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Defining math disability : the impact of using different cut offs when assessing the cognitive characteristics… Pinkerton, Neil A. 2011

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DEFINING MATH DISABILITY: THE IMPACT OF USING DIFFERENT CUT OFFS WHEN ASSESSING THE COGNITIVE CHARACTERISTICS OF MATH DISABLED PARTICIPANTS  by  NEIL A. PINKERTON B.A., University of Calgary, 2003 B.Ed., University of British Columbia, 2007  A THESIS IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF ARTS in THE FACULTY OF GRADUATE STUDIES (Mathematics Education)  The University of British Columbia (Vancouver)  December 2011 © Neil A. Pinkerton, 2011  ii ABSTRACT Mathematics disabilities (MD) can cause serious difficulties for children throughout their education. However, there is a lack of consensus in the literature regarding how MD should be defined. This makes it difficult to compare results across studies and to determine the overall cognitive profile of MD. The present study investigated the cognitive profiles of participants with MD defined using various cut off criteria and using different assessment measures. Further, this study endeavoured to investigate the differences between those who have MD at one time point and those who have MD at two time points (MD persistent). Over 700 participants were recruited from the North Vancouver school district to participate in this study. Participants were evaluated over two years (grades 2 and 3). Performance was measured using 16 different cognitive measures and 4 achievement tests. Scores on the achievement tests were used to assign MD status. A series of t-tests were conducted to determine whether there are differences between the cognitive profiles of those defined as MD using a 10th percentile cut off (MD10), a 25th percentile cut off (MD25), and a low achieving score between the 11th and 25th percentiles MD(11-25) when compared to typically achieving students (TA). Cut offs were based on performance on two separate math achievement tests: the WoodcockJohnson Third Edition (WJ III) Calculation test and the Woodcock-Johnson Third Edition Applied Problems test. This study concludes that MD10, MD25, and MD(11-25) all represent similar enough cognitive profiles when compared to the TA group that using a 25% cut off to define MD will suffice for future research. Furthermore, the results suggest that the MD persistent group has a similar enough profile to the terminus point group that additional caution should be used when studying MD persistent in the future.  iii The cognitive profiles of the MD groups are described and implications for research and practice are discussed.  iv PREFACE  Behavioural Ethics Research Board H06-80101  v TABLE OF CONTENTS Abstract ............................................................................................................................... ii Preface................................................................................................................................ iv Table of Contents .................................................................................................................v List of Tables .................................................................................................................... vii Acknowledgements .......................................................................................................... viii Dedication .......................................................................................................................... ix Chapter 1 Introduction .......................................................................................................1 1.1 Defining MD .......................................................................................1 1.1.1 Diagnosing math disability: IQ .............................................2 1.1.2 Diagnosing math disability: Tests used ................................4 1.1.3 Diagnosing math disability: Cut off criteria .........................5 1.1.4 Diagnosing math disability: Small sample size ....................6 1.1.5 Diagnosing math disability: Reading disabilities .................6 1.2 Characteristics of math disability .......................................................7 1.3 Working memory ................................................................................9 1.3.1 Working memory and math disabilities ..............................11 1.3.2 Working memory measures ................................................12 1.4 Other cognitive variables ..................................................................14 1.5 Present study .....................................................................................16 Chapter 2 Method ............................................................................................................17 2.1 Participants........................................................................................17 2.2 Materials ...........................................................................................17 2.2.1 Phase I. Calculation skills ...................................................17 2.2.2 Phase II. Visual spatial abilities ..........................................18 2.3 Procedure ..........................................................................................22 2.3.1 Classification ....................................................................22 Chapter 3 Results .............................................................................................................25 3.1 Comparing MD10 and MD25 ...........................................................26 3.1.1 MD10 vs MD25 applied problems .....................................27 3.1.2 MD10 vs MD25 calculation ...............................................29 3.1.3 MD10 vs MD25 both combined .........................................33 3.1.4 MD10 vs MD25 summary ..................................................34 3.2 Comparing MD10 and MD(11-25) ...................................................35 3.2.1 MD10 VS MD(11-25) summary ........................................41 3.3 MD Persistent ...................................................................................42 3.3.1 Comparing grade 2 MD and MD Persistent .......................42 3.3.2 Comparing grade 3 MD and MD Persistent .......................49  vi 3.3.3 Summary .............................................................................50 Chapter 4 Discussion .......................................................................................................51 4.1 MD10 vs. MD25 ...............................................................................51 4.2 MD10 vs. MD(11-25) .......................................................................52 4.3 Summary of MD10, MD25, and MD(11-25) ..................................54 4.4 Persistent MD compared to MD at grade 2 or grade 3 .....................55 4.5 Overall profile of MD25 ...................................................................56 4.5.1 Consistencies ......................................................................57 4.5.2 Inconsistencies ....................................................................60 Chapter 5  Implications ...................................................................................................62 5.1 Limitations .......................................................................................62 5.3 Research ...........................................................................................62 5.3 Practice ............................................................................................64  Chapter 6 Conclusions .....................................................................................................66 References ..........................................................................................................................67  vii LIST OF TABLES Table 1  MD Classifications and Grades Tested with Different Reading Control Cut Offs .............................................................................................................24  Table 2  Grade 3 Sample Sizes and Means for the MD10, MD25, and TA Groups Based on the WJ III Applied Problems Test Only ......................................27  Table 3  Grade 2 Sample Sizes and Means for the MD10, MD25 and TA Groups Based on the WJ III Calculation Test Only ................................................29  Table 4  Grade 3 Sample Sizes and Means for the MD10, MD25, and TA Groups Based on the WJ III Calculation Test Only ................................................32  Table 5  Grade 3 Sample Sizes and Means for the MD10, MD25 and TA Groups Based on Both WJ III Applied Problems and WJ III Calculation Tests.....34  Table 6  Grade 2 Sample Sizes and Means for the MD10, MD(11-25), and TA Groups Based on the WJ III Calculation Test Only ...................................36  Table 7  Grade 3 Sample Sizes and Means for the MD10, MD(11-25), and TA Groups Based on the WJ III Applied Problems Test Only .........................38  Table 8  Grade 3 Sample Sizes and Means for the MD10, MD(11-25), and TA Groups Based on the WJ III Calculation Test Only ...................................40  Table 9  MD25 Sample Sizes and Means for the Grade 2 MD25, Grade 3 MD25, and Persistent Groups Based on the WJ III Calculation Subtest Only ......44  Table 10  MD25 Sample Sizes and Means for the Grade 2 MD25, Grade 3 MD25, and Persistent Groups Based on the WJ III Applied Problems Subtest Only ............................................................................................................46  Table 11  MD25 Sample Sizes and Means for the Grade 2 MD25, Grade 3 MD25, and Persistent Groups Based on Both the WJ III Calculation and Applied Problems Subtests ......................................................................................48  viii ACKNOWLEDGEMENTS I would first like to thank Ann Anderson, whose support has been unwavering through my entire graduate studies and whose dedication to my work, and my person, has kept me on the right path at all times. Thank you for also always having a good laugh when I needed it most. I would also like to thank Linda Siegel whose contribution to this work cannot be understated. You believed in me from the moment we met and your generous offer to use the data in this study provided me with a project I can truly be proud of. Thank you for also pushing me not to settle for anything less than the best and for always encouraging me to test my own limits. I also feel a deep sense of gratitude towards my colleagues and my students who were most patient with me throughout this entire process. Thank you for all your support and understanding. Lastly I would like to thank my Jennifer without whom absolutely none of this would be possible and words cannot begin to describe the patience and amazing assistance you have provided throughout this journey.  ix DEDICATION  To those students who struggle with mathematics disabilities.  1 Chapter 1  Introduction Having a learning disability of any kind can cause serious difficulties for children as they progress through their schooling. It is estimated that mathematics disabilities (MD) affect approximately 5 – 8% of the school-aged population (Badian, 1999; Geary, 1993; Geary, 2004). This number is similar to the number affected by reading disabilities (RD; Gersten, Jordan, & Flojo, 2005). However, the study of MD has received comparatively little attention until recently (Gersten, Jordan, & Flojo, 2005; Swanson & Jerman, 2006). Two distinct avenues of research have focused on the definition of MD and an aspect of cognition that has been seen as problematic in MD, working memory (WM). There has been little consensus across studies on how to define MD and the role WM and other aspects of cognition play in MD. The present study was designed to address this issue by exploring the relationships between MD and several relevant cognitive processes, including WM, and how these relationships vary depending on which definition of MD is used. 1.1 Defining math disability Defining MD has not been a straightforward task. There has been a significant amount of disagreement on how to define MD, exacerbated by a distinct lack of agreement on what terminology to use when describing the difficulties faced by these children. Mazzocco (2005, p. 319) perhaps described this best by indicating that “(w)ithin the field, there are studies of children with mathematics difficulties (Gersten et al., 2005; Hanich, Jordan, Kaplan, & Dick, 2001; Russell & Ginsburg, 1984), mathematics disabilities (Geary, 1993, 2004), dyscalculia (Shalev & Gross-Tsur, 2001),  2 and poor math achievement (Mazzocco & Myers, 2003)”. The terms arithmetic learning disability (Jimenez & Garcia, 1999) and developmental dyscalculia (Shalev, 2004) used in research studies. Unfortunately, it is unclear whether these terms are intended to describe the same disability or whether they are intended to investigate different forms of MD (Mazzocco, 2005). There is clearly a lack of agreement on this issue. However, as much of the research involving WM and MD uses the classification mathematics disability, this paper will also identify MD as mathematics disabilities. Of additional concern is the issue of how one can differentiate between MD and simply poor performance in mathematics (Ginsberg, 1997). In one such effort, recent studies have begun distinguishing between MD, defined as scoring below the 10th percentile on some measure of mathematical ability, and those that have difficulties or are low achieving MD(11-25) in mathematics, scoring between 10% and 25%, for various reasons (Chong & Siegel, 2008; Geary, Hoard, Byrd-Craven, Nugent, & Numtee, 2007). Further complicating this matter, however, are the children who display MD at one time period but have this designation removed later on (Ginsberg, 1997). The label MD has not been uniformly applied, and prior to any further discussion on labelling MD, it should be noted that there is a lack of agreement on how to diagnose MD in general. 1.1.1 Diagnosing math disability: IQ. Traditionally, diagnosing MD has involved identifying a lower than expected achievement result on a standard test (i.e., one or two standard deviations below expected) when compared to a child’s full scale IQ (Geary, 2004). The precedence for using IQ in determining learning disability (LD) status is noted by Landerl, Bevan, and Butterworth (2004) as a traditional designation in the Diagnostic and statistical manual of mental disorders IV (American Psychiatric Association, 1994). This states that to be LD “… the child must substantially  3 underachieve on a standardized test relative to the level expected given age, education and intelligence, and must experience disruption to academic achievement or daily living” (Landerl et al., 2004, p. 100). In other words, performance is much lower on the standardized test than would be expected when compared to their IQ score (Geary, 2004). Indeed, this is one method of diagnosing MD and several researchers (e.g., Fuchs, Compton, Fuchs, Paulsen, Bryant, & Hamlett, 2005; Geary, 2004; Geary et al., 2007; Jordan, Kaplan, & Hanich, 2002; Landerl et al., 2004; Murphy, Mazzocco, Hanich, & Early, 2007; Share, Moffitt, & Silva, 1988; Swanson, 1993; 1994) have used this to identify MD participants. This designation, however, has posed problems in the research as it has been determined that IQ comparison is not the best predictor of any form of LD, MD or otherwise (Siegel, 1989). Additionally, eliminating low IQ scores could potentially eliminate from the study those that perform poorly as a result of their disability (Siegel, 1989). Ginsberg (1997) noted that IQ as an achievement predictor also overlooks other social and environmental factors such as inadequate instruction and thus is not a comprehensive measure. Siegel (1989) demonstrated that the IQ tests also discriminate against individuals from different backgrounds. As performance on the IQ test can be confounded by the LD present, this is not the best metric for determining LD status (Siegel 1989). Furthermore, Landerl et al. (2004) found that while IQ was often used for initial classification, it added little information to the study of MD. In addition, Alloway (2009) determined that IQ was less important than WM as a predictor of math achievement longitudinally. Jimenez and Garcia (1999) found that those identified as MD using IQ discrepancy displayed performance similar to those who were identified as having difficulties with mathematics but that did not meet the IQ discrepancy cut off.  4 They concluded that discriminating between participants based on IQ was irrelevant to the study of MD. 1.1.2 Diagnosing math disability: Tests used. Regardless of whether or not IQ is used to diagnose MD, there is an additional problem in that there is a lack of a single diagnostic tool designed to identify MD (Geary, 2004; Murphy et al., 2007). This lack of a single tool has lead to several theories and different approaches for defining and identifying MD. According to Murphy et al. (2007), the most typical assessment tools used to diagnose MD appear to be the Woodcock-Johnson Third Edition, Tests of Achievement (WJ III ACH; Woodcock, Mather & McGrew 2001a) and the Wide Range Achievement Test-Revised (WRAT-R; Jastak & Wilkinson, 1984). Standardized tests are further restricted in their selection of subtests. This may confine the participant to certain tasks, such as paper and pencil tests that may limit their performance (Mazzocco, 2005). This would be particularly evident if they have a comorbid deficit in a specific area which could potentially indicate LD when none is present (i.e., a written output disorder). In addition, these subtests are geared towards formal mathematics instruction and do not take into account the informal mathematics knowledge that the students may have (Mazzocco, 2005). Mazzocco (2005) also noted that there are typically a small number of items in many subtests. This could lead to an over representation of LD if a student were to make simple mistakes on a few items, as it would significantly reduce their score. As well, the definition of math achievement may differ between tests (Landerl et al., 2004). As an example, when comparing tools that have been used (e.g., the WRAT and the WJ III ACH), performance on the WJ III ACH was typically higher than the WRAT (Fuchs et al., 2005). This would likely influence rates of MD diagnosis.  