mathematics intervention for first- and second-grade students with mathematics difficulties: the...

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2008 29: 20Remedial and Special EducationDiane Pedrotty Bryant, Brian R. Bryant, Russell Gersten, Nancy Scammacca and Melissa M. Chavez

Effects of Tier 2 Intervention Delivered as Booster LessonsMathematics Intervention for First- and Second-Grade Students With Mathematics Difficulties: The

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Mathematics Intervention for First- and Second-Grade Students With Mathematics DifficultiesThe Effects of Tier 2 Intervention Delivered as Booster Lessons

Diane Pedrotty BryantBrian R. BryantUniversity of Texas at Austin

Russell GerstenInstructional Research Group, Long Beach, California

Nancy ScammaccaUniversity of Texas at Austin

Melissa M. ChavezUniversity of Texas Elementary School, Austin

This study sought to examine the effects of Tier 2 intervention in a multitiered model on the performance of first- and second-grade students who were identified as having mathematics difficulties. A regression discontinuity design was utilized.Participants included 126 (Tier 2, n = 26) first graders and 140 (Tier 2, n = 25) second graders. Tier 2 students received 15-min intervention booster lessons for 18 weeks in early mathematics skills and concepts. Results showed a significant inter-vention effect for second-grade Tier 2 students on the Texas Early Mathematics Inventories–Progress Monitoring (TEMI-PM)total standard score. The effect was not significant for first-grade Tier 2 students.

Keywords: Tier 2 intervention; response-to-intervention; mathematics intervention; mathematics difficulties

Preventing learning problems through the identifica-tion of students who are at risk for academic diffi-

culties and providing evidence-based multitieredintervention at an early age are widely endorsed in read-ing and are gaining interest in mathematics (Chard et al.,2005; Fuchs et al., 2005; Fuchs & Fuchs, 2001; Gersten,Jordan, & Flojo, 2005; Vaughn & Fuchs, 2003). Briefly,Tier 1 consists of evidence-based core instruction for allstudents. Tier 2 includes supplemental intervention andongoing progress monitoring for identified strugglingstudents. Instructional delivery in Tier 2 is characterizedby flexible groupings, informed instructional decisionmaking, and evidence-based interventions. Tier 3 isreserved for those students who are struggling to theextent that they require intensive intervention, whichmight include additional instructional time, smallinstructional grouping, adapted instructional content,and different materials. In the area of early reading, atiered-model approach to prevention and intervention

has shown educational benefits (e.g., Vaughn, Linan-Thompson, & Hickman-Davis, 2003). Generalizing fromearly reading to mathematics, we can surmise that with-out early identification, intervention, and progress moni-toring to determine students’ response to intervention,many young students with mathematics difficulties maynot develop a level of mathematics automaticity that isnecessary for becoming proficient in mathematics.

There has been an increasing interest in early mathemat-ics difficulties. We know that 5% to 10% of school-agechildren exhibit mathematics disabilities (Fuchs et al.,

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Authors’ Note: The work on this article was supported in part by GrantNo. 076600157110004 from the Texas Education Agency. Statementsdo not support the position or policy of this agency, and no officialendorsement should be inferred. The authors wish to thank BeckyNewman-Gonchar for her insightful feedback on an earlier draft of thisarticle. Address correspondence to Diane P. Bryant, University of Texasat Austin, College of Education SZB 408, 1 University Station D 4900,Austin, TX 78712-0365; e-mail: [email protected].

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Bryant et al. / Tier 2 Intervention as Booster Lessons 21

2005; Fuchs, Fuchs, & Hollenbeck, 2007; Gross-Tsur,Manor, & Shalev, 1996; Lewis, Hitch, & Walker, 1994;Ostad, 1998) and that mathematics disabilities (i.e., in cal-culations and problem solving) are a recognized type oflearning disability (Individuals with Disabilities EducationImprovement Act of 2004). The growing body of researchon young children’s mathematics cognition has greatly con-tributed to our understanding of the early numeracy skillsthat prove problematic for students at risk for mathematicsdisabilities. To help prevent mathematics disabilities, itstands to reason that the research efforts in early reading canbe replicated in early mathematics identification and inter-vention to prevent mathematics disabilities that result frominadequate instruction. However, there is a need for vali-dated interventions to address early mathematics difficul-ties. We can draw from the research on the nature ofmathematics difficulties to inform the development ofappropriate interventions.

An Overview of the Nature of Mathematics Difficulties

In examining the nature of cognitive development relatedto mathematics difficulties of young students, researchers(e.g., Geary, 1990, 1993; Griffin, Case, & Siegler, 1994;Jordan, Kaplan, Oláh, & Locuniak, 2006) have studied theelements of number sense, which is defined as “movingfrom the initial development of basic counting techniques tomore sophisticated understandings of the size of numbers,number relationships, patterns, operations, and place value”(National Council of Teachers of Mathematics [NCTM],2000, p. 79). Jordan et al. (2006) identified key elements ofnumber sense, which include counting (e.g., cardinality andstable order principles), number knowledge (e.g., quantitydiscrimination, counting sequences), number transforma-tion (e.g., addition and subtraction, verbal and nonverbalcalculations), and estimation (e.g., of group size). An under-standing of how young students at risk for mathematics dis-abilities perform on these elements compared to typicallyachieving students can inform mathematics intervention forprevention and remediation.

