DOCTORATE OF PHILOSOPHY Dec 2008 Committee: Richard B. Anderson, Advisor Dale Klopfer Yiwei Chen Barbara Moses

Transcription

1 THE INFLUENCE OF REPRESENTATIONAL PROCESSES ON THE NUMERICAL DISTANCE EFFECT Neil D. Berg A Dissertation Submitted to the Graduate College of Bowling Green State University in partial fulfillment of the requirements for the degree of DOCTORATE OF PHILOSOPHY Dec 2008 Committee: Richard B. Anderson, Advisor Dale Klopfer Yiwei Chen Barbara Moses

2 ii ABSTRACT Richard B. Anderson, Advisor Research has clearly established that the numerical distance between two Arabic numerals affects reaction times for tasks that involve determining the relative numerical magnitude of those numerals. The present study tested two characteristics of models that explain the effect of numerical distance: whether distance effect-sensitive representational processes occur with the presentation of a single comparate (i.e. a to-be-compared digit) or require two comparates to operate, and whether distance-effect-sensitive representations are static or dynamic. The present research examined the data from three number-comparison experiments that manipulated the asynchronous presentation of the comparates using stimulus onset asynchrony (SOA) and a between-trial repeat of a single comparate. Both manipulations of asynchronous presentation were intended to provide subjects with a head start in encoding the first of a pair of to-be-compared digits without providing a head start for the actual comparison of the digits thus yielding information about the onset of distance processing in relation to comparate presentation and also to identify whether relevant representations are dynamic or static. Experiments 1 and 2 also included a probe-response task in which subjects selected a numerical probe after being presented with a numerical prime. SOA in the probe-response task was found to moderate the distance effect (thus suggesting that onset of distance effect processing occurred with the first comparate), but SOA did not moderate the distance effect on the number-comparison task (thus suggesting that onset of distance effect processing occurred with the second comparate). This inconsistency, together with evidence from Experiment 3

3 iii (below), was interpreted as evidence that each task type used a separate representational pathway. Experiment 3 did not contain a probe-response task, but instead included an additional number-comparate task condition in which the comparate that the participant selected on the current trial was repeated as one of the comparates presented in the subsequent trial. The distance effect in Experiment 3 was stronger for trials without a repeated comparate than it was for trials with a repeated comparate. This was interpreted as evidence that the comparison process changed the representation of the repeated comparate and that retaining this changed representation reduces the amount of distance effect processing on the subsequent trial. Experiment 3 also yielded evidence that SOA moderated the effect of distance for both comparate-repeat and comparate-non-repeat trials. This moderation was interpreted as evidence that distance effect processing occurs with the onset of the first comparate for whichever representational pathway is given precedence. Thus, it was concluded that the probe-response task was given precedence over the number-comparison task for Experiments 1 and 2 (these experiments contained both task types) but that the number-comparison task had precedence for Experiment 3 (Experiment 3 contained only the number-comparison task). Both the comparaterepeat manipulation results and the SOA manipulation results are consistent with the explanation that dynamic, but not static, representation gives rise to the distance effect. The fact that both the comparate-repeat manipulation and the SOA manipulation moderated the distance effect suggests that the onset of the representational change that gives rise to the distance effect occurs with the presentation of the first comparate, but that substantial processing occurs with the second comparate as well. Overall results from all three experiments suggest that

4 iv representational change plays a role in creating the distance effect and that this change occurs along task-specific pathways.

5 v TABLE OF CONTENTS Page Literature Review... 1 General Models of Numerical Representation Magnitudinal Representation within a Number-Comparison Task... 6 Evidence that Semantic Distance Affects Task Performance... 7 Evidence of non-numerical semantic distance... 7 Evidence of numerical semantic distance in probe-response tasks... 9 Modeling semantic distance in number-comparison tasks An Accumulator Mechanism for Number Comparison Buckley and Gillman model Augmented Buckley and Gillman model Simulation of Accumulator Mechanism Models Simulation Simulation Experiment Predictions Method Subjects... 37

6 vi Apparatus and stimuli Procedure Data preparation Results Discussion Experiment Predictions Method Subjects Apparatus, stimuli, and procedure Data preparation Results Discussion Experiment Code Discrimination Mechanism Model Differentiation Using Asynchronous Presentation Conditions Predictions Method Subjects Apparatus, stimuli, and procedure

7 vii Data preparation Results Discussion General Discussion General Framework for Distance Effect Models Onset of Distance Effect Processing Dynamic Representation General Implications for the Study of Semantic Distance References Appendix Fits for Experiment 1: Number-Comparison Task, RT Fits for Experiment 1: Number-Comparison Task, Error Rate Fits for Experiment 1: Probe-Response Task, RT Fits for Experiment 1: Probe-Response Task, Error Rate Fits for Experiment 2: Number-Comparison Task, RT Fits for Experiment 2: Number-Comparison Task, Error Rate Fits for Experiment 2: Probe-Response Task,, RT Fits for Experiment 2: Probe-Response Task,, Error Rate Fits for Experiment 3: RT Fits for Experiment 3: Error Rate

