mcglaughlin differentiating

DIFFERENTIATING STUDENTS WITHMATHEMATICS DIFFICULTY IN COLLEGE:

MATHEMATICS DISABILITIES VS. NO DIAGNOSIS

Sean M. McGlaughlin, Andrew J. Knoop, and Gregory A. HoUiday

Abstract. Difficulties with college algebra can be the gate-keeper for earning a degree. Students struggle with algebra formany reasons. The focus of study was to examine students strug-gling with entry-level algebra courses and differentiate betweenthose who were identified as having a mathematics disability andthose who were not. Variables related to working memory, mathfluency, nonverbal/visual reasoning, attention, and reading wereanalyzed using a MANOVA and separate ANOVAs. Significant dif-ferences were found on all but attention, supporting the findingsof research on students in elementary and secondary education.Implications include a focus on techniques that help to remediatethese specific deficits.

SEAN M. MCGLAUGHLIN, M.A., is a doctoral student. University of Missouri-Columbia.ANDREW J. KNOOP, Ph.D., is a clinical assistant professor, University of Missouri-Columbia.

GREGORY A. HOLLIDAY, Ph.D., is a clinical associate professor. University of Missouri-Columbia.

By understanding students struggling with mathemat-ics at the postsecondary level, professionals can offer het-ter assistance both during and before college and canhelp identify appropriate remediation techniques.Approximately 5-6% of students have significant diffi-culty in mathematics (Fleischner & Manheimer, 1997),and many struggling students are not identified asrequiring special services for math during secondaryschool. It is becoming increasingly evident that studentsneed help understanding mathematics, especially withthe world rapidly evolving scientifically and mathema-tically. Many college students encounter mathematicsdifficulties, which can eventually act as a gatekeeper toearning a college degree.

Little research has been published on college studentsexperiencing mathematics difficulties compared to ele-mentary and secondary students having difficultieswith mathematics (Strawser & Miller, 2001; Zawaisa &

Gerber, 1993). This is especially the case for instruc-tional or remediation interventions. Consequently,researchers often present results involving elementaryand secondary students as part of their literaturereviews, generalizing this information to students atthe college level. Because enrollment in postsecondarysettings by students with documented learning diffi-culties is increasing, researchers must begin to focus onthe needs of this population of students (Mercer, 1997).

STUDENTS WITH MATHEMATICSDISABILITY

Research on students with mathematics disabilitiesoften consists of comparisons with other disabilities.That is, students with reading disabilities (RD), disordersof written expression (WD), and mathematics disabili-ties (MD) and combined reading/mathematics disorders(RD/MD) are compared and contrasted between and

Volume 28, Summer 2005 223

among disability groups to establish shared and dis-parate attributes (Benton, 2001; Greene, 2001; Rourke,1993; Shafrir & Siegel, 1994). Commonly, these studiesfocus on elementary and secondary students, highlight-ing the need for research at the postsecondary level.

Normally achieving (NA) students often make up afourth group in such comparisons. From the compar-isons made, several sources of individual differencescan be extracted, including deficiencies in (a) readingability, (b) nonverbal/visual skills, (c) working memory,(d) poor math fluency, and (e) attention and/or hyper-activity.Reading Comprehension and MD

Fleischner and Manheimer (1997) identified two pri-mary reasons why students struggle with mathematics.First, some students exhibit difficulty in reading com-prehension. Comprehension difficulties seem to exacer-bate difficulties in mathematics, especially for solvingword or story problems. Examining NA students andstudents with MD, RD, and MD/RD, Jordan and Hanich(2000) demonstrated that when asked to solve fourtypes of problems (number facts, story problems, placevalue, and written calculation), the students withMD/RD performed significantly lower than NA stu-dents. Further, the students with MD only performedlower on complex story problems compared to the NAstudents. Students with RD performed similarly to theNA students. Overall, students with MD/RD and MDsignificantly underperformed the students with RD andNA students on mathematics-related tasks.Nonverbal Reasoning and MD

Fleischner and Manheimer (1997) also asserted thatsome students exhibit difficulties in nonverbal reason-ing and/or primary mathematics knowledge. Rourkeand his colleagues (Rourke, 1993; Rourke & Del Dotto,1994; Rourke & Fuerst, 1996) have made significantcontributions to the field of learning disabilities bystudying nonverbal learning disability and its relation-ship to reading and mathematics disorders from a neu-rological perspective.

