Early Identification and Interventions for Students With Mathematics Difficulties

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<ul><li><p>JOURNAL OF LEARNING DISABILITIESVOLUME 38, NUMBER 4, JULY/AUGUST 2005, PAGES 293304</p><p>Early Identification andInterventions for Students WithMathematics Difficulties</p><p>Russell Gersten, Nancy C. Jordan, and Jonathan R. Flojo</p><p>Abstract</p><p>This article highlights key findings from the small body of research on mathematics difficulties (MD) relevant to early identification andearly intervention. The research demonstrates that (a) for many children, mathematics difficulties are not stable over time; (b) the pres-ence of reading difficulties seems related to slower progress in many aspects of mathematics; (c) almost all students with MD demon-strate problems with accurate and automatic retrieval of basic arithmetic combinations, such as 6 + 3. The following measures appear tobe valid and reliable indicators of potential MD in kindergartners: (a) magnitude comparison (i.e., knowing which digit in a pair is larger),(b) sophistication of counting strategies, (c) fluent identification of numbers, and (d) working memory (as evidenced by reverse digitspan). These are discussed in terms of the components of number sense. Implications for early intervention strategies are explored.</p><p>Currently, the field of learningdisabilities (LD) possesses anumber of valid measures thatcan predict relatively well which stu-dents are likely to have trouble withlearning to read. These measures areincreasingly used for screening pur-poses, and they enable schools to pro-vide additional support and relevantearly intervention for children in kin-dergarten and first grade. The evolu-tion of reliable, sound screening mea-sures was the result of more than 20years of interplay between researchand theory (Liberman, Shankweiler, &amp;Liberman, 1989). When the first authorentered the field of research 25 yearsago, the best predictive validity forreading readiness measures was .27;they now are routinely in the accept-able range of .6 to .7 (Schatschneider,Carlson, Francis, Foorman, &amp; Fletcher,2002).</p><p>Measurement of reading readi-ness evolved from a dynamic inter-action among theoretical studies (Schat-schneider et al., 2002) and interventionstudies (OConnor, Notari-Syverson, &amp;</p><p>Vadasy, 1996; Vellutino et al., 1996).Theories of early reading developmentwere tested using a variety of mea-sures, and, as the field evolved, find-ings from validation studies indicatedwhich measures were the most power-ful predictors, and which measuresmade the most sense to administer tochildren in kindergarten and firstgrade.</p><p>On the other hand, research onvalid early screening measures of sub-sequent mathematics proficiency is inits infancy. The small set of studies con-ducted to date involves various disci-plines, such as cognitive psychology,child development, and curriculum-based assessment (also known as gen-eral outcome measurement).</p><p>Our goal in this article is to high-light key findings from the diverse ap-proaches taken by the researchers inthe area of mathematics learning. Inparticular, we present what we cur-rently know concerning (a) the natureof mathematics difficulties; (b) the roleof number sense in young children; (c) valid screening measures for early</p><p>detection of potential difficulties inmathematics; and (d) early interven-tion and instruction.</p><p>Although we allude to some ofthe theory that guided the empirical re-search, we do not intend this to be atheory-building article. In some re-spects, the field of mathematics in-struction has been plagued with toomuch theory and theorizing and fartoo little programmatic, empirical re-search (Gersten et al., in press; Na-tional Research Council, 2001; Wood-ward &amp; Montague, 2002). At this point,there is enough empirical research tosuggest valid screening instrumentsfor determining who is likely to needextensive support in learning mathe-matics. Moreover, there is enough con-vergence in findings to begin to under-stand the trajectories of students withmathematics difficulties (MD) and tounderstand areas where these studentsneed intensive support. Both thesebodies of knowledge can inform thenature of instruction provided to youngchildren who are struggling with learn-ing mathematical concepts. Our hope</p><p> at Linkoping University Library on December 18, 2014ldx.sagepub.comDownloaded from </p><p>http://ldx.sagepub.com/</p></li><li><p>JOURNAL OF LEARNING DISABILITIES294</p><p>is to stimulate research that examinesthe effectiveness of various approachesto interventions for young studentslikely to have trouble in mathematics.</p><p>The Nature of MathematicalDifficulties</p><p>In the present article, we use the termmathematics difficulties rather thanmathematics disabilities. Children whoexhibit mathematics difficulties in-clude those performing in the low av-erage range (e.g., at or below the 35thpercentile) as well as those performingwell below average (Fuchs, Fuchs, &amp;Prentice, 2004; Hanich, Jordan, Kaplan,&amp; Dick, 2001). Using higher percentilecutoffs increases the likelihood thatyoung children who go on to have se-rious math problems will be picked upin the screening (Geary, Hamson, &amp;Hoard, 2000; Hanich et al., 2001). More-over, because mathematics achieve-ment tests are based on many differenttypes of items, specific deficits mightbe masked. That is, children might per-form at an average level in some areasof mathematics but have deficits inothers.