
FACILITATION AND INTERFERENCE COMPONENTS OF THE SIZE CONGRUITY EFFECT IN COLLEGE STUDENTS A Dissertation by TRINA GEYE Submitted to the Office of Graduate Studies of Texas A&M UniversityCommerce in partial fulfillment of the requirements for the degree of DOCTOR OF PHILOSOPHY December 2016 FACILITATION AND INTERFERENCE COMPONENTS OF THE SIZE CONGRUITY EFFECT IN COLLEGE STUDENTS A Dissertation by TRINA GEYE Approved by: Advisor: Tracy B. Henley Committee: Steven E. Ball Thomas J. Faulkenberry Raymond J. Green Head of Department: Jennifer L. Schroeder Dean of the College: Timothy D. Letzring Dean of Graduate Studies: Mary Beth Sampson iii Copyright © 2016 Trina Len Geye iv Dedicated to my grandmothers: Frankie Louise (Hicks) Geye Mary Alice (Reed) Chancellor v ABSTRACT FACILITATION AND INTERFERENCE COMPONENTS OF THE SIZE CONGRUITY EFFECT IN COLLEGE STUDENTS Trina Geye, PhD Texas A&M UniversityCommerce, 2016 Advisor: Tracy Henley, PhD The present study examined the facilitation and interference components of the size congruity effect in college students. In Experiment 1, computer mousetracking was used to capture the trajectories of responses during a numerical size congruity task in which participants were asked to select the physically larger of two digits. Response time and trajectory data supported the conclusion that numerical magnitude is automatically processed, and that the decision process continues through the motor response, consistent with the late interaction model (Faulkenberry, Cruise, Lavro, & Shaki, 2016; Santens & Verguts, 2011; Sobel, Puri, & Faulkenberry, 2016; Sobel, Puri, Faulkenberry, & Dague, in press). In Experiment 2, participants completed the Calculation Fluency Test (Sowinsky, Dunbar, & LeFevre, 2014) and the Brief Mathematics Assessment – 3 (Steiner & Ashcraft, 2012) as measures of math achievement and Form A of the abbreviated Standard Progressive Matrices test (Bilker et al., 2012) as a measure of fluid intelligence in addition to the experimental task used in Experiment 1. There were no significant findings related to fluid intelligence. Individuals with a previous diagnosis of learning vi disorder or attention deficit hyperactivity disorder had lower scores on measures of math achievement. Analysis of the size congruity effect replicated the findings of Experiment 1, again providing support for the automatic representation of numerical magnitude and the late interaction model. Neither facilitation nor interference was found to be significant in predicting math achievement. Facilitation and interference were evident in trajectory measures even when undiscernible in movement and initiation time, which demonstrates the value of computer mousetracking over response time alone in examining covert cognitive processes. vii ACKNOWLEDGEMENTS I extend heartfelt thanks to my committee chair, Dr. Tracy Henley, not only for the academic mentorship throughout this process, but also for knowing when to fan the flame and when to douse it. Thanks to Dr. Ray Green and Dr. Steve Ball for good naturedly supporting me through the maze of topic selection; I dare not guess how much time each of you spent with me discussing potential projects that never quite lifted off. To Dr. Tom Faulkenberry, without your relocation, this literally would not have happened. The opportunity to work with you provided me with the environment I needed for a “robust” dissertation experience, which has significantly influenced my growth as a researcher. I appreciate the support of my Tarleton family, especially Brenda Faulkner and the rest of the SSMI crew, for motivating me and keeping me accountable. Thanks to Liz Podany for falling in love with psychology with me; you will always be a part of my story. Haley Oyler, you have been my fiercest, most loyal friend since we were kids; thank you for being a better amiga than I deserve. The late Roger Hoon rooted for “Dr. T” from day one; I gave ‘em a little hell for you. Trisha and Scott, my siblings and two of my closest friends, you probably have no idea how many times you unknowingly talked me out of calling it quits. I also recognize that any good I have done in my life has been accomplished while standing on the shoulders of my parents. Thank you—all four of you—for having high expectations of me and for giving me every opportunity you possibly could to fulfill them. To my husband, Jake, and our boys, Gavin and Ian, thank you for your willingness to share me with my research. You made room in our family for this project, and chose to view it as an opportunity rather than an obligation. My cup runneth over. viii TABLE OF CONTENTS LIST OF TABLES ......................................................................................................................... x LIST OF FIGURES ...................................................................................................................... xi CHAPTER 1. INTRODUCTION .........................................................................................................1 Size Congruity Effect ..............................................................................................6 Individual Differences ............................................................................................9 Computer Mousetracking ......................................................................................11 2. EXPERIMENT 1 .........................................................................................................15 Method ..................................................................................................................16 Participants ................................................................................................16 Procedure ..................................................................................................16 Results ...................................................................................................................17 Response Time Analyses ..........................................................................18 Trajectory Analyses ..................................................................................19 Facilitation and Interference .....................................................................21 Discussion .............................................................................................................24 3. EXPERIMENT 2 .........................................................................................................26 Method ..................................................................................................................26 Participants ................................................................................................26 Materials and Procedure ...........................................................................26 Results ...................................................................................................................28 Response Time Analyses ..........................................................................29 ix Trajectory Analyses ..................................................................................31 Facilitation and Interference .....................................................................33 Predicting Math Achievement ..................................................................37 Discussion .............................................................................................................41 4. GENERAL DISCUSSION ..........................................................................................43 Future Directions ..................................................................................................46 Conclusions ...........................................................................................................47 REFERENCES .............................................................................................................................48 VITA ...........................................................................................................................................57 x LIST OF TABLES 1. Mean MT and IT by congruity, distance, and trajectory direction ...................................18 2. Mean AUC and MD by congruity, distance, and trajectory direction ..............................20 3. Mean MT and IT by congruity and trajectory direction (includes neutral) ......................22 4. Order of tasks ....................................................................................................................28 5. Mean MT and IT by congruity, distance, and trajectory direction ...................................29 6. Mean AUC and MD by congruity, distance, and trajectory direction ..............................32 7. Mean MT and IT by congruity and trajectory direction (includes neutral) ......................34 8. Mean math achievement and ASPM scores ......................................................................38 9. Correlation coefficients for math achievement by independent variable .........................39 10. Predictors of math achievement ........................................................................................40 11. Predictors of math achievement with age as interaction term ...........................................41 xi LIST OF FIGURES FIGURE 1. Mean MT as a function of condition and distance ............................................................19 2. Mean response trajectories by direction, congruity, and distance ....................................21 3. Mean MT by congruity .....................................................................................................23 4. Mean Response trajectories by congruity (with neutral) ..................................................24 5. Mean MT by congruity and distance ................................................................................30 6. Mean IT by diagnosis and distance ...................................................................................31 7. Mean response trajectories by direction, congruity, and distance ....................................33 8. Mean MT by congruity .....................................................................................................35 9. Mean IT by congruity and diagnosis ................................................................................36 10. Mean response trajectories by congruity (with neutral) ...................................................37 11. Predicted values with confidence band .............................................................................40 1 Chapter 1 INTRODUCTION The necessity for adults in the United States to have basic numeracy skills, or the ability to conceptualize and use numbers and mathematics, is incontrovertible. Economic opportunity and the capacity to engage in many aspects of daily life is negatively impacted for those individuals who cannot reason numerically at the level expected of eighth graders (Geary, Hoard, Nugent, & Bailey, 2013). The Organization for Economic Cooperation and Development’s (OECD) Survey of Adult Skills revealed that almost one third of the 5,010 individuals surveyed were unable to solve problems requiring multiple steps, discern frequently encountered decimals, fractions, and percentages, understand measurement and spatial representation, make estimations, and interpret tables, graphs, and data presented in text (OECD, 2013). The capability to evaluate the relative magnitude of symbols representing numerical values is fundamental to the development of numeracy (Girelli, Lucangeli, & Butterworth, 2000). Primates, both human and nonhuman, possess the capacity to comprehend and mentally manipulate approximate quantities. The mental representation of number is not a human invention, but appears to be the result of brain evolution (Dehaene, Piazza, Pinel, & Cohen, 2003). Several studies have provided evidence for the ability to differentiate quantities in various animal species. Rugani, Regolin, & Vallotigara (2008) found that newborn chicks will associate with the larger set of the object upon which they imprinted. Individual mosquito fish will choose to join a larger school, which is beneficial in the avoidance of predators (Agrillo, Dada, Serena, & Bisazza, 2009). Cantlon and Brannon (2007) found that monkeys will attend to object numerosity, even when it is irrelevant to the current task. 2 According to Gelman and Gallistel (1978), animals, including humans, form internal magnitude representations through a preverbal counting model, establishing a structure onto which external numerosities are mapped. Very early on, children understand the essential constraints on counting. Gallistel and Gelman (1992) describe these limits on counting in terms of five inherent rules: one to one correspondence, stable order, cardinality, abstraction, and order irrelevance. Repeated counting of stimuli in the external environment results in the development of a mental representation of the cardinal, or “how many,” associated with quantities. Humans, as opposed to their nonhuman counterparts, can also use this approximate number system for the representation of magnitudes associated with symbols (see Budgen, DeWind, & Brannon, 2016). Weber’s Law states that the just noticeable difference in a changing stimulus can be calculated as a constant ratio of the primary stimulus. Moyer and Landauer (1967) substantiated their hypothesis that numerical magnitude is processed in the same way as other psychophysical stimuli, such as length or pitch, and therefore subject to Weber’s Law, by replicating the results found in a line comparison task wherein participants are instructed to identify the longer of two presented lines. Reaction times were faster when the lines were comparatively small and when the disparity between the two lines was large. Moyer and Landauer similarly presented participants with two numerical digits and asked them to simply choose the digit with the larger value. As expected, participants identified the larger number more rapidly when the values were numerically farther apart. This finding has been termed the numerical distance effect and was an intriguing discovery because it was in opposition to then extant explanations of number comparison. The distance effect could not be predicted if number comparisons were made by counting from 1 to the smaller number, from one of the presented numbers, or via direct retrieval (Seigler & Opfer, 2003). The numerical distance effect is considered to be evidence that 3 numerals are converted into an analog representation of magnitude, much like other physical stimuli. Numbers seem to be represented in a domainspecific array according to cardinal magnitude as opposed to a more general way, such as, for example, according to frequency or association (Butterworth, Zorzi, Girelli, & Jonckheere, 2001). Restle (1970) describes a mental number line resulting from the ongoing association of external quantities with internal magnitude representations through practice as an analogy for the mental representation of magnitude associated with numbers. According to this model, numbers correspond to positions along a string, right to left, and a “next” marker associated with each number indicates directionality towards larger positions (Resnick, 1983). Dehaene, Bossini, and Giraux (1993) presented a mental number line model that preserved the features of the earlier version, but incorporated an explanation of differences in the level of activation. According to this model, numbers one through three are activated distinctly, or subitized. Subitizing is a process that allows for the quick and accurate assessment of numerosity without serial counting, and is possible only for small quantities (Atkinson, Campbell, & Francis, 1976; Feigenson, Dehaene, & Spelke, 2004). However, for four and beyond, magnitude representation becomes increasingly fuzzy, with overlapping Gaussian curves representing the range of activation. These overlaps in activation result in increasing interference, or concurrent partial activation, from other numbers. For example, when the number five is activated, four and six are also activated, although to a lesser degree, with the range of activation increasing with numerical value. Early examinations of numerosity in rats illustrate the intensity of activation according to the mental number line model. Mechner (1958) and Platt and Johnson (1971) created an environment in which rats were required to press one 4 lever a certain number of times, then press a separate lever once to release food. Attempts with too few presses would be punished, resulting in trial and error learning to discern the number of presses necessary to receive food. Even with considerable training, the number of presses remained imprecise; the number of presses for a target of four, for example, ranged from three to seven. As predicted by the mental number line model, the width of the response range increased with the target value, suggesting less precise activation of larger numerical magnitudes. A natural number line, along which numbers are spaced logarithmically (with lower values farther apart), is practical in the situations involving small magnitudes typically encountered by animals, such as when making decisions regarding foraging, because differences between two small numerosities can be quickly compared. Siegler and Opfer (2003) found that a logarithmictolinear shift occurs developmentally between kindergarten and second grade, allowing for use of a wider range of numbers. As they develop the mental number line, children first categorize small numbers into categories for comparison and eventually use this process in the comparison of larger numbers (Seigler & Robinson, 1982). This results in the adult classification of numbers as either "small" (less than five) or "large" (more than five), allowing for efficient memory retrieval (Tzelgov, Meyer, & Henik, 1992). The numerical distance effect, garnering robust support in the literature, was found to exist in the comparison of two digit numbers. Hinrichs, Yurko, & Hu (1981) displayed a number with two digits and instructed participants to judge it against a standard of either 50 or 55. Although the larger number was often identifiable using only the tens value, the numerical distance effect was still present; reaction times decreased with an increase in distance between compared numbers. The ones place was only ignored when the comparison standard was at a decade boundary (50) and the distance between the compared numbers was less than 10. These 5 results were replicated by Dehaene, Dupoux, and Mehler (1990) who interpreted them as support for the mental number line model, which explains the effect as a result of activation of overlapping ranges for two numbers located close together. Dehaene et al. (1990) asked participants to compare a presented number (between 1 and 99, excluding 55) to the reference of 55 using a button to make the selection of larger or smaller. Reaction times were faster on those trials when the button corresponding to smaller was on the left for choose smaller conditions, and faster when the button corresponding to larger was on the right for choose larger conditions. In a later study, Dehaene et al. (1993) instructed participants to indicate whether a number (from zero to nine) was even or odd. For smaller numbers (less than five) reaction times were faster when the corresponding response button was on the left. For larger numbers (more than five) reaction times were faster when the corresponding response button was on the right. In order to rule out association with a limb rather than spatial orientation, participants were asked to complete the task with their arms crossed, pressing the left button with the right hand and vice versa; the results were the same. This finding is known as the spatialnumerical association of response codes (SNARC) effect, and indicates that the mental number line is spatially oriented, with numbers increasing in value from left to right. The SNARC effect is influenced by several variables. The directional association of a digit is dependent upon its value in relation to other digits. In a range of zero to five, the number five is rightward associated, but is leftward associated in a range of five to nine (Dehaene et al., 1993). Negative numbers have been found to be leftward associated, while positive numbers are rightward associated (Fischer, 2003). The SNARC effect appears to be present in individuals who read and write in line with the Western tradition of left to right (Zebian, 2005). Shaki, Fischer, and Petrusic (2009) found the SNARC effect in monolingual participants who read from 6 left to right, and found a reverse SNARC effect in monolingual right to left readers. In the reverse SNARC effect, numbers representing larger magnitude are associated with the right, and numbers representing smaller magnitude are associated with the left. No reliable SNARC effect was present for participants who were able to read and write both types of languages. In a study in which participants were instructed to consider digits to be compared to either a ruler or to a clock, the SNARC effect was reversed on clock comparison trials (Bachtold, Baumuller, & Brugger, 1998). In addition to substantiation that numerical magnitude is represented along a spatially oriented mental number line, support that the processing of magnitude is irrepressible, or automatic, has emerged. Size Congruity Effect Automatic processes are those that occur without intention or effort across circumstances, therefore preserving limited cognitive resources, specifically working memory. Working memory briefly stores and processes information relevant to the current task. It has three components: the central executive, responsible for managing processes and allocating attention; the visuospatial sketchpad, which maintains visualspatial information; and the phonological loop, which maintains auditory information (Baddeley, 1996). Communication between the visuospatial sketchpad and phonological loop is facilitated by the episodic buffer (Baddeley, 2002). Spatial and temporal processes, such as those that permit humans to orient to the events in their environment, are automatic due to genetic predisposition, but other processes can become automatic with repetition (Hasher & Zacks, 1978). According to Girelli et al. (2000), automatization of the processing of numerical magnitude is a result of practice in children, allowing for cognitive resources to be efficiently allocated as numerical ability progresses. 