
small (250x250 max)
medium (500x500 max)
Large
Extra Large
large ( > 500x500)
Full Resolution

THE DYNAMICS OF NUMBER REPRESENTATION IN COLLEGE STUDENTS WITH LOW MATH ACHIEVEMENT A Thesis 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 MASTER OF SCIENCE December 2015 THE DYNAMICS OF NUMBER REPRESENTATION IN COLLEGE STUDENTS WITH LOW MATH ACHIEVEMENT A Thesis by TRINA GEYE Approved by: Advisor: Tracy Henley Committee: Steve Ball Tom Faulkenberry Ray Green Head of Department: Jennifer Schroeder Dean of the College: Timothy Letzring Dean of Graduate Studies: Arlene Horne iii ABSTRACT THE DYNAMICS OF NUMBER REPRESENTATION IN COLLEGE STUDENTS WITH LOW MATH ACHIEVEMENT Trina Geye, MS Texas A&M UniversityCommerce, 2015 Advisor: Tracy Henley, PhD Magnitude representation is essential to understanding individual differences in math achievement (Henik, Rubinsten, & Ashkenazi, 2011). The tendency for numerical magnitude to interfere with comparisons of physical size is termed the size congruity effect (Henik & Tzelgov, 1982). In the present study, the realtime dynamics of the size congruity effect were analyzed in both low and typically math achieving college students using computer mousetracking. Participants selected the physically larger of two presented numbers, ignoring numerical value, by using a computer mouse to make the selection. A larger area under the curve for incongruent trials indicates competition from the activation of the irrelevant numerical magnitude representation. The low math achieving group demonstrated more complex trajectories than the typical math achieving group, regardless of congruency condition. Interestingly, there were no significant betweengroup differences for reaction time, suggesting that computer mousetracking is a useful tool for identifying individual differences in numerical cognition beyond performance measures. iv ACKNOWLEDGEMENTS I would like to thank my advisor, Dr. Tracy Henley, for the continual encouragement throughout the program and for taking on a project that was so far outside of the realm of his typical research interests; and also for the soup. Thanks to Dr. Ray Green and Dr. Steve Ball for second and third chances, and for sticking with me through advisor changes, topic reboots, and flooded work spaces. Special thanks to Dr. Tom Faulkenberry for taking me under his wing and convincing me that learning all about numerical cognition and R at the same time would be a great idea. His guidance, instruction, and generous sharing of resources have been invaluable throughout this process. I am especially grateful to my friends and family for the laughs, the tissues, and the pushes, and to my parents for instilling in me the value of an education (and dealing with my journeys off the beaten path!) Finally, to the rest of the HoonGeye Posse – my husband and our boys – I could have finished this program years ago without you, but what fun would that have been? Thank you for your unabated patience and love. v TABLE OF CONTENTS LIST OF TABLES .................................................................................................................... vii LIST OF FIGURES ................................................................................................................. viii CHAPTER 1. INTRODUCTION .................................................................................................... 1 Representation of Numerical Magnitude ............................................................. 3 Automaticity of Numerical Magnitude Representation ...................................... 5 Individual Differences in Automatic Representation of Numerical Magnitude ... 6 Computer Mousetracking ...................................................................................... 8 2. METHOD ................................................................................................................. 10 Participants ......................................................................................................... 10 Materials ............................................................................................................ 10 Procedure and Design ........................................................................................ 10 3. RESULTS ................................................................................................................. 13 Reaction Time Analyses .................................................................................... 13 Trajectory Analyses ............................................................................................ 16 4. GENERAL DISCUSSION ....................................................................................... 19 Limitations .......................................................................................................... 21 Future Directions ................................................................................................ 21 Conclusions ......................................................................................................... 22 REFERENCES ......................................................................................................................... 23 VITA ...................................................................................................................................... 29 vi LIST OF TABLES TABLE 1. CFT and BMA correlation matrix ................................................................................... 11 2. Mean performance measures, in milliseconds, by group and condition ......................... 13 3. Mean trajectory measures by group and condition ......................................................... 18 vii LIST OF FIGURES FIGURE 1. Graphical representation of a mental number line ............................................................ 4 2. Mean RT as a function of condition (congruent, incongruent) and distance (1, 2, 3, 4) for rightward trajectories ................................................................................................. 14 3. Mean RT as a function of condition (congruent, incongruent) and distance (1, 2, 3, 4) for leftward trajectories ................................................................................................... 15 4. Mean response trajectories as a function of direction (right versus left trajectory), condition (congruent versus incongruent), and numerical distance (1, 2, 3, and 4) ....... 17 1 Chapter 1 INTRODUCTION In the United States, many adults do not master the level of mathematics expected of an eighth grader, resulting in a lack of preparedness for participation in our modern economy and completion of routine daily living tasks requiring quantitative skills (Geary, Hoard, Nugent, & Bailey, 2013) Although it is improbable that every individual struggling with numeracy skills as an adult has a domain specific learning disability in mathematics, the ambiguity of the diagnosis highlights both the heterogeneity of mathematical impairment and the lack of understanding of individual differences in numeracy in the general population (Kaufman et al., 2013). Despite a consensus in the field that a yet to be identified central nervous system disorder is the key to understanding domain specific deficits in learning, the differential diagnosis continues to favor identification in an academic setting, although the impact of those deficits certainly reaches beyond the classroom, particularly in adulthood (Scanlon, 2013). Even the recent revisions in the Diagnostic and Statistical Manual of Mental Disorders (DSM5) are a reflection of the needs of the field of education instead of psychology or neurology. Significant changes were made to the conceptualization of learning disability and attention deficit hyperactivity disorder (see Tannock, 2013). The chapters are sequenced according to lifespan development, with those diagnoses typically applying to children first. Among the disorders included in the neurodevelopmental chapter is specific learning disability, which has been collapsed into a single diagnosis rather than the previous four (reading disorder, disorder of written expression, mathematics disorder, and learning disorder not otherwise specified); practitioners are directed to include the affected domain(s) as a specifier (e.g., with impairment in mathematics). Specific learning disorder is described as “a neurodevelopmental 2 disorder with a biological origin that is the basis for abnormalities at a cognitive level that are associated with behavioral signs of the disorder” (American Psychiatric Association, 2013, p. 68). Although a biological origin is conceded in the discussion of diagnostic features, the academic skills dependent upon the affected underlying processes are attributed to intentional teaching rather than emergence during brain maturation. The new definition of learning disability is responsive to evidence that the most recent model is not an effective approach, in isolation, but is still problematic for identifying qualified individuals. The diagnostic criterion requiring that learning difficulties begin during schoolage years is a barrier in the identification of adults who have no history of educational impact. Additionally, adults not in a school setting may not have access to intervention services that would rule out lack of appropriate instruction. Scanlon (2013) argues that the classification of learning disability as a neurodevelopmental disorder effectively limits the diagnosis to adults who experienced academic difficulties as school age children, whereas subsuming the diagnosis under neurocognitive disorders, as the category currently exists, would require documenting a decline. Diagnosis of specific learning disability is based upon the need to identify individuals requiring services in an educational setting, not upon adherence to an operational definition of impairment for use by researchers exploring the underlying CNS disorder (Kavale, Holdnack, & Mostert, 2005). This ambiguity is compounded for those interested in math deficits, specifically, due to the paucity of research. The vast majority of empirical studies in the field of learning disability concern reading disability (Mazzocco & Myers, 2003), as do the reviews and recommendations for application (Aaron, 1997). Potential for the use of basic cognitive numeracy tasks in the assessment of and intervention for domain specific deficits is promising, 3 and is responsive to the call made by Kaufman et al. (2013) to utilize alternatives to purely curricular assessment. The diagnosis of specific learning disability relies on the assumption that domain specific deficits can exist independently. When two domains are shown to be dissociated by continued access to one domain while another becomes inaccessible, presumably the processes rely upon distinct physiological networks. For example, the existence of two systems for number representation (one for exact arithmetic and one for approximate arithmetic) has been supported by evidence obtained from fMRI. Results suggest that representation of numerical magnitude is dependent upon portions of the parietal lobes associated with visuospatial ability (Dehaene, Spelke, Pinel, Stanescu, & Tsivkin, 1999). It follows that deficiencies in these systems could manifest in the observable difficulties with math that are collectively labeled mathematics learning disability (MLD). Representation of Numerical Magnitude Evidence that humans process magnitude in much the same way that they process other physical stimuli, and not through recall from memory or counting, was presented by Moyer and Landauer (1967), who applied the same principles for examining comparison of visual and auditory stimuli to number representation. In making judgments of difference between two Arabic numerals, faster reaction times were associated with smaller values, and with larger difference between values. For example, when presented with two lines and asked to judge which is the longer, responses are faster when both lines are relatively small, and when the length difference is relatively great. In their experiment, Moyer and Landauer (1967) asked participants to simply choose the larger of two single digit numbers when presented. As expected, they found a negative relationship between numerical difference and reaction time; participants were able to more quickly identify the larger number when the numerical distance 4 was greater. This robust numerical distance effect was supported by Hinrichs, Yurko, and Hu (1981), who found that participants were able to choose the larger of 2 twodigit numbers more quickly the further apart they were in numerical value, even if comparison of the