8 viii LIST OF FIGURES Figure Page 1 Example of the creation of the distance effect within the Buckley and Gillman model Shape of the probability distributions across time within the Buckley and Gillman model, given a comparison task with an asynchronous presentation of stimuli Representational change across time within the Augmented Buckley and Gillman model, given a comparison task with an asynchronous presentation of stimuli Diagram of the key characteristics of interest in Experiment Predictions based upon the Augmented Buckley and Gillman model Predictions based upon the Buckley and Gillman model Non-degraded versions of letters that were excluded from the experiment, paired with a number of similar shape The procedure for Experiment 1 and The effects of numerical distance and SOA on mean of median RT and mean error rate for the number-comparison and probe-response tasks in Experiment Experiment 2 results for comparison and probe-response tasks for mean of median RT lines and mean error rate lines Diagram of the key characteristics of interest in Experiments 1, 2, and Pattern of distance by RT (or error rate) results for Experiment 3 predicted by the Buckley and Gillman model of the distance effect... 67

9 ix 13 Pattern of distance by RT (or error rate) results for Experiment 3 predicted by the Banks Code-Discrimination and Augmented Buckley and Gillman models of the distance effect Experiment 3 results for the mean of median RTs (top) and mean error rates The effects of numerical distance and SOA on comparate-non-repeat trials and comparate-repeat trials in Experiment

10 x LIST OF TABLES Table Page 1 The mean number of iterations necessary at several between-number distances in a simulated version of the Buckley and Gillman model of distance effect processing that includes equal and narrowed standard deviation conditions The threshold values necessary to create a.02 error rate and the mean number of iterations needed to exceed that threshold value... 27

11 1 Literature Review People are frequently asked to label, order, and perform mathematics on various kinds of numerical data. For example, a simple task like purchasing coffee involves applying a label to the weight of coffee that is requested, finding a package which most closely meets that weight, and then computing the cost of that coffee package relative to other packages of coffee. Given the complexity of even such common numerical undertakings, the task of determining the relative ordering of two Arabic numerals (i.e., which is higher or lower) seems simple. However simple the subjective experience of performing a numerical comparison may seem, the performance of this task is likely influenced by many complex factors. One such factor that has been extensively studied is the numerical distance between the numbers. Although many studies have demonstrated this effect, there is not yet a definitive model of numerical cognition that clearly explains the effect. To further elucidate what produces this effect, the present studies compare two characteristics of models of the number comparison process. The present document reviews the literature related to models of numerical cognition and then focuses on general and numerical models related to semantic distance. Next, the empirical literature related to representation within a numerical comparison task is reviewed, and the relationships of the numerical distance effect to two models that are based on an accumulator mechanism (e.g., Buckley & Gillman, 1974) will be described. Two experiments are then described that evaluate the relation of the distance effect to the time-course of the presentation of the comparates (i.e., the to-be-compared stimuli) in order to discriminate between these two accumulator-based models. Then, a code-discrimination mechanism (as posited by Banks, 1977) and an third experiment will be presented that contrasts this mechanism with one of the models that uses an accumulator mechanism. Finally, the implications of all three present experiments

12 2 for models of the distance effect will be discussed in terms of how those models explain the onset of distance effect processing and whether such processing involves dynamic or static representations of number. General Models of Numerical Representation In general, the numerical literature has concerned itself with two types of mental representations. One type is symbolic, and contains unambiguous word-like information; the other is driven by numerical value and contains information about approximate magnitude. The present review will focus on the latter type and show that a variety of techniques have confirmed the uniquely magnitudinal qualities of numbers (e.g., Moyer & Landauer, 1967, 1973; Dehaene & Akhavein, 1995, Dehaene, Bossini, & Giraux, 1993; Reynovet & Brysbaert, 1999; Pavese & Umiltà, 1998, 1999; Tzelgov, Meyer, & Henik, 1992). Several general models of numerical representation have been offered to explain numerical task performance (for general reviews, see Dehaene, 1992; McCloskey, 1992; and Gallistel & Gelman, 1992). Of these, two models that have been well-established and welldeveloped within the numerical cognition literature are the preverbal representation model (Gallistel & Gelman, 1978, 2000) and the triple-code model (Dehaene 1992; Dehaene & Changeux, 1993). These models are valuable because they provide a necessary general background for studying number-specific cognitive effects. Understanding the functional origin of adult human numerical cognition is central to Gallistel and Gelman s (1992) bidirectional mapping hypothesis, which posits that people represent numbers using two systems: a preverbal system that represents magnitude and a verbal system that represents orthographic information. Although their model focuses mainly on

