early identification and interventions for students with mathematics difficulties

JOURNAL OF LEARNING DISABILITIESVOLUME 38, NUMBER 4, JULY/AUGUST 2005, PAGES 293–304

Early Identification andInterventions for Students WithMathematics Difficulties

Russell Gersten, Nancy C. Jordan, and Jonathan R. Flojo

Abstract

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.

Currently, the field of learningdisabilities (LD) possesses anumber of valid measures that

can 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, &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, & Fletcher,2002).

Measurement of reading readi-ness evolved from a dynamic inter-action among theoretical studies (Schat-schneider et al., 2002) and interventionstudies (O’Connor, Notari-Syverson, &

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.

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).

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

detection of potential difficulties inmathematics; and (d) early interven-tion and instruction.

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 & 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

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is to stimulate research that examinesthe effectiveness of various approachesto interventions for young studentslikely to have trouble in mathematics.

The Nature of MathematicalDifficulties

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, &Prentice, 2004; Hanich, Jordan, Kaplan,& 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, &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.

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 child’s 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, & 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.

The seminal researcher in MDwas David Geary, although a gooddeal of subsequent research has been

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.

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, & Kaplan,2003).

Fluency and Mastery of Arithmetic Combinations

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, & Mertz,1988; Hasselbring, Goin, & 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 &Carper, 1943) is used, because basicproblems involving addition and sub-traction can be solved in a variety ofways and are not always retrieved as“facts” (National Research Council,2001). A major tenet in this body of re-search was that “the general concept of

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 & Pellegrino, 1987, p. 145).

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 thesechildren’s 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.

Maturity and Efficiency ofCounting Strategies

Another major finding is the link be-tween MD and efficient, effective count-



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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.

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.

Siegler’s research reminds us thata basic arithmetic combination, such as2 + 9, is at some time in a person’s lifea complex, potentially intriguing prob-lem to be solved. Only with repeateduse does it become a routine “fact” thatcan be easily recalled (Siegler & 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

counting creates few situations forlearning these principles.

Geary (1990) extended Siegler’sline 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 to“understand 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).

These studies suggest that matu-rity and efficiency of counting strate-gies are valid predictors of students’ability to profit from traditional math-

ematics instruction. Thus, they are ofgreat importance for understandingmethods for early screening of stu-dents for potential MD and the natureof potential interventions.

Mathematics Difficulties and Reading Difficulties

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,Jordan’s 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 & Montani,1997).

A distinguishing characteristic ofthis line of research involves the do-mains of mathematical cognition thatare assessed. Much of the early re-search on children with MD was nar-rowly focused, emphasizing only onearea of mathematical competence,namely arithmetic computation. Prob-lem solving and number sense receivedless attention. Because mathematics hasmultiple cognitive requirements (Na-tional Research Council, 2001), Jordanet al. (2003) hypothesized that abilitieswould be uneven across areas of math-ematical competence, especially amongchildren with MD only.

Hanich et al. (2001) identifiedchildren with MD only, MD + RD, RDonly, and typical achievement (TA) atthe beginning of second grade. In asubsequent study with the same sam-ple, Jordan, Kaplan, and Hanich (2002)examined each group’s achievementgrowth in reading and in mathematicson the Woodcock-Johnson Psycho–educational Battery–Revised (Wood-cock & Johnson, 1990) over a 2-yearperiod. MD and RD were defined by




performance at or below the 35th per-centile, based on national norms. TheMD-only group started at about thesame math level as the MD + RD group(mean percentiles of 22 and 21, respec-tively) and the RD-only group at aboutthe same math level as the TA group(mean percentile scores of 60 and 68,respectively).

Achievement growth of childrenwith MD only was significantly morerapid than that of children with MD +RD. It is important to note that this dif-ference persisted even when adjustedfor IQ and income level. The RD-onlygroup performed at about the samelevel in mathematics as children withMD only and at a lower level than TAchildren at the end of third grade. Dif-ficulties in reading seem to have anegative influence on children’s de-velopment in general mathematicsachievement. In contrast, mathematicsabilities did not influence readinggrowth. On reading measures, chil-dren with RD only achieved at thesame rate as children with MD + RDwhen IQ and income level were heldconstant. Reading difficulties identi-fied in second grade remained steadythroughout the test period, regardlessof whether they were specific (RDonly) or general (MD + RD).

Gender and ethnicity did not pre-dict growth in math or in read-ing. However, income level predictedgrowth in math but not in reading. Notsurprisingly, special services weremuch more likely to be provided inreading than in math during secondand third grades. Jordan et al. (2003)speculated that early reading interven-tions level the playing field for chil-dren with RD, making income less of a factor in primary school readinggrowth.

Over a 16-month period, Jordanand colleagues also investigated thedevelopment of specific mathematicalcompetencies in the same group ofchildren with MD only and MD + RD(Hanich et al., 2001; Jordan et al., 2003).The researchers looked at areas ofmathematical cognition that are di-rectly related to the teaching of mathe-

matics (as opposed to more generalcognitive competencies), includingbasic calculation, problem solving, andbase-10 concepts. Children were as-sessed twice in second grade and twicein third grade. Children’s ending levelof performance was examined, as weretheir growth rates.

Jordan et al. (2003) showed thatthird graders with MD only had an ad-vantage over their MD + RD peers intwo areas of mathematical cognition:accuracy on arithmetic combinationsand story problems. However, perfor-mance differences between the twoMD groups in arithmetic combinationsdisappeared when IQ, gender, ethnic-ity, and income level were considered.The MD-only group performed betterthan the MD + RD group at the end ofthird grade on calculation principles(e.g., understanding of the relations be-tween addition and subtraction andthe commutativity principle), irrespec-tive of predictor variables. The tasksthat did not differentiate the MD-onlygroup from the MD + RD group weretimed number combinations, estima-tion, place value, and written com-putation. The two MD groups did notdiffer in their growth rates (frommid–second grade to the end of thirdgrade) on any of the math tasks.

