Original Article
Operational Momentum in Numerosity Production Judgments of Multi-Digit Number Problems Oliver Lindemann and Michael D. Tira Donders Institute for Brain, Cognition and Behaviour, Radboud University, Nijmegen, The Netherlands Abstract. The current study demonstrates a numerosity production task and investigates approximate mental calculations with two-digit numbers. Participants were required to produce random dot patterns to indicate the size of two-digit numbers and the results of addition and subtraction problems. The stimuli in the calculation task consisted of problems requiring a carry operation (e.g., 24 + 18) or no-carry problems (e.g., 24 + 53) or zero problems (e.g., 24 + 0). Our analysis revealed that the outcomes of additions were estimated to be larger than the outcomes of subtractions. Interestingly, this judgment bias was present for no-carry and zero problems but not for carry problems. Taken together, the presented data provide empirical support for the presence of an operational momentum effect (OM effect) in multi-digit number arithmetic. These findings and the dissociation of the OM effect for carry and no-carry problems are discussed in the context of recent models on multi-digit number processing. Keywords: multi-digit number processing, operational momentum effect, numerosity production task, mental arithmetic
Recent research has shown that number representations and mental arithmetic are grounded on an analog approximate system for magnitude information that is independent of language (e.g., Dehaene, 2009). Evidence for this notion comes, for example, from the finding of a cognitive bias in numerical estimations after mental calculation – an effect that was first described by McCrink, Dehaene, and Dehaene-Lambertz (2007) for the processing of nonsymbolic numerosities. Participants viewed moving dot patterns being added or subtracted from one another and indicated whether the numerosity of a final set of dots was correct or incorrect. Surprisingly, in the case of addition, the subjects’ estimated outcomes tended to be larger than the actual outcomes, whereas the estimations tended to be smaller than the actual outcomes with subtraction. McCrink and colleagues (2007) used the metaphor of a mental number line for analog magnitude representations (Dehaene, 1997) to explain their finding. They speculated that mental calculations are functionally equivalent to movements along the spatial-numerical continuum and assumed that the overestimation after addition and the underestimation after subtraction reflect the subjects’ tendency to move ‘‘too far’’ to either the right or left side. Since the observed effects are reminiscent of a perceptual phenomenon called representational momentum (Freyd & Finke, 1984), which represents the tendency of subjects to misjudge the stopping point of a moving object, McCrink and colleagues (2007) labeled the observed judgment bias after mental calculation the operational momentum effect (OM effect). Interestingly, a recent study of Pinhas and Fischer (2008) demonstrated that spatial response biases also emerge after mental calculations with exact numbers and provided thus first direct empirical evidence that the OM effect generalizes Zeitschrift fu¨r Psychologie / Journal of Psychology 2011; Vol. 219(1):50–57 DOI: 10.1027/2151-2604/a000046
to symbolic arithmetic. Participants viewed addition and subtraction problems with Arabic digits and indicated the result by pointing to corresponding locations on a visually presented line that represented the numerical interval from 0 to 10 (see Siegler & Opfer, 2003, for a similar method). The analysis of the pointing end locations revealed that motor responses were systematically biased to the left side after subtracting and to the right side after adding. The finding of an OM effect for number processing has been interpreted as evidence that each approximate mental calculation, even when the input magnitude information is presented symbolically as Arabic numeral, relies on the same analog magnitude code as the processing of nonsymbolic numerosity information. This notion implies that mental arithmetic with multidigit numbers results in the same judgment bias as found for single-digit calculations. However, empirical evidence for an OM effect for multi-digit numbers is controversial. Knops, Viarouge, and Dehaene (2009), for instance, presented two-digit number problems and required their participants to select a dot pattern that represented the most plausible result. Although a very small difference between addition and subtraction was revealed in a first experiment, a second experiment showed that this effect vanished when controlling the average outcome of the addition and subtraction problems. Knops et al. (2009) argued that since the outcome in Experiment 1 was systematically larger for addition than for subtraction, it is possible that the subjects were spatially biased to the right by the larger numerosities (a simple variant of the spatial-numerical association of response codes; see, e.g., Hubbard, Piazza, Pinel, & Dehaene, 2005). It therefore seems likely that the observed differences 2011 Hogrefe Publishing