5 It is important to note that there is also lack of consistency on the tasks provided to the participants in research studies following the initial diagnosis. The assessments used to gage math ability during research studies use a wide range of tools; these have included tasks taken from the British Ability Scales (Bull & Johnson, 1997), the Curriculum-Based Measurement, CBM (Fuchs et al., 2005), the WJ III ACH Calculation test as well as individually created story problems (Fuchs et al., 2005) among others. 1.1.3 Diagnosing math disability: Cut off criteria. In using standardized tests to determine MD status, a variety of cut off criteria have been used to diagnose MD (Murphy et al., 2007). The cut off criteria for assigning MD status has ranged from the 10th to 45th percentile relative to normative samples throughout most of the research, with a typical value being 35 (Hanich et al., 2001; Murphy et al., 2007). This wide range of cut off values causes problems for identifying MD, as those towards the upper limit may not actually be displaying MD (Landerl et al., 2004). The resulting problem is that of a Type I error, of misidentifying someone as MD, which becomes much more likely the higher the cut off that is used. It has been noted that the higher values are not consistent with the estimated prevalence of MD in the population, which lies at between 5 to 8% (Geary, 2004; Murphy et al., 2007). Other researchers have argued that a higher cut off criteria should not be used to identify MD, but to capture those students who are low achieving or who are only transiently MD (Chong & Siegel, 2008; Geary et al., 2007; Vukovic & Siegel 2010). Nevertheless, such high cut offs have been used in the past as a sole identifier of MD, with the two predominant reasons for using such high cut off criteria being to catch as much of the MD population as possible or to ensure a large enough sample size for analysis (Hanich et al., 2001).  6 It has been argued that using lower cut off criteria would result in a better sample of students who truly display MD and are not low achieving (Mazzocco & Myers, 2003). With this knowledge, researchers have recently chosen to differentiate the cut off criteria into MD, the lowest percentiles (10th percentile and lower), and MD(11-25) as those with higher percentiles (between the 11th and 25th percentile) (Chong & Siegel, 2008; Geary et al., 2007; Geary, Hoard, Nugent, & Bailey, 2011). Indicating that MD has two categories, MD10 and MD(11-25), could allow for greater sensitivity to the severity of the math problems faced by these children (Geary et al., 2007). However, as there are so few studies that have investigated this, it is inconclusive at this time as to whether there are any significant differences between these groups. 1.1.4 Diagnosing math disability: Small sample size. Another significant issue in studies of MD is the typically small sample sizes that have been used in the past. Geary (1990), Geary, Brown, and Samaranayake (1991) and Jordan and Hanich (2000) are but a small sample of the research studies that have used only a few students to draw comparisons. A number of studies have even drawn conclusions after studying fewer than 20 MD participants (e.g., Cencebella & Noel, 2008; Geary et al., 1991; Geary et al., 2007 Geary et al., 2011; Hitch & McAuley, 1991; Landerl et al., 2004; Schuchardt, Maehler, & Hasselhorn, 2008). Much more reliable data would be achieved if larger sample sizes were used. However, as noted above, the cut offs used to identify MD have an impact on sample size, with stricter cut off criteria leading to smaller samples. 1.1.5 Diagnosing math disability: Reading disabilities. Reading problems in students with MD have also been noted to confound the results presented (Landerl et al., 2004). Some studies have removed low readers from their analysis (Bull & Johnson, 1997; Geary, Hoard, Byrd-Craven, & Desoto, 2004; Passolunghi & Siegel 2001) whereas  7 others have not (Keeler & Swanson, 2001). However, when reading was controlled by Bull and Johnson (1997), the significant results of their study disappeared, leading one to speculate on exactly how important reading is to the assessment of WM and MD. As reading problems could potentially affect performance on mathematical tasks, influencing the scores of these students, controlling for or acknowledging reading problems in the participants is a critical aspect of studying MD. 1.2 Characteristics of math disability Knowing that there are difficulties with identifying MD, several studies have examined the defining characteristics of MD. The most obvious indicator is the lack of mathematics achievement of students with MD. It must, however, be noted that mathematics in itself is not a straightforward subject and many different skills and concepts are required for success (Dowker 2005; Ginsberg, 1997). This is problematic, as even from the very start, mathematics has many complexities involving language use and it is difficult to dissociate the numerical information and skills from other cognitive aspects (Landerl, et al., 2004). Typically, for comparison with other students, only arithmetic problems have been examined. Arithmetic, it must be noted, is itself not as simple as it may seem and is a task requiring many pieces of knowledge to be present for effective calculation to occur (Dowker, 2005). However, as there is not yet an overall agreement on the full nature of arithmetic, very little work has focused on more complex math, let alone the other domains of mathematics (Geary, 2004). Mathematical achievement difficulties discussed above has been divided into subgroups for convenient analysis by several researchers. These include calculation, fact fluency deficits, mathematical procedural deficits, and problem solving deficits (Bull,  8 Epsy, & Wiebe, 2008; Chong & Siegel, 2008; Gersten et al., 2005). Although problem solving is noted as an important aspect of mathematics, much of the MD literature has focused on student’s calculations, fact fluency skills, and basic number sense, (e.g., Geary 1990; 1993; Geary et al., 1991). Indeed, it has been assumed that speed and accuracy with addition (i.e. fact fluency) is a good way of determining MD (Geary, 1990). Such fluency with facts has been noted to be slower in students with MD than more typically achieving (TA) peers, resulting is longer reaction times (Bull & Johnson, 1997). Studies have also shown that although fluency improves over the years, students with MD do not catch up to TA students (Chong & Siegel, 2008). In addition to fact fluency, the use of strategies to solve simple problems, when the fact cannot be recalled from memory, is less developed and less mature in students with MD (Geary 1990; 1993; Geary, et al., 2007; Gersten et al., 2005). The strategies used by all students have been split into several categories of complexity and maturity (Geary, 1990). TA students will tend to rely on more mature strategies with greater effectiveness as they age (Bull & Johnson, 1997; Geary et al., 1991), whereas students with MD typically count on their fingers, an immature strategy, for much longer than their TA peers (Bull & Johnson, 1997). Students with MD are also more error prone when using their strategies (Geary et al., 2004). In addition, procedural skills for all students have been shown to grow rapidly in the early elementary years (Chong & Siegel, 2008). However, when compared to MD(11-25) and TA children, students with MD10 grew at a slower rate (Chong & Siegel, 2008). Students with MD also had significantly more difficulty detecting errors than their peers (Geary et al., 2007), an important aspect of understanding procedural work. Work by Morgan, Farkas, and Wu (2009) has shown that growth of ability across both  9 stable and transient MD students was similar, but they did not catch up to TA peers over a 5 year time period (Morgan et al., 2009). Indeed, they remained 2 standard deviations below their TA peers (Morgan et al., 2009). There is also evidence to suggest that, when looking at MD overall, students generally fail to catch up over time (Morgan et al., 2009). In analyzing performance in problem solving tasks, the strategies involved can be quite complex and can vary with the specific problems. This important issue is often ignored in the research. Problem solving itself, as an important aspect of mathematics, has been shown to be difficult for students with MD (Jordan & Hanich, 2000), although the reasons for this are not entirely clear. Reports have indicated that one cause may be working memory deficits (Mabbot & Bisanz, 2008). Indeed, several cognitive components and deficits have been linked to MD, such as working memory, which will be discussed in further sections. It is not entirely clear within the literature exactly what it means to have MD and what defines this learning disability. Subsequent to this, the criteria used to identify those with MD have been inconsistent throughout the research, with variability in the cut off values used and the way that MD is classified. With the knowledge that there is little consensus regarding the definition of MD in mind, let us turn our attention to one of the possible cognitive aspects that may significantly affect MD: working memory (WM). 1.3 Working memory Despite the variability in MD definitions used in the literature, research has identified a number of cognitive processes that are relevant to MD, with research conducted over the last two decades linking WM and MD. The work of Siegel and Ryan (1989) and other research groups (e.g., Geary et al., 1991; Hitch & McAuley, 1991; Swanson, 1993; 1994) has examined WM and how it is related to achievement in  10 mathematics and MD. This work has led to further research into the cognitive roots of MD and an increasing body of research has developed. There seems to be a general agreement among researchers that WM is an important cognitive mechanism in math achievement and MD. However, there does not seem to be agreement on exactly how WM has an impact on MD (Raghbur, Barnes, & Hecht, 2010). WM has been defined as an aspect of cognition that “stands at the crossroads between memory, attention, and perception” (Baddeley, 1992, p. 31). It has been defined and refined over the last 35 years and research has typically followed the model developed by Baddeley & Hitch (1974). This model originally defined WM as the central executive, combined together with two slave systems: the phonological loop and the visual spatial sketch pad (Baddeley, 1992; 2003). Recently, in response to the elucidation of certain aspects of memory and attention which did not fit the original model, such as how long term memory was accessed, a third slave system, the episodic buffer, has been introduced (Baddeley, 2000). This fourth component to the original model has modified the overall picture of WM and has led to a more complete description of WM. Even so, several studies continue to make observations exclusively based around the phonological loop and the visual spatial sketchpad, handling verbal and visual information respectively, as these have been noted as easier to assess and document (Baddeley, 1992; 2000; 2003). It is important to note that research has identified some notable consequences to having a poor WM (Gathercole et al, 2008). In their work, Gathercole et al. (2008) found that a reduced WM can co-occur with poor attention. Their work demonstrated that those students labelled as having difficulty with inattention in a typical classroom also displayed poor performance on WM tasks (Gathercole et al., 2008). They postulated that  11 the poor WM capacity of the students would lead to an overload in their capacity much sooner, and thus a loss of focus and attention on more complex tasks. Having such a deficit affected not only their attention and behaviour but their academic learning as well (Gathercole et al., 2008). 1.3.1 Working memory and math disabilities. The meta-analysis of Swanson and Jerman (2006) indicates that there appears to be an overall deficit in verbal WM, the phonological system, in students with MD when compared to their TA peers. This conclusion is the culmination of an analysis of 85 articles on MD that these authors felt were relevant to the discussion. Even though this conclusion seems to reflect the general assessment of the work conducted at that time, it must be noted that relatively little work has been done on WM and MD (Swanson & Jerman, 2006). This relatively small body of research in the field of WM and MD has been wide ranging in focus. Some researchers have examined performance on math skills in general (Geary et al., 2007; Vukovic & Siegel, 2010). Some have focused only on the deficits in WM and how it relates to MD (Schuchardt et al., 2008). Others have focused on more specific skills or aspects of mathematics and how WM affects performance on these tasks. These include multiplication (Mabbott & Bisanz, 2008), problem solving (Passolunghi & Siegel, 2001; Swanson, Jerman, & Zheng, 2008), arithmetic (Wu et al., 2008), and the procedural deficits of MD participants (Chong & Siegel, 2008). There is also a disparity in the number of WM tasks used throughout these studies. Some studies have used only one WM measure (e.g., Chong & Siegel, 2008; Fuchs et al., 2005; Geary et al., 2004; Vukovic & Siegel, 2010), some have used two WM measures (e.g., Keeler & Swanson 2001; Mabbott & Bisanz, 2008), some nine (Geary et al., 2007; Geary, Hoard, Nugent, & Byrd-Craven, 2008), some eleven (Swanson, 1993) and one study used sixteen measures  12 of WM (Schuchardt et al., 2008). It is also important to note that only a few studies (i.e., Chong & Siegel, 2008; Geary et al., 2007; Geary, Bailey, Littlefield, Wood, Hoard, & Nugent, 2009: Geary et al., 2011; Mabbott & Bisanz, 2008; Wu et al., 2008;) considered potential differences between MD definitions (i.e., low achieving and MD). As discussed previously, there is a body of converging evidence that suggests that a single MD label might not accurately reflect the nature of MD in students. Reviewing the literature on WM and MD, however, only six studies (i.e., Chong & Siegel, 2008; Geary et al., 2007; Geary et al., 2009: Geary et al., 2011; Mabbott & Bisanz, 2008; Wu et al., 2008;) appear to have differentiated between the MD and low achieving groups. As mentioned before, these researchers were not all observing the same aspect of mathematics, nor were these studies necessarily uniform in their use of WM measures. 