Assisting Students Struggling With Mathematics Response to Intervention RtI for Elementary and Middle Schools

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    IES PRACTICE GUIDE

    NCEE 2009-4060U.S. DEPARTMENT OF EDUCATION

    WHAT WORKS CLEARINGHOUSE

    Assisting Students Struggling with

    Mathematics: Response to Intervention

    (RtI) for Elementary and Middle Schools

    Assisting Students Struggling with

    Mathematics: Response to Intervention

    (RtI) for Elementary and Middle Schools

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    The Institute o Education Sciences (IES) publishes practice guides in educationto bring the best available evidence and expertise to bear on the types o systemicchallenges that cannot currently be addressed by single interventions or programs.Authors o practice guides seldom conduct the types o systematic literature searchesthat are the backbone o a meta-analysis, although they take advantage o such workwhen it is already published. Instead, authors use their expertise to identiy the

    most important research with respect to their recommendations, augmented by asearch o recent publications to ensure that research citations are up-to-date.

    Unique to IES-sponsored practice guides is that they are subjected to rigorous exter-nal peer review through the same oce that is responsible or independent reviewo other IES publications. A critical task or peer reviewers o a practice guide is todetermine whether the evidence cited in support o particular recommendations isup-to-date and that studies o similar or better quality that point in a dierent di-rection have not been ignored. Because practice guides depend on the expertise otheir authors and their group decisionmaking, the content o a practice guide is notand should not be viewed as a set o recommendations that in every case depends

    on and ows inevitably rom scientic research.

    The goal o this practice guide is to ormulate specic and coherent evidence-basedrecommendations or use by educators addressing the challenge o reducing thenumber o children who struggle with mathematics by using response to interven-tion (RtI) as a means o both identiying students who need more help and provid-ing these students with high-quality interventions. The guide provides practical,clear inormation on critical topics related to RtI and is based on the best availableevidence as judged by the panel. Recommendations in this guide should not beconstrued to imply that no urther research is warranted on the eectiveness oparticular strategies used in RtI or students struggling with mathematics.

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    Assisting Students Struggling

    with Mathematics: Response to

    Intervention (RtI) for Elementary

    and Middle Schools

    April 2009

    Panel

    Russell Gersten (Chair)

    InstructIonal research Group

    Sybilla Beckmann

    unIversItyof GeorGIa

    Benjamin Clarke

    InstructIonal research Group

    Anne Foegen

    Iowa state unIversIty

    Laurel Marsh

    howard county publIc school system

    Jon R. Star

    harvard unIversIty

    Bradley Witzel

    wInthrop unIversIty

    Staf

    Joseph DiminoMadhavi Jayanthi

    Rebecca Newman-Gonchar

    InstructIonal research Group

    Shannon Monahan

    Libby Scott

    mathematIca polIcy research

    NCEE 2009-4060

    U.S. DEPARTMENT OF EDUCATION

    IES PRACTICE GUIDE

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    This report was prepared or the National Center or Education Evaluation and RegionalAssistance, Institute o Education Sciences under Contract ED-07-CO-0062 by the WhatWorks Clearinghouse, which is operated by Mathematica Policy Research, Inc.

    Disclaimer

    The opinions and positions expressed in this practice guide are the authors and donot necessarily represent the opinions and positions o the Institute o Education Sci-ences or the U.S. Department o Education. This practice guide should be reviewedand applied according to the specic needs o the educators and education agencyusing it, and with ull realization that it represents the judgments o the reviewpanel regarding what constitutes sensible practice, based on the research availableat the time o publication. This practice guide should be used as a tool to assist indecisionmaking rather than as a cookbook. Any reerences within the documentto specic educational products are illustrative and do not imply endorsement othese products to the exclusion o other products that are not reerenced.

    U.S. Department o EducationArne DuncanSecretary

    Institute o Education SciencesSue BetkaActing Director

    National Center or Education Evaluation and Regional AssistancePhoebe CottinghamCommissioner

    April 2009

    This report is in the public domain. Although permission to reprint this publicationis not necessary, the citation should be:

    Gersten, R., Beckmann, S., Clarke, B., Foegen, A., Marsh, L., Star, J. R., & Witzel,B. (2009). Assisting students struggling with mathematics: Response to Interven-tion (RtI) for elementary and middle schools(NCEE 2009-4060). Washington, DC:National Center for Education Evaluation and Regional Assistance, Institute ofEducation Sciences, U.S. Department of Education. Retrieved from http://ies.

    ed.gov/ncee/wwc/publications/practiceguides/.

    This report is available on the IES website at http://ies.ed.gov/ncee and http://ies.ed.gov/ncee/wwc/publications/practiceguides/.

    Alternative ormatsOn request, this publication can be made available in alternative ormats, such asBraille, large print, audiotape, or computer diskette. For more inormation, call theAlternative Format Center at 2022058113.

    http://ies.ed.gov/ncee/wwc/publications/practiceguides/http://ies.ed.gov/ncee/wwc/publications/practiceguides/http://ies.ed.gov/nceehttp://ies.ed.gov/ncee/wwc/publications/practiceguides/http://ies.ed.gov/ncee/wwc/publications/practiceguides/http://ies.ed.gov/ncee/wwc/publications/practiceguides/http://ies.ed.gov/ncee/wwc/publications/practiceguides/http://ies.ed.gov/nceehttp://ies.ed.gov/ncee/wwc/publications/practiceguides/http://ies.ed.gov/ncee/wwc/publications/practiceguides/
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    ( iii )

    Assisting Students Struggling with Mathematics:Response to Intervention (RtI) orElementary and Middle Schools

    ContentsItrodctio 1

    Te Wat Wors Cearigose stadards ad teir reevace to tis gide 3

    Overview 4

    Smmar o te Recommedatios 5

    Scope o te practice gide 9

    Cecist or carrig ot te recommedatios 11

    Recommedatio 1. Scree a stdets to ideti tose at ris

    or potetia matematics dicties ad provide itervetiosto stdets idetied as at ris. 13

    Recommedatio 2. Istrctioa materias or stdets receivigitervetios sod ocs itese o i-dept treatmet o woembers i idergarte trog grade 5 ad o ratioa mbers igrades 4 trog 8. Tese materias sod be seected b committee. 18

    Recommedatio 3. Istrctio drig te itervetio sod be expicitad sstematic. Tis icdes providig modes o prociet probem sovig,verbaizatio o togt processes, gided practice, corrective eedbac,ad reqet cmative review. 21

    Recommedatio 4. Itervetios sod icde istrctio o sovigword probems tat is based o commo derig strctres. 26

    Recommedatio 5. Itervetio materias sod icde opportitiesor stdets to wor wit visa represetatios o matematica ideas aditervetioists sod be prociet i te se o visa represetatios omatematica ideas. 30

    Recommedatio 6. Itervetios at a grade eves sod devote abot10 mites i eac sessio to bidig fet retrieva o basic aritmetic acts. 37

    Recommedatio 7. Moitor te progress o stdets receivig sppemetaistrctio ad oter stdets wo are at ris. 41

    Recommedatio 8. Icde motivatioa strategies i tier 2 ad tier 3itervetios. 44

    Gossar o terms as sed i tis report 48

    Appedix A. Postscript rom te Istitte o Edcatio Scieces 52

    Appedix B. Abot te ators 55

    Appedix C. Discosre o potetia coficts o iterest 59

    Appedix D. Tecica iormatio o te stdies 61

    Reereces 91

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    ( i )

    ASSISTInG STuDEnTS STRuGGlInG WITh MAThEMATICS: RESPOnSE TO InTERvEnTIOn (RTI)OR ElEMEnTARy AnD MIDDlE SChOOlS

    List o tables

    Tabe 1. Istitte o Edcatio Scieces eves o evidece or practice gides 2

    Tabe 2. Recommedatios ad correspodig eves o evidece 6

    Tabe 3. Sesitivit ad specicit 16

    Tabe D1. Stdies o itervetios tat icded expicit istrctioad met WWC Stadards (wit ad witot reservatios) 69

    Tabe D2. Stdies o itervetios tat tagt stdets to discrimiateprobem tpes tat met WWC stadards (wit or witot reservatios) 73

    Tabe D3. Stdies o itervetios tat sed visa represetatiostat met WWC stadards (wit ad witot reservatios) 7778

    Tabe D4. Stdies o itervetios tat icded act fec practicestat met WWC stadards (wit ad witot reservatios) 83

    List o examples

    Exampe 1. Cage probems 27

    Exampe 2. Compare probems 28

    Exampe 3. Sovig dieret probems wit te same strateg 29

    Exampe 4. Represetatio o te cotig o strateg sig a mber ie 33

    Exampe 5. usig visa represetatios or mtidigit additio 34

    Exampe 6. Strip diagrams ca ep stdets mae sese o ractios 34

    Exampe 7. Maipatives ca ep stdets derstad tat or mtipiedb six meas or grops o six, wic meas 24 tota objects 35

    Exampe 8. A set o matced cocrete, visa, ad abstract represetatiosto teac sovig sige-variabe eqatios 35

    Exampe 9: Commtative propert o mtipicatio 48

    Exampe 10: Mae-a-10 strateg 49

    Exampe 11: Distribtive propert 50

    Exampe 12: nmber decompositio 51

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    ( 1 )

    Introduction

    Students struggling with mathematics may

    benet rom early interventions aimed at

    improving their mathematics ability and

    ultimately preventing subsequent ailure.This guide provides eight specic recom-

    mendations intended to help teachers,

    principals, and school administrators use

    Response to Intervention (RtI) to identiy

    students who need assistance in mathe-

    matics and to address the needs o these

    students through ocused interventions.