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in the judgments after multi-digit arithmetic reflected merely an effect of the different problem sizes rather than an impact of the mental operation. Multi-digit number calculations, in contrast to processing of single digits, require knowledge about elementary mathematical facts such as the decimal place-value system and involve learned computation strategies such as the separate combination of units and tens (see Nuerk, Moeller, Klein, Willmes, and Fischer, 2011 (this issue)). It has therefore been argued that multi-digit number arithmetic is based on verbally decomposed representations of symbolic numbers and does thus not depend on analog codes (Kalaman & Lefevre, 2007; see also Nuerk & Willmes, 2005). Taking into account that the putative origin of the OM effect is in the approximate number system (McCrink et al., 2007; Pinhas & Fischer, 2008), it can be argued that the verbal and less analog nature of mental operations with multi-digit numbers is the reason for the lack of response biases for additions and subtractions in the study of Knops and colleagues (2009). An alternative explanation might be that their force-choice method, which yields merely ordinal data, does not have the power to detect cognitive effects such as the contentious measure of approximate number representations used by Pinhas and Fischer (2008). It can consequently not be excluded that the dissociation of the OM effect for single- and multi-digit number processing might be the result of this methodological difference. The aim of the present study was to investigate the OM effect in multi-digit arithmetic. Since it is still unclear whether the judgment biases after processing of nonsymbolic and symbolic quantities are indeed driven by the same cognitive effect, we aimed to employ a method that combines both types of numerosity information and that provides us with a continuous nonspatial measure of approximate number representations. We therefore developed a new magnitude production method, in which participants are required to match the numerosity of a random dot pattern with the size of a target numeral. Before we used the numerosity production task to study judgment biases after mental arithmetic (Experiment 2), we first tested the appropriateness and validity of our approach (Experiment 1).
the numerosity production task indeed on an approximate estimation of the amount of dots concurrently displayed on the screen and not on motor feedback reflecting, for instance, the angle of the performed knob rotation, we would expect to find that the response functions become steeper for dot arrays with lower luminance.
Experiment 1
Materials
The aim of Experiment 1 was to establish a new method of numerosity production. We asked participants to indicate the numbers 10–99 by producing random dot patterns. Participants controlled the number of presented dots by turning a knob in a clock- or counterclockwise direction. To test the appropriateness of our method to investigate psychological effects in number cognition, we varied the luminance of presented dots systematically. It has been argued by several authors that low-level perceptual features such as texture density and luminance have an impact on perceived numerosity (e.g., Allik & Tuulmets, 1991; Durgin, 1995). It could be therefore assumed that an array of dark objects is perceived as less numerous than an array of bright objects of the same numerosity. If participants base their responses in
The random dot patterns were presented in three different gray colors on a black background. The colors had a photometric luminance of either 5.48 lux (high luminance) 1.59 lux (middle luminance) or 0.03 lux (low luminance). All Arabic two-digit numbers served as targets for the numerosity production task. Numerals were presented in white sans-serif font (height: 4 mm, width: 3–7 mm).
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Method Participants Twenty-eight undergraduate students from the Radboud University Nijmegen (nine males, mean age of 24.2 years) participated in the experiment in return for 5 Euros or course credits. Apparatus A keyboard, a computer screen, and a custom-made knob of 5.20 cm in diameter and 2.80 cm in height mounted on a small box (9 cm · 15 cm · 6.50 cm) was placed on a table. The knob was connected to a computer via a USB interface and could be rotated 180 clockwise. Its rotation axis was parallel to the Cartesian z-axis. A spring inside the device ensured that the knob switched back to its initial position when the participant let go of the knob. We developed a Delphi program that processed the knob input and controlled the stimulus presentation. The further the knob was rotated in a clockwise direction, the more dots were presented on the screen. Importantly, the spatial pattern of the appearance or disappearance of the dots was unpredictable. Precisely, with every 0.9 of clockwise rotation one additional dot (2 mm in diameter) was presented at a randomly chosen free position within an unmarked circular target area of 70 mm in diameter centered on the screen center. Counterclockwise rotation deleted randomly selected dots from the display. The minimum distance between two dots was 0.25 mm. The maximum number of dots was limited to 200. Each production phase started with an empty screen (i.e., zero dots).
