Assisting Students Struggling with Mathematics

Contributed by:

Jonathan James Mon, Jan 10, 2022 11:16 PM UTC

Students struggling with mathematics may benefit from early interventions aimed at improving their mathematics ability and ultimately preventing subsequent failure. This guide provides eight specific recommendations intended to help teachers, principals, and school administrators use Response to Intervention (RtI) to identify students who need assistance in mathematics and to address the needs of these students through focused interventions. The guide provides suggestions on how to carry out each recommendation and explains how educators can overcome potential roadblocks to implementing the recommendations.

1. IES PRACTICE GUIDE WHAT WORKS CLEARINGHOUSE
Assisting Students Struggling with
Mathematics: Response to Intervention
(RtI) for Elementary and Middle Schools
NCEE 2009-4060
U.S. DEPARTMENT OF EDUCATION

2. The Institute of Education Sciences (IES) publishes practice guides in education
to bring the best available evidence and expertise to bear on the types of systemic
challenges that cannot currently be addressed by single interventions or programs.
Authors of practice guides seldom conduct the types of systematic literature searches
that are the backbone of a meta-analysis, although they take advantage of such work
when it is already published. Instead, authors use their expertise to identify the
most important research with respect to their recommendations, augmented by a
search of 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 office that is responsible for independent review
of other IES publications. A critical task for peer reviewers of a practice guide is to
determine whether the evidence cited in support of particular recommendations is
up-to-date and that studies of similar or better quality that point in a different di-
rection have not been ignored. Because practice guides depend on the expertise of
their authors and their group decisionmaking, the content of a practice guide is not
and should not be viewed as a set of recommendations that in every case depends
on and flows inevitably from scientific research.
The goal of this practice guide is to formulate specific and coherent evidence-based
recommendations for use by educators addressing the challenge of reducing the
number of children who struggle with mathematics by using “response to interven-
tion” (RtI) as a means of both identifying students who need more help and provid-
ing these students with high-quality interventions. The guide provides practical,
clear information on critical topics related to RtI and is based on the best available
evidence as judged by the panel. Recommendations in this guide should not be
construed to imply that no further research is warranted on the effectiveness of
particular strategies used in RtI for students struggling with mathematics.

3. IES PRACTICE GUIDE
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
University of 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
Staff
Joseph Dimino
Madhavi Jayanthi
Rebecca Newman-Gonchar
Instructional Research Group
Shannon Monahan
Libby Scott
Mathematica Policy Research
NCEE 2009-4060
U.S. DEPARTMENT OF EDUCATION

4. This report was prepared for the National Center for Education Evaluation and Regional
Assistance, Institute of Education Sciences under Contract ED-07-CO-0062 by the What
Works Clearinghouse, which is operated by Mathematica Policy Research, Inc.
The opinions and positions expressed in this practice guide are the authors’ and do
not necessarily represent the opinions and positions of the Institute of Education Sci-
ences or the U.S. Department of Education. This practice guide should be reviewed
and applied according to the specific needs of the educators and education agency
using it, and with full realization that it represents the judgments of the review
panel regarding what constitutes sensible practice, based on the research available
at the time of publication. This practice guide should be used as a tool to assist in
decisionmaking rather than as a “cookbook.” Any references within the document
to specific educational products are illustrative and do not imply endorsement of
these products to the exclusion of other products that are not referenced.
U.S. Department of Education
Arne Duncan
Institute of Education Sciences
Sue Betka
Acting Director
National Center for Education Evaluation and Regional Assistance
Phoebe Cottingham
April 2009
This report is in the public domain. Although permission to reprint this publication
is 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 of
Education Sciences, U.S. Department of Education. Retrieved from http://ies.
This report is available on the IES website at http://ies.ed.gov/ncee and http://ies.
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5. Assisting Students Struggling with Mathematics:
Response to Intervention (RtI) for
Elementary and Middle Schools
Introduction 1
The What Works Clearinghouse standards and their relevance to this guide 3
Overview 4
Summary of the Recommendations 5
Scope of the practice guide 9
Checklist for carrying out the recommendations 11
Recommendation 1. Screen all students to identify those at risk
for potential mathematics difficulties and provide interventions
to students identified as at risk. 13
Recommendation 2. Instructional materials for students receiving
interventions should focus intensely on in-depth treatment of whole
numbers in kindergarten through grade 5 and on rational numbers in
grades 4 through 8. These materials should be selected by committee. 18
Recommendation 3. Instruction during the intervention should be explicit
and systematic. This includes providing models of proficient problem solving,
verbalization of thought processes, guided practice, corrective feedback,
and frequent cumulative review. 21
Recommendation 4. Interventions should include instruction on solving
word problems that is based on common underlying structures. 26
Recommendation 5. Intervention materials should include opportunities
for students to work with visual representations of mathematical ideas and
interventionists should be proficient in the use of visual representations of
mathematical ideas. 30
Recommendation 6. Interventions at all grade levels should devote about
10 minutes in each session to building fluent retrieval of basic arithmetic facts. 37
Recommendation 7. Monitor the progress of students receiving supplemental
instruction and other students who are at risk. 41
Recommendation 8. Include motivational strategies in tier 2 and tier 3
interventions. 44
Glossary of terms as used in this report 48
Appendix A. Postscript from the Institute of Education Sciences 52
Appendix B. About the authors 55
Appendix C. Disclosure of potential conflicts of interest 59
Appendix D. Technical information on the studies 61
References 91
( iii )

6. ASSISTING STUDENTS STRUGGLING WITH MATHEMATICS: RESPONSE TO INTERVENTION (RTI)
FOR ELEMENTARY AND MIDDLE SCHOOLS
List of tables
Table 1. Institute of Education Sciences levels of evidence for practice guides 2
Table 2. Recommendations and corresponding levels of evidence 6
Table 3. Sensitivity and specificity 16
Table D1. Studies of interventions that included explicit instruction
and met WWC Standards (with and without reservations) 69
Table D2. Studies of interventions that taught students to discriminate
problem types that met WWC standards (with or without reservations) 73
Table D3. Studies of interventions that used visual representations
that met WWC standards (with and without reservations) 77–78
Table D4. Studies of interventions that included fact fluency practices
that met WWC standards (with and without reservations) 83
List of examples
Example 1. Change problems 27
Example 2. Compare problems 28
Example 3. Solving different problems with the same strategy 29
Example 4. Representation of the counting on strategy using a number line 33
Example 5. Using visual representations for multidigit addition 34
Example 6. Strip diagrams can help students make sense of fractions 34
Example 7. Manipulatives can help students understand that four multiplied
by six means four groups of six, which means 24 total objects 35
Example 8. A set of matched concrete, visual, and abstract representations
to teach solving single-variable equations 35
Example 9: Commutative property of multiplication 48
Example 10: Make-a-10 strategy 49
Example 11: Distributive property 50
Example 12: Number decomposition 51
( iv )

