people whose interaction with graphs suggests that they are using an analogical ... for the ability to comprehend graphs, whilst an answer to the second question ...... skill. European Journal of Cognitive Psychology Vol. 11 (1999) 473-498. 15.
Individual Differences in Graphical Reasoning Aidan Feeney, John Adams, Lara Webber & Michael Ewbank Department of Psychology, University of Durham Queen’s Campus, Thornaby, Stocktonon-Tees TS17 6BH, United Kingdom {aidan.feeney, l.j.webber, michael.ewbank, j.w.adams}@durham.ac.uk
Abstract. People sometimes appear to build analogical representations in order to reason about graphical information. In this paper we consider the extent to which the tendency to represent information analogically calls on spatial resources. We also examine whether people who represent graphical information analogically also represent numerical information using a spatial number line. Forty-eight adult participants carried out a series of graphical reasoning, number judgement and spatial working memory tasks. Evidence was found to suggest that people were forming analogical representations in both the number judgement and graphical reasoning tasks. Performance on the spatial memory task was positively associated with a measure of the tendency to use analogical representations on graph task. In addition, measures of the use of analogical representations for the graph and number tasks were associated. We interpret our results as providing further evidence that people build analogical representations of graphical information. We conclude with a discussion of whether the use of such analogical representations is confined to any one task or is instead a general representational strategy employed by people high in spatial ability.
1. Introduction Much everyday cognition appears to involve representations that are analogical in nature. For example, people often appear to represent information about relationships between objects in the world analogically. These analogical representations can form the basis for subsequent relational inferences [for a review, see 1]. Recently, we have applied some of the techniques previously used to examine analogical representations for relational inference to demonstrate that people represent graphical information by analogically [2; 3; 4]. These findings (which we will review in more detail below) suggest that accounts of graph comprehension that assume wholly propositional representational systems are incomplete [5; 6]. They also support accounts of graph comprehension that stress the importance of spatial transformations and mental model representations [7; 8]. Analogies to space have been implicated in people’s representation of number. It has been suggested that numbers may be represented by analogy to a spatial mental number line [9] which (in native English speakers) runs left-to-right with smaller numbers on the left and larger numbers on the right. However, there is debate as to whether the number line is a core component of the representation of number [10]
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Our goal in this paper is to examine individual differences in people’s use of analogical representations for thinking about graphs. We wish to examine whether people whose interaction with graphs suggests that they are using an analogical representational strategy tend to be higher in spatial ability than people who do not appear to be using analogical representations for graphical reasoning. In addition, we wish to investigate whether the tendency to use an analogical reasoning strategy in graphical reasoning is predicted by the tendency to adopt a spatial representation of number. The answer to the first of these questions has a bearing on how we account for the ability to comprehend graphs, whilst an answer to the second question will help us to understand the role played by space in abstract cognition. In addition, our findings may have a bearing on questions about whether spatial representations can be domain-specific [11] or constitute a domain-general resource that may be adapted to particular tasks via spatial mappings. 1.1 Analogical Representations of Graphical Information In a series of experiments [2; 4] we have attempted to demonstrate that, at least sometimes, people construct analogical representations for thinking about graphs. For example, Webber and Feeney presented participants with series of pairs of line graphs or premise displays (see Figure 1). Each of these graphs described two referents one of which was common to both graphs. Once participants had examined these graphs they vanished from the screen and were replaced by a third, conclusion graph describing the relationship between two of the referents from the previous screen. We Premise Displays
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Fig. 1. Sample materials from graphical relational reasoning task.
