In four experiments, we explore the role that verbal WM plays in numerical comparison. Experiment 1 demonstrates that verbal WM load differentially impacts the two most common variants of numerical comparison tasks, evidenced by distinct modulation of the size of the numerical distance effect (NDE). Specifically, when comparing one Arabic digit to a standard, the size of the NDE increases as a function of increased verbal WM load; however, when comparing two simultaneously presented Arabic digits, the size of the NDE decreases (and here is eliminated) as a function of an increased verbal WM load. Experiment 2, using the same task structure but different stimuli (physical size judgments), provides support for the notion that this pattern of results is unique to tasks employing numerical stimuli. Experiment 3 demonstrates that the patterns observed in Experiment 1 are not an artifact of the stimulus pairs used. Experiment 4 provides evidence that the differing pattern of results observed between Experiment 1 and Experiment 2 are due to differences in stimuli (numerical vs. non-numerical) rather than to other differences between tasks. These results are discussed in terms of current theories of numerical comparison.
Working memory (WM) has been demonstrated to play a central role in various types of mathematical processing (
The ability to compare two numbers is a fundamental numerical ability that has been the subject of much empirical research (
The most often studied phenomenon in numerical comparison is the numerical distance effect (NDE). When comparing two numerical quantities, participants are faster and more accurate at indicating which of two numbers is larger when the numerical distance separating the two numbers is relatively large (e.g., 2 vs. 9), compared to when it is small (e.g., 8 vs. 6;
One of the most prominent accounts of the NDE in numerical comparison posits that the NDE is due to the placement of numbers along an internal mental number line (
In contrast to the
More recently, a third account of the NDE has been put forth.
Neither the representational overlap account nor the response competition account make explicit predictions about the potential contribution of verbal WM to the NDE. This is likely due in large part to the aforementioned preponderance of studies examining the role of WM in higher level mathematical processing rather than in lower level numerical processing. That said, evidence that WM is involved in other low-level numerical processes suggests that there is good reason to expect that WM may play an important role in numerical comparison. For example,
The aim of Experiment 1 was to examine the relation between WM and the NDE in two of the most popular tasks used to measure the NDE (i.e., simultaneous comparison and comparison to a standard). The motivation to use both simultaneous comparison and comparison to a standard, rather than the more typical approach of using only one, is based on research demonstrating that the NDEs in these tasks may not be measuring the same underlying process. Specifically,
Thus, in Experiment 1, participants completed both the comparison to a standard task and the simultaneous presentation task with Arabic digit stimuli under two blocked WM load conditions (zero load and high load). The WM manipulation was verbal in nature and consisted of either 6 filler stimuli (in the zero load condition) or 6 letters (in the high load condition). We elected to use a verbal WM load as it is thought to be an integral component in at least one other basic numerical process (counting;
Fifty-three undergraduate students from a Canadian university participated and were granted experimental credit towards a course. Four participants failed to complete both tasks leaving forty-nine participants.
The data were collected on a Pentium 4 PC computer running E-Prime 1.1 (
In the simultaneous comparison task, each trial began with a fixation point that remained on the screen for 500 ms. In the “zero” WM load condition, a display containing 6 hash marks (######) was presented for 3 seconds. In the “high” WM load condition a display containing 6 letters was presented for 3 seconds. Letters were chosen from the set of F, H, J, K, L, N, P, Q, R, S, T, Y as used in
In the comparison to a standard task each trial began with a fixation point that remained on the screen for 500 ms. Participants were given the same working memory loads as in the simultaneous comparison task. After a 3-second display of the to-be-remembered letters, a display containing one Arabic digit was presented. Numbers were 1, 4, 6, or 9. Thus, the numerical distance between the stimuli and 5 was 1 or 4. Participants were told to identify whether the number was lower than 5 or higher than 5 by pressing the “A” key if the number was lower and the “L” key if the number was higher. Following their judgment, participants were prompted to type in the letters presented at the beginning of the trial. There were two blocks each with 160 trials. Stimuli composition was identical for both blocks (but appeared in a randomized order).