7. Introduction In some cases, recommendations reflect
evidence-based practices that have been
Students struggling with mathematics may demonstrated as effective through rigor-
benefit from early interventions aimed at ous research. In other cases, when such
improving their mathematics ability and evidence is not available, the recommen-
ultimately preventing subsequent failure. dations reflect what this panel believes are
This guide provides eight specific recom- best practices. Throughout the guide, we
mendations intended to help teachers, clearly indicate the quality of the evidence
principals, and school administrators use that supports each recommendation.
Response to Intervention (RtI) to identify
students who need assistance in mathe- Each recommendation receives a rating
matics and to address the needs of these based on the strength of the research evi-
students through focused interventions. dence that has shown the effectiveness of a
The guide provides suggestions on how recommendation (table 1). These ratings—
to carry out each recommendation and strong, moderate, or low—have been de-
explains how educators can overcome fined as follows:
potential roadblocks to implementing the
recommendations. Strong refers to consistent and generaliz-
able evidence that an intervention pro-
The recommendations were developed by gram causes better outcomes.1
a panel of researchers and practitioners
with expertise in various dimensions of Moderate refers either to evidence from
this topic. The panel includes a research studies that allow strong causal conclu-
mathematician active in issues related sions but cannot be generalized with as-
to K–8 mathematics education, two pro- surance to the population on which a
fessors of mathematics education, sev- recommendation is focused (perhaps be-
eral special educators, and a mathematics cause the findings have not been widely
coach currently providing professional de- replicated)—or to evidence from stud-
velopment in mathematics in schools. The ies that are generalizable but have more
panel members worked collaboratively to causal ambiguity than offered by experi-
develop recommendations based on the mental designs (such as statistical models
best available research evidence and our of correlational data or group comparison
expertise in mathematics, special educa- designs for which the equivalence of the
tion, research, and practice. groups at pretest is uncertain).
The body of evidence we considered in de- Low refers to expert opinion based on rea-
veloping these recommendations included sonable extrapolations from research and
evaluations of mathematics interventions theory on other topics and evidence from
for low-performing students and students studies that do not meet the standards for
with learning disabilities. The panel con- moderate or strong evidence.
sidered high-quality experimental and
quasi-experimental studies, such as those
meeting the criteria of the What Works
Clearinghouse (http://www.whatworks.
ed.gov), to provide the strongest evidence
of effectiveness. We also examined stud-
ies of the technical adequacy of batteries
1. Following WWC guidelines, we consider a posi-
of screening and progress monitoring tive, statistically significant effect or large effect
measures for recommendations relating size (i.e., greater than 0.25) as an indicator of
to assessment. positive effects.
(1)

8. Introduction
Table 1. Institute of Education Sciences levels of evidence for practice guides
In general, characterization of the evidence for 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 of the range
of participants and settings on which the recommendation is focused to support the conclu-
sion that the results can be generalized to those participants and settings). Strong evidence
for this practice guide is operationalized as:
• A systematic review of research that generally meets the standards of the What Works
Clearinghouse (WWC) (see http://ies.ed.gov/ncee/wwc/) and supports the effectiveness of
Strong a program, practice, or approach with no contradictory evidence of similar quality; OR
• Several well-designed, randomized controlled trials or well-designed quasi-experiments
that generally meet the standards of WWC and support the effectiveness of a program,
practice, or approach, with no contradictory evidence of similar quality; OR
• One large, well-designed, randomized controlled, multisite trial that meets WWC standards
and supports the effectiveness of a program, practice, or approach, with no contradictory
evidence of similar quality; OR
• For assessments, evidence of reliability and validity that meets the Standards for Educa-
tional and Psychological Testing.a
In general, characterization of the evidence for 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 from
studies that support strong causal conclusions but when generalization is uncertain, or stud-
ies that support the generality of a relationship but when the causality is uncertain. Moderate
evidence for this practice guide is operationalized as:
• Experiments or quasi-experiments generally meeting the standards of WWC and sup-
porting the effectiveness of a program, practice, or approach with small sample sizes
and/or other conditions of implementation or analysis that limit generalizability and
no contrary evidence; OR
Moderate • Comparison group studies that do not demonstrate equivalence of groups at pre-
test and therefore do not meet the standards of WWC but that (a) consistently show
enhanced outcomes for participants experiencing a particular program, practice, or
approach and (b) have no major flaws related to internal validity other than lack of
demonstrated equivalence at pretest (e.g., only one teacher or one class per condition,
unequal amounts of instructional time, highly biased outcome measures); OR
• Correlational research with strong statistical controls for selection bias and for dis-
cerning influence of endogenous factors and no contrary evidence; OR
• For assessments, evidence of reliability that meets the Standards for Educational and
Psychological Testingb but with evidence of validity from samples not adequately rep-
resentative of the population on which the recommendation is focused.
In general, characterization of the evidence for a recommendation as low means that the
recommendation is based on expert opinion derived from strong findings or theories in
Low related areas and/or expert opinion buttressed by direct evidence that does not rise to
the moderate or strong levels. Low evidence is operationalized as evidence not meeting
the standards for the moderate or high levels.
a. American Educational Research Association, American Psychological Association, and National Council on
Measurement in Education (1999).
b. Ibid.
(2)

9. Introduction
The What Works Clearinghouse Following the recommendations and sug-
standards and their relevance to gestions for carrying out the recommen-
this guide dations, Appendix D presents information
on the research evidence to support the
The panel relied on WWC evidence stan- recommendations.
dards to assess the quality of evidence
supporting mathematics intervention pro- The panel would like to thank Kelly Hay-
grams and practices. The WWC addresses mond for her contributions to the analysis,
evidence for the causal validity of instruc- the WWC reviewers for their contribution
tional programs and practices according to to the project, and Jo Ellen Kerr and Jamila
WWC standards. Information about these Henderson for their support of the intricate
standards is available at http://ies.ed.gov/ logistics of the project. We also would like
ncee/wwc/references/standards/. The to thank Scott Cody for his oversight of the
technical quality of each study is rated and overall progress of the practice guide.
placed into one of three categories:
Dr. Russell Gersten
• Meets Evidence Standards—for random- Dr. Sybilla Beckmann
ized controlled trials and regression Dr. Benjamin Clarke
discontinuity studies that provide the Dr. Anne Foegen
strongest evidence of causal validity. Ms. Laurel Marsh
Dr. Jon R. Star
• Meets Evidence Standards with Reser- Dr. Bradley Witzel
vations—for all quasi-experimental
studies with no design flaws and ran-
domized controlled trials that have
problems with randomization, attri-
tion, or disruption.
• Does Not Meet Evidence Screens—for
studies that do not provide strong evi-
dence of causal validity.
(3)