manipulated whether the points in the premise displays ascended (see top line of Fig. 1) or descended (see bottom line of Fig.1) left-to-right and the position of the repeated referent. In half our trials the repeated term separated the non-repeated terms (see top half of Fig. 1) whilst in the other half the occurrences of the repeated term were separated by the non-repeated terms (see bottom half of Fig. 1). We manipulated whether the conclusion graph followed logically from the premise displays and
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whether the order of the referents in the conclusion graph was consistent with their order in the premise displays. If participants use analogical representations in order to reason about the relationship between the premise and conclusion graphs, then we would expect to see reordering effects. We predicted these reordering effects on the grounds that an analogical representation of the information in the graphical premises is likely to be (a) integrated and (b) order the terms in the premises by size. Problems where the repeated term separates the non-repeated terms (separated trials) should not require reordering whereas problems where the non-repeated terms occur together (together trials) should require reordering. We expected inspection times – time taken to view the premise graphs - to reflect the need for reordering. We found that inspection times for together premise graphs were reliably longer than for separated premise graphs. In the work that follows, when this difference is positive we will assume that people have used a reordering strategy that is diagnostic of analogical representation. We also predicted that referent order in the premise graphs would interact with consistency between graph and conclusion displays in terms of order. For separate trials we predicted that consistent conclusions would be verified more quickly and lead to fewer errors than inconsistent conclusions. However, for together trials we predicted that reordering would often result in representations in which the order of terms was inconsistent with their order on the screen. For example, the order BCAB on the screen was predicted to be reordered as ABBC and the order BACB as CBBA. Thus, we predicted longer verification times and more errors for consistent conclusions than for inconsistent conclusions. We found the predicted effects in both error rates and verification times. Although we interpret these findings, and others [2], as evidence that people are building analogical representations for graphical reasoning, we have no evidence that these representations are analogical with respect to space or that they call on spatial resources. The first aim of the experiment to be described here was to provide some such evidence. 1.2 Spatial Strategies for Thinking There is substantial evidence in the literature suggesting that people, at least for some tasks, use spatial strategies for thinking. For example, the results of studies using a secondary task methodology, where people are asked to perform a primary reasoning task whilst carrying out a secondary task designed to tap into verbal or spatial resources, suggest that spatial strategies are used for tasks with spatial or temporal content [12; 13; 14; 15]. Much of this work has required people to reason about spatial or temporal relationships and is thus highly relevant to our own graphical reasoning task, which requires people to verify relationships between graphical referents. Another methodology commonly used to study the representations and processing resources used in the course of thinking involves individual differences. This methodology requires that participants’ spatial or verbal ability be measured independently of the primary task of interest. The strength of the statistical association between performance on an ability measure and performance on the reasoning task is
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taken to reflect the degree to which the resource measured by the ability task is involved in the primary task. Individual differences studies have shown that some individuals use spatial resources and some use verbal resources when verifying pictures against sentences [16] and a similar argument has recently been made about syllogistic reasoning [17]. In the study to be reported here we hoped to provide evidence for the involvement of spatial resources in relational reasoning about premises presented graphically. Such a finding would support the claim that people form representations by analogy to space when reasoning about graphs [3; 4]. Importantly we predicted that performance on an index of the construction of analogical representations for graphical reasoning would be positively associated with a measure of spatial resources. We did not predict that overall correct solution rates or overall processing times would be associated with the use of a spatial strategy. This prediction is motivated by work on reasoning which suggests that spatial or verbal strategies may be equally efficient for the accomplishment of high-level cognitive tasks [16; 18] Our final research question concerned relationships between the tendency to use analogical representations for graphical reasoning and number judgement. We predicted that people who showed evidence of building analogical representations of the information contained in graphs would also display evidence of analogical representations of number. Although there is now a substantial literature on the neuroanatomy of number representation [see 11] much of the behavioural evidence for the analogical representation of number comes from number judgment tasks where a variety of effects have been demonstrated. For example, people appear to find it easier to compare two numbers the further apart they are in magnitude [19]. In addition, when making parity or relative size judgments about numbers, native speakers of languages written left-to-right show a reaction time advantage when they are required to respond to larger numbers with their right hand and smaller numbers with their left-hand compared to when the response code is switched. This latter effect is known as the SNARC effect and it is interpreted as evidence for a mental number line [9]. In this paper we will test for an association between the SNARC effect and the tendency to build analogical representations for graphical reasoning.
2. Experiment
2.1 Method Participants: 48 participants recruited on the University of Durham’s Queen’s Campus took part in this experiment. Materials: Each participant attempted four tasks. One of these - a line bisection task – is not relevant to our current purposes and will not be discussed further. The other three tasks were a graph comprehension task, a number judgement task and a task measuring complex spatial span [20]. Participants completed these tasks individually
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and each task was presented using an IBM clone computer, monitor and keyboard. Task order was counterbalanced. We will now describe each task in turn. Graph Comprehension Task: The materials for this task comprised 112 trials. Each trial consisted of two visual displays, a premise display and a conclusion display, which appeared in that order. The premise display contained two simple bar graphs, which participants were told always depicted the sales figures for three employees over a month The most successful employee was represented by a bar 900mm in height, the next most successful by a bar 600mm in height and the least successful by a bar 300mm in height.