Research has been conducted to identify the predictivevariables of mathematics difficulties (Fuchs et al., 2005;Geary, 1990; Geary, Hamson, & Hoard, 2000; Jordan et al.,2006; Jordan, Hanich, & Kaplan, 2003). Overall, findingshave shown that cognitive development problems are man-ifested in difficulties with understanding number knowl-edge and relationships (e.g., magnitude, sequencing, base10), solving word problems, and using efficient countingand calculation strategies (e.g., counting on, doubles + 1) tosolve arithmetic combinations (i.e., number facts). Findings

from studies in these areas informed the design of the pre-ventative intervention practices described in this article,specifically, in the area of number knowledge and relation-ships and arithmetic combinations.

Number Knowledge and Relationships

Key early indicators are discerning magnitude ofnumbers, understanding the base-10 system, and comput-ing mental calculations (Gersten & Chard, 1999; Jordan et al., 2006). Jordan et al. (2006) found that at the end ofkindergarten, there existed a group of students whoseresponse to typical kindergarten instruction was identifiedas low and flat, suggesting that going into first grade, thesestudents would require an intervention to prevent furtherinstructional delay.

The base-10 system (i.e., place value, computation)arguably is one of the most important concepts studentsmust fully grasp (Van de Walle, 2004). Place-valueunderstanding helps students conceptualize numericalrelationships and the how and why of computational pro-cedures. Unfortunately, limited core mathematics instruc-tional time is devoted to teaching and practicing base-10concepts (Bryant, Smith, & Bryant, 2008). As a result,research results suggest that young students demonstrateproblems in the area of place-value tasks.

For example, Jordan et al. (2003) conducted a longitu-dinal study of 180 students in second grade and followedthem to third grade. They were administered a battery oftests designed to assess performance on a variety of earlymathematics tasks that included place value. Place-valuetasks involved problems with standard (e.g., 43 = four 10sand three 1s) and nonstandard (e.g., 43 = three 10s andthirteen 1s) place-value and digit representations (e.g., 43:Show with concrete models what 4 stands for; count out 40chips). Jordan et al. found that after time, students withmathematics difficulties scored lower on place-value taskscompared to average students. These findings suggest thatstudents with mathematics difficulties require sustainedinstructional time in place-value concepts in the earlygrades.

Arithmetic Combinations

Researchers (e.g., Geary, 1990, 2004; Jordan et al., 2003)have found that persistent deficits in the retrieval of arith-metic combinations among elementary-age students areassociated with mathematics difficulties. These students typ-ically display difficulties in mastering arithmetic combina-tions because of immature counting strategies (e.g., countingall, counting on fingers), which contribute to difficulties indeveloping computational fluency. It appears, then, that dif-ficulties with arithmetic combinations are a defining feature



of students with mathematics difficulties (Gersten et al.,2005; Hanich, Jordan, Kaplan, & Dick, 2001; Jordan,Kaplan, & Hanich, 2002) and are an important contributor tostudents’ ability to solve whole-number computation andword problems (Fuchs et al., 2005). If we surmise that theuse of mature and efficient counting strategies contributes tothe development of computational fluency, then we canhypothesize that teaching effective strategies to enhancemastery and fluency of arithmetic combinations should bepart of an intervention for students at risk for mathematicsdisabilities.

In sum, overall findings from research on predictors ofmathematics difficulties of young students are beginning tocontribute to the development of early mathematics inter-vention. We know that many students demonstrate prob-lems with number-sense tasks, arithmetic combinations,and word problems (Fuchs et al., 2005; Fuchs & Fuchs,2005). Overall research findings have shown that generally,students with mathematics difficulties demonstrate devel-opmentally different characteristics that remain persistentacross the grades (Geary, 1993; Jordan et al., 2002).Continued longitudinal studies on the predictive nature ofnumber-sense tasks and other mathematics skills and con-cepts (e.g., algebra, measurement, geometry) on the math-ematics achievement of at-risk students may offerimplications for early intervention development.

Early Mathematics Intervention

Compared to early reading intervention research,there is a paucity of research on early (first- and second-grade) Tier 2 intervention to prevent mathematics dis-abilities in struggling students and to address thedifficulties young students are experiencing in mathe-matics instruction (Gersten et al., 2005). The majority ofmathematics studies that are available focuses onresearch for older students with math disabilities or lowachievement (e.g., Fuchs et al., 2003).

A few studies were identified that conducted whole-class instruction, typically Tier 1 instruction withyounger students. For example, whole-class interventionprograms, such as Number Worlds (Griffin et al., 1994)and Peer Assisted Learning Strategies (Fuchs, Fuchs, &Karns, 2001; Fuchs, Fuchs, Yazdian, & Powell, 2002),provide effective practices for teaching numeracy skillsto low-achieving students.

Additionally, Fuchs et al. (2006) conducted a pilotstudy on instruction within the context of the generaleducation classroom (student pairs or whole class in acomputer lab). The study focused on the effects of com-puter-assisted instruction (CAI) on the acquisition and

transfer of number combination (i.e., addition and subtrac-tion facts with sums or minuends to 18) of 16 first-gradestudents with concurrent risk for mathematics and readingdifficulties. CAI was employed in 10-min sessions for atotal of 50 sessions (average 46.44, SD = 2.76, sessionsfor mathematics) across 18 weeks. Findings showed thatstudents benefited from CAI when learning addition com-binations but that similar benefits were not demonstratedfor subtraction combinations. Importantly, effects did nottransfer (generalize) to students’ accuracy in solving wordproblems. The authors recommended conducting addi-tional research with a larger sample, increasing the numberof weekly sessions, incorporating pictorial representationsinto the CAI program, and providing more follow-up tasksto facilitate transfer.