10. Assisting Students concluded that all students should receive
Struggling with preparation from an early age to ensure
their later success in algebra. In particular,
Mathematics: Response the report emphasized the need for math-
to Intervention (RtI) ematics interventions that mitigate and
prevent mathematics difficulties.
for Elementary and
Middle Schools This panel believes that schools can use an
RtI framework to help struggling students
prepare for later success in mathemat-
Overview ics. To date, little research has been con-
ducted to identify the most effective ways
Response to Intervention (RtI) is an early de- to initiate and implement RtI frameworks
tection, prevention, and support system that for mathematics. However, there is a rich
identifies struggling students and assists body of research on effective mathematics
them before they fall behind. In the 2004 interventions implemented outside an RtI
reauthorization of the Individuals with Dis- framework. Our goal in this practice guide
abilities Education Act (PL 108-446), states is to provide suggestions for assessing
were encouraged to use RtI to accurately students’ mathematics abilities and imple-
identify students with learning disabilities menting mathematics interventions within
and encouraged to provide additional sup- an RtI framework, in a way that reflects
ports for students with academic difficul- the best evidence on effective practices in
ties regardless of disability classification. mathematics interventions.
Although many states have already begun to
implement RtI in the area of reading, RtI ini- RtI begins with high-quality instruction
tiatives for mathematics are relatively new. and universal screening for all students.
Whereas high-quality instruction seeks to
Students’ low achievement in mathemat- prevent mathematics difficulties, screen-
ics is a matter of national concern. The re- ing allows for early detection of difficul-
cent National Mathematics Advisory Panel ties if they emerge. Intensive interventions
Report released in 2008 summarized the are then provided to support students
poor showing of students in the United in need of assistance with mathematics
States on international comparisons of learning.4 Student responses to interven-
mathematics performance such as the tion are measured to determine whether
Trends in International Mathematics and they have made adequate progress and (1)
Science Study (TIMSS) and the Program for no longer need intervention, (2) continue
International Student Assessment (PISA).2 to need some intervention, or (3) need
A recent survey of algebra teachers as- more intensive intervention. The levels of
sociated with the report identified key intervention are conventionally referred
deficiencies of students entering algebra, to as “tiers.” RtI is typically thought of as
including aspects of whole number arith- having three tiers.5 Within a three-tiered
metic, fractions, ratios, and proportions.3 RtI model, each tier is defined by specific
The National Mathematics Advisory Panel characteristics.
2. See, for example, National Mathematics Ad- 4. Fuchs, Fuchs, Craddock et al. (2008).
visory Panel (2008) and Schmidt and Houang 5. Fuchs, Fuchs, and Vaughn (2008) make the
(2007). For more information on the TIMSS, see case for a three-tier RtI model. Note, however,
http://nces.ed.gov/timss/. For more information that some states and school districts have imple-
on PISA, see http://www.oecd.org. mented multitier intervention systems with more
3. National Mathematics Advisory Panel (2008). than three tiers.
(4)

11. Overview
• Tier 1 is the mathematics instruction student performance data is critical in
that all students in a classroom receive. this tier. Typically, specialized person-
It entails universal screening of all stu- nel, such as special education teachers
dents, regardless of mathematics profi- and school psychologists, are involved
ciency, using valid measures to identify in tier 3 and special education services.14
students at risk for future academic However, students often receive rele-
failure—so that they can receive early vant mathematics interventions from a
intervention.6 There is no clear consen- wide array of school personnel, includ-
sus on the characteristics of instruction ing their classroom teacher.
other than that it is “high quality.”7
Summary of the Recommendations
• In tier 2 interventions, schools provide
additional assistance to students who This practice guide offers eight recom-
demonstrate difficulties on screening mendations for identifying and supporting
measures or who demonstrate weak students struggling in mathematics (table
progress.8 Tier 2 students receive sup- 2). The recommendations are intended to
plemental small group mathematics be implemented within an RtI framework
instruction aimed at building targeted (typically three-tiered). The panel chose to
mathematics proficiencies.9 These in- limit its discussion of tier 1 to universal
terventions are typically provided for screening practices (i.e., the guide does
20 to 40 minutes, four to five times each not make recommendations for general
week.10 Student progress is monitored classroom mathematics instruction). Rec-
throughout the intervention.11 ommendation 1 provides specific sugges-
tions for conducting universal screening
• Tier 3 interventions are provided to effectively. For RtI tiers 2 and 3, recom-
students who are not benefiting from mendations 2 though 8 focus on the most
tier 2 and require more intensive as- effective content and pedagogical prac-
sistance.12 Tier 3 usually entails one- tices that can be included in mathematics
on-one tutoring along with an appropri- interventions.
ate mix of instructional interventions.
In some cases, special education ser- Throughout this guide, we use the term
vices are included in tier 3, and in oth- “interventionist” to refer to those teach-
ers special education is considered an ing the intervention. At a given school, the
additional tier.13 Ongoing analysis of interventionist may be the general class-
room teacher, a mathematics coach, a spe-
6. For reviews see Jiban and Deno (2007); Fuchs, cial education instructor, other certified
Fuchs, Compton et al. (2007); Gersten, Jordan, school personnel, or an instructional as-
and Flojo (2005). sistant. The panel recognizes that schools
7. National Mathematics Advisory Panel (2008); rely on different personnel to fill these
National Research Council (2001).
roles depending on state policy, school
8. Fuchs, Fuchs, Craddock et al. (2008); Na- resources, and preferences.
tional Joint Committee on Learning Disabilities
9. Fuchs, Fuchs, Craddock et al. (2008).
Recommendation 1 addresses the type of
10. For example, see Jitendra et al. (1998) and
screening measures that should be used in
Fuchs, Fuchs, Craddock et al. (2008). tier 1. We note that there is more research
11. National Joint Committee on Learning Dis- on valid screening measures for students in
abilities (2005).
12. Fuchs, Fuchs, Craddock et al. (2008).
Fuchs, Fuchs, Craddock et al. ��
(2008); National 14. National Joint Committee on Learning Dis-
Joint Committee on Learning Disabilities (2005). abilities (2005).
(5)

12. Table 2. Recommendations and corresponding levels of evidence
Recommendation Level of evidence
Tier 1
1. Screen all students to identify those at risk for potential mathematics
Moderate
difficulties and provide interventions to students identified as at risk.
Tiers 2 and 3
2. Instructional materials for students receiving interventions should
focus intensely on in-depth treatment of whole numbers in kindergar-
Low
ten through grade 5 and on rational numbers in grades 4 through 8.
These materials should be selected by committee.
3. Instruction during the intervention should be explicit and systematic.
This includes providing models of proficient problem solving, verbal-
Strong
ization of thought processes, guided practice, corrective feedback, and
frequent cumulative review.
4. Interventions should include instruction on solving word problems
Strong
that is based on common underlying structures.
5. Intervention materials should include opportunities for students to
work with visual representations of mathematical ideas and interven-
Moderate
tionists should be proficient in the use of visual representations of
mathematical ideas.
6. Interventions at all grade levels should devote about 10 minutes in each
Moderate
session to building fluent retrieval of basic arithmetic facts.
7. Monitor the progress of students receiving supplemental instruction
Low
and other students who are at risk.
8. Include motivational strategies in tier 2 and tier 3 interventions. Low
Source: Authors’ compilation based on analysis described in text.
(6)