Figure 2: Two simple bar graphs with separate end terms and descending slope.
We manipulated the position of the non-repeated terms, so that they were either adjacent (see Figure 3 below) or separated by the middle term (as in Figure 2 above), and the slope of the graphs either ascended left-to-right (Figure 3) or was descending (Figure 2).
Figure 3: Two bar graphs with adjacent end terms and ascending slope.
The second display contained one simple bar graph, the conclusion graph (see Figure 4), specifying a relationship between two of the employees. For the integrated trials this graph consisted of two bars representing just the least and most successful salesmen, which were exactly the same height as in the premise graphs. Reversing the order of the bars and/or reversing the order of the labels of the bars generated four possible conclusion graphs. Of these two were valid and two invalid, two had labels that were consistent with the order in the premises and two that were inconsistent with the order. There were a total of 4 premise graphs, each with 4 possible conclusions to form 16 conditions. Four sets of materials were constructed for each condition by using different employees. This resulted in a total of 64 integrated trials.
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The premise graph displays used for the non-integrated trials were the same as those in the integrated trials. However, the conclusions differed to those presented in integrated trials. Each conclusion was one of the previously shown premise graphs and depicted a relationship between either the short non-repeated term and the middle term or the tall non-repeated term and the middle term. For these trials, we manipulated conclusion validity but not consistency. There were 16 (8 valid and 8 invalid) conclusion graphs that contained the tallest non-repeated term (tall conclusions) and 16 matched trials containing the shortest non-repeated term (short conclusions). To ensure that integrated conclusions did not vastly outnumber nonintegrated conclusions, we also included a further 16 trials with short conclusions.
Figure 4: A bar graph representing the valid and consistent conclusion that follows from the premises depicted in Figure 1.
For each trial, a fixation cross was presented for 1000ms. This was followed by a premise display visible until participants pressed the space bar. Once the space bar was pressed a conclusion graph appeared which remained visible until participants responded ‘yes’ or ‘no’ using the keyboard. We recorded the time taken by participants to inspect the premise graphs as well as the time they took to respond to the conclusion. In addition, we recorded the number of errors made by each participant. Number Judgement Task: In the number judgement task participants were required to indicate whether a target number, presented on a computer screen, was greater or less than a referent. The numbers 1 – 4 and 6 - 9 were the targets whilst the number 5 was the referent. Target numbers appeared in the centre of the computer screen one at a time and in random order. Each participant attempted two blocks of trials. In the first block participants had to press the right button on a button box to indicate that the target was bigger than the referent and the left button to indicate that the target was smaller than the referent. In the second block of trials a greater than response was made with the left hand and a smaller than response was made with the right. Participants saw each of the target numbers ten times in each block. Thus, each participant made 160 judgements in total. The order of blocks was counterbalanced. Complex Spatial Span: In the complex spatial span task participants were shown sets of capital letters and their mirror images that were rotated in different orientations. Participants had to decide whether each letter was in its canonical or mirror-imaged form whilst keeping track of the orientation of each of the letters in the set. At the end
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of each set participants attempted to recall the orientation of each letter in the set in the order that they had appeared. Trials were computer presented in PowerPoint. Each set was based on one of the following letters: F, J, P, L and R. Letters were presented one at a time and appeared in one of 7 orientations. The orientations ranged from 45-degrees through to 315 (excluding 0-degrees) and increased in 45-degree increments. The presentation of the letters was constrained within sets so that no orientation appeared more than once and opposing orientations were not presented successively. This resulted in 70 possible letter stimuli (letter x orientation x normal/mirror image). The task consisted of 25 letter sets, 5 sets at each level (one of each letter). Sets increased in difficulty by virtue of the number of letters they contained. There were five sets of two letters, five sets of three letters and so on up to 6 letters. Each of the 70 possible letter stimuli was used at least once in the experiment. As our design called for 100 letter stimuli, 30 of the stimuli were repeated. This repeated set of 30 contained six randomly selected examples of each of the letter types, three of which were reversed and three of which were not. For each set participants were require to verify out loud whether the letter was normal or mirror-imaged. After they responded the next letter appeared on the screen, when they had responded to all the letters in a set a recall screen appeared. The recall screen consisted of a ring of 8 black circles representing each orientation (including 0-degrees), participants were required to indicate the orientation of each of the letters within a set in the order they appeared by pointing to the corresponding black dot on the recall screen. After they had indicated the appropriate number of orientations the first letter of the next trial appeared. Trials were presented until the participant made three or more errors in recall on a given level. 