Two participants were removed because their accuracy was below 20% on the WM task, meaning that they did not have enough trials to calculate a reliable score in those cells. This left us with 47 participants. RTs and errors were analyzed across participants with Task, Numerical Distance, and WM load as within-subject factors. Trials on which participants responded incorrectly to the WM load (12.0%) and trials on which there was an incorrect response in the number comparison task (3.5%) were removed prior to RT analysis. Remaining RTs were submitted to an outlier removal procedure using a 2.5 standard deviation cutoff in each cell resulting in the removal of an additional 3.8% of the data.
WM Load | Comparison to a Standard |
Simultaneous Comparison |
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Distance 1 | Distance 4 | Distance 1 | Distance 4 | |
Low Load | 631 (4.1) | 572 (2.9) | 617 (2.1) | 567 (3.6) |
High Load | 899 (4.0) | 819 (2.1) | 790 (3.5) | 775 (5.0) |
The numerical distance effect under zero and high working memory loads in Experiment 1 calculated as RTs at distance 1 minus RTs at distance 4. Error bars represent +/- 1 standard error of the mean.
A 2 (Task: comparison to a standard vs. simultaneous comparison) x 2 (WM load: zero vs. high) x 2 (Numerical Distance: 1 vs. 4) ANOVA conducted on the RT data yielded no main effect of Task,
A parallel analysis on the error data yielded a main effect of Numerical Distance,
In Experiment 1, we assessed the role of verbal WM in numerical comparison by examining the conjoint effects of verbal WM load and numerical distance in the context of two classic numerical comparison tasks – the comparison to a standard and simultaneous numerical comparison tasks. Critically, the influence of verbal WM load was qualitatively different across these two variants. Specifically, in the simultaneous comparison task we found an interaction between verbal WM load and the NDE whereby the size of the NDE was
The increase in the NDE as a function of increased verbal WM load in the comparison to a standard task is inconsistent with the findings of
Experiment 1 has yielded a complex pattern of results relating WM load to the NDE in two different numerical comparison tasks. Specifically, increasing verbal WM load decreases the size of the NDE in the simultaneous presentation task, but increases the size of the NDE in the comparison to a standard task. The entire pattern of results is not predicted by any existing account of numerical comparison and demonstrates that the contribution of WM to numerical comparison is not in any way straightforward (at least provided current assumptions in the numerical cognition literature). This complex pattern thus provides a unique opportunity for novel insights into numerical comparison in general and the NDE in particular. In Experiment 2 we begin the attempt to decompose this complex pattern of results.
Numerical comparison tasks consist of two independent components: the “numerical” aspect (i.e., the stimuli) and the “comparison” aspect. In Experiment 2 we set out to test the hypothesis that the dissociation between simultaneous presentation and comparison to a standard arises because of differences in the structure of the two comparison tasks, rather than to the numerical aspects of the task per se. Specifically, we wanted to know if the difference arises because of a differential contribution of verbal WM to comparing a stimulus to a standard and comparing two simultaneously presented stimuli. Thus we kept the task instructions the same but changed the to-be-compared stimuli. Specifically, rather than compare Arabic digits, participants compared squares which varied in physical size. Comparing the size of shapes has been shown to elicit a distance effect (
Thus, in Experiment 2, participants performed both comparison tasks under zero and high verbal WM loads using single squares that varied in physical size. In the comparison to a standard task, participants were first shown a “standard” square which was smaller than half of the stimuli and larger than half of the stimuli. Participants identified whether each subsequently presented square was smaller than or larger than the standard. In the simultaneous square comparison task, participants saw two simultaneously presented squares and identified the larger of the two squares. The same verbal WM load conditions that were used in Experiment 1 were used here. If the dissociation observed in Experiment 1 arises because of the nature of the comparison task, then we should observe a similar dissociation between the two comparison tasks as a function of verbal WM load.
Sixty undergraduate students from a Canadian university participated and were granted experimental credit towards a course. Seven participants were excluded because they failed to complete both tasks; 6 participants were removed because their accuracy was below 20% on the WM task. Finally, 4 participants were removed for having too low accuracy on the comparison tasks. These participants each had accuracies below chance in one cell of either comparison task. This left us with 43 participants. Excluding these participants did not change the overall pattern of the results.