Although the number of studies is limited, researchfindings and intervention syntheses (e.g., Baker, Gersten,& Lee, 2002; Kroesbergen & Van Luit, 2003; Swanson,Hoskyn, & Lee, 1999) offer insight into practices andmaterials that hold promise for teaching young studentswho struggle with early mathematics. Researchers haveoffered recommendations for prevention and intervention,including peer-assisted tutors (Baker et al., 2002; Fuchset al., 2001), verbalizations of cognitive strategies (Fuchs& Fuchs, 2001), and physical (concrete) and visual (picto-rial) representations of number concepts (Fuchs et al.,2001; Gersten et al., 2005). Findings from studies onstudents with mathematics difficulties support the use ofexplicit, strategic instruction in teaching procedural andconceptual concepts (e.g., calculations principles, com-mutative property of addition, counting strategies; Bakeret al., 2002; Butler, Miller, Crehan, Babbit, & Pierce,2003; Gersten et al., 2005; Swanson et al., 1999).

Explicit and Strategic Instruction

Swanson et al. (1999), in their meta-analysis on aca-demic treatment outcome for students with learning dis-abilities, found that studies that used explicit andstrategic instructional procedures had larger effect sizescompared to other instructional approaches. Instructionalprocedures include sequencing instruction, providinginstructional routines (e.g., presentation of subject mat-ter, guided practice, and independent practice), focusingon massed practice, teaching to criterion, and evaluatingstudent learning on a regular basis (Swanson, 2001).Small-group instruction is also recommended as aneffective grouping practice to provide multiple opportu-nities for students to practice and receive immediate cor-rective or positive feedback from the teacher (Vaughn,Moody, & Schumm, 1998). Research findings on thebenefits of explicit, strategic instructional procedures are

22 Remedial and Special Education




well documented in the mathematics disability literature(Baker et al., 2002; Butler et al., 2003; Gersten et al.,2005; Swanson et al., 1999). It stands to reason thatthese same procedures can be employed with youngstudents who are identified as at risk for mathematics dis-abilities. Equally important is the instructional content.

Early Intervention Instructional Content

In thinking about mathematics skills and concepts totarget for early intervention, we must be mindful of find-ings from predictive studies about the mathematics char-acteristics of young, struggling students (e.g., difficultieswith number-sense tasks and arithmetic combinations).We based our intervention work on the early numeracyliterature (e.g., Jordan et al., 2003, 2006) that describesskills that are problematic for students with mathematicsdifficulties.

Clements and Sarama (2004) suggest that on the basisof the NCTM’s (2000) Principles and Standards forSchool Mathematics, “number and operations isarguably the most important of the area” (p. 16). Thus,we chose to focus our Tier 2 intervention on specificskills and concepts taken from the Number, Operation,and Quantitative Reasoning Skills and Concepts stan-dard for Grades K–2 (refer to www.nctm.org for adescription of the standards and to www.nctm.org/focal-points/downloads.asp for the NCTM curricula focalpoints). We chose Tier 2 as our focus rather than Tier 3,because the intent is to identify students who are demon-strating mathematics difficulties and to provide interven-tion to prevent future learning difficulties.

We know a great deal about effective instructionalprocedures for teaching struggling students, and there iscompelling evidence to focus on foundation skills inearly mathematics instruction at the primary level. Theemerging database about early mathematics interventionis promising but remains limited in scope compared tothe availability of research-validated early reading inter-vention. Thus, there is a need for studies to validate earlymathematics Tier 2 intervention for students at risk formathematics disabilities. The purpose of this study wasto determine the effects of Tier 2 intervention on thenumber, operation, and quantitative reasoning perfor-mance of students in first and second grades who wereidentified as having mathematics difficulties.

Method

Participants

This study took place in a primary-level elementaryschool in a major suburban school district in centralTexas. The primary elementary school serves students inGrade Pre-K through second grade. Participants in thisstudy included 126 students in first grade and 140students in second grade who had returned parent orguardian permission slips and participated in pre- andposttest assessments.

The school district provided demographic characteristicsof the sample with regard to key variables (see Table 1).Only students who spoke and understood English partic-ipated in this study, but the school-reported demograph-ics showed that only 8% of the student body (Grades

Table 1 Demographic Information for At-Risk and Not-At-Risk First- and

Second-Grade Participants (in percentages)

Number of Participants by Grade

Grade 1 Grade 2

At Risk Not At Risk At Risk Not At RiskVariable n = 26 n = 100 n = 25 n = 115

Gender Male 54 52 62 56Female 46 48 38 44

Race African American 17 26 37 27Hispanic 40 28 7 39White 34 34 44 25Asian/Pacific Islander 09 11 11 9

Qualified for free or reduced lunch Yes 37 33 37 42No 63 67 63 58




K–2) were English learners (individual data on this char-acteristic were not available from the district).

Design

The effects of early intervention were assessed usingthe regression-discontinuity design (RD). This quasi-experimental design is a particularly strong alternative toa randomized experiment when the goal is to evaluate theefficacy of a program (Shadish, Cook, & Campbell,2002; Trochim, 1984). It is appropriate to use RD whenthe group receiving intervention and the control groupare purposely selected to differ in preintervention abilityas assessed by a quantitative measure prior to the intro-duction of the intervention. The key to a successful RDis that the researchers enforce the cutoff criterion strictly.When this is done, RD provides a robust alternative to a ran-domized experimental design, with the added benefit of nothaving to construct a control group by denying the inter-vention to those who need it. The likelihood of extraneousvariables causing any observed effect is miniscule if a strictquantitative criterion is used to assign students to interven-tion and control conditions.