decades digit was sufficient for decision making; for example, it would be faster to compare 21 with 35 than 29 with 35. Dehaene, S., Dupoux, E., & Mehler, J. (1990) replicated the work of Hinrichs, Yurko, and Hu (1981) using a French sample and with varying controls for the comparison standard. The existence of mental number line (Figure 1) was proposed to explain why numerical comparisons appear to occur in much the same way as physical comparisons, such as size or length. According this model, numerical magnitudes are represented along a logarithmic mental number line. Numbers 1, 2, and 3 are distinct, and the associated magnitudes are activated quickly and accurately. Beginning with 4, however, magnitude representation begins to become fuzzy, with number 4 receiving the most activation, but 3 and 5 activating to a lesser extent. As magnitude increases, the number line becomes more compressed, with increasing overlap in activation. Figure 1. Graphical representation of a mental number line with overlapping activation curves peaking above each number (adapted from www.panamath.org) 5 The existence of the mental number line was supported by Dehaene, Bossini, and Giraux (1993). In this study, participants were asked to choose which of two numbers was odd or even by pressing a button corresponding to the selected choice (right or left). Regardless of parity, responses were faster for larger numbers with the right hand, and for smaller numbers with the left hand, suggesting that numbers are internally represented from left to right. This effect was termed the spatialnumerical association of response codes (SNARC). Subsequent work controlling for relative magnitude (Dehaene & Mehler, 1992) suggests that the activation of numerical magnitude was not only subject to psychophysical rules, but also irrepressible. Automaticity of Numerical Magnitude Representation In order to preserve limited cognitive resources for complex processes, some basic operations likely function across circumstances, and are therefore considered automatic. Automatic processes require minimal amounts of processing, and, as such, reduce the likelihood of cognitive overload. Some automatic processes are inherent, and others develop with practice (Hasher & Zacks, 1978). Automaticity is indicated when a stimulus is processed even when irrelevant to the task at hand. Stroop (1935) laid the groundwork for research into the automatic processing of the irrelevant features of stimuli in the classic experiment in which the names of colors were presented to participants in different font colors, some congruent (e.g., the word “red” in a red font color) and some incongruent (e.g., the word “red” in a blue font color). Participants were asked to simply say the color of the font, which they were able to do much easier in congruent trials. Results indicated that we continue to process the printed name of a color when asked only to focus on the color of the font. Girelli, Lucangeli, and Butterworth (2000) proposed that the process applied to compare magnitudes may be automatized in children until categorization allows for retrieval from 6 memory. In studies where participants have been asked to compare the physical size of numbers, difficulty in suppressing the numerical magnitude results in a size congruity effect (SCE; Henik & Tzelgov, 1982). The SCE occurs when two numbers are displayed and participants are asked to compare the physical size. When the number that is larger in magnitude is also in a larger font, reaction times are faster. This effect has also been obtained for symbols not previously associated with magnitude, such as Gibson figures (Tzelgov, Yehene, Kotler, & Alon, 2000). Applied to the study of numerical magnitude, the analogous presentation of symbolic numbers with the manipulation of physical features reveals Strooplike effects. When asked to choose the physically larger (in font size) of two numbers, reaction times are faster when the physically larger number is also the number with the largest magnitude. The internal magnitude associated with a numerical symbol is activated even when irrelevant to the task. This automatic processing of magnitude is essential for proficiency in numerical reasoning (Henik, Rubinsten, & Ashkenazi, 2011), and therefore critical to the understanding of variation in the ability to process numbers. Individual Differences in Automatic Representation of Numerical Magnitude Holloway and Ansari (2009) sought to explore the processing of numerical magnitude in individuals with low math ability in order to examine the potential relationship between basic numerical processing and math achievement. In their study, children aged 6 years through 8 years were instructed to select the numerically larger of two single digit Arabic numerals or the more numerous array of squares. The NDE was present only for the symbolic trials, not for the physical representations, suggesting that accessing the meaning of Arabic numerals may be key to understanding individual differences. They found a larger NDE to be associated with lower scores on a standardized test of mathematics, and propose that “noisier” mapping of symbols to magnitude can result in difficulty accessing exact magnitude, and, therefore, problems in 7 calculation. Notably, Rousselle and Noelle (2007) found an opposite relationship in MLD children, with reduced NDE associated with impairment in math. Rubinsten and Henik (2005) conducted the first of the few published studies examining SCE in MLD adults. Their design expanded upon the traditional Strooplike task used in these types of studies to include height and grayness of stimuli in addition to size as dimensions. As predicted, the control group exhibited a larger congruity effect for size, height, and grayness, with the SCE being the largest difference. These results were supported in a subsequent study by Ashkenazi, Rubinsten, and Henik (2009) who also explored the role of cognitive load in SCE. Participants were asked to compare the size of the two digits presented using a key press, and, for the load condition, were also asked to verbally compare two peripheral figures. While the noload condition replicated the previous results, nonMLD participants exhibited a diminished SCE in a load condition, suggesting that inability to recruit attention contributes to magnitude processing deficits in MLD. Although a diminished NDE and SCE in MLD has been obtained in experimental studies, extant literature does not provide any consensus in explaining why there appears to be a deficit in the automatic processing of magnitude in individuals with a domain specific math deficit. Rubenstin and Henik (2005) point out that the SCE observed in MLD adults was similar to the SCE found in first grade children and propose that schooling and cognitive maturation are likely associated. However, the questions of what, exactly, is learned and what processes mature remain. Proposed explanations include noisier mapping of magnitude onto symbols (Holloway & Ansari, 2009), difficulty inhibiting the natural logarithmic number line (Geary, Hoard, Nugent, & ByrdCraven, 2008), and problems in recruiting attention (Ashkenazi, Rubinsten & Henik, 2009; Risko, Maloney, & Fugelsang, 2013). 8 The collection of data beyond performance measures is essential to understanding and detecting the cognitive processes, and will provide the multidimensional examination critical heterogeneous disorder such as MLD. An alternative method of data collection that is more sensitive to the dynamics of underlying cognitive processes is possible through the use of mouse tracking software. Computer Mousetracking A method for collecting robust data representing cognitive processes in real time was introduced by Spivey, Grosjean, and Knoblich (2005) in their study of spoken word recognition. Participants were asked to identify which of two words was heard by clicking the selected response on the screen with a computer mouse. Hand trajectories were tracked by continuously recording the coordinates of the mouse. Results indicated that, when two words were phonetically similar, the trajectory was pulled towards the incorrect response early on in the trial. This “pull” towards the incorrect response reflects the interference caused by the activation of two competing mental representations. Freeman and Ambady (2010) presented the MouseTracker software as a resource for researchers interested in using this method to compute realtime processing data. Subsequently, computer mouse tracking has been recognized as having the potential to provide access to cognitive processes as they happen by relying on the intimate relationship between mental and motor dynamics (Freeman, Dale, & Farmer, 2011). Several recent studies have successfully applied hand tracking to the study of numerical cognition (Marghetis & Nunez, 2013; Santens, Goossens, & Verguts, 2011; Song & Nakayama, 2008). Faulkenberry (2014) proposed that decision making in a numerical parity (odd/even) task involves the activation of competing representations, providing a model for the application of the methodology in other numerical cognition tasks requiring forced choice comparison. This method has generated significant 9 interest and continues to undergo refinement and critique (see Fischer & Hartmann, 2014), but has demonstrated significant potential in revealing the nature of competing processes through the examination of previously inaccessible information (Faulkenberry & Rey, 2014). I hypothesized that participants with low math achievement would exhibit a smaller size congruity effect than those in the typically achieving group. Specifically, I predicted low math achievers would have smaller differences in reaction time between congruent and incongruent trials, and would experience less competing activation as evidenced by Area Under the Curve (AUC) and maximum deviation (MD) than their typically achieving peers. 10 Chapter 2 METHOD Participants 165 participants (mean age 19.9 years, 63% female) were recruited from Tarleton State University, primarily through courses in which faculty were willing to provide credit for research participation, including undergraduate psychology courses and first year seminar courses. From those, 25 participants meeting the criteria (see below) for low math achievement (LMA) were, along with matched typical math achievers (TMA), selected for inclusion in the study. Of the 50 selected, nine reported a previous diagnosis of ADHD or learning disability. Of those nine, eight were identified LMA, and only one was TMA. Materials Participants completed the Calculation Fluency Test (CFT; Sowinski, Dunbar, & LeFevre, 2014), which contains 60 twodigit addition, 60 twodigit subtraction, and 60 twodigit by onedigit multiplication problems. Participants also completed the Brief Mathematics Assessment3 (BMA; Steiner & Ashcraft, 2012), which consists of 10 increasingly difficult problems extracted from the Wide Range Achievement Test: Third Edition (WRAT3), including fractions and algebraic factoring. Additionally, Standard Progressive Matrices (Raven, Raven, & Court, 2000) were administered to those participants selected to complete the experimental task. Procedure and Design Participants completed an informed consent and a demographic questionnaire requesting information such as sex, age, handedness, and university math placement. It was explained that participation in the second phase of the study was to be scheduled at a later time for those selected. Both the CFT and the BMA were scored by totaling the number of correct responses 11 per page. Because the three CFT subtests (addition, subtraction, and multiplication; see Table 1) were highly correlated (Cronbach’s alpha = .93), a total CFT score was calculated. Table 1 CFT and BMA correlations CFT Add CFT Sub CFT Mult CFT Tot CFT Add CFT Sub .71 CFT Mult .62 .61 CFT Tot .90 .88 .84 BMA .36 .43 .34 .43 Total CFT and BMA scores were converted to zscores and added together in order to yield a composite math achievement score. Those scores falling in the 25th percentile (z ≤ 1.64) were assigned to the LMA group, while those who were at or above the mean (z ≥ 0.0) were considered to be TMA. There was a significant difference in the composite math achievement scores between groups, t(48) = 14.7, p < .001. Equally sized groups were created by randomly selecting 25 each of LMA and TMA participants (n = 50, 74% female), matched based on sex. Administration of the Standard Progressive Matrices (SPM) and the experimental task were scheduled and conducted with each participant individually. Raw SPM scores were calculated by totaling the number of correct responses for each participant. For the experimental task, participants were presented with a succession of pairs of numbers in differing font sizes displayed simultaneously in the topright and topleft corners of a 20 inch iMac desktop screen with a resolution of 1280 x 1024 pixels. For this study, stimuli were Arabic numerals 2, 3, 4, 5, 6, 7, and 8 in Arial font with point sizes 22 (small) and 28 (large). 