2.2 Results Graph Task: The overall error rate for the graph task was 6.41% (S.D. = 15.82%). We removed the data of four participants with an error rate greater than this mean plus one standard deviation. Amongst trials where the conclusion presented followed logically from the premises, the remaining 44 participants had an error rate of .36%. The mean error rate for trials where the conclusion did not follow from the premises was 4.19%. In the analyses that follow, we examined reaction times from logically valid trials only. Inspection Times: We removed the data for all trials where participants had given an incorrect response. In addition we removed all trials with a response less than 100 ms or more than two standard deviations above the mean for the entire experiment (mean = 3349ms, S.D. = 2852ms). These trimming procedures resulted in the removal of 4.2% of trials. We carried out a 2 (Slope: Ascending vs. Descending) x2 (Distance: End terms together vs. End terms separate) within participants Anova on inspection times. None of the effects in this analysis were significant although the effect of End Terms (F(1, 43) = 1.11, MSE = 425295, p = .30), previously found to be significant in experiments on bar and line graphs [4], was in the direction predicted. The mean inspection time
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for premise sets where the end terms were apart was shorter than the mean for premise sets where the end terms were together (2855ms vs. 2958ms). Verification Times: All trials to which a correct response was given with a latency of between 100 ms and the mean (1985ms) plus two standard deviations (1226ms) were included in this analysis. This meant that 3.61% of the data was not included. A 2 (Consistency) x (Slope) x (End Terms) within participants Anova was carried out on this data. The analysis revealed a significant main effect of Slope (F(1, 43) = 5.78, MSE = 93477, p < .03) and a significant interaction between Consistency and End Terms (F(1, 43) = 4.54, MSE = 98026, p < .04). Although this finding replicates the figural effect previously found in graphical reasoning [3; 4], it is qualified by a marginally significant three-way interaction (F(1, 43 = 3.55, MSE = 141458, p < .06). Table 1: Mean verification times by Slope, End Terms and Consistency. End Terms: Consistent Inconsistent
Descending Separate Together 1738 1960
1918 1847
Ascending Separate Together 1787 1805
1765 1792
Tests for simple interaction effects carried out on the means involved in the threeway interaction (see Table 1) revealed a significant interaction between Consistency and End Terms for Descending trials (F(1, 43) = 6.77, MSE = 139849, p < .02) but not for Ascending trials (F(1, 43) < .01). Whilst the figural effect is evidence for analogical representation [see 3; 4] it is somewhat unclear as to why it should be observed for descending trials only. People prefer to build analogical representations from the top down [21; 22]. This preference may have made trials where the premise graphs descend left-to-right particularly amenable to solution via the construction and interrogation of a spatial array. Thus, participants may have been more likely to build and interrogate analogical representations when the premise graphs had a descending slope that made use of such a strategy easy. Certainly the verification times in Table 1 suggest that participants used different strategies depending on the slope of the premise graphs. Number Judgement: The mean error rate for this part of the experiment was 2.15% and the highest individual error rate was 12.5%. A 2 (block) x 8 (number) entirely within participants Anova was carried out on participants’ mean judgement times for correct trials only. The means involved in this Anova are presented in Figure 5. All of the effects tested by the ANOVA were significant. Of most interest here is the significant main effect of Block (F(1, 47) = 12.66, MSE = 45555, p < .001) and the interaction between Block and Number (F(7, 329) = 3.28, MSE = 3748.3, p .5) task. Nor was it associated with mean overall inspection times for the graphical reasoning task (r = -.08, p > .6) or mean overall verification times for the graph task (r = -.17, p > .2). Thus, complex spatial span is associated with the tendency to use an analogical representational strategy in the course of attempting the graph task rather than people’s ability on the task. Finally, we observed a moderate correlation between scores on the GAI ad NAI indices (r = .34, p < .04). Even when we controlled for the effects of complex spatial span the correlation remained marginally significant (r = .29, p < .07). This result suggests that although there is a positive association between performances on the analogical indices from each of these tasks, that association is not due to the demands that each task places on the ability to temporarily store and concurrently process information in spatial memory. We will return to this issue below.