7 The hallmark method of measuring task irrelevant processing in order to discern automaticity was introduced by Stroop (1935). Participants were presented with the names of colors presented in various text colors and instructed to simply verbalize the color of the text in which the word was presented for 100 color words. Participants took longer to complete this task than when asked to simply name the color of 100 boxes. This suggests that the semantic meaning of the word was processed even though it was not relevant to the task at hand. Posner (1978) suggests that this Stroop effect is comprised of two components: facilitation and inhibition. Facilitation refers to the beneficial processing of an irrelevant but congruent stimulus feature to improve response over neutral conditions, while inhibition represents the extent to which the processing of an irrelevant and incongruent stimulus feature is detrimental to response when compared to neutral conditions. In an argument against equating tasks with processes, Jacoby (1991) posits that automaticity, which he synonymizes with familiarity, is revealed by facilitation, whereas interference is impacted by the division of attention in an intentional process. Henik and Tzelgov (1982) sought to explore how irrelevant stimulus dimensions are processed in a comparison of two dimensional (physical size and numerical value) stimuli created using Arabic numerals 1–9 and manipulating the congruity between physical size feature dimension and numerical value dimension. Congruent trials had a target that was both physically and numerically larger (5 3), incongruent trials had a target that was numerically smaller (5 3), and neutral trials had digits varying only on the designated dimension (e.g., 5 3 for a numerical comparison task; 5 5 for the physical comparison task). Participants viewed these digit pairs on a computer monitor, and were asked to select which number was larger on the relevant dimension (physical size or numerical 8 value). Among their results was the finding that congruence affects performance; reaction times were quicker on trials containing congruent pairs than on trials containing incongruent pairs, quicker on trials containing congruent than neutral trials in numerical value comparisons, and slower on trials containing incongruent than neutral trials. Additionally, the differences in reaction time increased with the distance between the numerical values of digits presented for comparison. This finding of the size congruity effect (Paivio, 1975) in numerical comparison tasks suggests that physical and numerical size comparisons occur in parallel, or at the same time, as opposed to serially, or one at a time. Also, the influence of irrelevant dimension of numerical value on physical size decisions provides evidence that the processing of numerical magnitude occurs automatically. Various models have emerged to explain at what point during processing this interference from the irrelevant stimulus feature happens. The early interaction model (Schwarz & Heinze, 1998) posits that relevant and irrelevant features are processed concurrently prior to the response stage, with information from both features mapped onto a shared analog representation. The decision is then based upon a single representation, with any interference from the irrelevant feature occurring prior to the motor response. As such, congruent stimulus features resolve into a single representation faster than incongruent stimulus features, causing the size congruity effect. In the late interaction model (Faulkenberry et al., 2016; Santens & Verguts, 2011; Sobel et al., 2016; Sobel et al., in press), relevant and irrelevant features are processed in parallel but independent channels, resulting in competing responses; interference from the irrelevant stimulus features happens during the response stage (Schwarz & Heinze, 1998). In an expansion upon the late interaction model, the Shared Decisions Account proposes that the resultant benefit of congruity and detriment of incongruity results from competing motor responses activated by 9 opposing response codes (Santens & Verguts, 2011). For a congruent stimulus pair, such as 3 7, the “right larger” response code is activated by both the relevant and irrelevant feature. For in incongruent stimulus pair, such as 3 7, “left larger” and “right larger” response codes are activated simultaneously, and the resulting competition between motor responses manifests as the size congruity effect. Individual Differences The ability to process numerical magnitude is critical to the development of proficient numeracy (Henik, Rubinsten, & Ashkenazi, 2011), and, consequently, provides a window into individual differences in numeracy. In order to examine the processing of numerical magnitude as it relates to mathematical achievement in individuals exhibiting diminished math ability, Holloway and Ansari (2009) asked a group of six to eight year old children to complete number comparison tasks. Mathematical ability was determined using scores on relevant subtests of the WoodcockJohnson III Tests of Achievement. In a symbolic condition, the children were directed to choose the larger of two Arabic numerals (one to nine); in a nonsymbolic condition, they selected the larger of two arrays of squares. There was a negative correlation between scores for mathematical fluency and the numerical distance effect as measured by the difference in response times between small and large numerical distances. Participants with lower mathematical fluency had a greater difference in response times between the two conditions, suggesting a noisier mapping of magnitudes onto respective Arabic numerals. However, Rousselle and Noel (2007) found a larger numerical distance effect in individuals with higher math achievement in a study using mathematics learning disability, not simply low achievement, as a factor. Holloway and Ansari (2009) reconcile the studies by considering them a demonstration of the differences between individuals who have a diminished efficiency in the 10 mapping of symbols and magnitudes versus those who have a domain specific deficit requiring alternative strategies. In addition to the representation of numerical magnitude, individual differences have also been identified in the ability to process numerical magnitude automatically as evinced in the size congruity effect. To distinguish between the facilitation and interference components comprising the size congruity effect in individuals with mathematics learning disability, Rubinsten and Henik (2005, 2006) incorporated the use of a neutral stimulus. On congruent trials, the number in the larger font point size was also the numerically larger (5 3), while on incongruent trials, the digit in the bigger font point size was the numerically smaller (5 3). Neutral trials were those in which the presented digits varied only in the relevant dimension (5 5). Facilitation of the response was determined by contrasting the reaction times on congruent trials against those on the neutral trials, and interference was determined by contrasting the reaction times on incongruent trials against those on the neutral trials. Interference was found in both the mathematics learning disability group and the control group, but facilitation occurred only in the control group. In other words, congruency did not speed up reaction times for individuals with low math ability. The results were interpreted to suggest that individuals with mathematics learning disability did not automatically process numerical magnitude, and were consistent with findings in individuals with acquired difficulties in mathematics (Ashkenazi, Henik, Ifergane, & Shelef, 2008). The diminished size congruity effect in individuals with mathematics learning disability cannot be presumed to result from domain specific deficits in mathematics, and may be related to domain general differences in processing. Risko, Maloney, and Fugelsang (2013) posit that attention may play a more significant role in the size congruity effect than explained by existing 11 models based upon their findings that the temporal onset of stimuli can reduce the interference. Further evidence that attention plays a role in the size congruity effect was presented by Sobel et al. (2016), who found that manipulating the top down processing of the stimuli impacted the size congruity effect. Ashkenazi, Rubinsten, and Henik (2009), interested in the influence of attention on the size congruity effect, designed an experiment in which participants, again assigned to groups based upon mathematical ability, were presented with a secondary stimulus in addition to a numerical and physical size comparison task. Two partial circles were concurrently depicted on the peripheries of the display for all conditions. In load conditions, participants were asked to report whether or not the two partial circles were the same. The noload condition, in which the partial circles were present but no comparison was required, replicated previous findings for physical size comparison, with the interference component found in both groups, but facilitation only in the control group. Load removed the facilitation effect in both groups in the physical size comparison task, while load increased interference in the numerical comparison task for both groups, but only for participants with low math ability during the physical size comparison task. The authors posit that individuals with low math ability have difficulty recruiting attention in numerical tasks, and that the lack of findings related to load and inhibition point to a maxing out of cognitive resources in attending to the task at hand. Computer Mousetracking Previous studies have relied on performance measures, such as accuracy and response time, to explore the automatic representation of numerical magnitude. However, performance measures do not allow for the capture of dynamic data, which is more useful in exploring questions such as those prompted by debate between the early and late interaction models (Faulkenberry et al., 2016; Santens & Verguts, 2011; Schwarz & Heinze 1998; Sobel et al., 12 2016; Sobel et al., in press). Computer mousetracking provides a unique method of collecting data that allows for the study of covert cognitive processes. In an analysis of spoken word recognition, participants were directed to listen to an audibly presented word, and then select between two pictures the one corresponding with what was heard (Spivey, Grosjean, & Knoblich, 2005). This selection task was completed using a computer mouse, which allowed for the tracking of hand trajectories through the capture of mouse position coordinates during the motor response. Analysis of the trajectories revealed that on trials in which the two presented words were phonetically similar, the response pattern veered in the direction of the incorrect phonological competitor. This finding was interpreted as online evidence of two competing mental representations during a motor response. Subsequent to the development of the MouseTracker software introduced by Freeman and Ambady (2010), computer mousetracking has been recognized by Freeman, Dale, and Farmer (2011) as offering a window into realtime cognitive processes via the association between mental and motor dynamics. This method of hand tracking has been fruitfully employed in the study of numerical cognition in a number of studies (Faulkenberry, 2014; Marghetis & Núñez, 2013; Marghetis, Núñez, & Bergen, 2014; Santens, Goossens, & Verguts, 2011; Song & Nakayama, 2008). Although computer mousetracking has engendered criticism, the opportunity to examine previously concealed information about competing processes has been valuable. Fischer & Hartmann (2014) raised legitimate concerns about potential differences in apparatus and settings that can be resolved through the specific reporting of, for example, mouse settings. In addition to these simply mediated criticisms, however, the authors claimed that interference from a distractor is only evident in those cases in which the computer mouse is moved completely onto the side of the computer screen associated with the incorrect response. 13 Faulkenberry and Rey (2014) responded to these concerns with support from the continuous cognition framework (Spivey, 2007), indicating that any movement towards the incorrect response can be interpreted as interference, even accounting for task difficulty. Computer mousetracking has been recently applied to the examination of the size congruity effect. Faulkenberry et al. (2016) found that response trajectories were pulled towards the incorrect response, with greater competition from the irrelevant feature evinced in the difference between the actual and ideal trajectories in incongruent trials. These results indicate that the competition between stimuli continues through the response process, lending support to the late interaction model. Similarly, data from an unpublished manuscript (Geye, 2015) reveals greater curvature towards the incorrect response, regardless of congruity, in individuals with low math achievement, echoing previous interest in the role played by attention in the size congruity effect (Ashkenazi et al., 2009). This finding was particularly interesting because there were not any significant differences in response times between groups. Computer mousetracking has also been used to examine the classic Stroop effect; Yamamoto, Incera, and McLennan (2016) found interference by comparing incongruent to neutral trials in the examination of trajectories in a task requiring participants to click on the color word corresponding to the presented stimulus. The dynamic data available via computer mousetracking provides an opportunity to refine the lens through which basic numerical processes are examined. Kaufman et al. (2013) raises the concern that individual differences in basic numerical abilities may be overlooked using current methods. A fine grained examination of dynamic responses may be an instrumental tool in teasing out discrepancies not evident in performance data, such as response time, and inform ongoing efforts to develop the assessment of numeracy through noncurricular means. 14 The intention of this project was to explore the facilitation and interference elements of the size congruity effect using the trajectory data available through computer mousetracking, and to investigate the relationship of the components to overall math achievement. In Experiment 1, which served as a calibration, participants completed a comparison task typical to size congruity effect research. A neutral stimulus was introduced, allowing for the differentiation between facilitation and interference components. I predicted that trajectories would be more complex in incongruent trials than congruent, with greater difference between the ideal and actual trajectories as measured by area under the curve, maximum deviation, and movement time. I also hypothesized that facilitation would be evident in the comparison of congruent and neutral trials, with lower area under the curve, maximum deviation, and movement time in congruent than neutral trials. Additionally, I predicted that, due to interference, incongruent trials would have greater area under the curve, maximum deviation, and movement time than neutral trials. In the second experiment, math achievement was introduced as a variable. I hypothesized that math achievement would increase in relation to facilitation, as measured by the difference in area under the curve, maximum deviation, and movement time between congruent and neutral trials. 15 Chapter 2 EXPERIMENT 1 In Experiment 1, I sought to investigate the size congruity effect in college students using computer mousetracking during a physical size comparison task with numerical digits. Additionally, as the first application of computer mousetracking in the examination of the size congruity effect using neutral trials, it served as a calibration in which to test the stimuli prior to Experiment 2. The size congruity effect is consistently supported in the literature (Ashkenazi et al., 2009; Henik & Tzelgov, 1982), and results related to response times have been echoed using computer mouse trajectory data (Faulkenberry, Cruise, & Shaki, 2016; Geye, 2015), providing support for the late interaction account (Santens & Verguts, 2011). I predicted that movement times would be faster for congruent than incongruent trials, and that computer mouse trajectory, as measured by area under the curve (AUC) and maximum deviation (MD), would be correspondingly less complex. Digits are easier to compare when numerically far apart (Moyer & Landauer, 1967), and, correspondingly, irrelevant magnitudes seem to be more difficult to suppress with increased distance. Therefore, I expected the differences between the congruent and incongruent trials to increase with numerical distance, which is consistent with previous findings (Faulkenberry et al., 2016). Because Rubenstein and Henik (2005, 2006) were able to discern the facilitation and interference components through the comparison of response times on congruent and incongruent trials to neutral trials, I predicted that the comparison of trajectory data would yield similar results. Specifically, I hypothesized that participants would exhibit faster, less complex responses on congruent trials than neutral trials due to facilitation, and slower, more complex responses on incongruent than neutral trials due to interference. 16 Method Participants Thirty students attending a midsized public university in the Southwest were recruited for participation. The mean age of the participants was 21.7 (SD = 3.76), with an age range of 1830. Most of the participants were enrolled in undergraduate psychology or first year seminar courses for which credit was awarded for research participation. All of the participants were right handed, and two reported having previously received a diagnosis of attention deficit hyperactivity disorder (ADHD) or specific learning disorder. Procedure Each participant completed the experimental task in a quiet laboratory environment in an approximately 15 minute timeframe by appointment. Each participant was informed that the study’s purpose was to examine how college students think about numbers. They signed a hard copy informed consent form and completed a brief questionnaire containing demographic questions. For the experimental task, a series of simultaneously displayed pairs of single digit numbers were shown on a 20 inch iMac desktop monitor with a 1280 x 1024 pixel screen resolution in the upper right hand and upper left hand corners. The pairs were created using the Arabic numerals 1, 2, 8, 9, which were displayed in Arial font in point sizes 22 for small trials and 28 for large trials. Two numerical distance categories were created: close (1 – 2, 8 – 9) and far (1 – 8, 2 – 8, 1 – 9, 2 – 9). Congruent trials were those in which the correct response (larger physical size) was also the numerically larger digit (e.g., 2 8), and incongruent trials were those in which the correct response was the numerically smaller digit (e.g., 2 8). Neutral trials were generated using digits differing only in the relevant feature, physical size (e.g., 2 2, 8 8). Each of 17 the six potential pairings appeared in a randomized order eight times — two congruent and two incongruent for each correct answer trajectory (right or left) — in each of the 5 blocks. Additionally, each block contained 3 randomized presentations of each of the neutral pairs, resulting in 360 stimuli. MouseTracker was used for both stimulus presentation and data collection. It is a software program designed by Freeman and Ambady (2010) and it is offered for free download at http://mousetracker.jbfreeman.net. I instructed those participating in the study to quickly select the digit that was larger in physical size, disregarding the value of the numbers, using a computer mouse. In order to activate each trial, a START button in the bottom center of the screen was clicked. Responses initiated after 400 ms were followed by warning, and incorrect responses were followed by a 2000 ms display of an “X.” Seventy x and ycoordinate pairs along the mouse trajectory were recorded each second. Results Participants completed a combined total of 10,800 trials. Fifty two (0.48%) of the trials were excluded due to the selection of the incorrect response, as were 157 (0.15%) with response speeds beyond 3 standard deviations from the mean for correct selections. R, which is freely available for download at www.Rproject.org, was used for subsequent analyses on the remaining 10,591 trials MouseTracker calculates two response time measures: initiation time (IT) and reaction time (RT). IT is the amount of time taken to begin moving the mouse after the presentation of a stimulus, and RT is the total amount of time taken to complete the response. Because RT includes IT, the time spent in motion towards the correct response was calculated by subtracting IT from RT for each trial to obtain the movement time (MT). For analysis, both MT and IT were 18 aggregated and averaged by condition for each participant (see Table 1). Neutral trials, or those in which differing sizes of the same numerical digit were presented, were excluded in any analyses using numerical distance as a factor. Table 1 Mean movement times (MT) and initiation times (IT), in milliseconds, by congruity, distance, and trajectory direction. Leftward Trajectories Rightward Trajectories Numerical Distance Close Far Close Far Congruent MT 1181 1183 1195 1192 IT 128 128 132 123 Incongruent MT 1214 1213 1206 1241 IT 132 136 139 129 Response Time Analyses A 2 (physicalnumerical size congruity: congruent vs. incongruent) x 2 (numerical distance: close vs. far) x 2 (trajectory direction: leftward vs. rightward) repeated measures analysis of variance was conducted for MT. A significant main effect of physicalnumerical size congruity revealed that participants completed movement on congruent trials 31 ms faster than incongruent trials, F(1,29) = 9.83, p = .004, ηp2 = .25 (see Figure 1). No significant main effect of numerical distance emerged, F(1,29) = 1.85, p = .18, or trajectory direction, F(1,29) = 1.02, p = .32, and no significant interaction of congruity and numerical distance, F(1,29) = 1.76, p = .20, or congruity and trajectory direction, F(1,29) = .002, p = .97. 