13. Overview
kindergarten through grade 2,15 but there providing professional development for
are also reasonable strategies to use for stu- interventionists.
dents in more advanced grades.16 We stress
that no one screening measure is perfect Next, we highlight several areas of re-
and that schools need to monitor the prog- search that have produced promising find-
ress of students who score slightly above or ings in mathematics interventions. These
slightly below any screening cutoff score. include systematically teaching students
about the problem types associated with
Recommendations 2 though 6 address the a given operation and its inverse (such as
content of tier 2 and tier 3 interventions problem types that indicate addition and
and the types of instructional strategies subtraction) (recommendation 4).18 We also
that should be used. In recommendation 2, recommend practices to help students
we translate the guidance by the National translate abstract symbols and numbers
Mathematics Advisory Panel (2008) and into meaningful visual representations
the National Council of Teachers of Math- (recommendation 5).19 Another feature
ematics Curriculum Focal Points (2006) that we identify as crucial for long-term
into suggestions for the content of inter- success is systematic instruction to build
vention curricula. We argue that the math- quick retrieval of basic arithmetic facts
ematical focus and the in-depth coverage (recommendation 6). Some evidence exists
advocated for proficient students are also supporting the allocation of time in the in-
necessary for students with mathematics tervention to practice fact retrieval using
difficulties. For most students, the content flash cards or computer software.20 There
of interventions will include foundational is also evidence that systematic work with
concepts and skills introduced earlier in properties of operations and counting
the student’s career but not fully under- strategies (for younger students) is likely
stood and mastered. Whenever possible, to promote growth in other areas of math-
links should be made between founda- ematics beyond fact retrieval.21
tional mathematical concepts in the inter-
vention and grade-level material. The final two recommendations address
other considerations in implementing tier
At the center of the intervention recom- 2 and tier 3 interventions. Recommenda-
mendations is that instruction should be tion 7 addresses the importance of moni-
systematic and explicit (recommendation toring the progress of students receiving
3). This is a recurrent theme in the body
of valid scientific research.17 We explore
18. Jitendra et al. (1998); Xin, Jitendra, and Deat-
the multiple meanings of explicit instruc- line-Buchman (2005); Darch, Carnine, and Gersten
tion and indicate which components of (1984); Fuchs et al. (2003a); Fuchs et al. (2003b);
explicit instruction appear to be most re- Fuchs, Fuchs, Prentice et al. (2004); Fuchs, Fuchs,
lated to improved student outcomes. We and Finelli (2004); Fuchs, Fuchs, Craddock et al.
(2008) Fuchs, Seethaler et al. (2008).
believe this information is important for
19. Artus and Dyrek (1989); Butler et al. (2003);
districts and state departments to have
Darch, Carnine, and Gersten (1984); Fuchs et
as they consider selecting materials and al. (2005); Fuchs, Seethaler et al. (2008); Fuchs,
Powell et al. (2008); Fuchs, Fuchs, Craddock et
15. Gersten, Jordan, and Flojo (2005); Gersten, al. (2008); Jitendra et al. (1998); Walker and Po-
Clarke, and Jordan (2007). teet (1989); Wilson and Sindelar (1991); Witzel
16. Jiban and Deno (2007); Foegen, Jiban, and (2005); Witzel, Mercer, and Miller (2003); Wood-
Deno (2007). ward (2006).
17. Darch, Carnine, and Gersten (1984); Fuchs 20. Beirne-Smith (1991); Fuchs, Seethaler et al.
et al. (2003a); Jitendra et al. (1998); Schunk and (2008); Fuchs et al. (2005); Fuchs, Fuchs, Hamlett
Cox (1986); Tournaki (2003); Wilson and Sindelar et al. (2006); Fuchs, Powell et al. (2008).
(1991). 21. Tournaki (2003); Woodward (2006).
(7)

14. interventions. Specific types of formative and those with scores slightly above or
assessment approaches and measures are below the cutoff score on screening mea-
described. We argue for two types of ongo- sures with broader measures of mathemat-
ing assessment. One is the use of curricu- ics proficiency. This information provides
lum-embedded assessments that gauge how the school with a sense of how the overall
well students have learned the material in mathematics program (including tier 1, tier
that day’s or week’s lesson(s). The panel 2, and tier 3) is affecting a given student.
believes this information is critical for in-
terventionists to determine whether they Recommendation 8 addresses the impor-
need to spend additional time on a topic. It tant issue of motivation. Because many of
also provides the interventionist and other the students struggling with mathematics
school personnel with information that have experienced failure and frustration
can be used to place students in groups by the time they receive an intervention,
within tiers. In addition, we recommend we suggest tools that can encourage active
that schools regularly monitor the prog- engagement of students and acknowledge
ress of students receiving interventions student accomplishments.
(8)

15. Scope of the The scope of this guide does not include
practice guide recommendations for special education
referrals. Although enhancing the valid-
ity of special education referrals remains
Our goal is to provide evidence-based sug- important and an issue of ongoing discus-
gestions for screening students for mathe- sion23 and research,24 we do not address
matics difficulties, providing interventions it in this practice guide, in part because
to students who are struggling, and moni- empirical evidence is lacking.
toring student responses to the interven-
tions. RtI intentionally cuts across the bor- The discussion of tier 1 in this guide re-
ders of special and general education and volves only around effective screening, be-
involves school-wide collaboration. There- cause recommendations for general class-
fore, our target audience for this guide in- room mathematics instruction were beyond
cludes teachers, special educators, school the scope of this guide. For this reason,
psychologists and counselors, as well as studies of effective general mathematics
administrators. Descriptions of the ma- instruction practices were not included in
terials and instructional content in tier 2 the evidence base for this guide.25
and tier 3 interventions may be especially
useful to school administrators selecting The studies reviewed for this guide in-
interventions, while recommendations cluded two types of comparisons among
that relate to content and pedagogy will groups. First, several studies of tier 2 in-
be most useful to interventionists.22 terventions compare students receiving
multicomponent tier 2 interventions with
The focus of this guide is on providing students receiving only routine classroom
RtI interventions in mathematics for stu- instruction.26 This type of study provides
dents in kindergarten through grade 8. This evidence of the effectiveness of providing
broad grade range is in part a response tier 2 interventions but does not permit
to the recent report of the National Math- conclusions about which component is
ematics Advisory Panel (2008), which em- most effective. The reason is that it is not
phasized a unified progressive approach possible to identify whether one particular
to promoting mathematics proficiency for component or a combination of compo-
elementary and middle schools. Moreover, nents within a multicomponent interven-
given the growing number of initiatives tion produced an effect. Second, several
aimed at supporting students to succeed
in algebra, the panel believes it essential
23. Kavale and Spaulding (2008); Fuchs, Fuchs,
to provide tier 2 and tier 3 interventions to and Vaughn (2008); VanDerHeyden, Witt, and
struggling students in grades 4 through 8. Gilbertson (2007).
Because the bulk of research on mathemat- 24. Fuchs, Fuchs, Compton et al. (2006).
ics interventions has focused on students 25. There were a few exceptions in which general
in kindergarten through grade 4, some rec- mathematics instruction studies were included in
ommendations for students in older grades the evidence base. When the effects of a general
mathematics instruction program were specified
are extrapolated from this research.
for low-achieving or disabled students and the
intervention itself appeared applicable to teach-
ing tier 2 or tier 3 (e.g., teaching a specific opera-
tional strategy), we included them in this study.
Note that disabled students were predominantly
22. Interventionists may be any number of school learning disabled.
personnel, including classroom teachers, special 26. For example, Fuchs, Seethaler et al. (2008)
educators, school psychologists, paraprofession- examined the effects of providing supplemen-
als, and mathematics coaches and specialists. tal tutoring (i.e., a tier 2 intervention) relative to
The panel does not specify the interventionist. regular classroom instruction (i.e., tier 1).
(9)

16. Scope of the practice guide
other studies examined the effects of two or estimation), concepts (knowledge of
methods of tier 2 or tier 3 instruction.27 properties of operations, concepts involv-
This type of study offers evidence for the ing rational numbers, prealgebra con-
effectiveness of one approach to teaching cepts), problem solving (word problems),
within a tier relative to another approach and measures of general mathematics
and assists with identifying the most ben- achievement. Measures of fact fluency
eficial approaches for this population. were also included because quick retrieval
of basic arithmetic facts is essential for
The panel reviewed only studies for prac- success in mathematics and a persistent
tices that sought to improve student math- problem for students with difficulties in
ematics outcomes. The panel did not con- mathematics.28
sider interventions that improved other
academic or behavioral outcomes. Instead, Technical terms related to mathematics
the panel focused on practices that ad- and technical aspects of assessments (psy-
dressed the following areas of mathematics chometrics) are defined in a glossary at the
proficiency: operations (either computation end of the recommendations.
27. For example, Tournaki (2003) examined the
effects of providing supplemental tutoring in an
operations strategy (a tier 2 intervention) relative
to supplemental tutoring with a drill and practice 28. Geary (2004); Jordan, Hanich, and Kaplan
approach (also a tier 2 intervention). (2003).
( 10 )