    The guide provides suggestions on how

    to carry out each recommendation and

    explains how educators can overcome

    potential roadblocks to implementing the

    recommendations.

    The recommendations were developed by

    a panel o researchers and practitioners

    with expertise in various dimensions o

    this topic. The panel includes a research

    mathematician active in issues related

    to K8 mathematics education, two pro-

    essors o mathematics education, sev-

    eral special educators, and a mathematics

    coach currently providing proessional de-

    velopment in mathematics in schools. Thepanel members worked collaboratively to

    develop recommendations based on the

    best available research evidence and our

    expertise in mathematics, special educa-

    tion, research, and practice.

    The body o evidence we considered in de-

    veloping these recommendations included

    evaluations o mathematics interventions

    or low-perorming students and students

    with learning disabilities. The panel con-sidered high-quality experimental and

    quasi-experimental studies, such as those

    meeting the criteria o the What Works

    Clearinghouse (http://www.whatworks.

    ed.gov), to provide the strongest evidence

    o eectiveness. We also examined stud-

    ies o the technical adequacy o batteries

    o screening and progress monitoring

    measures or recommendations relating

    to assessment.

    In some cases, recommendations reect

    evidence-based practices that have been

    demonstrated as eective through rigor-

    ous research. In other cases, when such

    evidence is not available, the recommen-

    dations reect what this panel believes arebest practices. Throughout the guide, we

    clearly indicate the quality o the evidence

    that supports each recommendation.

    Each recommendation receives a rating

    based on the strength o the research evi-

    dence that has shown the eectiveness o a

    recommendation (table 1). These ratings

    strong, moderate, or lowhave been de-

    ned as ollows:

    Strongreers to consistent and generaliz-

    able evidence that an intervention pro-

    gram causes better outcomes.1

    Moderate reers either to evidence rom

    studies that allow strong causal conclu-

    sions but cannot be generalized with as-

    surance to the population on which a

    recommendation is ocused (perhaps be-

    cause the ndings have not been widely

    replicated)or to evidence rom stud-

    ies that are generalizable but have morecausal ambiguity than oered by experi-

    mental designs (such as statistical models

    o correlational data or group comparison

    designs or which the equivalence o the

    groups at pretest is uncertain).

    Lowreers to expert opinion based on rea-

    sonable extrapolations rom research and

    theory on other topics and evidence rom

    studies that do not meet the standards or

    moderate or strong evidence.

    1. Following WWC guidelines, we consider a posi-tive, statistically signicant eect or large eectsize (i.e., greater than 0.25) as an indicator opositive eects.

    http://www.whatworks.ed.gov/http://www.whatworks.ed.gov/http://www.whatworks.ed.gov/http://www.whatworks.ed.gov/
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    InTRODuCTIOn

    ( 2 )

    Table 1. Institute o Education Sciences levels o evidence or practice guides

    Strong

    In general, characterization o the evidence or a recommendation as strong requires both

    studies with high internal validity (i.e., studies whose designs can support causal conclusions)

    and studies with high external validity (i.e., studies that in total include enough o the range

    o participants and settings on which the recommendation is ocused to support the conclu-

    sion that the results can be generalized to those participants and settings). Strong evidenceor this practice guide is operationalized as:

    A sys tematic review o research that generally meets the standards o the What Works

    Clearinghouse (WWC) (see http://ies.ed.gov/ncee/wwc/) and supports the eectiveness o

    a program, practice, or approach with no contradictory evidence o similar quality; OR

    Several well-designed, randomized controlled trials or well-designed quasi-experiments

    that generally meet the standards o WWC and support the eectiveness o a program,

    practice, or approach, with no contradictory evidence o similar quality; OR

    One large, well-designed, randomized controlled, multisite trial that meets WWC standards

    and supports the eectiveness o a program, practice, or approach, with no contradictory

    evidence o similar quality; OR

    For assessments, evidence o reliability and validity that meets the Standards or Educa-

    tional and Psychological Testing.a

    Moderate

    In general, characterization o the evidence or a recommendation as moderate requires stud-

    ies with high internal validity but moderate external validity, or studies with high external

    validity but moderate internal validity. In other words, moderate evidence is derived rom

    studies that support strong causal conclusions but when generalization is uncertain, or stud-

    ies that support the generality o a relationship but when the causality is uncertain. Moderate

    evidence or this practice guide is operationalized as:

    Experiments or quasi-experiments generally meeting the standards o WWC and sup-

    porting the eectiveness o a program, practice, or approach with small sample sizes

    and/or other conditions o implementation or analysis that limit generalizability and

    no contrary evidence; OR

    Comparison group studies that do not demonstrate equivalence o groups at pre-

    test and thereore do not meet the standards o WWC but that (a) consistently showenhanced outcomes or participants experiencing a particular program, practice, or

    approach and (b) have no major aws related to internal validity other than lack o

    demonstrated equivalence at pretest (e.g., only one teacher or one class per condition,

    unequal amounts o instructional time, highly biased outcome measures); OR

    Correlational research with strong statistical controls or selection bias and or dis-

    cerning inuence o endogenous actors and no contrary evidence; OR

    For assessments, evidence o reliability that meets the Standards or Educational and

    Psychological Testingb but with evidence o validity rom samples not adequately rep-

    resentative o the population on which the recommendation is ocused.

    Low

    In general, characterization o the evidence or a recommendation as low means that the

    recommendation is based on expert opinion derived rom strong ndings or theories in

    related areas and/or expert opinion buttressed by direct evidence that does not rise tothe moderate or strong levels. Low evidence is operationalized as evidence not meeting

    the standards or the moderate or high levels.

    a. American Educational Research Association, American Psychological Association, and National Council on

    Measurement in Education (1999).

    b. Ibid.

    http://ies.ed.gov/ncee/wwc/http://ies.ed.gov/ncee/wwc/
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    InTRODuCTIOn

    ( 3 )

    The What Works Clearinghousestandards and their relevance tothis guide

    The panel relied on WWC evidence stan-

    dards to assess the quality o evidencesupporting mathematics intervention pro-

    grams and practices. The WWC addresses

    evidence or the causal validity o instruc-

    tional programs and practices according to

    WWC standards. Inormation about these

    standards is available at http://ies.ed.gov/

    ncee/wwc/reerences/standards/. The

    technical quality o each study is rated and

    placed into one o three categories:

    Meets Evidence Standards or random-

    ized controlled trials and regression

    discontinuity studies that provide the

    strongest evidence o causal validity.

    Meets Evidence Standards with Reser-

    vationsor all quasi-experimental

    studies with no design aws and ran-

    domized controlled trials that have

    problems with randomization, attri-

    tion, or disruption.

    Does Not Meet Evidence Screens orstudies that do not provide strong evi-

    dence o causal validity.

    Following the recommendations and sug-

    gestions or carrying out the recommen-

    dations, Appendix D presents inormation

    on the research evidence to support the

    recommendations.

    The panel would like to thank Kelly Hay-

    mond or her contributions to the analysis,

    the WWC reviewers or their contribution

    to the project, and Jo Ellen Kerr and Jamila

    Henderson or their support o the intricate

    logistics o the project. We also would like

    to thank Scott Cody or his oversight o the

    overall progress o the practice guide.

    Dr. Russell Gersten

    Dr. Sybilla Beckmann

    Dr. Benjamin Clarke

    Dr. Anne Foegen

    Ms. Laurel Marsh

    Dr. Jon R. Star

    Dr. Bradley Witzel

    http://ies.ed.gov/ncee/wwc/references/standards/http://ies.ed.gov/ncee/wwc/references/standards/http://ies.ed.gov/ncee/wwc/references/standards/http://ies.ed.gov/ncee/wwc/references/standards/
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    ( 4 )

    Assisting StudentsStruggling withMathematics: Responseto Intervention (RtI)or Elementary andMiddle Schools

    Overview

    Response to Intervention (RtI) is an early de-

    tection, prevention, and support system that

    identies struggling students and assists

    them beore they all behind. In the 2004

    reauthorization o the Individuals with Dis-

    abilities Education Act (PL 108-446), states

    were encouraged to use RtI to accurately

    identiy students with learning disabilities

    and encouraged to provide additional sup-

    ports or students with academic dicul-

    ties regardless o disability classication.

    Although many states have already begun to

    implement RtI in the area o reading, RtI ini-

    tiatives or mathematics are relatively new.