Rourke (1993) summarized several of these studiesconducted with children with learning disabilities (LD)using a variety of assessment measures, including theWechsler Intelligence Scale for Children (WISC) andWoodcock-Johnson Tests of Achievement (WJ). Amongthe three main clinical groups, Rourke and his col-leagues identified differences between students with RDand MD, noting that students with RD typically haveverbal deficits whereas students with MD typically havenonverbal deficits, as measured by the discrepancybetween VIQ and PIQ measures on the WISC. "Qldergroup [RD] children exhibit normal levels of perform-ance on visual-spatial-organizational, psychomotor.

and tactile-perceptual tasks; group [MD] children haveoutstanding difficulties on such tasks" (p. 220).

Additional findings were presented by Rourke andFuerst (1996) and Rourke and Del Dotto (1994). Forspecifics on the range of studies that make up the pre-ceding three studies, the interested reader is referred toRourke and colleagues' publications on students with RDand MD. Rourke originally referred to MD as MD, butlater changed to NLD (nonverbal learning disabilities).For the purposes of this article, Rourke's NLD group isreferred to by his initial identification of MD.

According to Silver, Pennett, Black, Fair, and Balise(1999), visuo-spatial deficits may be the core deficit forstudents with isolated mathematics disorders, yet sub-type stability is poorer for the arithmetic-only group. Ina literature review, Jordan (1995) found three subtypesof mathematics disabilities, including one identified ashaving visuo-spatial deficits, similar to the nonverbaldeficit discussed by Rourke.

Geary (1993) examined visuo-spatial deficits as oneof three core difficulties (along with semantic memoryand procedural deficiencies) experienced by studentswith calculation difficulties. Cirino, Morris, and Morris(2002) further explored Geary's theory on a sampleof college students using a comprehensive batteryof assessment instruments, Wechsler Adult Intelli-gence Scale-Revised and Woodcock-Johnson Tests ofAchievement-Revised, along with a variety of otherneuropsychological instruments. They determined thatvisuo-spatial deficits did not contribute to the calcula-tion difficulties experienced by this sample. However,they did find evidence that Geary's other two cate-gories of deficiency - semantic memory deficits andprocedural deficits - contributed to the calculation dif-ficulties of college students.Working Memory, Math Fluency, and MD

Geary (1993) described semantic memory deficitsand procedural deficits, arguing that they could berelated to an underlying memory deficit. Semantic mem-ory deficits include difficulties in fact retrieval and inlearning mathematic facts to the point that theybecome automatic or retrieved directly from long-termmemory. Students with this deficit continue to men-tally calculate the answers rather than obtain themthrough direct retrieval. Procedural deficits are consid-ered more problematic and pervasive, causing faultyprocedures that are learned with errors. Students withthis deficit are more likely to use inappropriate devel-opmental procedures because they appear to strugglewith remembering the appropriate ways to completecertain problems. An oversimplified example of a faultyprocedure might be to learn that a plus sign means sub-tracting instead of adding.

Learning Disability Quarterly 224

Swanson (1993, 1994) investigated working memoryin children with and without LD. He defined workingmemory as "the system of limited capacity for the tem-porary maintenance and manipulation of information"(Swanson, 1994, p. 190). Students were operationallydefined as having an LD if they had scores that wereone half standard deviation below the mean or havingscores on the Wide Range Achievement Test-Revised(WRAT-R) that were below the 25th percentile for theirage. Normally achieving students were identified asthose having scores equal to or above the 50th per-centile for their age. Students were excluded in bothcases if their scores were deemed due to "retardation,poor teaching, or cultural deprivation" (Swanson, 1994,p. 192). Using a variety of memory measures, Swansonfound that students with LD had particular difficultiesin several areas of working memory compared to NAstudents. Further, he (1994) demonstrated that differ-ences in verbal working memory were not as discrepantfrom NA students as were students considered to beslow learners, but were discrepant nonetheless. McLeanand Hitch (1999) found that students with arithmeticdifficulties were impaired on spatial working memoryand executive processing.