The underlying assumption behind RD is that if therewere no treatment effect, the relationship between thepretest criterion score (used to select students for interven-tion) and the posttest outcome score would be the same forall students (those who did and those who did not qualifyfor intervention). In other words, the null hypothesis is thatthe regression line for pretest and posttest scores for thestudents who scored above the cutoff (and received nointervention) would accurately predict posttest scores forstudents who did receive the intervention (i.e., scoredbelow the cutoff). An effective intervention would signifi-cantly raise the scores of students who scored below thecutoff. Data analysis examines the degree to which theactual regression line for the students who received inter-vention differs from the expected line and determineswhether this is likely to be due to chance or a systematicincrease above projected scores. A statistically significantdiscontinuity between these two regression lines indicatesthat the intervention had a significant effect.

RD can be used to determine both main effects and inter-actions between pretest score and treatment. An interactioneffect indicates whether the intervention is particularlyeffective with a subgroup of the students who receivedintervention (typically, those who score highest or lowest onthe pretest).

The students were administered the four subtests com-posing the Texas Early Mathematics Inventory–ProgressMonitoring (TEMI-PM), described in the Measures sec-tion. Although different methods and cut scores have

been used to identify mathematics difficulties (e.g.,Geary, 1990; Hanich et al., 2001; Parmar, Cawley, &Frazita, 1996), for the purposes of this study, studentswho scored at or below the 25th percentile (total stan-dard score of 90 or below) at the start of the school yearon the TEMI-PM were assigned to the Tier 2 treatmentgroup. Students who scored above 90 on the TEMI-PMat the start of the school year received no intervention butdid take the TEMI-PM posttest measure at the same timeas the intervention students.

Measures

During the academic year 2005–2006, all participat-ing students were administered a set of researcher-designed experimental mathematics measures, the TexasEarly Mathematics Inventories–Progress Monitoring, inthe fall (September), winter (January), and spring (lateApril–early May). The mathematics subtests from theStanford Achievement Test–10th Edition (SAT-10;Harcourt Assessment, 2003) were also administered inthe spring of 2006. First graders were administered thePrimary I level, consisting of Mathematics Procedures(MP) and Mathematics Problem Solving (MPS).Mathematics Procedures consists of addition and sub-traction items. Mathematics Problem Solving is com-posed of items that assess a variety of mathematics skills(e.g., numeration, operations, word problems, statistics,and probability). A Total Mathematics score (TMS) isalso available. For second grade, the Primary 2 is admin-istered; the same scores are available as for the PrimaryI level. The SAT-10 internal consistency reliability coef-ficients for our sample were computed using the coeffi-cient alpha technique (first grade MP = .85, MPS = .87,TMS = .92; second grade MP = .89, MPS = .88,TMS = .93). Concurrent validity of the TEMI-PMscores was assessed by correlations with the SAT-10and is reported by subtest.

The TEMI-PM consists of four forms (A, B, C, D);only Form A was administered in both fall and spring.There are four subtests: Magnitude Comparison (MC),Number Sequences (NS), Place Value (PV), andAddition/Subtraction Combinations (ASC). An aggre-gate total score of the four subtests was used to measurepre- and posttest student performance in the RD analysisbecause it is the most robust indicator of performance of thefour constructs (B. R. Bryant et al., 2007). Descriptions ofthe TEMI-PM subtests and the total score are providedbelow. Included at the end of each test description aredata concerning reliability and validity. These data pro-vide beginning evidence for the construct validity of theTEMI-PM scores reported in this study.




Magnitude Comparison. This subtest assesses a child’sability to differentiate the bigger or smaller of two numbersthat are shown side by side within a box. The measure issimilar to that used by Clarke and Shinn (2004) and Chardet al. (2005) in their Quantity Discrimination measure.

For first-grade students, MC is individually adminis-tered. Targeted numerals range from 0 through 99; studentsare told to name the smaller of the two numbers. If twonumbers are equal (e.g., 35, 35), they are told to say equal.

For second-grade students, the test is group adminis-tered. Targeted numerals range from 0 through 999, andstudents identify which of the two numbers is “less” by cir-cling their answer (i.e., the smaller number) on the stimu-lus sheet. If two numbers are of equal magnitude, studentsdraw a circle around both numbers. The use of the wordless is designed to match vocabulary that should have beenmastered by the end of the first grade. The students state orcircle their answer within each two-numeral set in left-to-right order for 1 min, and the amount of correctly identifiednumbers constitutes the raw score.

Immediate test–retest with alternate-forms reliabilitycoefficients for Form A with Forms B, C, and D rangedfrom .88 to .90 (median = .88) for first grade and from.80 to .87 (median = .86) for second grade. Correlatingthe spring Form A MC score with the Total Mathematicsscore from the SAT-10 yielded a concurrent validitycoefficient of .63 for first grade and .49 for second grade.

Number Sequences. This experimental measureassesses a child’s ability to identify a missing number froma sequence of three numbers. The missing number couldappear in any of three positions: the first number, the secondnumber, or the third number. This measure is adapted fromone Clark and Shinn (2004) used in their Missing NumberVaried 1-20 test.

For first-grade students, targeted numerals ranged from 0through 99; for second-grade students, targeted numeralsranged from 0 through 999. Two different numerals and oneblank (the missing number was represented by a blank)constitute a set and appear in boxed rows of a student stim-ulus worksheet, several boxed sets to a row. The first-gradechildren name the missing number during individualadministration; they have 1 min to do as many items as theycan, and the amount of correctly identified missing numbersis summed to constitute the raw score.