</p><p>In the past, approaches to study-ing children with mathematics diffi-culties (MD) often assessed their per-formance at a single point in time.Contemporary approaches determinea childs growth trajectory throughlongitudinal research, which is funda-mental to understanding learning dif-ficulties and essential for setting thestage for critical intervention targets(Francis, Shaywitz, Steubing, Shay-witz, &amp; Fletcher, 1994). Measurementof growth through longitudinal inves-tigations has been a major focus in thestudy of reading difficulties since theseminal research of Juel (1988). Al-though less longitudinal research hasbeen devoted to mathematics difficul-ties than to reading difficulties, severalstudies have shown the advantages oflongitudinal approaches.</p><p>The seminal researcher in MDwas David Geary, although a gooddeal of subsequent research has been</p><p>conducted by Nancy Jordan and hercolleagues and other researchers suchas Snorre Ostad (1998). These ambi-tious lines of longitudinal researchstudies have attempted to reveal thenature and types of mathematics diffi-culties that students experience in theelementary grades and to examine theextent to which these difficulties per-sist or change over time. For example,Geary et al. (2000) found that for manychildren, mathematics difficulties arenot stable over time, identifying a groupof variable children who showedmathematics difficulties on a standard-ized test in first grade but not in secondgrade. It is likely that some of thesechildren outgrew their developmentaldelays, whereas others were misidenti-fied to begin with.</p><p>Typically, the researchers in theMD area have examined longitudinaltrajectories of students over periods of2 to 5 years on different measures ofmathematical proficiency. These stud-ies explored the relationship betweenMD and reading difficulties and spec-ify the nature of the deficits that un-derlie the various types of MD (Gearyet al., 2000; Jordan, Hanich, &amp; Kaplan,2003).</p><p>Fluency and Mastery of Arithmetic Combinations</p><p>The earliest theoretical research on MDfocused on correlates of students identi-fied as having a learning disability in-volving mathematics. A consistent find-ing (Goldman, Pellegrino, &amp; Mertz,1988; Hasselbring, Goin, &amp; Bransford,1988) was that students who struggledwith mathematics in the elementarygrades were unable to automaticallyretrieve what were then called arith-metic facts, such as 4 + 3 = 7 or 9 8 =72. Increasingly, the term arithmetic (ornumber) combinations (Brownell &amp;Carper, 1943) is used, because basicproblems involving addition and sub-traction can be solved in a variety ofways and are not always retrieved asfacts (National Research Council,2001). A major tenet in this body of re-search was that the general concept of</p><p>automaticity . . . is that, with extendedpractice, specific skills can reach a levelof proficiency where skill execution israpid and accurate with little or noconscious monitoring . . . attentionalresources can be allocated to othertasks or processes, including higher-level executive or control function(Goldman &amp; Pellegrino, 1987, p. 145).</p><p>In subsequent longitudinal re-search, Jordan et al. (2003) comparedtwo groups of children: those who pos-sessed low mastery of arithmetic combi-nations at the end of third grade, andthose who showed full mastery ofarithmetic combinations at the end ofthird grade. They then looked at thesechildrens development on a variety ofmath tasks between second and thirdgrades over four time points. When IQ,gender, and income level were heldconstant, children in both groups per-formed at the same level in solving un-timed story problems involving simpleaddition and subtraction operationsand progressed at the same rate. Chil-dren who possessed combination mas-tery increased at a steady rate on timednumber combinations, whereas chil-dren with low combination masterymade almost no progress over the 2-year period. What the research seemsto indicate is that although studentswith MD often make good strides interms of facility with algorithms andprocedures and simple word problemswhen provided with classroom in-struction, deficits in the retrieval ofbasic combinations remain (Geary,2004; Hanich et al., 2001); this seems toinhibit their ability to understandmathematical discourse and to graspthe more complex algebraic conceptsthat are introduced. Our interpretationis that failure to instantly retrieve abasic combination, such as 8 + 7, oftenmakes discussions of the mathematicalconcepts involved in algebraic equa-tions more challenging.