Geary, Hoard, and Hamson (1999) investigated thedigit-span scores of first graders and found that "averageIQ children with low arithmetic achievement scoreshave difficulties retaining information in workingmemory while engaging in a counting task" (p. 235).Further, they demonstrated that students with MD/RDand students with RD committed higher numbers ofretrieval errors than the students with RD alone.However, the authors acknowledged that this may notnecessarily be due to a working memory deficit, buta deficit in attentional capacity.

Retrieval and reaction time can be linked to workingmemory as well. For example, Hayes, Hynd, andWisenbaker (1986) found that students with LD haddifficulty building facts into math fluency. Thus, com-pared to normally achieving peers, groups of studentswith LD had more difficulty in recalling basic facts andthus made more errors. In addition, these students weremore variable when making those errors.

Geary, Brown, and Samaranayake (1991) found simi-lar results in groups of flrst and second graders. Whencomparing NA students and students with MD, the NAstudents were not as likely to rely on counting proce-dures, a less than optimum strategy when buildingmath fluency, and were more likely to rely on memoryfor recalling basic facts. Students with MD still reliedon the counting procedures and not on memory.According to Geary et al. (1991), this poor workingmemory not only affects retrieval, it can lead to errorsin procedure such as counting errors, which can lead

to memorization of wrong facts (e.g., 7 + 6 = 12). Allof these compounded can cause further difficulty forstudents struggling with math.

In studies designed to look at the effects of time onstudents with and without MD, Alster (1997) andJordan and Mantani (1997) demonstrated that studentswith MD in grades 3 through 12 did better on taskswhen their retrieval ability was not tested by time. Thus,both studies demonstrated that when doing algebraicand word problems, students with MD performedpoorer than students without MD when time wasinvolved, but were similar to students without MDwhen there was no time limit. Both studies inferred thatshort-term memory and semantic memory difficultiesmay be the reason that timed tests have an effect.Jordan and Mantani (1997) found that this is especiallytrue for people with speciflc mathematics difficultiesrather than general academic difficulties.AD/HD and MD

When examining LD subtypes, some researchersinclude students with attention-deficit disorders in theiranalyses and discussion (Hurley, 1996; Marshall,Schafer, O'Donnell, Elliott, & Handwerk, 1999; Rourke,1993). While 3% of children in the United States haveLD, approximately 35% of children with AD/HD havelearning difficulties (Woodrich, 2000). Mayes, Calhoun,and Crowell (2000) studied a sample of 8- to 16-year-olds with and without AD/HD. The group was set up tobe as similar as possible to those referred to clinical set-tings. These researchers found that the proportion ofLD among AD/HD students was significantly higherthan for the sample of students without AD/HD.Mathematics had the second highest comorbidity, withapproximately 33% of the students with AD/HD alsohaving MD diagnoses.

Research is sometimes conducted with students withAD/HD to compare their profiles to those of studentswith LD; at other times, studies address the comorbidityof these two disorders. Riccio, Gonzalez, and Hynd(1994) suggested that the comorbidity of AD/HD andLD is so high that the two may be indistinguishable.Attention-related concerns affect achievement andmust be explored when analyzing students struggling inmath.Postsecondary Intervention

Thus far, this literature review has focused on com-mon difficulties among students with MD in order toaid practitioners in identifying students with MD aswell as begin to focus on remediation. Again, empiricalevidence for effective interventions for postsecondarystudents struggling with mathematics is, at best, scarce,with much of the intervention literature focusing onelementary and secondary education. Thus, Strawser


and Miller (2001) noted that "the literature is silent" onthe utility of generalizing to other populations thoseintervention techniques that have been empiricallydemonstrated to be successful with elementary and sec-ondary students.Current Study

The following analysis explored the characteristics ofcollege students with mathematics difficulties who weresubsequently identified as having mathematics disabili-ties. Because lower-than-average abilities on measures ofmathematics achievement were expected among thispopulation, mathematics achievement scores were notexplored in depth.