12 The numerals were presented in pairs chosen to create numerical distance categories: 2  3, 3  4, 4  5 (distance 1); 2  4, 3  5, 4  6 (distance 2); 2  5, 3  6, 4  7 (distance 3); 2  6, 3  7, 4  8 (distance 4). The 12 possible pairs were presented in a randomized order two times in eight blocks, resulting in 192 stimuli. Congruent trials were those in which the numerically larger number was also in the larger font point size (e.g., 1 9), while in incongruent trials, the numerically smaller number was in the larger font point size (e.g., 1 9). For stimulus presentation and trajectory recording, MouseTracker was used, a software program developed by Freeman & Ambady (2010) and available for free download (http://mousetracker.jbfreeman.net). Participants were instructed to select the physically larger number, ignoring numerical size, as quickly as possible by clicking on the selected answer using a computer mouse. Clicking a START button before each trial activated the trial, and the streaming x and ycoordinates of mouse movement were recorded at a rate of 70 points per second. The intention was to capture trajectories representing online cognitive processes, not movements made postdecision. In order to encourage participants to begin moving towards their selection as soon as possible, movement initiated after 400 ms resulted in a warning. Incorrect responses resulted in the display of an “X” for 2000 ms. Participants were debriefed and thanked for their involvement in the study. A 2 (group; LMA vs. TMA) x 2 (congruity; congruent or incongruent) x 4 (distance; 1, 2, 3, 4) factorial design was used. Dependent variables included reaction time (comprised of initial time and movement time), average trajectory, and measures of dynamic complexity of hand movements (maximum deviation and Area Under the Curve). Maximum deviation (MD) is the largest perpendicular distance between the actual trajectory and ideal, straightline trajectory for all time steps, and (AUC) is the geometric area between the actual and ideal trajectories. 13 Chapter 3 RESULTS Participants completed a total of 19,200 trials for each trajectory direction (left and right). One participant did not successfully respond to any incongruent trials, and was therefore excluded, as were 61 incorrect trials (.32%) and 98 for which reaction times were more than 3 standard deviations beyond the mean for all correct response reaction times (.51%). Subsequent analyses were conducted using the remaining 18,657 trials; mean reaction times (RTs) and initiation times (ITs) are displayed in Table 2. Right trajectory and left trajectory analyses were conducted separately for all measures and will be presented independently. An analysis of SPM revealed higher scores for TMA than LMA participants, t(48) = 4.65, p < .001. Table 2 Mean reaction times (RT) and initiation times (IT), in milliseconds, by group (LMA, TMA) and condition Congruent Trials Incongruent Trials Numerical Distance 1 2 3 4 1 2 3 4 Rightward Trajectories LMA RT 1454 1428 1414 1394 1436 1445 1472 1450 IT 106 106 104 107 105 110 114 106 TMA RT 1327 1336 1325 1314 1354 1369 1382 1391 IT 127 128 130 129 123 126 134 125 Leftward Trajectories LMA RT 1427 1444 1438 1408 1454 1478 1482 1475 IT 101 108 97 101 103 108 106 104 TMA RT 1349 1372 1337 1346 1381 1371 1406 1403 IT 131 129 131 121 128 139 127 129 14 Reaction Time Analyses A 2 (group: LMA vs. TMA) x 2 (physicalnumerical size congruity: congruent vs. incongruent) x 4 (numerical distance: 1, 2, 3, 4) repeated measures analysis of variance was conducted for both measures. There was a significant main effect of physicalnumerical size congruity, for right trajectory RTs, with participants taking 39 ms longer on incongruent than congruent trials F(1,47) = 31.02, p < .001, p2 = .40. For numerical distance, there was no significant main effect, F(3, 141) = .666, p = .574. However, there was a significant interaction of physicalnumerical size congruity and distance, F(3, 141) = 5.87, p < .001, p2 = .11. As is evident in Figure 2, the difference in RTs between congruent and incongruent trials increased Figure 2. Mean RT as a function of condition (congruent, incongruent) and distance (1, 2, 3, 4) for rightward trajectories 15 with the numeral distance between the presented pairs. I tested for a linear contrast on the interaction term in the ANOVA model by performing a within subjects contrast, F(1,47) = 15.537, p = .009, p2 =.127. The main effect of group approached significance, with LMA taking longer to respond than TMA, F(1,47) = 3.20, p = .08. There were no differences for initiation times between conditions on any factor (all Fratios were less than 0.64). To examine left trajectory performance measures, a 2 x 2 x 4 repeated measures analysis of variance (see above) was conducted for RT and initiation times. There was a significant main effect of physicalnumerical size congruity on RT, with participants taking 41 ms longer on incongruent trials, F(1,47) = 29.08, p < .001, p2 = .38. Similar to the right trajectory results, no significant main effect of numerical distance was obtained, F(3,141) = .72, p = .54. There was a significant interaction of physicalnumerical size congruity and distance, F(3,141) = 2.80, p = .043 (see Figure 3). In order to test for a linear contrast on the interaction term in the ANOVA Figure 3. Mean RT as a function of condition (congruent, incongruent) and distance (1, 2, 3, 4) for leftward trajectories 16 model, I conducted a withinsubjects contrast, F(1,47) = 6.82, p = .011, p2 = .245. Again, RT differences between the LMA and TMA groups did not reach significance, F(1,47) = 2.12, p = .15. There were no significant effects for initiation times (all Fratios were less than 0.59). Trajectory Analyses To enable comparison among trajectories of different lengths, RTs were equalized by creating 100 x and y coordinate pairs for each trial. Mousetracker accomplishes this normalization by using a default of 101 time steps in order to permit 100 equal spaces between coordinate pairs. These normalized trajectories for both conditions (congruent and incongruent) were crossed with numerical distances 1, 2, 3, and 4 (see Figure 4). Area Under the Curve A 2 (group: LMA vs. TMA) x 2 (physicalnumerical size congruity: congruent vs. incongruent) x 4 (numerical distance: 1, 2, 3, 4) repeated measures analysis of variance was used to analyze the average AUC values for each participant. For right trajectories, a significant main effect of group emerged, with a higher AUC for LMA than TMA participants (see Table 3 for means), F(1, 47) = 4.73, p = .035, p2 = .09. There was a significant effect of physicalnumerical size congruity, with higher AUC on incongruent (m = 0.83) than congruent (m = 0.57) trials, F(1, 47) = 65.83, p < .001, p2 = .58. There was also a significant interaction effect of condition and distance, F(3,141) = 4.37, p < .006, p2 = .09. These results indicate that trajectories are significantly pulled towards the incorrect, numerically larger digit in incongruent trials. 17 Figure 4. Mean response trajectories as a function of direction (right versus left trajectory), condition (congruent versus incongruent), and numerical distance (1, 2, 3, and 4) 18 Table 3 Mean trajectory measures by group (LMA, TMA) and condition Results of the analyses of left trajectories were similar to the right trajectories. There was a significant main effect of group, with higher AUC for LMA than TMA, F(1, 47) = 4.38, p =.042, p2 = .09. There was also a main effect of condition (congruent vs. incongruent), F(1, 47) = 51.34, p < .001, p2 = .52. Again, there was a significant interaction effect for condition and distance, F(3, 141) = 4.43, p = .005. Maximum Deviation MD was also examined using the same 2 (group: LMA vs. TMA) x 2 (physicalnumerical size congruity: congruent vs. incongruent) x 4 (numerical distance: 1, 2, 3, 4) repeated measures analysis of variance model applied to AUC. For right trajectories, the main effect of group approached significance between LMA (m = 0.40) and TMA (m = 0.30), (F(1, 47) = 3.61, p = .06. There was a main effect of physicalnumerical size congruity, with greater MD for incongruent (m = 0.40) than congruent trials (m = 0.29), F(1, 47) = 75.04, p < .001, p2 = .58. Congruent Trials Incongruent Trials Numerical Distance 1 2 3 4 1 2 3 4 Rightward Trajectories LMA AUC 0.78 0.77 0.69 0.70 0.87 0.95 0.98 1.00 MD 0.38 0.37 0.33 0.35 0.42 0.45 0.44 0.46 TMA AUC 0.45 0.43 0.39 0.39 0.62 0.70 0.72 0.83 MD 0.24 0.23 0.23 0.21 0.32 0.35 0.37 0.41 Leftward Trajectories LMA AUC 0.83 0.74 0.76 0.71 0.94 0.98 1.05 1.02 MD 0.42 0.39 0.41 0.38 0.46 0.48 0.49 0.49 TMA AUC 0.53 0.56 0.42 0.50 0.70 0.71 0.80 0.82 MD 0.30 0.31 0.24 0.27 0.36 0.38 0.42 0.42 19 No main effect for distance was found, F (3, 141) = 0.46, p = .71. There was an interaction for condition and distance, with larger MDs associated with larger numerical distance, F(3, 141) = 3.71, p = .013, p2 = .82. The same pattern of results was found for left trajectories, with a main effect of group approaching significance between LMA (m = 0.44) and TMA (m = 0.34), F(1, 47) = 3.73, p = .06. There was a main effect of physicalnumerical size congruity, with greater MD for incongruent (m = 0.44) than congruent trials (m = 0.34), F(1, 47) = 63.69, p < .001, p2 = .58. No main effect of distance was found, F(3, 141) = 0.149, p = .93. Like the right trajectories, there was an interaction for condition and distance, F(3, 141) = 4.64, p = .004, p2 = .69. 20 Chapter 4 GENERAL DISCUSSION Because previous studies have not addressed the dynamics in the underlying cognitive processes associated with math ability with data beyond performance measures, the current study sought to investigate whether differences in the size congruity effect (SCE) would be evident in LMA and TMA participants using robust data available from the use of computer mousetracking software. Participants were assigned to either group (TMA or LMA) based upon scores on the CFT and the BMA. As expected, RTs were longer for trials in which the numerical value and physical size were incongruent than in congruent trials, indicating a SCE for both TMA and LMA participants. Analyses of computer mouse trajectories were consistent with RT results, with a larger AUC and MD for incongruent versus congruent trials, indicating a pull towards the incorrect alternative. Additionally, the differences in RT between congruent and incongruent trials increased as a function of numerical distance. Participants took longer to choose the correct number on incongruent as the difference between the two numbers increased. This discovery of a more complex trajectory for incongruent presentations is theoretically consistent with the SCE. Girelli, Lucangeli, and Butterworth (2000) proposed that the SCE represented an automatic representation of magnitude that gradually develops with improved numerical skill. The more automatic the processing of magnitude becomes, the more difficult it is to suppress that information. In the present study, both performance measures and trajectory analyses indicate that the incorrect response is activated during online processing of information related to the decision. 