    Students low achievement in mathemat-

    ics is a matter o national concern. The re-cent National Mathematics Advisory Panel

    Report released in 2008 summarized the

    poor showing o students in the United

    States on international comparisons o

    mathematics perormance such as the

    Trends in International Mathematics and

    Science Study (TIMSS) and the Program or

    International Student Assessment (PISA).2

    A recent survey o algebra teachers as-

    sociated with the report identiied key

    deciencies o students entering algebra,including aspects o whole number arith-

    metic, ractions, ratios, and proportions.3

    The National Mathematics Advisory Panel

    2. See, or example, National Mathematics Ad-visory Panel (2008) and Schmidt and Houang(2007). For more inormation on the TIMSS, seehttp://nces.ed.gov/timss/. For more inormationon PISA, see http://www.oecd.org.

    3. National Mathematics Advisory Panel (2008).

    concluded that all students should receive

    preparation rom an early age to ensure

    their later success in algebra. In particular,

    the report emphasized the need or math-

    ematics interventions that mitigate and

    prevent mathematics diculties.

    This panel believes that schools can use an

    RtI ramework to help struggling students

    prepare or later success in mathemat-

    ics. To date, little research has been con-

    ducted to identiy the most eective ways

    to initiate and implement RtI rameworks

    or mathematics. However, there is a rich

    body o research on eective mathematics

    interventions implemented outside an RtI

    ramework. Our goal in this practice guide

    is to provide suggestions or assessing

    students mathematics abilities and imple-

    menting mathematics interventions within

    an RtI ramework, in a way that reects

    the best evidence on eective practices in

    mathematics interventions.

    RtI begins with high-quality instruction

    and universal screening or all students.

    Whereas high-quality instruction seeks to

    prevent mathematics diculties, screen-

    ing allows or early detection o dicul-ties i they emerge. Intensive interventions

    are then provided to support students

    in need o assistance with mathematics

    learning.4 Student responses to interven-

    tion are measured to determine whether

    they have made adequate progress and (1)

    no longer need intervention, (2) continue

    to need some intervention, or (3) need

    more intensive intervention. The levels o

    intervention are conventionally reerred

    to as tiers. RtI is typically thought o ashaving three tiers.5 Within a three-tiered

    RtI model, each tier is dened by specic

    characteristics.

    4. Fuchs, Fuchs, Craddock et al. (2008).

    5. Fuchs, Fuchs, and Vaughn (2008) make thecase or a three-tier RtI model. Note, however,that some states and school districts have imple-mented multitier intervention systems with morethan three tiers.

    http://nces.ed.gov/timss/http://www.oecd.org/http://www.oecd.org/http://nces.ed.gov/timss/
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    OvERvIEW

    ( 5 )

    Tier 1 is the mathematics instruction

    that all students in a classroom receive.

    It entails universal screening o all stu-

    dents, regardless o mathematics pro-

    ciency, using valid measures to identiy

    students at risk or uture academicailureso that they can receive early

    intervention.6 There is no clear consen-

    sus on the characteristics o instruction

    other than that it is high quality.7

    In tier 2 interventions, schools provide

    additional assistance to students who

    demonstrate diculties on screening

    measures or who demonstrate weak

    progress.8 Tier 2 students receive sup-

    plemental small group mathematics

    instruction aimed at building targeted

    mathematics prociencies.9 These in-

    terventions are typically provided or

    20 to 40 minutes, our to ve times each

    week.10 Student progress is monitored

    throughout the intervention.11

    Tier 3 interventions are provided to

    students who are not beneting rom

    tier 2 and require more intensive as-

    sistance.12 Tier 3 usually entails one-

    on-one tutoring along with an appropri-ate mix o instructional interventions.

    In some cases, special education ser-

    vices are included in tier 3, and in oth-

    ers special education is considered an

    additional tier.13 Ongoing analysis o

    6. For reviews see Jiban and Deno (2007); Fuchs,Fuchs, Compton et al. (2007); Gersten, Jordan,and Flojo (2005).

    7. National Mathematics Advisory Panel (2008);National Research Council (2001).

    8. Fuchs, Fuchs, Craddock et al. (2008); Na-tional Joint Committee on Learning Disabilities(2005).

    9. Fuchs, Fuchs, Craddock et al. (2008).

    10. For example, see Jitendra et al. (1998) andFuchs, Fuchs, Craddock et al. (2008).

    11. National Joint Committee on Learning Dis-abilities (2005).

    12. Fuchs, Fuchs, Craddock et al. (2008).

    13. Fuchs, Fuchs, Craddock et al. (2008); NationalJoint Committee on Learning Disabilities (2005).

    student perormance data is critical in

    this tier. Typically, specialized person-

    nel, such as special education teachers

    and school psychologists, are involved

    in tier 3 and special education services.14

    However, students oten receive rele-vant mathematics interventions rom a

    wide array o school personnel, includ-

    ing their classroom teacher.

    Summary o the Recommendations

    This practice guide oers eight recom-

    mendations or identiying and supporting

    students struggling in mathematics (table

    2). The recommendations are intended to

    be implemented within an RtI ramework

    (typically three-tiered). The panel chose to

    limit its discussion o tier 1 to universal

    screening practices (i.e., the guide does

    not make recommendations or general

    classroom mathematics instruction). Rec-

    ommendation 1 provides specic sugges-

    tions or conducting universal screening

    eectively. For RtI tiers 2 and 3, recom-

    mendations 2 though 8 ocus on the most

    eective content and pedagogical prac-

    tices that can be included in mathematics

    interventions.

    Throughout this guide, we use the term

    interventionist to reer to those teach-

    ing the intervention. At a given school, the

    interventionist may be the general class-

    room teacher, a mathematics coach, a spe-

    cial education instructor, other certied

    school personnel, or an instructional as-

    sistant. The panel recognizes that schools

    rely on dierent personnel to ll these

    roles depending on state policy, schoolresources, and preerences.

    Recommendation 1 addresses the type o

    screening measures that should be used in

    tier 1. We note that there is more research

    on valid screening measures or students in

    14. National Joint Committee on Learning Dis-abilities (2005).

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    OvERvIEW

    ( 6 )

    Table 2. Recommendations and corresponding levels o evidence

    Recommendation Level o evidence

    Tier 1

    1. Screen all students to identiy those at risk or potential mathematics

    diculties and provide interventions to students identied as at risk.Moderate

    Tiers 2 and 3

    2. Instructional materials or students receiving interventions should

    ocus intensely on in-depth treatment o whole numbers in kindergar-

    ten through grade 5 and on rational numbers in grades 4 through 8.

    These materials should be selected by committee.

    Low

    3. Instruction during the intervention should be explicit and systematic.

    This includes providing models o procient problem solving, verbal-ization o thought processes, guided practice, corrective eedback, and

    requent cumulative review.

    Strong

    4. Interventions should include instruction on solving word problems

    that is based on common underlying structures.Strong

    5. Intervention materials should include opportunities or students to

    work with visual representations o mathematical ideas and interven-

    tionists should be procient in the use o visual representations o

    mathematical ideas.

    Moderate

    6. Interventions at all grade levels should devote about 10 minutes in each

    session to building uent retrieval o basic arithmetic acts.Moderate

    7. Monitor the progress o students receiving supplemental instruction

    and other students who are at risk.Low

    8. Include motivational strategies in tier 2 and tier 3 interventions. Low

    Source: Authors compilation based on analysis described in text.

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    OvERvIEW

    ( 7 )

    kindergarten through grade 2,15 but there

    are also reasonable strategies to use or stu-

    dents in more advanced grades.16 We stress

    that no one screening measure is perect

    and that schools need to monitor the prog-

    ress o students who score slightly above orslightly below any screening cuto score.

    Recommendations 2 though 6 address the

    content o tier 2 and tier 3 interventions

    and the types o instructional strategies

    that should be used. In recommendation 2,

    we translate the guidance by the National

    Mathematics Advisory Panel (2008) and

    the National Council o Teachers o Math-

    ematics Curriculum Focal Points (2006)

    into suggestions or the content o inter-

    vention curricula. We argue that the math-

    ematical ocus and the in-depth coverage

    advocated or procient students are also

    necessary or students with mathematics

    diculties. For most students, the content

    o interventions will include oundational

    concepts and skills introduced earlier in

    the students career but not ully under-

    stood and mastered. Whenever possible,

    links should be made between ounda-

    tional mathematical concepts in the inter-

    vention and grade-level material.

    At the center o the intervention recom-

    mendations is that instruction should be

    systematic and explicit (recommendation

    3). This is a recurrent theme in the body

    o valid scientic research.17 We explore

    the multiple meanings o explicit instruc-

    tion and indicate which components o

    explicit instruction appear to be most re-

    lated to improved student outcomes. We

    believe this inormation is important ordistricts and state departments to have

    as they consider selecting materials and

    15. Gersten, Jordan, and Flojo (2005); Gersten,Clarke, and Jordan (2007).

    16. Jiban and Deno (2007); Foegen, Jiban, andDeno (2007).

    17. Darch, Carnine, and Gersten (1984); Fuchset al. (2003a); Jitendra et al. (1998); Schunk andCox (1986); Tournaki (2003); Wilson and Sindelar(1991).

    providing proessional development or

    interventionists.