17. Checklist for carrying out the in-depth coverage of rational numbers as
recommendations well as advanced topics in whole number
arithmetic (such as long division).
Recommendation 1. Screen all
students to identify those at risk for  Districts should appoint committees,
potential mathematics difficulties and including experts in mathematics instruc-
provide interventions to students tion and mathematicians with knowledge
identified as at risk. of elementary and middle school math-
ematics curricula, to ensure that specific
 As a district or school sets up a screen- criteria are covered in-depth in the cur-
ing system, have a team evaluate potential riculum they adopt.
screening measures. The team should se-
lect measures that are efficient and reason- Recommendation 3. Instruction during
ably reliable and that demonstrate predic- the intervention should be explicit and
tive validity. Screening should occur in the systematic. This includes providing
beginning and middle of the year. models of proficient problem solving,
verbalization of thought processes,
 Select screening measures based on guided practice, corrective feedback,
the content they cover, with an emphasis and frequent cumulative review.
on critical instructional objectives for each
grade.  Ensure that instructional materials are
systematic and explicit. In particular, they
 In grades 4 through 8, use screen- should include numerous clear models of
ing data in combination with state testing easy and difficult problems, with accom-
results. panying teacher think-alouds.
 Use the same screening tool across a  Provide students with opportunities
district to enable analyzing results across to solve problems in a group and commu-
schools. nicate problem-solving strategies.
Recommendation 2. Instructional  Ensure that instructional materials in-
materials for students receiving clude cumulative review in each session.
interventions should focus intensely
on in-depth treatment of whole Recommendation 4. Interventions
numbers in kindergarten through should include instruction on solving
grade 5 and on rational numbers in word problems that is based on
grades 4 through 8. These materials common underlying structures.
should be selected by committee.
 Teach students about the structure of
 For students in kindergarten through various problem types, how to categorize
grade 5, tier 2 and tier 3 interventions problems based on structure, and how to
should focus almost exclusively on prop- determine appropriate solutions for each
erties of whole numbers and operations. problem type.
Some older students struggling with
whole numbers and operations would  Teach students to recognize the com-
also benefit from in-depth coverage of mon underlying structure between famil-
these topics. iar and unfamiliar problems and to transfer
known solution methods from familiar to
 For tier 2 and tier 3 students in grades unfamiliar problems.
4 through 8, interventions should focus on
( 11 )

18. Checklist for carrying out the recommendations
Recommendation 5. Intervention  Teach students in grades 2 through
materials should include opportunities 8 how to use their knowledge of proper-
for students to work with visual ties, such as commutative, associative,
representations of mathematical and distributive law, to derive facts in
ideas and interventionists should their heads.
be proficient in the use of visual
representations of mathematical ideas. Recommendation 7. Monitor the
progress of students receiving
 Use visual representations such as supplemental instruction and other
number lines, arrays, and strip diagrams. students who are at risk.
 If visuals are not sufficient for develop-  Monitor the progress of tier 2, tier 3,
ing accurate abstract thought and answers, and borderline tier 1 students at least once
use concrete manipulatives first. Although a month using grade-appropriate general
this can also be done with students in upper outcome measures.
elementary and middle school grades, use
of manipulatives with older students should  Use curriculum-embedded assess-
be expeditious because the goal is to move ments in interventions to determine
toward understanding of—and facility whether students are learning from the
with—visual representations, and finally, to intervention. These measures can be used
the abstract. as often as every day or as infrequently as
once every other week.
Recommendation 6. Interventions at
all grade levels should devote about  Use progress monitoring data to re-
10 minutes in each session to building group students when necessary.
fluent retrieval of basic arithmetic facts.
Recommendation 8. Include
 Provide about 10 minutes per ses- motivational strategies in tier 2 and
sion of instruction to build quick retrieval tier 3 interventions.
of basic arithmetic facts. Consider using
technology, flash cards, and other materi-  Reinforce or praise students for their
als for extensive practice to facilitate au- effort and for attending to and being en-
tomatic retrieval. gaged in the lesson.
 For students in kindergarten through  Consider rewarding student accom
grade 2, explicitly teach strategies for ef- plishments.
ficient counting to improve the retrieval of
mathematics facts.  Allow students to chart their progress
and to set goals for improvement.
( 12 )

19. Recommendation 1. aspects of what is often referred to as
Screen all students to number sense.30 They assess various as-
pects of knowledge of whole numbers—
identify those at risk for properties, basic arithmetic operations,
potential mathematics understanding of magnitude, and applying
mathematical knowledge to word prob-
difficulties and provide lems. Some measures contain only one
interventions to aspect of number sense (such as magni-
tude comparison) and others assess four
students identified to eight aspects of number sense. The sin-
as at risk. gle-component approaches with the best
ability to predict students’ subsequent
mathematics performance include screen-
The panel recommends that schools ing measures of students’ knowledge of
and districts systematically use magnitude comparison and/or strategic
universal screening to screen all counting.31 The broader, multicomponent
students to determine which students measures seem to predict with slightly
have mathematics difficulties and greater accuracy than single-component
require research-based interventions. measures.32
Schools should evaluate and select
screening measures based on their Effective approaches to screening vary in
reliability and predictive validity, with efficiency, with some taking as little as 5
particular emphasis on the measures’ minutes to administer and others as long
specificity and sensitivity. Schools as 20 minutes. Multicomponent measures,
should also consider the efficiency of which by their nature take longer to ad-
the measure to enable screening many minister, tend to be time-consuming for
students in a short time. administering to an entire school popu-
lation. Timed screening measures33 and
Level of evidence: Moderate untimed screening measures34 have been
shown to be valid and reliable.
The panel judged the level of evidence sup-
porting this recommendation to be mod- For the upper elementary grades and mid-
erate. This recommendation is based on a dle school, we were able to locate fewer
series of high-quality correlational studies studies. They suggest that brief early
with replicated findings that show the abil- screening measures that take about 10
ity of measures to predict performance in minutes and cover a proportional sam-
mathematics one year after administration pling of grade-level objectives are reason-
(and in some cases two years).29 able and provide sufficient evidence of reli-
ability.35 At the current time, this research
Brief summary of evidence to area is underdeveloped.
support the recommendation
A growing body of evidence suggests that 30. Berch (2005); Dehaene (1999); Okamoto and
there are several valid and reliable ap- Case (1996); Gersten and Chard (1999).
proaches for screening students in the pri- 31. Gersten, Jordan, and Flojo (2005).
mary grades. All these approaches target 32. Fuchs, Fuchs, Compton et al. (2007).
33. For example, Clarke and Shinn (2004).
29. For reviews see Jiban and Deno (2007); Fuchs, 34. For example, Okamoto and Case (1996).
Fuchs, Compton et al. (2007); Gersten, Jordan, 35. Jiban and Deno (2007); Foegen, Jiban, and
and Flojo (2005). Deno (2007).
( 13 )