The apparatus and program used were identical to that of Experiment 1 with minor exceptions. Rather than comparing Arabic digits, participants compared the physical size of squares. In one task participants judged whether a presented square was larger than a pre-determined standard, and in the other task judged which of two simultaneously presented squares were larger. In the comparison to a standard task the standard square was presented for 1000 ms before each WM load display to remind participants of its size. Participants completed both tasks and tasks were presented in a counterbalanced order.
The size of the squares used here was parametrically varied so as to create differences that paralleled the differences in the numeric tasks. There were 9 squares in total, the area of each ranged from 1 cm2 to 9 cm2 with a square mapping onto each of the intervening integers within that range (2 cm2 to 8 cm2). This set of squares was then used to create stimulus sets with physical differences in area of 1 (e.g., 3 cm2 to 4 cm2) and 4 (e.g., 5 cm2 to 9 cm2). Within the comparison to a standard task, a 5 cm2 square was used as the standard, and the presented stimuli were either 1 cm2, 4 cm2, 6 cm2, or 9 cm2 and in the 2-item comparison task the full set was used. In both tasks there were two blocks each with 160 trials. Trial composition was identical for both blocks (but was presented in a randomized order).
RTs and errors were analyzed across participants, with Physical Distance, WM load, and Task variant as within-subject factors. Trials on which participants responded incorrectly to the WM load (10.6% in the comparison to a standard task and 9.7% in the simultaneous presentation) or made an incorrect response (9.4% and 1.8%, respectively) were removed prior to RT data analysis. Remaining RTs were submitted to a 2.5 standard deviation outlier removal procedure resulting in the removal of an additional 3.0% and 2.9% of trials, respectively.
WM Load | Comparison to a Standard |
Simultaneous Comparison |
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---|---|---|---|---|
Distance 1 | Distance 4 | Distance 1 | Distance 4 | |
Low Load | 1227 (13.5) | 784 (4.5) | 1024 (1.4) | 739 (0.6) |
High Load | 1363 (16.7) | 1044 (4.2) | 1145 (2.5) | 906 (2.8) |
The distance effects under zero and high working memory loads for each task in Experiment 2. Error bars represent +/- 1 standard error of the mean.
A 2 (Task: comparison to a standard vs. 2-item comparison) x 2 (WM load: zero vs. high) x 2 (Physical Distance: 1 vs. 4) ANOVA conducted on the RT data yielded a main effect of Task,
A parallel analysis on the error data yielded a main effect of Task,
In Experiment 2 we assessed whether the dissociation that was observed in Experiment 1 was driven by an inherent difference between the two comparison tasks by removing the numerical components of the tasks. The results of Experiment 2 are inconsistent with this idea. When squares were used as stimuli, the size of the distance effect decreased as a function of increased WM capacity in both of versions of the comparison tasks. If the act of comparing one item to a standard or comparing two simultaneously presented items made differential demands on WM, then we should have observed a similar pattern as that observed in Experiment 1. Specifically, we should have observed an increase in the distance effect in the comparison to a standard task and a decrease in the distance effect in the 2-item comparison task as a function of increased verbal WM load. This was not the case. Thus, it appears that the dissociation observed in Experiment 1 derives from an interaction between the particular comparison task and the to-be-compared stimuli (i.e., Arabic digits).
While Experiment 2 indicated that the structure of the two tasks cannot account for the differential effect of verbal WM load, Experiment 3 served to test another hypothesis. Specifically, Experiment 3 served to test whether differences in the to-be-compared stimuli could account for the interaction that was seen in Experiment 1. In the comparison to a standard task that was used in Experiment 1, the digits 1, 4, 6, and 9 were compared to 5 (as a standard held in memory). Thus, pairs with numerical distance 1 and 4 were used. In the simultaneous comparison task from Experiment 1, the number pairs were generated from the numbers 1-4 and 6-9, and only those among them and with distances 1 and 4 were included in the stimulus set. It follows that the specific pairs used were different in the two tasks. In Experiment 3 we tested the hypothesis that the differential effect of verbal WM load on the distance effects that arise in the comparison to a standard task and the simultaneous presentation task arises due to the difference in stimuli between the two tasks. To test this hypothesis, we conducted a version of the simultaneous comparison task in which we paired the numbers 1, 4, 6, and 9 with the number 5. That is, the number 5 was present on every trial, making the number pairings identical to those used in the comparison to a standard task. Importantly, if the specific stimulus set that was used in the comparison to a standard task from Experiment 1 was the reason for the overadditive interaction with verbal WM load, then we should also expect to see an increased NDE in the simultaneous comparison task with these stimuli. However, if it was not the specific number pairings but rather something about the nature of comparing two simultaneously presented Arabic digits, then we should expect a decrease in the NDE as a function of increased verbal WM load despite these specific number pairings.