    Next, we highlight several areas o re-

    search that have produced promising nd-

    ings in mathematics interventions. Theseinclude systematically teaching students

    about the problem types associated with

    a given operation and its inverse (such as

    problem types that indicate addition and

    subtraction) (recommendation 4).18 We also

    recommend practices to help students

    translate abstract symbols and numbers

    into meaningul visual representations

    (recommendation 5).19 Another eature

    that we identiy as crucial or long-term

    success is systematic instruction to build

    quick retrieval o basic arithmetic acts

    (recommendation 6). Some evidence exists

    supporting the allocation o time in the in-

    tervention to practice act retrieval using

    ash cards or computer sotware.20 There

    is also evidence that systematic work with

    properties o operations and counting

    strategies (or younger students) is likely

    to promote growth in other areas o math-

    ematics beyond act retrieval.21

    The nal two recommendations addressother considerations in implementing tier

    2 and tier 3 interventions. Recommenda-

    tion 7 addresses the importance o moni-

    toring the progress o students receiving

    18. Jitendra et al. (1998); Xin, Jitendra, and Deat-line-Buchman (2005); Darch, Carnine, and Gersten(1984); Fuchs et al. (2003a); Fuchs et al. (2003b);Fuchs, Fuchs, Prentice et al. (2004); Fuchs, Fuchs,and Finelli (2004); Fuchs, Fuchs, Craddock et al.(2008) Fuchs, Seethaler et al. (2008).

    19. Artus and Dyrek (1989); Butler et al. (2003);Darch, Carnine, and Gersten (1984); Fuchs etal. (2005); Fuchs, Seethaler et al. (2008); Fuchs,Powell et al. (2008); Fuchs, Fuchs, Craddock etal. (2008); Jitendra et al. (1998); Walker and Po-teet (1989); Wilson and Sindelar (1991); Witzel(2005); Witzel, Mercer, and Miller (2003); Wood-ward (2006).

    20. Bernie-Smith (1991); Fuchs, Seethaler et al.(2008); Fuchs et al. (2005); Fuchs, Fuchs, Hamlettet al. (2006); Fuchs, Powell et al. (2008).

    21. Tournaki (2003); Woodward (2006).

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    OvERvIEW

    ( 8 )

    interventions. Specic types o ormative

    assessment approaches and measures are

    described. We argue or two types o ongo-

    ing assessment. One is the use o curricu-

    lum-embedded assessments that gauge how

    well students have learned the material inthat days or weeks lesson(s). The panel

    believes this inormation is critical or in-

    terventionists to determine whether they

    need to spend additional time on a topic. It

    also provides the interventionist and other

    school personnel with inormation that

    can be used to place students in groups

    within tiers. In addition, we recommend

    that schools regularly monitor the prog-

    ress o students receiving interventions

    and those with scores slightly above or

    below the cuto score on screening mea-

    sures with broader measures o mathemat-

    ics prociency. This inormation provides

    the school with a sense o how the overall

    mathematics program (including tier 1, tier2, and tier 3) is aecting a given student.

    Recommendation 8 addresses the impor-

    tant issue o motivation. Because many o

    the students struggling with mathematics

    have experienced ailure and rustration

    by the time they receive an intervention,

    we suggest tools that can encourage active

    engagement o students and acknowledge

    student accomplishments.

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    Scope o thepractice guide

    Our goal is to provide evidence-based sug-

    gestions or screening students or mathe-matics diculties, providing interventions

    to students who are struggling, and moni-

    toring student responses to the interven-

    tions. RtI intentionally cuts across the bor-

    ders o special and general education and

    involves school-wide collaboration. There-

    ore, our target audience or this guide in-

    cludes teachers, special educators, school

    psychologists and counselors, as well as

    administrators. Descriptions o the ma-

    terials and instructional content in tier 2

    and tier 3 interventions may be especially

    useul to school administrators selecting

    interventions, while recommendations

    that relate to content and pedagogy will

    be most useul to interventionists.22

    The ocus o this guide is on providing

    RtI interventions in mathematics or stu-

    dents in kindergarten through grade 8. This

    broad grade range is in part a response

    to the recent report o the National Math-

    ematics Advisory Panel (2008), which em-phasized a unied progressive approach

    to promoting mathematics prociency or

    elementary andmiddle schools. Moreover,

    given the growing number o initiatives

    aimed at supporting students to succeed

    in algebra, the panel believes it essential

    to provide tier 2 and tier 3 interventions to

    struggling students in grades 4 through 8.

    Because the bulk o research on mathemat-

    ics interventions has ocused on students

    in kindergarten through grade 4, some rec-ommendations or students in older grades

    are extrapolated rom this research.

    22. Interventionists may be any number o schoolpersonnel, including classroom teachers, specialeducators, school psychologists, paraproession-als, and mathematics coaches and specialists.The panel does not speciy the interventionist.

    The scope o this guide does not include

    recommendations or special education

    reerrals. Although enhancing the valid-

    ity o special education reerrals remains

    important and an issue o ongoing discus-

    sion23

    and research,24

    we do not addressit in this practice guide, in part because

    empirical evidence is lacking.

    The discussion o tier 1 in this guide re-

    volves only around eective screening, be-

    cause recommendations or general class-

    room mathematics instruction were beyond

    the scope o this guide. For this reason,

    studies o eective general mathematics

    instruction practices were not included in

    the evidence base or this guide.25

    The studies reviewed or this guide in-

    cluded two types o comparisons among

    groups. First, several studies o tier 2 in-

    terventions compare students receiving

    multicomponent tier 2 interventions with

    students receiving only routine classroom

    instruction.26This type o study provides

    evidence o the eectiveness o providing

    tier 2 interventions but does not permit

    conclusions about which component is

    most eective. The reason is that it is notpossible to identiy whether one particular

    component or a combination o compo-

    nents within a multicomponent interven-

    tion produced an eect. Second, several

    23. Kavale and Spaulding (2008); Fuchs, Fuchs,and Vaughn (2008); VanDerHeyden, Witt, andGilbertson (2007).

    24. Fuchs, Fuchs, Compton et al. (2006).

    25. There were a ew exceptions in which general

    mathematics instruction studies were included inthe evidence base. When the eects o a generalmathematics instruction program were speciedor low-achieving or disabled students and theintervention itsel appeared applicable to teach-ing tier 2 or tier 3 (e.g., teaching a specic opera-tional strategy), we included them in this study.Note that disabled students were predominantlylearning disabled.

    26. For example, Fuchs, Seethaler et al. (2008)examined the eects o providing supplemen-tal tutoring (i.e., a tier 2 intervention) relative toregular classroom instruction (i.e., tier 1).

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    SCOPE O ThE PRACTICE GuIDE

    ( 10 )

    other studies examined the eects o two

    methods o tier 2 or tier 3 instruction.27

    This type o study oers evidence or the

    eectiveness o one approach to teaching

    within a tier relative to another approach

    and assists with identiying the most ben-ecial approaches or this population.

    The panel reviewed only studies or prac-

    tices that sought to improve student math-

    ematics outcomes. The panel did not con-

    sider interventions that improved other

    academic or behavioral outcomes. Instead,

    the panel ocused on practices that ad-

    dressed the ollowing areas o mathematics

    prociency: operations(either computation

    27. For example, Tournaki (2003) examined theeects o providing supplemental tutoring in anoperations strategy (a tier 2 intervention) relativeto supplemental tutoring with a drill and practiceapproach (also a tier 2 intervention).

    or estimation), concepts(knowledge o

    properties o operations, concepts involv-

    ing rational numbers, prealgebra con-

    cepts), problem solving(word problems),

    and measures ogeneral mathematics

    achievement. Measures oact luencywere also included because quick retrieval

    o basic arithmetic acts is essential or

    success in mathematics and a persistent

    problem or students with diculties in

    mathematics.28

    Technical terms related to mathematics

    and technical aspects o assessments (psy-

    chometrics) are dened in a glossary at the

    end o the recommendations.

    28. Geary (2004); Jordan, Hanich, and Kaplan(2003).

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    ( 11 )

    i-dept coerage o ratioa mbers as

    we as adaced topics i woe mber

    aritmetic (sc as og diisio).

    Districts sod appoit committees,

    icdig eperts i matematics istrc-tio ad matematicias wit owedge

    o eemetar ad midde scoo mat-

    ematics crrica, to esre tat specic

    criteria are coered i-dept i te cr-

    ricm te adopt.

    Recommendation 3. Instruction duringthe intervention should be explicit andsystematic. This includes providingmodels o procient problem solving,verbalization o thought processes,guided practice, corrective eedback,and requent cumulative review.

    Esre tat istrctioa materias aresstematic ad epicit. I particar, te

    sod icde meros cear modes o

    eas ad dict probems, wit accom-

    paig teacer ti-aods.

    Proide stdets wit opportitiesto soe probems i a grop ad comm-

    icate probem-soig strategies.

    Esre tat istrctioa materias i-cde cmatie reiew i eac sessio.

    Recommendation 4. Interventionsshould include instruction on solvingword problems that is based oncommon underlying structures.

    Teac stdets abot te strctre o

    arios probem tpes, ow to categorizeprobems based o strctre, ad ow to

    determie appropriate sotios or eac

    probem tpe.

    Teac stdets to recogize te com-mo derig strctre betwee ami-

    iar ad amiiar probems ad to traser

    ow sotio metods rom amiiar to

    amiiar probems.