For second graders, NS is group administered; studentswrite the missing number in the blank that substitutes for themissing number in the series. The amount of correctly writ-ten missing numbers obtained in 1 min is summed tocompute the raw score. Immediate test–retest with alternate-forms reliability coefficients for Form A with Forms B, C,and D ranged from .88 to .90 (median = .89) for first grade

and from .74 to .88 (median = .78) for second grade. Springintercorrelations between Form A NS scores and SAT-10Total Mathematics score resulted in concurrent validity coef-ficients of .58 for first grade and .60 for second grade.

Place Value. This experimental test is designed toassess first and second graders’ knowledge of place value.The test uses a similar format to what is commonly seenin early mathematics textbooks (e.g., Charles et al., 1999:Scott Foresman—AddisonWesley; Bell et al., 2001:SRA/McGraw-Hill). For first graders, values range from 1to 99. For second graders, values range from 1 to 999.

Students in first grade are shown boxes of figuresdepicting 10s and 1s during individual administration.For example, the number 34 is depicted by showing threegroups of 10s and four 1s. The student provides an oralresponse stating how many there are in all. A 1-min timelimit is imposed.

Students in the second grade see boxes of figuresdepicting 100s, 10s, and 1s. For example, the number134 is depicted by showing one group of 100, threegroups of 10s, and four 1s. The student has to write thequantity shown (i.e., how many), in this case, 134.

Intercorrelation coefficients between Form A and FormsB, C, and D ranged from .81 to .85 (median = .83) for firstgraders and from .71 to .78 (median = .72) for secondgraders. Correlating Form A with the Total Mathematicsscore from the SAT-10 yielded a validity coefficient of .64for first grade and .63 for second grade.

Addition/Subtraction Combinations. This experimen-tal measure assesses first and second graders’ ability tocorrectly write the answers to addition and subtractionfacts (sums or minuends ranging from 0 to 18). Itemsappear eight to a row, with five rows of problems in all.Students in both grades compute and write their answersto as many items as they can in 1 min. The total numberof correctly computed problems written correctly (e.g.,with no reversals or gross illegibility) is summed to pro-duce the raw score. Immediate test–retest with alternate-forms reliability coefficients for Form A with Forms B,C, and D ranged from .72 to .93 (median = .83) for firstgraders and from .72 to .84 (median = .83) for studentsin second grade. We correlated the results of Form AASC with the Total Mathematics score from the SAT-10,which resulted in a validity coefficient of .55 for firstgrade and .59 for second grade.

Total score (TOT). Local normative scores (standardscores having a mean of 10 and standard deviation of 3)were computed at each grade level. Total scores werederived using the sum-of-standard-scores procedure,




wherein the standard scores for MC, NS, PV, and ASCwere summed and the results across students averaged togenerate a total standard score having a mean of 100 andstandard deviation of 15. Using the median reliabilitycoefficients for MC, NS, PV, and ASC, we usedGuilford’s (1954) formula to compute a reliability coef-ficient for the first-grade and second-grade composite.The resulting reliability coefficients for the Total scoresfor first and second grades, respectively, were .95 and.92. We also correlated the spring Form A TOT with theTotal Mathematics score of the SAT-10; the resultingconcurrent validity coefficient for first grade was .72 andfor second grade was .73.

Data Collection Procedures

For group testing, project staff administered the teststo intact classrooms of students who had returned signed,affirmative permission slips. A testing monitor was alsopresent during testing to help keep students on task, andclassroom teachers were asked to remain in the room forbehavior-management purposes. For individual testing,examiners were given a class list to share. For both gradelevels, timers were used to ensure accuracy in theamount of time allotted for testing.

Training.A 3-hr training session occurred in late summer2005, with refresher training conducted prior to each testadministration. A training session on all measures was con-ducted for the project testers, who were either undergradu-ate or graduate students in general education, specialeducation, or educational psychology. All students hadtaken a basic assessment course. During training, adminis-tration and scoring procedures for each of the experimentaltests and the SAT-10 were explained and modeled. Projectstaff served as test administrator and “student,” providingexamples of effective test administration. Following model-ing, questions were posed and answered; then the prospec-tive testers were provided scripted tests to practice givingthe tests to one another and recording item responses.Testers were observed and monitored while administeringand scoring the practice items. The process continued untilprospective testers responded to the testing situation with95% agreement and accuracy.

Examiner fidelity. To ensure fidelity of administrationand scoring in first grade, examiners were observed byone of three members of the project staff. Each projectstaff member observed a student being evaluated andscored the student’s performance on a second protocol.Before evaluating interscorer reliability between a testerand the primary tester, the rating consistency among the

project staff members was computed, and agreement wasfound to be better than 97%.

For first-grade testing, reliability checks occurred formore than 20% of each grade’s data on each test. Theraters achieved more than 97% agreement across all themeasures during each testing period (fall, winter, spring).The second raters checked all testers at least three times.The ratings were conducted across 4 days during eachtesting period. Interrater reliability was calculated bydividing the number of agreements by the total number ofstudent trials and multiplying the quotient by 100. Resultsfor the fall indicated a range of agreement from 96.2% to100% (median = 98.7%) for the fall testing, from 97.7%to 100% (median = 99.0%) for winter testing, and from98.6% to 100% (median = 99.5%) for spring testing.

Fidelity for second-grade testing and SAT-10 testing(all grades) was examined differently because all mea-sures were group administered. Trained examiners readaloud the group-administration instructions to thestudents while an additional trained examiner served asmonitor. In this role, the monitor was directed to stop theexaminer if any instructions were conveyed incorrectlyand to report any discrepancies during testing to the siteassessment coordinator. The monitors reported success-ful implementation of the test instructions.