Difficulties of children identifiedas having MD + RD in early secondgrade are pervasive and steady oversecond and third grades, even whenpredictors such as IQ are held constant.Despite their weaknesses, however,children with MD + RD achieved at arate that was similar to that of childrenin the other groups. Children with MDonly appeared to have consistent diffi-culties with calculation fluency. Theyperformed as low as children with MD + RD when required to respond tonumber combinations quickly and re-lied on their fingers as much as chil-dren with MD + RD, even at the end ofthird grade. However, children withMD only used finger counting strate-gies more accurately than childrenwith MD + RD, suggesting better facil-ity with counting procedures in theformer group.

Jordan et al. (2003) speculated thatweaknesses in spatial representationsrelated to numerical magnitudes (ratherthan weaknesses in verbal representa-tions) underpin rapid fact retrievaldeficits. Such children might have dif-ficulties manipulating visual (non-verbal) representations on a numberline—a skill that may be critical forsolving addition and subtraction prob-lems. In fact, it was found that childrenwith poor mastery of number combi-nations performed worse than childrenwith solid mastery on nonverbal blockmanipulation and pattern recognitiontasks. In contrast, the groups per-formed at about the same level on ver-bal cognitive tasks.

In summary, deficits in calcula-tion fluency appear to be a hallmark ofmathematics difficulties, specific orotherwise. The achievement growthpatterns reported by Hanich et al.(2001) for students below the 35th per-centile held even for children with ex-tremely low performance (i.e., childrenwho fell below the 15th percentile inachievement in mathematics; Jordan etal., 2003). In other words, when MDand RD were defined as below the 15thpercentile, students with MD + RDtended to acquire mathematical com-petence at a slower rate than studentswith MD only. The deficits occurred inonly two areas of arithmetic: accuracyon arithmetic combinations and storyproblems.

Number SenseA very different developmental re-search tradition may also have impli-cations for early identification andearly intervention. This body of re-search emanates primarily from psy-chologists who are interested in thecognitive development of children; theyfocus on the concept of number sense(Bereiter & Scardamalia, 1981; De-haene, 1997; Greeno, 1991; Okamoto &Case, 1996). Unfortunately, no two re-searchers have defined number sensein precisely the same fashion.

Even though in 1998, Case wrotethat “number sense is difficult to de-




fine but easy to recognize” (p. 1), hedid attempt to operationalize this con-cept (Okamoto & Case, 1996) in one ofhis last writings on the topic (Kalch-man, Moss, & Case, 2001), noting that

the characteristics of good numbersense include: (a) fluency in estimat-ing and judging magnitude, (b) abil-ity to recognize unreasonable results,(c) flexibility when mentally com-puting, (d) ability to move among dif-ferent representations and to use the most appropriate representation. (p. 2)

Case, Harris, and Graham (1992)found that when middle-income kin-dergartners are shown two groups ofobjects (e.g., a group of 5 chips and agroup of 8 chips), most are able to iden-tify the “bigger group” and know thatthe bigger group has more objects. Yetonly those with well-developed num-ber sense would know that 8 is 3 bigger than 5. Likewise, only thosewith developed number sense wouldknow that 12 is a lot bigger than 3,whereas 5 is just a little bit bigger than3. These findings highlight the need fordifferentiated instruction in mathemat-ics in kindergarten.

In an attempt to identify the fea-tures of number sense, Okamoto (2000,cited in Kalchman et al., 2001, p. 3) con-ducted a factor analysis on students’performance and identified two dis-tinct factors in children’s mathematicsproficiency in kindergarten. The firstfactor related to counting, a key indica-tor of the digital, sequential, verbalstructure; the second to quantity dis-crimination (e.g., tell me which is more,5 or 3?). For example, Okamoto andCase (1996) found that some studentswho could count to 5 without error hadno idea which number was bigger, 4 or2. They concluded that in children thisage, the two key components of num-ber sense are not well linked. Implicitin their argument is that these two fac-tors are precursors of the other compo-nents of number sense specified previ-ously, such as estimation and ability tomove across representational systems.These abilities can be developed as stu-

dents become increasingly fluent incounting and develop increasingly so-phisticated counting strategies.

Further support for the impor-tance of assessing quantity discrimina-tion and its potential importance as anearly screening measure emanatesfrom the research of Griffin, Case, andSiegler (1994), who found that studentsentering kindergarten differed in theirability to answer quantity discrimina-tion questions such as, “Which numberis bigger, 5 or 4?” even when they con-trolled for student abilities in countingand simple computation. Another im-portant finding was that the high–socioeconomic status (SES) childrenanswered the question correctly 96% ofthe time, compared to low-SES chil-dren, who answered correctly only18% of the time. This finding suggeststhat aspects of number sense develop-ment may be linked to the amount ofinformal instruction that students re-ceive at home on number concepts andthat some students, when providedwith appropriate instruction in pre-school, kindergarten, or first grade inthe more complex aspects such asquantity discrimination, may quicklycatch up with their peers. Geary’s(1990) research with primarily low-income students found a substantialgroup of such students in the firstgrade. It may be that intervention in ei-ther pre-K or K would be very benefi-cial for this group.

In summary, number sense is, ac-cording to Case et al. (1992), a concep-tual structure that relies on many linksamong mathematical relationships,mathematical principles (e.g., commu-tativity), and mathematical proce-dures. The linkages serve as essentialtools for helping students to thinkabout mathematical problems and todevelop higher order insights whenworking on mathematical problems.Number sense development can be en-hanced by informal or formal instruc-tion prior to entering school. At least in5- and 6-year-olds, the two compo-nents of number sense (counting/simple computation and sense ofquantity/use of mental number lines)

are not well linked. Creating such link-ages early may be critical for the de-velopment of proficiency in mathemat-ics. Children who have not acquiredthese linkages probably require inter-vention that builds such linkages. Thetwo components of number sense alsoprovide a framework for determiningthe focus of a screening battery, espe-cially if some measure of workingmemory is also included.

Early Detection of MD and Potential ScreeningMeasures

Using this framework, Baker, Gersten,Flojo, et al. (2002) explored the predic-tive validity of a set of measures thatassess students’ number sense andother aspects of number knowledgethat are likely to predict subsequentperformance in arithmetic. A battery ofmeasures was administered to morethan 200 kindergartners in two urbanareas. Performance on these measureswas correlated with subsequent per-formance on a standardized measureof mathematics achievement—the twomathematics subtests of the StanfordAchievement Test–Ninth Edition (SAT-9),Procedures and Problem Solving (Har-court Educational Measurement, 2001).