    Checklist or carrying out therecommendations

    Recommendation 1. Screen allstudents to identiy those at risk or

    potential mathematics diculties andprovide interventions to studentsidentied as at risk.

    As a district or scoo sets p a scree-ig sstem, ae a team eaate potetia

    screeig measres. Te team sod se-

    ect measres tat are eciet ad reaso-

    ab reiabe ad tat demostrate predic-

    tie aidit. Screeig sod occr i te

    begiig ad midde o te ear.

    Seect screeig measres based ote cotet te coer, wit a empasis

    o critica istrctioa objecties or eac

    grade.

    I grades 4 trog 8, se scree-ig data i combiatio wit state testig

    rests.

    use te same screeig too across adistrict to eabe aazig rests across

    scoos.

    Recommendation 2. Instructionalmaterials or students receivinginterventions should ocus intenselyon in-depth treatment o wholenumbers in kindergarten throughgrade 5 and on rational numbers ingrades 4 through 8. These materialsshould be selected by committee.

    or stdets i idergarte troggrade 5, tier 2 ad tier 3 iteretiossod ocs amost ecsie o prop-

    erties o woe mbers ad operatios.

    Some oder stdets strggig wit

    woe mbers ad operatios wod

    aso beeit rom i-dept coerage o

    tese topics.

    or tier 2 ad tier 3 stdets i grades4 trog 8, iteretios sod ocs o

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    ChECklIST OR CARRyInG OuT ThE RECOMMEnDATIOnS

    ( 12 )

    Recommendation 5. Interventionmaterials should include opportunitiesor students to work with visualrepresentations o mathematicalideas and interventionists should

    be procient in the use o visualrepresentations o mathematical ideas.

    use isa represetatios sc asmber ies, arras, ad strip diagrams.I isas are ot sciet or deeop-ig accrate abstract togt ad aswers,

    se cocrete maipaties rst. Atog

    tis ca aso be doe wit stdets i pper

    eemetar ad midde scoo grades, se

    o maipaties wit oder stdets sod

    be epeditios becase te goa is to moe

    toward derstadig oad aciit

    witisa represetatios, ad a, to

    te abstract.

    Recommendation 6. Interventions atall grade levels should devote about10 minutes in each session to buildingfuent retrieval o basic arithmetic acts.

    Proide abot 10 mites per ses-sio o istrctio to bid qic retrieao basic aritmetic acts. Cosider sig

    tecoog, fas cards, ad oter materi-

    as or etesie practice to aciitate a-

    tomatic retriea.

    or stdets i idergarte troggrade 2, epicit teac strategies or e-

    ciet cotig to improe te retriea o

    matematics acts.

    Teac stdets i grades 2 trog8 ow to se teir owedge o proper-

    ties, sc as commtatie, associatie,

    ad distribtie aw, to derie acts i

    teir eads.Recommendation 7. Monitor theprogress o students receivingsupplemental instruction and otherstudents who are at risk.

    Moitor te progress o tier 2, tier 3,ad borderie tier 1 stdets at east oce

    a mot sig grade-appropriate geera

    otcome measres.use crricm-embedded assess-mets i iteretios to determie

    weter stdets are earig rom te

    iteretio. Tese measres ca be sed

    as ote as eer da or as ireqet as

    oce eer oter wee.

    use progress moitorig data to re-grop stdets we ecessar.

    Recommendation 8. Includemotivational strategies in tier 2 and

    tier 3 interventions.

    Reiorce or praise stdets or teireort ad or attedig to ad beig e-

    gaged i te esso.

    Cosider rewardig stdet accom-pismets.

    Aow stdets to cart teir progressad to set goas or improemet.

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    ( 13 )

    Recommendation 1.Screen all students toidentiy those at risk orpotential mathematicsdiculties and provideinterventions tostudents identiedas at risk.

    Te pae recommeds tat scoosad districts sstematica seiversa screeig to scree astdets to determie wic stdetsave matematics dicties adreqire researc-based itervetios.Scoos sod evaate ad seectscreeig measres based o teirreiabiit ad predictive vaidit, witparticar empasis o te measresspecicit ad sesitivit. Scoossod aso cosider te eciec ote measre to eabe screeig mastdets i a sort time.

    Level o evidence: Moderate

    The panel judged the level o evidence sup-

    porting this recommendation to be mod-

    erate. This recommendation is based on a

    series o high-quality correlational studies

    with replicated ndings that show the abil-

    ity o measures to predict perormance in

    mathematics one year ater administration

    (and in some cases two years).29

    Brie summary o evidence tosupport the recommendation

    A growing body o evidence suggests that

    there are several valid and reliable ap-

    proaches or screening students in the pri-

    mary grades. All these approaches target

    29. For reviews see Jiban and Deno (2007); Fuchs,Fuchs, Compton et al. (2007); Gersten, Jordan,and Flojo (2005).

    aspects o what is oten reerred to as

    number sense.30 They assess various as-

    pects o knowledge o whole numbers

    properties, basic arithmetic operations,

    understanding o magnitude, and applying

    mathematical knowledge to word prob-lems. Some measures contain only one

    aspect o number sense (such as magni-

    tude comparison) and others assess our

    to eight aspects o number sense. The sin-

    gle-component approaches with the best

    ability to predict students subsequent

    mathematics perormance include screen-

    ing measures o students knowledge o

    magnitude comparison and/or strategic

    counting.31 The broader, multicomponent

    measures seem to predict with slightly

    greater accuracy than single-component

    measures.32

    Eective approaches to screening vary in

    eciency, with some taking as little as 5

    minutes to administer and others as long

    as 20 minutes. Multicomponent measures,

    which by their nature take longer to ad-

    minister, tend to be time-consuming or

    administering to an entire school popu-

    lation. Timed screening measures33 and

    untimed screening measures34 have beenshown to be valid and reliable.

    For the upper elementary grades and mid-

    dle school, we were able to locate ewer

    studies. They suggest that brie early

    screening measures that take about 10

    minutes and cover a proportional sam-

    pling o grade-level objectives are reason-

    able and provide sucient evidence o reli-

    ability.35 At the current time, this research

    area is underdeveloped.

    30. Berch (2005); Dehaene (1999); Okamoto andCase (1996); Gersten and Chard (1999).

    31. Gersten, Jordan, and Flojo (2005).

    32. Fuchs, Fuchs, Compton et al. (2007).

    33. For example, Clarke and Shinn (2004).

    34. For example, Okamoto and Case (1996).

    35. Jiban and Deno (2007); Foegen, Jiban, andDeno (2007).

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    RECOMMEnDATIOn 1. SCREEn All STuDEnTS TO IDEnTIy ThOSE AT RISk

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    How to carry out thisrecommendation

    1. As a district or scoo sets p a scree-

    ig sstem, ave a team evaate potetia

    screeig measres. Te team sod seectmeasres tat are eciet ad reasoab

    reiabe ad tat demostrate predictive va-

    idit. Screeig sod occr i te begi-

    ig ad midde o te ear.

    The team that selects the measures should

    include individuals with expertise in mea-

    surement (such as a school psychologist or

    a member o the district research and eval-

    uation division) and those with expertise in

    mathematics instruction. In the opinion o

    the panel, districts should evaluate screen-

    ing measures on three dimensions.

    Predictivevalidity is an index o how

    well a score on a screening measure

    earlier in the year predicts a students

    later mathematics achievement. Greater

    predictive validity means that schools

    can be more condent that decisions

    based on screening data are accurate.

    In general, we recommend that schools

    and districts employ measures withpredictive validity coeicients o at

    least .60 within a school year.36

    Reliabilityis an index o the consistency

    and precision o a measure. We recom-

    mend measures with reliability coe-

    cients o .80 or higher.37

    Eciencyis how quickly the universal

    screening measure can be adminis-

    tered, scored, and analyzed or all thestudents. As a general rule, we suggest

    that a screening measure require no

    36. A coecient o .0 indicates that there is norelation between the early and later scores, anda coecient o 1.0 indicates a perect positiverelation between the scores.

    37. A coecient o .0 indicates that there is norelation between the two scores, and a coe-cient o 1.0 indicates a perect positive relationbetween the scores.

    more than 20 minutes to administer,

    which enables collecting a substantial

    amount o inormation in a reasonable

    time rame. Note that many screening

    measures take ve minutes or less.38 We

    recommend that schools select screen-ing measures that have greater ei-

    ciency i their technical adequacy (pre-

    dictive validity, reliability, sensitivity,

    and specicity) is roughly equivalent

    to less ecient measures. Remember

    that screening measures are intended

    or administration to all students in a

    school, and it may be better to invest

    more time in diagnostic assessment o

    students who perorm poorly on the

    universal screening measure.

    Keep in mind that screening is just a means

    o determining which students are likely to

    need help. I a student scores poorly on a

    screening measure or screening battery

    especially i the score is at or near a cut

    point, the panel recommends monitoring

    her or his progress careully to discern

    whether extra instruction is necessary.