1.3.2 Working memory measures. In order to measure WM, a task must be designed that “…require[s] the simultaneous processing and storage of information” (Vukovic & Siegel, 2010, p. 29). At present, there is no overall consensus as to how WM is best measured, and no standard set of tasks seems to be accepted by all researchers. Several measures of WM have been designed over the years. For example there is the Working Memory Test Battery for Children (WMTB-C; Pickering & Gathercole, 2001), the Automated Working Memory Assessment (Alloway, 2009), and the Swanson Cognitive Processing Test. (S-CPT; Swanson, 1995; Keeler & Swanson, 2001). These measures are made up of a variety of tasks which are meant to assess different subsystems of WM, as well as WM overall. There has, however, been considerable inconsistency in the tasks used to measure the same cognitive aspects when studying WM and MD. Part of the reason for this is that the researchers are not always using the tasks as measures of the same aspect of WM. Two examples will illustrate this.  13 The forward digit span has been used as a measure of WM by Bull et al. (2008), Geary et al. (1991), Geary et al. (2008), Passolunghi and Cornoldi (2008), Schulhart (2008), and Wu et al. (2008). In contrast to this, Vukovic and Siegel (2010), Swanson et al. (2008), and Passolunghi and Siegel (2001; 2004) have used the forward digit span in their work to measure short term memory. These studies have yielded differing conclusions, with some indicating that this measure is significantly associated with MD (Krajewski & Schneider, 2009) while other conflicting reports indicate otherwise (Vukovic & Siegel, 2010). Furthermore, some studies have collapsed this measure into other measures, such as the forward word span, and have included this measure in a group of measures assessing verbal WM (Geary et al., 1991). There is also a disparity of views concerning what the backward digit span assesses (Raghbur et al., 2010). Bull et al. (2008), Geary et al. (2007), Krajewski and Schneider (2009), Schulhart (2008), and Wu et al. (2008) have used this measure to assess the central executive. Passolunghi and Siegel (2004), on the other hand, used this to measure what they described as simple WM, while Passolunghi and Siegel (2001), Landerl et al. (2004), and Swanson et al. (2008) used this as a measure of short term memory and did not consider it a WM measure. Complicating this issue further, Bull et al. (2008) eliminated this measure from their growth models and several researchers, including Geary et al. (1991), Krajewski and Schneider (2009), Passolunghi and Siegel (2001), and Swanson et al. (2008), have grouped this task together with the forward digit span or the backward word span task for analysis. The use of these two specific measures of WM, forward digit span and backward digit span, provides an illustration of the inconsistent use of and analysis of measures that makes it difficult to compare the results of these studies against one another.  14 1.4 Other cognitive variables Although much of the research into the cognitive characteristics of MD has focused on WM, there are other aspects of achievement and cognition that may be relevant to MD. Processing speed has been linked to general quantitative knowledge in more typical students (Taub, Floyd, Keith, & McGrew, 2008). However, decreased processing speed has been linked to MD in various capacities by several researchers (Bull & Johnston, 1997: Chong & Siegel 2008: Vukovic & Siegel 2010). Bull and Johnston (1997) found that this aspect of cognition was significantly lower in those participants whom they labelled as having difficulty with arithmetic. Chong and Siegel (2008) also found persistent deficits in processing speed for their low achieving and MD groups when compared to their TA counterparts. Consistent with this, Vukovic and Siegel (2010) were able to use processing speed to distinguish those who had MD consistently over time from their TA and more transient MD counterparts, but only at two of their four time points. In contrast, Anderson (2010) found that processing speed was not significantly different for MD participants (relative to TA), although it was significant when participants had both MD combined with reading difficulties. In addition to processing speed, Taub et al. (2008) used several other tasks to determine predictors of quantitative reasoning with a large sample of individuals ranging in age from 5 to 19. Commenting on math achievement in general in relation to typical students, Taub et al. (2008) was not specifically interested in how these aspects of cognition related to MD in particular. However, these variables may be important factors in distinguishing MD from TA and may be related to the differences between the MD and low achieving subgroups. These cognitive variables have included, but are not limited to, categorical reasoning, language vocabulary development, fluid reasoning, and auditory  15 processing. Categorical reasoning in particular was found to be related to overall quantitative reasoning by Taub et al. (2008) and problem solving by Fuch et al. (2006). Similar to categorical reasoning, fluid reasoning was significantly related to quantitative understanding, while language vocabulary development was only moderately related (Taub et al., 2008). In contrast, auditory processing was not found to be related to quantitative reasoning at all. These last three aspects of cognition, fluid reasoning, language vocabulary development, and auditory processing, do not appear to have been addressed in the MD literature to date. It is important to consider them, however, given that reading difficulties have been known to be associated with MD for some time and there is potential that language development and auditory processing may represent areas of deficit for those who have MD. In summary, as there has been a growing body of research conducted in the area of MD, it is surprising that there is little consensus on how MD is identified and defined. Of concern is the identification of MD and the wide range of cut offs which have been used to designate MD. With such disparate cut offs found in the literature, it is difficult for any conclusive statements to be made. This is an important issue, as it is becoming more common for a low achieving group to be studied by researchers and these participants may constitute an important and often missed subgroup of MD. It is also clear that WM has a significant impact on learning and the education of children, and WM deficits are present in students with MD. With the multitude of tasks designed to assess WM at present, there is, however, a lack of consensus on how WM affects MD. There are also additional cognitive characteristics that may relate to MD, including processing speed and categorical reasoning among others, which have not often been investigated. These factors make it difficult to compare findings across MD studies.  16 1.5 Present study The purpose of this present study is to examine different cut off criteria used to assign MD status. Further to this, the present study investigated how several different cognitive abilities, including WM, were affected by the different cut off values when compared to TA peers. In particular, this researcher set out to answer three key questions. First, does it matter if MD is assigned at 25% or at 10%, and does it matter which types of tests are used to identify MD (i.e., calculations, problem solving, or both combined)? Are there enough cognitive and WM differences between these two cut off designations (i.e., 25% vs. 10%) when compared to their TA peers to warrant further investigation of the 10% cut off? Second, how does the identification of a low achieving group (i.e., 1125%) affect this comparison? Are they different enough from their TA peers and the other definitions of MD to warrant further consideration? Third, do participants who have persistent MD, whatever the cut off, have cognitive profiles that are different from those who have MD at only one of two time points?  17 Chapter 2  Method 2.1 Participants Participants were drawn from a larger longitudinal study conducted by Linda Siegel’s research group at the University of British Columbia in 2006 to 2008. As part of the longitudinal study, participants were recruited from 28 elementary schools in the North Vancouver School district. Participants were recruited in grade 1 and tested each year from grades 1 – 3, with new participants allowed to join each year.1 There were no restrictions placed on participation in the study. The current study investigated only the grade 2 and grade 3 participants. The data collection for these grades took place in two phases each year, approximately 1 month apart. The grade 2 sample consisted of a total of 868 participants assessed during phase 1 (49.42% female; Mage = 7.48), and 222 participants (50.90% female; Mage = 7.57) assessed during phase 2. In grade 3, a total of 923 participants were assessed during phase 1 (49.51% female, Mage = 8.43) while 441 (49.43% female, Mage = 8.54) were tested during phase 2. 2.2 Materials 2.2.1 Phase I. Calculation skills. Calculation skills were assessed with the Calculation test of the WJ III ACH (Woodcock et al., 2001a). This task is a measure of the participant’s ability to perform arithmetic calculations and requires the participants to complete a series of mathematical calculations of increasing difficulty on paper. The calculation questions are not read to the participants and they are required, in their own  1  The grade 1 data was analyzed separately. The present study focused on the grade 2 and 3 data to ensure that the majority of the novelty effects associated with the assessment battery had expired.  18 time, to complete as many as they can. In the current study, percentile scores are analyzed. Practical problem solving. Practical problem solving was assessed with the Applied Problems test of the WJ III ACH (Woodcock et al., 2001a). This task is a measure of quantitative reasoning and requires the participants to solve practical mathematical problems of increasing difficulty as they are read aloud by the examiner (e.g., Maggie had $6. She bought a kite for $3.35 and a pencil for $1.05. How much money did she have left?). The participants had access to paper and pencil and were able to see the problem. Percentile scores are analyzed. Word Reading. Word reading was measured using the Letter-Word Identification test of the WJ III ACH (Woodcock et al., 2001a). This task requires the participants to read individual letters and isolated sight words of increasing complexity. Percentile scores are analyzed. Word Decoding. Decoding was assessed using the Word Attack test of the WJ III ACH (Woodcock et al., 2001a). This task requires the participants to decode and read aloud a series of nonsense words that are presented to them. Percentile scores are analyzed. 2.2.2 Phase II. Visual spatial abilities. Visual spatial abilities were assessed using the Spatial Relations test of the Woodcock-Johnson Third Edition Tests of Cognitive Abilities (WJ III COG; Woodcock, McGrew & Mather, 2001b). This task requires the participant to identify which pieces are used to form a complete shape. Percentile scores are analyzed. Visual spatial abilities were also assessed with the Block Rotation test of the WJ III: Diagnostic Supplement to the Tests of Cognitive Abilities (WJ III-DS; Woodcock McGrew, Mather & Schranck, 2003). In this task the participants  19 are required to mentally manipulate 3 dimensional objects. After being presented with a stimulus object, the participant is required to define which 2 of the 4 items presented represent the object after it has been rotated in space. Percentile scores are analyzed. Fluid Reasoning. Fluid reasoning and inductive reasoning was assessed using the Concept Formation test of the WJ III COG (Woodcock et al., 2001b). This task requires that the participant examine increasingly difficult sets of items and formulate a rule that categorizes all the items. Percentile scores are analyzed. Fluid reasoning and general deductive abilities were also assessed using the Analysis and Synthesis test of the WJ III COG (Woodcock et al., 2001b). In this task, participants are asked to examine and solve a series of increasingly complex puzzles with instructions provided to facilitate their performance. Percentile scores are analyzed. Quantitative reasoning. Quantitative reasoning was measured using the Quantitative Concepts test of the WJ III ACH (Woodcock et al., 2001a) and included the concepts and number series subtests. The test is comprised of two subtests. The concepts subtest requires the participants to identify shapes, identify basic numerical information, and identify formulae. The number series subtest requires the participants to provide missing numbers from increasingly complex pattern series. Percentile scores are analyzed. Perceptual reasoning. Perceptual reasoning was assessed using the Visual Matching 2 test of the WJ III COG (Woodcock et al., 2001b). The test requires the participants to locate and circle two identical numbers in a row of six within a three minute time limit. The numbers increased in complexity from single digit up to 3 digits. Percentile scores are analyzed.  20 Language vocabulary development. Language development was assessed using the Verbal Comprehension test of the WJ III COG (Woodcock et al., 2001b).which is comprised of four subtests (Picture Vocabulary, Synonyms, Antonyms, Analogies). Picture Vocabulary requires the participants to identify a series of pictures. Synonyms requires the participants to hear a word and provide a synonym for that word, while Antonyms is a similar task requiring an antonym for the words. The Analogies subtest presents the participant with three words of an analogy and they are required to fill in the fourth analogous word. Percentile scores are analyzed. Processing speed. Processing speed was measured using the Rapid Picture Naming test of the WJ III COG (Woodcock et al., 2001b). This task requires the participant to quickly name a series of pictures within a time limit of two minutes. Percentile scores are analyzed. A second measure of processing speed that was included was the Rapid Number Naming task of Denckla and Rudel (1974). This task presents the participants with a 5 x 5 grid of single digit numbers. In this task, participants are required to read the numbers in order horizontally, by row, as fast as possible. Time, rounded to the nearest second, is analyzed. Auditory processing. Auditory processing was assessed using the Incomplete Words test of the WJ III COG (Woodcock et al., 2001b). This task requires the participants to listen to an audio recording of words in which one or more phonemes are missing. The participants are required to complete the word and say it orally. Percentile scores are analyzed. Short term memory. Short term memory was assessed using the Digits Forward test of the Wechsler Intelligence Scale for Children-Fourth Edition (WISC-IV; Wechsler,  21 2003). This task requires the participants to repeat a series of numbers verbatim after they have been read aloud by the examiner. Raw scores are analyzed. Working memory for numbers. Working memory for numbers was assessed using the Digits Backward test of the (WISC –IV; Wechsler, 2003). This task requires the participants to repeat a series of numbers in reverse order after they had been read aloud by the examiner. Raw scores are analyzed. Mathematical working memory. Mathematical working memory capacity was assessed through the Counting Span task (Case, Kurland, & Goldberg, 1982; Siegel & Ryan, 1989). This task presents the participant with a series of index cards containing blue and yellow dots. The participant is required to count out loud the yellow dots and then recall the number of dots for each item, in order, at the end of the trial. The dots are arranged in a random order and the number varies from one to nine. Raw scores are analyzed. Verbal working memory. Verbal working memory was assessed using the Working Memory for Words test which was developed by Siegel and Ryan (1989), modified from Daneman and Carpenter (1980). This task requires the participant to listen to a series of short sentences in which the last word is missing. The participant must complete the sentence and then recall the words at the end of a trial. Complexity increases from 2 sentences to 5 sentences. Raw scores are analyzed. Phonological working memory. Phonological working memory was assessed using the Letter Number Sequence test of the WISC –IV (Wechsler, 2003). This task requires the participants to listen to increasingly complex sequences of letters and numbers. They are required to recall the letters in alphabetical order followed by the numbers in ascending order. Raw scores are analyzed.  