19 A 2 (physicalnumerical size congruity: congruent vs. incongruent) x 2 (numerical distance: close vs. far) x 2 (trajectory direction: leftward vs. rightward) repeated measures analysis of variance was conducted for IT. There were no significant main effects of congruity, F(1,29) = 2.66, p = .11, numerical distance, F(1,29) = 2.41, p=.13, or trajectory direction, F(1,29) = .04, p = .85. There were no significant interactions of congruity and numerical distance, F(1,29) = .10, p = .75, or congruity and trajectory direction, F(1,29) = 5.8, p = .92. Figure 1. Mean MT as a function of condition and distance. Trajectory Analyses In order to compare trajectories of various lengths, MouseTracker normalizes response measures by creating x and y coordinates using 100 default time steps. The software provides two measures of trajectory complexity: area under the curve (AUC) and maximum deviation (MD). AUC is a geometric measure of the difference in area between the ideal, straight line 20 trajectory between the starting point and the correct response and the actual trajectory of the motor response during selection. MD is the longest perpendicular distance connecting the path of the actual trajectory and the perfect, straight line trajectory. Table 2 contains mean AUC and MD values. 21 Table 2 Mean AUC and MD by congruity, distance, and trajectory direction. Leftward Trajectories Rightward Trajectories Numerical Distance Close Far Close Far Congruent AUC .65 .56 .65 .58 MD .36 .33 .32 .30 Incongruent AUC .86 .93 .79 .90 MD .45 .48 .40 .44 I conducted a 2 (physicalnumerical size congruity: congruent vs. incongruent) x 2 (numerical distance: close vs. far) x 2 (trajectory direction: leftward vs. rightward) repeated measures analysis of variance for AUC using each participants average AUC value. There was a significant main effect of congruity, with larger mean AUC values for incongruent than congruent trials, F(1, 29) = 27.93, p < .001, ηp2 = .49. There was no significant main effect of distance, F(1, 29) = .004, p = .95, or trajectory direction, F(1,29) = .09, p = .76. The interaction between congruity and distance was significant, F(1,29) = 5.28, p = .03, ηp2 = .15, with larger differences in AUC on congruent and incongruent trials with increased numerical distance (see Figure 2). A 2 (physicalnumerical size congruity: congruent vs. incongruent) x 2 (numerical distance: close vs. far) x 2 (trajectory direction: leftward vs. rightward) repeated measures analysis of variance was conducted for MD using the average values for each participant. As with AUC, a significant main effect of congruity was found, with larger MD for incongruent than congruent trials, F(1,29) = 37.13, p < .001, ηp2 = .56. There was no significant main effect of distance F(1,29) = .05, p = .83, or trajectory direction, F(1,29) = 1.35, p = .26. There was a 22 marginally significant interaction between congruity and distance, with larger MD on far trials, F(1,29) = 4.88, p = .04, ηp2 = .14. Figure 2. Mean response trajectories by trajectory direction, congruity, and numerical distance Facilitation and Interference In order to differentiate the contribution of facilitation and interference to the size congruity effect found in the initial analyses, analyses including the neutral trials were conducted. Table 3 contains the means for all response time and trajectory measures by congruity and trajectory direction. 23 Table 3 Mean movement times (MT) and initiation times (IT), in milliseconds, by congruity and trajectory direction. Congruent Neutral Incongruent Trajectory Direction Left Right Left Right Left Right MT (ms) 1182 1193 1189 1191 1213 1229 IT (ms) 128 126 132 129 134 132 AUC .59 .60 .75 .67 .91 .86 MD .34 .31 .41 .34 .47 .43 Response Time. A 3 (physicalnumerical size congruity: congruent vs. neutral vs. incongruent) x 2 (trajectory direction: leftward vs. rightward) repeated measures analysis of variance was conducted for MT. There was an overall main effect of congruity on MT, F(2,58) = 10.34, p < .001, ηp2 = .26. Planned contrasts among congruity conditions revealed a significant difference in mean MT between incongruent versus neutral trials, F(1, 58) = 20.645, p <.001, ηp2 =.26, but not between congruent versus neutral trials, F(1, 58) = .034, p = .85. Participants completed congruent trials the fastest, followed by neutral and incongruent, respectively (see Figure 3). There was no significant main effect of trajectory direction, F(2,58) = .63, p = .54. A 3 (physicalnumerical size congruity: congruent vs. neutral vs. incongruent) x 2 (trajectory direction: leftward vs. rightward) repeated measures analysis of variance was conducted for IT. There was no overall main effect of congruity, F(2,58) = 2.35, p = .10, or trajectory direction, F(2,58) = .004, p = .99. 24 Figure 3. Mean MT by congruity. Trajectory. A 3 (physicalnumerical size congruity: congruent vs. neutral vs. incongruent) x 2 (trajectory direction: leftward vs. rightward) repeated measures analysis of variance was conducted for AUC. A significant overall effect of congruity for AUC emerged, F(2,58) = 29.61, p < .001, ηp2 = .51 (see Figure 4). Planned contrasts revealed a significant difference in AUC between congruent versus neutral trials, F(1,58) = 8.96, p = .004, ηp2 = .13, and between incongruent versus neutral trials, F(1,58) = 50.23, p < .001, ηp2 = .46. No significant main effect of trajectory direction emerged, F(2,58) = 1.55, p = .22. 25 Figure 4. Mean response trajectories as a function of congruity. A 3 (physicalnumerical size congruity: congruent vs. neutral vs. incongruent) x 2 (trajectory direction: leftward vs. rightward) repeated measures analysis of variance was conducted for MD. There was a significant overall effect of congruity, F(2,58) = 38.28, p < .001, ηp2 = .57. Planned contrasts revealed significant differences between congruent versus neutral trials, F(1,58) = 13.56, p < .001, ηp2 = .19, and incongruent versus neutral trials, F(1,58) = 62.99, p < .001, ηp2 = .52. There was no significant main effect of trajectory direction, F(2,58) = 1.12, p = .334. Discussion The results provide confirmation for the predictions in Experiment 1. Participants completed the response motion faster for congruent than incongruent trials, and exhibited less pull towards the distractor as measured by AUC and MD. The difference between congruent and 26 incongruent trajectories also increased as a function of distance, suggesting that there is more distinct activation of the numerically greater digit’s associated magnitude when the compared digits are farther apart. This is consistent with Moyer and Landauer’s (1967) numerical distance effect. With the introduction of the neutral trials, facilitation was evident in AUC and MD, and interference was evident in MT, AUC, and MD. It is interesting to note that had the trajectory data not been available, I would not have been able to identify facilitation using only the performance measure data. In addition to providing additional evidence of the facilitation and interference components of the size congruity effect, these findings also support the late interaction model (Faulkenberry et al., 2016; Santens & Verguts, 2011; Sobel et al., 2016; Sobel et al., in press). As is evident in Figures 2 and 4, the response trajectories are smooth, without sudden corrective movements. The trajectory analyses provide evidence that competition continues throughout the motor response, which cannot be explained by an early interaction account in which the decision is made prior to physical selection. 27 Chapter 3 EXPERIMENT 2 The goal of Experiment 2 was to replicate the findings of Experiment 1, and to explore the value of facilitation and interference in predicting individual differences in math achievement. I predicted that the comparison of congruent and incongruent trials would reveal the size congruity effect, and that the differences would increase with numerical distance. I hypothesized that facilitation and interference would be evident in the measures of movement time and trajectory when comparisons were made among congruent, neutral, and incongruent trials. I also predicted that facilitation, or the beneficial effect of congruity on responses, would be associated with higher math achievement score. This prediction is consistent with Rubinsten and Henik (2005, 2006), who did not find facilitation in individuals with lower mathematical fluency. Method Participants As in Experiment 1, students were recruited primarily from classes at Tarleton State University for which research participation is a requirement or extra credit option. Seventy participants (76% female), with a mean age 24 years (SD = 9.8) and range of 1856, completed the experimental task and the additional measures in a 3040 minute individual session. A historical diagnosis of either ADHD or a specific learning disorder was reported by 8 participants, with 3 indicating multiple diagnoses, and 84% were right handed. Materials and Procedure Sowinski, Dunbar, & LeFevre’s (2014) Calculation Fluency Test (CFT) and Steiner and Ashcraft’s (2012) Brief Mathematics Assessment – 3 (BMA) were used to assess math 28 achievement. The CFT is comprised of 60 each of three problem types: one page of two digit addition, one page of two digit subtraction, and one page of two digit by onedigit multiplication problems. Participants were instructed to complete as many as they correctly could on each page in one minute without skipping any problems. The CFT was scored by counting correct responses for each problem type. The BMA contains 10 problems obtained from the Wide Range Achievement Test: Third Edition (WRAT3; Jastak Associates, 1993), ranging from mathematical operations and fractions to algebraic factoring. Participants were given 10 minutes to complete as many problems as they could, in order, and were instructed to stop if they reached one they were unable to solve. The BMA is also scored by simply counting correct responses (mean score 6.3). Scores on the CFT and the BMA had high internal consistency (Cronbach’s alpha = .81), so a single math achievement score was calculated for each participant. For each participant, the scores for both the CFT and the BMA were transformed to zscores; the resulting zscores were averaged in order to obtain a single value for math achievement. Form A of the abbreviated Standard Progressive Matrices (ASPM) test (Bilker et al., 2012) was used as a measure of fluid intelligence. It contains 9 items from the original Standard Progressive Matrices (Raven, Raven, & Court, 2000), each of which requires that participants identify the missing piece of an image from the presented options. The raw score for each participant was obtained by counting the correct responses (mean score 6.5). To control for order effects, a balanced Latin square design was used to identify 4 possible task sequences (A, B, C, D; see Table 4). Twenty packets of materials were created for each of the possible sequences and a random number was assigned to each in a spreadsheet. The spreadsheet was then sorted based on the random number, and a participant number was assigned 29 based upon ordinal position. A sort based on task order allowed for the assignment of participant numbers to the 4 sets of packets, which were then combined and ordered by participant number. Table 4 Order of tasks for each of the sequence groups (A, B, C, D). First Task Second Task Third Task Fourth Task A CFT BMA ASPM Experiment B BMA Experiment CFT ASPM C Experiment ASPM BMA CFT D ASPM CFT Experiment BMA The procedure for the experimental task in Experiment 2 was the same as that used in Experiment 1. The stimuli were created using the Arabic numerals 1, 2, 8, and 9 presented in pairs in differing font point sizes: 22 (small) and 28 (large). Numerical distance was manipulated by creating close (1 – 2, 8 – 9) and far (1 – 8, 2 – 8, 1 – 9, 2 – 9) pairs. Each possible pair was presented twice for each congruity condition (congruent, incongruent, and neutral) for each trajectory direction (right or left) in 5 randomized blocks. With the 3 randomized presentation of each neutral pair per block, each participant was presented with a total of 360 trials. Results Seventy participants completed a combined total of 25,200 trials. One participant responded incorrectly on all incongruent trials with leftward trajectories and far numerical distance, and was excluded from analysis. Ninetynine (0.40%) of the remaining 24,840 trials were excluded due to the selection of the incorrect response, as were 165 (0.66%) with reaction times beyond 3 standard deviations from the mean for correct selections. Subsequent analyses were performed on the remaining 24,576 trials. 30 Response Time Analyses A 2 (physicalnumerical size congruity: congruent vs. incongruent) x 2 (numerical distance: close vs. far) x 2 (trajectory direction: leftward vs. rightward) x 2 (sex: female vs. male) x 2 (history of diagnosis: diagnosis vs. no diagnosis) analysis of variance was conducted for MT and IT. Table 5 contains the mean MT and IT values. Table 5 Mean movement times (MT) and initiation times (IT), in milliseconds, by congruity, distance, and trajectory direction Leftward Trajectories Rightward Trajectories Numerical Distance Close Far Close Far Congruent MT 1202 1200 1237 1207 IT 125 121 126 125 Incongruent MT 1226 1259 1261 1286 IT 126 124 122 124 For MT, there were no between subjects main effects of sex, F(1,65) = .34, p = .56, or diagnosis, F(1,65) = 3.46, p = .07. A significant main effect of physicalnumerical size congruity was found, with completion of movement on congruent trials 46 ms faster than incongruent trials, F(1,65) = 56.97, p < .001, ηp2 = .47 (see Figure 5). The main effect of trajectory direction was also significant, F(1,65) = 8.93, p = .004, ηp2 = .20, with participants completing leftward trajectory trials 25 ms faster than rightward, regardless of congruity. There was no impact of distance on MT, F(1,65) = 2.86, p = .10. A significant interaction emerged between congruity 31 and distance, F(1,65) = 22.63, p < .001, ηp2 = .26, with a larger effect of congruity on trials with greater numerical distance. Figure 5. Mean MT as a function of congruity and distance. For IT, there was a significant between subjects main effect of sex, F(1,65) = 6.87, p = .01, ηp2 = .10, with males initiating movement 32 ms faster than females. The main effect of congruity was not significant, F(1,65) = 0.00, p = .99, but there was a significant interaction between congruity and diagnosis, F(1,65) = 7.82, p = .007, ηp2 = .11. Participants with a history of diagnosis initiated movement faster than those without a history of diagnosis, and had a greater difference in reaction time between congruent and incongruent trials (see Figure 6). No significant main effects of distance, F(1,65) = .90, p = .35, or trajectory direction, F(1,65) = .10, 32 p = .75 were found, but there was a significant interaction between congruity and trajectory direction, F(1,65) = 4.40, p = .04, ηp2 = .06. On leftward trajectory trials, the initiation time was 3 ms faster for congruent trials; this was reversed on rightward trajectories, with 3 ms faster IT on trials with incongruent stimuli. Figure 6. Mean IT as a function of history of diagnosis and distance. Trajectory Analyses A 2 (physicalnumerical size congruity: congruent vs. incongruent) x 2 (numerical distance: close vs. far) x 2 (trajectory direction: leftward vs. rightward) x 2 (sex: female vs. male) x 2 (history of diagnosis: diagnosis vs. no diagnosis) repeated measures analysis of variance was conducted for AUC and MD. Table 6 contains mean AUC and MD values. 33 Table 6 Mean AUC and MD by congruity, distance, and trajectory direction. Leftward Trajectories Rightward Trajectories Numerical Distance Close Far Close Far Congruent AUC .66 .61 .62 .53 MD .37 .35 .32 .28 Incongruent AUC .91 1.01 .88 .96 MD .49 .53 .43 .46 There were no between subjects main effects of sex, F(1,65)=.08, p=.78, or diagnosis, F(1,65)=.91, p=.35. Congruity did impact AUC, with higher mean values for incongruent than congruent trials, F(1, 65) = 102.76, p < .001, ηp2 = .61. Significant main effects of distance, F(1, 65) = 0.16, p = .69, or trajectory direction, F(1,65) = .78, p = .38, were not found. The interaction between of congruity and distance was significant, F(1,65) = 17.17, p < .001, ηp2 = .21, with greater differences in trajectory complexity for congruent and incongruent trials with increased numerical distance (see Figure 7). A 2 (physicalnumerical size congruity: congruent vs. incongruent) x 2 (numerical distance: close vs. far) x 2 (trajectory direction: leftward vs. rightward) x 2 (sex: female vs. male) x 2 (history of diagnosis: diagnosis vs. no diagnosis) repeated measures analysis of variance was conducted for MD. There were no significant between subject main effects of sex, F(1,65) = .16, p = .70, or diagnosis, F(1,65) = .31, p = .58. Like AUC, a significant main effect of congruity was found, with larger MD for trials with incongruent stimuli than trials with congruent trials, F(1,65) = 154.16, p < .001, ηp2 = .70. There was also a significant main effect of trajectory direction, F(1,65) = 5.52, p = .02, ηp2 =.08, with more complex responses on leftward 34 trajectories. While there was no significant main effect of distance F(1,65) = .04, p = .85, an interaction between congruity and distance, F(1,65) = 16.72, p < .001, ηp2 = .20, was found. Figure 7. Mean response trajectories by trajectory direction, congruity, and numerical distance. Facilitation and Interference Neutral trials were included for comparison with the purpose of further exploring the overall size congruity effect observed in the initial analyses. Difference between trials with neutral stimuli and trials with congruent stimuli represent facilitation, while difference between trials with neutral stimuli and trials with incongruent stimuli represent interference. Table 7 contains the means for all dependent measures by congruity and trajectory direction. 35 Table 7 Mean movement times (MT) and initiation times (IT), in milliseconds, by congruity and trajectory direction. Congruent Neutral Incongruent Trajectory Direction Left Right Left Right Left Right MT (ms) 1201 1217 1223 1232 1248 1278 IT (ms) 122 126 124 129 125 123 AUC .63 .56 .77 .65 .98 .93 MD .36 .29 .42 .33 .51 .45 Response Time. I performed a 3 (physicalnumerical size congruity: congruent vs. neutral vs. incongruent) x 2 (trajectory direction: leftward vs. rightward) x 2 (sex: female vs. male) x 2 (history of diagnosis: diagnosis vs. no diagnosis) repeated measures analysis of variance for both MT and IT using the average values for each participant. MT was not impacted by sex, F(1,65) = .41, p = .52, or diagnosis, F(1,65) = 3.52, p = .07. There was an overall main effect of congruity on MT, F(2,130) = 48.98, p < .001, ηp2 =.43. Planned contrasts between congruity conditions revealed a difference in mean MT between congruent versus neutral trials, F(1, 130) = 11.99, p < .001, ηp2 =.8, with participants completing congruent trials faster than neutral trials, and between incongruent versus neutral trials, F(1,130) = 85.95, p < .001, ηp2 =.40 , with participants completing neutral trials faster than incongruent trials (see Figure 8). 