The importance of this study lies in its ability to addsupporting or refuting evidence of the presence of pre-viously identified sources of individual differences instudents with MD by analyzing similar assessmentmeasures and examining the results to determine if thedifficulties that exist in younger populations also existat the postsecondary level. A preliminary discussion onremediation techniques based upon the results is alsoincluded.

METHODParticipants

Participants consisted of undergraduate studentsattending a public research university in the midwestwith a student body representative of the general mid-western population. All participants reported experienc-ing significant difficulties when attempting to completethe mathematics course requirements for their degree.Participants were full-time students who, at the time ofthe evaluation, needed to earn course credit in at leastone college-level mathematics course to fulfill theirdegree requirements.

Potential participants were referred by their academicadvisors or instructors, or were self-referred through"word of mouth." To qualify as a research participant,each student underwent and passed an initial screeninginterview conducted by the researchers to document (a)the student's need for additional course credit in a col-lege-level mathematics course, and (b) significant evi-dence of current and previous mathematics difficulty,such as low grades in high school mathematics courses,low ACT mathematics scores, and/or significant self-reported difficulties in understanding mathematics-related concepts compared to concepts in other subjectareas. Students who successfully completed the screen-ing process were asked to volunteer as participants;those who agreed were administered the psychoeduca-tional assessment battery.

In all, 205 students completed at least some of thepsychoeducational battery. A total of 129 participants

were removed because their diagnostic categories didnot match the identified groups used in this study - MDand no diagnosis (ND) - or because they did not com-plete significant portions of the evaluation. This left 76participants for the analysis.

Overall, 34 participants were identified as MD and 42as ND. The database included MD participants rangingfrom 18 to 23 years of age, with a mean age of 20.6years, and ND participants ranging from 18 to 43 yearsof age, with a mean age of 22.9 years. The meanattained ACT composite score for MD was 21.6 and 21.7for ND, with ranges of 14 (scores from 15 to 29) and12 (scores from 16 to 28), respectively. The databaseconsisted of 53 females (27 ND; 26 MD) and 23 males(15 ND; 8 MD), who identified themselves as Caucasian,68 (36 ND; 32 MD), African-American, 7 (5 ND; 2 MD),and Hispanic, 1 (1 ND; 0 MD). Table 1 shows participantcharacteristics.Instrumentation

To investigate and describe the characteristics of col-lege students with math difficulties, the researchersadministered a complete psychoeducational evaluationbattery to each of the participants: structured interview,the Wechsler Adult Intelligence Scale-Third Edition(WAIS-III) (Wechsler, 1997a), the Wechsler MemoryScale-Third Edition (WMS-III) (Wechsler, 1997b), theWoodcock-Johnson-III Tests of Achievement (WJ-III)(Woodcock, McGrew & Mather, 2001), and the ConnersAdult ADHD Rating Scale (CAARS) (Conners, Erhardt, &Sparrow, 1998).

WAIS-III. The WAIS-III is an appropriate tool formeasuring general levels of cognitive functioning forindividuals aged 16 to 89. A complete administration ofthe WAIS-III yields three composite IQ scores (Full Scale,Verbal, and Performance) and four index scores (VerbalComprehension, Perceptual Organization, WorkingMemory, and Processing Speed). The WAIS-III consistsof 14 subtests, 7 each in the Verbal and Performancescales (Wechsler, 1997a).