22 2.3 Procedure The present study examined students’ data from grades 2 and 3. Data was collected in two phases each year. The first phase consisted of several tests of academic abilities, while the second phase consisted of an assessment of the cognitive abilities of the participants. Although there was no initial restriction on testing, those students who were identified as speaking English as a second language (ESL; i.e., those who reported that their first language was not English) were excluded from the analyses. This left 799 (50.06% female) participants in grade 2 phase 1, 209 (52.15% female) in grade 2 phase 2, 843 (49.82% female) in grade 3 phase 1, and finally 422 (50.00% female) in grade 3 phase 2. The participants were all tested in the spring of the school year, from April to June. The first phase of testing was completed in all schools, followed by the second phase beginning after phase 1 had been completed. This was repeated for grade 3. Trained graduate and undergraduate student research assistants (RAs) administered all tasks individually to each participant. Individual testing for Phase I was conducted in a quiet hallway that was separate from the children’s regular classroom activities; only selected students (i.e., those with significant issues with mathematics as well as those selected as comparison TA participants) were selected for individual testing for Phase II. RAs were trained and supervised in the administration of the tasks by a graduate student from the University of British Columbia. Each session lasted approximately 45 minutes. 2.3.1 Classification. As the purpose of this study was to examine the differences in cognitive profiles that might be present given different classifications of MD, the participants were classified as MD using a variety of criteria. MD was defined using the Calculation test and the Applied Problems test of the WJ III ACH (Woodcock et al.,  23 2001a). MD was defined using the 25th percentile, inclusive, with the cut off criteria first applied to the Calculation test and then to the Applied Problems test. A third classification of MD identified those who scored at or below 25% on both tests. This classification approach was repeated for the Grade 3 participants. In addition, groups defined as scoring 25% or below on the tests at both time periods, Grades 2 and 3, were labelled as MD persistent (MDp). These classifications were further repeated using 10% as a cut off. Further, an MD(11-25) group was created by identifying those participants who scored above 10%, yet not above the 25% inclusive. To ensure that the participants under investigation did not display significant reading problems, reading ability was controlled for by ensuring that those labelled as MD had scores on both the Letter-Word Identification and Word Attack tests of the WJ III ACH higher than 35%. Finally, all groups were reclassified on reading ability using a 40% cut off on the reading tests. A comparison TA group was identified as scoring above 35% on all measures of reading and mathematics (i.e., Word Attack, Letter-Word Identification, Applied Problems, and Calculation). For comparison, TA groups based on a 40% cut off were also identified. This created 54 separate groups for analysis (see Table 1). The 35% and 40% cut offs were used to demarcate typical performance as it was felt that this would represent a sample of those students whose performance would be considered typical of a classroom with both students that excel and students who require additional assistance but are not facing overwhelming struggles with either reading or mathematics.  24 Table 1 MD Classifications and Grades Tested with Different Reading Control Cut Offs. MD classification / grade Reading % cut off Reading % cut off MD both 25% Grade 2 35 40 MD app 25% Grade 2 35 40 MD calc 25% Grade 2 35 40 MD both 10% Grade 2 35 40 MD app 10% Grade 2 35 40 MD calc 10% Grade 2 35 40 MD both 25% Grade 3 35 40 MD app 25% Grade 3 35 40 MD calc 25% Grade 3 35 40 MD both 10% Grade 3 35 40 MD app 10% Grade 3 35 40 MD calc 10% Grade 3 35 40 MD both 25% Persistent 35 40 MD app 25% Persistent 35 40 MD calc 25% Persistent 35 40 MD both 10% Persistent 35 40 MD app 10% Persistent 35 40 MD calc 10% Persistent 35 40 MD(11-25) app 10.01 - 25% Grade 2 35 40 MD(11-25) calc 10.01 – 25% Grade 2 35 40 MD(11-25) both 10.01 – 25% Grade 2 35 40 MD(11-25) app 10.01 - 25% Grade 3 35 40 MD(11-25) calc 10.01 – 25% Grade 3 35 40 MD(11-25) Both 10.01 – 25% Grade 3 35 40 MD(11-25) app 10.01 - 25% Persistent 35 40 MD(11-25) calc 10.01 – 25% Persistent 35 40 MD(11-25) both 10.01 – 25% Persistent 35 40  25 Chapter 3  Results A series of t-tests were calculated to determine whether there were significant differences, α = 0.05, in the average performance on each of the variables between the different MD groups and the TA groups. The Grade 2 comparisons were made with a Grade 2 TA group while the Grade 3 comparisons were made with a Grade 3 TA group. The persistent TA group was composed of those participants who had TA scores in both grades. The analyses examined performance on the 15 different outcome measures, with the three different definitions of MD (i.e., MD based on Applied Problems, Calculation, or both measures) at the two different time points compared to the relevant TA group. In addition to the comparisons made between the different definitions of MD and the respective TA groups, additional t-tests compared the MD(11-25) definitions to the relevant TA group and the MD10 to the MD(11-25) group for each grade. The results from these analyses, including means, standard errors, and sample sizes, are presented in Tables 2 through 11. Significant group differences are identified using superscript letters and bolded text, as indicated.2 Given the sheer number of analyses performed and the relatively underdeveloped state of the literature on this topic, no correction for multiple comparisons was applied. This was to ensure that potentially meaningful results are not overlooked due to power limitations.3  2  Full results are available from the author upon request. Given the small sample sizes in the MD10 groups it was apparent that the majority of the contrasts involving this group would have low statistical power. As the present study focused on examining general similarities and differences between the MD groups, it was decided that applying a correction for multiple comparisons would only further reduce the power for these comparisons. This decision was made with the understanding that this may affect the type I error rate in this study. 3  26 The results presented in the tables reflect the use of the 35% cut off for the reading difficulties control criteria and the TA groups. It should be noted that, although the analyses were repeated using the 40% cut off (as per Table 1) there were only a few differences in the findings when comparing the results obtained using the 35% cut off and the 40% cut off. Based on this, it was determined that using the 40% cut off would not substantially affect the results obtained. These additional analyses will not be reported here.4 It should be noted that there were no participants in Grade 2 who met the criteria for MD using the MD10 Applied Problems and MD10 combined definitions. There were also no participants in the Persistent group who met the criteria for MD10, nor were there participants who met the criteria for the Persistent MD(11-25) combined group. Also, the Grade 2 MD25 Applied Problems and the Grade 2 MD(11-25) Applied Problems groups represent the same participants, as there were no MD10 participants for this sample. These factors did not affect the results obtained. The results were organized based upon the study’s main research questions. 3.1 Comparing MD10 and MD25 The first research question investigated whether there were there enough cognitive and WM differences between the MD10 and MD25 cut off designations when compared to their TA peers to warrant further investigation of the use of the MD10 cut off for research purposes.  4  Verbal comprehension Grade 3 MD 25 both, Verbal comprehension Grade 2 MD25 calc, Concept Formation Grade 3 MD10 both, Visual Matching Grade 2 MD25 calc, Letter Number Sequence Grade 3 MD10 calc, Spatial relations MDp25 calc, Visual Matching MDp25 calc, Rapid number naming MDp25 calc, and Digit Span Forward MDp25 calc were the 9 instances (out of 315) where there was a difference in the results between the 35% cut off and the 40% cut off.  27 3.1.1 MD10 and MD25 applied problems. When examining the Applied Problems definition of MD and comparing the MD10 and the MD25 cut off points, it is important to note that it was not possible to examine the Grade 2 data as no MD10 participants were identified using this definition. When considering the Grade 3 groups, there were only three instances in which the MD10 (N = 4) profile and the MD25 (N = 19) profile were different from one another (Table 2): Verbal Comprehension, Visual Matching, and Block Rotation. In all three cases, the results indicated that the MD10 group was not significantly different from the TA group, while the MD25 group was. However, this appears to be attributable to low statistical power. When the mean values (M) and standard errors (SE) for the groups are examined, a more complete picture emerges. Each measure will be discussed in turn. Table 2 Grade 3 Sample Sizes and Means for the MD10, MD25, and TA Groups Based on the WJ III Applied Problems Test Only (Standard Errors in Parentheses). MD10 MD25 TA Variable* N Mean N Mean N Mean Verbal Comprehension Composite 4 59.75 (10.01) 19 53.21 (5.35) 211 72.49 (1.18) Spatial Relations 4 36.25 (8.14) 19 42.89 (4.45) 209 64.38 (1.32) Concept formation 4 42.25 (8.78) 19 41.05 (5.00) 208 72.56 (1.58) Visual Matching 2 4 44.38 (21.62) 18 37.58 (6.66) 210 53.39 (1.93) Incomplete Words 4 71.25 (6.17) 19 54.58 (6.09) 211 57.10 (1.87) Analysis and Synthesis 4 20.00 (10.78) 19 40.47 (4.99) 210 66.95 (1.59) Rapid Picture Naming 4 54.25 (11.01) 19 40.68 (6.00) 210 48.74 (1.66) Counting Span Task 4 19 4.79 (0.39) 211 5.95 (0.15) 3.50 (0.65) Working memory for Words 4 2.75 (0.48) 19 3.37 (0.33) 212 4.41 (0.24) Rapid Number Naming 4 11.75 (1.38) 19 11.74 (0.59) 211 11.70 (0.17) Digit Span Forward 4 19 7.37 (0.34) 212 8.33 (0.13) 6.75 (0.25) Digit Span Backwards 4 4.50 (0.29) 19 5.26 (0.26) 212 6.72 (0.43) Letter Number Sequence 4 10.75 (1.65) 19 11.89 (0.86) 212 16.02 (0.20) Quantitative Concepts 4 21.25 (7.72) 19 29.11 (5.41) 211 67.33 (1.43) Block Rotation 4 47.50 (15.65) 19 48.11 (7.00) 203 67.93 (2.12) Note: *Variables that are significantly different when compared to the TA group are highlighted bold.  28 For Verbal Comprehension, examination of the mean values and standard errors show that the means for MD10 (M = 59.75, SE = 10.01) and MD25 (M = 53.21, SE = 5.35) are quite similar. It appears that the rather large standard error of the MD10 group is driving the non significant result when compared to the TA group (M = 72.49, SE = 1.18; Table 2). The standard errors for Visual Matching are even larger than Verbal Comprehension, with the MD10 group in this case displaying an SE of 21.62 which is extraordinarily large compared to the SE of 6.66 for the MD25 group. This is important as, while the mean values are further apart in this case (MD10: M = 44.38, MD25: M = 37.58), both means are quite different from the TA mean (M = 53.39, SE = 1.93). The pattern of means and standard errors for these variables suggests that the different significance values for these comparisons are likely not meaningful (Table 2), reflecting power limitations rather than actual differences in the MD10 and MD25 comparisons. When Block Rotation was considered, the mean values of the MD10 (M =47.50) and the MD25 (M = 48.11) definitions were almost identical, suggesting that the larger standard error in the MD10 group (SE = 15.65) likely explains why the comparison to TA is not significant in the MD10 group. Based on these results for Grade 3 Applied Problems, there do not appear to be any meaningful differences in the cognitive profiles of the MD10 and the MD25 groups. These results indicate that the MD10 and MD25 groups displayed significantly poorer performance relative to the TA group on 10 of the 15 tasks: Verbal Comprehension, Concept Formation, Spatial Relations, Visual Matching 2, Analysis and Synthesis, Counting Span Task, Digit Span Forward, Letter Number Sequence, Quantitative Concepts, and Block Rotation (Table 2).  29 3.1.2 MD10 and MD25 calculation. Using the results of performance on the Calculation test to define MD provided the most participants in the MD10 group of the three different definitions, Applied Problems, Calculation, and both combined (Grade 2 MD10: N = 11; Grade 3 MD10: N = 10). When the two grades were examined separately, the Grade 2 (MD25: N = 40) results indicated that the cognitive profiles of the MD10 and MD25 groups were quite similar when compared to the TA group. There were only four instances in which the cognitive profiles for these two cut-offs differed: Concept Formation, Spatial Relations, WM for Words, and Verbal Comprehension (Table 3). In the first three cases, as was the case with the Applied Problems definition, the results were significantly different for the MD25 group but not for the MD10 group (Table 3). As with Applied Problems, this seemed to be counter intuitive and may also be explained by low statistical power and by considering the pattern of mean values and standard errors across these variables (Table 3). Table 3 Grade 2 Sample Sizes and Means for the MD10, MD25 and TA Groups Based on the WJ III Calculation Test Only (Standard Errors in Parentheses). MD10 MD25 TA Variable* N Mean N Mean N Mean Verbal Comprehension Composite 11 41 61.49 (3.95) 95 69.14 (1.95) 54.45 (7.34) Spatial Relations 11 55.00 (6.25) 41 55.10 (3.41) 95 63.80 (1.91) Concept formation 11 62.18 (8.95) 40 62.30 (3.81) 95 72.65 (2.33) Visual Matching 2 11 34.18 (4.67) 40 41.35 (3.85) 94 53.92 (2.75) Incomplete Words 11 59.18 (7.72) 40 62.35 (4.10) 95 63.44 (2.56) Analysis and Synthesis 11 53.18 (8.98) 40 55.70 (3.92) 95 69.61 (2.17) Rapid Picture Naming 11 50.36 (7.99) 40 51.37 (4.33) 95 54.18 (2.35) Counting Span Task 11 4.36 (0.61) 40 4.13 (0.31) 95 4.69 (0.23) Working memory for Words 11 3.09 (0.37) 40 95 3.26 (0.17) 2.58 (0.23) Rapid Number Naming 11 13.45 (1.15) 40 14.03 (0.68) 95 12.81 (0.35) Digit Span Forward 11 40 95 9.34 (0.21) 7.27 (0.51) 7.73 (0.32) Digit Span Backwards 11 5.18 (0.48) 40 5.55 (0.21) 95 5.65 (0.15) Letter Number Sequence 11 10.64 (1.22) 40 11.80 (0.59) 95 14.66 (0.34) Quantitative Concepts 11 30.82 (8.19) 40 39.00 (4.21) 95 69.44 (2.40) Block Rotation 11 53.32 (10.62) 39 56.44 (5.72) 93 66.11 (3.24)  30 Note: *Variables that are significantly different when compared to the TA group are highlighted bold.  For Concept Formation, the mean values of the MD10 (M = 62.18) and MD25 (M = 62.30) definitions were almost identical and when you consider the SE of the MD10 group (SE = 8.95) relative to the MD25 group (SE = 3.81), the reason for the lack of significance in the TA comparison for the MD10 group became clear (Table 3). This same situation was repeated for the Spatial Relations measure in which the MD10 mean (M = 55.00) was virtually identical to the MD25 mean (M = 55.10) while the standard errors differed (SE = 6.25 vs. SE = 3.41). Verbal Comprehension and WM for Words presented a different case. The MD10 results for Verbal Comprehension (M = 54.45, SE = 7.34) presented a smaller degree of overlap with the MD25 results when standard errors and the confidence intervals for the means were considered (M = 61.49, SE = 3.95), allowing us to consider the idea that differences in significance between the MD10 and MD25 definitions may be meaningful for this measure. The results for the WM for measure, MD10 (M =3.09, SE = 0.37) and the MD25 (M =2.58, SE = 0.23) groups also showed a very little overlap in their results (based on the standard errors and confidence intervals for the means) and this may indicate a meaningful difference between these two groups. Overall, the results for Grade 2 Calculation indicated that the MD groups displayed significantly poorer performance relative to the TA group on 9 of the 15 tasks: Verbal Comprehension, Concept Formation, Spatial Relations, Visual Matching 2, Analysis and Synthesis, WM for Words, Digit Span Forward, Letter Number Sequence, and Quantitative Concepts (Table 3).  