The primary measure in the pre-dictive battery was the Number Knowl-edge Test, developed by Okamoto andCase (1996). An item response theoryreliability was conducted on the mea-sure based on a sample of 470 students.Using a two-parameter model, the reli-ability was .93. The Number KnowledgeTest is an individually administeredmeasure that allows the examiner notonly to appraise children’s knowledgeof basic arithmetic concepts and oper-ations, but also to assess their depth ofunderstanding through a series ofstructured probes that explore stu-dents’ understanding of magnitude,the concept of “bigger than,” and thestrategies they use in counting. Table 1presents sample items from severallevels of the Number Knowledge Test.

Baker, Gersten, Flojo, et al. (2002)also included a series of measures of




specific skills and proficiencies devel-oped by Geary et al. (2000). Theyincluded measures of quantity dis-crimination (magnitude comparison),counting knowledge, number identifi-cation, and working memory. Brief de-scriptions of these measures are pro-vided in Table 2.

The Number Knowledge Test, themeasure with the most breadth, wasthe best predictor in the set of mea-sures of SAT-9 Procedures and Prob-lem Solving. The zero-order predictive

significant for each of the screeningmeasures at p < .01 and in the moder-ate range, with the exception of rapidautomatized naming and letter nam-ing fluency, which are smaller in mag-nitude and significant at the .05 level.As might be expected, the measuresthat involve mathematics tend to pre-dict better than those that are unrelatedto mathematics—Phoneme Segmenta-tion (a measure of phonemic aware-ness), letter naming fluency, and abil-ity to name pictures and colors. Inparticular, the magnitude comparisontask and the digit span backward taskseem particularly promising for ascreening battery. The reader shouldnote that both of these measures wereuntimed.

We next attempted to examinewhich combination of measures ap-peared to be the best predictor of themost reliable criterion measure—theSAT-9 Total Mathematics score 1 yearlater. The number of predictor vari-ables that can be included in the multi-ple regressions is limited by samplesizes (Frick, Lahey, Christ, Loeber, &Green, 1991). Because of the small sam-ple sizes, two predictors were chosento be included in the regression analy-sis. The second best predictor wasDigit Span Backward, a measure ofworking memory. Working memory(for abstract information, such as a se-quence of numbers) seems to be re-lated to many arithmetic operations.Several other researchers have alsofound that problems with numericaldigit span have been identified withMD (Geary & Brown, 1991; Siegel & Ryan, 1988; Swanson & Beebe-Frankenberger, 2004). From a prag-matic perspective, the predictive abil-ity of the set of two measures over a12-month period is impressive, R = .74;F(2, 64) = 38.50, p < .01.

Use of Rate Measures as EarlyPredictors of MD

In 1999, Gersten and Chard noted thattwo good indicators of number sense in

validity correlations were .73 for TotalMathematics on the SAT-9, .64 for theProcedures subtest, and .69 for Prob-lem Solving. All correlations weremoderately strong and significant, p <.01.

Table 3 presents the predictive va-lidity correlations for each of the mea-sures administered in kindergarten forSAT-9 Total Mathematics and the Num-ber Knowledge Test, which was re-administered in first grade. Note thatthe predictive validity correlations are

TABLE 1Sample Items From the Number Knowledge Test

Sample item Level

I’m going to show you some counting chips. 0Would you count these for me?

Here are some circles and triangles. Count just the triangles and tell me 0how many there are.

How much is 8 take away 6? 1

If you had 4 chocolates and someone gave you 3 more, how many 1chocolates would you have altogether?

Which is bigger: 69 or 71? 2

Which is smaller: 27 or 32? 2

What number comes 9 numbers after 999? 3

Which difference is smaller: the difference between 48 and 36 or the 3difference between 84 and 73?

TABLE 2Description of Selected Early Math Measures

Subtest Description

Geary (2003) Digit Span Student repeats a string of numbers either forwards or backwards.

Magnitude comparison From a choice of four visually or verbally presented numbers, student chooses the largest.

Numbers from Dictation From oral dictation, student writes numbers.

Clarke & Shinn (2004)Missing Number Student names a missing number from a sequence of numbers

between 0 and 20.

Number Identification Student identifies numbers between 0 and 20 from printed numbers.

Quantity Discrimination Given two printed numbers, student identifies which is larger.




young children were (a) quantity dis-crimination and (b) identifying a miss-ing number in a sequence. A group ofresearchers explored the predictive va-lidity of timed assessments of thesetwo components. Both studies exam-ined predictive validity from fall tospring, Clarke and Shinn (2004) onlyfor first grade, and Chard et al. (inpress) for both kindergarten and firstgrade. All predictive measures usedwere timed measures and were easy toadminister. Brief descriptions of someof the measures are in Table 2. The re-searchers also included a measure ofrapid automatized number naming,analogous to the rapid letter namingmeasures that are effective screeningmeasures for beginning readers (Baker,Gersten, & Keating, 2000).

The screening measures used byBaker et al. (2000) were untimed mea-sures. These included instruments de-veloped by Okamoto and Case (1996)and Geary (2003) in their lines of re-search. In the area of reading, re-searchers have found that timed mea-sures of letter naming and phonemesegmentation administered to kinder-gartners and children entering firstgrade are solid predictors of readingachievement in Grades 1 and 2 (Dy-namic Indicators of Basic Early Liter-acy Skills [DIBELS]; available at http://idea.uoregon.edu/~dibels). In particu-lar, the ability to rapidly name letters ofthe alphabet seems to consistently pre-dict whether a student will experiencedifficulty in learning to read in the pri-mary grades (Adams, 1990; Schat-schneider, Fletcher, Francis, Carlson, &Foorman, 2004). The three measuresdeveloped by Clarke and Shinn (2004),although brief, proved to be reason-ably reliable, with test–retest reliabili-ties ranging from .76 to .86. The pre-dictive validity of these three measuresfor first grade (Chard et al., in press;Clarke & Shinn, 2004) and kinder-garten (Chard et al., in press) are pre-sented in Table 4. For both studies,number knowledge was used as a cri-terion measure. Clarke and Shinn usedthe Woodcock-Johnson Tests of Achieve-

ment (Woodcock & Mather, 1989) Ap-plied Problems test as a second cri-terion measure. Each of the three mea-sures appears to be a reasonably validpredictor of future performance. Theyare presented in Table 4, in the rightcolumn.