WMS III. The WMS-III is an individually adminis-tered test of verbal and nonverbal memory for adoles-cents and adults aged 16 to 89. It is useful in educationalsettings to (a) assess the degree to which an individual'smemory functioning impacts academic performance,(b) assist in developing individualized education pro-grams, or (c) recommend external strategies or memorytraining to assist those with learning disorders. Thetest consists of 11 subtests, 6 primary subtests, and 5optional subtests (Wechsler, 1997b).

WJ-III. The WJ-III is an individually administered,norm-referenced test of academic achievement that canbe administered to individuals from age 2 to over 90.When combined with other pertinent information, the


Table 1Sample CharacteristicsMathematics

Numbers:MaleFemaleTotal

Age:MeanRange

Race/Ethnicity:CaucasianAfrican-AmericanHispanicTotal

Overall Academic Achievement''MeanSDRange

"ACT scores.

Disabilities (MD) vs. No

Mathematics Disability

82634

20.618-23

322034

21.63.3

15-29

Diagnosis (ND)

No Diagnosis

152742

22.918-43

3651

42

21.73.1

16-28

results of the WJ-III aid in developing educational pro-gramming and appropriate services for reading, writing,oral language, or math skills. The test consists of 12 sub-tests within the Standard Battery, yielding compositestandard scores in Broad Reading, Broad Mathematics,Broad Written Language, and Oral Language Cluster(Mather & Woodcock, 2001).

CAARS. The CAARS is a self-report rating scale thatprovides a quantitative measure of ADHD sjonptoms foradults over the age of 18. The CAARS consists of 66 itemspresented in a 4-point Likert-style format that contributeto nine empirically derived scales. The CAARS includesfour factor-derived subscales: Inattention/ Memory Prob-lems, Hyperactivity/Restlessness, Impulsivity/EmotionalLability, and Problems with Self-Concept. Additionally,three ADHD symptom subscales are generated, includingInattentive Symptoms, Total ADHD Symptoms, and Hy-peractive-Impulsive Symptoms. The CAARS may be uti-lized to help identify, assess, and treat adults with ADHD-related symptoms and behaviors (Conners et al., 1998).

From this battery of tests, six individual variables wereselected to analyze differences between students withMD and ND. These variables, representing the sourcesof individual differences identified by previous research,were as follows: (a) the WMS-III Working MemoryIndex score (WM), (b) the WJ-III Math Fluency (MF)subtest score, (c) the WJ-III Passage Comprehension(PC) subtest score, (d) the WAIS-III PerformanceIntelligence Quotient (PIQ), (e) the WAIS-III VerbalIntelligence score (VIQ) minus the PIQ score, and (f) theCAARS DSM-IV Total Symptoms Index score (DSM-IVTSI).Procedure

To test the significance of observed differences in themean scores of the two groups on each of the six vari-ables that represent sources of individual differences, amultiple analysis of variance (MANOVA) was used.MANQVA is an extension of the ANQVA method tocover cases where there is more than one dependentvariable.


Conducting the MANOVA reduced the chances of aType I statistical error. Assumptions of MANOVA mustbe met before analysis was completed; namely (a) inde-pendence of observations, (b) multivariate normality,and (c) a check of covariance among the variables. First,independence of observations was assumed for the data.Due to the difficulty of checking for multivariate nor-mality in SPSS (Stevens, 1999), the variables were testedfor univariate normality. The assumption of univariatenormality was met for each of the variables; thus, thesecond assumption was met. Because Box's test isaffected by non-normality, it can be a good indicator ofviolations of multivariate data. Also, it helps to deter-mine any covariance among the three variables, and isthe third assumption of MANOVA. Box's test deter-mined covariance among the variables and was not sig-nificant. Thus, the third assumption was also met(Stevens, 1999).

With Type I statistical error ruled out, a one-wayanalysis of variance (ANOVA) was used to test the sig-nificance of observed differences in the mean scoresbetween the two groups on each of the six variables thatrepresent sources of individual differences. ANOVA is astatistical analysis that utilizes one categorical inde-pendent variable (diagnostic group) and one continu-ous variable (source of individual difference).