21 The between groups analyses, however, did not yield the expected results. It was hypothesized that a smaller SCE would be found in LMA participants when compared with TMA as measured by RT, and that LMA would have less competition from the incorrect response as measured by MD and AUC; this was not the case. There was no significant difference between groups for RT, although results approaching significance do suggest a trend of longer RTs for LMA. The results for AUC and MD are the opposite of the hypothesis, with a significantly larger AUC for LMA participants than TMA participants and a suggested trend towards similar results for MD. There was no significant difference in AUC between conditions with group as a factor. Previous research (Ashkenazi, Rubinsten, & Henik, 2009; Rubinsten & Henik, 2005) has found a smaller SCE in MLD adults through analysis of RT. Although the between groups analyses did not reveal significant differences for RT, the trend is reversed. Taken in isolation, this evidence might be considered an anomaly; however, when paired with the trajectory analyses, the results suggest that the dynamics of the physical comparison task are more complex than anticipated. LMA participants experienced more distraction from the incorrect response, regardless of congruity, suggesting that between group differences may not be due to differences in automatic representation of magnitude, but to attentional issues. Posner (1978) and Lindsay and Jacoby (1994) propose that the Strooplike effect found in cognitive tasks actually comprises two components  facilitation and inhibition  with facilitation associated with automaticity and inhibition associated with attentional processing. Rubinsten and Henik (2005) found only an interference component in a group of adults with developmental dyscalculia, whereas the control group exhibited a facilitation component in addition to interference. Our results may represent the interference component in the LMA participants as 22 supported by a significantly larger AUC. Subsequent studies using mousetracking to examine the SCE should include a neutral stimulus in which the numeral value is constant. This creates an opportunity for comparison to the incongruent stimulus for the interference component and the congruent stimulus for the facilitation component. The possibility of an interference component is consistent with the findings of Ashkenazi, Rubinsten, and Henik (2009) and Risko, Maloney, and Fugelsang (2013), who suggest a research focus on the recruitment of attention in physical size comparison tasks. In addition to providing direction for future research in the representation of numerical magnitude, this study also highlights the value of computer mousetracking as a research tool. The robust data collected through this method provided a unique opportunity to discover individual differences through the analysis of dynamic trajectories that would not have emerged through more traditional means. This tool provides the opportunity to reexamine previous efforts to understand individual differences in a variety of domains, collecting data beyond reaction time. Scanlon (2013) described our understanding of LD as “evolving” and the construct as being defined prematurely. MLD, specifically, is underrepresented in the literature and often assumed to be homogenous. Kaufman et al. (2013) propose that educational tests are inadequate to examine the basic numerical processing deficits associated with MLD, and that deficits in subdomains may be present in individuals who do not meet the diagnostic criteria. The current response to intervention model (RTI) is largely experimental (Kavale, Holdnack, & Mostert, 2005). Studies such as the present may be critical to identifying fine grain individual differences that will be key to moving forward in a definition of MLD that is responsive to research in cognitive processes rather than academic skill performance. Identifying individual differences in 23 basic numerical cognition processes will be useful in defining the presumed underlying CNS disorder and understanding the role of particular regions of the brain. Future Directions As there as a significant difference in the AUC for LMA participants, further research is warranted. Additional studies using this mousetracking method should replicate previous research using only performance measures to see if the robust data provided through the analysis of trajectories provides more insight. Specifically, research should explore the interference component of the SCE and the recruitment of attention in physical size comparison tasks, including the impact of stimulus presentation timing (Risko, Maloney, & Fugelsang, 2013), cognitive load (Ashkenazi, Rubinsten, & Henik, 2009), and attentional training (Askenazi & Henik, 2012) Conclusions The results of this study appear to suggest that the role of attention in magnitude comparison will be important to understanding individuals with domain specific deficits in math. Additionally, these results highlight the value of mousetracking as an alternative to traditional measures of performance; the difference in trajectories between LMA and TMA participants would have gone unnoticed measuring only reaction time. Further research is needed to replicate these results using neutral stimuli to allow for the isolation of facilitation and interference components of SCE. Continuing work in this area will be important to the evaluation of the emerging alternative explanation of SCE set forth by Risko, Maloney, and Fugelsang (2013), who propose that attention, rather shared representation, explains the difference in congruent versus incongruent stimuli. 24 REFERENCES Aaron, P. G. (1997). The impending demise of the discrepancy formula. Review of Educational Research, 67(4). doi: 10.3102/00346543067004461 American Psychiatric Association. (2013). Diagnostic and statistical manual of mental disorders (5th ed.). Arlington, VA: American Psychiatric Publishing. Ashkenazi, S., Rubinsten, O., & Henik, A. (2009). Attention, automaticity, and developmental dyscalculia. Neuropsychology, 23(4), 535540. doi: 10.1037/a0015347 Dehaene, S., Bossini, S., & Giraux, P. (1993). The mental representation of parity and number magnitude. Journal of Experimental Psychology, 122(3), 371396. doi: 10.1037/00963445.122.3.371 Dehaene, S., Dupoux, E., & Mehler, J. (1990). Is numerical comparison digital Analogical and symbolic effects in twodigit number comparison. Journal of Experimental Psychology: Human Perception and Performance, 16, 626641. doi: 10.1037/00961523.16.3.626 Dehaene, S., & Mehler, J. (1992). Crosslinguistic regularities in the frequency of number words. Cognition, 43(1), 129. doi: 10.1016/00100277(92)90030L Dehaene, S., Spelke, E., Pinel, P., Stanescu, R., & Tsivkin, S. (1999). Sources of mathematical thinking: Behavioral and brainimaging evidence. Science, 284, 970–974. doi: 10.1126/science.284.5416.970 Faulkenberry, T. J. (2014). Hand movements reflect competitive processing in numerical cognition. Canadian Journal of Experimental Psychology, 68(3), 147151. doi: 10.1037/cep0000021 25 Faulkenberry, T. J., & Rey, Amandine (2014). Extending the reach of mousetracking in numerical cognition: A comment on Fischer and Hartmann (2014). Frontiers in Psychology, 5, 1436. doi: 10.3389/fpsyg.2014.01436 Fischer, M. H., & Hartmann, M. (2014). Pushing forward in embodied cognition: May we mouse the mathematical mind? Frontiers in Psychology, 5, 1315. doi: 10.3389/fpsyg.2014 .01315 Freeman, J. B., & Ambady, N. (2010). MouseTracker: Software for studying realtime mental processing using a computer mousetracking method. Behavior Research Methods, 42(1), 226241. doi: 10.3758/BRM.42.1.226 Freeman, J. B., Dale, R., & Farmer, T. A. (2011). Hand in motion reveals mind in motion. Frontiers in Psychology, 2, 16. doi: 10.389/fpsyg.2011.00059 Geary, D. C., Hoard, M. K. Nugent, L., & ByrdCraven, J. (2008). Development of number line representations in children with mathematical learning disability. Developmental Neuropsychology, 33(3), 277299. doi: 10.1080/87565640801982361 Geary, D. C., Hoard, M. K., Nugent, L., & Bailey, D. H. (2013). Adolescents’ functional numeracy is predicted by their school entry number system knowledge. PLoS ONE, 8(1): e54651. doi:10.1371/journal.pone.0054651 Girelli, L., Lucangeli, D., & Butterworth, B. (2000). The development of automaticity in accessing number magnitude. Journal of Experimental Child Psychology, 76, 104122. doi: 10.1006/jecp.2000.2564 Hasher, L., & Zacks, R. T. (1978). Automatic and effortful processes in memory. Journal of Experimental Psychology, 108(3), 356388. doi: 10.1037/00963445.108.3.356 26 Henik, A., Rubinsten, O., Ashkenazi, S. (2011). The “where” and “what” in developmental dyscalculia. The Clinical Neuropsychologist, 25(6), 9891008. doi: 10.1080/13854046 .2011.599820 Henik, A., & Tzelgov, J. (1982). Is three greater than five: The relation between physical and semantic size in comparison tasks. Memory & Cognition, 10(4), 389395. doi: 10.3758/BF03202431 Hinrichs, J. V., Yurko, D. S, & Hu, J. M. (1981). Twodigit number comparison: Use of place information. Journal of Experimental Psychology: Human Perception and Performance, 7, 890901. doi: 10.1037/00961523.7.4.890 Holloway, I. D., & Ansari, D. (2009). Mapping numerical magnitudes onto symbols: The numerical distance effect and individual differences in children’s mathematics achievement. Journal of Experimental Child Psychology, 103, 1729. doi: 10.1016 /j.jecp.2008.04.001 Kaufman, L., Mazzocco, M. M., Dowker, A., von Aster, M., Goebel, S., Grabner, R., … Nuerk, H. (2013). Dyscalculia from a developmental and differential perspective. Frontiers in Psychology, 4, 516. doi: 10.3389/fpsyg.2013.00516 Kavale, K. A., Holdnack, J. A., & Mostert, M. P. (2005). Responsiveness to intervention and the identification of specific learning disability: a critique and alternative proposal. Learning Disability Quarterly, 28, 216. doi: 10.2307/4126970 Lindsay, D. S., & Jacoby, L. L. (1994). Stroop process dissociations: The relationship between facilitation and interference. Journal of Experimental Psychology: Human Perception and Performance, 20(2), 219234. doi: 10.1037/00961523.20.2.219 27 Marghetis, T., & Nunez, R. (2013). The motion behind the symbols: A vital role for dynamism in the conceptualization of limits and continuity in expert mathematics. Topics in Cognitive Science, 5, 299316. doi: 10.1111/tops.12013 Mazzocco, M. M. M., & Myers, G. F. (2003). Complexities in identifying and defining mathematic learning disability in the primary schoolage years. Annals of Dyslexia, 53, 218253. doi: 10.1007/s1188100300117 Moyer, R. S., & Landauer, T. K. (1967). Time required for judgments of numerical inequality. Nature, 215, 15191520. doi: 10.1038/2151519a0 Posner, M. I. (1978). Chronometric explorations of mind. Hillsdale, NJ: Erlbaum. Raven, J., Raven, J. C., & Court, J. H. (2000). Standard Progressive Matrices. Oxford: Oxford Psychologists Press. Risko, E. F., Maloney, E. A., & Fugelsang, J. A. (2013). Paying attention to attention: evidence for an attentional contribution to the size congruity effect. Attention, Perception, and Psychophysics, 75(6), 11371147. doi: 10.3758/s1341401304772 Rousselle, L., & Noel, M. (2007). Basic numerical skills in children with mathematics learning disabilities: A comparison of symbolic vs nonsymbolic number magnitude processing. Cognition, 102, 361395. doi: 10.1016/j.cognition.2006.01.005 Rubinsten, O., & Henik, A. (2005). Automatic activation of internal magnitudes: A study of developmental dyscalculia. Neuropsychology, 19(5), 641648. doi: 10.1037/08944105.19.5.641 Santens, S., Goossens, S., & Verguts, T. (2011). Distance in motion: Response trajectories reveal the dynamics of number comparison. PLoS ONE, 6(9), 16. doi: 10.1371/journal.pone.0025429 28 Scanlon, D. (2013). Specific learning disability and its newest definition: Which is comprehensive? and which is insufficient? Journal of Learning Disabilities, 46(1), 2633. doi: 10.1177/0022219412464342 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 Song, J., & Nakayama, K. (2008). Numeric comparison in a visuallyguided manual reaching task. Cognition, 106(2), 9941003. 10.1016/j.cognition.2007.03.014 Sowinski, C., Dunbar, K., & LeFevre, J. (2014). Calculation Fluency Test, unpublished technical report, Math Lab, Carleton University, Ottawa, Canada. Retrieved from http://carleton.ca/cacr/wpcontent/uploads/CalculationFluencyTestDescription _MathLabTechnicalReport.pdf Steiner, E. T., & Ashcraft, M. H. (2012). Three brief assessments of math achievement. Behavior Research Methods, 44(4), 11011107. doi: 10.3758/s1342801101856 Stroop, J. R. (1935). Studies of interference in serial verbal reactions. Journal of Experimental Psychology, 18(6), 643662. doi: 10.1037/h0054651 Tannock, R. (2013). Rethinking ADHD and LD in DSM5: Proposed changes in diagnostic criteria. Journal of Learning Disabilities, 46(1), 525. doi: 10.1177/00022219412464341 Tzelgov, J., Yehene, V., Kotler, L., & Alon, A. (2000). Automatic comparisons of artificial digits never compared: Learning linear ordering relations. Journal of Experimental Psychology: Learning, Memory, and Cognition, 26(1), 103–120. doi: 10.1037/02787393.26.1.103 29 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 comprises the testing center and 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; and Academic Advising Council). She also 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. Her research interests include the individual differences in the mental representation of numbers. Department of Psychological Sciences, Box T0820, Stephenville, Texas 76402 geye@tarleton.edu
Click tabs to swap between content that is broken into logical sections.
Rating  
Title  The Dynamics of Number Representation in College Students with Low Math Achievement 
Author  Geye, Trina 
Subject  Psychology; Educational psychology 