Procedure and Design Numerosity Production Task. Participants were required to produce a random dot pattern that corresponded to a previously presented target number by rotating the knob with Zeitschrift fu¨r Psychologie / Journal of Psychology 2011; Vol. 219(1):50–57
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O. Lindemann & M. D. Tira: Operational Momentum in Multi-Digit Numbers
Figure 1. Results of the logarithmic regression analyses of the numerosity production judgments in Experiment 1: (a) The resulting individual nonlinear regressions. (b) and (c) depict the average produced quantities together with the within- or between-subjects variance. The x-axis represents the asked quantity. The dotted lines indicate a correct match between numbers and estimations. their right hand. If participants had the feeling that the numerosity of the dots was equivalent to the requested number, they finished their estimation with a keypress with their other hand. Participants were instructed to respond as quickly and accurately as possible. Each trial started with a centrally presented attentional cue (‘‘#’’ symbol), which was replaced after 500 ms by a two-digit target number. Participants’ task was to indicate the number magnitude by numerosity production as described above. As soon as the knob was turned, the number disappeared and the random dot pattern started to appear. The numerosity productions always started from zero. Once participants made their judgment and let go of the knob, the next trial started after an intertrial interval of 500 ms. Before the actual experiment started, participants were familiarized with the numerosity production method in a short training session (20 randomly chosen trials of middle luminance). In contrast to the experimental session, written feedback about the amount of produced dots was provided after each trial. This training served as a calibration of numerosity judgments and minimized the variability caused by interindividual differences in the perception of nonsymbolic numerosities (see Izard & Dehaene, 2008). Each two-digit numeral (10–99) was presented once for each luminance condition (low, middle, high), resulting in a total of 270 trials. The order of trials was randomized. The experiment lasted about 30 min.
Results and Discussion Response Latencies The average latency to perform a number judgment was 2,432 ms. Importantly, the latencies were not correlated with the number size, r = .03. The quick responses and the fact 1
that the response times did not increase with number size clearly show that participants did not use a counting strategy and rather based their judgments on approximate numerosity estimations. Accuracy The shape of the response functions in magnitude estimation tasks is generally best described by a power function, R = a*nb, with an exponent b < 1 (Izard & Dehaene, 2008; see also Siegler & Opfer, 2003).1 In order to test the shape of the response function of the numerosity production task, we performed a logarithmic regression analysis (see, for instance, Seber & Wild, 2003) on the estimations of each number averaged across subjects and luminance conditions. Remarkably, the regression of the averaged data fitted almost perfectly, r2 = .996 (see Figure 1b). As the resulting response function, y = 1.12* NUMEROSITY1.08, indicates, the relation between number and estimation was nearly linear. Participants produced on average too many dots. This numerosity overproduction, which increases with number size, is in line with the empirically well-established tendency of participants to underestimate perceived nonsymbolic numerosities (e.g., Izard & Dehaene, 2008). We explored the precision of numerosity production judgments in greater detail by performing individual logarithmic regression analyses. The regressions fitted the estimations very well for almost all participants, median r2 = .91 (ranging between r2 = .78 and .99). However, as illustrated in Figure 1a, the participants varied in their response functions with exponents ranging between 0.82 and 1.40. That is, some participants showed a logarithmically compressed relationship between asked and produced quantity, while others showed an expanded relationship. To analyze the variability of the estimations, we calculated
An exponent smaller or larger than 1 indicates that the response function is logarithmically compressed or expanded, respectively. An exponent close to 1 suggests that the relation between numbers and responses is linear.