Chard et al. (in press) exploredthe predictive validity of the set ofthree screening measures developedby Clarke and Shinn (2004) using alarger sample. Their study differedfrom that of Clarke and Shinn in that

(a) they correlated fall and spring per-formance for both kindergarten andfirst grade, and (b) they used theNumber Knowledge Test as a criterionmeasure rather than the Woodcock-Johnson. Rs were significant at p < .01and moderately large (.66 for kinder-garten and .68 for first grade).

When we attempt to put togetherthe information gained from thesestudies, it appears that the strongestscreening measure is the relativelycomprehensive Number Knowledge Test

TABLE 3Predictive Validity of Number Sense Measures

Spring of First Grade

Spring of Kindergarten Predictor SAT-9a NKTb

Mathematical MeasuresNKT .72** .72**Digit Span Backward .47** .60**Numbers From Dictation .47** .48**Magnitude Comparison .54** .45**

Nonmathematical measuresPhonemic Segmentation .42** .34**Letter Naming Fluency .43** .27*Rapid Automatized Naming (Colors and Pictures) .34** .31*

Note. SAT-9 = Stanford Achievement Test, ninth edition (Harcourt Educational Measurement, 2001); NKT =Number Knowledge Test (Okamoto & Case, 1996).an = 65. bn = 64.*p < .05. **p < .01.

TABLE 4Predictive Validity Correlations of Early Math Measures

Spring Scoresa

Fall pretest Chard et al. (in press) Clarke & Shinn (2004)

Kindergarten samplen 436Number Identification .58Quantity Discrimination .53Missing Number .61

First-grade samplen 483 52Number Identification .58 .72Quantity Discrimination .53 .79Missing Number .61 .72

Note. All correlations significant at p < .05.aCriterion measure for Chard et al. (in press) was the Number Knowledge Test (Okamoto & Case, 1996);criterion measure for Clarke and Shinn (2004) was the Woodcock-Johnson Tests of Achievement (Wood-cock & Mather, 1989) Applied Problems subtest.




(Okamoto & Case, 1996). However,three relatively brief measures appearto be quite promising: (a) of quantitydiscrimination or magnitude comparison;(b) identifying the missing number in asequence, a measure of countingknowledge; (c) some measure of num-ber identification. It also appears that ameasure of rapid automatized naming(such as Clarke & Shinn’s NumberIdentification) and a measure of work-ing memory for mathematical informa-tion (such as reverse digit span) alsoseem to be valid predictors. Note thatwith the exception of working mem-ory, all of these measures can be linkedto instructional objectives that are ap-propriate at these grade levels.

We envision several critical nextsteps in this line of research. Thesesteps can and should be concurrent.From a technical point of view, weneed to continue to study the long- andshort-term predictive validity of thesetypes of number sense measures.Equally important would be researchthat determines whether measures ad-ministered to kindergartners and firstgraders should be timed or untimed.Although there is good evidence tosuggest that the rate of retrieval of basicarithmetic combinations is a criticalcorrelate of mathematical proficiency,to date there is no evidence suggesting,for example, that the speed with whichstudents can identify which number oftwo is the biggest, or count backwardsfrom a given number, is an importantvariable to measure.

Instructional Implications

There is a paucity of research on earlyinterventions to prevent MD in strug-gling students. Only two researchershave described early intervention re-search (Fuchs, Fuchs, Karns, Hamlett,& Katzaroff, 1999; Griffin, Case, &Siegler, 1994), and these have beenstudies of whole-class instruction.Therefore, in discussing possible inter-ventions for children indicating a po-tential for MD in kindergarten or first

grade, we primarily rely on interpola-tions from longitudinal and develop-mental research and on a recent syn-thesis of the intervention research forolder students with LD or low achieve-ment in mathematics (Baker, Gersten,& Lee, 2002; Gersten et al., in press)and, to some extent, on the thinkingabout early intervention of Fuchs,Fuchs, and Karns (2001) and Robinson,Menchetti, and Torgesen (2002).

We have a good sense of some ofthe goals of early intensive interven-tion. Certainly, one goal is increasedfluency and accuracy with basic arith-metic combinations. Other, relatedgoals are the development of more ma-ture and efficient counting strategiesand the development of some of thefoundational principles of numbersense—in particular, magnitude com-parison and ability to use some type ofnumber line. It is quite likely that otheraspects of number sense are equallyimportant goals.

The evidence does not suggestone best way to build these proficien-cies in young children; in fact, the con-sistent finding that there are at leasttwo different types of MD (i.e., MDonly vs. MD + RD) indicates that cer-tain approaches might work better forsome subgroups of students (Fuchs et al., 2004). The need for differentiatedintervention is also highlighted by thefinding (Okamoto & Case, 1996) that inyoung children, counting ability andnumber sense develop relatively inde-pendently. Thus, there is no clear “bestway” to reach this goal, but there aresome promising directions.

Early Intervention Research on Arithmetic Combinations

Pellegrino and Goldman (1987) andHasselbring et al. (1988) provided in-tensive interventions to older elemen-tary students with MD. However, webelieve that their research raises im-portant implications for interventionin earlier grades. Both groups of re-searchers directly targeted fluency and

accuracy with basic arithmetic combi-nations. They attempted to create situ-ations that helped students with MDstore arithmetic combinations in mem-ory in such a manner that the childrencould retrieve the fact “quickly, effort-lessly and without error” (Hasselbringet al., 1988, p. 2). Hasselbring et al.designed a computer program thatcreated individually designed practicesets consisting of a mix of combina-tions that the student could fluently re-call and those that were difficult forthat student. The program used con-trolled response times to force studentsto rely on retrieval rather than count-ing. This was a clear attempt to some-what forcefully restructure students’strategies toward retrieval rather thaninefficient finger counting. Practicecontinued until the student consis-tently used retrieval. This procedurewas effective for the majority but notall of the students diagnosed as havingMD. Students who relied only on fin-ger counting did not benefit at all. Itappears that as students begin to ma-ture in counting strategy use, sometype of individualized practice oncombinations may be useful in en-hancing fluency. The use of technologyto create sensible, individualized prac-tice sets also seems a viable alternativeto worksheets or whole-class practice.