13 3 the development of numerical representation in early childhood, there are also formal characteristics of their model that apply to adult human cognition. They posit that the preverbal system is innate, and that most mammals (and many non-mammals) use this system to understand the world and to operate within it. This is evidenced by numerosity-sensitive behavior in organisms ranging from rats (Fernandes & Church, 1982) and pigeons (Brannon, Wusthoff, Gallistel, & Gibbon, 2001) to chimpanzees (Matsuzawa, 1985) and human infants (Gallistel & Gelman, 1978). Fernandes and Church trained rats to choose one of two levers based strictly upon the number of events (noises or flashes) they were presented. Numerosity was determined to be the relevant cue when alternative temporal cues of stimulus duration, interval duration, and total sequence duration were controlled. Using a different technique, Brannon et al. (2001) trained pigeons to recognize the number of light flashes (range 1-7) that occurred near a food bin. The pigeons had to execute that number of taps on a middle key and could then choose either to tap that number of times again on a standard key or to tap eight minus that number on number-remaining key. The behavior of interest was whether the pigeons would economize their responses by selecting the standard key for low numbers of flashes and the number-remaining key for high numbers of flashes. Indeed, it was found that the pigeons tended to use the most economical response key. This suggests that not only are pigeons sensitive to number through a nonverbal system, but that they are able to use this system to perform a rudimentary form of subtraction. The capacity to understand number symbols was demonstrated by Matsuzawa s successful training of a chimpanzee to recognize quantities of 1 to 6 objects and apply the appropriate Arabic numeral labels. This study raised the question of whether the human capacity to name numbers arises through a similar mapping of a non-verbal system to an Arabic symbol system. The notion that number-to-name mapping occurs in humans

14 4 is supported by Wynn, Bloom, and Chang s (2002) finding that such a non-verbal system exists within human infants. They trained five-month-old infants on sets of either two or four threeitem collections and found that looking times decreased as infants became habituated to the items. They then presented all infants with two collections of four objects and four collections of two objects. They found that infants who had habituated to looking at collections of four spent longer at test looking at the collections of two whereas those infants who had habituated to looking at two collections spent longer at test looking at collections of four. This was interpreted as evidence that infants extract quantity information from stimuli and that this capacity relies solely upon the number of objects that are salient to the infant. Thus, there is substantial evidence that non-verbal number-sensitive processes exist and that they effectively guide behavior across a number of organisms and tasks. Based on the types of findings described above, Gallistel and Gelman (1992) posited that the adult human understanding of number is learned by mapping a symbol-based system to this preverbal system. They posit that the foundation of numerical understanding is the preverbal system, and that this system is unavoidably accessed during numerical tasks even for tasks that are verbally-oriented. However, the representations that this preverbal representational system produces are theorized to be inherently inaccurate. The most relevant aspect of this model to the present study is that the preverbal system influences the time that it takes to discriminate between numbers because the comparates (the to-be-compared numbers) are represented not just by precise verbal representations, but by imprecise preverbal magnitudes as well. Dehaene s (1992; Dehaene & Akhavein, 1995; Dehaene, Bossini, & Giraux, 1993) triplecode model is similar to Gallistel and Gelman s (1992) preverbal representation model in many respects. However, it differs in the formatting that it posits for the number representations.

15 5 Instead of emphasizing an always-present preverbal mapping of verbal representations, the triple-code model emphasizes the different uses for each of three coding formats. The three codes of mental number representation auditory verbal word frame, visual Arabic number form, and analog magnitude are each tied to specific input and output codes, and each is emphasized to a varying degree depending upon the type of task that is being performed. The auditory word frame is posited to arise from general-purpose language processing modules that are used to handle the input of spoken and written language-based numbers. It is used for verbal counting and addition/multiplication table look-ups. The visual Arabic number form is posited to consist of Arabic numeral representations that extend spatially across the representational medium and are used for Arabic numeral reading and writing, as well as for parity and multi-digit operations. The analog magnitude representation is posited to represent quantity as distributions of activation over an analog number line. It receives input using the subitization and estimation of quantity and is used for comparison and approximate calculation tasks. Because many tasks require using attributes of multiple representations, it is posited that each format has an internal mapping to each of the other representational formats. For instance, the Arabic numeral input from a multiplication problem (e.g., 3 x 4) would be translated from Arabic numeral format into a verbal format; the verbal format would be used in a table look-up of the answer, which could then be sent to the auditory word frame for a verbal output. The relevance of the triple-code model to the present study is that the analog magnitudinal code is posited to play an important role in performing numerical comparison tasks. Specifically, Dehaene states that in number comparison, Arabic numeral input is translated into a magnitudinal representation before a comparison can be made. The triple-code