36 Figure 8. Mean MT by congruity. IT was impacted by sex, F(1,65) = 6.55, p = .01, ηp2 = .09, with males initiating movement 31 ms faster than females. There was no main effect of diagnosis, F(1,65) = .39, p = .53, or of congruity, F(2,130) = 1.85, p = .16. However, planned contrasts between congruity conditions did reveal an interaction of diagnosis, with participants with a history of diagnosis initiating movement 3 ms slower on incongruent than on neutral trials, F(1,130) = 10.26, p = .002, ηp2 = .07 (Figure 9). There was a marginally significant overall main effect of trajectory direction, F(1,130) = 3.22, p = .04, ηp2 =.05, with leftward trajectory movement initiating 2 ms faster than rightward trajectory movement. 37 Figure 9. Mean IT by congruity and history of diagnosis. Trajectory. A 3 (physicalnumerical size congruity: congruent vs. neutral vs. incongruent) x 2 (trajectory direction: leftward vs. rightward) x 2 (sex: female vs. male) x 2 (history of diagnosis: diagnosis vs. no diagnosis) repeated measures analysis of variance was conducted for AUC and MD using the average values for each participant. For AUC, there were no between subjects main effects of sex, F(1,65) = .16, p = .69, or diagnosis, F(1,65) = .87, p = .36. A significant overall effect of congruity for AUC was found, F(2,130) = 95.67, p < .001, ηp2 = .60 (see Figure 10). Planned contrasts revealed a significant difference in AUC between congruent versus neutral trials, F(1,130) = 18.66, p < .001, ηp2 = .13, and between incongruent versus neutral trials, F(1,130) = 172.68, p = .001, ηp2=.57. There was no significant main effect of side, and no significant interactions (all F values less than 2.4). 38 The results for MD echoed those for AUC, with a significant overall effect of congruity, F(2,130) = 135.53, p < .001, ηp2 = .68, a difference between congruent versus neutral trials, F(1,130) = 30.63, p < .001, ηp2 = .19, and incongruent versus neutral trials F(1,130) = 240.42, p < .001, ηp2 = .65. There were no significant main effect of trajectory direction, and no interactions (all F values less than 1.6). Figure 10. Mean response trajectories as a function of congruity. Predicting Math Achievement A 2 (sex: female vs. male) x 2 (handedness: right vs. other) x 2 (diagnosis: yes vs. no) x 4 (order: A, B, C, D) was conducted for both math achievement and standard progressive matrices scores. For math achievement, there was a significant main effect of diagnosis, F(1,47) = 6.54, p = .01, ηp2 = .12; participants with no reported history of ADHD or LD diagnosis scored higher than those with a history of diagnosis (see Table 8). There were no significant main effects of 39 sex, F(1, 47) = 1.32, p = .26, handedness, F(1,47) = 2.02, p = .14, or task order, F(1,47) = 2.04, p = .12, and no significant interactions. For scores on ASPM, there were no main effects and no significant interactions (all F ratios were less than 1.96). Table 8 Mean math achievement (zscores) and ASPM. Math Achievement ASPM Overall .018 6.51 Sex Female .104 6.57 Male .424 6.31 Handedness Right .196 6.51 Left .858 6.30 Both .725 7.03 Diagnosis ADHD or LD .159 6.58 No Diagnosis 1.37 5.94 Task Order A .210 6.44 B .639 6.38 C .343 6.40 D .548 6.83 One of the goals of Experiment 2 was to identify variables that may be valuable in the prediction of math achievement so Pearson’s productmoment correlation (r) was calculated for each of the potential predictors and math achievement. Table 9 contains the bivariate correlation values for each of the predictor variables and math achievement. 40 Table 9 Correlation coefficients for math achievement by variable (p < .05 = *, p < .01 = **, p < .001 = ***). Sex Diagnosis Task Order r F M Yes No A B C D Age .42*** .50*** .46 .65 .46*** .34 .55* .57** .47 ASPM .24* .30* .14 .16 .19 .18 .01 .42 .49* Facilitation MT .07 .17 .27 .27 .14 .40 .45 .20 .11 IT .00 .04 .05 .29 .12 .13 .12 .21 .08 Trajectory .02 .04 .05 .36 .02 .21 .26 .26 .35 Interference MT .03 .17 .45 .05 .05 .26 .13 .11 .08 IT .11 .15 .03 .37 .00 .27 .03 .06 .02 Trajectory .01 .07 .32 .47 .02 .08 .10 .01 .13 In order to screen extreme independent variable values in preparation for regression analyses, Mahalanobis scores (Mahalanobis, 1936; Stevens, 1984) were calculated for each participant; no participants were beyond the cut off Mahalanobis score. Five participants had more than expected leverage on the dependent variable as indicated by elevated hat values, and 5 participants were identified as having significant influence, or leverage and discrepancy, according to Cook’s values. Two participants meeting more than 1 indicator of extreme scores were excluded. In order to ensure that there was not considerable multicollinearity among independent variables, a correlation matrix was created. None of the correlations between the variables were excessively high (r < .70), so none were excluded. The assumptions of normality, linearity, homogeneity, and homoscedasticity were upheld upon inspection of relevant plots. A simultaneous multiple regression was conducted to determine if math achievement could be predicted by age, sex, history of diagnosis, task order, ASPM, facilitation (MT, IT, and trajectory), and interference (MT, IT, and trajectory). The model did explain a significant amount 41 of the variance present in scores of math achievement, F(15,51) = 3.82, p = .0002, R2 = .53, R2Adjusted = .39 (see Figure 11). In the overall model, none of the measures of facilitation or interference were significant predictors of math achievement. Table 10 contains the significant results of the simultaneous multiple regression. Table 10 Predictors of math achievement Variable b t β p Age .04 4.39 .49 <.001 Left Handedness .74 2.97 .36 .005 History of Diagnosis 1.13 3.91 1.48 <.001 Figure 11. Predicted math scores with confidence band (95%) as compared to actual data points. 42 In order to determine if the effects of facilitation (MT, IT, and trajectory) or interference (MT, IT, and trajectory) were modified by age, handedness, sex, diagnostic history, order, or ASPM, 5 regressions were conducted using each of the stated variables in the model. The model predicting math achievement using age as an interaction term was significant, F(13,55) = 2.44 p = .01, R2 = .37, R2Adjusted = .22 (see Table 11). Table 11 Predictors of math achievement with age as interaction term Variable b t β p Facilitation IT .11 2.17 1.36 .04 Age ~ Facilitation IT .005 2.19 .24 .04 Discussion The predictions related to the overall size congruity effect based upon a comparison of congruent and incongruent trials were confirmed. Response times were faster on congruent trials, and computer mouse trajectories were more complex on incongruent trials. Additionally, numerical distance impacted differences in response time and AUC. With the introduction of neutral trials, facilitation was discernable in congruent trials in MT, AUC, and MD, and interference was apparent in both measures of trajectory and MT for incongruent trials. The analysis of IT trajectories did not reveal facilitation, but did indicate interference in individuals with a history of diagnosis. Unanticipated findings included faster initiation times in males than females, and more complex trajectories on leftward responses as measured by MD. Experiment 2, as predicted, replicated the results of Experiment 1, providing support for the utility of computer mousetracking in examining the facilitation and interference components of the size congruity effect. Additionally, the results of Experiment 2 lend support to the late 43 interaction effect, which predicts that competition between two choices will continue throughout the motor response. This is precisely what was observed in an analysis of the trajectories. In an examination of individual differences in math achievement, participants with no history of diagnosis did have higher combined CFT and BMA scores. This difference was not apparent in the abbreviated ASPM scores; this finding suggests that the between group discrepancies in math achievement were not attributed to a difference in fluid intelligence. Although significant findings did emerge in the first regression analysis, facilitation was not a predictor of math achievement, as hypothesized. The most significant factors in the overall model were age, left handedness, and history of diagnosis. However, the interaction of age and IT facilitation was significant when age was introduced as a modifier in a regression model containing only the measures of facilitation and interference. 44 Chapter 4 GENERAL DISCUSSION The ability to compare the values of numerical symbols is integral to numeracy in humans, and develops out of an innate approximate number system (Girelli et al., 2000).With practice, the repeated pairing of a symbol with its corresponding magnitude will result in the automatic processing of the magnitude when presented with the symbol, even when it is irrelevant, as is evident in the size congruity effect. Experimentally, the size congruity effect was first demonstrated by Henik and Tzelgov (1982), who found that participants would more quickly identify the physically larger of two presented numbers when the correct choice was also the numerically larger value, or both features were congruent. According to Hasher and Zacks (1978), automatic processes are beneficial because they preserve limited cognitive resources. In terms of mental arithmetic, the automatic processing of numerical magnitude is valuable because it frees working memory for use in more complex problem solving. The size congruity effect has been supported by numerous studies, and the results of both Experiment 1 and 2 align with the literature. Participants spent less time in motion towards the correct selection during the motor response for congruent trials than incongruent trials. In Experiment 2, leftward trajectory responses were completed 25 ms faster than rightward trajectory responses. Participants also exhibited more complex movement towards the response on leftward trajectories, as measured by MD, than rightward. These two findings are inconsistent with one another; it is unlikely that a theoretical basis for both faster and more complex leftward trajectories exists. However, Faulkenberry et al. (2016) found a small effect on initiation time for leftward trajectories, so further exploration of the impact of trajectory direction on all of the measures would be valuable. 45 Lindsay and Jacoby (1994) posit that the size congruity effect is not attributable solely to the benefit of automatic processing, or facilitation. The irrelevant stimulus feature draws attention, therefore interfering with the response. These two components—facilitation and interference — can be examined through the comparison of congruent and incongruent trials, respectively, with neutral trials. The difference between congruent and neutral trials measures facilitation, while the difference between incongruent and neutral trials measures interference (Ashkenazi et al., 2008, Rubinsten & Henik, 2005, 2006). Again, the results of both Experiment 1 and 2 contribute to the growing body of research concerning the components of the size congruity effect. Responses were more efficient for congruent trials when compared to neutral trials, and less efficient for incongruent trials when compared to neutral trials. The unique contribution this study makes to the understanding of the components of the size congruity effect is the application of computer mousetracking in conjunction with the use of neutral stimuli. This relatively new data collection method has been used to examine the size congruity effect (Faulkenberry, Cruise, & Shaki, 2016; Geye, 2015), demonstrating more complex trajectories on incongruent than congruent trials. In the current study, results from both experiments replicated these findings; AUC and MD were larger on incongruent than congruent trials. This provides evidence for ongoing competition between the two choices throughout the motor response, which is consistent with the late interaction account (Santens & Vergus, 2011; Schwarz & Heinze, 1998). The late interaction model posits that the competition of possible choices continues through the motor response, whereas the early interaction model argues that the competition between two the two possible choices is resolved prior to the inception of the motor response. If this were the case, a significant difference in initiation times would have been observed in the comparison of congruent versus incongruent and neutral trials. 46 It can be argued that the instructions to quickly begin movement to respond, reinforced by the 400 ms cutoff for initiation, prematurely ends the representational stage described by the early interaction model. Faulkenberryet al. (2016) removed the speeded initiation component of the instructions, and found that the size congruity effect was still evident through continued competition in the response stage. No previous study has used computer mousetracking in a design containing neutral trials, allowing for the examination of facilitation and interference using trajectory data. As expected, the neutral trials were flanked by the congruent and incongruent trials on all measures: MT, IT, AUC, and MD. In Experiment 1, there was a significant difference between congruent and neutral trials (facilitation) for trajectory measures only, although MT did join AUC and MD in Experiment 2. Oddly, IT facilitation in Experiment 2 was significant for those with a history of diagnosis. Interference was significant for MT, AUC, and MD both Experiment 1 and in Experiment2. The trajectory data was uniquely useful in identifying facilitation, particularly in Experiment 1. These results underscore the usefulness of computer mousetracking as a measure of dynamic response. Geye (2015) found more complex trajectories for individuals in a low math achieving group than those in a typically math achieving group, regardless of congruity, which is inconsistent with the results of this study. Overall trajectory complexity was not correlated with math achievement, and neither facilitation nor interference, on any of the measures, predicted math achievement. It is possible that this lack of significance in predicting math achievement could be due to homogeneity of the population. When modified by age in the regression model, facilitation IT did become a significant predictor of math achievement, so perhaps a study with larger variability in age would yield more robust findings. Additionally, although there was 47 variation in math achievement within the current sample, there were no participants with a reported mathematics learning disability. Obtaining a sample with a significant number of individuals with very low math achievement may require sampling outside of the University setting; partnering with an openaccess community college or vocational training program may become necessary. Although math achievement was not predicted, as expected, by facilitation, other between subjects findings did emerge in Experiment 2. Participants with a history of diagnosis initiated movement faster, and had a greater difference between IT on congruent and incongruent trials than those without a history of diagnosis. Facilitation was evident in IT for those with a history of diagnosis, as well. This finding is likely an anomaly due to the small number of participants with a history of a diagnosis, but subsequent work may explore the possibility of a difference in processing. A few participants reported choosing to look at only one side of the screen in order to identify the correct response. Varying the font point sizes would help to mediate the impact of this approach to the task, although it is unlikely that results would differ substantially. In a recent study, Sobel, Puri, and Faulkenberry (2016) incorporated a visual search paradigm into the examination of the size congruity effect and found that top down processing does have a role in the size congruity effect that is not inconsistent with the late interaction effect. Future Directions This study is the first to employ computer mousetracking to explore the facilitation and interference components of the size congruity effect, and was therefore largely exploratory in nature. Future research should seek to recruit a more diversified sample in order to test more participants with very low math achievement. Efforts should also be made to more closely 48 examine the differences in results obtained for individuals with a history of diagnosis in order to discern whether the differences are truly attributable to diagnosis or if they would be better explained by sex. The prevalence of ADHD worldwide is estimated to between 3% to 7%, with a 3:1 maletofemale ratio in the population and an even more pronounced proportionality in clinical samples (Skogli, Teicher, Andersen, Hovik, & Øie, 2013). It is possible that a closer examination of this data would reveal that a maxing out of attentional resources would occur earlier for individuals with a history of a diagnosis than those without. In a study involving a driving simulation, Reimer, D’Ambrosio, Coughlin, Fried, and Biederman (2007) found that adults with ADHD succumbed to the monotony of a repetitive task more quickly than those in the control group. Conclusions These results highlight the usefulness of trajectory analysis in the examination of the size congruity effect. The dynamic data provided through this method allowed for a more fine grained study, leading to significant findings not possible through response time measures. The combination of computer mousetracking with the neutral stimulus provides the foundation for useful work related to the size congruity effect, specifically the role of attention (Risko et al., 2013). 49 REFERENCES Agrillo, C., Dadda, M., Serena, G., & Bisazza, A. (2009). Use of number by fish. Plos ONE, 4(3). doi: 10.1371/journal.pone.0004786 Ashkenazi, S., Henik, A., Ifergane, G., & Shelef, I. (2008). Basic numerical processing in left intraparietal sulcus (IPS) acalculia. Cortex, 44, 439448. doi: 0.1016/j.cortex.2007.08.008 Ashkenazi, S., Rubinsten, O., & Henik, A. (2009). Attention, automaticity, and developmental dyscalculia. 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Retrieved from http://carleton.ca/cacr/wpcontent/uploads/CalculationFluencyTestDescription_MathLabTechnicalReport.pdf 57 Spivey, M. (2007). The continuity of mind. New York: Oxford University Press. Spivey, M. J., Grosjean, M., & Knoblich, G. (2005). Continuous attraction toward phonological competitors. Proceedings of the National Academy of Sciences of the United States of America, 102, 1039310398. doi: 10.1073/pnas.0503903102 Steiner, E. T., & Ashcraft, M. H. (2012). Three brief assessments of math achievement. Behavior Research Methods, 44, 11011107. doi: 10.3758/s1342801101856 Stevens, J. P. (1984). Outliers and influential data points in regression analysis. Psychological Bulletin, 95(2), 334344. doi:10.1037/00332909.95.2.334 Stroop, J. R. (1935). Studies of interference in serial verbal reactions. Journal of Experimental Psychology, 18, 643662. doi: 10.1037/h0054651 Tzelgov, J., Meyer, J., & Henik, A. (1992). Automatic and intentional processing of numerical information. Journal of Experimental Psychology: Learning, Memory, and Cognition, 18, 166179. doi:10.1037/02787393.18.1.166 Yamamoto, N., Incera, S., & McLennan, C. T. (2016). A reverse Stroop Task with mouse tracking. Frontiers in Psychology, 7, 670. doi: 10.3389/fpsyg.2016.00670 Zebian, S. (2005). Linkages between number concepts, spatial thinking, and directionality of writing: The SNARC effect and the reverse SNARC effect in English and Arabic monoliterates, biliterates, and illiterate Arabic speakers. Journal of Cognition and Culture, 5(12), 165190. doi: 10.1163/1568537054068660 58 VITA Trina Geye earned her Bachelor of Science degree in Psychology from Tarleton State University in 1999, and her Master of Science degree in Counseling Psychology from Tarleton State University in 2002. As a Licensed Professional Counselor, Trina’s early career was in nonprofit and included case management for individuals with developmental disabilities seeking employment, counseling at risk youth and their families, and serving as Executive Director of a child advocacy agency. In 2006, Trina transitioned to higher education, becoming Coordinator of Student Success Programs at Tarleton. She was promoted to Director of Student Disability Services and Instructor of Psychology in 2007, and subsequently assumed responsibility over Student Assessment Services 2012, and Academic Support Center in 2014. As the Director of Academic Support Centers, she manages the Center for Access and Academic Testing, which is comprised of the testing center and office providing services to students with disabilities, and the Academic Resource Center, which provides programs such as Supplemental Instruction and peer tutoring. She also assists with curriculum development for the First Year Seminar, and serves on numerous committees (i.e., Campus Assessment, Response, and Evaluation; University Committee on Diversity, Access, and Equity; Developmental Education Advisory Committee; Reimaging the First Year; and Academic Advising Council). She continues to teach for the Department of Psychological Sciences. Trina was accepted into the PhD program in Psychology at Texas A&M UniversityCommerce in 2006. She married Jake Hoon in 2008; they have two sons, Ian and Gavin. Department of Psychological Sciences, Box T0820, Stephenville, Texas 76402 geye@tarleton.edu
