, M., Girelli, L., & Jonckheere, A. R. (2001). Storage and retrieval of addiction fact; the role of number comparison. Quarterly Journal of Experimental Psychology, 54A, 1005 – 1029. Campbell, J. I. D. (1997). Reading-based interference in cognitive arithmetic. Canadian Journal of Experimental Psychology, 51, 74 - 81. Campbell, J. I. D. (1999). Division by multiplication. Memory and Cognition, 27, 791 – 802. Campbell, J. I. D., & Graham, D. J. (1985). Mental multiplication skill: structure, process and acquisition. Canadian Journal of Psychology, 39, 338 – 366. Campbell, J. I. D., & Gunter, R. (2002). Calculation, culture, and the repeated operand effect. Cognition, 86, 71 – 96. Campbell, J. I. D., & Xue, Q. (2001). Cognitive arithmetic across cultures. Journal of Experimental Psychology: General, 130, 299 – 315. Gallistel, C. R., & Gelman, R. (1992). Verbal and preverbal counting and computation. Cognition, 4, 43 – 74. Graham, D. J., & Campbell, J. I. D. (1992). Network interference and number-fact retrieval: evidence from children's alphaplication. Canadian Journal of Psychology, 46, 65 – 91.

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Groen, G. J., & Parkman, J. M. (1972). A chronometric analysis of simple addition. Psychological Review, 79, 329 – 343. Hamann, M. S., & Ashcraft, M. H. (1985). Simple and complex mental addition across development. Journal of Experimental Child Psychology, 40, 49 - 70. Hamann, M. S. & Ashcraft, M. H. (1986). Textbook presentations of the basic addition facts. Cognition and Instruction, 3, 173 – 192. Koechlin, E., Naccache, L, Block, E., & Dehaene, S. (1999). Primed numbers: exploring the modularity of numerical representation with masked and unmasked priming. Journal of Experimental Psychology: Human, Perception and Performance, 25, 1882 – 1905. LeFevre, J., Bisanz, J., Daley, K. E., Buffone, L., Greenham, S. L., & Sadesky, G. S. (1996). Multiple routes to solution of single-digit multiplication problems. Journal of Experimental Psychology: General, 125, 284 – 306. LeFevre, J., Sadesky, G. S., & Bisanz, J. (1996). Selections of procedures in mental addition; reassessing the problem size effect in adults. Journal of Experimental Psychology: Learning, Memory, and Cognition, 22, 216 – 230. Lemaire, P., & Fayol, M. (1995). When plausibility judgements supersede fact retrieval: the example of the odd-even effect on production verification. Memory and Cognition, 23 (1), 34 – 48. Mauro, D. G., LeFevre, J., & Morris, J. (2003). Effects of problem format on division and multiplication performance: division facts are mediated via multiplicationbased representations. Journal of Experimental Psychology: Learning, Memory and Cognition, 29 (2), 163 – 170.

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Miller, K. F., Perlmutter, M., & Keating, D. (1984). Cognitive arithmetic: Comparison of operations. Journal of Experimental Psychology: Learning, Memory, and Cognition. 1l0, 46 - 60. Reynvoet, B., & Brysbaert, M. (2004). Cross-notation number priming investigate at different stimulus onset asynchronies in parity and naming tasks. Experimental Psychology, 51, 81 – 90. Siegler, R. S. (1998). Strategy choice procedures and development of multiplication skill. Journal of Experimental Psychology: General, 117, 258 – 275.

Ties Effect Author Note We would like to thank J. I. D. Campbell for his comments on an early version of this paper.

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Ties Effect Footnotes 1

It is worth noticing that the use of strategies based on divisions is still likely to imply

mediation of multiplication-based representations (Mauro, LeFevre & Morris, 2003).

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Ties Effect Tables

Table 1 Reaction times (ms) and error rate ("false negative" reponses only) for tie and non-tie problems. Ties

Non-Ties

Reaction time (ms)

705

781

Errors

7.6%

4.8%

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Ties Effect Figure Caption Figure 1. Mean correct reaction times in the production and the memory tasks of Experiment 1.

Figure 2. Reaction times for correct responses for tie and non-tie problems as a function of table (from 2x2 to 9x9).

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Ties Effect Figure 1

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Ties Effect Figure 2

1300 1200 Reaction Times (ms)

1100 1000 900 800 700 600 500 Ties

400

NO ties

300 2

3

4

5 6 Tables

7

8

9

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