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O. Lindemann & M. D. Tira: Operational Momentum in Multi-Digit Numbers
for each number the between-subjects deviation, defined as the standard deviation of the individual mean judgments, and correlated the parameter with the number magnitude. Another source of variance is the error variability of the estimations of one individual subject. To investigate this, we calculated for each number the within-subject deviation, defined as the average of the individual standard deviations from the correct response. The variability between subjects was highly correlated with number magnitude, r2 = .93 (see also Figure 1a). This was also true for the estimation variability within subjects, r2 = .89 (see Figure 1b), reflecting that the precision of judgments decreases with number size. Other studies on nonverbal approximate number processing have previously reported that variability estimates increase proportionally to number magnitude, a property called ‘‘scalar variability’’ (Cordes, Gelman, Gallistel, & Whalen, 2001; Izard & Dehaene, 2008; Whalen, Gallistel, & Gelman, 1999). The numerosity production judgments in the present study confirm the notion that accuracy of number representations decreases with number size and provide thus new support for the scalar variability of internal number representations. The regression analyses above show that the average response function of numerosity production judgments is linear. The effects of the different dot luminance conditions can be consequently tested by linear regression analyses of repeated-measures data as suggested by Lorch and Myers (1990). That is, linear regressions were performed for each participant and each luminance level and compared statistically. The one-way repeated-measures analysis of variance (ANOVA) on the regression coefficients revealed an effect of luminance level, F(2, 54) = 5.85, p < .001, gp2 = .18. Post hoc t tests indicated that regression slopes were less steep in the high luminance condition (y = 1.55* NUMEROSITY-2.27) as compared to the middle luminance (y = 1.61*NUMEROSITY-2.73), t(27) = 2.11, p < .05, Cohen’s dz = .40,2 and low luminance condition (y = 1.65* NUMEROSITY-6.76), t(27) = 3.33, p < .01, dz = .63. Thus, the overproduction of dots was less pronounced when the random dot pattern was presented in a bright color, suggesting that stimulus luminance is a property that affects numerosity production judgments. This finding is thus in line with psychophysical models of numerosity perception assuming that the processing of nonsymbolic numerosity information makes use of several low-level perceptual features like texture density, occupied area, and averaged luminance (Allik & Tuulmets, 1991; Durgin, 1995). Taken together, Experiment 1 establishes a new numerosity production task. The good fits of the estimations with the correct numbers suggest that participants had no difficulties with our methods. Importantly, the luminance effect demonstrates the sufficient sensitivity of numerosity productions for the detection of cognitive effects and corroborates that participants used visual feedback and the perceived number of dots to give their responses. We consequently conclude that the method of numerosity productions is 2
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suitable to study analog representations of numbers. Moreover, the short reaction times and the effect of stimulus luminance confirm that participants did not use a counting or motor strategy to produce the judgments and rather based their responses as instructed on an approximate estimation of the number of produced dots.
Experiment 2 Experiment 2 used the method described above to investigate the presence of the OM effect when participants are required to indicate the sum or difference between two two-digit numbers. It has been shown that mental calculations with carry-overs over the decade break are performed slower, are more error prone (Deschuyteneer, De Rammelaere, & Fias, 2005), and result in more working memory load as compared to multi-digit calculations without carry-overs (DeStefano & LeFevre, 2004; Imbo, Vandierendonck, & De Rammelaere, 2007). Since additions and subtractions with carry or borrowing involve apparently different cognitive processes than calculations without carry, we varied the requirement to perform carry operations systematically. We additionally included problems with the zero as second operand. These zero problems have been suggested to provide a measurement of the ‘‘pure’’ operational momentum effect (Pinhas & Fischer, 2008) without activation of a second magnitude, and also address the methodological dilemma that operands of addition and subtraction problems always differ when problem size is kept constant. Experiment 1 revealed large interindividual differences in the shapes of the nonlinear response functions (Figure 1a). To compensate for this between-subject variance and for the different nonlinear relations between asked and produced quantity, we estimated the individual logarithmic response functions from data collected in an initial number estimations session and used them to adjust the judgments in the subsequent calculation task.