Abstract  THE DYNAMICS OF NUMBER REPRESENTATION IN COLLEGE STUDENTS WITH LOW MATH ACHIEVEMENT A Thesis 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 MASTER OF SCIENCE December 2015 THE DYNAMICS OF NUMBER REPRESENTATION IN COLLEGE STUDENTS WITH LOW MATH ACHIEVEMENT A Thesis by TRINA GEYE Approved by: Advisor: Tracy Henley Committee: Steve Ball Tom Faulkenberry Ray Green Head of Department: Jennifer Schroeder Dean of the College: Timothy Letzring Dean of Graduate Studies: Arlene Horne iii ABSTRACT THE DYNAMICS OF NUMBER REPRESENTATION IN COLLEGE STUDENTS WITH LOW MATH ACHIEVEMENT Trina Geye, MS Texas A&M UniversityCommerce, 2015 Advisor: Tracy Henley, PhD Magnitude representation is essential to understanding individual differences in math achievement (Henik, Rubinsten, & Ashkenazi, 2011). The tendency for numerical magnitude to interfere with comparisons of physical size is termed the size congruity effect (Henik & Tzelgov, 1982). In the present study, the realtime dynamics of the size congruity effect were analyzed in both low and typically math achieving college students using computer mousetracking. Participants selected the physically larger of two presented numbers, ignoring numerical value, by using a computer mouse to make the selection. A larger area under the curve for incongruent trials indicates competition from the activation of the irrelevant numerical magnitude representation. The low math achieving group demonstrated more complex trajectories than the typical math achieving group, regardless of congruency condition. Interestingly, there were no significant betweengroup differences for reaction time, suggesting that computer mousetracking is a useful tool for identifying individual differences in numerical cognition beyond performance measures. iv ACKNOWLEDGEMENTS I would like to thank my advisor, Dr. Tracy Henley, for the continual encouragement throughout the program and for taking on a project that was so far outside of the realm of his typical research interests; and also for the soup. Thanks to Dr. Ray Green and Dr. Steve Ball for second and third chances, and for sticking with me through advisor changes, topic reboots, and flooded work spaces. Special thanks to Dr. Tom Faulkenberry for taking me under his wing and convincing me that learning all about numerical cognition and R at the same time would be a great idea. His guidance, instruction, and generous sharing of resources have been invaluable throughout this process. I am especially grateful to my friends and family for the laughs, the tissues, and the pushes, and to my parents for instilling in me the value of an education (and dealing with my journeys off the beaten path!) Finally, to the rest of the HoonGeye Posse – my husband and our boys – I could have finished this program years ago without you, but what fun would that have been? Thank you for your unabated patience and love. v TABLE OF CONTENTS LIST OF TABLES .................................................................................................................... vii LIST OF FIGURES ................................................................................................................. viii CHAPTER 1. INTRODUCTION .................................................................................................... 1 Representation of Numerical Magnitude ............................................................. 3 Automaticity of Numerical Magnitude Representation ...................................... 5 Individual Differences in Automatic Representation of Numerical Magnitude ... 6 Computer Mousetracking ...................................................................................... 8 2. METHOD ................................................................................................................. 10 Participants ......................................................................................................... 10 Materials ............................................................................................................ 10 Procedure and Design ........................................................................................ 10 3. RESULTS ................................................................................................................. 13 Reaction Time Analyses .................................................................................... 13 Trajectory Analyses ............................................................................................ 16 4. GENERAL DISCUSSION ....................................................................................... 19 Limitations .......................................................................................................... 21 Future Directions ................................................................................................ 21 Conclusions ......................................................................................................... 22 REFERENCES ......................................................................................................................... 23 VITA ...................................................................................................................................... 29 vi LIST OF TABLES TABLE 1. CFT and BMA correlation matrix ................................................................................... 11 2. Mean performance measures, in milliseconds, by group and condition ......................... 13 3. Mean trajectory measures by group and condition ......................................................... 18 vii LIST OF FIGURES FIGURE 1. Graphical representation of a mental number line ............................................................ 4 2. Mean RT as a function of condition (congruent, incongruent) and distance (1, 2, 3, 4) for rightward trajectories ................................................................................................. 14 3. Mean RT as a function of condition (congruent, incongruent) and distance (1, 2, 3, 4) for leftward trajectories ................................................................................................... 15 4. Mean response trajectories as a function of direction (right versus left trajectory), condition (congruent versus incongruent), and numerical distance (1, 2, 3, and 4) ....... 17 1 Chapter 1 INTRODUCTION In the United States, many adults do not master the level of mathematics expected of an eighth grader, resulting in a lack of preparedness for participation in our modern economy and completion of routine daily living tasks requiring quantitative skills (Geary, Hoard, Nugent, & Bailey, 2013) Although it is improbable that every individual struggling with numeracy skills as an adult has a domain specific learning disability in mathematics, the ambiguity of the diagnosis highlights both the heterogeneity of mathematical impairment and the lack of understanding of individual differences in numeracy in the general population (Kaufman et al., 2013). Despite a consensus in the field that a yet to be identified central nervous system disorder is the key to understanding domain specific deficits in learning, the differential diagnosis continues to favor identification in an academic setting, although the impact of those deficits certainly reaches beyond the classroom, particularly in adulthood (Scanlon, 2013). Even the recent revisions in the Diagnostic and Statistical Manual of Mental Disorders (DSM5) are a reflection of the needs of the field of education instead of psychology or neurology. Significant changes were made to the conceptualization of learning disability and attention deficit hyperactivity disorder (see Tannock, 2013). The chapters are sequenced according to lifespan development, with those diagnoses typically applying to children first. Among the disorders included in the neurodevelopmental chapter is specific learning disability, which has been collapsed into a single diagnosis rather than the previous four (reading disorder, disorder of written expression, mathematics disorder, and learning disorder not otherwise specified); practitioners are directed to include the affected domain(s) as a specifier (e.g., with impairment in mathematics). Specific learning disorder is described as “a neurodevelopmental 2 disorder with a biological origin that is the basis for abnormalities at a cognitive level that are associated with behavioral signs of the disorder” (American Psychiatric Association, 2013, p. 68). Although a biological origin is conceded in the discussion of diagnostic features, the academic skills dependent upon the affected underlying processes are attributed to intentional teaching rather than emergence during brain maturation. The new definition of learning disability is responsive to evidence that the most recent model is not an effective approach, in isolation, but is still problematic for identifying qualified individuals. The diagnostic criterion requiring that learning difficulties begin during schoolage years is a barrier in the identification of adults who have no history of educational impact. Additionally, adults not in a school setting may not have access to intervention services that would rule out lack of appropriate instruction. Scanlon (2013) argues that the classification of learning disability as a neurodevelopmental disorder effectively limits the diagnosis to adults who experienced academic difficulties as school age children, whereas subsuming the diagnosis under neurocognitive disorders, as the category currently exists, would require documenting a decline. Diagnosis of specific learning disability is based upon the need to identify individuals requiring services in an educational setting, not upon adherence to an operational definition of impairment for use by researchers exploring the underlying CNS disorder (Kavale, Holdnack, & Mostert, 2005). This ambiguity is compounded for those interested in math deficits, specifically, due to the paucity of research. The vast majority of empirical studies in the field of learning disability concern reading disability (Mazzocco & Myers, 2003), as do the reviews and recommendations for application (Aaron, 1997). Potential for the use of basic cognitive numeracy tasks in the assessment of and intervention for domain specific deficits is promising, 3 and is responsive to the call made by Kaufman et al. (2013) to utilize alternatives to purely curricular assessment. The diagnosis of specific learning disability relies on the assumption that domain specific deficits can exist independently. When two domains are shown to be dissociated by continued access to one domain while another becomes inaccessible, presumably the processes rely upon distinct physiological networks. For example, the existence of two systems for number representation (one for exact arithmetic and one for approximate arithmetic) has been supported by evidence obtained from fMRI. Results suggest that representation of numerical magnitude is dependent upon portions of the parietal lobes associated with visuospatial ability (Dehaene, Spelke, Pinel, Stanescu, & Tsivkin, 1999). It follows that deficiencies in these systems could manifest in the observable difficulties with math that are collectively labeled mathematics learning disability (MLD). Representation of Numerical Magnitude Evidence that humans process magnitude in much the same way that they process other physical stimuli, and not through recall from memory or counting, was presented by Moyer and Landauer (1967), who applied the same principles for examining comparison of visual and auditory stimuli to number representation. In making judgments of difference between two Arabic numerals, faster reaction times were associated with smaller values, and with larger difference between values. For example, when presented with two lines and asked to judge which is the longer, responses are faster when both lines are relatively small, and when the length difference is relatively great. In their experiment, Moyer and Landauer (1967) asked participants to simply choose the larger of two single digit numbers when presented. As expected, they found a negative relationship between numerical difference and reaction time; participants were able to more quickly identify the larger number when the numerical distance 4 was greater. This robust numerical distance effect was supported by Hinrichs, Yurko, and Hu (1981), who found that participants were able to choose the larger of 2 twodigit numbers more quickly the further apart they were in numerical value, even if comparison of the decades digit was sufficient for decision making; for example, it would be faster to compare 21 with 