RESULTSThe results of the MANOVA indicated significant dif-

ferences among the variables. Thus, all four of the sta-tistics produced by SPSS were significant. The Pillai'sTrace, Wilk's Lambda, Hotelling's Trace, and Roy'sLargest Roots all had the same statistical value, signifi-cance level, p value, effect size, and power. The statisticwas F(6, 69) = 8.499, p < .000, with an effect size of .425,indicated by a partial eta squared and a power of 1.000.

Separate ANOVAs were run to determine which of thevariables were significantly different between the twogroups. Results showed that the PIQ, WM, MF, and PCwere all significantly different when comparing stu-dents with MD and ND. Results of the ANOVA for PIQindicated an f(l, 74) = 6.960, p < .01, with an effectsize, indicated by a partial eta squared, of .086 and apower indicated at .740. Results of the ANOVA for WMindicated an f (1, 74) = 16.050, p < .000, with an effectsize, indicated by a partial eta squared, of .178 and apower of .977. Results of the ANOVA for PC indicatedan f(l, 74) = 3.991, p < .049, with an effect size, indi-cated by a partial eta squared, of .051 and a power of.505. Results of the ANOVA for MF indicated an F{1, 74)= 33.939, p< .000, with an effect size, indicated by apartial eta squared, of .314 and a power of 1.000. The

Table 2Descriptive StatisticsMathematics Disabilities (MD) vs.

Instrument

PIQ"*VIQPIQDFWJPASCOM"

WJMTHFLU"**

WMSWM"**

DSMIVADHD"

'n=42.''n=34.^ Normative sample mean = 100; SD = 15.""Normative sample mean = 50; SD = 10.*p

results for the other two variables (VIQ/PIQ differenceand ADHD Total Symptoms) were not significant. Table2 shows the mean and standard deviations for studentswith MD and ND on each of the variables analyzed.

DISCUSSIONThe results of this study suggest that students with

mathematics disabilities at the college level tend to mir-ror research findings for students identified with math-ematics disabilities at the elementary and secondarylevels.

Among the similarities, college students identifiedwith mathematics disabilities demonstrate significantweaknesses in reading comprehension, nonverbal rea-soning, working memory, and math fluency. Interest-ingly, attention difficulties, which have been identifiedas a source of individual differences in elementary andsecondary students, were found not to be a significantdistinguishing factor between students with MD andND in this study.Reading Comprehension

The results support previous findings that studentswith MD have difficulty with reading comprehension(Fleischner & Manheimer, 1997; Jordan & Hanich,2000). For example, on the WJ-III Passage Comprehen-sion score, students with MD scored lower than stu-dents with ND, indicating that they had more difficultyunderstanding contextual clues relating to readingcomprehension.Nonverbal Reasoning

Rourke (1993), Rourke and Fuerst (1996), and Rourkeand Del Dotto (1994) discussed nonverbal/visual-spatialdiscrepancies among students with MD. Our resultsindicate that when compared to students with ND, thestudents with MD had significantly lower WAIS-III PIQsubtest scores; however, the VIQPIQDF scores were notsignificantly different. This provides some supportingevidence to findings of visual-spatial deficits and non-verbal difficulties for students with MD at the collegelevel.Working Memory and Math Fluency

Previous researchers have concluded that memory,especially working memory, is a deficit area for stu-dents with MD (Geary, 1993; Geary et al., 1991; McLean& Hitch, 1999; Swanson, 1993, 1994). In the currentstudy the WMS-III measure demonstrated that collegestudents diagnosed with mathematics disorders haddifficulty with working memory above and beyondthat of students who struggle with mathematics whoare not diagnosed.

Math fluency scores based upon the WJ-III MathFluency subtest score were, on average, one standarddeviation lower for students with MD than for students

with ND. In fact, on average, their score nearly fell inthe low instead of low-average range. This is consistentwith past findings showing math fluency to be prob-lematic (Geary 1993; Geary et al., 1991; Hayes et al.,1986; Jordan & Mantani, 1997).