3. Discussion The results described in this paper support the claim that some people, at least some of the time, build analogical representations for reasoning about graphs. This is in contrast to the assumptions of several recent models of graph comprehension [5; 6]. For graphs with a descending slope only, our data shows evidence of the reordering strategy that we argue is diagnostic of the use of spatial analogies for representation. In addition, the tendency to inspect graphs that require reordering for longer than graphs that do not is positively associated with complex spatial span, a measure of spatial storage and processing. People who inspect graphs that require reordering for longer than graphs that do not, have significantly higher complex spatial span scores than people who do not display the inspection time difference. This finding suggests that the representational medium used for the construction and manipulation of the analogical representations used by some people for graphical reasoning, is spatial in nature. Some people appear to represent graphical information by analogy to space. The second finding of interest in this paper is our replication of the SNARC effect usually interpreted as indicating that one aspect of number representation is position along an analogical number line. We have shown that there are individual differences in people’s tendency to represent numerical information by analogy to space and that people who represent number by analogy to space also tend to represent graphical information in the same manner. We will return to the significance of this finding for claims that have been made about the use of spatial representations for mathematical cognition [11]. Before considering the implications of our results, it is very important to note that the relationship between spatial ability and performance on our graphical task is unlikely to be due to a higher-order relationship between general intellectual ability and task performance. The GAI that we used as a measure of the tendency to represent information analogically was not associated with the number of errors made on the task, overall inspection times or overall verification times. Neither was complex spatial span significantly associated with measures on any of these variables.
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Only the tendency to reorder as indexed by the GAI is associated with complex spatial span. 3.1 Individual Differences in Graph Comprehension Although we have observed an association between spatial abilities and the tendency to reorder the information contained in graphical premises in a manner consistent with the use of analogical representations for reasoning, we have not found associations between spatial ability or reordering and overall performance on the task. This result suggests that although some people may use analogical representations for thinking about graphs, the use of such representations does not lead to better performance. This is in contrast with previous work on other diagrammatic tasks [e.g. 23] showing that people low in spatial ability make more errors than people high in spatial ability. This difference in error rates due to spatial ability increased as the diagrammatic task became more complex. An explanation of these results may be given in terms of capacity limits [24] where the more complex problems are more likely to exceed the capacity limitations of people low in spatial ability thus leading to greater differences due to spatial ability as the problems get more difficult. This capacity explanation might be applied to our finding that the association between spatial ability and error rates in graph comprehension is not statistically significant. Perhaps the graphical reasoning problems that we used were too simple to reach the capacity limits of our participants. Hence they did not discriminate between people of differing spatial ability and no association was found between error rates and ability. An alternative account of our results is that the graph task may be accomplished using spatial or verbal resources. The GAI is a measure of the extent to which people use a particular representational strategy in order to perform the task and scores on the GAI allow us to separate out people who use a predominately spatial strategy on the task. However, neither a verbal nor a spatial strategy leads to fewer errors or faster inspection and verification times. Interestingly, recent research on syllogistic reasoning [18] has classified people, based on their external representations for reasoning, as using spatial or verbal representations. There were no differences in reasoning performance between the two groups. These results, and our own, leave open the possibility that some people may efficiently carry out some graph-based tasks using predominantly verbal resources. 3.2 Representation by Analogy to Space – A Domain-General Ability? The question that we would like to consider for the remainder of this paper is whether the representation of abstract concepts by analogy to space is a domain-general strategy for abstract thought or whether, in the course of cognition, we build a number of similar but distinct and domain-specific spatial representations. The former position might be attributed to proponents of mental model theory [25] who assume that the same analogical representations underlie transitive reasoning about space and time. Similarly the notion that spatial schemas [26] underlie many cognitive abilities