35 than 29 with 35. Dehaene, S., Dupoux, E., & Mehler, J. (1990) replicated the work of Hinrichs, Yurko, and Hu (1981) using a French sample and with varying controls for the comparison standard. The existence of mental number line (Figure 1) was proposed to explain why numerical comparisons appear to occur in much the same way as physical comparisons, such as size or length. According this model, numerical magnitudes are represented along a logarithmic mental number line. Numbers 1, 2, and 3 are distinct, and the associated magnitudes are activated quickly and accurately. Beginning with 4, however, magnitude representation begins to become fuzzy, with number 4 receiving the most activation, but 3 and 5 activating to a lesser extent. As magnitude increases, the number line becomes more compressed, with increasing overlap in activation. Figure 1. Graphical representation of a mental number line with overlapping activation curves peaking above each number (adapted from www.panamath.org) 5 The existence of the mental number line was supported by Dehaene, Bossini, and Giraux (1993). In this study, participants were asked to choose which of two numbers was odd or even by pressing a button corresponding to the selected choice (right or left). Regardless of parity, responses were faster for larger numbers with the right hand, and for smaller numbers with the left hand, suggesting that numbers are internally represented from left to right. This effect was termed the spatialnumerical association of response codes (SNARC). Subsequent work controlling for relative magnitude (Dehaene & Mehler, 1992) suggests that the activation of numerical magnitude was not only subject to psychophysical rules, but also irrepressible. Automaticity of Numerical Magnitude Representation In order to preserve limited cognitive resources for complex processes, some basic operations likely function across circumstances, and are therefore considered automatic. Automatic processes require minimal amounts of processing, and, as such, reduce the likelihood of cognitive overload. Some automatic processes are inherent, and others develop with practice (Hasher & Zacks, 1978). Automaticity is indicated when a stimulus is processed even when irrelevant to the task at hand. Stroop (1935) laid the groundwork for research into the automatic processing of the irrelevant features of stimuli in the classic experiment in which the names of colors were presented to participants in different font colors, some congruent (e.g., the word “red” in a red font color) and some incongruent (e.g., the word “red” in a blue font color). Participants were asked to simply say the color of the font, which they were able to do much easier in congruent trials. Results indicated that we continue to process the printed name of a color when asked only to focus on the color of the font. Girelli, Lucangeli, and Butterworth (2000) proposed that the process applied to compare magnitudes may be automatized in children until categorization allows for retrieval from 6 memory. In studies where participants have been asked to compare the physical size of numbers, difficulty in suppressing the numerical magnitude results in a size congruity effect (SCE; Henik & Tzelgov, 1982). The SCE occurs when two numbers are displayed and participants are asked to compare the physical size. When the number that is larger in magnitude is also in a larger font, reaction times are faster. This effect has also been obtained for symbols not previously associated with magnitude, such as Gibson figures (Tzelgov, Yehene, Kotler, & Alon, 2000). Applied to the study of numerical magnitude, the analogous presentation of symbolic numbers with the manipulation of physical features reveals Strooplike effects. When asked to choose the physically larger (in font size) of two numbers, reaction times are faster when the physically larger number is also the number with the largest magnitude. The internal magnitude associated with a numerical symbol is activated even when irrelevant to the task. This automatic processing of magnitude is essential for proficiency in numerical reasoning (Henik, Rubinsten, & Ashkenazi, 2011), and therefore critical to the understanding of variation in the ability to process numbers. Individual Differences in Automatic Representation of Numerical Magnitude Holloway and Ansari (2009) sought to explore the processing of numerical magnitude in individuals with low math ability in order to examine the potential relationship between basic numerical processing and math achievement. In their study, children aged 6 years through 8 years were instructed to select the numerically larger of two single digit Arabic numerals or the more numerous array of squares. The NDE was present only for the symbolic trials, not for the physical representations, suggesting that accessing the meaning of Arabic numerals may be key to understanding individual differences. They found a larger NDE to be associated with lower scores on a standardized test of mathematics, and propose that “noisier” mapping of symbols to magnitude can result in difficulty accessing exact magnitude, and, therefore, problems in 7 calculation. Notably, Rousselle and Noelle (2007) found an opposite relationship in MLD children, with reduced NDE associated with impairment in math. Rubinsten and Henik (2005) conducted the first of the few published studies examining SCE in MLD adults. Their design expanded upon the traditional Strooplike task used in these types of studies to include height and grayness of stimuli in addition to size as dimensions. As predicted, the control group exhibited a larger congruity effect for size, height, and grayness, with the SCE being the largest difference. These results were supported in a subsequent study by Ashkenazi, Rubinsten, and Henik (2009) who also explored the role of cognitive load in SCE. Participants were asked to compare the size of the two digits presented using a key press, and, for the load condition, were also asked to verbally compare two peripheral figures. While the noload condition replicated the previous results, nonMLD participants exhibited a diminished SCE in a load condition, suggesting that inability to recruit attention contributes to magnitude processing deficits in MLD. Although a diminished NDE and SCE in MLD has been obtained in experimental studies, extant literature does not provide any consensus in explaining why there appears to be a deficit in the automatic processing of magnitude in individuals with a domain specific math deficit. Rubenstin and Henik (2005) point out that the SCE observed in MLD adults was similar to the SCE found in first grade children and propose that schooling and cognitive maturation are likely associated. However, the questions of what, exactly, is learned and what processes mature remain. Proposed explanations include noisier mapping of magnitude onto symbols (Holloway & Ansari, 2009), difficulty inhibiting the natural logarithmic number line (Geary, Hoard, Nugent, & ByrdCraven, 2008), and problems in recruiting attention (Ashkenazi, Rubinsten & Henik, 2009; Risko, Maloney, & Fugelsang, 2013). 8 The collection of data beyond performance measures is essential to understanding and detecting the cognitive processes, and will provide the multidimensional examination critical heterogeneous disorder such as MLD. An alternative method of data collection that is more sensitive to the dynamics of underlying cognitive processes is possible through the use of mouse tracking software. Computer Mousetracking A method for collecting robust data representing cognitive processes in real time was introduced by Spivey, Grosjean, and Knoblich (2005) in their study of spoken word recognition. Participants were asked to identify which of two words was heard by clicking the selected response on the screen with a computer mouse. Hand trajectories were tracked by continuously recording the coordinates of the mouse. Results indicated that, when two words were phonetically similar, the trajectory was pulled towards the incorrect response early on in the trial. This “pull” towards the incorrect response reflects the interference caused by the activation of two competing mental representations. Freeman and Ambady (2010) presented the MouseTracker software as a resource for researchers interested in using this method to compute realtime processing data. Subsequently, computer mouse tracking has been recognized as having the potential to provide access to cognitive processes as they happen by relying on the intimate relationship between mental and motor dynamics (Freeman, Dale, & Farmer, 2011). Several recent studies have successfully applied hand tracking to the study of numerical cognition (Marghetis & Nunez, 2013; Santens, Goossens, & Verguts, 2011; Song & Nakayama, 2008). Faulkenberry (2014) proposed that decision making in a numerical parity (odd/even) task involves the activation of competing representations, providing a model for the application of the methodology in other numerical cognition tasks requiring forced choice comparison. This method has generated significant 9 interest and continues to undergo refinement and critique (see Fischer & Hartmann, 2014), but has demonstrated significant potential in revealing the nature of competing processes through the examination of previously inaccessible information (Faulkenberry & Rey, 2014). I hypothesized that participants with low math achievement would exhibit a smaller size congruity effect than those in the typically achieving group. Specifically, I predicted low math achievers would have smaller differences in reaction time between congruent and incongruent trials, and would experience less competing activation as evidenced by Area Under the Curve (AUC) and maximum deviation (MD) than their typically achieving peers. 10 Chapter 2 METHOD Participants 165 participants (mean age 19.9 years, 63% female) were recruited from Tarleton State University, primarily through courses in which faculty were willing to provide credit for research participation, including undergraduate psychology courses and first year seminar courses. From those, 25 participants meeting the criteria (see below) for low math achievement (LMA) were, along with matched typical math achievers (TMA), selected for inclusion in the study. Of the 50 selected, nine reported a previous diagnosis of ADHD or learning disability. Of those nine, eight were identified LMA, and only one was TMA. Materials Participants completed the Calculation Fluency Test (CFT; Sowinski, Dunbar, & LeFevre, 2014), which contains 60 twodigit addition, 60 twodigit subtraction, and 60 twodigit by onedigit multiplication problems. Participants also completed the Brief Mathematics Assessment3 (BMA; Steiner & Ashcraft, 2012), which consists of 10 increasingly difficult problems extracted from the Wide Range Achievement Test: Third Edition (WRAT3), including fractions and algebraic factoring. Additionally, Standard Progressive Matrices (Raven, Raven, & Court, 2000) were administered to those participants selected to complete the experimental task. Procedure and Design Participants completed an informed consent and a demographic questionnaire requesting information such as sex, age, handedness, and university math placement. It was explained that participation in the second phase of the study was to be scheduled at a later time for those selected. Both the CFT and the BMA were scored by totaling the number of correct responses 11 per page. Because the three CFT subtests (addition, subtraction, and multiplication; see Table 1) were highly correlated (Cronbach’s alpha = .93), a total CFT score was calculated. Table 1 CFT and BMA correlations CFT Add CFT Sub CFT Mult CFT Tot CFT Add CFT Sub .71 CFT Mult .62 .61 CFT Tot .90 .88 .84 BMA .36 .43 .34 .43 Total CFT and BMA scores were converted to zscores and added together in order to yield a composite math achievement score. Those scores falling in the 25th percentile (z ≤ 1.64) were assigned to the LMA group, while those who were at or above the mean (z ≥ 0.0) were considered to be TMA. There was a significant difference in the composite math achievement scores between groups, t(48) = 14.7, p < .001. Equally sized groups were created by randomly selecting 25 each of LMA and TMA participants (n = 50, 74% female), matched based on sex. Administration of the Standard Progressive Matrices (SPM) and the experimental task were scheduled and conducted with each participant individually. Raw SPM scores were calculated by totaling the number of correct responses for each participant. For the experimental task, participants were presented with a succession of pairs of numbers in differing font sizes displayed simultaneously in the topright and topleft corners of a 20 inch iMac desktop screen with a resolution of 1280 x 1024 pixels. For this study, stimuli were Arabic numerals 2, 3, 4, 5, 6, 7, and 8 in Arial font with point sizes 22 (small) and 28 (large). 