While math fluency may be closely related to workingmemory capacity, it may also be affected by other vari-ables. Students with higher levels of math fluency (orautomaticity with math facts) are more likely to utilizetheir finite working memory capacity for other calcula-tion processes, or "chunking" more relevant content atone time. However, the level of math fluency can alsobe negatively impacted by inattention, depression, andhigh or low degrees of anxiety, as well as other factorsthat are not directly related to working memory.AD/HD

The results of the AD/HD measure did not distinguishthe two groups. Thus, our results with a college studentpopulation do not support Riccio and colleagues' (1994)assertion that the comorbidity of AD/HD and LD is sohigh that the two may be indistinguishable. However,this does not preclude the likelihood that attention-related symptoms are important factors that need to beaddressed. We strongly advocate that further researchbe conducted to examine the effect of AD/HD symp-toms on students with mathematics disabilities and stu-dents in general at the college level.Intervention

We now turn to remediation and/or accommodation.While not meant to be comprehensive, this discussionis intended to spark further research into effective reme-diation and intervention strategies suitable for collegestudents with mathematics difficulties.

Based upon our results, it appears there are factorsother than IQ-achievement discrepancies that affectclassroom performance. When faced with trying tomeet the needs of students with LD in mathematics,universities have traditionally emphasized academicsupports, such as classroom accommodations (e.g.,extended time to complete examinations) and in-creased access to tutoring and individualized assis-tance. While these supports are aimed at increasing theperformance of students with documented disabilities,they do little to address the wider variety of differingabilities of students who do not have documented dis-abilities but experience significant difficulty when per-forming math tasks.

One emerging trend in providing support for alllearners, regardless of disability status, is to implementclassroom presentation methods that engender con-cepts aligned with the universal design for learning(UDL), as advocated by Rose, Meyer, Strangman, andRappolt (2002). The three principles of the UDL frame-


work promote classroom instruction that provides for(a) multiple, flexible methods of content presentation;(b) multiple, flexible methods of content expression;and (c) multiple, flexible options for individual engage-ment in the material being presented. Within the UDLperspective, existing curriculum materials can be pre-sented with the necessary flexibility to support thediverse learning needs of a variety of students. UDLinstructional concepts intentionally provide multi-modal learning support for all students in a classroom,allowing each student to assimilate new content in waysthat are most efficacious for the individual learner, asopposed to a traditional, single instructional method.

Students with MD not only need remediation withspecific mathematics techniques, they also need reme-diation techniques specifically designed to addressdeficits in reading comprehension, nonverbal/visualskills, working memory, and math fluency. Levine(1999) identified academic strategies that can be used toprovide learning support for these deficits. With regardto math fluency deficits, Levine suggested that studentsutilize flashcards to develop automaticity for basicarithmetic and more advanced algebraic concepts, cal-culations, and manipulations. Mathematics-relatedcomputer software will help hold a student's attentionwhile rehearsing common calculations. Finally, stu-dents may choose to rehearse, or automatize, commoncalculation methods and multiple-step processes withinan overall solution.

For working memory deficits, Levine (1999) advo-cated that students concentrate on performing one taskor subtask at a time, develop definite and consistentstepwise approaches to problem solving as opposed totackling the entire problem at once, use self-monitoringtechniques (talking their way through a problem) to aidin problem solution, and practice extended arithmeticproblems in their head (such as 23 x 22, or 4 x 7 x 3) todevelop working memory capacity. Students may alsofind it beneficial to summarize complex instructions orsolution processes before attempting the solution.

For nonverbal reasoning deficits,' Levine (1999) rec-ommended the use of graph paper to aid in the spatialplacement of numbers and as well as the use of manip-ulatives or concrete applications of the math concept asan example of the problem to be solved. Instructionaltechniques should incorporate a highly verbal approachof explaining mathematical or geometric conceptsrather than relying solely on algebraic notations andgraphical plots.