20. Recommendation 1. Screen all students to identify those at risk
How to carry out this more than 20 minutes to administer,
recommendation which enables collecting a substantial
amount of information in a reasonable
1. As a district or school sets up a screen- time frame. Note that many screening
ing system, have a team evaluate potential measures take five minutes or less.38 We
screening measures. The team should select recommend that schools select screen-
measures that are efficient and reasonably ing measures that have greater effi-
reliable and that demonstrate predictive va- ciency if their technical adequacy (pre-
lidity. Screening should occur in the begin- dictive validity, reliability, sensitivity,
ning and middle of the year. and specificity) is roughly equivalent
to less efficient measures. Remember
The team that selects the measures should that screening measures are intended
include individuals with expertise in mea- for administration to all students in a
surement (such as a school psychologist or school, and it may be better to invest
a member of the district research and eval- more time in diagnostic assessment of
uation division) and those with expertise in students who perform poorly on the
mathematics instruction. In the opinion of universal screening measure.
the panel, districts should evaluate screen-
ing measures on three dimensions. Keep in mind that screening is just a means
of determining which students are likely to
• Predictive validity is an index of how need help. If a student scores poorly on a
well a score on a screening measure screening measure or screening battery—
earlier in the year predicts a student’s especially if the score is at or near a cut
later mathematics achievement. Greater point, the panel recommends monitoring
predictive validity means that schools her or his progress carefully to discern
can be more confident that decisions whether extra instruction is necessary.
based on screening data are accurate.
In general, we recommend that schools Developers of screening systems recom-
and districts employ measures with mend that screening occur at least twice
predictive validity coefficients of at a year (e.g., fall, winter, and/or spring).39
least .60 within a school year.36 This panel recommends that schools alle-
viate concern about students just above or
• Reliability is an index of the consistency below the cut score by screening students
and precision of a measure. We recom- twice during the year. The second screen-
mend measures with reliability coeffi- ing in the middle of the year allows another
cients of .80 or higher.37 check on these students and also serves to
identify any students who may have been at
• Efficiency is how quickly the universal risk and grown substantially in their mathe-
screening measure can be adminis- matics achievement—or those who were on-
tered, scored, and analyzed for all the track at the beginning of the year but have
students. As a general rule, we suggest not shown sufficient growth. The panel
that a screening measure require no considers these two universal screenings
to determine student proficiency as distinct
36. A coefficient of .0 indicates that there is no from progress monitoring (Recommenda-
relation between the early and later scores, and tion 7), which occurs on a more frequent
a coefficient of 1.0 indicates a perfect positive
relation between the scores.
37. A coefficient of .0 indicates that there is no 38. Foegen, Jiban, and Deno (2007); Fuchs, Fuchs,
relation between the two scores, and a coeffi- Compton et al. (2007); Gersten, Clarke, and Jordan
cient of 1.0 indicates a perfect positive relation (2007).
between the scores. 39. Kaminski et al. (2008); Shinn (1989).
( 14 )

21. Recommendation 1. Screen all students to identify those at risk
basis (e.g., weekly or monthly) with a select these grade levels, districts, county offices,
group of intervention students in order to or state departments may need to develop
monitor response to intervention. additional screening and diagnostic mea-
sures or rely on placement tests provided
2. Select screening measures based on the by developers of intervention curricula.
content they cover, with an emphasis on crit-
ical instructional objectives for each grade. 4. Use the same screening tool across a district
to enable analyzing results across schools.
The panel believes that content covered
in a screening measure should reflect the The panel recommends that all schools
instructional objectives for a student’s within a district use the same screening
grade level, with an emphasis on the most measure and procedures to ensure ob-
critical content for the grade level. The Na- jective comparisons across schools and
tional Council of Teachers of Mathematics within a district. Districts can use results
(2006) released a set of focal points for from screening to inform instructional de-
each grade level designed to focus instruc- cisions at the district level. For example,
tion on critical concepts for students to one school in a district may consistently
master within a specific grade. Similarly, have more students identified as at risk,
the National Mathematics Advisory Panel and the district could provide extra re-
(2008) detailed a route to preparing all sources or professional development to
students to be successful in algebra. In the that school. The panel recommends that
lower elementary grades, the core focus of districts use their research and evaluation
instruction is on building student under- staff to reevaluate screening measures an-
standing of whole numbers. As students nually or biannually. This entails exam-
establish an understanding of whole num- ining how screening scores predict state
bers, rational numbers become the focus testing results and considering resetting
of instruction in the upper elementary cut scores or other data points linked to
grades. Accordingly, screening measures instructional decisionmaking.
used in the lower and upper elementary
grades should have items designed to as- Potential roadblocks and solutions
sess student’s understanding of whole and
rational number concepts—as well as com- Roadblock 1.1. Districts and school person-
putational proficiency. nel may face resistance in allocating time re-
sources to the collection of screening data.
3. In grades 4 through 8, use screening data
in combination with state testing results. Suggested Approach. The issue of time
and personnel is likely to be the most sig-
In the panel’s opinion, one viable option nificant obstacle that districts and schools
that schools and districts can pursue is to must overcome to collect screening data.
use results from the previous year’s state Collecting data on all students will require
testing as a first stage of screening. Students structuring the data collection process to
who score below or only slightly above a be efficient and streamlined.
benchmark would be considered for sub-
sequent screening and/or diagnostic or The panel notes that a common pitfall is
placement testing. The use of state testing a long, drawn-out data collection process,
results would allow districts and schools with teachers collecting data in their class-
to combine a broader measure that covers rooms “when time permits.” If schools are
more content with a screening measure that allocating resources (such as providing an
is narrower but more focused. Because of intervention to students with the 20 low-
the lack of available screening measures at est scores in grade 1), they must wait until
( 15 )

22. Recommendation 1. Screen all students to identify those at risk
all the data have been collected across high on the previous spring’s state as-
classrooms, thus delaying the delivery sessment, additional screening typically
of needed services to students. Further- is not required.
more, because many screening measures
are sensitive to instruction, a wide gap Roadblock 1.3. Screening measures may
between when one class is assessed and identify students who do not need services
another is assessed means that many stu- and not identify students who do need
dents in the second class will have higher services.
scores than those in the first because they
were assessed later. Suggested Approach. All screening mea-
sures will misidentify some students as
One way to avoid these pitfalls is to use data either needing assistance when they do
collection teams to screen students in a not (false positive) or not needing assis-
short period of time. The teams can consist tance when they do (false negative). When
of teachers, special education staff includ- screening students, educators will want to
ing such specialists as school psychologists, maximize both the number of students
Title I staff, principals, trained instructional correctly identified as at risk—a measure’s
assistants, trained older students, and/or sensitivity—and the number of students
local college students studying child devel- correctly identified as not at risk—a mea-
opment or school psychology. sure’s specificity. As illustrated in table 3,
screening students to determine risk can
Roadblock 1.2. Implementing universal result in four possible categories indicated
screening is likely to raise questions such by the letters A, B, C, and D. Using these
as, “Why are we testing students who are categories, sensitivity is equal to A/(A + C)
doing fine?” and specificity is equal to D/(B + D).
Suggested Approach. Collecting data Table 3. Sensitivity and specificity
on all students is new for many districts
STUDENTS
and schools (this may not be the case for ACTUALLY AT RISK
elementary schools, many of which use
Yes No
screening assessments in reading).40 But
STUDENTS Yes A (true B (false
screening allows schools to ensure that all
IDENTIFIED positives) positives)
students who are on track stay on track
AS BEING No C (false D (true
and collective screening allows schools to AT RISK negatives) negatives)
evaluate the impact of their instruction
on groups of students (such as all grade
2 students). When schools screen all stu-
dents, a distribution of achievement from The sensitivity and specificity of a mea-
high to low is created. If students consid- sure depend on the cut score to classify
ered not at risk were not screened, the children at risk.41 If a cut score is high
distribution of screened students would (where all students below the cut score are
consist only of at-risk students. This could considered at risk), the measure will have
create a situation where some students at a high degree of sensitivity because most
the “top” of the distribution are in real- students who truly need assistance will be
ity at risk but not identified as such. For
upper-grade students whose scores were
41. Sensitivity and specificity are also influenced
by the discriminant validity of the measure and
40. U.S. Department of Education, Office of Plan- its individual items. Measures with strong item
ning, Evaluation and Policy Development, Policy discrimination are more likely to correctly iden-
and Program Studies Service (2006). tify students’ risk status.
( 16 )