Forty-five undergraduate students from a Canadian university participated and were granted experimental credit towards a course. One participant was removed for failing to complete the task.
The apparatus and program used were identical to the simultaneous comparison task that was used in Experiment 1. However, unlike in Experiment 1, here the number pairs were composed of the numbers 1, 4, 6, and 9 paired with the number 5. That is, the number 5 was present on every trial. On half of the trials the number 5 appeared to the left of the fixation and on half it appeared to the right.
RTs and errors were analyzed across participants, with Numerical Distance and WM load as within-subject factors. Trials on which participants responded incorrectly to the WM load (13.4%) or made an incorrect response (3.8%) were removed prior to RT data analysis. Remaining RTs were submitted to a 2.5 standard deviation outlier removal procedure resulting in the removal of an additional 3.0% of trials. An additional 2 participants were removed for having NDEs that were greater than 2.5 standard deviations from the mean.
WM Load | Distance 1 | Distance 4 |
---|---|---|
Low Load | 683 (3.0) | 634 (2.2) |
High Load | 903 (3.6) | 891 (3.4) |
The numerical distance effect under zero and high working memory loads in Experiment 3 calculated as RTs at distance 1 minus RTs at distance 4. Error bars represent +/- 1 standard error of the mean.
A 2 (WM load: zero vs. high) x 2 (Numerical Distance: 1 vs. 4) ANOVA conducted on the RT data yielded a main effect of WM load,
Experiment 3 served to test the account that the dissociation that was observed in Experiment 1 resulted from the exact number pairings that were used. Specifically, in Experiment 1, because there were a greater number of possible number pairs in the simultaneous comparison task than in the comparison to a standard task, it is possible that the stimuli themselves could have caused the observed pattern. Thus, in Experiment 3 we ran the simultaneous comparison task using the same number pairings that were used in the comparison to a standard task in Experiment 1. If it was the specific stimuli that were used that was causing the overadditivity in the comparison to a standard task (relative to the underadditivity that is observed in the simultaneous comparison task) then we would expect to see overadditivity in Experiment 3. We did not. Rather, the pattern that was observed in the simultaneous comparison variant used in Experiment 3 replicated that observed for the same variant of the task in Experiment 1. As such, we can conclude that the differential effect of verbal WM load on these two task variants is not dependent upon the specific number pairing used (at least within the 1-9 range of numbers).
The results from Experiment 2 supported the idea that the structure of the two tasks cannot, in and of itself, account for the differential effect of verbal WM load on the NDE produced under the simultaneous comparison and the comparison to a standard tasks. However, in the comparison to a standard condition in Experiment 2, the standard square, to which all other squares were to be compared, was presented at the beginning of each trial and was thus not required to be maintained in WM. This was done because pilot testing indicated that participants were unable to hold the exact size of the square in WM. In contrast, in the parallel task in Experiment 1, participants were told at the beginning of the task that they were to compare each subsequently presented number to the number 5 and were not reminded of the standard (5) on each trial. To address this confound, in Experiment 4, participants again were told to decide whether the numbers presented were higher than or lower than the number 5. Importantly, the number 5 was presented at the beginning of every trial to exactly parallel what was done in Experiment 2. If the presentation of the standard stimuli on each trial is what caused the NDE to decrease under increased verbal WM load in Experiment 2, then in Experiment 4, we should also expect the NDE to decrease under increased verbal WM load. However, if this confound between Experiments 1 and 2 cannot explain the underadditivity in Experiment 2, then we should not expect underadditivity of the NDE here in Experiment 4.