    Developers o screening systems recom-

    mend that screening occur at least twicea year (e.g., all, winter, and/or spring).39

    This panel recommends that schools alle-

    viate concern about students just above or

    below the cut score by screening students

    twice during the year. The second screen-

    ing in the middle o the year allows another

    check on these students and also serves to

    identiy any students who may have been at

    risk and grown substantially in their mathe-

    matics achievementor those who were on-

    track at the beginning o the year but havenot shown suicient growth. The panel

    considers these two universal screenings

    to determine student prociency as distinct

    rom progress monitoring (Recommenda-

    tion 7), which occurs on a more requent

    38. Foegen, Jiban, and Deno (2007); Fuchs, Fuchs,Compton et al. (2007); Gersten, Clarke, and Jordan(2007).

    39. Kaminski et al. (2008); Shinn (1989).

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    RECOMMEnDATIOn 1. SCREEn All STuDEnTS TO IDEnTIy ThOSE AT RISk

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    basis (e.g., weekly or monthly) with a select

    group o intervention students in order to

    monitor response to intervention.

    2. Seect screeig measres based o te

    cotet te cover, wit a empasis o crit-ica istrctioa objectives or eac grade.

    The panel believes that content covered

    in a screening measure should reect the

    instructional objectives or a students

    grade level, with an emphasis on the most

    critical content or the grade level. The Na-

    tional Council o Teachers o Mathematics

    (2006) released a set o ocal points or

    each grade level designed to ocus instruc-

    tion on critical concepts or students to

    master within a specic grade. Similarly,

    the National Mathematics Advisory Panel

    (2008) detailed a route to preparing all

    students to be successul in algebra. In the

    lower elementary grades, the core ocus o

    instruction is on building student under-

    standing o whole numbers. As students

    establish an understanding o whole num-

    bers, rational numbers become the ocus

    o instruction in the upper elementary

    grades. Accordingly, screening measures

    used in the lower and upper elementarygrades should have items designed to as-

    sess students understanding o whole and

    rational number conceptsas well as com-

    putational prociency.

    3. I grades 4 trog 8, se screeig data

    i combiatio wit state testig rests.

    In the panels opinion, one viable option

    that schools and districts can pursue is to

    use results rom the previous years statetesting as a rst stage o screening. Students

    who score below or only slightly above a

    benchmark would be considered or sub-

    sequent screening and/or diagnostic or

    placement testing. The use o state testing

    results would allow districts and schools

    to combine a broader measure that covers

    more content with a screening measure that

    is narrower but more ocused. Because o

    the lack o available screening measures at

    these grade levels, districts, county oces,

    or state departments may need to develop

    additional screening and diagnostic mea-

    sures or rely on placement tests provided

    by developers o intervention curricula.

    4. use te same screeig too across a district

    to eabe aazig rests across scoos.

    The panel recommends that all schools

    within a district use the same screening

    measure and procedures to ensure ob-

    jective comparisons across schools and

    within a district. Districts can use results

    rom screening to inorm instructional de-

    cisions at the district level. For example,

    one school in a district may consistently

    have more students identied as at risk,

    and the district could provide extra re-

    sources or proessional development to

    that school. The panel recommends that

    districts use their research and evaluation

    sta to reevaluate screening measures an-

    nually or biannually. This entails exam-

    ining how screening scores predict state

    testing results and considering resetting

    cut scores or other data points linked to

    instructional decisionmaking.

    Potential roadblocks and solutions

    Roadblock 1.1. Districts and school person-

    nel may ace resistance in allocating time re-

    sources to the collection o screening data.

    Suggested Approach. The issue o time

    and personnel is likely to be the most sig-

    nicant obstacle that districts and schools

    must overcome to collect screening data.

    Collecting data on all students will requirestructuring the data collection process to

    be ecient and streamlined.

    The panel notes that a common pitall is

    a long, drawn-out data collection process,

    with teachers collecting data in their class-

    rooms when time permits. I schools are

    allocating resources (such as providing an

    intervention to students with the 20 low-

    est scores in grade 1), they must wait until

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    all the data have been collected across

    classrooms, thus delaying the delivery

    o needed services to students. Further-

    more, because many screening measures

    are sensitive to instruction, a wide gap

    between when one class is assessed andanother is assessed means that many stu-

    dents in the second class will have higher

    scores than those in the rst because they

    were assessed later.

    One way to avoid these pitalls is to use data

    collection teams to screen students in a

    short period o time. The teams can consist

    o teachers, special education sta includ-

    ing such specialists as school psychologists,

    Title I sta, principals, trained instructional

    assistants, trained older students, and/or

    local college students studying child devel-

    opment or school psychology.

    Roadblock 1.2. Implementing universal

    screening is likely to raise questions such

    as, Why are we testing students who are

    doing ne?

    Suggested Approach. Collecting data

    on all students is new or many districts

    and schools (this may not be the case orelementary schools, many o which use

    screening assessments in reading).40 But

    screening allows schools to ensure that all

    students who are on track stay on track

    and collective screening allows schools to

    evaluate the impact o their instruction

    on groups o students (such as all grade

    2 students). When schools screen all stu-

    dents, a distribution o achievement rom

    high to low is created. I students consid-

    ered not at risk were not screened, thedistribution o screened students would

    consist only o at-risk students. This could

    create a situation where some students at

    the top o the distribution are in real-

    ity at risk but not identied as such. For

    upper-grade students whose scores were

    40. U.S. Department o Education, Oce o Plan-ning, Evaluation and Policy Development, Policyand Program Studies Service (2006).

    high on the previous springs state as-

    sessment, additional screening typically

    is not required.

    Roadblock 1.3. Screening measures may

    identiy students who do not need servicesand not identiy students who do need

    services.

    Suggested Approach. All screening mea-

    sures will misidentiy some students as

    either needing assistance when they do

    not (alse positive) or not needing assis-

    tance when they do (alse negative). When

    screening students, educators will want to

    maximize both the number o students

    correctly identied as at riska measures

    sensitivityand the number o students

    correctly identied as not at riska mea-

    sures specicity. As illustrated in table 3,

    screening students to determine risk can

    result in our possible categories indicated

    by the letters A, B, C, and D. Using these

    categories, sensitivity is equal to A/(A + C)

    and specicity is equal to D/(B + D).

    Table 3. Sensitivity and specicity

    STUDENTS

    ACTUALLYAT RISK

    Yes No

    STUDENTS

    IDENTIFIED

    AS BEING

    AT RISK

    Yes A (true

    positives)

    B (alse

    positives)

    No C (alse

    negatives)

    D (true

    negatives)

    The sensitivity and specicity o a mea-

    sure depend on the cut score to classiy

    children at risk.41

    I a cut score is high(where all students below the cut score are

    considered at risk), the measure will have

    a high degree o sensitivity because most

    students who truly need assistance will be

    41. Sensitivity and specicity are also inuencedby the discriminant validity o the measure andits individual items. Measures with strong itemdiscrimination are more likely to correctly iden-tiy students risk status.

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    RECOMMEnDATIOn 1. SCREEn All STuDEnTS TO IDEnTIy ThOSE AT RISk

    ( 17 )

    identied as at risk. But the measure will

    have low specicity since many students

    who do not need assistance will also be

    identied as at risk. Similarly, i a cut score

    is low, the sensitivity will be lower (some

    students in need o assistance may not beidentied as at risk), whereas the specic-

    ity will be higher (most students who do

    not need assistance will not be identied

    as at risk).

    Schools need to be aware o this tradeo

    between sensitivity and speciicity, and

    the team selecting measures should be

    aware that decisions on cut scores can be

    somewhat arbitrary. Schools that set a cut

    score too high run the risk o spending re-

    sources on students who do not need help,

    and schools that set a cut score too low run

    the risk o not providing interventions to

    students who are at risk and need extra in-

    struction. I a school or district consistently

    nds that students receiving intervention

    do not need it, the measurement team

    should consider lowering the cut score.

    Roadblock 1.4. Screening data may iden-

    tiy large numbers o students who are at

    risk and schools may not immediately havethe resources to support all at-risk students.

    This will be a particularly severe problem

    in low-perorming Title I schools.

    Suggested Approach. Districts and

    schools need to consider the amount o

    resources available and the allocation o

    those resources when using screening

    data to make instructional decisions. Dis-

    tricts may nd that on a nationally normed

    screening measure, a large percentage o

    their students (such as 60 percent) will be

    classied as at risk. Districts will have todetermine the resources they have to pro-

    vide interventions and the number o stu-

    dents they can serve with their resources.

    This may mean not providing interven-

    tions at certain grade levels or providing

    interventions only to students with the

    lowest scores, at least in the rst year o

    implementation.

    There may also be cases when schools

    identiy large numbers o students at risk

    in a particular area and decide to pro-

    vide instruction to all students. One par-

    ticularly salient example is in the area o

    ractions. Multiple national assessments

    show many students lack prociency in

    ractions,42 so a school may decide that,

    rather than deliver interventions at the

    individual child level, they will provide a

    school-wide intervention to all students. A

    school-wide intervention can range rom a

    supplemental ractions program to proes-

    sional development involving ractions.

    42. National Mathematics Advisory Panel(2008); Lee, Grigg, and Dion (2007).

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    ( 18 )

    Recommendation 2.Instructional materialsor students receivinginterventions shouldocus intensely onin-depth treatmento whole numbers inkindergarten throughgrade 5 and on rationalnumbers in grades4 through 8. These

    materials should beselected by committee.