23. Recommendation 1. Screen all students to identify those at risk
identified as at risk. But the measure will those resources when using screening
have low specificity since many students data to make instructional decisions. Dis-
who do not need assistance will also be tricts may find that on a nationally normed
identified as at risk. Similarly, if a cut score screening measure, a large percentage of
is low, the sensitivity will be lower (some their students (such as 60 percent) will be
students in need of assistance may not be classified as at risk. Districts will have to
identified as at risk), whereas the specific- determine the resources they have to pro-
ity will be higher (most students who do vide interventions and the number of stu-
not need assistance will not be identified dents they can serve with their resources.
as at risk). This may mean not providing interven-
tions at certain grade levels or providing
Schools need to be aware of this tradeoff interventions only to students with the
between sensitivity and specificity, and lowest scores, at least in the first year of
the team selecting measures should be implementation.
aware that decisions on cut scores can be
somewhat arbitrary. Schools that set a cut There may also be cases when schools
score too high run the risk of spending re- identify large numbers of students at risk
sources on students who do not need help, in a particular area and decide to pro-
and schools that set a cut score too low run vide instruction to all students. One par-
the risk of not providing interventions to ticularly salient example is in the area of
students who are at risk and need extra in- fractions. Multiple national assessments
struction. If a school or district consistently show many students lack proficiency in
finds that students receiving intervention fractions,42 so a school may decide that,
do not need it, the measurement team rather than deliver interventions at the
should consider lowering the cut score. individual child level, they will provide a
school-wide intervention to all students. A
Roadblock 1.4. Screening data may iden- school-wide intervention can range from a
tify large numbers of students who are at supplemental fractions program to profes-
risk and schools may not immediately have sional development involving fractions.
the resources to support all at-risk students.
This will be a particularly severe problem
in low-performing Title I schools.
Suggested Approach. Districts and
schools need to consider the amount of 42. National Mathematics Advisory Panel
resources available and the allocation of (2008); Lee, Grigg, and Dion (2007).
( 17 )

24. Recommendation 2. Level of evidence: Low
Instructional materials The panel judged the level of evidence
for students receiving supporting this recommendation to be low.
interventions should This recommendation is based on the pro-
fessional opinion of the panel and several
focus intensely on recent consensus documents that reflect
in-depth treatment input from mathematics educators and re-
search mathematicians involved in issues
of whole numbers in related to kindergarten through grade 12
kindergarten through mathematics education.44
grade 5 and on rational Brief summary of evidence to
numbers in grades support the recommendation
4 through 8. These The documents reviewed demonstrate a
materials should be growing professional consensus that cov-
erage of fewer mathematics topics in more
selected by committee. depth and with coherence is important
for all students.45 Milgram and Wu (2005)
suggested that an intervention curriculum
The panel recommends that individuals for at-risk students should not be over-
knowledgeable in instruction and simplified and that in-depth coverage of
mathematics look for interventions that key topics and concepts involving whole
focus on whole numbers extensively numbers and then rational numbers is
in kindergarten through grade 5 and critical for future success in mathematics.
on rational numbers extensively in The National Council of Teachers of Math-
grades 4 through 8. In all cases, the ematics (NCTM) Curriculum Focal Points
specific content of the interventions will (2006) called for the end of brief ventures
be centered on building the student’s into many topics in the course of a school
foundational proficiencies. In making year and also suggested heavy emphasis on
this recommendation, the panel is instruction in whole numbers and rational
drawing on consensus documents numbers. This position was reinforced by
developed by experts from mathematics the 2008 report of the National Mathematics
education and research mathematicians Advisory Panel (NMAP), which provided de-
that emphasized the importance of tailed benchmarks and again emphasized
these topics for students in general.43 in-depth coverage of key topics involving
We conclude that the coverage of whole numbers and rational numbers as
fewer topics in more depth, and crucial for all students. Although the latter
with coherence, is as important, and two documents addressed the needs of all
probably more important, for students students, the panel concludes that the in-
who struggle with mathematics. depth coverage of key topics is especially
44. National Council of Teachers of Mathemat-
ics (2006); National Mathematics Advisory Panel
(2008); Milgram and Wu (2005).
45. National Mathematics Advisory Panel (2008);
43. National Council of Teachers of Mathemat- Schmidt and Houang (2007); Milgram and Wu
ics (2006); National Mathematics Advisory Panel (2005); National Council of Teachers of Math-
(2008). ematics (2006).
( 18 )

25. Recommendation 2. Instructional materials for students receiving interventions
important for students who struggle with the reasoning that underlies algorithms for
mathematics. addition and subtraction of whole num-
bers, as well as solving problems involv-
How to carry out this ing whole numbers. This focus should in-
recommendation clude understanding of the base-10 system
(place value).
1. For students in kindergarten through
grade 5, tier 2 and tier 3 interventions should Interventions should also include materi-
focus almost exclusively on properties of als to build fluent retrieval of basic arith-
whole numbers46 and operations. Some metic facts (see recommendation 6). Ma-
older students struggling with whole num- terials should extensively use—and ask
bers and operations would also benefit from students to use—visual representations of
in-depth coverage of these topics. whole numbers, including both concrete
and visual base-10 representations, as well
In the panel’s opinion, districts should as number paths and number lines (more
review the interventions they are con- information on visual representations is
sidering to ensure that they cover whole in recommendation 5).
numbers in depth. The goal is proficiency
and mastery, so in-depth coverage with 2. For tier 2 and tier 3 students in grades 4
extensive review is essential and has through 8, interventions should focus on in-
been articulated in the NCTM Curriculum depth coverage of rational numbers as well
Focal Points (2006) and the benchmarks as advanced topics in whole number arith-
determined by the National Mathematics metic (such as long division).
Advisory Panel (2008). Readers are recom-
mended to review these documents.47 The panel believes that districts should
review the interventions they are consid-
Specific choices for the content of interven- ering to ensure that they cover concepts
tions will depend on the grade level and involving rational numbers in depth. The
proficiency of the student, but the focus focus on rational numbers should include
for struggling students should be on whole understanding the meaning of fractions,
numbers. For example, in kindergarten decimals, ratios, and percents, using visual
through grade 2, intervention materials representations (including placing fractions
would typically include significant atten- and decimals on number lines,48 see recom-
tion to counting (e.g., counting up), num- mendation 5), and solving problems with
ber composition, and number decomposi- fractions, decimals, ratios, and percents.
tion (to understand place-value multidigit
operations). Interventions should cover the In the view of the panel, students in
meaning of addition and subtraction and grades 4 through 8 will also require ad-
ditional work to build fluent retrieval of
46. Properties of numbers, including the associa- basic arithmetic facts (see recommenda-
tive, commutative, and distributive properties. tion 6), and some will require additional
47. More information on the National Mathemat- work involving basic whole number top-
ics Advisory Panel (2008) report is available at
ics, especially for students in tier 3. In the
index.html. More information on the National opinion of the panel, accurate and fluent
Council of Teachers of Mathematics Curricu-
lum Focal Points is available at www.nctm.org/
focalpoints. Documents elaborating the National 48. When using number lines to teach rational
Council of Teachers of Mathematics Curriculum numbers for students who have difficulties, it is
Focal Points are also available (see Beckmann et important to emphasize that the focus is on the
al., 2009). For a discussion of why this content is length of the segments between the whole num-
most relevant, see Milgram and Wu (2005). ber marks (rather than counting the marks).
( 19 )