Intervention

Tutoring sessions were delivered in same-ability, smallinstructional groups consisting of three to four studentswithin a grade level (first or second grade) from acrossclasses. Taking absences into consideration, a median of 64fifteen-min tutoring sessions for first graders and a medianof 62 fifteen-min tutoring sessions for second graders wereconducted across 18 weeks.

Four tutors were identified to conduct the Tier 2 inter-vention booster lessons. One full-time tutor and three grad-uate research assistants (GRAs) were trained by the firstauthor to implement the lessons. The full-time tutor was aformer kindergarten teacher with 17 years of classroomexperience. Two of the GRAs were full-time doctoralstudents in the Department of Education Psychology, andone was a full-time doctoral student in the School of SocialWork; they all tutored for the project 20 hr per week. TheGRAs had previous teaching or tutoring experience.

Tutor training. At the beginning of the program, initialtraining consisted of (a) an explanation of the program, (b)a description of the lessons, and (c) an explanation of pro-cedures for explicit, systematic instruction. Tutors weregiven time to practice the initial lessons. Once the programbegan, additional tutor training consisted of reviewing new



lessons and making adjustments to lessons based on tutors’feedback.

Tutoring program. The focus of this Tier 2 interventionprogram was on the development and implementation ofbooster lessons. The intent was to “boost” student learningin the area of number, operation, and quantitative reason-ing by providing systematic, explicit intervention in smallgroups during the school day. These lessons composedTier 2 intervention and were supplemental to core mathe-matics instruction, which ranged from 45 min to 60 min ofinstruction on the designated skill areas (e.g., measure-ment, problem solving) for the week. Content for thebooster lessons was based on the number, operation, andquantitative-reasoning skills and concepts from the TexasEssential Knowledge and Skills (TEKS) standards. Themathematics TEKS are rooted in the Principles andStandards for School Mathematics from the NationalCouncil of Teachers of Mathematics (2000). Table 2 showsthe conceptual framework used to guide the development ofbooster lessons.

The booster lessons were designed on the basis of prac-tices identified as critical for students who are at risk for orhave identified mathematics difficulties with the followingfeatures. First, a combined instructional approach employ-ing explicit, systematic teaching procedures and strategic

instruction was implemented. Procedures including model-ing, “thinking aloud,” guided practice, pacing, and errorcorrection were used to deliver the scripted booster lessons.During modeling, tutors demonstrated the processes orsteps needed to solve problems or provided explanations ofhow to perform skills. Strategic instruction focused onteaching students specific strategies for learning additionand subtraction combinations. Guided practice activitiesconsisted of multiple opportunities to practice skills andconcepts including reading, writing, and “making”numbers within a given number range (described below).

Second, the instructional content was controlled during a2-week instructional time period. Within an instructional 2-week time period, booster lessons were provided on skillsand concepts related to number concepts, base 10 and placevalue, and addition and subtraction combinations. Particu-larly, emphasis was placed on those number concepts thatprove problematic for students with mathematics difficul-ties (e.g., teen numbers, zero as a place holder). Also, withineach 2-week time period, a designated number range wastargeted that represented a smaller unit of the identifiednumber range for that grade level. For example, numbers 0through 99 were taught in first grade, so a unit of instruc-tional content for first grade for a 2-week period mightinclude numbers 10 through 20. For second grade, numbers0 through 999 composed the curriculum; the instructionalnumber range for a 2-week period might include numbers100 through 200. Instructional content for the 2-weekbooster lessons, then, focused on the designated numberrange across lessons that dealt with number concepts andrelationships (i.e., magnitude, number identification andwriting, numeric sequencing and counting), base 10, andplace value. Instructional content for addition and subtrac-tion combinations included sums or minuends rangingfrom 0 to 18 and sequenced from easier (e.g., addition facts+ 0, + 1, + 2; subtraction facts – n, – 0, – 1) to more diffi-cult (e.g., addition facts doubles + 1 [6 + 7], make 10 + more [9 + 7]; subtraction inverse facts) combinations.Addition and subtraction combinations were taughtseparately, with the easier combinations for both pre-ceding the more difficult ones.

Third, as appropriate, the concrete–semiconcrete–abstract(CSA) approach (Butler et al., 2003; Mercer, Jordan, &Miller, 1996) was used to teach number concept and rela-tionships, base 10 and place value, and addition and sub-traction combinations. We used a variety of instructionalmaterials with this approach. For example, at the concretelevel, base-10 models and counters were used to teachconcepts and skills. For the semiconcrete level, we usednumber lines and 100s charts. Finally, the abstract levelfocused on the manipulation of numerals. Also, the levelswere mixed so that students might manipulate base-10


Table 2Conceptual Framework for First- and Second-Grade Number, Operation, and Quantitative

Reasoning

Number concepts Counting: rote, rational, counting up or back, skip (2, 5, 10)Number recognition and writing: 0–99 (first grade); 0–999

(second grade) Comparing and grouping numbers: number relationships of

more, less; relationships of one and two more than or less than, anchoring numbers to 5 and 10 frames; part-part-whole relationships (e.g., ways to represent numbers)

Numeric sequencingBase 10 and place value

Making and counting: groups of 10s and 1s (first grade); groups of 100s, 10s, and 1s (second grade)

Using base-10 language (three 100s, zero 10s, six 1s) and standard language (306) to describe place value

Reading and writing numbers to represent base-10 modelsNaming the place value held by digits in numbers

Addition and subtraction combinationsIdentity element and propertiesFact familiesCounting and decomposition strategies (e.g., addition: count

on [+ 0, + 1, + 2], doubles, doubles +1, make 10 + more; subtraction: count down [–0, –1, –2, –3], count up; number bonds)




models to make numbers, read numbers, and writenumbers of concrete model representations.