Method Participants A total of 26 students of the Radboud University Nijmegen (eight males, mean age 22.3 years) participated in the experiment in return for 7.50 Euros or course credits. Apparatus and Materials The same apparatus and method as introduced in Experiment 1 were used for the numerosity production judgments. Dots were always presented in the middle gray color. The number task (first block) comprised the numbers 10–99. For the calculation task (second block), we compiled a list of 24 addition and 24 subtraction problems (see Table 1).
Effect sizes were calculated using the G*Power 3 program (Faul, Erdfelder, Lang, & Buchner, 2007).
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O. Lindemann & M. D. Tira: Operational Momentum in Multi-Digit Numbers
Table 1. Addition and subtraction problems used in Experiment 2. The results were not presented during the experiment Addition problem* 13 21 12 14 13 29 18 12 18 24 19 41 19 14 16 53 16 28 47 29 51 63 12 52
+ + + + + + + + + + + + + + + + + + + + + + + +
(N)
21 14(N) 24(N) 23(N) 29(C) 14(C) 29(C) 41(N) 36(C) 32(N) 38(C) 17(N) 43(C) 49(C) 48(C) 14(N) 57(C) 46(C) 28(C) 47(C) 32(N) 21(N) 73(N) 34(N)
Subtraction problem* 57 69 98 58 91 61 74 74 72 68 71 89 81 82 82 79 92 92 91 93 97 97 98 98
Addition zero problem
(N)
23 34(N) 62(N) 21(N) 49(C) 18(C) 27(C) 21(N) 18(C) 12(N) 14(C) 31(N) 19(C) 19(C) 18(C) 12(N) 19(C) 18(C) 16(C) 17(C) 14(N) 13(N) 13(N) 12(N)
34 35 36 37 42 43 47 53 54 56 57 58 62 63 64 67 73 74 75 76 83 84 85 86
+ + + + + + + + + + + + + + + + + + + + + + + +
Subtraction zero problem
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
34 35 36 37 42 43 47 53 54 56 57 58 62 63 64 67 73 74 75 76 83 84 85 86
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
Result 34 35 36 37 42 43 47 53 54 56 57 58 62 63 64 67 73 74 75 76 83 84 85 86
Note. *Letters in parentheses indicate the problem type. C = carry problem, N = no-carry problem.
Each problem comprised two operands. No decade numbers or tie numbers (e.g., 22, 33) occurred as operand or result. Operands and result of one problem never had the identical decade or unit digits. For additions and subtractions, half of the first and half of the second operands were odd numbers. The outcomes of addition and subtraction problems were matched. Half of the problems required a carry operation; the other half were no-carry problems. The mean result of the carry problems was 59.5 (STD = 16.6) and 60.0 (STD = 11.8) for the no-carry problems (see Table 1). We added furthermore 48 zero problems by generating for each addition and subtraction problem a corresponding problem with the same result but with zero as second operand. Procedure and Design Every participant performed first a number task in which they indicated the approximate magnitude of two-digit numbers. Each two-digit number was presented once. The calculation task comprised 144 trials in total. Instead of a single number, we presented an addition or subtraction problem and instructed the participant to indicate the result of the presented problem as quickly and accurately as possible. To balance the amount of trials in the three problem type conditions (carry, no-carry, zero), we presented the 48 carry and no-carry problems twice. The trial order was randomized in both blocks. The experiment lasted about 45 min. Zeitschrift fu¨r Psychologie / Journal of Psychology 2011; Vol. 219(1):50–57
Results and Discussion Response Latencies Responses in the calculation task were given relatively quickly (M = 3,174 ms across all responses). Latencies correlated only very weakly with problem size, r = .28, suggesting that participants used approximate estimation to indicate the calculated outcome of the problems. A repeated-measures ANOVA with the factors Problem Type (no-carry problem, carry problem, zero problem) and Operation (Plus, Minus) revealed slower responses for subtraction problems (3,281 ms) than for addition problems (3,06 ms), F(2, 50) = 10.94, p < .01, gp2 = .30. The main effect of the factor Problem Type, F(2, 50) = 15.99, p < .001, gp2 = .39, indicated that it took longer to solve a carry problem (3,676 ms) than a no-carry problem (3,305 ms), t(25) = 3.21, p < .01, dz = .32, suggesting that carry problems are more difficult to solve than no-carry problems. No-carry problems were solved faster than zero problems (2,537 ms), t(25) = 4.18, p < .001, dz = .82. Also the interaction between the two factors reached significance, F(2, 50) = 6.79, p < .01, gp2 = .21. Post hoc comparisons revealed a significant reaction time difference between minus and plus operations for carry problems (3,758 ms vs. 3,086 ms), t(25) = 4.57, p < .001, dz = .86, but no effect of the operation for no-carry problems (2,563 ms vs. 2,511 ms), t(25) < S1, and zero problems (3,758 ms vs. 3,594 ms), t(25) = 1.59, p > .10. 2011 Hogrefe Publishing