Other Options for Building Fluency

Because fluency and accuracy witharithmetic combinations require theuse of mature strategies, instructionand guidance in strategy use seemscritical. It seems logical that some stu-dents will need direct instruction instrategy use that may be unnecessaryfor their peers. Siegler (1988) describedthe intuitive benefits of this type of dif-ferential instruction as follows: “Teach-ing children to execute backup strate-gies more accurately affords themmore opportunities to learn the correctanswer [and] . . . reduces the likelihoodof associating incorrect answers, pro-




duced by faulty execution of backupstrategies, with the problem” (p. 850).

Shrager and Siegler (1998) dem-onstrated that the generalization ofstrategy use proceeds slowly in youngchildren. Adults often underestimatethe time it takes a child to use a newlylearned mathematical strategy consis-tently. This observation is important tokeep in mind as preventative interven-tions are designed. Shrager and Siegleralso found that at least for very basicarithmetic strategies, generalization in-creases with the presentation of chal-lenging problems (i.e., problems thatare very difficult to solve without theuse of a strategy and fairly easy tosolve with a strategy). This finding,too, would seem to have important in-structional implications for the designof interventions.

Recently, some researchers (Jor-dan et al., 2003; Robinson et al., 2002)have argued that children can deriveanswers quickly and with minimalcognitive effort by employing calcula-tion principles or “shortcuts,” such asusing a known number combination toderive an answer (2 + 2 = 4, so 2 + 3 =5), relations among operations (6 + 4 =10, so 10 − 4 = 6), n + 1, commutativity,and so forth. This approach to instruc-tion is consonant with recommenda-tions by the National Research Council(2001). Instruction along these linesmay be much more productive thanrote drill without linkage to countingstrategy use. Rote drill places heavydemands on associative memory, anarea of weakness in many childrenwith LD in mathematics (Geary, 1994).As children become more proficient inapplying calculation principles, theyrely less on their fingers and eventu-ally master many combinations (Jor-dan, Levine, & Huttenlocher, 1994).Mental calculation shortcuts have theadvantage of developing children’snumber sense as well as their fluency.Research has shown that even adultswho are competent in math use mentalshortcuts instead of automatic retrievalon some number combinations (Le-Fevre et al., 1996).

Interventions to Build NumberSense and ConceptualUnderstanding

Robinson et al. (2002) proposed that in-terventions for students with poormastery of arithmetic combinationsshould include two aspects: (a) inter-ventions to help build more rapid re-trieval of information, and (b) con-certed instruction in any and all areasof number sense or arithmetic conceptsthat are underdeveloped in a child(e.g., principles of commutativity, or“counting on” from the larger ad-dend.) This seems a logical frameworkfor tailoring interventions to ensurethat students develop increasingly ma-ture and efficient counting strategy usethat is linked to fluent retrieval of com-binations, yet continues to solidifytheir knowledge of mathematics con-cepts and algorithms. Tailoring canand should be linked to the assess-ments discussed previously in this ar-ticle.

In the view of Robinson et al.(2002), number sense is “a skill or kindof knowledge about numbers ratherthan an intrinsic process” (p. 86). Thusnumber sense, like phonemic aware-ness, should be teachable. This is sup-ported by Case and Griffin’s (1990)finding that number sense is linked, inpart, to the amount of informal or for-mal instruction on number conceptsprovided at home.

In terms of developing interven-tions to develop number sense, we drawinferences from existing research. Oneseemingly promising approach is theNumber Worlds curriculum, developedby Griffin (2004) and evaluated asRightstart. However, based on imple-mentation research (Gersten, Chard,Griffiths, Katz, & Bryant, 2003), wewould only recommend the whole-class activities, such as practice in“counting on,” practice in listening tocoins being dropped in a box andcounting, practice in counting back-wards, practice in linking adding andsubtracting to the manipulation of ob-jects. These could easily be done with

small groups of children and appearedto be helpful in building a sense ofnumber in students who needed workin this area. In contrast, the games thatcomposed much of the curriculumproved extraordinarily difficult to im-plement in a typical classroom. Wetherefore envision the use of the con-ceptual framework developed by Grif-fin and Case (1997) serving as a basisfor structured small-group learning ac-tivities. We also think that the goals ofnumber sense games—to encourageyoung students in kindergarten andfirst grade to learn how to talk aboutnumbers and relationships—can bebetter realized in the context of activi-ties that accompany explicit, small-group instruction. As Milgram (2004)noted, students can learn to use so-phisticated strategies by viewing mod-els of proficient performance andbeing provided with steps that helpthem solve a problem. This practice isstrongly supported by meta-analysesof research on effective approaches forteaching students with MD (Baker etal., 2002; Gersten et al., in press). Stu-dents will need to learn the very ab-stract language of mathematics beforethey can express their ideas; thus,learning the vocabulary of mathemat-ics seems another critical piece of earlyintervention.