31 The Grade 3 (MD25: N = 68) data present a similar case to the Grade 2 groups. For Grade 3, 8 measures displayed consistent results across the MD10 and MD25 definitions when compared to the TA group indicating that there were strong similarities between the cognitive profiles of the two groups. These results indicated that Visual Matching 2, Analysis and Synthesis, Counting Span, Quantitative Concepts, and Block Rotation all showed significant differences between the MD and TA groups, with the TA group outperforming the MD groups. In contrast, performance on the Incomplete Words, WM for Words, and Backward Digit Span measures was not significantly different when compared to the TA group, indicating that these variables were not deficient in the MD groups. In contrast 6 of the 15 measures displayed different significance values across the MD10 and MD25 definitions compared to the TA group, with 5 of these (Concept Formation, Spatial Relations, Verbal Comprehension, Digit Span Forward, and Letter Number Sequence) displaying the same counter intuitive results of having the MD10 group demonstrate non-significant results when compared to the TA group while the MD25 group demonstrated significant results. Only the Rapid Number Naming task results indicated that a significant difference for the MD10 group but not for the MD25 group. Three of the stated measures, Spatial Relations, Digit Span Forward, and Letter Number Sequence, displayed a pattern in which the mean values are very similar for the MD10 and MD25 groups and it is likely the large standard errors that have caused the discrepancy in the results (Spatial Relations MD10: M = 56.10, SE = 9.03 vs. MD25: M = 54.18, SE = 2.65; Digit Span Forward MD10: M = 7.20, SE =0.36 vs. MD25: M = 7.64, SE = 0.20; Letter Number Sequence MD10: M = 13.10, SE = 1.43 vs. MD25: M = 13.46, SE = 0.47; see Table 4). In the other three instances, the comparability of the results is  32 questionable. Interestingly, in two of the cases, the MD10 group outperforms the MD25 group (Concept Formation MD10: M = 67.40, SE = 7.16 vs. MD25: M = 53.87, SE = 3.12; Verbal Comprehension MD10: M = 74.40, SE = 5.15 vs. MD25: M = 64.47, SE = 2.44; Rapid Number Naming MD10: M = 13.30, SE = 0.67 vs. MD25: M = 12.00, SE = 0.31). Although it is unclear why the MD10 group would perform better than the MD25 group on these measures, it appears that the lack of significance in the MD10 group may reflect an observable difference from the TA group in these cases.  Table 4 Grade 3 Sample Sizes and Means for the MD10, MD25, and TA Groups Based on the WJ III Calculation Test Only (Standard Errors in Parentheses). MD10 MD25 TA Variable* N Mean N Mean N Mean Verbal Comprehension Composite 10 74.40 (5.15) 68 64.47 (2.44) 211 72.49 (1.18) Spatial Relations 10 56.10 (9.03) 68 54.18 (2.65) 209 64.38 (1.32) Concept formation 10 67.40 (7.16) 68 53.87 (3.12) 208 72.56 (1.58) Visual Matching 2 10 19.15 (7.26) 68 32.73 (3.28) 210 53.39 (1.93) Incomplete Words 9 50.22 (8.73) 68 56.54 (3.22) 211 57.10 (1.87) Analysis and Synthesis 10 50.00 (9.15) 69 50.93 (3.00) 210 66.95 (1.59) Rapid Picture Naming 10 63.20 (4.99) 68 48.44 (3.03) 210 48.74 (1.66) Counting Span Task 10 4.50 (0.62) 68 4.78 (0.25) 211 5.95 (0.15) Working memory for Words 10 3.90 (0.50) 68 3.66 (0.18) 212 4.41 (0.24) Rapid Number Naming 10 13.30 (0.67) 68 12.00 (0.31) 211 11.70 (0.17) Digit Span Forward 10 7.20 (0.36) 69 7.64 (0.20) 212 8.33 (0.13) Digit Span Backwards 10 4.80 (0.61) 69 5.67 (0.16) 212 6.72 (0.43) Letter Number Sequence 10 13.10 (1.43) 69 13.46 (0.47) 212 16.02 (0.20) Quantitative Concepts 10 39.00 (7.31) 68 42.01 (2.79) 211 67.33 (1.43) Block Rotation 10 44.70 (7.30) 67 54.67 (4.13) 203 67.93 (2.12) Note: *Variables that are significantly different when compared to the TA group are highlighted bold. Overall, the Grade 3 Calculation results further indicated that the MD groups displayed significantly poorer performance relative to the TA group on 8 of the 15 tasks: Spatial Relations, Visual Matching 2, Analysis and Synthesis, Counting Span Task, Digit Span Forward, Letter Number Sequence, Quantitative Concepts, and Block Rotation (Table 4).  33 In summary, when using the Calculation test to define MD at the 25% and the 10% cut offs, there were only a few small differences between the cognitive profiles for these definitions. When Grade 2 is considered Verbal Comprehension and the WM for Words measures results which suggest that there were differences between the MD10 and MD25 groups. The Grade 3 results indicate that the Concept Formation, Verbal Comprehension, and Rapid Number Naming measures represent differences between the profiles of the MD10 and MD25 groups. Given that, of the 15 measures used, only 4 demonstrate differences between the definitions and that there is only one consistency across grades, Verbal Comprehension, the cognitive profiles of the MD10 and MD25 groups are similar enough to warrant considering the 25% cut off when defining MD for future studies. 3.1.3 MD10 and MD25 both combined. Using the Applied Problems and Calculation tests together as a means of identifying MD proved to be problematic in this study when considering that only one participant met the criteria for MD10 in Grade 3 and there were no participants meeting these criteria in Grade 2, it is not possible to draw meaningful conclusions from this data. While the comparisons between MD25 and MD10 using the Grade 3 results (Table 5) indicated that the profiles of MD10 and MD25 are still quite similar (with Verbal Comprehension, Visual Matching 2, and Counting Span being the only measures showing different levels of significance across the two groups) these results should be interpreted with caution and should not be used to draw any strong conclusions.  34 Table 5 Grade 3 Sample Sizes and Means for the MD10, MD25 and TA Groups Based on Both WJ III Applied Problems and WJ III Calculation Tests (Standard Errors in Parentheses). MD10 MD25 TA Variable* N Mean N Mean N Mean Verbal Comprehension Composite 1 88.00 7 58.14 (11.53) 211 72.49 (1.18) Spatial Relations 1 16.00 7 42.14 (8.14) 209 64.38 (1.32) Concept formation 1 26.00 7 27.86 (6.23) 208 72.56 (1.58) Visual Matching 2 1 0.50a 6 27.42 (11.72) 210 53.39 (1.93) Incomplete Words 1 60.00 7 46.71 (9.08) 211 57.10 (1.87) Analysis and Synthesis 1 7 31.57 (8.34) 210 66.95 (1.59) 2.00 Rapid Picture Naming 1 47.00 7 33.71 (7.32) 210 48.74 (1.66) Counting Span Task 1 3.00 7 211 5.95 (0.15) 3.71 (0.68) Working memory for Words 1 2.00 7 2.71 (0.57) 212 4.41 (0.24) Rapid Number Naming 1 15.00 7 12.57 (1.04) 211 11.70 (0.17) Digit Span Forward 1 6.00 7 7.86 (0.70) 212 8.33 (0.13) Digit Span Backwards 1 4.00 7 4.86 (0.34) 212 6.72 (0.43) Letter Number Sequence 1 7 212 16.02 (0.20) 7.00 9.43 (1.25) Quantitative Concepts 1 10.00 7 17.00 (4.86) 211 67.33 (1.43) Block Rotation 1 66.00 7 47.71 (14.58) 203 67.93 (2.12) Note:*Variables that are significantly different when compared to the TA group are highlighted bold.  3.1.4 MD10 vs MD25 summary. When considering the three different definitions Applied Problems, Calculation, and both combined, using the MD25 and the MD10 cut off points, Applied Problems yielded 12 cognitive and WM variables that showed identical patterns of significance relative to the TA group across the two cut off values. Calculation yielded 11 cognitive and WM variables with identical patterns for Grade 2 and 9 cognitive and WM variables with identical patterns of significance relative to the TA group at Grade 3. With the exception of 4 cases, all of the differences in significance values can be explained by examining the individual mean values and standard errors. This indicates that, while there may be a few small differences between the groups, these differences were far outweighed by the similarities between the two cut off values. Overall, there appears to be few meaningful differences between the MD10 and MD25 cut offs of MD.  35 3.2 Comparing MD10 and MD(11 –25) In addition to discussing the general pattern of results across the MD10 and MD25 groups, it is important to discuss the differences between the MD10 and MD(1125) groups. Since the MD10 participants are encompassed in the MD25 groups, a complete picture would benefit from investigating the MD10 group in isolation from the remainder of those who scored less than 25%. Thus, the MD(11-25) group (11-25%) was compared with the MD10 group. These groups were compared not only to the TA groups, but also to one another to determine whether there were meaningful differences between the MD10 and MD(11-25) groups that would recommend separating these two groups in future research. When comparing the performance of these two groups, only 4 defining tests could be examined, as the remainder of the groups did not have any participants that fell into the MD10 groups. Those considered included Grade 2 MD10 Calculation, Grade 3 MD10 Applied Problems, Grade 3 MD10 Calculation, and Grade 3 MD10 Both tests combined. Unfortunately the Grade 3 combined Calculation and Applied Problems group had only 1 participant for MD10 which was not enough for accurate analysis. Comparing the results for the MD groups at Grade 2, Grade 2 MD10 Calculation had the largest group of any MD10 cut off with N = 11. Comparing the MD10 and MD(11-25) groups to the TA group, different patterns of significant differences were found for Verbal Comprehension, Spatial Relations, Concept Formation, Visual Matching 2, and WM for Words (Table 6). It is important to note that, when the Calculation test was used to define MD, there were no significant differences between the MD10 and MD(11-25) groups on any of these variables (Table 6). In addition, any differences in significance in the MD10 vs. TA and MD(11-25) vs. TA comparisons  36 appear to be attributable primarily to the small sample size and large SE found in the MD10 groups. The results indicated that there are comparable means between the MD(11-25) and MD10 groups, suggesting few differences between the groups. For example, Spatial Relations has a rather large SE for the MD10 group (SE = 6.25), while the means are extremely similar (MD10: M =55.00 vs. MD(11-25): M = 55.13; Table 6). Concept Formation was similar in that the mean values for MD10 (M = 62.18) and MD(11-25) (M = 62.34) were also almost identical while the SE for MD10 (SE = 8.95) greatly exceeds the SE for MD(11-25; SE = 4.14) which was significantly different from the TA group. Table 6 Grade 2 Sample Sizes and Means for the MD10, MD(11-25), and TA Groups Based on the WJ III Calculation Test Only (Standard Errors in Parentheses).* MD10 MD(11-25) Variable** N Mean N Mean N ac a Verbal Comprehension Composite 11 54.45 (7.34) 30 64.07 (4.67) 95 Spatial Relations 11 55.00 (6.25)ab 30 55.13 (4.12)a 95 Concept formation 11 62.18 (8.95)ab 29 62.34 (4.14)a 95 Visual Matching 2 11 34.18 (4.67)a 29 44.07 (4.96)ac 94 Incomplete Words 11 59.18 (7.72)a 29 63.54 (4.90)a 95 Analysis and Synthesis 11 53.18 (8.98)a 29 56.66 (4.29)a 95 Rapid Picture Naming 11 50.36 (7.99)a 29 51.74 (5.23)a 95 Counting Span Task 11 4.36 (0.61)a 29 4.03 (0.36)a 95 ab Working memory for Words 11 3.09 (0.37) 29 2.38 (0.27)a 95 Rapid Number Naming 11 13.45 (1.15)a 29 14.24 (0.83)a 95 Digit Span Forward 11 29 7.90 (0.40)a 95 7.27 (0.51)a a Digit Span Backwards 11 5.18 (0.48) 29 5.69 (0.23)a 95 Letter Number Sequence 11 10.64 (1.22)a 29 12.24 (0.67)a 95 Quantitative Concepts 11 30.82 (8.19)a 29 42.10 (4.87)a 95 Block Rotation 11 53.32 (10.62)a 28 57.66 (6.89)a 93 Note: *Variables that are not significantly different across the groups are identified by the same superscript letter. **Variables that are significantly different when compared to the TA group are highlighted bold.  TA Mean 69.14 (1.95)c 63.80 (1.91)b 72.65 (2.33)b 53.92 (2.75)c 63.44 (2.56)a 69.61 (2.17) 54.18 (2.35)a 4.69 (0.23)a 3.26 (0.17)b 12.81 (0.35)a 9.34 (0.21) 5.65 (0.15)a 14.66 (0.34) 69.44 (2.40) 66.11 (3.24)a  Upon closer examination, Verbal Comprehension, Visual Matching, and WM for Words show different patterns of significant differences across the MD10 and MD(11-25)  37 groups relative to the TA group for this definition of MD. Verbal Comprehension has a mean of 54.45 and a SE of 7.34 in the MD10 group which was significantly different from the TA group. This is compared with the MD(11-25) group (M = 64.07, SE = 4.67) which appears to be different, but was not statistically significant when compared to the TA group (Table 6). When examining WM for Words, the MD10 group displays a mean of 3.09 (SE = 0.37) that is divergent from the MD(11-25) group (M = 2.38, SE = 0.27) and yet is not significantly different at p = 0.05 (Table 6). Visual Matching also has mean values between the groups (MD10: M = 34.18 vs. MD(11-25): M = 44.07) but these differences were not statistically significant (Table 6). Based on these results, the cognitive and WM profiles of the MD10 and MD(11-25) groups using the Grade 2 Calculation definition of MD are similar enough to not warrant making a distinction between the groups. This profile indicates that the MD10 and MD(11-25) groups displayed significantly poorer performance relative to the TA group on 8 of the 15 tasks: Concept Formation, Spatial Relations, Visual Matching 2, Analysis and Synthesis, WM for Words, Digit Span Forward, Letter Number Sequence, and Quantitative Concepts (Table 6). As there were some differences between the significance of these measures compared to TA, future research may wish to focus on those few areas identified as potentially different between MD10 and MD(11-25) (i.e., Verbal Comprehension, and Visual Matching) When comparing the results of the MD10 and MD(11-25) groups for the Grade 3 Applied Problems definition of MD, the Grade 3 MD10 had only 4 participants which severely limited our ability to draw meaningful conclusions. Nevertheless, 5 measures, Verbal Comprehension, Visual Matching 2, Counting Span, Digit Span Forward, and Block Rotation displayed significantly different results when comparing the MD10 and  38 MD(11-25) groups to the TA group (Table 7). However, when comparing the MD10 and MD(11-25) groups with one another, no group differences were found for these measures. As with the Grade 2 Calculation definition, the SE was substantially larger in the MD10 group for four of the measures compared to the MD(11-25) group which was likely the cause of the discrepancy in significance (i.e., Verbal Comprehension: MD 10 SE = 10.01 vs. MD(11-25) SE = 6.31; Visual Matching 2: MD10 SE = 21.62 vs. MD(11-25) SE= 6.51; Counting Span: MD10 (SE = 0.65) vs. MD(11-25; SE= 0.43); Block Rotation: MD10 (SE = 15.65) vs. MD(11-25; SE= 8.11; Table 7). In addition, the mean value of Digit Span Forward, in the MD10 group (M = 6.75) is comparable to the mean for the MD(11-25) group (M=7.53; Table 7). Table 7 Grade 3 Sample Sizes and Means for the MD10, MD(11-25), and TA Groups Based on the WJ III Applied Problems Test Only (Standard Errors in Parentheses).