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Title  Facilitation and Interference Components of the Size Congruity Effect in College Students 
Author  Geye, Trina Len 
Subject  Psychology; Educational psychology; Mathematics education 
Abstract  FACILITATION AND INTERFERENCE COMPONENTS OF THE SIZE CONGRUITY EFFECT IN COLLEGE STUDENTS A Dissertation by TRINA GEYE Submitted to the Office of Graduate Studies of Texas A&M UniversityCommerce in partial fulfillment of the requirements for the degree of DOCTOR OF PHILOSOPHY December 2016 FACILITATION AND INTERFERENCE COMPONENTS OF THE SIZE CONGRUITY EFFECT IN COLLEGE STUDENTS A Dissertation by TRINA GEYE Approved by: Advisor: Tracy B. Henley Committee: Steven E. Ball Thomas J. Faulkenberry Raymond J. Green Head of Department: Jennifer L. Schroeder Dean of the College: Timothy D. Letzring Dean of Graduate Studies: Mary Beth Sampson iii Copyright © 2016 Trina Len Geye iv Dedicated to my grandmothers: Frankie Louise (Hicks) Geye Mary Alice (Reed) Chancellor v ABSTRACT FACILITATION AND INTERFERENCE COMPONENTS OF THE SIZE CONGRUITY EFFECT IN COLLEGE STUDENTS Trina Geye, PhD Texas A&M UniversityCommerce, 2016 Advisor: Tracy Henley, PhD The present study examined the facilitation and interference components of the size congruity effect in college students. In Experiment 1, computer mousetracking was used to capture the trajectories of responses during a numerical size congruity task in which participants were asked to select the physically larger of two digits. Response time and trajectory data supported the conclusion that numerical magnitude is automatically processed, and that the decision process continues through the motor response, consistent with the late interaction model (Faulkenberry, Cruise, Lavro, & Shaki, 2016; Santens & Verguts, 2011; Sobel, Puri, & Faulkenberry, 2016; Sobel, Puri, Faulkenberry, & Dague, in press). In Experiment 2, participants completed the Calculation Fluency Test (Sowinsky, Dunbar, & LeFevre, 2014) and the Brief Mathematics Assessment – 3 (Steiner & Ashcraft, 2012) as measures of math achievement and Form A of the abbreviated Standard Progressive Matrices test (Bilker et al., 2012) as a measure of fluid intelligence in addition to the experimental task used in Experiment 1. There were no significant findings related to fluid intelligence. Individuals with a previous diagnosis of learning vi disorder or attention deficit hyperactivity disorder had lower scores on measures of math achievement. Analysis of the size congruity effect replicated the findings of Experiment 1, again providing support for the automatic representation of numerical magnitude and the late interaction model. Neither facilitation nor interference was found to be significant in predicting math achievement. Facilitation and interference were evident in trajectory measures even when undiscernible in movement and initiation time, which demonstrates the value of computer mousetracking over response time alone in examining covert cognitive processes. vii ACKNOWLEDGEMENTS I extend heartfelt thanks to my committee chair, Dr. Tracy Henley, not only for the academic mentorship throughout this process, but also for knowing when to fan the flame and when to douse it. Thanks to Dr. Ray Green and Dr. Steve Ball for good naturedly supporting me through the maze of topic selection; I dare not guess how much time each of you spent with me discussing potential projects that never quite lifted off. To Dr. Tom Faulkenberry, without your relocation, this literally would not have happened. The opportunity to work with you provided me with the environment I needed for a “robust” dissertation experience, which has significantly influenced my growth as a researcher. I appreciate the support of my Tarleton family, especially Brenda Faulkner and the rest of the SSMI crew, for motivating me and keeping me accountable. Thanks to Liz Podany for falling in love with psychology with me; you will always be a part of my story. Haley Oyler, you have been my fiercest, most loyal friend since we were kids; thank you for being a better amiga than I deserve. The late Roger Hoon rooted for “Dr. T” from day one; I gave ‘em a little hell for you. Trisha and Scott, my siblings and two of my closest friends, you probably have no idea how many times you unknowingly talked me out of calling it quits. I also recognize that any good I have done in my life has been accomplished while standing on the shoulders of my parents. Thank you—all four of you—for having high expectations of me and for giving me every opportunity you possibly could to fulfill them. To my husband, Jake, and our boys, Gavin and Ian, thank you for your willingness to share me with my research. You made room in our family for this project, and chose to view it as an opportunity rather than an obligation. My cup runneth over. viii TABLE OF CONTENTS LIST OF TABLES ......................................................................................................................... x LIST OF FIGURES ...................................................................................................................... xi CHAPTER 1. INTRODUCTION .........................................................................................................1 Size Congruity Effect ..............................................................................................6 Individual Differences ............................................................................................9 Computer Mousetracking ......................................................................................11 2. EXPERIMENT 1 .........................................................................................................15 Method ..................................................................................................................16 Participants ................................................................................................16 Procedure ..................................................................................................16 Results ...................................................................................................................17 Response Time Analyses ..........................................................................18 Trajectory Analyses ..................................................................................19 Facilitation and Interference .....................................................................21 Discussion .............................................................................................................24 3. EXPERIMENT 2 .........................................................................................................26 Method ..................................................................................................................26 Participants ................................................................................................26 Materials and Procedure ...........................................................................26 Results ...................................................................................................................28 Response Time Analyses ..........................................................................29 ix Trajectory Analyses ..................................................................................31 Facilitation and Interference .....................................................................33 Predicting Math Achievement ..................................................................37 Discussion .............................................................................................................41 4. GENERAL DISCUSSION ..........................................................................................43 Future Directions ..................................................................................................46 Conclusions ...........................................................................................................47 REFERENCES .............................................................................................................................48 VITA ...........................................................................................................................................57 x LIST OF TABLES 1. Mean MT and IT by congruity, distance, and trajectory direction ...................................18 2. Mean AUC and MD by congruity, distance, and trajectory direction ..............................20 3. Mean MT and IT by congruity and trajectory direction (includes neutral) ......................22 4. Order of tasks ....................................................................................................................28 5. Mean MT and IT by congruity, distance, and trajectory direction ...................................29 6. Mean AUC and MD by congruity, distance, and trajectory direction ..............................32 7. Mean MT and IT by congruity and trajectory direction (includes neutral) ......................34 8. Mean math achievement and ASPM scores ......................................................................38 9. Correlation coefficients for math achievement by independent variable .........................39 10. Predictors of math achievement ........................................................................................40 11. Predictors of math achievement with age as interaction term ...........................................41 xi LIST OF FIGURES FIGURE 1. Mean MT as a function of condition and distance ............................................................19 2. Mean response trajectories by direction, congruity, and distance ....................................21 3. Mean MT by congruity .....................................................................................................23 4. Mean Response trajectories by congruity (with neutral) ..................................................24 5. Mean MT by congruity and distance ................................................................................30 6. Mean IT by diagnosis and distance ...................................................................................31 7. Mean response trajectories by direction, congruity, and distance ....................................33 8. Mean MT by congruity .....................................................................................................35 9. Mean IT by congruity and diagnosis ................................................................................36 10. Mean response trajectories by congruity (with neutral) ...................................................37 11. Predicted values with confidence band .............................................................................40 1 Chapter 1 INTRODUCTION The necessity for adults in the United States to have basic numeracy skills, or the ability to conceptualize and use numbers and mathematics, is incontrovertible. Economic opportunity and the capacity to engage in many aspects of daily life is negatively impacted for those individuals who cannot reason numerically at the level expected of eighth graders (Geary, Hoard, Nugent, & Bailey, 2013). The Organization for Economic Cooperation and Development’s (OECD) Survey of Adult Skills revealed that almost one third of the 5,010 individuals surveyed were unable to solve problems requiring multiple steps, discern frequently encountered decimals, fractions, and percentages, understand measurement and spatial representation, make estimations, and interpret tables, graphs, and data presented in text (OECD, 2013). The capability to evaluate the relative magnitude of symbols representing numerical values is fundamental to the development of numeracy (Girelli, Lucangeli, & Butterworth, 2000). Primates, both human and nonhuman, possess the capacity to comprehend and mentally manipulate approximate quantities. The mental representation of number is not a human invention, but appears to be the result of brain evolution (Dehaene, Piazza, Pinel, & Cohen, 2003). Several studies have provided evidence for the ability to differentiate quantities in various animal species. Rugani, Regolin, & Vallotigara (2008) found that newborn chicks will associate with the larger set of the object upon which they imprinted. Individual mosquito fish will choose to join a larger school, which is beneficial in the avoidance of predators (Agrillo, Dada, Serena, & Bisazza, 2009). Cantlon and Brannon (2007) found that monkeys will attend to object numerosity, even when it is irrelevant to the current task. 2 According to Gelman and Gallistel (1978), animals, including humans, form internal magnitude representations through a preverbal counting model, establishing a structure onto which external numerosities are mapped. Very early on, children understand the essential constraints on counting. Gallistel and Gelman (1992) describe these limits on counting in terms of five inherent rules: one to one correspondence, stable order, cardinality, abstraction, and order irrelevance. Repeated counting of stimuli in the external environment results in the development of a mental representation of the cardinal, or “how many,” associated with quantities. Humans, as opposed to their nonhuman counterparts, can also use this approximate number system for the representation of magnitudes associated with symbols (see Budgen, DeWind, & Brannon, 2016). Weber’s Law states that the just noticeable difference in a changing stimulus can be calculated as a constant ratio of the primary stimulus. Moyer and Landauer (1967) substantiated their hypothesis that numerical magnitude is processed in the same way as other psychophysical stimuli, such as length or pitch, and therefore subject to Weber’s Law, by replicating the results found in a line comparison task wherein participants are instructed to identify the longer of two presented lines. Reaction times were faster when the lines were comparatively small and when the disparity between the two lines was large. Moyer and Landauer similarly presented participants with two numerical digits and asked them to simply choose the digit with the larger value. As expected, participants identified the larger number more rapidly when the values were numerically farther apart. This finding has been termed the numerical distance effect and was an intriguing discovery because it was in opposition to then extant explanations of number comparison. The distance effect could not be predicted if number comparisons were made by counting from 1 to the smaller number, from one of the presented numbers, or via direct retrieval (Seigler & Opfer, 2003). The numerical distance effect is considered to be evidence that 3 numerals are converted into an analog representation of magnitude, much like other physical stimuli. Numbers seem to be represented in a domainspecific array according to cardinal magnitude as opposed to a more general way, such as, for example, according to frequency or association (Butterworth, Zorzi, Girelli, & Jonckheere, 2001). Restle (1970) describes a mental number line resulting from the ongoing association of external quantities with internal magnitude representations through practice as an analogy for the mental representation of magnitude associated with numbers. According to this model, numbers correspond to positions along a string, right to left, and a “next” marker associated with each number indicates directionality towards larger positions (Resnick, 1983). Dehaene, Bossini, and Giraux (1993) presented a mental number line model that preserved the features of the earlier version, but incorporated an explanation of differences in the level of activation. According to this model, numbers one through three are activated distinctly, or subitized. Subitizing is a process that allows for the quick and accurate assessment of numerosity without serial counting, and is possible only for small quantities (Atkinson, Campbell, & Francis, 1976; Feigenson, Dehaene, & Spelke, 2004). However, for four and beyond, magnitude representation becomes increasingly fuzzy, with overlapping Gaussian curves representing the range of activation. These overlaps in activation result in increasing interference, or concurrent partial activation, from other numbers. For example, when the number five is activated, four and six are also activated, although to a lesser degree, with the range of activation increasing with numerical value. Early examinations of numerosity in rats illustrate the intensity of activation according to the mental number line model. Mechner (1958) and Platt and Johnson (1971) created an environment in which rats were required to press one 4 lever a certain number of times, then press a separate lever once to release food. Attempts with too few presses would be punished, resulting in trial and error learning to discern the number of presses necessary to receive food. Even with considerable training, the number of presses remained imprecise; the number of presses for a target of four, for example, ranged from three to seven. As predicted by the mental number line model, the width of the response range increased with the target value, suggesting less precise activation of larger numerical magnitudes. A natural number line, along which numbers are spaced logarithmically (with lower values farther apart), is practical in the situations involving small magnitudes typically encountered by animals, such as when making decisions regarding foraging, because differences between two small numerosities can be quickly compared. Siegler and Opfer (2003) found that a logarithmictolinear shift occurs developmentally between kindergarten and second grade, allowing for use of a wider range of numbers. As they develop the mental number line, children first categorize small numbers into categories for comparison and eventually use this process in the comparison of larger numbers (Seigler & Robinson, 1982). This results in the adult classification of numbers as either "small" (less than five) or "large" (more than five), allowing for efficient memory retrieval (Tzelgov, Meyer, & Henik, 1992). The numerical distance effect, garnering robust support in the literature, was found to exist in the comparison of two digit numbers. Hinrichs, Yurko, & Hu (1981) displayed a number with two digits and instructed participants to judge it against a standard of either 50 or 55. Although the larger number was often identifiable using only the tens value, the numerical distance effect was still present; reaction times decreased with an increase in distance between compared numbers. The ones place was only ignored when the comparison standard was at a decade boundary (50) and the distance between the compared numbers was less than 10. These 5 results were replicated by Dehaene, Dupoux, and Mehler (1990) who interpreted them as support for the mental number line model, which explains the effect as a result of activation of overlapping ranges for two numbers located close together. Dehaene et al. (1990) asked participants to compare a presented number (between 1 and 99, excluding 55) to the reference of 55 using a button to make the selection of larger or smaller. Reaction times were faster on those trials when the button corresponding to smaller was on the left for choose smaller conditions, and faster when the button corresponding to larger was on the right for choose larger conditions. In a later study, Dehaene et al. (1993) instructed participants to indicate whether a number (from zero to nine) was even or odd. For smaller numbers (less than five) reaction times were faster when the corresponding response button was on the left. For larger numbers (more than five) reaction times were faster when the corresponding response button was on the right. In order to rule out association with a limb rather than spatial orientation, participants were asked to complete the task with their arms crossed, pressing the left button with the right hand and vice versa; the results were the same. This finding is known as the spatialnumerical association of response codes (SNARC) effect, and indicates that the mental number line is spatially oriented, with numbers increasing in value from left to right. The SNARC effect is influenced by several variables. The directional association of a digit is dependent upon its value in relation to other digits. In a range of zero to five, the number five is rightward associated, but is leftward associated in a range of five to nine (Dehaene et al., 1993). Negative numbers have been found to be leftward associated, while positive numbers are rightward associated (Fischer, 2003). The SNARC effect appears to be present in individuals who read and write in line with the Western tradition of left to right (Zebian, 2005). Shaki, Fischer, and Petrusic (2009) found the SNARC effect in monolingual participants who read from 6 left to right, and found a reverse SNARC effect in monolingual right to left readers. In the reverse SNARC effect, numbers representing larger magnitude are associated with the right, and numbers representing smaller magnitude are associated with the left. No reliable SNARC effect was present for participants who were able to read and write both types of languages. In a study in which participants were instructed to consider digits to be compared to either a ruler or to a clock, the SNARC effect was reversed on clock comparison trials (Bachtold, Baumuller, & Brugger, 1998). In addition to substantiation that numerical magnitude is represented along a spatially oriented mental number line, support that the processing of magnitude is irrepressible, or automatic, has emerged. Size Congruity Effect Automatic processes are those that occur without intention or effort across circumstances, therefore preserving limited cognitive resources, specifically working memory. Working memory briefly stores and processes information relevant to the current task. It has three components: the central executive, responsible for managing processes and allocating attention; the visuospatial sketchpad, which maintains visualspatial information; and the phonological loop, which maintains auditory information (Baddeley, 1996). Communication between the visuospatial sketchpad and phonological loop is facilitated by the episodic buffer (Baddeley, 2002). Spatial and temporal processes, such as those that permit humans to orient to the events in their environment, are automatic due to genetic predisposition, but other processes can become automatic with repetition (Hasher & Zacks, 1978). According to Girelli et al. (2000), automatization of the processing of numerical magnitude is a result of practice in children, allowing for cognitive resources to be efficiently allocated as numerical ability progresses. 