Finally, daily activity-level progress monitoring was con-ducted. Students were given four either oral or written prob-lems to determine their response to instruction on eachbooster lesson taught that day. The majority of students inthe group had to demonstrate accuracy on three out of fourof the problems to consider the lessons as successful foreach day.

Fidelity of Implementation

An expert trainer (project coordinator) observed treat-ment sessions for each tutor for three sessions for the 18-week intervention to assess fidelity or quality ofimplementation of specific performance indicators.Quality performance indicators included (a) followingthe scripted lessons for the content, (b) implementing theinstructional procedures, and (c) managing studentbehavior and materials. Performance indicators wererated on a 0- to 2-point scale where 0 = indicator notimplemented, 1 = indicator implemented on a limitedbasis, and 2 = indicator implemented on a regular basis.Results were shared with the tutors, and areas in need offurther training were addressed. Results on the qualityperformance indicators showed (a) for following thescripted lessons for the content, a median of 1.83 with arange of 1.66 to 2.00; (b) for implementing the instruc-tional procedures, a median of 2.00 with a range of 1.86to 2.00; and (c) for managing student behavior, a medianof 2.0. These results across both grades show that acrossthe tutors overall, there was a high degree of fidelity inthe implementation of the booster lessons.

Results

First Grade

Complete pretest and posttest data were available for100 students who did not qualify for intervention and didnot receive intervention and for 26 students who qualifiedfor and received intervention. For the spring (posttest)TEMI-PM total standard score, RD analysis revealed thatno significant effect was observed among first-gradestudents (b = .04). Figure 1 shows a scatter plot of scoresand regression lines for Tier 1 and Tier 2 students. On theplot, the cutoff score has been adjusted to zero (by sub-tracting 90 from each student’s pretest score) to betterdepict the data. Note that at zero on the x-axis, there is littleto no discontinuity between the regression line for the Tier2 (at-risk) group and the Tier 1 (not-at-risk) group.

Second Grade

Complete pretest and posttest data were available for115 students who did not qualify for intervention and didnot receive intervention and for 25 students who quali-fied for and received intervention. The RD analysisshowed that a significant main effect (b = .19, p = .018)was observed for the spring TEMI-PM total standardscore, indicating a positive effect for the interventionwith second-grade students.

Figure 2 shows a scatter plot of scores and regressionlines for Tier 1 and Tier 2 students. At zero on the x-axis,there is a discontinuity (a gap) between the regressionline for the Tier 2 (at-risk) group and the Tier 1 (not-at-risk) group. This discontinuity demonstrates the positiveeffect of the program on Grade 2 at-risk students. If at-risk students had not received intervention, it is expectedthat their posttest scores would be equal to the values onthe regression line for the not-at-risk students at zero onthe x-axis and to the left of zero (because zero is the cut-off point for selection for intervention, all at-riskstudents have scores that fall at zero or below on the x-axis). Because the regression line for the actual

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posttest scores for at-risk students is above the regressionline for not-at-risk students at zero on the x-axis and tothe left of zero, we conclude that the posttest scores of at-risk students were significantly higher than expected,resulting in the finding of a main effect for the intervention.

Regression discontinuity analyses at the subtest levelindicated that a significant main effect existed for theAddition/Subtraction subtest (b = .21, p < .05), whereasresults for the remaining three subtests showed no sig-nificant effects.

Our finding is similar to other studies (e.g., Fuchs et al., 2005) showing that although Tier 2 intervention(tutoring) can improve mathematics achievement, espe-cially for second graders in our sample, students’ perfor-mance still remains below that of their typicallyachieving peers. Thus, identifying more powerful inter-ventions is necessary to help close the performance gapbetween typically performing and Tier 2 students.

Discussion

Increasing attention is being focused on the identificationof young students with mathematics difficulties. Research to

validate early mathematics Tier 2 interventions is important,because multitiered instruction is based on the premise thatevidence-based interventions are being provided to preventmathematics disabilities. We know a great deal about whatconstitutes effective instructional procedures (e.g., model-ing, strategic instruction) that produce positive academicoutcomes for struggling students (Swanson et al., 1999).These instructional procedures should be incorporated intomathematics intervention. Furthermore, a case can be madethat Tier 2 intervention content should focus on mathematicsskills such as number, operation, and quantitative reasoningbecause they are predictive of mathematics problems(Jordan et al., 2002, 2003). Thus, the purpose of this studywas to determine the effects of Tier 2 intervention on thenumber, operation, and quantitative-reasoning performanceof students in first and second grades who were identified ashaving mathematics difficulties. The TEMI-PM total score,which comprised four subtests, was used to identify (per-forming at or below the 25th percentile; TEMI-PM totalstandard score of 90 or below) students who were strugglingin critical early mathematics skills and to monitor theirprogress during Tier 2 intervention.

A total of 26 first-grade and 25 second-grade studentsparticipated in the Tier 2 intervention delivered as boosterlessons. Fifteen-min intervention sessions were conducted3 to 4 days per week for 18 weeks. Students receivedinstruction in small groups by trained tutors. The interven-tion consisted of explicit, strategic instructional proce-dures; instructional content (e.g., numbers 0–99 for firstgrade) for the designated grade level (first or second); andbooster lessons in number concepts, base 10 and placevalue, and addition and subtraction combinations withsums or minuends to 18, respectively.