16 6 model provides a framework for understanding numerical cognition, but it is not sufficiently specific to provide testable predictions concerning the issues that are relevant to the present study of the mechanisms and comparison-dependence involved in the creation of the distance effect. Magnitudinal Representation within a Number-Comparison Task Number-comparison tasks involve subjects either classifying whether a single comparate is larger or smaller than a memorized numerical standard or selecting which of two comparates is larger or smaller in value. Number-comparison tasks have been shown to produce several magnitude-related effects, but the nature of these effects is determined in part by the type of task that gives rise to them (Dehaene, 1989). The two experimental tasks that are most commonly used for number comparison experiments are classification, in which a single number must be classified as being numerically greater or lesser than a memorized standard (e.g., Dehaene et al., 1998), and selection, in which the numerically greater or lesser number must be selected from a pair of numbers (e.g., Buckley & Gillman, 1974). The present study will deal solely with number-selection tasks because selection tasks (but not classification tasks) allow for the asynchronous stimulus presentation that is necessary for the present experimental manipulations (this point will be expanded upon later). To minimize the ambiguity given the task types used in the present study (one of which involves selecting a number but not comparison), the term number comparison will be used in the present study to refer to the number-selection task. One of the most interesting variables that influences performance in a numbercomparison task is semantic distance (see Dehaene, 1992, for a review of other factors that can affect number comparison performance). Semantic distance is usually defined as the numerical difference between the comparate digits. Moyer and Landauer (1967) found that as the semantic

17 7 distance between two simultaneously-presented digits increased, the time that it took subjects to determine the numerically larger of the two digits decreased. This led them to suggest that the representations of the digits were not verbal in nature but rather that they had been converted to analog magnitudes. As a variant on Moyer and Landauer s (1967) work, Brysbaert (1995; Experiment 3) conducted a two-digit number comparison experiment with SOAs (stimulus onset asynchronies) of 0, 200, 400, and 600 ms between the comparates. All subjects were to choose the smaller of the two numbers. Within decades (i.e., when the tens digit was the same for the first and second comparate), he found an inverse relationship between numerical distance and response time (RT), though no such relationship existed when there was a break in decades. No analysis was performed that addressed whether SOA moderated the numerical distance effect. The effects of semantic distance are apparent in both RTs and error rates. When experimental conditions result in different levels of task difficulty, then accuracy would be expected to change when subjects attempt to keep RTs constant across conditions, and RT would be expected to change when subjects attempt to keep accuracy constant across conditions. It is assumed that these effects are all due to an underlying mechanism that is able to trade speed for accuracy. Thus, both RT and error rate must be analyzed because effects may occur in either or both of the measures. Evidence that Semantic Distance Affects Task Performance Evidence of non-numerical semantic distance. Semantic distance is a general concept that can be found both inside and outside of the numerical cognition literature. Therefore, it is important to note how semantic distance affects task performance with non-numerical stimuli as

18 8 well as with numerical stimuli. Moyer (1973) and Paivio (1975) each found a semantic distance effect when they asked subjects to use the written names of common physical objects (e.g., bee or ant ) to mentally represent and then compare the physical sizes of those objects. Judgments of the size-differences between the objects were found, like with numerical magnitude, to be inversely related to the RT to compare them. Using a similar procedure, Holyoak (1977) gave subjects a task in which they were to visualize the size of a standard stimulus (a common object) as larger-than-normal, smaller-than-normal, or normal in size. Subjects then decided whether a subsequently-presented comparison stimulus was physically larger or smaller than the normalsized standard (even if they had visualized the standard as smaller as or larger than normal). Consistent with the distance effect, Holyoak found that RTs were influenced by the real-world size difference between the stimulus items that were to be visualized at their normal size. More interestingly, he found that subjects were faster to respond when they were instructed to visualize the standard stimulus (i.e., the first stimulus in the pair) at a normal size than when they were instructed to visualize it as either larger or smaller than normal size even if the standard image size that they had visualized was directionally-consistent with the correct response (e.g., when the standard image was visualized as larger than usual and it was subsequently judged, correctly, to be larger than the second stimulus). Thus, Holyoak found that semantic distance influenced task performance and that this influence was not susceptible to manipulations of the magnitudes of the mental stimuli. To test whether the distance effect occurs in other semantic contexts, Holyoak and Walker (1976) had subjects perform a comparison task that involved scales of time (e.g., seconds, days), quality (e.g., bad, good), and temperature (e.g., cold, hot). Half of the subjects were to choose which of the two words represented a shorter, worse, or colder concept. The