7 The hallmark method of measuring task irrelevant processing in order to discern automaticity was introduced by Stroop (1935). Participants were presented with the names of colors presented in various text colors and instructed to simply verbalize the color of the text in which the word was presented for 100 color words. Participants took longer to complete this task than when asked to simply name the color of 100 boxes. This suggests that the semantic meaning of the word was processed even though it was not relevant to the task at hand. Posner (1978) suggests that this Stroop effect is comprised of two components: facilitation and inhibition. Facilitation refers to the beneficial processing of an irrelevant but congruent stimulus feature to improve response over neutral conditions, while inhibition represents the extent to which the processing of an irrelevant and incongruent stimulus feature is detrimental to response when compared to neutral conditions. In an argument against equating tasks with processes, Jacoby (1991) posits that automaticity, which he synonymizes with familiarity, is revealed by facilitation, whereas interference is impacted by the division of attention in an intentional process. Henik and Tzelgov (1982) sought to explore how irrelevant stimulus dimensions are processed in a comparison of two dimensional (physical size and numerical value) stimuli created using Arabic numerals 1–9 and manipulating the congruity between physical size feature dimension and numerical value dimension. Congruent trials had a target that was both physically and numerically larger (5 3), incongruent trials had a target that was numerically smaller (5 3), and neutral trials had digits varying only on the designated dimension (e.g., 5 3 for a numerical comparison task; 5 5 for the physical comparison task). Participants viewed these digit pairs on a computer monitor, and were asked to select which number was larger on the relevant dimension (physical size or numerical 8 value). Among their results was the finding that congruence affects performance; reaction times were quicker on trials containing congruent pairs than on trials containing incongruent pairs, quicker on trials containing congruent than neutral trials in numerical value comparisons, and slower on trials containing incongruent than neutral trials. Additionally, the differences in reaction time increased with the distance between the numerical values of digits presented for comparison. This finding of the size congruity effect (Paivio, 1975) in numerical comparison tasks suggests that physical and numerical size comparisons occur in parallel, or at the same time, as opposed to serially, or one at a time. Also, the influence of irrelevant dimension of numerical value on physical size decisions provides evidence that the processing of numerical magnitude occurs automatically. Various models have emerged to explain at what point during processing this interference from the irrelevant stimulus feature happens. The early interaction model (Schwarz & Heinze, 1998) posits that relevant and irrelevant features are processed concurrently prior to the response stage, with information from both features mapped onto a shared analog representation. The decision is then based upon a single representation, with any interference from the irrelevant feature occurring prior to the motor response. As such, congruent stimulus features resolve into a single representation faster than incongruent stimulus features, causing the size congruity effect. In the late interaction model (Faulkenberry et al., 2016; Santens & Verguts, 2011; Sobel et al., 2016; Sobel et al., in press), relevant and irrelevant features are processed in parallel but independent channels, resulting in competing responses; interference from the irrelevant stimulus features happens during the response stage (Schwarz & Heinze, 1998). In an expansion upon the late interaction model, the Shared Decisions Account proposes that the resultant benefit of congruity and detriment of incongruity results from competing motor responses activated by 9 opposing response codes (Santens & Verguts, 2011). For a congruent stimulus pair, such as 3 7, the “right larger” response code is activated by both the relevant and irrelevant feature. For in incongruent stimulus pair, such as 3 7, “left larger” and “right larger” response codes are activated simultaneously, and the resulting competition between motor responses manifests as the size congruity effect. Individual Differences The ability to process numerical magnitude is critical to the development of proficient numeracy (Henik, Rubinsten, & Ashkenazi, 2011), and, consequently, provides a window into individual differences in numeracy. In order to examine the processing of numerical magnitude as it relates to mathematical achievement in individuals exhibiting diminished math ability, Holloway and Ansari (2009) asked a group of six to eight year old children to complete number comparison tasks. Mathematical ability was determined using scores on relevant subtests of the WoodcockJohnson III Tests of Achievement. In a symbolic condition, the children were directed to choose the larger of two Arabic numerals (one to nine); in a nonsymbolic condition, they selected the larger of two arrays of squares. There was a negative correlation between scores for mathematical fluency and the numerical distance effect as measured by the difference in response times between small and large numerical distances. Participants with lower mathematical fluency had a greater difference in response times between the two conditions, suggesting a noisier mapping of magnitudes onto respective Arabic numerals. However, Rousselle and Noel (2007) found a larger numerical distance effect in individuals with higher math achievement in a study using mathematics learning disability, not simply low achievement, as a factor. Holloway and Ansari (2009) reconcile the studies by considering them a demonstration of the differences between individuals who have a diminished efficiency in the 10 mapping of symbols and magnitudes versus those who have a domain specific deficit requiring alternative strategies. In addition to the representation of numerical magnitude, individual differences have also been identified in the ability to process numerical magnitude automatically as evinced in the size congruity effect. To distinguish between the facilitation and interference components comprising the size congruity effect in individuals with mathematics learning disability, Rubinsten and Henik (2005, 2006) incorporated the use of a neutral stimulus. On congruent trials, the number in the larger font point size was also the numerically larger (5 3), while on incongruent trials, the digit in the bigger font point size was the numerically smaller (5 3). Neutral trials were those in which the presented digits varied only in the relevant dimension (5 5). Facilitation of the response was determined by contrasting the reaction times on congruent trials against those on the neutral trials, and interference was determined by contrasting the reaction times on incongruent trials against those on the neutral trials. Interference was found in both the mathematics learning disability group and the control group, but facilitation occurred only in the control group. In other words, congruency did not speed up reaction times for individuals with low math ability. The results were interpreted to suggest that individuals with mathematics learning disability did not automatically process numerical magnitude, and were consistent with findings in individuals with acquired difficulties in mathematics (Ashkenazi, Henik, Ifergane, & Shelef, 2008). The diminished size congruity effect in individuals with mathematics learning disability cannot be presumed to result from domain specific deficits in mathematics, and may be related to domain general differences in processing. Risko, Maloney, and Fugelsang (2013) posit that attention may play a more significant role in the size congruity effect than explained by existing 11 models based upon their findings that the temporal onset of stimuli can reduce the interference. Further evidence that attention plays a role in the size congruity effect was presented by Sobel et al. (2016), who found that manipulating the top down processing of the stimuli impacted the size congruity effect. Ashkenazi, Rubinsten, and Henik (2009), interested in the influence of attention on the size congruity effect, designed an experiment in which participants, again assigned to groups based upon mathematical ability, were presented with a secondary stimulus in addition to a numerical and physical size comparison task. Two partial circles were concurrently depicted on the peripheries of the display for all conditions. In load conditions, participants were asked to report whether or not the two partial circles were the same. The noload condition, in which the partial circles were present but no comparison was required, replicated previous findings for physical size comparison, with the interference component found in both groups, but facilitation only in the control group. Load removed the facilitation effect in both groups in the physical size comparison task, while load increased interference in the numerical comparison task for both groups, but only for participants with low math ability during the physical size comparison task. The authors posit that individuals with low math ability have difficulty recruiting attention in numerical tasks, and that the lack of findings related to load and inhibition point to a maxing out of cognitive resources in attending to the task at hand. Computer Mousetracking Previous studies have relied on performance measures, such as accuracy and response time, to explore the automatic representation of numerical magnitude. However, performance measures do not allow for the capture of dynamic data, which is more useful in exploring questions such as those prompted by debate between the early and late interaction models (Faulkenberry et al., 2016; Santens & Verguts, 2011; Schwarz & Heinze 1998; Sobel et al., 12 2016; Sobel et al., in press). Computer mousetracking provides a unique method of collecting data that allows for the study of covert cognitive processes. In an analysis of spoken word recognition, participants were directed to listen to an audibly presented word, and then select between two pictures the one corresponding with what was heard (Spivey, Grosjean, & Knoblich, 2005). This selection task was completed using a computer mouse, which allowed for the tracking of hand trajectories through the capture of mouse position coordinates during the motor response. Analysis of the trajectories revealed that on trials in which the two presented words were phonetically similar, the response pattern veered in the direction of the incorrect phonological competitor. This finding was interpreted as online evidence of two competing mental representations during a motor response. Subsequent to the development of the MouseTracker software introduced by Freeman and Ambady (2010), computer mousetracking has been recognized by Freeman, Dale, and Farmer (2011) as offering a window into realtime cognitive processes via the association between mental and motor dynamics. This method of hand tracking has been fruitfully employed in the study of numerical cognition in a number of studies (Faulkenberry, 2014; Marghetis & Núñez, 2013; Marghetis, Núñez, & Bergen, 2014; Santens, Goossens, & Verguts, 2011; Song & Nakayama, 2008). Although computer mousetracking has engendered criticism, the opportunity to examine previously concealed information about competing processes has been valuable. Fischer & Hartmann (2014) raised legitimate concerns about potential differences in apparatus and settings that can be resolved through the specific reporting of, for example, mouse settings. In addition to these simply mediated criticisms, however, the authors claimed that interference from a distractor is only evident in those cases in which the computer mouse is moved completely onto the side of the computer screen associated with the incorrect response. 13 Faulkenberry and Rey (2014) responded to these concerns with support from the continuous cognition framework (Spivey, 2007), indicating that any movement towards the incorrect response can be interpreted as interference, even accounting for task difficulty. Computer mousetracking has been recently applied to the examination of the size congruity effect. Faulkenberry et al. (2016) found that response trajectories were pulled towards the incorrect response, with greater competition from the irrelevant feature evinced in the difference between the actual and ideal trajectories in incongruent trials. These results indicate that the competition between stimuli continues through the response process, lending support to the late interaction model. Similarly, data from an unpublished manuscript (Geye, 2015) reveals greater curvature towards the incorrect response, regardless of congruity, in individuals with low math achievement, echoing previous interest in the role played by attention in the size congruity effect (Ashkenazi et al., 2009). This finding was particularly interesting because there were not any significant differences in response times between groups. Computer mousetracking has also been used to examine the classic Stroop effect; Yamamoto, Incera, and McLennan (2016) found interference by comparing incongruent to neutral trials in the examination of trajectories in a task requiring participants to click on the color word corresponding to the presented stimulus. The dynamic data available via computer mousetracking provides an opportunity to refine the lens through which basic numerical processes are examined. Kaufman et al. (2013) raises the concern that individual differences in basic numerical abilities may be overlooked using current methods. A fine grained examination of dynamic responses may be an instrumental tool in teasing out discrepancies not evident in performance data, such as response time, and inform ongoing efforts to develop the assessment of numeracy through noncurricular means. 14 The intention of this project was to explore the facilitation and interference elements of the size congruity effect using the trajectory data available through computer mousetracking, and to investigate the relationship of the components to overall math achievement. In Experiment 1, which served as a calibration, participants completed a comparison task typical to size congruity effect research. A neutral stimulus was introduced, allowing for the differentiation between facilitation and interference components. I predicted that trajectories would be more complex in incongruent trials than congruent, with greater difference between the ideal and actual trajectories as measured by area under the curve, maximum deviation, and movement time. I also hypothesized that facilitation would be evident in the comparison of congruent and neutral trials, with lower area under the curve, maximum deviation, and movement time in congruent than neutral trials. Additionally, I predicted that, due to interference, incongruent trials would have greater area under the curve, maximum deviation, and movement time than neutral trials. In the second experiment, math achievement was introduced as a variable. I hypothesized that math achievement would increase in relation to facilitation, as measured by the difference in area under the curve, maximum deviation, and movement time between congruent and neutral trials. 15 Chapter 2 EXPERIMENT 1 In Experiment 1, I sought to investigate the size congruity effect in college students using computer mousetracking during a physical size comparison task with numerical digits. Additionally, as the first application of computer mousetracking in the examination of the size congruity effect using neutral trials, it served as a calibration in which to test the stimuli prior to Experiment 2. The size congruity effect is consistently supported in the literature (Ashkenazi et al., 2009; Henik & Tzelgov, 1982), and results related to response times have been echoed using computer mouse trajectory data (Faulkenberry, Cruise, & Shaki, 2016; Geye, 2015), providing support for the late interaction account (Santens & Verguts, 2011). I predicted that movement times would be faster for congruent than incongruent trials, and that computer mouse trajectory, as measured by area under the curve (AUC) and maximum deviation (MD), would be correspondingly less complex. Digits are easier to compare when numerically far apart (Moyer & Landauer, 1967), and, correspondingly, irrelevant magnitudes seem to be more difficult to suppress with increased distance. Therefore, I expected the differences between the congruent and incongruent trials to increase with numerical distance, which is consistent with previous findings (Faulkenberry et al., 2016). Because Rubenstein and Henik (2005, 2006) were able to discern the facilitation and interference components through the comparison of response times on congruent and incongruent trials to neutral trials, I predicted that the comparison of trajectory data would yield similar results. Specifically, I hypothesized that participants would exhibit faster, less complex responses on congruent trials than neutral trials due to facilitation, and slower, more complex responses on incongruent than neutral trials due to interference. 