Seventy-eight undergraduate students from a Canadian university participated and were granted experimental credit towards a course. One participant was removed as they answered fewer than 20% of the WM trials correctly.
The apparatus and program used were identical to the comparison to a standard task that was used in Experiment 1 with the exception that, unlike in Experiment 1, here the number 5 was presented at the beginning of each trial for 1000ms to parallel the design of Experiment 2.
RTs and errors were analyzed across participants, with Numerical Distance and WM load as within-subject factors. Trials on which participants responded incorrectly to the WM load (11.8%) or made an incorrect response (0.7%) were removed prior to RT data analysis. Remaining RTs were submitted to a 2.5 standard deviation outlier removal procedure resulting in the removal of an additional 2.9% of trials.
WM Load | Distance |
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1 | 2 | 3 | 4 | |
Low Load | 742 (1.4) | 724 (0.9) | 697 (0.8) | 695 (0.9) |
High Load | 935 (0.7) | 938 (0.4) | 899 (0.3) | 895 (0.3) |
The numerical distance effect under zero and high working memory loads in Experiment 4 calculated as RTs at distance 1 minus RTs at distance 4. Error bars represent +/- 1 standard error of the mean.
A 2 (WM load: zero vs. high) x 2 (Numerical Distance: 1 through 4) ANOVA conducted on the RT data yielded a main effect of WM load,
Experiment 4 served to test whether the decrease in the DE under increased WM load that was observed in Experiment 2 could be explained by the fact that, in Experiment 2, the standard was presented on each trial. Here, when the standard (5) was presented on each trial, the NDE did not decrease under increased WM load. Therefore, we can conclude that the difference in the patterns observed on the comparison to a standard task in Experiments 1 and 2 (i.e., an overadditive interaction with numerical stimuli and an underadditive interaction with squares) cannot be explained by the presentation of the standard stimulus at the beginning of each trial in Experiment 2 but not in Experiment 1. In Experiment 4, when the standard (5) was presented on each trial, the NDE neither increased nor decreased under increased verbal WM load. This pattern is further unpacked in the General Discussion.
The present investigation of the involvement of verbal WM in numerical comparison has yielded a number of important insights. The results from four experiments converge strongly on the notion that verbal WM is involved in symbolic numerical comparison, one of the most basic numerical abilities (
The interaction between the NDE and WM load provides important new challenges for theories of the distance effect, specifically, and numerical representation in general. Here we have demonstrated in two experiments that in the simultaneous presentation task, an increased WM load leads to the elimination of the NDE. There are at least two theoretically distinct ways that one could explain the elimination of the NDE under a high verbal WM load. The first is a disruption account, which explains the elimination by recourse to the idea that the mechanism that gives rise to the NDE is in some fashion disrupted under a verbal WM load and thus there is no NDE produced (for a similar argument with the SNARC effect see
The second account of the elimination of the NDE under high verbal WM load in the simultaneous comparison task is that the NDE is absorbed into slack created by the need to maintain the verbal WM load (
One strategy for distinguishing between the disruption and absorption accounts of the elimination of the NDE in the simultaneous presentation condition is to use the time-course of the NDE. According to an absorption account, the NDE should decrease in magnitude as overall RT increases (for the basis of this logic see work on the Simon effect;
The numerical distance effects under zero and high working memory loads as a function of overall RTs divided into quartiles.
Results of this analysis indicate that in the zero WM load condition there was a main effect of Distance,
One interesting issue that arises when considering the idea that a high verbal WM load disrupts the processes that produce the NDE is that participants could still accurately compare the two numbers despite this disruption. Specifically, existing accounts of numerical comparison do not offer a mechanism by which numerical comparison can occur without producing an NDE. Thus, if one accepts a disruption account as the explanation for the WM-induced reduction in the NDE, then a complete theory of numerical comparison would require the existence of a separate type of representation or mode of access that could support numerical comparison but not produce a distance effect. This is because participants in the present study are able to perform the numerical comparison task under high verbal load without demonstrating an NDE. One potential explanation could be based on the fact that in Experiments 1, 3, and 4 we used single digits 1-9. Perhaps in addition to storing numerical information on a mental number line we also have a mental store of basic “number facts” for numbers 1 through 9, such as “9 is larger than 5”. Akin to performing basic addition (e.g., 2+2=4) through direct retrieval, accessing these facts would presumably be independent of WM. In addition, these “number facts” need not have stored with them magnitude information per se, but rather semantic knowledge that “9 is larger than 5”, thus, when comparison is completed via recourse to these “number facts” an NDE would not arise.