Findings showed differential effects by grade level.For first-grade Tier 2 students, the regression discontinu-ity analysis did not reveal a program effect, whereas forsecond-grade Tier 2 students, RD analyses showed a sig-nificant main effect, indicating a positive program effect.In attempting to explain the differential effects, it is pos-sible that first graders need more intervention time tolearn number-sense tasks. In particular, the teen numbersand 11 and 12 are often troublesome for strugglingstudents to name. Numerical concepts (e.g., quantity dis-crimination, number sequences) and place value alsorequire sustained instructional time that may not havebeen sufficient in our 15-min sessions. Also, arithmeticcombinations (e.g., fact families, sums or minuends to18), especially subtraction, take time to develop accuracyas well as fluency (Fuchs et al., 2006).

On the other hand, findings from second grade areencouraging regarding Tier 2 students’ ability to improvetheir performance with number-sense tasks, place value, and


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arithmetic combinations. Lessons specifically on number-sense tasks (e.g., number concepts) and fluency buildingwith arithmetic combinations apparently provided studentsthe added “boost” they needed to become more proficient inthese areas. However, refinement of the booster lessons isneeded to help close the achievement gap. Thus, future stud-ies should examine additional tutoring features to help Tier2 students achieve more commensurate with their peers.

The limitations of this study should be addressed infuture research. Methodologically, a larger sample size isneeded, and student performance on individual subtestsshould be examined. Also, instructional content shouldencompass word problem solving, given the issuesstudents with mathematics difficulties manifest in this area(Fuchs & Fuchs, 2005; Jordan et al., 2003).

Future studies are warranted to strengthen the effectsof the intervention for both first- and second-grade Tier2 intervention. First, additional intervention time may beneeded to allow students more practice opportunitieswith different representations (e.g., pictorial, abstract)and to provide more fluency-building time, especiallywith arithmetic combinations. This proposed change isin line with findings from other research. For example,Fuchs et al. (2006) recommend increasing the numberof weekly intervention sessions, which was described intheir original study as 18 weeks of CAI intervention. Wewould suggest increasing the number of sessions and theamount of daily intervention time to 20 min. Given thatmultiple tasks are taught, increasing daily time from 15-to 20-min sessions could provide more meaningful prac-tice time that Tier 2 students require but oftentimes donot receive as part of Tier 1 teaching. Second, instructionin word problem solving is imperative. In our study, wechose to limit the number of skills because of the need todevelop multiple booster lessons. However, instructionin the components of word problem solving (e.g., typesof problems, extraneous information, multiple steps,contextualized problems) and near and far transfer,which is difficult to effect among struggling students, isespecially important for struggling students (D. P.Bryant, Bryant, & Hammill, 2000; Fuchs & Fuchs, 2005;Woodward & Baxter, 1997). Finally, future researchmust examine the identification of Tier 3 students andspecific, individualized interventions to address theirlearning needs. In reviewing the RD data in Figures 1and 2, clearly there is a group of students whose perfor-mance remained low compared to other Tier 2 students.

In sum, Tier 2 intervention for students with mathematicsdifficulties holds promise for improving mathematics per-formance in number-sense tasks and arithmetic combina-tions. Practitioners can implement small-group interventionfocusing on number, operation, and quantitative reasoning

tasks with some assurance that a group of their strugglingstudents will benefit from this instruction. Finally, instruc-tion that includes explicit, strategic procedures along withmaterials that engage students in representing numericalconcepts, place value, and arithmetic combinations shouldbe included as part of Tier 2 intervention.

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Diane Pedrotty Bryant, PhD, is a professor of special education inthe Learning Disabilities/Behavior Disorders Program in theDepartment of Special Education in the College of Education at theUniversity of Texas (UT) at Austin and a fellow in the Mollie VilleretDavis Professorship in Learning Disabilities. She also serves as a pro-ject director for the development of an early mathematics model inthe Vaughn Gross Center for Reading and Language Arts at UT. Hermajor research interests include studying effective practices for earlymathematics intervention and reading instruction for students withlearning disabilities.

Brian R. Bryant, PhD, lives and works in Austin, Texas. He is alecturer in the Department of Special Education in the College of




Education at the University of Texas (UT) at Austin. He also servesas the project coordinator for the early mathematics assessment andresponse-to-intervention model in the Vaughn Gross Center at UT.His research interests include investigating the development andvalidation of early mathematics measures and interventions and theidentification of learning disabilities.

Russell Gersten, PhD, is a director of the Instructional Research Group,a nonprofit educational research institute in Long Beach, California. Heis also a professor emeritus in the College of Education at the Universityof Oregon. His major research interests are the study of the process ofeffectively translating research into classroom practice, instructional

research on English-language learners, reading comprehension, andteaching history to students receiving special education services.

Nancy Scammacca, PhD, is a research associate at the Vaughn GrossCenter for Reading and Language Arts at the University of Texas atAustin. Her interests include advanced statistical applications, programevaluation, and meta-analytic research.

Melissa M. Chavez, MEd, is the assistant principal and director ofspecial education at the University of Texas Elementary School inAustin. Her research interests include studying effective interventionsfor students with learning difficulties.



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Errata

Bryant, D. P., Bryant, B. R., Gersten, R., Scammacca, N., & Chavez, M. M. (2008). Mathematics intervention for first- and second-grade students with mathematics difficulties:The effects of tier 2 intervention delivered as booster lessons. Remedial and Special Education29(1), 20-32. (Original DOI: 10.1177/0741932507309712).

I n this article, a line was inadvertently dropped from “Figure 2: Scatter Plot of Grade 2 Scores With RegressionLines” (p. 29). Below is how the figure should have appeared.

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