Finally, for reading comprehension deficits, Levine(1999) suggested that students incorporate visual mod-els to accompany written explanations. Students shouldalso develop an individualized mathematics glossary ofterms and concepts used in class so they can be

reminded of these definitions as they complete home-work assignments. Math-related computer software canincrease mathematical conceptual understanding, andthe development of good estimating skills can serve asa self-check mechanism. Further, Levine suggested thatinstruction, including ways to translate specific wordsinto numerical symbols or processes, would support astudent's mastery of word problems.Implications for Practice

For practitioners working with students with MD, twoimportant pieces of information are gained from thisresearch. First, once students have been identified ashaving MD, it is important to focus on each individualto determine his or her specific strengths and weak-nesses. Although we present common characteristics forthe sake of simplicity, it is important to recognize thatstudents with MD are a heterogeneous group of indi-viduals with different weaknesses that contribute totheir difficulties in math (Fleischner & Manheimer,1997; Parmar & Cawley, 1997; Strawser & Miller, 2001).Information gained from this study can aid in creatinginitial hypotheses to gain insight into the strengths andweaknesses of individual students. Specific remediationtechniques must subsequently be explored.

Second, it is important to recognize the need for tech-niques that not only focus on intervention of specificmathematics weaknesses, but also on intervention inareas where students have other identified weaknesses.For example, students struggling with mathematics whoare identified as having working memory deficitsshould receive instruction in mathematics gearedtoward the difficulty in math together with the workingmemory deficit to attempt to maximize their ability togain information. Such interventions must be studiedand examined empirically to identify their utility.Further research needs to address the effects of specificstrategies and techniques already identified as usefulinterventions at elementary and secondary levels todetermine their usefulness at the postsecondary level.Limitations

It is important to note that the students who made upthe two groups in this study all presented with mathe-matics difficulties. Thus, a major limitation in under-standing students with MD is that there is no purelyrandom comparison group, and the design is not trulyexperimental. Future research is needed to comparethese findings with a normative group, which wouldallow for the data to be treated as a true experimentaldesign.

Another limitation is the presence of comorbidity inthe sample. For the purposes of this study, students withmathematics difficulty were included and the differencebetween the diagnoses of MD and ND was the focus.


However, in 14 cases comorbidity was an issue. Futureresearch is needed to focus on students with MD alonecompared with students with MD and other disorders.

CONCLUSIONSWith the growing demand for mathematics compe-

tency, it is becoming increasingly important to under-stand what differentiates postsecondary students withmathematics disorders/difficulties from their peers. Pastresearchers have focused on identification and remedia-tion of students with MD in elementary and secondarygrades. This article examined information from thesestudies to determine if the same weaknesses exist in asample of college students identified as MD. This article isthe beginning of what we hope will become an extensivebody of research on individuals with LD in mathematicsat the college level. Future research must not only focuson what weaknesses these students have, but also ondetermining which remediation techniques are useful.

Overall, these analyses demonstrated that workingmemory, math fluency, reading comprehension, andvisual/spatial/nonverbal ability weaknesses are signifi-cant contributors to difficulties for college students withmathematics disabilities compared to students who arestruggling in math but are not diagnosed. Academicsupports for these deficits must not only address specificmathematics difficulty areas but also the individualunderlying difficulties that exacerbate poor math per-formance. For example, if a student has a relative and/ornormative weakness in working memory and is strug-gling with mathematics, it is not enough to focus onhelping the student learn the importance of a certainmathematical skill or concept. Instructors must alsosupport the working memory deficit that is a contribut-ing factor to the mathematics difficulty. By supportingboth the math difficulty and the underlying processingdeficits that exacerbate poor math performance,instructors and tutors will significantly increase stu-dents' ability to assimilate and utilize the mathematicsconcepts and skills needed to meet college-level mathe-matics requirements.

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AUTHOR NOTEThis material is based on work supported by the National ScienceFoundation under Grant 0099216. We thank the individuals whoparticipated in the study and all those who graciously aided theprogress of this manuscript in its many stages.

Requests for reprints should be addressed to: Sean McGlaughlin,205 Lewis Hall, University of Missouri, Columbia, MO 65211.Email: [email protected]


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