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Individual logarithmic regressions were computed for the judgments in the number task. Again, the regressions fitted the judgments of all participants nicely, median r2 = .95 (ranging between r2 = .87 and r2 = .98). Again, the fits of the average estimations revealed a nearly linear response function, y = 0.91*NUMEROSITY1.09. Accuracy To investigate the effects of mental arithmetic on numerosity judgments, we first adjusted the responses in the calculation task by calculating individual logarithmic regression equations resulting from judgments in the number task. These response functions were inverted to yield the number size as a function of the response and then applied to the judgments in the calculation task. Afterwards, we calculated for each participant the constant judgment error, defined as the average error between correct outcomes and adjusted responses, and the variable error, defined as the standard deviation of the estimations from the subject’s mean response (cf. Schutz & Roy, 1973). Both variables were submitted to separate two-way repeated-measures ANOVAs with the factors Problem Type (no-carry problem, carry problem, zero problem) and Operation (Plus, Minus). The analysis of variable errors revealed only an effect of the factor Problem Type, F(2, 50) = 3.20, p = .049, gp2 = .05, reflecting that responses to zero problems (10.49 dots) were less variable than to no-carry problems (11.67 dots), t(25) = 2.38, p < .05, dz = .46. Importantly, however, variable errors for carry (11.21 dots) and no-carry problems did not differ, |t| < 1, showing that both problem types were solved with the same accuracy. The constant errors, which reflect systematic judgment biases, were also affected by the factor Problem Type, F(2, 50) = 7.47, p < .001, gp2 = .23, showing that the tendency to produce too many dots was smaller when participants indicated the results of no-carry problems (8.52 dots) as compared to carry problems (10.24 dots), t(25) = 3.38, p < .05, dz = .66, or zero problems (10.81 dots), t(25) = 3.74, p < .05, dz = .92. An explanation for this stronger overproduction tendency for carry problems is speculative at this point, but one might argue that it reflects an effect of greater task demands and the use of a cognitive heuristic that task complexity and problem size are associated. Constant errors in the carry and zero problem condition did not differ, t(25) < 1. Most importantly, there was a significant interaction between the factors Problem Type and Operation, F(2, 50) = 4.28, p < .05, gp2 = .15. Post hoc t tests indicated that the overall overproduction tendency was significantly smaller after subtraction than after addition if the arithmetic operation did not require a carry operation (7.57 vs. 9.45 dots), t(25) = 2.38, p < .05, dz = .47 (see Figure 2). These differences reflect the presence of an OM effect in multi-digit number processing, because overall judgments were smaller after solving a subtraction problem than after solving an addition problem. Furthermore, a similar difference between addition and subtraction was observed for the zero problems. That is, estimations were smaller after subtracting a zero (10.20 dots) than after adding a zero 2011 Hogrefe Publishing
Figure 2. Mean number judgment biases in the arithmetic task of Experiment 2 as a function of the factors Problem Type and Operation. Asterisks represent a significance level of p < .05. (11.41 dots), t(25) = 2.61, p < .05, dz = .51. Our data thus also support a ‘‘pure’’ operational momentum effect. Interestingly, however, when the arithmetic problem required a carry operation we found no evidence for OM effects, t(25) = 1.24, p = .22, and differences in the estimations between subtraction and addition problems with carrying even reversed descriptively (10.94 vs. 9.54 dots). Since the statistical effects size parameters in the nocarry and zero problem condition were similar in size (dz = .47 and dz = .51), it can be assumed that the OM effects did not differ between the two conditions. This interpretation was confirmed by an additional t test that compared the individual OM effects (i.e., the difference between mean estimations of additions and subtractions for each individual) for carry and zero problems, t(25) < 1.