Siegler (1988) precisely demon-strated that procedural and conceptualknowledge in mathematics are fre-quently integrated, and parsing themout is difficult. Therefore, embeddingthe teaching of concepts in work onprocedures seems the ideal route.Siegler demonstrated that for youngchildren, very simple arithmetic com-putations are, at some point in their de-velopment, extremely difficult prob-lems to solve, requiring a good deal ofeffort and orchestration of severalknowledge bases (counting knowledge,number–symbol correspondence, or-der irrelevance). Thus, the creation oflearning situations where students areexplicitly shown how to orchestrate allthese concepts when doing arithmeticwould seem ideal. Exactly how to




reach that goal is much less clear. In-terventions should follow the findingof Rittle-Johnson, Siegler, and Alibali(2001), who noted that procedural andconceptual abilities in mathematics“lie on a continuum and cannot alwaysbe separated” (p. 346). They also foundthat conceptual knowledge is “implicitor explicit understanding of the princi-ples that govern a domain. . . . Thisknowledge is flexible and not tied tospecific problem types and is thereforegeneralizable. . . . Furthermore it mayor may not be verbalizable” (p. 346). Itseems that we need to develop a farbetter understanding of when it makessense for students to verbalize their ap-proach and when this is not a helpfulinstructional procedure. It is possiblethat students with MD only may bemuch more amenable to this com-monly advocated approach than stu-dents with MD + RD, who may lack theverbal skill to articulate either their un-derstandings or their confusions.

The meta-analyses (Baker, Ger-sten, & Lee, 2002; Gersten et al., inpress) have indicated several promis-ing directions: using structured peerwork, using visuals and multiple rep-resentations, providing students withstrategies that could then serve as a“hook” for them as they solve prob-lems. Many of these studies have dealtonly with arithmetic computations orsimple arithmetic word problems. Wehave a good deal to learn about how touse visuals—how best to model andthink aloud so that students can inter-nalize and truly understand some ofthe concepts that they only partiallyunderstand.

Summary and Conclusions

We see a major goal of early mathe-matics interventions to be the develop-ment of fluency and proficiency withbasic arithmetic combinations and theincreasingly accurate and efficient useof counting strategies. We believe thatone reason why the fast retrieval ofarithmetic combinations is critical isthat students cannot really comprehend

any type of dialogue about numberconcepts or various problem-solvingapproaches unless they automaticallyknow that 6 + 4 is 10, that doubling 8makes 16, and so forth. In other words,students who are still slowly usingtheir fingers to count a combinationsuch as 7 + 8 or 3 × 2 are likely to be to-tally lost when teachers assume theycan effortlessly retrieve this informa-tion and use this assumption as a basisfor explaining concepts essential forproblem solving or for understandingwhat division means.

It thus appears that lack of flu-ency with arithmetic combinations re-mains a critical correlate of MD andneeds to be a goal of intervention ef-forts for many children. Furthermore,teachers need to be aware which stu-dents have not mastered basic combi-nations and note that these studentsmay well need additional time to un-derstand the concepts and operationsthat are explained or discussed.

From a conceptual point of view,we need to work more on linking spe-cific measures that can be used forearly screening and identification ofMD with the theories that have beendeveloped about MD. For example, inearly elementary school, the shift fromconcrete to mental representationsseems critical for developing calcula-tion fluency (Jordan & Hanich, 2003;Jordan et al., 2003). Early screeningmeasures should examine children’scalculation strategies on different typesof problems to determine whether chil-dren are making this transition. We be-lieve that further advances in develop-ing valid measures for screening andearly detection will need to be basedon more refined, better operational-ized definitions of number sense. Inorder to accomplish this goal, we needto understand in depth the specificskills, strategies, and understandingsthat predict subsequent problems inbecoming proficient in mathematics.This understanding will help the fieldto refine the nature of measures devel-oped for early identification and helpto shape the nature of effective early in-tervention programs.

ABOUT THE AUTHORS

Russell Gersten, PhD, is executive director ofa nonprofit educational research institute andprofessor emeritus in the College for Educationat the University of Oregon. Dr. Gersten is a na-tionally recognized expert in both quantitativeand qualitative research and evaluation method-ologies, with an emphasis on translating re-search into classroom practice. He has con-ducted synthesis of intervention research onteaching mathematics, teaching expressive writ-ing and teaching reading comprehension to low-achieving students and students with learningdisabilities. Nancy C. Jordan, EdD, is a pro-fessor in the School of Education at the Uni-versity of Delaware. Her interests includemathematics difficulties, relationships betweenmathematics difficulties and reading difficul-ties, assessment, and cognitive development.Jonathan R. Flojo, BA, is a research associateat Instructional Research Group and will com-plete his doctorate in counseling psychology atUniversity of Oregon in the summer of 2005.His areas of expertise include statistics and mea-surement, including hierarchical linear model-ing. Address: Nancy C. Jordan, School of Edu-cation, University of Delaware, Newark, DE19716; e-mail: [email protected]

AUTHORS’ NOTE

We wish to dedicate this article and special issueto the memory of the late Dr. Michelle Gersten.We also wish to express our appreciation toSusan Unok Marks for her expert editorial andorganizational feedback and to Michelle Spear-man for her assistance in putting all the piecestogether in a coherent fashion. Finally, wewould like to thank Rollanda O’Connor andDoug Carnine for providing very helpful feed-back on an earlier version of this article.

REFERENCES

Adams, M. J. (1990). Beginning to read:Thinking and learning about print. Cam-bridge, MA: MIT Press.

Baker, S., Gersten, R., Flojo, J., Katz, R.,Chard, D., & Clarke, B. (2002). Preventingmathematics difficulties in young children:Focus on effective screening of early numbersense delays (Tech. Rep. No. 0305). Eugene,OR: Pacific Institutes for Research.

Baker, S., Gersten, R., & Keating, T. J. (2000).When less may be more: A 2-year longi-tudinal evaluation of a volunteer tutoringprogram requiring minimal training.Reading Research Quarterly, 35, 494–519.




Baker, S., Gersten, R., & Lee, D. (2002). Asynthesis of empirical research on teach-ing mathematics to low-achieving stu-dents. The Elementary School Journal, 103,51–73.

Bereiter, C., & Scardamalia, M. (1981). Fromconversation to composition: The role ofinstruction in a developmental process.In R. Glaser (Ed.), Advances in instruc-tional psychology (Vol. 2, pp. 132–165).Hillsdale, NJ: Erlbaum.

Brownell, W. A., & Carper, D. V. (1943).Learning multiplication combinations. Dur-ham, NC: Duke University Press.

Case, R., & Griffin, S. (1990). Child cogni-tive development: The role of central con-ceptual structures in the development ofscientific and social thoughts. In C. A.Hauert (Ed.), Advances in psychology–Developmental psychology: Cognitive, per-ception–motor and neurological perspectives.Amsterdam: North Holland.