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seems to presuppose a domain-general representational ability. However, the contrary position appears to have been adopted particularly with respect to number cognition. For example, Dehaene and his colleagues [11] have claimed that we have evolved a domain-specific system located in the parietal cortex, for the spatial representation of number. They interpret evidence that sensitivity to number is present very early in development, and in other species, as supporting their argument that spatial representation for number is evolutionarily inherited. They further claim that the representation of number is abstract. This claim is based on the insensitivity of several of the signature effects in the literature to variations in stimulus input, as well as on the results of lesion and imaging studies. We are unconvinced by arguments for domain-specificity. One problem for the argument is our finding that roughly one third of the participants in our study did not display a positive score on the NAI. These participants were worse off when responding to high numbers with their right rather than their left hand. If an abstract spatial representation for number is domain-specific and evolutionarily determined then we might expect almost everyone to display the SNARC effect. This finding suggests, as has been acknowledged in the reasoning literature [16; 25], that different spatial or verbal representational strategies may be used for the accomplishment of high-level cognitive tasks such as number comparison. In addition, we observed a positive correlation between the number of errors made by participants on the task and the NAI index (r = .38, p < .02). That is, a large SNARC effect is associated with higher errors on the task. We are unsure as to why an evolved and domain-specific representation for number would result in error. It is perhaps, more likely that some people use a general spatial strategy for representing number and that this strategy, whilst useful in many contexts, may not be appropriate for some tasks. A second potential problem for the domain-specificity argument is the overlap that exists between effects found in the literature on numerical cognition and those found in the literature on relational reasoning. For example, the finding that it is easier to compare two numbers the further apart they are in magnitude has a direct analogue in the reasoning literature. That is, the further apart are two referents in a spatial representation of some relational premises, the faster people are to verify a conclusion concerning them [27]. Given this overlap, we were unsurprised by the association we observed between performance on the indices of an analogical representational strategy taken from the graph and number tasks. Although not everyone appears to use spatial representation for graphical reasoning and number judgment, the tendency to do so on one task is associated with the tendency to also do so on the other. Of course, an association does not necessitate a common cause. Analysis of the processes involved in the graph and number tasks suggests that although they share a number of common features, such as a requirement that a spatial or verbal representation be constructed, maintained and interrogated, they differ in several respects. For example, a spatial representational strategy on the graphical reasoning tasks calls for a mapping between actual and representational space and reordering of referents [see 28]. Our GAI may have been particularly sensitive to these aspects of the task and the fact that they place concurrent demands on spatial storage and processing may explain the association we observed between complex spatial span and performance on the GAI. Intuitively, the SNARC task often appears to call for the inhibition of one response over another particularly when participants must respond to
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small numbers with their right hand. Scores on the NAI may partly reflect ability to inhibit one response in favour of another. In addition, the tasks may place very different demands on spatial storage and processing resources, which may explain why, when we control for complex spatial span the association between performance on the analogical indices is still relatively strong. In short, the tasks are sufficiently complex and different in order for there to be difficulties in interpreting the statistical association observed between them. Despite the complexities of interpretation that we have outlined above, we speculate that there are two possible explanations for the association between the graph and number tasks. Recently, Dehaene and colleagues [29] have suggested that the processing of number has three components, each of which relies on separate parietal circuitry. One of these components is domain-general and is responsible for attentional orientation onto a variety of spatial dimensions. Perhaps this attentional process is involved in both the graph and number tasks. Its involvement in both tasks would account for the association we observed between them. Another possibility is that a general magnitude system underlies much of our spatial and temporal cognition [30]. This system may partly support both graphical reasoning and number comparisons. Interestingly, the first of these explanations preserves the Dehaene’s domain-specificity argument as the tripartite system that he suggests includes a parietal circuit specifically dedicated to domain-specific representation of number. The second explanation does not. As our data does not differentiate between these points of view, further work will be required to tease them apart. 3.4 Conclusions Our results suggest that some people represent information presented graphically by analogy to space. They appear to build analogical representations of graphical information and the time it takes them to reorder information in premises presented graphically is associated with their spatial capacity. We also observed that the size of the SNARC effect in number cognition is associated with people’s tendency to reorder information in graphical premises. This led us to consider whether people possess just one mechanism that allows them to represent abstract concepts by analogy to space or whether they possess a variety of domain-specific representational abilities. Although the data that we have reported here bears on this issue, additional experimental work is required to answer the question.
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