General Discussion We have introduced a numerosity production task which required matching of the numerosity of random dot pattern to symbolic numbers, in order to investigate approximate mental calculations with two-digit numbers. Our data show that participants produced systematically more dots when indicating the outcome of an addition problem than when indicating the result of a subtraction. Since the same judgment bias could be observed for zero problems, we interpret our findings as empirical support for the notion of an OM effect in multidigit number arithmetic. Interestingly, our data suggest that the judgment bias after mental arithmetic depends on the presence or absence of a carry or borrow operation. The present study replicates not only the finding of a response bias after symbolic arithmetic as demonstrated by Pinhas and Fischer (2008) but also substantially extends the available research on mental calculation. First, numerosity production judgments in Experiment 2 clearly show that the OM effect is not restricted to the processing of singledigit numerals. In general, the OM effect has been interpreted as support for an analog representation of symbolic and nonsymbolic numerical magnitude information Zeitschrift fu¨r Psychologie / Journal of Psychology 2011; Vol. 219(1):50–57
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(McCrink et al., 2007; Pinhas & Fischer, 2008). Our observation documents an involvement of analog representations in processing of multi-digit numbers. A previous study by Knops et al. (2009) used a force-choice paradigm to investigate approximate calculation with multi-digit number and reported no differences in the estimations of the result of additions and subtractions when controlling the problem size. However, the present study shows that when using a continuous parametric measure of approximate numerosity representation, participants’ judgments revealed the presence of an OM in multi-digit number processing. Considering the research of Pinhas and Fischer (2008), who also employed a continuous measure with their number-location task, we argue that the existing discrepancy in the empirical research on the symbolic OM effect is basically driven by methodological differences and variations in the data quality and does not reflect a difference in the cognitive processing or representation of single- and multi-digit numbers. Second, an OM effect was not present if the problems required a carry or borrow operation, showing that the judgment bias after multi-digit arithmetic is mediated by problem complexity. Importantly, carry operations affected the systematic judgment bias, while the variability of the responses and thus the general number acuity remained unaffected. It is therefore unlikely that the disappearance of OM for carry problems is the result of general task accuracy and might instead be driven by some specific processing requirements. The difference between addition and subtraction problems was furthermore not only diminished for carry problems but even reversed descriptively, which also suggests that qualitative differences in the processing of problems with and without carry operation account for the dissociation between the OM effects in the two conditions. As shown by previous research, each carry operation involves a decomposition of the place-value system and results in an increased load of phonological working memory resources (Deschuyteneer et al., 2005; DeStefano & LeFevre, 2004; Imbo et al., 2007; Kalaman & Lefevre, 2007). In order to explain the absence of the OM effect for carry problems, one might therefore assume that the processing of carry problems, which is strongly based on verbal processing strategies, engages fewer nonverbal analog representations of numerical magnitude information. An alternative explanation for the absence of the OM effect for carry problems could be that participants approximated the results of the carry problems and did not perform the required carry operations. However, in case of an addition, this heuristic would result in an underestimation of the outcome and in case of a subtraction in an overestimation. This type of error would abolish the OM effect, so that a response bias after mental arithmetic as found for no-carry problems could not be observed in the carry problem condition. While it is virtually undisputed that analog magnitude codes are activated automatically when processing single digits, the nature of semantic magnitude representations of multi-digit numbers is still debated. Nuerk and Willmes (2005) reviewed different models of symbolic number processing and distinguished between the holistic and decomposition hypothesis of multi-digit number representations. The notion of holistic magnitude coding assumes that a Zeitschrift fu¨r Psychologie / Journal of Psychology 2011; Vol. 219(1):50–57