* MD10 MD(11-25) TA Variable** N Mean N Mean N Mean ab a Verbal Comprehension Composite 4 59.75 (10.01) 15 51.47 (6.31) 211 72.49 (1.18)b Spatial Relations 4 36.25 (8.14)a 15 44.67 (5.24)a 209 64.38 (1.32) Concept formation 4 42.25 (8.78)a 15 40.73 (6.02)a 208 72.56 (1.58) Visual Matching 2 4 44.38 (21.62)ab 14 35.64 (6.51)a 210 53.39 (1.93)b Incomplete Words 4 71.25 (6.17)b 15 50.13 (7.18)c 211 57.10 (1.87)bc Analysis and Synthesis 4 20.00 (10.78) 15 45.93 (4.88) 210 66.95 (1.59) Rapid Picture Naming 4 54.25 (11.01)a 15 37.07 (6.87)a 210 48.74 (1.66)a Counting Span Task 4 15 5.13 (0.43)ac 211 5.95 (0.15)c 3.50 (0.65)a a Working memory for Words 4 2.75 (0.48) 15 3.53 (0.39)a 212 4.41 (0.24)a Rapid Number Naming 4 11.75 (1.38)a 15 11.73 (0.68)a 211 11.70 (0.17)a Digit Span Forward 4 15 7.53 (0.41)ac 212 8.33 (0.13)c 6.75 (0.25)a Digit Span Backwards 4 4.50 (0.29)a 15 5.47 (0.31)a 212 6.72 (0.43)a a Letter Number Sequence 4 10.75 (1.65) 15 12.20 (1.01)a 212 16.02 (0.20) Quantitative Concepts 4 21.25 (7.72)a 15 31.20 (6.54)a 211 67.33 (1.43) Block Rotation 4 47.50 (15.65)ab 15 48.27 (8.11)a 203 67.93 (2.12)b Note: *Variables that are not significantly different across the groups are identified by the same superscript letter. **Variables that are significantly different when compared to the TA group are highlighted bold.  39 Only two measures, Incomplete Words and Analysis and Synthesis, showed significant group differences when comparing MD10 and MD(11-25) for this definition. While the MD10 group was not significantly different from TA, the MD(11-25) group was significantly different from TA. Therefore, these results indicated that, when using Applied Problems to define MD, there are no meaningful differences between the MD10 and MD(11-25) groups in the Grade 3 data. The overall profile for the Grade 3 Applied Problems definition indicated that the MD10 and MD(11-25) groups displayed significantly poorer performance relative to the TA group on 10 of the 15 tasks: Verbal Comprehension, Concept Formation, Spatial Relations, Visual Matching 2, Analysis and Synthesis, Counting Span Task, Digit Span Forward, Letter Number Sequence, Quantitative Concepts and Block Rotation (Table 7). Turning our attention to the Calculation test, when comparing the results of the MD10 and MD(11-25) groups relative to TA using the Grade 3 Calculation definition of MD (Grade 3 MD10: N = 10), different results, in terms of significance, were found between these groups for only 6 of the 15 measures: Verbal Comprehension, Spatial Relations, Concept Formation, Rapid Number Naming, Digit Span Forward, and Letter Number Sequence. However, as with the Grade 2 Calculation definition, no significant differences were found between the MD10 and MD(11-25) groups for any of these measures (Table 8). In two cases, the SE in the MD10 group is substantially larger when compared to the MD(11-25) group (Spatial Relations MD10: M =56.10 SE = 9.03 vs. MD(11-25): M = 53.84 SE =2.73; Concept Formation MD10: M = 67.40 SE = 7.16 vs. MD(11-25): M = 51.53 SE = 3.37; Table 8). In a similar vein, the mean scores did not reflect notable differences between the MD10 and MD(11-25) groups in the case of the Digit Span Forward (MD10: M =7.20 vs. MD(11-25): M = 7.71) and Letter Number  40 Sequencing (MD10: M =13.10 vs. MD(11-25): M = 13.53) tasks. This supports minimal differences between the MD10 and MD(11-25) groups using this defining test. Table 8 Grade 3 Sample Sizes and Means for the MD10, MD(11-25), and TA Groups Based on the WJ III Calculation Test Only (Standard Errors in Parentheses).* MD10 MD(11-25) TA Variable** N Mean N Mean N Mean a a Verbal Comprehension Composite 10 74.40 (5.15) 58 62.76 (2.67) 211 72.49 (1.18) Spatial Relations 10 56.10 (9.03)a 58 53.84 (2.73)a 209 64.38 (1.32) Concept formation 10 67.40 (7.16)a 58 51.53 (3.37)a 208 72.56 (1.58) Visual Matching 2 10 19.15 (7.26)a 58 35.07 (3.57)a 210 53.39 (1.93) Incomplete Words 9 50.22 (8.73)a 59 57.50 (3.48)a 211 57.10 (1.87)a Analysis and Synthesis 10 50.00 (9.15)a 59 51.08 (3.18)a 210 66.95 (1.59) Rapid Picture Naming 10 63.20 (4.99)b 58 45.90 (3.35)c 210 48.74 (1.66)bc Counting Span Task 10 4.50 (0.62)a 58 4.83 (0.27)a 211 5.95 (0.15) Working memory for Words 10 3.90 (0.50)a 58 3.62 (0.20)a 212 4.41 (0.24)a Rapid Number Naming 10 13.30 (0.67)a 58 11.78 (0.34)a 211 11.70 (0.17) Digit Span Forward 10 7.20 (0.36)ab 59 7.71 (0.23)a 212 8.33 (0.13)b Digit Span Backwards 10 4.80 (0.61)b 59 5.81 (0.15)c 212 6.72 (0.43)bc Letter Number Sequence 10 13.10 (1.43)ab 59 13.53 (0.49)a 212 16.02 (0.20)b Quantitative Concepts 10 39.00 (7.31)a 58 42.53 (3.04)a 211 67.33 (1.43) Block Rotation 10 44.70 (7.30)a 57 56.42 (4.67)a 203 67.93 (2.12) Note: *Variables that are not significantly different across the groups are identified by the same superscript letter. **Variables that are significantly different when compared to the TA group are highlighted bold.  Of the 15 measures used, only Rapid Picture Naming and Digit Span Backwards were significantly different when MD10 and MD(11-25) were compared using Calculation to define MD. The MD(11-25) group performed at a higher level than the MD10 on the Digit Span Backwards while the MD10 group performed at a higher level than the MD(11-25) group on the Rapid Picture Naming. However, neither group showed significant differences from the TA group for the Digit Span Backwards measure, eliminating this as a meaningful difference in the cognitive profiles for these groups. For Rapid Picture Naming, MD10 was significantly different from the TA group while MD(11-25) was not, suggesting that meaningful group differences may be present for this  41 one measure with the more severe MD10 showing significant deficits relative to TA while the MD(11-25) group does not. Interestingly, for Verbal Comprehension, significant differences between the MD(11-25) and TA groups but not the MD10 and TA groups appear to be potentially meaningful. Although the MD10 group has a larger SE (5.15 relative to 2.67 for MD(1125) and 1.18 for TA), similar means were found in the MD10 and TA groups (M= 74.40 and 72.49 respectively), suggesting that a different process may be affecting the MD(1125) and MD10 groups for Verbal Comprehension. Replication will be necessary to determine whether this is a statistical artefact. This presents an overall profile for Grade 3 Calculation indicating that the MD10 and MD(11-25) groups displayed significantly poorer performance relative to the TA group on 10 of the 15 tasks: Verbal Comprehension, Concept Formation, Spatial Relations, Visual Matching 2, Analysis and Synthesis, Counting Span Task, Digit Span Forward, Letter Number Sequence, Quantitative Concepts and Block Rotation (Table 8). 3.2.1 MD 10 VS MD(11-25) summary. When comparing MD10 to MD(11-25), virtually all of the comparisons indicated a lack of meaningful differences between these groups. In only 3 instances (i.e., Rapid Picture Naming, Verbal Comprehension, and Rapid Number Naming) were group differences found, and in only one of these (i.e., Rapid Picture Naming using the Grade 3 Calculation definition of MD) does there appear to be any meaningful difference between the groups. When comparing the mean values for the groups, two measures (i.e., Verbal Comprehension and Rapid Number Naming) were also found to represent meaningful differences in the cognitive profiles for the groups for Grade 3 Calculation only, although it must be noted that there were no statistically significant differences between the MD(11-25) and the MD10 groups for  42 these variables. As there are only three areas in which there are potentially meaningful differences between the cognitive profiles of MD(11-25) and MD10 and two of these did not show statistically significant differences between the groups, it stands to reason that the cognitive profile of the MD(11-25) group is virtually identical to that of the MD 10 group. When considering these results along with the comparison between MD10 and MD25, a picture emerges suggesting that using the MD25 cut off would be appropriate when identifying MD for the purposes of research, as there are few differences between this group (and the MD(11-25) group) and the more restrictive MD10 group. 3.3 MD Persistent The third research question considered how the cognitive profiles of the MD definitions at Grades 2 and 3 compared to the cognitive profiles of those students who demonstrated MD at both time points (i.e., the Persistent group). When comparing the results of the Grade 2 MD, Grade 3 MD, and MD persistent groups, these comparisons were based only on the MD25 definition as the Persistent group did not yield any participants in the MD10 groups. Furthermore, the MD(11-25) Persistent groups had very small sample sizes and did not have any participants in the combined group, limiting the utility of the MD(11-25) groups in drawing comparisons. 3.3.1 Comparing grade 2 MD and MD Persistent. When comparing the results of the Grade 2 and Persistent MD groups, there were a number of instances in which the cognitive profiles were the same for all three defining tests Calculation, Applied Problems, and both combined (i.e., Verbal Comprehension, Concept Formation, Spatial Relations, Visual Matching 2, Analysis and Synthesis, Digit Span Forward, Letter Number Sequence, and Quantitative Concepts where the TA groups performed at a significantly higher level than the MD groups; Tables 9,10, and 11). However, there  43 were several differences between the patterns of results (relative to the TA group) and these differences will be the focus of this section. When Calculation is considered alone, the results indicated that twelve measures represented the same pattern of results across the Grade 2 and the Persistent groups and only three measures (Counting Span Task, Rapid Number Naming, and WM for Words) presented different patterns relative to the TA group. The Counting Span and Rapid Number Naming variables appear to become more significant for those who were Persistent MD, as the Grade 2 group was found not to display significantly different results from the TA Group, while differences in WM for words (relative to TA) appear to be less significant with the Persistent group when compared to the Grade 2 cases. Upon closer examination, the Counting Span task (Grade 2: M = 4.13, SE = 0.31; MDp: M = 4.24, SE =0.48) and the Rapid Number Naming task (Grade 2: M =14.03, SE = 0.68; MDp: M = 13.00, SE = 0.78) have very similar mean values for both the Grade 2 and Persistent groups which may indicate that there is actually no meaningful difference between these groups for these measures (Table 9). WM for Words displays a different pattern (Grade 2: M = 2.58, SE = 0.23; MDp: M = 3.35, SE =0.38), and when compared to the TA group, the results were sufficiently different to suggest that the Grade 2 and Persistent groups may differ on this variable with the Grade 2 participants demonstrating poorer performance.  44  Table 9 MD25 Sample Sizes and Means for the Grade 2 MD25, Grade 3 MD25, and Persistent Groups Based on the WJ III Calculation Subtest Only (Standard Errors in Parentheses). TA Means and Standard Errors are for the TA Persistent Group. Grade 2 MD25 Grade 3 MD25 MDp TA Variable* N Mean N Mean N Mean N Mean Verbal Comprehension Composite 41 61.49 (3.95) 68 17 63.24 (4.12) 175 71.88 (1.32) 64.47 (2.44) Spatial Relations 41 68 17 173 64.80 (1.43) 55.10 (3.41) 54.18 (2.65) 53.94 (4.27) Concept formation 40 68 17 173 72.26 (1.75) 62.30 (3.81) 53.87 (3.12) 59.53 (6.82) Visual Matching 2 40 68 17 174 54.46 (2.12) 41.35 (3.85) 32.73 (3.28) 35.85 (8.08) Incomplete Words 40 62.35 (4.10) 68 56.54 (3.22) 17 62.00 (4.47) 175 54.78 (1.99) Analysis and Synthesis 40 69 17 174 68.33 (1.74) 55.70 (3.92) 50.93 (3.00) 52.47 (5.41) Rapid Picture Naming 40 51.37 (4.33) 68 48.44 (3.03) 17 46.29 (6.89) 174 48.91 (1.83) Counting Span Task 40 4.13 (0.31) 68 17 175 6.02 (0.16) 4.78 (0.25) 4.24 (0.48) Working memory for Words 40 68 3.66 (0.18) 17 3.35 (0.38) 176 4.51 (0.28) 2.58 (0.23) Rapid Number Naming 40 14.03 (0.68) 68 12.00 (0.31) 17 175 11.58 (0.18) 13.00 (0.78) Digit Span Forward 40 69 17 176 8.34 (0.15) 7.73 (0.32) 7.64 (0.20) 7.18 (0.33) Digit Span Backwards 40 5.55 (0.21) 69 5.67 (0.16) 17 5.94 (0.31) 176 6.84 (0.52) Letter Number Sequence 40 69 17 176 16.16 (0.21) 11.80 (0.59) 13.46 (0.47) 13.12 (0.78) Quantitative Concepts 40 68 17 175 69.17 (1.53) 39.00 (4.21) 42.01 (2.79) 35.94 (5.12) Block Rotation 39 56.44 (5.72) 67 17 59.35 (8.38) 168 68.08 (2.35) 54.67 (4.13) Note: *Variables that are significantly different when compared to the TA group are highlighted bold.  45  When Applied Problems was considered alone, 12 measures presented a consistent pattern of results for both the Persistent and Grade 2 groups. Only 3 measures had different levels of significance relative to the TA group. All three (Rapid Picture Naming, WM for Words, and Digit Span Backwards) were found to be significant for the Grade 2 participants and not for the Persistent group, although both groups performed lower on these measures than the TA group. Upon closer examination, performance on the Rapid Picture Naming measure was virtually identical across the Grade 2 and Persistent groups (Grade 2: M = 37.80, SE = 4.03; MDp: M = 38.13, SE = 5.60 Tables 3 and 9). A similar picture is seen for the WM for Words measure (Grade: 2 M =2.05, SE = 0.29; MDp M = 2.38, SE = 0.46; Table 10). A slightly less straightforward case is presented for the Digit Span Backwards measure. Here the mean values of the Grade 2 and Persistent MD groups were less similar (Grade 2: M = 4.80 SE = 0.25; MDp: M = 5.50 SE = 0.50; Table 10). Since group differences from TA for these three measures were not significant for the Persistent group but were significant for the Grade 2 group, this may indicate that these factors (i.e., processing speed; Rapid Picture Naming and working memory; WM for Words, Digit Span Backwards) become less relevant as the participants age and develop compensatory strategies.  46  Table 10 MD25 Sample Sizes and Means for the Grade 2 MD25, Grade 3 MD25, and Persistent Groups Based on the WJ III Applied Problems Subtest Only (Standard Errors in Parentheses). TA Means and Standard Errors are for the TA Persistent Group. Grade 2 MD25 Grade 3 MD25 MDp TA Variable* N Mean N Mean N Mean N Mean Verbal Comprehension Composite 21 19 8 175 71.88 (1.32) 40.62 (6.00) 53.21 (5.35) 53.13 (7.54) Spatial Relations 21 19 8 173 64.80 (1.43) 48.24 (4.98) 42.89 (4.45) 41.00 (8.07) Concept formation 20 19 8 173 72.26 (1.75) 42.80 (6.26) 41.05 (5.00) 37.25 (6.94) Visual Matching 2 20 18 8 54.46 (2.12) 33.95 (5.04) 37.58 (6.66) 34.44 (12.00) 174 Incomplete Words 20 59.60 (5.16) 19 54.58 (6.09) 8 59.13 (6.86) 175 54.78 (1.99) Analysis and Synthesis 20 19 8 174 68.33 (1.74) 34.95 (5.34) 40.47 (4.99) 29.88 (7.50) Rapid Picture Naming 20 19 40.68 (6.00) 8 38.13 (5.60) 174 48.91 (1.83) 37.80 (4.03) Counting Span Task 20 19 8 175 6.02 (0.16) 3.60 (0.35) 4.79 (0.39) 3.88 (0.44) Working memory for Words 20 19 3.37 (0.33) 8 2.38 (0.46) 176 4.51 (0.28) 2.05 (0.29) Rapid Number Naming 20 14.30 (0.87) 19 11.74 (0.59) 8 11.13 (1.06) 175 11.58 (0.18) Digit Span Forward 20 8.45 (0.46) 19 8 176 8.34 (0.15) 7.37 (0.34) 6.75 (0.25) Digit Span Backwards 20 19 5.26 (0.26) 8 5.50 (0.50) 176 6.84 (0.52) 4.80 (0.25) Letter Number Sequence 20 19 8 176 16.16 (0.21) 9.85 (0.77) 11.89 (0.86) 12.13 (1.13) Quantitative Concepts 20 19 8 175 69.17 (1.53) 29.40 (4.21) 29.11 (5.41) 31.00 (6.86) Block Rotation 20 19 8 168 68.08 (2.35) 41.05 (7.96) 48.11 (7.00) 45.13 (7.95) Note: *Variables that are significantly different when compared to the TA group are highlighted bold.  47  The combined Calculation and Applied Problems definition presented the most different pattern of results between the Grade 2 and Persistent groups of the three definitions, with only 9 measures showing consistent results. It is difficult to draw strong conclusions based on this definition of MD, as N= 3 for the Persistent groups and N = 9 for the Grade 2 group (Table 11). Five measures (Verbal Comprehension, Visual Matching, WM for Words, Digit Span Forward, and Block Rotation) showed significant differences relative to TA when this definition was applied to the Grade 2 group but not for the Persistent group. Three of the measures, (Verbal Comprehension Grade 2: M =38.90, SE = 7.77; MDp: M = 68.67, SE = 14.19; Block Rotation Grade 2: M = 33.11, SE = 11.30; MDp: M = 38.67 SE = 13.78; Visual Matching Grade 2: M = 35.67, SE = 7.24; MDp: M = 29.17 SE = 25.49; Table 11) show a pattern of results suggesting that the Grade 2 and Persistent groups were not similar on these measures and this may reflect meaningful differences in the cognitive profiles for these two time points. Given that the MD Persistent group was assessed at Grade 3, these results suggest that these variables become less relevant to MD or that compensatory strategies develop with age. The results of the other two measures, however (WM for Words Grade 2: M = 2.00, SE = 0.44; MDp: M = 2.00; SE = 1.15; Digit Span Forward Grade 2: M = 7.44, SE = 0.60; MDp: M = 6.67, SE = 0.33; Table 11) displayed results indicating that performance on these measures was very similar regardless of the time period of the MD. Only one measure, the Counting Span Task, became significant for the Persistent group while it was not significant for Grade 2. This likely reflects a meaningful difference between the groups as the mean values are notably different when the SE is considered (Table 11).  