</p><p>Maturity and Efficiency ofCounting Strategies</p><p>Another major finding is the link be-tween MD and efficient, effective count-</p><p> at Linkoping University Library on December 18, 2014ldx.sagepub.comDownloaded from </p><p>http://ldx.sagepub.com/</p></li><li><p>VOLUME 38, NUMBER 4, JULY/AUGUST 2005 295</p><p>ing strategy use (Geary, 1993, 2003;Geary et al., 2000). Jordan et al. (2003)found that a significant area of differ-ence between students with numbercombination mastery and those with-out was the sophistication of theircounting strategies. The poor combina-tion mastery group continued to usetheir fingers to count on untimed prob-lems between second and third grades,whereas their peers increasingly usedverbal counting without fingers, whichled much more easily to the types ofmental manipulations that constitutemathematical proficiency.</p><p>Siegler and Shrager (1984) stud-ied the development of counting knowl-edge in young children and found thatchildren use an array of strategieswhen solving simple counting andcomputational problems. For example,when figuring out the answer to 3 + 8,a child using a very unsophisticated,inefficient strategy would depend onconcrete objects by picking out first 3and then 8 objects and then countinghow many objects there are all to-gether. A more mature but still ineffi-cient counting strategy is to begin at 3and count up 8. Even more maturewould be to begin with the larger ad-dend, 8, and count up 3, an approachthat requires less counting. Some chil-dren will simply have this combinationstored in memory and remember thatit is 11.</p><p>Sieglers research reminds us thata basic arithmetic combination, such as2 + 9, is at some time in a persons lifea complex, potentially intriguing prob-lem to be solved. Only with repeateduse does it become a routine fact thatcan be easily recalled (Siegler &amp; Shra-ger, 1984). If a child can easily retrievesome basic combinations (e.g., 6 + 6),then he or she can use this informationto help quickly solve another problem(e.g., 6 + 7) by using decomposition(e.g., 6 + 6 + 1 = 13). The ability to storethis information in memory and easilyretrieve it helps students build bothprocedural and conceptual knowledgeof abstract mathematical principles,such as commutativity and the asso-ciative law. Immature finger or object</p><p>counting creates few situations forlearning these principles.</p><p>Geary (1990) extended Sieglersline of research to MD by studying thecounting strategy use of students withMD in the first grade. He divided thesample into three groups: (a) thosewho began and ended the year at av-erage levels of performance, (b) thosewho entered with weak arithmetic skilland knowledge but benefited from in-struction and ended the year at orabove the average level, and (c) thosewho began and ended the year at lowlevels of number knowledge. The sec-ond group included students who hadnot received much in the way of infor-mal or formal instruction in numberconcepts and counting; the third groupof students were classified as havingMD. The students who entered withweak number knowledge but showedreasonable growth during the first yearof systematic mathematics instructiontended to perform at the same level asthe group that entered at averageachievement levels on the strategychoice measure. Those with minimalgains (i.e., the MD group) used slower,less mature strategies more often. Al-though first graders with MD tendedto use the same types of strategies onaddition combinations as average stu-dents, they made three or even fourtimes as many errors while using thestrategy. For example, when childrenin the MD group used their fingers tocount, they were wrong half the time.When they counted verbally, they werewrong one third of the time. In con-trast, the average-achieving studentsrarely made finger or verbal countingmistakes. Geary et al. (2000) noted thatstudents with MD on the whole tend tounderstand counting as a rote, me-chanical activity. More precisely, thesechildren appear to believe that count-ing . . . can only be executed in the standard way . . . from left to right, andpointing at adjacent objects in succes-sion. (p. 238).</p><p>These studies suggest that matu-rity and efficiency of counting strate-gies are valid predictors of studentsability to profit from traditional math-</p><p>ematics instruction. Thus, they are ofgreat importance for understandingmethods for early screening of stu-dents for potential MD and the natureof potential interventions.</p><p>Mathematics Difficulties and Reading Difficulties</p><p>Jordan and colleagues have devoted agood deal of effort to distinguishingsimilarities and differences betweenchildren with specific mathematics dif-ficulties (MD only) and those withboth mathematics and reading difficul-ties (MD + RD). In many earlier stud-ies, children with MD were defined asa single group of low achievers (e.g.,Geary, 1993; Ostad, 1998). However,Jordans work, as well as recent re-search by Fuchs et al. (2004), has sug-gested that children with MD who areadequate readers show a different pat-tern of cognitive deficits than childrenwith MD who are also poor readers(Jordan et al., 2003; Jordan &amp; Montani,1997).</p><p>A distinguishing characteristic ofthis line of research involves the do-mai...</p></li></ul>

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