multi-digit numeral is mapped onto one single magnitude representation (Zhang & Norman, 1995). On the other hand, the decomposition model holds that the magnitudes of all digits of a numeral are represented separately. Two-digit numbers are therefore assumed to be represented by two analog magnitude codes that are labeled with regard to their place value. Although the OM effect does not speak directly to the question of whether the base-10 structure is retained in analog number representations or not, our observation that the judgment biases differ for calculations with and without carry operation seems to suggest that the decomposition of the number and a separate magnitude processing of decades and units is not an obligatory process and rather depends on the concurrent task demands. The third advance made by the present study is the demonstration of a new method to investigate approximate magnitude representations. The response time analyses exclude the involvement of counting strategies in the present task. It is therefore plausible to assume that participants used their approximate estimation of the amount of dots displayed on the screen to control their responses in the numerosity production task. The interpretation that judgments were based on visual feedback received additional empirical support from the finding that the estimations in number estimation task of Experiment 1 were modulated by the luminance of the dot patterns. Both experiments confirm therefore our notion that the numerosity production task provides a valid measurement of approximate magnitudes that sheds new light on the cognitive processing of multi-digit numbers. However, despite the evidence in favor of a matching of perceived visual numerosities, we cannot exclude at this point that participants used additional nonvisual information, like, for instance, proprioceptive feedback about the performed rotation of the knob, to indicate the number magnitude. Such a possible involvement of motor magnitudes or posture-based response strategies does not affect the interpretation of the presence of an OM effect for multi-digit numbers, but suggests the possibility that the OM effect occurred at the level of motor programming instead at the level of analog numerosity representation. In other words, it might be possible that the overproduction of dots for additions compared to subtractions is due to the fact that the knob was turned ‘‘too far’’ in a clockwise direction even though the numerical computation itself was unbiased. We think that further research is required that aims to disentangle the impact of the visual and motor feedback in numerosity productions tasks. These types of studies might also help us to better understand the functional coupling between number representations and sensory-motor processes as observed in other tasks (e.g., Lindemann, Abolafia, Girardi, & Bekkering, 2007). The excellent fits between numbers and estimations in Experiment 1 demonstrate the validity of the numerosity production estimates. Importantly, our method allows a direct continuous measurement of numerosity without cognitive transformations to spatial codes. Although we observed interindividual difference in the nonlinear relations between asked and produced quantities, we found that the average produced quantities show an astonishing linear relationship with the asked number sizes. We observed additionally that the variability of the numerosity productions increases proportionally 2011 Hogrefe Publishing
O. Lindemann & M. D. Tira: Operational Momentum in Multi-Digit Numbers
to the number magnitude. Consequently, our findings provide support for the view that numbers are represented in terms of a linear magnitude with scalar variability (Whalen et al., 1999). This outcome is in line with other numerosity production tasks reported in the literature. For example, in the so-called nonverbal counting task, participants are required to produce an estimated number of key presses at very high rates so that any counting is impossible (Whalen et al., 1999). Compared to this technique, the numerosity productions can account for a much larger range of numbers. Another advantage of the present method is that magnitude estimations are not confounded with response duration. It can be argued that asking participants to point at number locations also represents a sort of magnitude production task (Pinhas & Fischer, 2008; Siegler & Opfer, 2003). Although we know that spatial-numerical associations are highly overlearned cognitive strategies (Fischer, 2006; Hubbard et al., 2005), the way that numbers are mapped onto space is rather flexible and dependent on the task context and the cultural background (Ba¨chtold, Baumu¨ller, & Brugger, 1998; Lindemann, Abolafia, Pratt, & Bekkering, 2008; Zebian, 2005). Note, the presented measurement of approximate number representations involves a matching of symbolic numbers and perceived visual numerosities and does not require a mapping of numbers onto locations in space. The numerosity production task is therefore independent from culturally acquired spatial-numerical associations and represents thus a promising tool for research on cognitive representations of multi-digit numbers and approximate mental arithmetic.
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[email protected]
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