    Te pae recommeds tat idividasowedgeabe i istrctio admatematics oo or itervetios tatocs o woe mbers extesivei idergarte trog grade 5 ado ratioa mbers extesive igrades 4 trog 8. I a cases, te

    specic cotet o te itervetios wibe cetered o bidig te stdetsodatioa prociecies. I maigtis recommedatio, te pae isdrawig o cosess docmetsdeveoped b experts rom matematicsedcatio ad researc matematiciastat empasized te importace otese topics or stdets i geera.43We cocde tat te coverage oewer topics i more dept, ad

    wit coerece, is as importat, adprobab more importat, or stdetswo strgge wit matematics.

    43. National Council o Teachers o Mathemat-ics (2006); National Mathematics Advisory Panel(2008).

    Level o evidence: Low

    The panel judged the level o evidence

    supporting this recommendation to be low.

    This recommendation is based on the pro-

    essional opinion o the panel and severalrecent consensus documents that reect

    input rom mathematics educators and re-

    search mathematicians involved in issues

    related to kindergarten through grade 12

    mathematics education.44

    Brie summary o evidence tosupport the recommendation

    The documents reviewed demonstrate a

    growing proessional consensus that cov-

    erage o ewer mathematics topics in more

    depth and with coherence is important

    or all students.45Milgram and Wu (2005)

    suggested that an intervention curriculum

    or at-risk students should not be over-

    simplied and that in-depth coverage o

    key topics and concepts involving whole

    numbers and then rational numbers is

    critical or uture success in mathematics.

    The National Council o Teachers o Math-

    ematics (NCTM) Curriculum Focal Points

    (2006) called or the end o brie venturesinto many topics in the course o a school

    year and also suggested heavy emphasis on

    instruction in whole numbers and rational

    numbers. This position was reinorced by

    the 2008 report o the National Mathematics

    Advisory Panel (NMAP), which provided de-

    tailed benchmarks and again emphasized

    in-depth coverage o key topics involving

    whole numbers and rational numbers as

    crucial or all students. Although the latter

    two documents addressed the needs o allstudents, the panel concludes that the in-

    depth coverage o key topics is especially

    44. National Council o Teachers o Mathemat-ics (2006); National Mathematics Advisory Panel(2008); Milgram and Wu (2005).

    45. National Mathematics Advisory Panel (2008);Schmidt and Houang (2007); Milgram and Wu(2005); National Council o Teachers o Math-ematics (2006).

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    RECOMMEnDATIOn 2. InSTRuCTIOnAl MATERIAlS OR STuDEnTS RECEIvInG InTERvEnTIOnS

    ( 19 )

    important or students who struggle with

    mathematics.

    How to carry out thisrecommendation

    1. or stdets i idergarte trog

    grade 5, tier 2 ad tier 3 itervetios sod

    ocs amost excsive o properties o

    woe mbers46 ad operatios. Some

    oder stdets strggig wit woe m-

    bers ad operatios wod aso beet rom

    i-dept coverage o tese topics.

    In the panels opinion, districts should

    review the interventions they are con-

    sidering to ensure that they cover whole

    numbers in depth. The goal is prociency

    and mastery, so in-depth coverage with

    extensive review is essential and has

    been articulated in the NCTM Curriculum

    Focal Points (2006) and the benchmarks

    determined by the National Mathematics

    Advisory Panel (2008). Readers are recom-

    mended to review these documents.47

    Specic choices or the content o interven-

    tions will depend on the grade level and

    prociency o the student, but the ocusor struggling students should be on whole

    numbers. For example, in kindergarten

    through grade 2, intervention materials

    would typically include signicant atten-

    tion to counting (e.g., counting up), num-

    ber composition, and number decomposi-

    tion (to understand place-value multidigit

    operations). Interventions should cover the

    meaning o addition and subtraction and

    46. Properties o numbers, including the associa-tive, commutative, and distributive properties.

    47. More inormation on the National Mathemat-ics Advisory Panel (2008) report is available atwww.ed.gov/about/bdscomm/list/mathpanel/index.html. More inormation on the NationalCouncil o Teachers o Mathematics Curricu-lum Focal Points is available at www.nctm.org/ocalpoints. Documents elaborating the NationalCouncil o Teachers o Mathematics CurriculumFocal Points are also available (see Beckmann etal., 2009). For a discussion o why this content ismost relevant, see Milgram and Wu (2005).

    the reasoning that underlies algorithms or

    addition and subtraction o whole num-

    bers, as well as solving problems involv-

    ing whole numbers. This ocus should in-

    clude understanding o the base-10 system

    (place value).

    Interventions should also include materi-

    als to build uent retrieval o basic arith-

    metic acts (see recommendation 6). Ma-

    terials should extensively useand ask

    students to usevisual representations o

    whole numbers, including both concrete

    and visual base-10 representations, as well

    as number paths and number lines (more

    inormation on visual representations is

    in recommendation 5).

    2. or tier 2 ad tier 3 stdets i grades 4

    trog 8, itervetios sod ocs o i-

    dept coverage o ratioa mbers as we

    as advaced topics i woe mber arit-

    metic (sc as og divisio).

    The panel believes that districts should

    review the interventions they are consid-

    ering to ensure that they cover concepts

    involving rational numbers in depth. The

    ocus on rational numbers should includeunderstanding the meaning o ractions,

    decimals, ratios, and percents, using visual

    representations (including placing ractions

    and decimals on number lines,48 see recom-

    mendation 5), and solving problems with

    ractions, decimals, ratios, and percents.

    In the view o the panel, students in

    grades 4 through 8 will also require ad-

    ditional work to build uent retrieval o

    basic arithmetic acts (see recommenda-tion 6), and some will require additional

    work involving basic whole number top-

    ics, especially or students in tier 3. In the

    opinion o the panel, accurate and uent

    48. When using number lines to teach rationalnumbers or students who have diculties, it isimportant to emphasize that the ocus is on thelength o the segments between the whole num-ber marks (rather than counting the marks).

    http://www.ed.gov/about/bdscomm/list/mathpanel/index.htmlhttp://www.ed.gov/about/bdscomm/list/mathpanel/index.htmlhttp://www.nctm.org/focalpointshttp://www.nctm.org/focalpointshttp://www.nctm.org/focalpointshttp://www.nctm.org/focalpointshttp://www.ed.gov/about/bdscomm/list/mathpanel/index.htmlhttp://www.ed.gov/about/bdscomm/list/mathpanel/index.html
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    RECOMMEnDATIOn 2. InSTRuCTIOnAl MATERIAlS OR STuDEnTS RECEIvInG InTERvEnTIOnS

    ( 20 )

    arithmetic with whole numbers is neces-

    sary beore understanding ractions. The

    panel acknowledges that there will be

    periods when both whole numbers and

    rational numbers should be addressed in

    interventions. In these cases, the balanceo concepts should be determined by the

    students need or support.

    3. Districts sod appoit committees, i-

    cdig experts i matematics istrctio

    ad matematicias wit owedge o e-

    emetar ad midde scoo matematics

    crricm, to esre tat specic criteria

    (described beow) are covered i dept i

    te crrica te adopt.

    In the panels view, intervention materials

    should be reviewed by individuals with

    knowledge o mathematics instruction and

    by mathematicians knowledgeable in el-

    ementary and middle school mathematics.

    They can oten be experts within the district,

    such as mathematics coaches, mathematics

    teachers, or department heads. Some dis-

    tricts may also be able to draw on the exper-

    tise o local university mathematicians.

    Reviewers should assess how well interven-tion materials meet our criteria. First, the

    materials integrate computation with solv-

    ing problems and pictorial representations

    rather than teaching computation apart

    rom problem-solving. Second, the mate-

    rials stress the reasoning underlying cal-

    culation methods and ocus student atten-

    tion on making sense o the mathematics.

    Third, the materials ensure that students

    build algorithmic prociency. Fourth, the

    materials include requent review or bothconsolidating and understanding the links

    o the mathematical principles. Also in the

    panels view, the intervention program

    should include an assessment to assist in

    placing students appropriately in the in-

    tervention curriculum.

    Potential roadblocks and solutions

    Roadblock 2.1. Some interventionists

    may worry i the intervention program

    is not aligned with the core classroom

    instruction.

    Suggested Approach. The panel believes

    that alignment with the core curriculum is

    not as critical as ensuring that instruction

    builds students oundational procien-

    cies. Tier 2 and tier 3 instruction ocuses

    on oundational and oten prerequisite

    skills that are determined by the students

    rate o progress. So, in the opinion o the

    panel, acquiring these skills will be neces-

    sary or uture achievement. Additionally,

    because tier 2 and tier 3 are supplemental,

    students will still be receiving core class-

    room instruction aligned to a school or

    district curriculum (tier 1).

    Roadblock 2.2. Intervention materials

    may cover topics that are not essential tobuilding basic competencies, such as data

    analysis, measurement, and time.

    Suggested Approach. In the panels opin-

    ion, it is not necessary to cover every topic

    in the intervention materials. Students will

    gain exposure to many supplemental top-

    ics (such as data analysis, measurement,

    and time) in general classroom instruc-

    tion (tier 1). Depending on the students

    age and prociency, it is most importantto ocus on whole and rational numbers in

    the interventions.