19 9 other half were to choose the word that represented a longer, better, or warmer concept. They found a semantic distance effect, e.g., that RTs were slower for comparing second to hour than for comparing second to day. These results suggest that representations of items on ordered non-numerical scales operate in a similar fashion to representations of items on numerical scales insofar as they appear to contain similar magnitude information. Also like numbers, the magnitudes of the non-numerical scalar items appeared to be represented monotonically (i.e., the magnitude of cold was always less than the magnitude of lukewarm which was always less than the magnitude of hot and these relationships never varied in order). Evidence of numerical semantic distance in probe-response tasks. A variety of methods have demonstrated effects of semantic distance on the performance of semantic association tasks by manipulating stimulus attributes along a magnitudinal dimension. For instance, Pavese and Umiltá (1998, 1999) found magnitude-to-numeral Stroop interference in a numeral counting task, and Tzelgov et al. (1992) found that physical size interfered with numerical magnitude in a task that involved comparing numerals of different physical sizes. Another important method for investigating the effect of numerical distance is to evaluate primed probe-responses. Such tasks typically involve the presentation of one stimulus (a prime) that does not require a response, followed by a second stimulus (a probe) that does require a response (but that does not require comparison to the prime). The difference in RTs (and/or error rates) between large prime-probe distances and small prime-probe distances is taken as evidence that distance effect processing influences primed probe-response tasks. Using a priming paradigm, Den Heyer and Briand (1986; also see Dehaene et al., 1998; Koechlin, Naccache, Block, & Dehaene, 1999; Naccache, Blandin, & Dehaene, 2002) found that

20 10 as the numerical distance between the prime and probe numbers increased, the amount of priming decreased. In other words, there was a negative relationship between distance and the capacity of one number to prime another number. Because priming shortens RTs, there is a positive relationship between distance and RT, and this is an opposite relationship to what is found in numerical comparison. Probe-response tasks also produce a positive relationship between error rate and distance again, opposite to what is found in numerical comparison. The effects of semantic distance on RT and error rate are, however, consistent with activation models that is, models that account for semantic effects as arising from the activation of nodes that represent semantic information. These are able to account for the numerical distance effect because when distance is small, probes have already been partially activated by the prime, therefore allowing for a rapid activation. But as prime-probe distance increases, the amount of activation decreases, thus lengthening RTs and/or increasing error rates. Den Heyer and Briand interpreted this effect of distance to indicate that the effect of activation decreases as a function of semantic distance along the number line. Marcel and Forrin (1974) generated similar results using a primed number-reading task. Their results (among other primed reading results, e.g., Brysbaert, 1995) suggest that even a task as simple as number reading is sensitive to priming via numerical magnitude. Marcel and Forrin interpret their results as supportive of a spreading activation model (see the section titled Augmented Buckley and Gillman Model, below). Modeling semantic distance in number-comparison tasks. Number-comparison tasks are another source of evidence of the influences of semantic distance. Many studies of these tasks suggest that distance-associated effects are due to the representation of numerical stimuli as mental analog magnitudes. In their seminal study, Moyer and Landauer (1967) concluded, from

21 11 the similarity of error rate and RT curves from a number-comparison task to the error rate and RT curves from a physical comparison task, that the displayed numerals are converted to analogue magnitudes, and the comparison is then made between these magnitudes in much the same way that comparisons are made between physical stimuli such as loudness or length of line (p. 1520). Note that the Moyer and Landauer analogy does not make mechanistic or representational assumptions about how such a comparison occurs, particularly with respect to when distance-effect-related processing is initiated and whether the representations that give rise to the effect are dynamic or static. The difference between dynamic and static representation models as they pertain to representational change (or lack thereof) is important for understanding the more general process that underlies the semantic distance effect. This is because such a distinction is central to defining the magnitude-sensitive component of the model. Dynamic representation models posit that the semantic distance effect is produced, at least in part, by a process that involves a progressive increase in the precision of the informational content of the semantic representation, across time. Static representation models posit that the semantic distance effect is wholly produced by processes that act upon semantic representations, and that the informational content of the representations is static across time. It should be noted that both models are dynamic in the sense that they are both process models, and thus both involve the processing of inputs to produce an output response. However, they differ in whether they posit that the magnitude representation is dynamic or static in nature. The dynamic/static characteristic is testable using a number-comparison task, wherein it can be addressed by whether processes that allow for the representation of a single number to change can influence the informational accuracy of that mental representation. In addition to allowing this style of testing, number-comparison tasks

22 12 have been attractive to researchers who are interested in determining the effect of general semantic distance because (a) numbers are symbolic and thus easy to measure, and (b) numbers are well-learned and contain relatively little representational variation between subjects, at least when compared with other types of semantically-ordered stimuli. Of equal importance to the question of static or dynamic representation is the question of whether distance effect processing is sensitive to possible representational processing that begins upon stimulus presentation (and is thus comparison-independent ), as one would expect with automatic activation of the numerical representations, or if distance effect processing only occurs during the comparison process (and is thus comparison-initiated ). The concept that comparison-independent representational change can moderate the distance effect obviously assumes that the representations change, and is thus incompatible with the assumptions of static representation models. However, it is not necessarily the case that all models that posit dynamic representation are comparison-independent. The question of comparison independence is addressed in the following experiments and thus the classifications of such representational models are described in detail in the following sections. An Accumulator Mechanism for Number Comparison An accumulator mechanism exists in two models that are described in detail in this section. In both models, an accumulator gathers information about the relative magnitudes of two comparates and enables a choice to be made about whether that information is sufficient to make a response about which of the comparates is higher or lower in value. Though the models have an accumulator mechanism in common, they employ different assumptions about number representation and timing dependencies.