Case, L. P., Harris, K. R., & Graham, S.(1992). Improving the mathematicalproblem-solving skills of students withlearning disabilities: Self-regulated strat-egy development. The Journal of SpecialEducation, 26, 1–19.

Chard, D., Clarke, B., Baker, B., Otterstedt,J., Braun, D., & Katz, R. (in press). Usingmeasures of number sense to screen fordifficulties in mathematics: Preliminaryfindings. Assessment Issues in Special Edu-cation.

Clarke, B., & Shinn, M. (2004). A prelimi-nary investigation into the identificationand development of early mathematicscurriculum-based measurement. SchoolPsychology Review, 33, 234–248.

Dehaene, S. (1997). The number sense: Howthe mind creates mathematics. New York:Oxford University Press.

Francis, D. J., Shaywitz, S., Steubing, K.,Shaywitz, B., & Fletcher, J. (1994). Themeasurement of change: Assessing be-havior over time and within a develop-mental context. In G. R. Lyon (Ed.),Frames of reference for the assessment oflearning disabilities: New views on measure-ment (pp. 29–58). Baltimore: Brookes.

Frick, P. J., Lahey, B. B., Christ, M. A. G.,Loeber, R., & Green, S. (1991). History ofchildhood behavior problems in biologi-cal relatives of boys with attention deficithyperactivity disorder and conduct dis-order. Journal of Clinical Child Psychology,20, 445-451.

Fuchs, L. S., Fuchs, D., & Karns, K. (2001).Enhancing kindergartners’ mathematicaldevelopment: Effects of peer-assisted

learning strategies. The Elementary SchoolJournal, 101, 495–510.

Fuchs, L. S., Fuchs, D., Karns, K., Hamlett,C. L., & Katzaroff, M. (1999). Mathemat-ics performance assessment in the class-room: Effects on teacher planning andstudent problem solving. American Edu-cational Research Journal, 36, 609–646.

Fuchs, L. S., Fuchs, D., & Prentice, K. (2004).Responsiveness to mathematical prob-lem-solving instruction: Comparing stu-dents at risk of mathematics disabilitywith and without risk of reading disabil-ity. Journal of Learning Disabilities, 37, 293–306.

Geary, D. (1990). A componential analysisof an early learning deficit in mathemat-ics. Journal of Experimental Child Psychol-ogy, 33, 386–404.

Geary, D. C. (1993). Mathematical disabili-ties: Cognitive, neuropsychological, andgenetic components. Psychological Bul-letin, 114, 345–362.

Geary, D. C. (1994). Children’s mathematicaldevelopment. Washington, DC: APA.

Geary, D. C. (2003). Learning disabilities inarithmetic: Problem solving differencesand cognitive deficits. In H. L. Swanson,K. Harris, & S. Graham (Eds.), Handbookof learning disabilities (pp. 199–212). NewYork: Guilford.

Geary, D. C. (2004). Mathematics and learn-ing disabilities. Journal of Learning Dis-abilities, 37, 4–15.

Geary, D. C., & Brown, S. C. (1991). Cogni-tive addition: Strategy choice and speed-of-processing differences in gifted, normal,and mathematically disabled children.Developmental Psychology, 27, 787–797.

Geary, D. C., Hamson, C. O., & Hoard, M. K. (2000). Numerical and arithmeticalcognition: A longitudinal study of pro-cess and concept deficits in children withlearning disability. Journal of ExperimentalChild Psychology, 77, 236–263.

Gersten, R., & Chard, D. (1999). Numbersense: Rethinking arithmetic instructionfor students with mathematical disabili-ties. The Journal of Special Education, 33(1),18–28.

Gersten, R., Chard, D., Baker, S., Jayanthi,M., Flojo, J., & Lee, D. (under review). Ex-perimental and quasi-experimental re-search on instructional approaches forteaching mathematics to students withlearning disabilities: A research synthe-sis. Review of Educational Research.

Gersten, R., Chard, D., Griffiths, R., Katz,R., & Bryant, D. (2003, April). Methods andmaterials for reaching students in early and

intermediate mathematics. Paper presentedat the Council for Exceptional Children,Seattle, WA.

Goldman, S. R., & Pellegrino, J. W. (1987).Information processing and educationalmicrocomputer technology: Where do wego from here? Journal of Learning Disabili-ties, 20, 144–154.

Goldman, S. R., Pellegrino, J. W., & Mertz,D. L. (1988). Extended practice of basicaddition facts: Strategy changes in learn-ing disabled students. Cognition and In-struction, 5, 223–265.

Greeno, J. (1991). Number sense as situatedknowing in a conceptual domain. Journalfor Research in Mathematics Education, 22,170–218.

Griffin, S. (1998, April). Fostering the devel-opment of whole number sense. Paper pre-sented at the annual meeting of theAERA, San Diego, CA.

Griffin, S. (2004). Building number sensewith number worlds: A mathematics pro-gram for young children. Early ChildhoodResearch Quarterly, 19(1), 173–180.

Griffin, S., & Case, R. (1997). Re-thinkingthe primary school math curriculum: Anapproach based on cognitive science. Is-sues in Education, 3(1), 1–49.

Griffin, S. A., Case, R., & Siegler, R. S. (1994).Rightstart: Providing the central concep-tual prerequisites for first formal learningof arithmetic to students at risk for schoolfailure. In K. McGilly (Ed.), Classroomlessons: Integrating cognitive theory andclassroom practice (pp. 24–49). Cambridge,MA: MIT Press.

Hanich, L., Jordan, N., Kaplan, D., & Dick,J. (2001). Performance across differentareas of mathematical cognition in chil-dren with learning disabilities. Journal ofEducational Psychology, 93, 615–626.

Harcourt Educational Measurement. (2001).Stanford achievement test (9th ed.). San An-tonio, TX: Author.

Hasselbring, T. S., Goin, L. I., & Bransford,J. D. (1988). Developing math automatic-ity in learning handicapped children: Therole of computerized drill and practice.Focus on Exceptional Children, 20(6), 1–7.