Unlike the NDE in the simultaneous presentation task, the NDE in the comparison to a standard task increases with increased WM load. The dissociation reported here is particularly surprising given that the literature assumes that these two tasks are tapping the same underlying mechanisms (e.g.
An alternative conceptualization of the dissociation between simultaneous presentation and comparison to a standard with numeric stimuli is that the tasks encourage different comparison strategies. For example,
In Experiment 4, participants completed a comparison to a standard task, however, the standard (5) was presented at the beginning of each trial. In this novel version of the task, the NDE did not change as a function of the increase in verbal WM load. While this pattern may, at first, seem counterintuitive, it is, in fact, consistent with the aforementioned theory. Specifically, by presenting the standard (5) at the beginning of every trial, participants are able to either perform the comparison in short-term memory or to adopt a more long-term comparison strategy wherein on each trial they retrieve the comparison digit (5) from long-term memory and carry out the comparison. If participants are sometimes making the comparison in short-term memory (which would yield a decrease in the NDE under high WM load) and sometimes making the comparison on long-term memory (which would yield an increase in the NDE under high verbal WM load), then these two effects would cancel each other out, resulting in an additive interaction. Indeed, this is what we see in Experiment 4.
The present results suggest not only that participants may be using different comparison strategies across tasks, but that those strategies are also differentially reliant on WM. We have already discussed the potential mechanisms mediating the WM by distance effect interactions in the simultaneous presentation tasks. With respect to the long-term comparison strategy, the distance effect may be arising, in part, during the retrieval of the standard from long-term memory. For example, on presentation of the probe digit, it would be harder to recall 5 when the probe is near (e.g., 6) than far (e.g., 9) from that standard. The interaction with WM can be understood from the standpoint of
We have assessed the role of verbal WM in numerical comparison by examining the effects of verbal WM load on the NDE in two classic numerical cognition tasks. We demonstrated that verbal WM does interact with the NDE and that the nature of the interaction is dependent upon the comparison task used. When comparing two simultaneously presented Arabic digits, the size of the NDE decreases (and here is eliminated) as a function of an increased verbal WM load. On the other hand, when comparing one Arabic digit to a standard, the size of the NDE increases as a function of increased WM load. We further demonstrated that this dissociation is not due to the task structure alone given that when both tasks are performed with squares as stimuli, the distance effect decreases in both tasks. Importantly, we present a novel account of how this dissociation of the NDE can be explained if participants adopt a long-term comparison strategy on the comparison to a standard task and a short-term comparison strategy on the simultaneous comparison task. This complex pattern has straightforward implications: (1) WM is intimately tied up with numerical comparison, (2) the NDE in simultaneous presentation and comparison to a standard are not equivalent, and (3) the NDE can be eliminated. These insights promise to generate further research.
The lead author does not have authorisation from their ethics board to release the data publicly. However, should readers wish to see the data, they should contact Erin Maloney (
This research was supported by Discovery Grants from the Natural Sciences and Engineering Research Council of Canada (NSERC) to EAM, EFR, and JAF.
Due to a programming error in the zero WM load condition there were three trials on which the distance was 2 where it should have been 4. These trials were removed from analyses.
Note that we have opted to remove 6 participants because their accuracy was below 20% on the WM task and 4 participants were removed for having too low accuracy on the comparison tasks. If we did not remove those who had too low accuracy on the comparison task (but still remove the 5/6 participants who scored below 20% on the WM task as there were cells in which they had no analyzable data), then the pattern of results does not change. Indeed, even when including these five additional participants, the critical three-way interaction that was observed in Experiment 1 is still not found here,
The authors have declared that no competing interests exist.
The authors have no support to report.