12 The numerals were presented in pairs chosen to create numerical distance categories: 2  3, 3  4, 4  5 (distance 1); 2  4, 3  5, 4  6 (distance 2); 2  5, 3  6, 4  7 (distance 3); 2  6, 3  7, 4  8 (distance 4). The 12 possible pairs were presented in a randomized order two times in eight blocks, resulting in 192 stimuli. Congruent trials were those in which the numerically larger number was also in the larger font point size (e.g., 1 9), while in incongruent trials, the numerically smaller number was in the larger font point size (e.g., 1 9). For stimulus presentation and trajectory recording, MouseTracker was used, a software program developed by Freeman & Ambady (2010) and available for free download (http://mousetracker.jbfreeman.net). Participants were instructed to select the physically larger number, ignoring numerical size, as quickly as possible by clicking on the selected answer using a computer mouse. Clicking a START button before each trial activated the trial, and the streaming x and ycoordinates of mouse movement were recorded at a rate of 70 points per second. The intention was to capture trajectories representing online cognitive processes, not movements made postdecision. In order to encourage participants to begin moving towards their selection as soon as possible, movement initiated after 400 ms resulted in a warning. Incorrect responses resulted in the display of an “X” for 2000 ms. Participants were debriefed and thanked for their involvement in the study. A 2 (group; LMA vs. TMA) x 2 (congruity; congruent or incongruent) x 4 (distance; 1, 2, 3, 4) factorial design was used. Dependent variables included reaction time (comprised of initial time and movement time), average trajectory, and measures of dynamic complexity of hand movements (maximum deviation and Area Under the Curve). Maximum deviation (MD) is the largest perpendicular distance between the actual trajectory and ideal, straightline trajectory for all time steps, and (AUC) is the geometric area between the actual and ideal trajectories. 13 Chapter 3 RESULTS Participants completed a total of 19,200 trials for each trajectory direction (left and right). One participant did not successfully respond to any incongruent trials, and was therefore excluded, as were 61 incorrect trials (.32%) and 98 for which reaction times were more than 3 standard deviations beyond the mean for all correct response reaction times (.51%). Subsequent analyses were conducted using the remaining 18,657 trials; mean reaction times (RTs) and initiation times (ITs) are displayed in Table 2. Right trajectory and left trajectory analyses were conducted separately for all measures and will be presented independently. An analysis of SPM revealed higher scores for TMA than LMA participants, t(48) = 4.65, p < .001. Table 2 Mean reaction times (RT) and initiation times (IT), in milliseconds, by group (LMA, TMA) and condition Congruent Trials Incongruent Trials Numerical Distance 1 2 3 4 1 2 3 4 Rightward Trajectories LMA RT 1454 1428 1414 1394 1436 1445 1472 1450 IT 106 106 104 107 105 110 114 106 TMA RT 1327 1336 1325 1314 1354 1369 1382 1391 IT 127 128 130 129 123 126 134 125 Leftward Trajectories LMA RT 1427 1444 1438 1408 1454 1478 1482 1475 IT 101 108 97 101 103 108 106 104 TMA RT 1349 1372 1337 1346 1381 1371 1406 1403 IT 131 129 131 121 128 139 127 129 14 Reaction Time Analyses A 2 (group: LMA vs. TMA) x 2 (physicalnumerical size congruity: congruent vs. incongruent) x 4 (numerical distance: 1, 2, 3, 4) repeated measures analysis of variance was conducted for both measures. There was a significant main effect of physicalnumerical size congruity, for right trajectory RTs, with participants taking 39 ms longer on incongruent than congruent trials F(1,47) = 31.02, p < .001, p2 = .40. For numerical distance, there was no significant main effect, F(3, 141) = .666, p = .574. However, there was a significant interaction of physicalnumerical size congruity and distance, F(3, 141) = 5.87, p < .001, p2 = .11. As is evident in Figure 2, the difference in RTs between congruent and incongruent trials increased Figure 2. Mean RT as a function of condition (congruent, incongruent) and distance (1, 2, 3, 4) for rightward trajectories 15 with the numeral distance between the presented pairs. I tested for a linear contrast on the interaction term in the ANOVA model by performing a within subjects contrast, F(1,47) = 15.537, p = .009, p2 =.127. The main effect of group approached significance, with LMA taking longer to respond than TMA, F(1,47) = 3.20, p = .08. There were no differences for initiation times between conditions on any factor (all Fratios were less than 0.64). To examine left trajectory performance measures, a 2 x 2 x 4 repeated measures analysis of variance (see above) was conducted for RT and initiation times. There was a significant main effect of physicalnumerical size congruity on RT, with participants taking 41 ms longer on incongruent trials, F(1,47) = 29.08, p < .001, p2 = .38. Similar to the right trajectory results, no significant main effect of numerical distance was obtained, F(3,141) = .72, p = .54. There was a significant interaction of physicalnumerical size congruity and distance, F(3,141) = 2.80, p = .043 (see Figure 3). In order to test for a linear contrast on the interaction term in the ANOVA Figure 3. Mean RT as a function of condition (congruent, incongruent) and distance (1, 2, 3, 4) for leftward trajectories 16 model, I conducted a withinsubjects contrast, F(1,47) = 6.82, p = .011, p2 = .245. Again, RT differences between the LMA and TMA groups did not reach significance, F(1,47) = 2.12, p = .15. There were no significant effects for initiation times (all Fratios were less than 0.59). Trajectory Analyses To enable comparison among trajectories of different lengths, RTs were equalized by creating 100 x and y coordinate pairs for each trial. Mousetracker accomplishes this normalization by using a default of 101 time steps in order to permit 100 equal spaces between coordinate pairs. These normalized trajectories for both conditions (congruent and incongruent) were crossed with numerical distances 1, 2, 3, and 4 (see Figure 4). Area Under the Curve A 2 (group: LMA vs. TMA) x 2 (physicalnumerical size congruity: congruent vs. incongruent) x 4 (numerical distance: 1, 2, 3, 4) repeated measures analysis of variance was used to analyze the average AUC values for each participant. For right trajectories, a significant main effect of group emerged, with a higher AUC for LMA than TMA participants (see Table 3 for means), F(1, 47) = 4.73, p = .035, p2 = .09. There was a significant effect of physicalnumerical size congruity, with higher AUC on incongruent (m = 0.83) than congruent (m = 0.57) trials, F(1, 47) = 65.83, p < .001, p2 = .58. There was also a significant interaction effect of condition and distance, F(3,141) = 4.37, p < .006, p2 = .09. These results indicate that trajectories are significantly pulled towards the incorrect, numerically larger digit in incongruent trials. 17 Figure 4. Mean response trajectories as a function of direction (right versus left trajectory), condition (congruent versus incongruent), and numerical distance (1, 2, 3, and 4) 18 Table 3 Mean trajectory measures by group (LMA, TMA) and condition Results of the analyses of left trajectories were similar to the right trajectories. There was a significant main effect of group, with higher AUC for LMA than TMA, F(1, 47) = 4.38, p =.042, p2 = .09. There was also a main effect of condition (congruent vs. incongruent), F(1, 47) = 51.34, p < .001, p2 = .52. Again, there was a significant interaction effect for condition and distance, F(3, 141) = 4.43, p = .005. Maximum Deviation MD was also examined using the same 2 (group: LMA vs. TMA) x 2 (physicalnumerical size congruity: congruent vs. incongruent) x 4 (numerical distance: 1, 2, 3, 4) repeated measures analysis of variance model applied to AUC. For right trajectories, the main effect of group approached significance between LMA (m = 0.40) and TMA (m = 0.30), (F(1, 47) = 3.61, p = .06. There was a main effect of physicalnumerical size congruity, with greater MD for incongruent (m = 0.40) than congruent trials (m = 0.29), F(1, 47) = 75.04, p < .001, p2 = .58. Congruent Trials Incongruent Trials Numerical Distance 1 2 3 4 1 2 3 4 Rightward Trajectories LMA AUC 0.78 0.77 0.69 0.70 0.87 0.95 0.98 1.00 MD 0.38 0.37 0.33 0.35 0.42 0.45 0.44 0.46 TMA AUC 0.45 0.43 0.39 0.39 0.62 0.70 0.72 0.83 MD 0.24 0.23 0.23 0.21 0.32 0.35 0.37 0.41 Leftward Trajectories LMA AUC 0.83 0.74 0.76 0.71 0.94 0.98 1.05 1.02 MD 0.42 0.39 0.41 0.38 0.46 0.48 0.49 0.49 TMA AUC 0.53 0.56 0.42 0.50 0.70 0.71 0.80 0.82 MD 0.30 0.31 0.24 0.27 0.36 0.38 0.42 0.42 19 No main effect for distance was found, F (3, 141) = 0.46, p = .71. There was an interaction for condition and distance, with larger MDs associated with larger numerical distance, F(3, 141) = 3.71, p = .013, p2 = .82. The same pattern of results was found for left trajectories, with a main effect of group approaching significance between LMA (m = 0.44) and TMA (m = 0.34), F(1, 47) = 3.73, p = .06. There was a main effect of physicalnumerical size congruity, with greater MD for incongruent (m = 0.44) than congruent trials (m = 0.34), F(1, 47) = 63.69, p < .001, p2 = .58. No main effect of distance was found, F(3, 141) = 0.149, p = .93. Like the right trajectories, there was an interaction for condition and distance, F(3, 141) = 4.64, p = .004, p2 = .69. 20 Chapter 4 GENERAL DISCUSSION Because previous studies have not addressed the dynamics in the underlying cognitive processes associated with math ability with data beyond performance measures, the current study sought to investigate whether differences in the size congruity effect (SCE) would be evident in LMA and TMA participants using robust data available from the use of computer mousetracking software. Participants were assigned to either group (TMA or LMA) based upon scores on the CFT and the BMA. As expected, RTs were longer for trials in which the numerical value and physical size were incongruent than in congruent trials, indicating a SCE for both TMA and LMA participants. Analyses of computer mouse trajectories were consistent with RT results, with a larger AUC and MD for incongruent versus congruent trials, indicating a pull towards the incorrect alternative. Additionally, the differences in RT between congruent and incongruent trials increased as a function of numerical distance. Participants took longer to choose the correct number on incongruent as the difference between the two numbers increased. This discovery of a more complex trajectory for incongruent presentations is theoretically consistent with the SCE. Girelli, Lucangeli, and Butterworth (2000) proposed that the SCE represented an automatic representation of magnitude that gradually develops with improved numerical skill. The more automatic the processing of magnitude becomes, the more difficult it is to suppress that information. In the present study, both performance measures and trajectory analyses indicate that the incorrect response is activated during online processing of information related to the decision. 