16 Method Participants Thirty students attending a midsized public university in the Southwest were recruited for participation. The mean age of the participants was 21.7 (SD = 3.76), with an age range of 1830. Most of the participants were enrolled in undergraduate psychology or first year seminar courses for which credit was awarded for research participation. All of the participants were right handed, and two reported having previously received a diagnosis of attention deficit hyperactivity disorder (ADHD) or specific learning disorder. Procedure Each participant completed the experimental task in a quiet laboratory environment in an approximately 15 minute timeframe by appointment. Each participant was informed that the study’s purpose was to examine how college students think about numbers. They signed a hard copy informed consent form and completed a brief questionnaire containing demographic questions. For the experimental task, a series of simultaneously displayed pairs of single digit numbers were shown on a 20 inch iMac desktop monitor with a 1280 x 1024 pixel screen resolution in the upper right hand and upper left hand corners. The pairs were created using the Arabic numerals 1, 2, 8, 9, which were displayed in Arial font in point sizes 22 for small trials and 28 for large trials. Two numerical distance categories were created: close (1 – 2, 8 – 9) and far (1 – 8, 2 – 8, 1 – 9, 2 – 9). Congruent trials were those in which the correct response (larger physical size) was also the numerically larger digit (e.g., 2 8), and incongruent trials were those in which the correct response was the numerically smaller digit (e.g., 2 8). Neutral trials were generated using digits differing only in the relevant feature, physical size (e.g., 2 2, 8 8). Each of 17 the six potential pairings appeared in a randomized order eight times — two congruent and two incongruent for each correct answer trajectory (right or left) — in each of the 5 blocks. Additionally, each block contained 3 randomized presentations of each of the neutral pairs, resulting in 360 stimuli. MouseTracker was used for both stimulus presentation and data collection. It is a software program designed by Freeman and Ambady (2010) and it is offered for free download at http://mousetracker.jbfreeman.net. I instructed those participating in the study to quickly select the digit that was larger in physical size, disregarding the value of the numbers, using a computer mouse. In order to activate each trial, a START button in the bottom center of the screen was clicked. Responses initiated after 400 ms were followed by warning, and incorrect responses were followed by a 2000 ms display of an “X.” Seventy x and ycoordinate pairs along the mouse trajectory were recorded each second. Results Participants completed a combined total of 10,800 trials. Fifty two (0.48%) of the trials were excluded due to the selection of the incorrect response, as were 157 (0.15%) with response speeds beyond 3 standard deviations from the mean for correct selections. R, which is freely available for download at www.Rproject.org, was used for subsequent analyses on the remaining 10,591 trials MouseTracker calculates two response time measures: initiation time (IT) and reaction time (RT). IT is the amount of time taken to begin moving the mouse after the presentation of a stimulus, and RT is the total amount of time taken to complete the response. Because RT includes IT, the time spent in motion towards the correct response was calculated by subtracting IT from RT for each trial to obtain the movement time (MT). For analysis, both MT and IT were 18 aggregated and averaged by condition for each participant (see Table 1). Neutral trials, or those in which differing sizes of the same numerical digit were presented, were excluded in any analyses using numerical distance as a factor. Table 1 Mean movement times (MT) and initiation times (IT), in milliseconds, by congruity, distance, and trajectory direction. Leftward Trajectories Rightward Trajectories Numerical Distance Close Far Close Far Congruent MT 1181 1183 1195 1192 IT 128 128 132 123 Incongruent MT 1214 1213 1206 1241 IT 132 136 139 129 Response Time Analyses A 2 (physicalnumerical size congruity: congruent vs. incongruent) x 2 (numerical distance: close vs. far) x 2 (trajectory direction: leftward vs. rightward) repeated measures analysis of variance was conducted for MT. A significant main effect of physicalnumerical size congruity revealed that participants completed movement on congruent trials 31 ms faster than incongruent trials, F(1,29) = 9.83, p = .004, ηp2 = .25 (see Figure 1). No significant main effect of numerical distance emerged, F(1,29) = 1.85, p = .18, or trajectory direction, F(1,29) = 1.02, p = .32, and no significant interaction of congruity and numerical distance, F(1,29) = 1.76, p = .20, or congruity and trajectory direction, F(1,29) = .002, p = .97. 19 A 2 (physicalnumerical size congruity: congruent vs. incongruent) x 2 (numerical distance: close vs. far) x 2 (trajectory direction: leftward vs. rightward) repeated measures analysis of variance was conducted for IT. There were no significant main effects of congruity, F(1,29) = 2.66, p = .11, numerical distance, F(1,29) = 2.41, p=.13, or trajectory direction, F(1,29) = .04, p = .85. There were no significant interactions of congruity and numerical distance, F(1,29) = .10, p = .75, or congruity and trajectory direction, F(1,29) = 5.8, p = .92. Figure 1. Mean MT as a function of condition and distance. Trajectory Analyses In order to compare trajectories of various lengths, MouseTracker normalizes response measures by creating x and y coordinates using 100 default time steps. The software provides two measures of trajectory complexity: area under the curve (AUC) and maximum deviation (MD). AUC is a geometric measure of the difference in area between the ideal, straight line 20 trajectory between the starting point and the correct response and the actual trajectory of the motor response during selection. MD is the longest perpendicular distance connecting the path of the actual trajectory and the perfect, straight line trajectory. Table 2 contains mean AUC and MD values. 21 Table 2 Mean AUC and MD by congruity, distance, and trajectory direction. Leftward Trajectories Rightward Trajectories Numerical Distance Close Far Close Far Congruent AUC .65 .56 .65 .58 MD .36 .33 .32 .30 Incongruent AUC .86 .93 .79 .90 MD .45 .48 .40 .44 I conducted a 2 (physicalnumerical size congruity: congruent vs. incongruent) x 2 (numerical distance: close vs. far) x 2 (trajectory direction: leftward vs. rightward) repeated measures analysis of variance for AUC using each participants average AUC value. There was a significant main effect of congruity, with larger mean AUC values for incongruent than congruent trials, F(1, 29) = 27.93, p < .001, ηp2 = .49. There was no significant main effect of distance, F(1, 29) = .004, p = .95, or trajectory direction, F(1,29) = .09, p = .76. The interaction between congruity and distance was significant, F(1,29) = 5.28, p = .03, ηp2 = .15, with larger differences in AUC on congruent and incongruent trials with increased numerical distance (see Figure 2). A 2 (physicalnumerical size congruity: congruent vs. incongruent) x 2 (numerical distance: close vs. far) x 2 (trajectory direction: leftward vs. rightward) repeated measures analysis of variance was conducted for MD using the average values for each participant. As with AUC, a significant main effect of congruity was found, with larger MD for incongruent than congruent trials, F(1,29) = 37.13, p < .001, ηp2 = .56. There was no significant main effect of distance F(1,29) = .05, p = .83, or trajectory direction, F(1,29) = 1.35, p = .26. There was a 22 marginally significant interaction between congruity and distance, with larger MD on far trials, F(1,29) = 4.88, p = .04, ηp2 = .14. Figure 2. Mean response trajectories by trajectory direction, congruity, and numerical distance Facilitation and Interference In order to differentiate the contribution of facilitation and interference to the size congruity effect found in the initial analyses, analyses including the neutral trials were conducted. Table 3 contains the means for all response time and trajectory measures by congruity and trajectory direction. 23 Table 3 Mean movement times (MT) and initiation times (IT), in milliseconds, by congruity and trajectory direction. Congruent Neutral Incongruent Trajectory Direction Left Right Left Right Left Right MT (ms) 1182 1193 1189 1191 1213 1229 IT (ms) 128 126 132 129 134 132 AUC .59 .60 .75 .67 .91 .86 MD .34 .31 .41 .34 .47 .43 Response Time. A 3 (physicalnumerical size congruity: congruent vs. neutral vs. incongruent) x 2 (trajectory direction: leftward vs. rightward) repeated measures analysis of variance was conducted for MT. There was an overall main effect of congruity on MT, F(2,58) = 10.34, p < .001, ηp2 = .26. Planned contrasts among congruity conditions revealed a significant difference in mean MT between incongruent versus neutral trials, F(1, 58) = 20.645, p <.001, ηp2 =.26, but not between congruent versus neutral trials, F(1, 58) = .034, p = .85. Participants completed congruent trials the fastest, followed by neutral and incongruent, respectively (see Figure 3). There was no significant main effect of trajectory direction, F(2,58) = .63, p = .54. A 3 (physicalnumerical size congruity: congruent vs. neutral vs. incongruent) x 2 (trajectory direction: leftward vs. rightward) repeated measures analysis of variance was conducted for IT. There was no overall main effect of congruity, F(2,58) = 2.35, p = .10, or trajectory direction, F(2,58) = .004, p = .99. 24 Figure 3. Mean MT by congruity. Trajectory. A 3 (physicalnumerical size congruity: congruent vs. neutral vs. incongruent) x 2 (trajectory direction: leftward vs. rightward) repeated measures analysis of variance was conducted for AUC. A significant overall effect of congruity for AUC emerged, F(2,58) = 29.61, p < .001, ηp2 = .51 (see Figure 4). Planned contrasts revealed a significant difference in AUC between congruent versus neutral trials, F(1,58) = 8.96, p = .004, ηp2 = .13, and between incongruent versus neutral trials, F(1,58) = 50.23, p < .001, ηp2 = .46. No significant main effect of trajectory direction emerged, F(2,58) = 1.55, p = .22. 25 Figure 4. Mean response trajectories as a function of congruity. A 3 (physicalnumerical size congruity: congruent vs. neutral vs. incongruent) x 2 (trajectory direction: leftward vs. rightward) repeated measures analysis of variance was conducted for MD. There was a significant overall effect of congruity, F(2,58) = 38.28, p < .001, ηp2 = .57. Planned contrasts revealed significant differences between congruent versus neutral trials, F(1,58) = 13.56, p < .001, ηp2 = .19, and incongruent versus neutral trials, F(1,58) = 62.99, p < .001, ηp2 = .52. There was no significant main effect of trajectory direction, F(2,58) = 1.12, p = .334. Discussion The results provide confirmation for the predictions in Experiment 1. Participants completed the response motion faster for congruent than incongruent trials, and exhibited less pull towards the distractor as measured by AUC and MD. The difference between congruent and 26 incongruent trajectories also increased as a function of distance, suggesting that there is more distinct activation of the numerically greater digit’s associated magnitude when the compared digits are farther apart. This is consistent with Moyer and Landauer’s (1967) numerical distance effect. With the introduction of the neutral trials, facilitation was evident in AUC and MD, and interference was evident in MT, AUC, and MD. It is interesting to note that had the trajectory data not been available, I would not have been able to identify facilitation using only the performance measure data. In addition to providing additional evidence of the facilitation and interference components of the size congruity effect, these findings also support the late interaction model (Faulkenberry et al., 2016; Santens & Verguts, 2011; Sobel et al., 2016; Sobel et al., in press). As is evident in Figures 2 and 4, the response trajectories are smooth, without sudden corrective movements. The trajectory analyses provide evidence that competition continues throughout the motor response, which cannot be explained by an early interaction account in which the decision is made prior to physical selection. 27 Chapter 3 EXPERIMENT 2 The goal of Experiment 2 was to replicate the findings of Experiment 1, and to explore the value of facilitation and interference in predicting individual differences in math achievement. I predicted that the comparison of congruent and incongruent trials would reveal the size congruity effect, and that the differences would increase with numerical distance. I hypothesized that facilitation and interference would be evident in the measures of movement time and trajectory when comparisons were made among congruent, neutral, and incongruent trials. I also predicted that facilitation, or the beneficial effect of congruity on responses, would be associated with higher math achievement score. This prediction is consistent with Rubinsten and Henik (2005, 2006), who did not find facilitation in individuals with lower mathematical fluency. Method Participants As in Experiment 1, students were recruited primarily from classes at Tarleton State University for which research participation is a requirement or extra credit option. Seventy participants (76% female), with a mean age 24 years (SD = 9.8) and range of 1856, completed the experimental task and the additional measures in a 3040 minute individual session. A historical diagnosis of either ADHD or a specific learning disorder was reported by 8 participants, with 3 indicating multiple diagnoses, and 84% were right handed. Materials and Procedure Sowinski, Dunbar, & LeFevre’s (2014) Calculation Fluency Test (CFT) and Steiner and Ashcraft’s (2012) Brief Mathematics Assessment – 3 (BMA) were used to assess math 28 achievement. The CFT is comprised of 60 each of three problem types: one page of two digit addition, one page of two digit subtraction, and one page of two digit by onedigit multiplication problems. Participants were instructed to complete as many as they correctly could on each page in one minute without skipping any problems. The CFT was scored by counting correct responses for each problem type. The BMA contains 10 problems obtained from the Wide Range Achievement Test: Third Edition (WRAT3; Jastak Associates, 1993), ranging from mathematical operations and fractions to algebraic factoring. Participants were given 10 minutes to complete as many problems as they could, in order, and were instructed to stop if they reached one they were unable to solve. The BMA is also scored by simply counting correct responses (mean score 6.3). Scores on the CFT and the BMA had high internal consistency (Cronbach’s alpha = .81), so a single math achievement score was calculated for each participant. For each participant, the scores for both the CFT and the BMA were transformed to zscores; the resulting zscores were averaged in order to obtain a single value for math achievement. Form A of the abbreviated Standard Progressive Matrices (ASPM) test (Bilker et al., 2012) was used as a measure of fluid intelligence. It contains 9 items from the original Standard Progressive Matrices (Raven, Raven, & Court, 2000), each of which requires that participants identify the missing piece of an image from the presented options. The raw score for each participant was obtained by counting the correct responses (mean score 6.5). To control for order effects, a balanced Latin square design was used to identify 4 possible task sequences (A, B, C, D; see Table 4). Twenty packets of materials were created for each of the possible sequences and a random number was assigned to each in a spreadsheet. The spreadsheet was then sorted based on the random number, and a participant number was assigned 29 based upon ordinal position. A sort based on task order allowed for the assignment of participant numbers to the 4 sets of packets, which were then combined and ordered by participant number. Table 4 Order of tasks for each of the sequence groups (A, B, C, D). First Task Second Task Third Task Fourth Task A CFT BMA ASPM Experiment B BMA Experiment CFT ASPM C Experiment ASPM BMA CFT D ASPM CFT Experiment BMA The procedure for the experimental task in Experiment 2 was the same as that used in Experiment 1. The stimuli were created using the Arabic numerals 1, 2, 8, and 9 presented in pairs in differing font point sizes: 22 (small) and 28 (large). Numerical distance was manipulated by creating close (1 – 2, 8 – 9) and far (1 – 8, 2 – 8, 1 – 9, 2 – 9) pairs. Each possible pair was presented twice for each congruity condition (congruent, incongruent, and neutral) for each trajectory direction (right or left) in 5 randomized blocks. With the 3 randomized presentation of each neutral pair per block, each participant was presented with a total of 360 trials. Results Seventy participants completed a combined total of 25,200 trials. One participant responded incorrectly on all incongruent trials with leftward trajectories and far numerical distance, and was excluded from analysis. Ninetynine (0.40%) of the remaining 24,840 trials were excluded due to the selection of the incorrect response, as were 165 (0.66%) with reaction times beyond 3 standard deviations from the mean for correct selections. Subsequent analyses were performed on the remaining 24,576 trials. 30 Response Time Analyses A 2 (physicalnumerical size congruity: congruent vs. incongruent) x 2 (numerical distance: close vs. far) x 2 (trajectory direction: leftward vs. rightward) x 2 (sex: female vs. male) x 2 (history of diagnosis: diagnosis vs. no diagnosis) analysis of variance was conducted for MT and IT. Table 5 contains the mean MT and IT values. Table 5 Mean movement times (MT) and initiation times (IT), in milliseconds, by congruity, distance, and trajectory direction Leftward Trajectories Rightward Trajectories Numerical Distance Close Far Close Far Congruent MT 1202 1200 1237 1207 IT 125 121 126 125 Incongruent MT 1226 1259 1261 1286 IT 126 124 122 124 For MT, there were no between subjects main effects of sex, F(1,65) = .34, p = .56, or diagnosis, F(1,65) = 3.46, p = .07. A significant main effect of physicalnumerical size congruity was found, with completion of movement on congruent trials 46 ms faster than incongruent trials, F(1,65) = 56.97, p < .001, ηp2 = .47 (see Figure 5). The main effect of trajectory direction was also significant, F(1,65) = 8.93, p = .004, ηp2 = .20, with participants completing leftward trajectory trials 25 ms faster than rightward, regardless of congruity. There was no impact of distance on MT, F(1,65) = 2.86, p = .10. A significant interaction emerged between congruity 31 and distance, F(1,65) = 22.63, p < .001, ηp2 = .26, with a larger effect of congruity on trials with greater numerical distance. Figure 5. Mean MT as a function of congruity and distance. For IT, there was a significant between subjects main effect of sex, F(1,65) = 6.87, p = .01, ηp2 = .10, with males initiating movement 32 ms faster than females. The main effect of congruity was not significant, F(1,65) = 0.00, p = .99, but there was a significant interaction between congruity and diagnosis, F(1,65) = 7.82, p = .007, ηp2 = .11. Participants with a history of diagnosis initiated movement faster than those without a history of diagnosis, and had a greater difference in reaction time between congruent and incongruent trials (see Figure 6). No significant main effects of distance, F(1,65) = .90, p = .35, or trajectory direction, F(1,65) = .10, 32 p = .75 were found, but there was a significant interaction between congruity and trajectory direction, F(1,65) = 4.40, p = .04, ηp2 = .06. On leftward trajectory trials, the initiation time was 3 ms faster for congruent trials; this was reversed on rightward trajectories, with 3 ms faster IT on trials with incongruent stimuli. Figure 6. Mean IT as a function of history of diagnosis and distance. Trajectory Analyses A 2 (physicalnumerical size congruity: congruent vs. incongruent) x 2 (numerical distance: close vs. far) x 2 (trajectory direction: leftward vs. rightward) x 2 (sex: female vs. male) x 2 (history of diagnosis: diagnosis vs. no diagnosis) repeated measures analysis of variance was conducted for AUC and MD. Table 6 contains mean AUC and MD values. 33 Table 6 Mean AUC and MD by congruity, distance, and trajectory direction. Leftward Trajectories Rightward Trajectories Numerical Distance Close Far Close Far Congruent AUC .66 .61 .62 .53 MD .37 .35 .32 .28 Incongruent AUC .91 1.01 .88 .96 MD .49 .53 .43 .46 There were no between subjects main effects of sex, F(1,65)=.08, p=.78, or diagnosis, F(1,65)=.91, p=.35. Congruity did impact AUC, with higher mean values for incongruent than congruent trials, F(1, 65) = 102.76, p < .001, ηp2 = .61. Significant main effects of distance, F(1, 65) = 0.16, p = .69, or trajectory direction, F(1,65) = .78, p = .38, were not found. The interaction between of congruity and distance was significant, F(1,65) = 17.17, p < .001, ηp2 = .21, with greater differences in trajectory complexity for congruent and incongruent trials with increased numerical distance (see Figure 7). A 2 (physicalnumerical size congruity: congruent vs. incongruent) x 2 (numerical distance: close vs. far) x 2 (trajectory direction: leftward vs. rightward) x 2 (sex: female vs. male) x 2 (history of diagnosis: diagnosis vs. no diagnosis) repeated measures analysis of variance was conducted for MD. There were no significant between subject main effects of sex, F(1,65) = .16, p = .70, or diagnosis, F(1,65) = .31, p = .58. Like AUC, a significant main effect of congruity was found, with larger MD for trials with incongruent stimuli than trials with congruent trials, F(1,65) = 154.16, p < .001, ηp2 = .70. There was also a significant main effect of trajectory direction, F(1,65) = 5.52, p = .02, ηp2 =.08, with more complex responses on leftward 34 trajectories. While there was no significant main effect of distance F(1,65) = .04, p = .85, an interaction between congruity and distance, F(1,65) = 16.72, p < .001, ηp2 = .20, was found. Figure 7. Mean response trajectories by trajectory direction, congruity, and numerical distance. Facilitation and Interference Neutral trials were included for comparison with the purpose of further exploring the overall size congruity effect observed in the initial analyses. Difference between trials with neutral stimuli and trials with congruent stimuli represent facilitation, while difference between trials with neutral stimuli and trials with incongruent stimuli represent interference. Table 7 contains the means for all dependent measures by congruity and trajectory direction. 