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    ( 21 )

    Recommendation 3.Instruction during theintervention should beexplicit and systematic.This includes providingmodels o procientproblem solving,verbalization othought processes,guided practice,corrective eedback,

    and requentcumulative review.

    Te natioa Matematics AdvisorPae dees explicit instruction asoows (2008, p. 23):

    Teacers provide cear modes orsovig a probem tpe sig aarra o exampes.

    Stdets receive extesive practicei se o ew eared strategiesad sis.

    Stdets are provided witopportities to ti aod (i.e.,ta trog te decisios temae ad te steps te tae).

    Stdets are provided wit

    extesive eedbac.

    Te nMAP otes tat tis does ot meatat a matematics istrctio sodbe expicit. Bt it does recommed tatstrggig stdets receive some expicitistrctio regar ad tat someo te expicit istrctio esre tatstdets possess te odatioa sisad cocepta owedge ecessaror derstadig teir grade-eve

    matematics.49 Or pae spportstis recommedatio ad beievestat districts ad scoos sod seectmaterias or itervetios tat refecttis orietatio. I additio, proessioadeveopmet or itervetioists sodcotai gidace o tese compoetso expicit istrctio.

    Level o evidence: Strong

    Our panel judged the level o evidence

    supporting this recommendation to be

    strong. This recommendation is based on

    six randomized controlled trials that met

    WWC standards or met standards with

    reservations and that examined the e-

    ectiveness o explicit and systematic in-

    struction in mathematics interventions.50

    These studies have shown that explicit and

    systematic instruction can signicantly

    improve prociency in word problem solv-

    ing51 and operations52 across grade levels

    and diverse student populations.

    Brie summary o evidence to support

    the recommendation

    The results o six randomized controlledtrials o mathematics interventions show

    extensive support or various combina-

    tions o the ollowing components o ex-

    plicit and systematic instruction: teacher

    demonstration,53 student verbalization,54

    49. National Mathematics Advisory Panel(2008).

    50. Darch, Carnine, and Gersten (1984); Fuchset al. (2003a); Jitendra et al. (1998); Schunk and

    Cox (1986); Tournaki (2003); Wilson and Sindelar(1991).

    51. Darch, Carnine, and Gersten (1984); Jitendraet al. (1998); Fuchs et al. (2003a); Wilson and Sin-delar (1991).

    52. Schunk and Cox (1986); Tournaki (2003).

    53. Darch, Carnine, and Gersten (1984); Jitendraet al. (1998); Fuchs et al. (2003a); Schunk andCox (1986); Tournaki (2003); Wilson and Sindelar(1991).

    54. Jitendra et al. (1998); Fuchs et al. (2003a);Schunk and Cox (1986); Tournaki (2003).

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    RECOMMEnDATIOn 3. InSTRuCTIOn DuRInG ThE InTERvEnTIOn ShOulD BE ExPlICIT AnD SySTEMATIC

    ( 22 )

    guided practice,55 and corrective eed-

    back.56 All six studies examined interven-

    tions that included teacher demonstra-

    tions early in the lessons.57 For example,

    three studies included instruction that

    began with the teacher verbalizing aloudthe steps to solve sample mathematics

    problems.58 The eects o this component

    o explicit instruction cannot be evaluated

    rom these studies because the demonstra-

    tion procedure was used in instruction or

    students in both treatment and compari-

    son groups.

    Scaolded practice, a transer o control

    o problem solving rom the teacher to the

    student, was a component in our o the six

    studies.59Although it is not possible to parse

    the eects o scaolded instruction rom the

    other components o instruction, the inter-

    vention groups in each study demonstrated

    signicant positive gains on word problem

    prociencies or accuracy measures.

    Three o the six studies included opportu-

    nities or students to verbalize the steps

    to solve a problem.60 Again, although e-

    ects o the interventions were statistically

    signicant and positive on measures oword problems, operations, or accuracy,

    the eects cannot be attributed to a sin-

    gle component o these multicomponent

    interventions.

    55. Darch, Carnine, and Gersten (1984); Jiten-dra et al. (1998); Fuchs et al. (2003a); Tournaki(2003).

    56. Darch, Carnine, and Gersten (1984); Jitendraet al. (1998); Schunk and Cox (1986); Tournaki

    (2003).57. Darch, Carnine, and Gersten (1984); Fuchset al. (2003a); Jitendra et al. (1998); Schunk andCox (1986); Tournaki (2003); Wilson and Sindelar(1991).

    58. Schunk and Cox (1986); Jitendra et al. (1998);Darch, Carnine, and Gersten (1984).

    59. Darch, Carnine, and Gersten (1984); Fuchset al. (2003a); Jitendra et al. (1998); Tournaki(2003).

    60. Schunk and Cox (1986); Jitendra et al. (1998);Tournaki (2003).

    Similarly, our o the six studies included

    immediate corrective eedback,61 and the

    eects o these interventions were posi-

    tive and signicant on word problems and

    measures o operations skills, but the e-

    ects o the corrective eedback compo-nent cannot be isolated rom the eects o

    other components in three cases.62

    With only one study in the pool o six in-

    cluding cumulative review as part o the

    intervention,63 the support or this compo-

    nent o explicit instruction is not as strong

    as it is or the other components. But this

    study did have statistically signicant pos-

    itive eects in avor o the instructional

    group that received explicit instruction

    in strategies or solving word problems,

    including cumulative review.

    How to carry out thisrecommendation

    1. Esre tat istrctioa materias are

    sstematic ad expicit. I particar, te

    sod icde meros cear modes o

    eas ad dict probems, wit accompa-

    ig teacer ti-aods.

    To be considered systematic, mathematics

    instruction should gradually build pro-

    ciency by introducing concepts in a logical

    order and by providing students with nu-

    merous applications o each concept. For

    example, a systematic curriculum builds

    student understanding o place value in

    an array o contexts beore teaching pro-

    cedures or adding and subtracting two-

    digit numbers with regrouping.

    Explicit instruction typically begins with

    a clear unambiguous exposition o con-

    cepts and step-by-step models o how

    61. Darch, Carnine, and Gersten (1984); Jiten-dra et al. (1998); Tournaki (2003); Schunk andCox (1986).

    62. Darch, Carnine, and Gersten (1984); Jitendraet al. (1998); Tournaki (2003).

    63. Fuchs et al. (2003a).

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    RECOMMEnDATIOn 3. InSTRuCTIOn DuRInG ThE InTERvEnTIOn ShOulD BE ExPlICIT AnD SySTEMATIC

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    to perorm operations and reasons or

    the procedures.64 Interventionists should

    think aloud (make their thinking pro-

    cesses public) as they model each step o

    the process.65,66 They should not only tell

    students about the steps and proceduresthey are perorming, but also allude to the

    reasoning behind them (link to the under-

    lying mathematics).

    The panel suggests that districts select

    instructional materials that provide inter-

    ventionists with sample think-alouds or

    possible scenarios or explaining concepts

    and working through operations. A crite-

    rion or selecting intervention curricula

    materials should be whether or not they

    provide materials that help intervention-

    ists model or think through dicult and

    easy examples.

    In the panels view, a major aw in many

    instructional materials is that teachers are

    asked to provide only one or two models

    o how to approach a problem and that

    most o these models are or easy-to-solve

    problems. Ideally, the materials will also

    assist teachers in explaining the reason-

    ing behind the procedures and problem-solving methods.

    2. Provide stdets wit opportities to

    sove probems i a grop ad commicate

    probem-sovig strategies.

    For students to become procient in per-

    orming mathematical processes, explicit

    instruction should include scaolded prac-

    tice, where the teacher plays an active

    role and gradually transers the work to

    64. For example, Jitendra et al. (1998); Darch,Carnine, and Gersten (1984); Woodward (2006).

    65. See an example in the summary o Tournaki(2003) in appendix D.

    66. Darch, Carnine, and Gersten (1984); Jiten-dra et al. (1998); Fuchs et al. (2003a); Schunkand Cox (1986); Tournaki (2003); Wilson and Sin-delar (1991).

    the students.67 This phase o explicit in-

    struction begins with the teacher and the

    students solving problems together. As

    this phase o instruction continues, stu-

    dents should gradually complete more

    steps o the problem with decreasing guid-ance rom the teacher. Students should

    proceed to independent practice when

    they can solve the problem with little or

    no support rom the teacher.

    During guided practice, the teacher should

    ask students to communicate the strate-

    gies they are using to complete each step

    o the process and provide reasons or

    their decisions.68 In addition, the panel

    recommends that teachers ask students to

    explain their solutions.69 Note that not only

    interventionistsbut ellow studentscan

    and should communicate how they think

    through solving problems to the inter-

    ventionist and the rest o the group. This

    can acilitate the development o a shared

    language or talking about mathematical

    problem solving.70

    Teachers should give speciic eedback

    that claries what students did correctly

    and what they need to improve.71 Theyshould provide opportunities or students

    to correct their errors. For example, i a

    student has diculty solving a word prob-

    lem or solving an equation, the teacher

    should ask simple questions that guide the

    st