23 13 The first accumulator-based model that will be discussed is the Buckley and Gillman (1974) model, which states that stimulus values are processed in a way that makes their representations functionally static (at least insofar as representational dynamics could influence the distance effect). The second accumulator-based model that will be discussed is an augmented version of the Buckley and Gillman model, which states that representations of numerical activation automatically narrow across time and that these are processed and used as inputs to the accumulator mechanism. These models also involve specific predictions that relate the process to the presentation method and the comparison process. Buckley and Gillman model. Buckley and Gillman (1974) posited an accumulator mechanism in their model of the processes that give rise to the numerical distance effect. Within their model, two assumptions are made about the representation of the comparate stimuli. First, the process that transforms the two comparate values from external stimuli into internal representations yields the logarithm of those comparate values. Buckley and Gillman s second assumption is that there is error in the process of translating these logarithmic values into the mental representations that are used within the comparison process. They do not specify the way this translation process operates, but they do specify that each number is representable by a random variable, and the distributions of random variables representing two simultaneously presented stimuli may overlap (p. 1134). They assume that the random variable is the logarithm of the stimulus value plus or minus a random undefined amount of error. Note that this process that produces the representations only operates upon a single pair of translated comparate values at any point in time (this corresponds to one iteration of sampling and adding to the accumulator). However, it is useful for modeling and prediction purposes to discuss the values of the comparate representations in terms of probability distributions (i.e., theoretical

24 14 distributions in which all intervals are assigned a probability) that are centered on the comparate logarithm and that have variances large enough for the comparates individual distributions to overlap, especially if the comparates are close in value. The Buckley and Gillman (1974) model accounts for the distance effect by applying an accumulator mechanism to the sampled pairs of comparate representations. After representation, the signed difference between the values of the representations is added to an accumulator (i.e., a mental space used to monitor the sum of the numbers entered for a given sampling period). If the accumulator exceeds a predetermined threshold value after the first comparate sampling process, then a decision is made as to which comparate is larger or smaller in value and a response is made. If the threshold value has not been exceeded, then another iteration of sampling from the stimulus comparates is performed and the difference of the representation values is again added to the accumulator, which then evaluates the accumulated value against the threshold value. This process repeats until the threshold value is exceeded. The number of iterations that are necessary to exceed the threshold value is posited to be positively related to RT thus creating the effect of many iterations requiring a long time and few iterations requiring a minimal amount of time. The distance effect is posited to be a result of this process because large distances require relatively few iterations (yielding short RTs) and small distances require relatively many iterations (yielding long RTs). See Figure 1 for a demonstration of how this model can account for the distance effect. Note that choosing a low threshold value in this model minimizes the number of iterations required to respond, but does so at the expense of increasing the risk of producing incorrect responses. Thus, by relating the threshold value to a speed or accuracy bias, this model is able to account for the positive relationship not only between distance and RT but between distance and error rate as well.

25 15 In the Buckley and Gillman (1974) model, the probability distributions of the representations are posited to remain static across time. The distance effect is explained as a strict consequence of the number of iterations required to reach a threshold value. Consequently, representational change can have no role in the creation of the distance effect. It is possible to test the assumption of static distributions by experimentally manipulating the time interval that separates the presentation of the two comparate stimuli. If the representations are considered to be probability distributions that are unable to change across time then presenting one comparate prior to the other would be expected to result in a distance effect that is no different than what is produced with simultaneous comparate presentation. Figure 2 shows the theoretical shape of the comparate probability distributions across time given the static representational characteristics of the Buckley and Gillman model operating on asynchronously-presented stimuli. Note that advance processing of a single comparate (i.e., processing it in a non-comparative way prior to the presentation of the second number) does not influence the representation (i.e., its probability distribution) at a later stage of processing. Also note that the model does not explicitly address the possibility that the asynchrony may affect the likelihood of the subjects forgetting the first representation. If forgetting is an issue when SOA is introduced into the number-comparison task then error rates should be high for the task, especially at long SOAs.