48  Table 11 MD25 Sample Sizes and Means for the Grade 2 MD25, Grade 3 MD25, and Persistent Groups Based on Both the WJ III Calculation and Applied Problems Subtests (Standard Errors in Parentheses). TA Means and Standard Errors are for the TA Persistent Group. Grade 2 MD25 Grade 3 MD25 MDp TA Variable* N Mean N Mean N Mean N Mean Verbal Comprehension Composite 10 7 3 68.67 (14.19) 175 71.88 (1.32) 38.90 (7.77) 58.14 (11.53) Spatial Relations 10 7 3 173 64.80 (1.43) 49.80 (5.61) 42.14 (8.14) 30.33 (7.62) Concept formation 9 7 3 173 72.26 (1.75) 46.00 (10.83) 27.86 (6.23) 31.67 (4.26) Visual Matching 2 9 6 3 29.17 (25.49) 174 54.46 (2.12) 35.67 (7.24) 27.42 (11.72) Incomplete Words 9 56.56 (6.58) 7 46.71 (9.08) 3 51.33 (16.74) 175 54.78 (1.99) Analysis and Synthesis 9 7 3 68.33 (1.74) 26.11 (7.96) 31.57 (8.34) 31.00 (14.64) 174 Rapid Picture Naming 9 39.33 (6.89) 7 33.71 (7.32) 3 33.67 (11.39) 174 48.91 (1.83) Counting Span Task 9 3.78 (0.57) 7 3 175 6.02 (0.16) 3.71 (0.68) 2.67 (0.33) Working memory for Words 9 7 2.71 (0.57) 3 2.00 (1.15) 176 4.51 (0.28) 2.00 (0.44) Rapid Number Naming 9 13.56 (1.07) 7 12.57 (1.04) 3 12.00 (2.08) 175 11.58 (0.18) Digit Span Forward 9 7 7.86 (0.70) 3 6.67 (0.33) 176 8.34 (0.15) 7.44 (0.60) Digit Span Backwards 9 5.00 (0.41) 7 4.86 (0.34) 3 4.67 (0.67) 176 6.84 (0.52) Letter Number Sequence 9 7 3 176 16.16 (0.21) 10.56 (1.43) 9.43 (1.25) 11.00 (2.08) Quantitative Concepts 9 7 3 175 69.17 (1.53) 16.56 (5.89) 17.00 (4.86) 22.67 (9.39) Block Rotation 9 7 47.71 (14.58) 3 38.67 (13.78) 168 68.08 (2.35) 33.11 (11.30) Note: *Variables that are significantly different when compared to the TA group are highlighted bold.  49 3.3.2 Comparing grade 3 MD and MD Persistent. When comparing the Grade 3 and Persistent MD groups, the resulting profiles presented a more consistent picture. Here, 10 of the 15 measures represent identical profiles for both of these groups, with only 5 measures showing any differences in the cognitive profile of the Grade 3 and Persistent groups. The Applied Problems definition was the most consistent and presented an identical pattern for both the Grade 3 and Persistent groups (Table 10). This indicates that those who had persistent problems with problem solving displayed the same pattern of difficulties as those who were MD at the Grade 3 time point. For Calculation, there were only three differences between the profiles for these two groups. Block Rotation and Verbal Comprehension showed significant differences between the Grade 3 MD group and the TA group while presenting non-significant results for the Persistent cases (Table 9) Rapid Number Naming presented the opposite case. For Block Rotation and Verbal Comprehension, these results are likely driven by the large standard error of the Persistent group as the mean values for the groups were actually quite similar (Block Rotation Grade 3: M = 54.67, SE = 4.13; MDp: = 59.35, SE = 8.38; Verbal Comprehension Grade 3: M = 64.47, SE = 2.44; MDp: M = 63.24, SE = 4.12; Table 9). Rapid Number Naming presented a picture that may indicate meaningful differences between the groups, as the mean values do differ and the MDp group took longer to complete the task (Grade 3: M = 12.00, SE = 0.31; MDp: M = 13.00, SE = 0.78; Table 9). When both Calculation and Applied Problems were considered together, there were only two differences between the profiles for the Grade 3 and persistent MD groups. The variables affected were Verbal Comprehension and Visual Matching 2, both of which were found to be significant for the Grade 3 cases but not for the Persistent cases.  50 These results are curious, as the results suggest that the Persistent cases have fewer differences from the TA group than the Grade 3 cases. However, these results are highly suspect when considering that the Persistent group had a sample size of 3 while the Grade 3 group had a sample size of 7. For both of these measures, large standard errors in the smaller Persistent group likely led to the different significance values of the t-tests (Verbal Comprehension MDp: SE =14.19 vs. Grade 3: SE= 5.35; Visual Matching MDp: SE = 25.49 vs. Grade 3: SE= 6.66; Table 11). Further research would be required to establish whether there is a real difference in the Persistent and Grade 3 profiles for this definition or whether this is an anomaly. Based on the similarities of the overall profiles of the Grade 3 MD and MD Persistent groups, using the MD25 cut off, both groups displayed overall lower performance on 10 of the 15 tasks: Verbal Comprehension, Concept Formation, Spatial Relations, Visual Matching 2, Analysis and Synthesis, Counting Span Task, Digit Span Forward, Letter Number Sequence, Quantitative Concepts, and Block Rotation (tables 6 and 9). 3.3.3 Summary. The overall pattern of differences between the Grade 2, Grade 3, and the Persistent groups suggest that slight changes in the cognitive profile of MD occur over time and that the Persistent group displays an almost identical profile to the Grade 3, group. Although there were differences between the Grade 2 and the Persistent group, with the most notable being the WM for Words task, there were also a large number of measures that displayed a similar profile across the groups. Although it is difficult to draw strong conclusions as a result of the small sample sizes involved in the comparisons, it appears that there were more similarities than differences in the cognitive profiles of the MD groups, regardless of the time point considered and the persistence of the MD.  51 Chapter 4  Discussion The present study was designed to answer three questions. First, are there enough cognitive and WM differences between the MD groups identified using a 10% cut off and a 25% cut off, when compared to their TA peers, to warrant further investigation of the 10% cut off? Secondly, how does an MD(11-25) group compare with the MD10 group? Is this group different enough from their MD10 and TA peers to warrant differentiation between the MD10 and MD(11-25) groups? Third, do participants who experience Persistent MD have cognitive profiles that are different from those who have MD at only one of two time points? This discussion will address each of these questions in turn, as well as presenting the overall cognitive and WM profile of the MD25 group. 4.1 MD10 vs. MD25 In comparing MD25 to MD10, it was found that many apparent differences between these groups in their cognitive and WM profiles (relative to TA) were not actually meaningful and could be explained by examining the mean values and the standard errors for the individual comparisons. Typically the mean values were actually quite comparable between the MD10 and MD25 groups, with both groups’ results demonstrating divergent profiles when compared to the TA group. However, having such small sample sizes, the MD10 groups often had larger standard errors than the MD25 groups which limited the statistical significance of the t-tests in the MD10 groups.  In all  but four cases (Grade 2 Calculation WM for Words, where MD10 performed at a higher level than MD25; Grade 3 Calculation Verbal Comprehension and Concept Formation, where MD10 performed at a higher level than the MD25; and Grade 3 Calculation Rapid  52 Number Naming, where MD25 performed at a higher level than MD10), the results were reasonably consistent across the two MD cut-offs. Given that the present study elected to remove those participants who had significant reading difficulties from the analyses, these results indicate that students who score below 10 % and those that score below 25% on the Calculation or the Applied Problems tests showed a striking degree of similarity in terms of their overall cognitive and WM profiles. This suggests that these two groups are not substantively different from one another. In addition, it suggests that 25% should be a sufficient cut off point when identifying those participants who are experiencing MD in future research studies. As the current body of research has yet to agree on an overall cognitive profile of MD, further differentiation between MD25 and lower performing groups will not likely confer important research benefits. Rather, using the MD25 cut off point would benefit the literature by providing a stable cut off point with which to uncover results that will be interpretable and generalizable across studies. 4.2 MD10 vs. MD(11-25) As with the MD25 vs. MD10 comparisons, when the MD10 groups were compared with the MD(11-25) groups, there were only a few instances in which there were meaningful differences in their overall cognitive profiles. This suggests that there are not enough meaningful differences in the cognitive profiles to consider these to be separate groups for study. Although Verbal Comprehension, Rapid Number Naming, Concept Formation, and WM for Words were found to reflect potentially meaningful differences in the cognitive profiles of the MD10 and MD(11-25) groups, these differences are relatively minor when compared to the overall similarities. When the small sample sizes that were found using the MD10 definitions in this study (as in other  53 studies, i.e., Geary et al., 2007; Geary et al., 2011; Wu et al., 2008) are taken into account, it seems unlikely that investigating this group, particularly in relation to MD(11-25), will yield strong results in the future, despite small potential differences in the cognitive profiles. These results are inconsistent with work by Chong and Siegel (2008), Mabbott and Bisanz (2008), Wu et al. (2008), Geary et al. (2007), and Geary et al. (2009) suggesting that there may be differences between the MD profiles. There is, however, one very critical difference between the present study and those finding potential profile differences: reading difficulties. Many of the studies investigating MD have retained those participants who have reading difficulties within the MD sample (i.e., Chong & Siegel, 2008; Geary et al., 2007; Geary et al., 2009; Geary et al., 2011; Mabbott & Bisanz, 2008; Murphy, et al., 2007). While it is important to acknowledge that reading difficulties and MD do coincide (Landerl et al., 2004; Mabbott & Bisanz, 2008; Mazzocco, 2005), including co-morbid RD and MD makes it difficult to determine exactly how MD manifests and whether the underlying cognitive profile of the RD is affecting mathematical performance. Investigating RD and MD together is important for developing a comprehensive understanding of LD. However, it is also important to examine MD in isolation to clarify its specific cognitive profile. The decision to eliminate low readers from the MD sample may account for the discrepancy in the findings between the present study and other studies investigating the MD10 and MD(11-25) groups. However, it is important to acknowledge that this is not the first study to find marked similarity between the MD10 and MD(11-25) groups. Recent work by Geary et al. (2011) supports the argument that the MD(11-25) and MD10 groups are not sufficiently  54 different to warrant separating the groups by indicating similar growth trajectories in both the MD(11-25) and lower MD groups throughout the elementary years. Although it is critical to note that this study did not eliminate low readers from the sample, it was longitudinal and does lend further support to the notion that the MD10 definition may be unnecessary. In summary, the results of this study indicate that, while there may be a few small differences between the cognitive profiles of the MD10 and MD(11-25) groups, these differences are minor in comparison to the similarities between the groups and thus the results indicated that distinguishing between the MD10 and MD(11-25) groups when investigating MD is unnecessary. 4.3 Summary of MD10, MD25, and MD(11-25) MD25, MD10, and MD(11-25) all appear to represent a similar cognitive profile when compared to the TA group. Although there were a few inconsistencies across the MD25 vs. MD10 comparisons and the MD10 vs. MD(11-25) comparisons, several of the apparent group differences were actually eliminated when you consider the results across the two sets of analyses. As an illustrative example, while Rapid Picture Naming may represent a meaningful difference between the MD10 and MD(11-25) groups (as the two groups were significantly different from one another), when MD(11-25) and MD10 were compared to the TA groups, the MD(11-25) and MD10 groups actually produce an identical pattern of results, suggesting that the differences between the MD10 and MD(11-25) groups were not substantive. Using the 15 different cognitive and WM measures to uncover the pattern of deficits found when the different definitions of MD were used with different cut off values on several defining variables has led to the overall conclusion that the MD10,  55 MD(11-25), and MD25 groups present a similar cognitive profile and that using the MD25 definition is sufficient when defining MD for the purposes of future research. Future research should strongly consider the impact of reading when defining MD at the 25% cut off. Future research should also consider investigating those areas that showed anomalous patterns of results between the groups in this study to confirm whether or not they represent replicable differences between the MD(11-25) and MD10 profiles. 4.4 Persistent MD compared to MD at grade 2 or grade 3 With the notable exception of the WM for Words task, there were very few differences between the cognitive profiles of the Persistent MD groups and the Grade 2 and Grade 3 groups overall. Many of the differences between the groups were found when comparing the Grade 2 and the Persistent group. When considering the cognitive profiles of the Persistent and Grade 2 MD groups, there are several variables that show significant differences in the Grade 2 group relative to the TA group and non-significant results in the Persistent group relative to the TA group. This pattern was found for WM for Words, Rapid Picture Naming, Backwards Digit Span, Verbal Comprehension, and Digit Span Forward. These results could be indicative of developmental growth in the Persistent group, where participants develop either stronger abilities in these domains or compensatory strategies. This is consistent with the work of Morgan et al. (2009) who found that MD participants did develop throughout their study, albeit at a slower rate than their TA peers. In contrast, Rapid Number Naming (processing speed) and the Counting Span Task (mathematical WM) showed significant differences from TA when the Persistent group was considered, but not when the Grade 2 group was considered. These results are inconsistent with that of Rapid Picture Naming and Digit Span Backwards which also  56 measure processing speed and WM for numbers, presenting a more complicated picture. Although the work of Vukovic and Siegel (2010) indicated that working memory, processing speed, and numerical reasoning were deficient in the MD persistent groups, the results of the present study neither confirm nor deny these findings. Importantly, however, it is difficult to draw strong conclusions about differences in the cognitive profiles of the Persistent MD and Grade 2 groups given that the MD definition that showed the most frequent profile differences across the groups, the combined Calculation and Applied Problems definition, had a very small sample size. Further research will be required to determine the extent of the differences, if any are present, between these two groups. The Grade 3 group, regardless of the defining test, showed a remarkably similar profile to the Persistent group. This was not particularly surprising, as many of the students who fell under the Grade 3 MD definitions were also found in the Persistent group. As these groups represented very similar profiles when investigating MD Persistent, future research should consider using at least 3 or more time points (i.e., Geary et al., 2011; Morgan et al., 2009; Vukovic & Siegel, 2010) to draw comparisons as the MD Persistent group may display a similar cognitive profile to the final age group simply as a result of assessing these groups at the same point in time and this reduces the effectiveness of making a distinction between persistent MD and MD at a single time point. 4.5 Overall profile of MD25 Given that the findings of the present study suggest that there is little benefit in separating the MD10 and MD(11-25) groups, the overall cognitive and WM profile of MD will now be discussed using the MD25 group as the basis for all comparisons to the  57 TA group. Using the MD25 definition, it is possible to identify the cognitive and WM characteristics of the MD groups identified using the three different approaches to defining MD (i.e., Calculation, Applied Problems, or both combined) and to determine if there are any trends and differences to be found, based on the measures used to define MD. Several universal trends came across when reviewing the results of this study. 