Jordan, N., & Hanich, L. (2003). Character-istics of children with moderate mathe-matics deficiencies: A longitudinal per-spective. Learning Disabilities Research &Practice, 18, 213–221.

Jordan, N., Hanich, L., & Kaplan, D. (2003).A longitudinal study of mathematicalcompetencies in children with specificmathematics difficulties versus children




with comorbid mathematics and readingdifficulties. Child Development, 74, 834–850.

Jordan, N., Kaplan, D., & Hanich, L. (2002).Achievement growth in children withlearning difficulties in mathematics: Find-ings of a two-year longitudinal study.Journal of Educational Psychology, 94, 586–597.

Jordan, N., Levine, S., & Huttenlocher, J.(1994). Development of calculation abili-ties in middle and low income childrenafter formal instruction in school. Journalof Applied Developmental Psychology, 15,223–240.

Jordan, N., & Montani, T. (1997). Cognitivearithmetic and problem solving: A com-parison of children with specific and gen-eral mathematics difficulties. Journal ofLearning Disabilities, 30, 624–634.

Juel, C. (1988). Learning to read and write:A longitudinal study of 54 children fromfirst through fourth grades. Journal of Ed-ucational Psychology, 80, 437–447.

Kalchman, M., Moss, J., & Case, R. (2001).Psychological models for the develop-ment of mathematical understanding:Rational numbers and functions. In S.Carver & D. Klahr (Eds.), Cognition andinstruction (pp. 1–38). Mahwah, NJ: Erl-baum.

LeFevre, J., Bisanz, J., Daley, K., Buffone, L.,Greenham, S., & Sadesky, G. (1996). Mul-tiple routes to solution of single-digitmultiplication problems. Journal of Exper-imental Psychology: General, 125, 284–306.

Liberman, I. Y., Shankweiler, D., & Liber-man, A. M. (1989). The alphabetic princi-ple and learning to read. In D. Shank-weiler & I. Y. Liberman (Eds.), Phonologyand reading disability: Solving the readingpuzzle (pp. 1–33). Ann Arbor: Universityof Michigan Press.

Milgram, J. (2004, March 7–10). Assessingstudents’ mathematics learning: Issues, costs,and benefits. Paper presented at the MSRI(Math Science Research Institute) work-shop, Berkley, CA.

National Research Council. (2001). Adding itup: Helping children learn mathematics. JKilpatrick, J. Swafford, & B. Findell(Eds.), Mathematics Learning StudyCommittee, Center for Education, Divi-sion of Behavioral and Social Sciencesand Education. Washington, DC: Na-tional Academy Press.

O’Connor, R. E., Notari-Syverson, A., &Vadasy, P. F. (1996). Ladders to literacy:The effects of teacher-led phonologicalactivities for kindergarten children withand without learning disabilities. Excep-tional Children, 63, 117–130.

Okamoto, Y., & Case, R. (1996). Exploringthe microstructure of children’s centralconceptual structures in the domain ofnumber. Monographs of the Society for Re-search in Child Development, 61, 27–59.

Ostad, S. (1998). Developmental differencesin solving simple arithmetic word prob-lems and simple number-fact problems:A comparison of mathematically dis-abled children. Mathematical Cognition,4(1), 1–19.

Pellegrino, J. W., & Goldman, S. R. (1987).Information processing and elementarymathematics. Journal of Learning Disabili-ties, 20, 23–32.

Rittle-Johnson, B., Siegler, R., & Alibali, M.(2001). Developing conceptual under-standing and procedural skill in mathe-matics: An iterative process. Journal of Ed-ucational Psychology, 93, 346–362.

Robinson, C., Menchetti, B., & Torgesen, J.(2002). Toward a two-factor theory of onetype of mathematics disabilities. LearningDisabilities Research & Practice, 17, 81–89.

Schatschneider, C., Carlson, C. D., Francis,D. J., Foorman, B., & Fletcher, J. (2002).Relationships of rapid automatized nam-ing and phonological awareness in earlyreading development: Implications forthe double deficit hypothesis. Journal ofLearning Disabilities, 35, 245–256.

Schatschneider, C., Fletcher, J., Francis, D. J.,Carlson, C. D., & Foorman, B. (2004).Kindergarten prediction of reading skills:

A longitudinal comparative analysis. Jour-nal of Educational Psychology, 96, 265–282.

Shrager, J., & Siegler, R. S. (1998). SCADS: Amodel of children’s strategy choices andstrategy discoveries. Psychological Science,9, 405–422.

Siegel, L., & Ryan, E. (1988). Developmentof grammatical-sensitivity, phonological,and short-term memory skills in nor-mally achieving and learning disabledchildren. Developmental Psychology, 24, 28–37.

Siegler, R. (1988). Individual differences instrategy choices: Good students, not-so-good students, and perfectionists. ChildDevelopment, 59, 833–851.

Siegler, R. S., & Shrager, J. (1984). Strategychoice in addition and subtraction: Howdo children know what to do? In C.Sophian (Ed.), Origins of cognitive skills(pp. 229–293). Mahwah, NJ: Erlbaum.

Swanson, H. L., & Beebe-Frankenberger, M. E. (2004). The relationship betweenworking memory and mathematicalproblem solving in children at risk andnot at risk for serious math difficulties.Journal of Educational Psychology, 96, 471–491.

Vellutino, F., Scanlon, D. M., Sipay, E. R.,Small, S. G., Pratt, A., Chen, R., et al.(1996). Cognitive profiles of difficult-to-remediate and readily remediated poorreaders: Intervention as a vehicle fordistinguishing between cognitive and ex-perimental deficits as basic cause of spe-cific reading disability. Journal of Educa-tional Psychology, 88, 601–638.

Woodcock, R., & Johnson, M. (1990). Wood-cock-Johnson psycho-educational battery:Test of achievement. Chicago: Riverside.

Woodcock, R., & Mather, N. (1989). Wood-cock-Johnson Tests of Achievement Standardand Supplemental Batteries: Examiner’s man-ual. Allen, TX: DLM.

Woodward, J., & Montague, M. (2002).Meeting the challenge of mathematics re-form for students with learning disabili-ties. The Journal of Special Education, 36,89–101.



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