35 Table 7 Mean movement times (MT) and initiation times (IT), in milliseconds, by congruity and trajectory direction. Congruent Neutral Incongruent Trajectory Direction Left Right Left Right Left Right MT (ms) 1201 1217 1223 1232 1248 1278 IT (ms) 122 126 124 129 125 123 AUC .63 .56 .77 .65 .98 .93 MD .36 .29 .42 .33 .51 .45 Response Time. I performed a 3 (physicalnumerical size congruity: congruent vs. neutral vs. incongruent) x 2 (trajectory direction: leftward vs. rightward) x 2 (sex: female vs. male) x 2 (history of diagnosis: diagnosis vs. no diagnosis) repeated measures analysis of variance for both MT and IT using the average values for each participant. MT was not impacted by sex, F(1,65) = .41, p = .52, or diagnosis, F(1,65) = 3.52, p = .07. There was an overall main effect of congruity on MT, F(2,130) = 48.98, p < .001, ηp2 =.43. Planned contrasts between congruity conditions revealed a difference in mean MT between congruent versus neutral trials, F(1, 130) = 11.99, p < .001, ηp2 =.8, with participants completing congruent trials faster than neutral trials, and between incongruent versus neutral trials, F(1,130) = 85.95, p < .001, ηp2 =.40 , with participants completing neutral trials faster than incongruent trials (see Figure 8). 36 Figure 8. Mean MT by congruity. IT was impacted by sex, F(1,65) = 6.55, p = .01, ηp2 = .09, with males initiating movement 31 ms faster than females. There was no main effect of diagnosis, F(1,65) = .39, p = .53, or of congruity, F(2,130) = 1.85, p = .16. However, planned contrasts between congruity conditions did reveal an interaction of diagnosis, with participants with a history of diagnosis initiating movement 3 ms slower on incongruent than on neutral trials, F(1,130) = 10.26, p = .002, ηp2 = .07 (Figure 9). There was a marginally significant overall main effect of trajectory direction, F(1,130) = 3.22, p = .04, ηp2 =.05, with leftward trajectory movement initiating 2 ms faster than rightward trajectory movement. 37 Figure 9. Mean IT by congruity and history of diagnosis. Trajectory. A 3 (physicalnumerical size congruity: congruent vs. neutral vs. incongruent) x 2 (trajectory direction: leftward vs. rightward) x 2 (sex: female vs. male) x 2 (history of diagnosis: diagnosis vs. no diagnosis) repeated measures analysis of variance was conducted for AUC and MD using the average values for each participant. For AUC, there were no between subjects main effects of sex, F(1,65) = .16, p = .69, or diagnosis, F(1,65) = .87, p = .36. A significant overall effect of congruity for AUC was found, F(2,130) = 95.67, p < .001, ηp2 = .60 (see Figure 10). Planned contrasts revealed a significant difference in AUC between congruent versus neutral trials, F(1,130) = 18.66, p < .001, ηp2 = .13, and between incongruent versus neutral trials, F(1,130) = 172.68, p = .001, ηp2=.57. There was no significant main effect of side, and no significant interactions (all F values less than 2.4). 38 The results for MD echoed those for AUC, with a significant overall effect of congruity, F(2,130) = 135.53, p < .001, ηp2 = .68, a difference between congruent versus neutral trials, F(1,130) = 30.63, p < .001, ηp2 = .19, and incongruent versus neutral trials F(1,130) = 240.42, p < .001, ηp2 = .65. There were no significant main effect of trajectory direction, and no interactions (all F values less than 1.6). Figure 10. Mean response trajectories as a function of congruity. Predicting Math Achievement A 2 (sex: female vs. male) x 2 (handedness: right vs. other) x 2 (diagnosis: yes vs. no) x 4 (order: A, B, C, D) was conducted for both math achievement and standard progressive matrices scores. For math achievement, there was a significant main effect of diagnosis, F(1,47) = 6.54, p = .01, ηp2 = .12; participants with no reported history of ADHD or LD diagnosis scored higher than those with a history of diagnosis (see Table 8). There were no significant main effects of 39 sex, F(1, 47) = 1.32, p = .26, handedness, F(1,47) = 2.02, p = .14, or task order, F(1,47) = 2.04, p = .12, and no significant interactions. For scores on ASPM, there were no main effects and no significant interactions (all F ratios were less than 1.96). Table 8 Mean math achievement (zscores) and ASPM. Math Achievement ASPM Overall .018 6.51 Sex Female .104 6.57 Male .424 6.31 Handedness Right .196 6.51 Left .858 6.30 Both .725 7.03 Diagnosis ADHD or LD .159 6.58 No Diagnosis 1.37 5.94 Task Order A .210 6.44 B .639 6.38 C .343 6.40 D .548 6.83 One of the goals of Experiment 2 was to identify variables that may be valuable in the prediction of math achievement so Pearson’s productmoment correlation (r) was calculated for each of the potential predictors and math achievement. Table 9 contains the bivariate correlation values for each of the predictor variables and math achievement. 40 Table 9 Correlation coefficients for math achievement by variable (p < .05 = *, p < .01 = **, p < .001 = ***). Sex Diagnosis Task Order r F M Yes No A B C D Age .42*** .50*** .46 .65 .46*** .34 .55* .57** .47 ASPM .24* .30* .14 .16 .19 .18 .01 .42 .49* Facilitation MT .07 .17 .27 .27 .14 .40 .45 .20 .11 IT .00 .04 .05 .29 .12 .13 .12 .21 .08 Trajectory .02 .04 .05 .36 .02 .21 .26 .26 .35 Interference MT .03 .17 .45 .05 .05 .26 .13 .11 .08 IT .11 .15 .03 .37 .00 .27 .03 .06 .02 Trajectory .01 .07 .32 .47 .02 .08 .10 .01 .13 In order to screen extreme independent variable values in preparation for regression analyses, Mahalanobis scores (Mahalanobis, 1936; Stevens, 1984) were calculated for each participant; no participants were beyond the cut off Mahalanobis score. Five participants had more than expected leverage on the dependent variable as indicated by elevated hat values, and 5 participants were identified as having significant influence, or leverage and discrepancy, according to Cook’s values. Two participants meeting more than 1 indicator of extreme scores were excluded. In order to ensure that there was not considerable multicollinearity among independent variables, a correlation matrix was created. None of the correlations between the variables were excessively high (r < .70), so none were excluded. The assumptions of normality, linearity, homogeneity, and homoscedasticity were upheld upon inspection of relevant plots. A simultaneous multiple regression was conducted to determine if math achievement could be predicted by age, sex, history of diagnosis, task order, ASPM, facilitation (MT, IT, and trajectory), and interference (MT, IT, and trajectory). The model did explain a significant amount 41 of the variance present in scores of math achievement, F(15,51) = 3.82, p = .0002, R2 = .53, R2Adjusted = .39 (see Figure 11). In the overall model, none of the measures of facilitation or interference were significant predictors of math achievement. Table 10 contains the significant results of the simultaneous multiple regression. Table 10 Predictors of math achievement Variable b t β p Age .04 4.39 .49 <.001 Left Handedness .74 2.97 .36 .005 History of Diagnosis 1.13 3.91 1.48 <.001 Figure 11. Predicted math scores with confidence band (95%) as compared to actual data points. 42 In order to determine if the effects of facilitation (MT, IT, and trajectory) or interference (MT, IT, and trajectory) were modified by age, handedness, sex, diagnostic history, order, or ASPM, 5 regressions were conducted using each of the stated variables in the model. The model predicting math achievement using age as an interaction term was significant, F(13,55) = 2.44 p = .01, R2 = .37, R2Adjusted = .22 (see Table 11). Table 11 Predictors of math achievement with age as interaction term Variable b t β p Facilitation IT .11 2.17 1.36 .04 Age ~ Facilitation IT .005 2.19 .24 .04 Discussion The predictions related to the overall size congruity effect based upon a comparison of congruent and incongruent trials were confirmed. Response times were faster on congruent trials, and computer mouse trajectories were more complex on incongruent trials. Additionally, numerical distance impacted differences in response time and AUC. With the introduction of neutral trials, facilitation was discernable in congruent trials in MT, AUC, and MD, and interference was apparent in both measures of trajectory and MT for incongruent trials. The analysis of IT trajectories did not reveal facilitation, but did indicate interference in individuals with a history of diagnosis. Unanticipated findings included faster initiation times in males than females, and more complex trajectories on leftward responses as measured by MD. Experiment 2, as predicted, replicated the results of Experiment 1, providing support for the utility of computer mousetracking in examining the facilitation and interference components of the size congruity effect. Additionally, the results of Experiment 2 lend support to the late 43 interaction effect, which predicts that competition between two choices will continue throughout the motor response. This is precisely what was observed in an analysis of the trajectories. In an examination of individual differences in math achievement, participants with no history of diagnosis did have higher combined CFT and BMA scores. This difference was not apparent in the abbreviated ASPM scores; this finding suggests that the between group discrepancies in math achievement were not attributed to a difference in fluid intelligence. Although significant findings did emerge in the first regression analysis, facilitation was not a predictor of math achievement, as hypothesized. The most significant factors in the overall model were age, left handedness, and history of diagnosis. However, the interaction of age and IT facilitation was significant when age was introduced as a modifier in a regression model containing only the measures of facilitation and interference. 44 Chapter 4 GENERAL DISCUSSION The ability to compare the values of numerical symbols is integral to numeracy in humans, and develops out of an innate approximate number system (Girelli et al., 2000).With practice, the repeated pairing of a symbol with its corresponding magnitude will result in the automatic processing of the magnitude when presented with the symbol, even when it is irrelevant, as is evident in the size congruity effect. Experimentally, the size congruity effect was first demonstrated by Henik and Tzelgov (1982), who found that participants would more quickly identify the physically larger of two presented numbers when the correct choice was also the numerically larger value, or both features were congruent. According to Hasher and Zacks (1978), automatic processes are beneficial because they preserve limited cognitive resources. In terms of mental arithmetic, the automatic processing of numerical magnitude is valuable because it frees working memory for use in more complex problem solving. The size congruity effect has been supported by numerous studies, and the results of both Experiment 1 and 2 align with the literature. Participants spent less time in motion towards the correct selection during the motor response for congruent trials than incongruent trials. In Experiment 2, leftward trajectory responses were completed 25 ms faster than rightward trajectory responses. Participants also exhibited more complex movement towards the response on leftward trajectories, as measured by MD, than rightward. These two findings are inconsistent with one another; it is unlikely that a theoretical basis for both faster and more complex leftward trajectories exists. However, Faulkenberry et al. (2016) found a small effect on initiation time for leftward trajectories, so further exploration of the impact of trajectory direction on all of the measures would be valuable. 45 Lindsay and Jacoby (1994) posit that the size congruity effect is not attributable solely to the benefit of automatic processing, or facilitation. The irrelevant stimulus feature draws attention, therefore interfering with the response. These two components—facilitation and interference — can be examined through the comparison of congruent and incongruent trials, respectively, with neutral trials. The difference between congruent and neutral trials measures facilitation, while the difference between incongruent and neutral trials measures interference (Ashkenazi et al., 2008, Rubinsten & Henik, 2005, 2006). Again, the results of both Experiment 1 and 2 contribute to the growing body of research concerning the components of the size congruity effect. Responses were more efficient for congruent trials when compared to neutral trials, and less efficient for incongruent trials when compared to neutral trials. The unique contribution this study makes to the understanding of the components of the size congruity effect is the application of computer mousetracking in conjunction with the use of neutral stimuli. This relatively new data collection method has been used to examine the size congruity effect (Faulkenberry, Cruise, & Shaki, 2016; Geye, 2015), demonstrating more complex trajectories on incongruent than congruent trials. In the current study, results from both experiments replicated these findings; AUC and MD were larger on incongruent than congruent trials. This provides evidence for ongoing competition between the two choices throughout the motor response, which is consistent with the late interaction account (Santens & Vergus, 2011; Schwarz & Heinze, 1998). The late interaction model posits that the competition of possible choices continues through the motor response, whereas the early interaction model argues that the competition between two the two possible choices is resolved prior to the inception of the motor response. If this were the case, a significant difference in initiation times would have been observed in the comparison of congruent versus incongruent and neutral trials. 46 It can be argued that the instructions to quickly begin movement to respond, reinforced by the 400 ms cutoff for initiation, prematurely ends the representational stage described by the early interaction model. Faulkenberryet al. (2016) removed the speeded initiation component of the instructions, and found that the size congruity effect was still evident through continued competition in the response stage. No previous study has used computer mousetracking in a design containing neutral trials, allowing for the examination of facilitation and interference using trajectory data. As expected, the neutral trials were flanked by the congruent and incongruent trials on all measures: MT, IT, AUC, and MD. In Experiment 1, there was a significant difference between congruent and neutral trials (facilitation) for trajectory measures only, although MT did join AUC and MD in Experiment 2. Oddly, IT facilitation in Experiment 2 was significant for those with a history of diagnosis. Interference was significant for MT, AUC, and MD both Experiment 1 and in Experiment2. The trajectory data was uniquely useful in identifying facilitation, particularly in Experiment 1. These results underscore the usefulness of computer mousetracking as a measure of dynamic response. Geye (2015) found more complex trajectories for individuals in a low math achieving group than those in a typically math achieving group, regardless of congruity, which is inconsistent with the results of this study. Overall trajectory complexity was not correlated with math achievement, and neither facilitation nor interference, on any of the measures, predicted math achievement. It is possible that this lack of significance in predicting math achievement could be due to homogeneity of the population. When modified by age in the regression model, facilitation IT did become a significant predictor of math achievement, so perhaps a study with larger variability in age would yield more robust findings. Additionally, although there was 47 variation in math achievement within the current sample, there were no participants with a reported mathematics learning disability. Obtaining a sample with a significant number of individuals with very low math achievement may require sampling outside of the University setting; partnering with an openaccess community college or vocational training program may become necessary. Although math achievement was not predicted, as expected, by facilitation, other between subjects findings did emerge in Experiment 2. Participants with a history of diagnosis initiated movement faster, and had a greater difference between IT on congruent and incongruent trials than those without a history of diagnosis. Facilitation was evident in IT for those with a history of diagnosis, as well. This finding is likely an anomaly due to the small number of participants with a history of a diagnosis, but subsequent work may explore the possibility of a difference in processing. A few participants reported choosing to look at only one side of the screen in order to identify the correct response. Varying the font point sizes would help to mediate the impact of this approach to the task, although it is unlikely that results would differ substantially. In a recent study, Sobel, Puri, and Faulkenberry (2016) incorporated a visual search paradigm into the examination of the size congruity effect and found that top down processing does have a role in the size congruity effect that is not inconsistent with the late interaction effect. Future Directions This study is the first to employ computer mousetracking to explore the facilitation and interference components of the size congruity effect, and was therefore largely exploratory in nature. Future research should seek to recruit a more diversified sample in order to test more participants with very low math achievement. Efforts should also be made to more closely 48 examine the differences in results obtained for individuals with a history of diagnosis in order to discern whether the differences are truly attributable to diagnosis or if they would be better explained by sex. The prevalence of ADHD worldwide is estimated to between 3% to 7%, with a 3:1 maletofemale ratio in the population and an even more pronounced proportionality in clinical samples (Skogli, Teicher, Andersen, Hovik, & Øie, 2013). It is possible that a closer examination of this data would reveal that a maxing out of attentional resources would occur earlier for individuals with a history of a diagnosis than those without. In a study involving a driving simulation, Reimer, D’Ambrosio, Coughlin, Fried, and Biederman (2007) found that adults with ADHD succumbed to the monotony of a repetitive task more quickly than those in the control group. Conclusions These results highlight the usefulness of trajectory analysis in the examination of the size congruity effect. The dynamic data provided through this method allowed for a more fine grained study, leading to significant findings not possible through response time measures. The combination of computer mousetracking with the neutral stimulus provides the foundation for useful work related to the size congruity effect, specifically the role of attention (Risko et al., 2013). 49 REFERENCES Agrillo, C., Dadda, M., Serena, G., & Bisazza, A. (2009). Use of number by fish. Plos ONE, 4(3). doi: 10.1371/journal.pone.0004786 Ashkenazi, S., Henik, A., Ifergane, G., & Shelef, I. (2008). Basic numerical processing in left intraparietal sulcus (IPS) acalculia. Cortex, 44, 439448. doi: 0.1016/j.cortex.2007.08.008 Ashkenazi, S., Rubinsten, O., & Henik, A. (2009). Attention, automaticity, and developmental dyscalculia. 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As a Licensed Professional Counselor, Trina’s early career was in nonprofit and included case management for individuals with developmental disabilities seeking employment, counseling at risk youth and their families, and serving as Executive Director of a child advocacy agency. In 2006, Trina transitioned to higher education, becoming Coordinator of Student Success Programs at Tarleton. She was promoted to Director of Student Disability Services and Instructor of Psychology in 2007, and subsequently assumed responsibility over Student Assessment Services 2012, and Academic Support Center in 2014. As the Director of Academic Support Centers, she manages the Center for Access and Academic Testing, which is comprised of the testing center and office providing services to students with disabilities, and the Academic Resource Center, which provides programs such as Supplemental Instruction and peer tutoring. She also assists with curriculum development for the First Year Seminar, and serves on numerous committees (i.e., Campus Assessment, Response, and Evaluation; University Committee on Diversity, Access, and Equity; Developmental Education Advisory Committee; Reimaging the First Year; and Academic Advising Council). She continues to teach for the Department of Psychological Sciences. Trina was accepted into the PhD program in Psychology at Texas A&M UniversityCommerce in 2006. She married Jake Hoon in 2008; they have two sons, Ian and Gavin. Department of Psychological Sciences, Box T0820, Stephenville, Texas 76402 geye@tarleton.edu 
Date  2016 
Faculty Advisor  Henley, Tracy B. 
Committee Members 
Ball, Steven E. Faulkenberry, Thomas J. Green, Raymond J. 
University Affiliation  Texas A&M UniversityCommerce 
Department  PhD Educational Psychology 
Degree Awarded  Ph.D. 
Pages  69 
Type  Text 
Format  
Language  eng 
Rights  All rights reserved. 



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