21 The between groups analyses, however, did not yield the expected results. It was hypothesized that a smaller SCE would be found in LMA participants when compared with TMA as measured by RT, and that LMA would have less competition from the incorrect response as measured by MD and AUC; this was not the case. There was no significant difference between groups for RT, although results approaching significance do suggest a trend of longer RTs for LMA. The results for AUC and MD are the opposite of the hypothesis, with a significantly larger AUC for LMA participants than TMA participants and a suggested trend towards similar results for MD. There was no significant difference in AUC between conditions with group as a factor. Previous research (Ashkenazi, Rubinsten, & Henik, 2009; Rubinsten & Henik, 2005) has found a smaller SCE in MLD adults through analysis of RT. Although the between groups analyses did not reveal significant differences for RT, the trend is reversed. Taken in isolation, this evidence might be considered an anomaly; however, when paired with the trajectory analyses, the results suggest that the dynamics of the physical comparison task are more complex than anticipated. LMA participants experienced more distraction from the incorrect response, regardless of congruity, suggesting that between group differences may not be due to differences in automatic representation of magnitude, but to attentional issues. Posner (1978) and Lindsay and Jacoby (1994) propose that the Strooplike effect found in cognitive tasks actually comprises two components  facilitation and inhibition  with facilitation associated with automaticity and inhibition associated with attentional processing. Rubinsten and Henik (2005) found only an interference component in a group of adults with developmental dyscalculia, whereas the control group exhibited a facilitation component in addition to interference. Our results may represent the interference component in the LMA participants as 22 supported by a significantly larger AUC. Subsequent studies using mousetracking to examine the SCE should include a neutral stimulus in which the numeral value is constant. This creates an opportunity for comparison to the incongruent stimulus for the interference component and the congruent stimulus for the facilitation component. The possibility of an interference component is consistent with the findings of Ashkenazi, Rubinsten, and Henik (2009) and Risko, Maloney, and Fugelsang (2013), who suggest a research focus on the recruitment of attention in physical size comparison tasks. In addition to providing direction for future research in the representation of numerical magnitude, this study also highlights the value of computer mousetracking as a research tool. The robust data collected through this method provided a unique opportunity to discover individual differences through the analysis of dynamic trajectories that would not have emerged through more traditional means. This tool provides the opportunity to reexamine previous efforts to understand individual differences in a variety of domains, collecting data beyond reaction time. Scanlon (2013) described our understanding of LD as “evolving” and the construct as being defined prematurely. MLD, specifically, is underrepresented in the literature and often assumed to be homogenous. Kaufman et al. (2013) propose that educational tests are inadequate to examine the basic numerical processing deficits associated with MLD, and that deficits in subdomains may be present in individuals who do not meet the diagnostic criteria. The current response to intervention model (RTI) is largely experimental (Kavale, Holdnack, & Mostert, 2005). Studies such as the present may be critical to identifying fine grain individual differences that will be key to moving forward in a definition of MLD that is responsive to research in cognitive processes rather than academic skill performance. Identifying individual differences in 23 basic numerical cognition processes will be useful in defining the presumed underlying CNS disorder and understanding the role of particular regions of the brain. Future Directions As there as a significant difference in the AUC for LMA participants, further research is warranted. Additional studies using this mousetracking method should replicate previous research using only performance measures to see if the robust data provided through the analysis of trajectories provides more insight. Specifically, research should explore the interference component of the SCE and the recruitment of attention in physical size comparison tasks, including the impact of stimulus presentation timing (Risko, Maloney, & Fugelsang, 2013), cognitive load (Ashkenazi, Rubinsten, & Henik, 2009), and attentional training (Askenazi & Henik, 2012) Conclusions The results of this study appear to suggest that the role of attention in magnitude comparison will be important to understanding individuals with domain specific deficits in math. Additionally, these results highlight the value of mousetracking as an alternative to traditional measures of performance; the difference in trajectories between LMA and TMA participants would have gone unnoticed measuring only reaction time. Further research is needed to replicate these results using neutral stimuli to allow for the isolation of facilitation and interference components of SCE. Continuing work in this area will be important to the evaluation of the emerging alternative explanation of SCE set forth by Risko, Maloney, and Fugelsang (2013), who propose that attention, rather shared representation, explains the difference in congruent versus incongruent stimuli. 24 REFERENCES Aaron, P. G. (1997). The impending demise of the discrepancy formula. Review of Educational Research, 67(4). doi: 10.3102/00346543067004461 American Psychiatric Association. (2013). Diagnostic and statistical manual of mental disorders (5th ed.). Arlington, VA: American Psychiatric Publishing. Ashkenazi, S., Rubinsten, O., & Henik, A. (2009). Attention, automaticity, and developmental dyscalculia. Neuropsychology, 23(4), 535540. doi: 10.1037/a0015347 Dehaene, S., Bossini, S., & Giraux, P. (1993). The mental representation of parity and number magnitude. Journal of Experimental Psychology, 122(3), 371396. doi: 10.1037/00963445.122.3.371 Dehaene, S., Dupoux, E., & Mehler, J. (1990). Is numerical comparison digital Analogical and symbolic effects in twodigit number comparison. Journal of Experimental Psychology: Human Perception and Performance, 16, 626641. doi: 10.1037/00961523.16.3.626 Dehaene, S., & Mehler, J. (1992). Crosslinguistic regularities in the frequency of number words. Cognition, 43(1), 129. doi: 10.1016/00100277(92)90030L Dehaene, S., Spelke, E., Pinel, P., Stanescu, R., & Tsivkin, S. (1999). Sources of mathematical thinking: Behavioral and brainimaging evidence. Science, 284, 970–974. doi: 10.1126/science.284.5416.970 Faulkenberry, T. J. (2014). Hand movements reflect competitive processing in numerical cognition. Canadian Journal of Experimental Psychology, 68(3), 147151. doi: 10.1037/cep0000021 25 Faulkenberry, T. J., & Rey, Amandine (2014). Extending the reach of mousetracking in numerical cognition: A comment on Fischer and Hartmann (2014). Frontiers in Psychology, 5, 1436. doi: 10.3389/fpsyg.2014.01436 Fischer, M. H., & Hartmann, M. (2014). Pushing forward in embodied cognition: May we mouse the mathematical mind? Frontiers in Psychology, 5, 1315. doi: 10.3389/fpsyg.2014 .01315 Freeman, J. B., & Ambady, N. (2010). MouseTracker: Software for studying realtime mental processing using a computer mousetracking method. Behavior Research Methods, 42(1), 226241. doi: 10.3758/BRM.42.1.226 Freeman, J. B., Dale, R., & Farmer, T. A. (2011). Hand in motion reveals mind in motion. Frontiers in Psychology, 2, 16. doi: 10.389/fpsyg.2011.00059 Geary, D. C., Hoard, M. K. Nugent, L., & ByrdCraven, J. (2008). Development of number line representations in children with mathematical learning disability. Developmental Neuropsychology, 33(3), 277299. doi: 10.1080/87565640801982361 Geary, D. C., Hoard, M. K., Nugent, L., & Bailey, D. H. (2013). Adolescents’ functional numeracy is predicted by their school entry number system knowledge. PLoS ONE, 8(1): e54651. doi:10.1371/journal.pone.0054651 Girelli, L., Lucangeli, D., & Butterworth, B. (2000). The development of automaticity in accessing number magnitude. Journal of Experimental Child Psychology, 76, 104122. doi: 10.1006/jecp.2000.2564 Hasher, L., & Zacks, R. T. (1978). Automatic and effortful processes in memory. Journal of Experimental Psychology, 108(3), 356388. doi: 10.1037/00963445.108.3.356 26 Henik, A., Rubinsten, O., Ashkenazi, S. (2011). The “where” and “what” in developmental dyscalculia. The Clinical Neuropsychologist, 25(6), 9891008. doi: 10.1080/13854046 .2011.599820 Henik, A., & Tzelgov, J. (1982). Is three greater than five: The relation between physical and semantic size in comparison tasks. Memory & Cognition, 10(4), 389395. doi: 10.3758/BF03202431 Hinrichs, J. V., Yurko, D. S, & Hu, J. M. (1981). Twodigit number comparison: Use of place information. Journal of Experimental Psychology: Human Perception and Performance, 7, 890901. doi: 10.1037/00961523.7.4.890 Holloway, I. D., & Ansari, D. (2009). Mapping numerical magnitudes onto symbols: The numerical distance effect and individual differences in children’s mathematics achievement. Journal of Experimental Child Psychology, 103, 1729. doi: 10.1016 /j.jecp.2008.04.001 Kaufman, L., Mazzocco, M. M., Dowker, A., von Aster, M., Goebel, S., Grabner, R., … Nuerk, H. (2013). Dyscalculia from a developmental and differential perspective. Frontiers in Psychology, 4, 516. doi: 10.3389/fpsyg.2013.00516 Kavale, K. A., Holdnack, J. A., & Mostert, M. P. (2005). Responsiveness to intervention and the identification of specific learning disability: a critique and alternative proposal. Learning Disability Quarterly, 28, 216. doi: 10.2307/4126970 Lindsay, D. S., & Jacoby, L. L. (1994). Stroop process dissociations: The relationship between facilitation and interference. Journal of Experimental Psychology: Human Perception and Performance, 20(2), 219234. doi: 10.1037/00961523.20.2.219 27 Marghetis, T., & Nunez, R. (2013). The motion behind the symbols: A vital role for dynamism in the conceptualization of limits and continuity in expert mathematics. Topics in Cognitive Science, 5, 299316. doi: 10.1111/tops.12013 Mazzocco, M. M. M., & Myers, G. F. (2003). Complexities in identifying and defining mathematic learning disability in the primary schoolage years. Annals of Dyslexia, 53, 218253. doi: 10.1007/s1188100300117 Moyer, R. S., & Landauer, T. K. (1967). Time required for judgments of numerical inequality. Nature, 215, 15191520. doi: 10.1038/2151519a0 Posner, M. I. (1978). Chronometric explorations of mind. Hillsdale, NJ: Erlbaum. Raven, J., Raven, J. C., & Court, J. H. (2000). Standard Progressive Matrices. Oxford: Oxford Psychologists Press. Risko, E. F., Maloney, E. A., & Fugelsang, J. A. (2013). Paying attention to attention: evidence for an attentional contribution to the size congruity effect. Attention, Perception, and Psychophysics, 75(6), 11371147. doi: 10.3758/s1341401304772 Rousselle, L., & Noel, M. (2007). Basic numerical skills in children with mathematics learning disabilities: A comparison of symbolic vs nonsymbolic number magnitude processing. Cognition, 102, 361395. doi: 10.1016/j.cognition.2006.01.005 Rubinsten, O., & Henik, A. (2005). Automatic activation of internal magnitudes: A study of developmental dyscalculia. Neuropsychology, 19(5), 641648. doi: 10.1037/08944105.19.5.641 Santens, S., Goossens, S., & Verguts, T. (2011). Distance in motion: Response trajectories reveal the dynamics of number comparison. PLoS ONE, 6(9), 16. doi: 10.1371/journal.pone.0025429 28 Scanlon, D. (2013). Specific learning disability and its newest definition: Which is comprehensive? and which is insufficient? Journal of Learning Disabilities, 46(1), 2633. doi: 10.1177/0022219412464342 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 Song, J., & Nakayama, K. (2008). Numeric comparison in a visuallyguided manual reaching task. Cognition, 106(2), 9941003. 10.1016/j.cognition.2007.03.014 Sowinski, C., Dunbar, K., & LeFevre, J. (2014). Calculation Fluency Test, unpublished technical report, Math Lab, Carleton University, Ottawa, Canada. Retrieved from http://carleton.ca/cacr/wpcontent/uploads/CalculationFluencyTestDescription _MathLabTechnicalReport.pdf Steiner, E. T., & Ashcraft, M. H. (2012). Three brief assessments of math achievement. Behavior Research Methods, 44(4), 11011107. doi: 10.3758/s1342801101856 Stroop, J. R. (1935). Studies of interference in serial verbal reactions. Journal of Experimental Psychology, 18(6), 643662. doi: 10.1037/h0054651 Tannock, R. (2013). Rethinking ADHD and LD in DSM5: Proposed changes in diagnostic criteria. Journal of Learning Disabilities, 46(1), 525. doi: 10.1177/00022219412464341 Tzelgov, J., Yehene, V., Kotler, L., & Alon, A. (2000). Automatic comparisons of artificial digits never compared: Learning linear ordering relations. Journal of Experimental Psychology: Learning, Memory, and Cognition, 26(1), 103–120. doi: 10.1037/02787393.26.1.103 29 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 comprises the testing center and 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; and Academic Advising Council). She also 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. Her research interests include the individual differences in the mental representation of numbers. Department of Psychological Sciences, Box T0820, Stephenville, Texas 76402 geye@tarleton.edu 
Date  2015 
Faculty Advisor  Henley, Tracy 
Committee Members 
Green, Ray Ball, Steve 
University Affiliation  Texas A&M UniversityCommerce 
Department  MSPsychology 
Degree Awarded  M.S. 
Pages  36 
Type  Text 
Format  
Language  eng 
Rights  All rights reserved. 



A 

B 

C 

D 

E 

F 

G 

H 

J 

L 

M 

N 

P 

Q 

R 

S 

T 

V 

W 