4.5.1 Consistencies. Reasoning skills were found to represent a significant deficit in all of the different MD groups, regardless of which tests were used to define MD. Performance on all of the measures assessing reasoning skills (i.e., Concept Formation, Quantitative Concepts, Visual Matching 2, and Analysis and Synthesis) was significantly lower in the MD groups relative to the TA groups, and these differences are indicative of a deficit in MD. This applied to both Grades 2 and 3. When considering that Taub et al. (2008) found that fluid reasoning was an important predictor of mathematical knowledge, it appears that reasoning deficits are important to the study of MD. This has serious implications for MD, both in terms of classroom instruction and intervention when educating students with MD. Students with MD may show less development in their ability to reason and problem-solve relative to their TA peers. As a result, they will likely require more direct and explicit instruction in problem-solving. In addition, teachers may need to consider intervention strategies to assist these students in compensating for these deficits. Group differences based on MD status never emerged for auditory processing, as measured by the Incomplete Words test, and this does not represent a deficit in MD when compared to TA peers. These results are similar to those of Taub et al. (2008) who found that this measure did not predict quantitative knowledge in typical students. This factor might not be relevant to future MD research or to the instruction of students with MD.  58 Visual Spatial abilities were measured with both the Spatial Relations and Block Rotations measures. The Spatial Relations test was found to represent a significant deficit for students with MD, regardless of the tests used to define MD. This was found in both Grade 2 and Grade 3 and likely represents a stable deficit that is consistent over time. These results are consistent with the work of Holmes, Adams, and Hamilton (2007) and Schuchardt et al. (2008). The Block Rotation measure was found to reflect a performance deficit when the Applied Problems test was used to define MD. The other definitions did not present conclusive cases. Although there was a deficit in the Grade 3 Calculation group, this was not consistent with the Grade 2 findings. These findings may indicate a growth in this deficit as the participant’s age. This could also indicate that, as the numerical problems become more complex, deficits that were previously masked emerge and become more important. These results are contradictory to Holmes et al. (2007) who found that visual spatial skills were less impacted as participants age. When combined with the Spatial Relations results, it would appear that there is a deficit in visual spatial abilities in MD that is primarily evident when MD is defined as those participants who struggle with problem solving. When considering the instruction of and interventions with students with MD, this will be a critical area of concern as well, as visual spatial abilities have been linked with poor overall math performance in the past (Swanson & Sachse-Lee, 2001; c.f., Wu et al., 2008) and employing strategies that both strengthen visual spatial skills and mediate this deficit will be critical to the growth of students with MD. The Letter Number Sequence measure was found to indicate a deficit in MD regardless of the definition used and this was stable across Grades 2 and Grades 3. This is indicative of a deficit in phonological WM in MD and is consistent with the summary  59 provided by Swanson and Jerman (2006). While this measure was used in conjunction with the WM for Words task, the results for this latter measure indicated that deficits occur only in the Grade 2 group, regardless of the test used to define MD. Thus, although verbal WM was found to be deficient in students with MD, it appears that this may, in part, reflect a developmental delay, as some of these deficits appear to remit by Grade 3. In general the results of this study seem to support the overall deficit in phonological WM that has been identified in the body of research into MD (Swanson & Jerman, 2006). The results of the two processing speed measures, Rapid Number Naming and Rapid Picture Naming, indicated that processing speed was not a significant factor in MD in all but one case (i.e., when MD was defined using Applied Problems alone, the grade 2 participants were deficient in this area). This represents an anomaly in this study, as processing speed deficits were not found to be characteristic of MD for any of the other MD definitions and time periods, with the results indicating that there are generally no processing speed deficits in MD when compared to TA peers. These results are similar to those of Anderson (2010) who also found that processing speed was not a significant predictor of MD. However, decreased processing speed has been linked to MD in various capacities by several researchers (Bull & Johnston, 1997; Chong & Siegel, 2008; Vukovic & Siegel, 2010) and was found to be lower overall in MD in the meta-analysis of Swanson and Jerman (2006). Processing speed has also been linked to general quantitative knowledge in more typical students (Taub et al., 2008). Therefore, the results of this study, while clearly suggesting that processing speed is not an important factor in MD, stand in contrast with a significant body of previous research. The Language Development variable presented a clear picture in that there were significant deficits in language development for those students with MD defined by  60 Applied Problems and when both Calculation and Applied Problems are combined for both grades, but only when Calculation are considered for the Grade 3 group. This suggests that language development is most relevant to MD when decoding and interpreting problems is required as an aspect of mathematics performance and may also suggest that this variable becomes more relevant with age, as it was deficient for all of the Grade 3 definitions. Focused interventions involving the instruction of mathematical vocabulary may be required throughout the education of students with MD, as this appears to be a consistent deficit. This will be especially important as new mathematics curriculums stress more problem based approaches to learning which require a strong vocabulary in order to decode the information presented and make decisions. 4.5.2 Inconsistencies. Several measures demonstrated varying results depending on the test used to define MD and the time period of testing. The results of the short term memory measure, Digit Span Forward, indicated that short term memory deficits are found for those students with MD defined using the Calculation test at both grades. Additional deficits were found for Grade 2, but only for those with MD defined using both tests combined. Those with MD defined using the Applied Problems test showed deficiencies in short term memory in Grades 3 only. The results indicate that deficits in short term memory are found when Calculation are used to define MD, but when Applied Problems are a factor in the definition of MD, the results are not conclusive. These results are supported by the work of Passolunghi and Siegel (2001) who found similar deficits in short term memory for numerical information but not for linguistic information. When WM involved numerical information, the results are unclear. Separating this aspect of WM from other aspects of WM is important, as specific numerical WM difficulties were alluded to in early WM research (Siegel & Ryan, 1989). This research,  61 which was further supported by the later work of Landerl et al. (2004) and Passolunghi and Siegel (2001), indicates that verbal tasks that involve numbers are more challenging for students with MD than those tasks that have no numerical elements (Siegel & Ryan, 1989). Based on this, greater WM deficits would be expected when numerical information is used as part of the WM tasks. However, in this study, only the Backward Digit Span task reflected performance deficits associated with MD when Applied Problems were used to define MD and only for Grade 2. This is very interesting, as the Counting Span task displayed more substantive deficits for the MD groups. The Counting Span task was found to be deficient for all but the Grade 2 combined and Calculation groups. This may indicate that performance deficits on this measure become more pronounced as the participants age, becoming more relevant as operational tasks increase in complexity, although this is unclear. However, when combined with the results of the Backward Digit Span task, it is unclear whether a general WM problem involving numbers is represented in students with MD.  62 Chapter 5  Implications 5.1 Limitations This study has an important limitation that should be taken into consideration when interpreting the results. As previously indicated, the overall small sample sizes that were used in several of the MD comparison groups. This may have affected the results, although the researcher has done his best to ensure that accurate descriptions of the results and the comparisons within them have been presented.  5.2 Research There are several important contributions that this study has made to the growing body of MD research. Critically, the results of this study have led to the overall conclusion that the MD10, MD(11-25), and MD25 groups present a similar cognitive profile and that using the MD25 definition is sufficient when defining MD for the purposes of future research. Importantly, this study did not include any participants who displayed significant reading problems. Therefore, these results are indicative of those who have MD in relative isolation from other learning issues. This is a critical component of this study, as a number of researchers (e.g., Chong & Siegel, 2008; Geary et al., 2007; Geary et al., 2009; Mabbott & Bisanz, 2008; Murphy et al., 2007) have not excluded those with reading difficulties from their studies and, while their results did indicate that there were differences in performance between the MD10 and MD(11-25) groups, interpretation of their results must take this into account. This has important implications for MD research as it is possible that reading difficulties may affect the performance of MD participants and it could be the co-morbid effects of both disabilities  63 that are represented in the previous findings suggesting differences between the MD10 and MD(11-25) groups. Importantly, Bull and Johnson (1997) found that their results became non-significant when reading was controlled for in their study. Further, Schuchardt et al. (2008) concluded that participants with RD demonstrated different profiles from the MD group, and the RD+MD group. Controlling for reading difficulties provides a clearer picture of the deficits specific to MD and this should be considered in future research. When an MD group, with reading difficulties taken into account, is investigated, the present results suggest that using the MD25 cut off for research purposes will provide an appropriate sample of participants to compare to a TA group. Future studies should consider combining the MD(11-25) and MD10 groups for more robust results, unless there are sufficient numbers of participants to create a large enough MD10 group where the statistical power is sufficient for strong comparisons. While having small sample sizes is not an uncommon limitation in the MD literature (e.g., Cencebella & Noel, 2008; Geary et al., 1991; Geary et al., 2007; Geary et al., 2011; Hitch & McAuley, 1991; Landerl et al., 2004; Schuchardt, et al., 2008) it is this researcher’s opinion that efforts to develop a complete picture of a possible MD10 group do not seem practical at this point in time unless a very large sample of MD10 participants is available for study. The present study had over 800 participants and was still limited in the ability to use the MD10 group to make comparisons. Furthermore, one reason for the introduction of the MD10 group has been to address the concern that the higher cut off values will not reflect the estimated prevalence of MD in the population, which lies at between 5% - 8% (Badian, 1999; Geary, 1993; Geary, 2004). In this study, the largest group of MD25 was the Grade 3 Calculation group (N = 68). This group was drawn from a sample of 843  64 grade 3 participants and represents 8% of this sample, reflecting a similar prevalence rate to the overall population estimates. These results also suggest that there are no differences between the Persistent and Grade 3 (terminus) groups when Applied Problems were considered and only three differences between the profiles when Calculation were considered, and these differences may not be meaningful. This makes sense given that the Persistent groups were composed primarily of those who were labelled MD25 in Grade 3. Additional caution should be levied in future research when using the Persistent MD label as this group may show a similar cognitive profile to the highest grade or age cut off used in the study, as was the case here. Future research in this area would benefit from investigating at least three different time points to more clearly identify potential differences between those who are MD persistent and those that display MD at only one time point. When investigating MD25 in the future, the defining tests used to identify MD should also be considered carefully. The Calculation and Applied Problems tests did yield very similar results, presenting slightly different patterns on relatively few occasions. The most divergent of the groups were those who scored low on both of these tests combined. Although it would seem logical that these participants would be those most highly affected in an instructional setting, this study found it difficult to identify a large enough sample with noteworthy deficits on both variables to determine if any differences were meaningful. 5.3 Practice The results of this study suggest that, when confronted with students who have a profile of MD25, MD10, or MD(11-25), a practitioner would be well advised to consider them to be similar in terms of the educational approach they require for success in school.  65 This, of course, is restricted by the caveat that the students with MD in the present study did not present any additional problems with reading. Reading difficulties may confound students’ ability to perform and learn mathematics and students with both MD and RD may present a slightly different cognitive profile. Having acknowledged this, it appears that there are several very distinct areas that may require remediation when considering interventions for students with MD. These areas include reasoning skills, language development, and visual spatial skills. When reasoning skills were investigated, a significant deficit was uncovered in the MD groups. As noted above, this has implications for interventions when educating those students with MD and it would seem that instruction in this area should be considered when designing strategies to assist students with MD in compensating for these deficits. Future research would benefit from investigating reasoning skills and the consequences possible delays present when considering instructional practice. Interventions involving instruction in mathematical vocabulary might also benefit the education of students with MD, as this also appears to be an area of deficit. As noted above, recent changes to mathematics curriculums emphasizing problem based approaches to learning require a strong vocabulary. In light of this, interventions in this area might be important in assisting the progress of students with MD. Finally, visual spatial skills appear to be deficient in students with MD. It might be advantageous to identify strategies to help students compensate for these deficits. It may also be beneficial if additional instructional time was spent ensuring that related abilities, such as the development of a mental number line and the representation of numerical information in a visual fashion, are in place. This may help to facilitate the mathematical development of these students.  66 Chapter 6  Conclusion This study was designed to investigate several different definitions of MD using the cut off criteria of MD10, MD25, and MD(11-25). Based on the results of this study, it is reasonable to conclude that, while there might be some slight differences between the different definitions, these are, by comparison, relatively minor when considering the degree of similarity between the profiles. Therefore, it can be concluded that using the MD25 definition is sufficient when investigating the cognitive and WM deficits found in MD participants. Given that the MD literature is currently plagued with small sample sizes, distinguishing between MD10, MD25, and MD(11-25) does not seem to make practical sense at this point. Only after the full cognitive profile of MD is established, and sufficiently sensitive diagnostic tools have been designed to address the needs of diagnosing MD, should more specific and restrictive definitions be considered. 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