Assisting Students Struggling With Mathematics Response to Intervention RtI for Elementary and...
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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.
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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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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.
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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.
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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
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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.
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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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( 9 )
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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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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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
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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
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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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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
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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