26 16 Iteration 6 Iteration 5 Iteration 4 Iteration 3 Iteration 2 Iteration 1 Distance = 1 Iterations = 6 Comparates: 5 4 Sample taken 4 3 Sample taken 4 4 Sample taken 3 5 Sample taken 5 3 Sample taken 6 4 Sample taken 5 4 Accumulator (Threshold = 4) Was 0 Change +1 Now +1 Accumulator (Threshold = 4) Was +1 Change 0 Now +1 Accumulator (Threshold = 4) Was +1 Change -2 Now -1 Accumulator (Threshold = 4) Was -1 Change +2 Now +1 Accumulator (Threshold = 4) Was +1 Change +2 Now +3 Accumulator (Threshold = 4) Was +3 Change +1 Now +4 Distance = 2 Iterations = 4 Comparates: 5 3 Sample taken 4 2 Sample taken 4 3 Sample taken 3 4 Sample taken 5 2 Accumulator (Threshold = 4) Was 0 Change +2 Now +2 Accumulator (Threshold = 4) Was +2 Change +1 Now +3 Accumulator (Threshold = 4) Was +3 Change -1 Now +2 Accumulator (Threshold = 4) Was +2 Change +3 Now +5 Distance = 3 Iterations = 2 Comparates: 5 2 Sample taken 4 1 Sample taken 4 2 Accumulator (Threshold = 4) Was 0 Change +3 Now +3 Accumulator (Threshold = 4) Was +3 Change +2 Now +5 Figure 1. Example of the creation of the distance effect within the Buckley and Gillman model. The distance effect is explained by the fact that the number of iterations (and, consequently, RT) decreases as distance increases. Note that the variation from the mean in each sample is identical across distances for the purpose of illustration.

27 17 Figure 2. Shape of the probability distributions across time within the Buckley and Gillman (1974) model, given a comparison task with an asynchronous presentation of stimuli. The top curve represents the distribution of the hypothetical mental representation upon presentation of the first comparate. The middle curve shows that the distribution remains static as time progresses. The bottom left curve shows that the first representation has not changed and that it is joined by the second comparate representation (the bottom right curve), which is identical to the first comparate in terms of narrowness. The Buckley and Gillman model contains a characteristic that has important implications for the present study: If the numerical distance between the comparates is held constant, then the number of necessary iterations will tend will have no reason to shift systematically from trial to trial. This leads to the prediction that manipulations designed to change the comparate representations would not influence the effect of distance. Augmented Buckley and Gillman model. Here, I propose an augmented form of the Buckley and Gillman model. It is similar to the original Buckley and Gillman (1974) model,

28 18 except that it posits that stimulus magnitudes are represented as activation distributions rather than as probability distributions, and that the variances of the distributions automatically decrease (i.e. the distributions narrow) over time. It should be noted that this distinction affects the parameters that are used by the accumulator but do not affect the operation of the accumulator itself. By positing such automatic activation narrowing, this augmented version of the Buckley and Gillman model is related to general activation models in ways that the original model is not. A discussion of general activation models will be presented followed by a specific description of the Augmented Buckley and Gillman model. A general activation model that has accounted for the effects that one item in memory has on the processing of other items is the spreading activation model of Collins and Loftus (1975), which has been used primarily to explain the semantic effects found in the processing of words. This model posits that the activation of one concept facilitates the activation of related concepts, but that this facilitation wanes as the concepts become distant from one another. This distancesensitive characteristic of the model provides an explanation of priming in that the presentation of a prime facilitates the processing of related targets and that the activation of these targets dissipates with semantic distance from the prime. This is supported by evidence that the RTs to targets that are more closely related to the prime are shorter than the RTs to targets that are less closely related to the prime. Numerical priming studies support this spreading activation model by finding that both consciously (Den Heyer & Briand, 1986) and unconsciously (Dehaene et al., 1998) presented number primes affect number probe responses more strongly at small than at large numerical distances. Because the Augmented Buckley and Gillman model is based upon evidence of automatic single-number priming, it predicts (within the context of a numericalcomparison task) that representational change necessarily occurs while a number is being

29 19 represented (even in the absence of a second comparate being presented), that such processing of a single comparate occurs automatically, and that the magnitude of the distance effect will be positively related to the amount of representational change that results from this processing. For word priming tasks, automatic target activation decreases as the time between the prime and probe increases (Neely, 1977). This may be due to the fact that subjects are not required to keep the prime active in working memory after presentation thus, the primed representation is allowed to decay. In contrast, number comparison studies suggest that number representations become more precise as the time between the prime and probe increases. Instead of overall decay being able to take place, it is necessary for semantic activation of the first comparate to remain active. However, it is not necessary for peripherally-related activations along the number line to remain active. Thus, word and number priming tasks make different representational demands, and subjects are therefore likely to handle representations within each task differently. The action of such a process in number-comparison tasks is implied by Gallistel and Gelman (1992). Instead of activation simply spreading across time, they posit that numerical semantic activation is best characterized by a narrowing effect, across time, after the initial activation. Although not explicit in their model (their focus was on the counting process), this attribute can be inferred from the Arabic numeral-to-magnitude conversion of the bidirectional mapping hypothesis. This model states that there is an internal symbol-to-magnitude mapping system that translates digits into corresponding preverbal magnitudes and that this system contains a speed-accuracy tradeoff such that representations that are formed in early processing have a larger variability than do the representations that occur later in processing. Thus, Gallistel and Gelman s conception of numerical representation is consistent with a model that posits that

30 20 the processes that give rise to dynamic numerical representations yield an increase, across time, in the informational precision of numerical representations. The Augmented Buckley and Gillman model posits that a progressive decrease of the variability of each distribution of comparate representations will influence the distance effect. Thus, the accumulator mechanism in the Augmented Buckley and Gillman model is similar to the accumulator mechanism in the Buckley and Gillman (1974) model, except that instead of the accumulator input consisting of values sampled from stimuli according to static numberprobability distributions, the input consists of values sampled from representations consisting of dynamic activation distributions. This narrowing of the activation distributions occurs in both comparate representations until a decision is made by the accumulator mechanism. In the Augmented Buckley and Gillman model, the accumulator mechanism and the process of representational narrowing both contribute to the distance effect. Thus, the major structural difference between the two models is in the static versus dynamic nature of the distributions that contain the samples to be used by the accumulator mechanism. It should be noted that although the theoretical distinction between the models relates to a characteristic of the representations used in the model, the empirical difference that is being tested is whether the distance effect can be influenced by manipulating variables designed to affect comparison-independent processes. In my Augmented Buckley and Gillman model, numbers are represented as a distribution of activation along the number line. Thus, a broad code (i.e., a code that covers a large area of the number line) is represented by a distribution with a large variance, whereas a narrower code (i.e., a code that covers a small area of the number line) is represented by a distribution with a small variance. Thus, not considering changes in variation due to activation narrowing, the overlap between the distributions should be negatively related to the distance between the

31 21 numbers. Like the Buckley and Gillman (1974) model, the Augmented Buckley and Gillman model operates under the assumption that the distributions can overlap, and that an accumulator is necessary to handle such situations. The Augmented Buckley and Gillman model contains a characteristic that has important implications for the present study: If the numerical distance between the comparates is held constant, then the number of necessary iterations tends to be negatively related to the degree of specificity (i.e., narrowness) of the comparate representations. This effect is posited to occur due to a reduction in the threshold value needed to trigger a comparison response (this is explained in detail in the Predictions section). Because the purpose of the threshold is to allow for toleration of imprecise comparate values during sampling, it is logical to assume that this threshold decreases when the sampling distribution of one of the comparates narrows (and, thus, sampling becomes more precise). This leads to the prediction that an experimental manipulation that improves the specificity of the representations should require less magnitude-specific processing (due to a lowered threshold) prior to the trigger of the decision and should thus diminish the effect of distance. The influence of distribution narrowing on the distance effect is posited to occur solely due to its expected influence on the threshold value (see Simulation of Accumulator Mechanism Models for a simulation of this phenomenon). This is because the narrowing of the first comparate probability distribution is posited to affect its variance but not its mean. No function of the accumulator links the narrowness of the distributions to a change the average signed value that is accumulated on a given iteration therefore the value that the accumulator samples from a narrow probability distribution would be equally likely to be above the mean as below the mean and that this has no relation to the narrowness of the probability distributions. Rather, the only

32 22 characteristic of the sampled values that would change between narrow and broad probability distributions is that values sampled from the narrow distribution would, on average, deviate from the mean less than values sampled from the broad distribution. This decreased deviation allows for a lower threshold to be used without an increase in the likelihood of producing an error, therefore decreasing the effect of distance. Figure 3 shows the progression of representational change across time given the automatic narrowing of the activation distributions (as is posited by the Augmented Buckley and Gillman model). Note that the advance processing of one of the comparate numbers is posited to have the effect of decreasing the variance of the distribution of that number s representation. Thus, advance processing is predicted to decrease the variance of one of the distributions, which then decreases the decision threshold. Decreasing the decision threshold decreases the number of iterations that are necessary for the accumulator to reach its threshold value, which then decreases the distance effect. This model attributes this moderation of the distance effect to the decreasing values and variances of the individual comparates activation distributions. Though this is slightly more complicated than the Buckley and Gillman (1974) model, the complications are reasonable due to the theoretical performance advantage yielded by the refinement of the decision-making information that the accumulator receives. Note, though, that the accumulator mechanism alone is sufficient to give rise to a distance effect, and that distribution narrowing is a separate additional influence that also may influence the distance effect. Thus, the distance effect is predicted by both the Buckley and Gillman model (which explains that the functioning of accumulator alone gives rise to the distance effect) as well as the Augmented Buckley and Gillman model (which explains that the distance effect is related to both an accumulator mechanism and the narrowing of distributions).