26. Recommendation 2. Instructional materials for students receiving interventions
arithmetic with whole numbers is neces- panel’s view, the intervention program
sary before understanding fractions. The should include an assessment to assist in
panel acknowledges that there will be placing students appropriately in the in-
periods when both whole numbers and tervention curriculum.
rational numbers should be addressed in
interventions. In these cases, the balance Potential roadblocks and solutions
of concepts should be determined by the
student’s need for support. Roadblock 2.1. Some interventionists
may worry if the intervention program
3. Districts should appoint committees, in- is not aligned with the core classroom
cluding experts in mathematics instruction instruction.
and mathematicians with knowledge of el-
ementary and middle school mathematics Suggested Approach. The panel believes
curriculum, to ensure that specific criteria that alignment with the core curriculum is
(described below) are covered in depth in not as critical as ensuring that instruction
the curricula they adopt. builds students’ foundational proficien-
cies. Tier 2 and tier 3 instruction focuses
In the panel’s view, intervention materials on foundational and often prerequisite
should be reviewed by individuals with skills that are determined by the students’
knowledge of mathematics instruction and rate of progress. So, in the opinion of the
by mathematicians knowledgeable in el- panel, acquiring these skills will be neces-
ementary and middle school mathematics. sary for future achievement. Additionally,
They can often be experts within the district, because tier 2 and tier 3 are supplemental,
such as mathematics coaches, mathematics students will still be receiving core class-
teachers, or department heads. Some dis- room instruction aligned to a school or
tricts may also be able to draw on the exper- district curriculum (tier 1).
tise of local university mathematicians.
Roadblock 2.2. Intervention materials
Reviewers should assess how well interven- may cover topics that are not essential to
tion materials meet four criteria. First, the building basic competencies, such as data
materials integrate computation with solv- analysis, measurement, and time.
ing problems and pictorial representations
rather than teaching computation apart Suggested Approach. In the panel’s opin-
from problem-solving. Second, the mate- ion, it is not necessary to cover every topic
rials stress the reasoning underlying cal- in the intervention materials. Students will
culation methods and focus student atten- gain exposure to many supplemental top-
tion on making sense of the mathematics. ics (such as data analysis, measurement,
Third, the materials ensure that students and time) in general classroom instruc-
build algorithmic proficiency. Fourth, the tion (tier 1). Depending on the student’s
materials include frequent review for both age and proficiency, it is most important
consolidating and understanding the links to focus on whole and rational numbers in
of the mathematical principles. Also in the the interventions.
( 20 )

27. Recommendation 3. mathematics.49 Our panel supports
this recommendation and believes
Instruction during the that districts and schools should select
intervention should be materials for interventions that reflect
this orientation. In addition, professional
explicit and systematic. development for interventionists should
This includes providing contain guidance on these components
of explicit instruction.
models of proficient
problem solving, Level of evidence: Strong
verbalization of Our panel judged the level of evidence
thought processes, supporting this recommendation to be
guided practice, strong. This recommendation is based on
six randomized controlled trials that met
corrective feedback, WWC standards or met standards with
and frequent reservations and that examined the ef-
fectiveness of explicit and systematic in-
cumulative review. struction in mathematics interventions.50
These studies have shown that explicit and
systematic instruction can significantly
The National Mathematics Advisory improve proficiency in word problem solv-
Panel defines explicit instruction as ing51 and operations52 across grade levels
follows (2008, p. 23): and diverse student populations.
• “Teachers provide clear models for Brief summary of evidence to support
solving a problem type using an the recommendation
array of examples.”
The results of six randomized controlled
• “Students receive extensive practice trials of mathematics interventions show
in use of newly learned strategies extensive support for various combina-
and skills.” tions of the following components of ex-
plicit and systematic instruction: teacher
• “Students are provided with demonstration,53 student verbalization,54
opportunities to think aloud (i.e.,
talk through the decisions they
49. National Mathematics Advisory Panel
make and the steps they take).”
(2008).
50. Darch, Carnine, and Gersten (1984); Fuchs
• “Students are provided with et al. (2003a); Jitendra et al. (1998); Schunk and
extensive feedback.” Cox (1986); Tournaki (2003); Wilson and Sindelar
(1991).
The NMAP notes that this does not mean 51. Darch, Carnine, and Gersten (1984); Jitendra
that all mathematics instruction should et al. (1998); Fuchs et al. (2003a); Wilson and Sin-
delar (1991).
be explicit. But it does recommend that
52. Schunk and Cox (1986); Tournaki (2003).
struggling students receive some explicit
instruction regularly and that some 53. Darch, Carnine, and Gersten (1984); Jitendra
et al. (1998); Fuchs et al. (2003a); Schunk and
of the explicit instruction ensure that Cox (1986); Tournaki (2003); Wilson and Sindelar
students possess the foundational skills (1991).
and conceptual knowledge necessary 54. Jitendra et al. (1998); Fuchs et al. (2003a);
for understanding their grade-level Schunk and Cox (1986); Tournaki (2003).
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28. Recommendation 3. Instruction during the intervention should be explicit and systematic
guided practice,55 and corrective feed- Similarly, four of the six studies included
back.56 All six studies examined interven- immediate corrective feedback,61 and the
tions that included teacher demonstra- effects of these interventions were posi-
tions early in the lessons.57 For example, tive and significant on word problems and
three studies included instruction that measures of operations skills, but the ef-
began with the teacher verbalizing aloud fects of the corrective feedback compo-
the steps to solve sample mathematics nent cannot be isolated from the effects of
problems.58 The effects of this component other components in three cases.62
of explicit instruction cannot be evaluated
from these studies because the demonstra- With only one study in the pool of six in-
tion procedure was used in instruction for cluding cumulative review as part of the
students in both treatment and compari- intervention,63 the support for this compo-
son groups. nent of explicit instruction is not as strong
as it is for the other components. But this
Scaffolded practice, a transfer of control study did have statistically significant pos-
of problem solving from the teacher to the itive effects in favor of the instructional
student, was a component in four of the six group that received explicit instruction
studies.59 Although it is not possible to parse in strategies for solving word problems,
the effects of scaffolded instruction from the including cumulative review.
other components of instruction, the inter-
vention groups in each study demonstrated How to carry out this
significant positive gains on word problem recommendation
proficiencies or accuracy measures.
1. Ensure that instructional materials are
Three of the six studies included opportu- systematic and explicit. In particular, they
nities for students to verbalize the steps should include numerous clear models of
to solve a problem.60 Again, although ef- easy and difficult problems, with accompa-
fects of the interventions were statistically nying teacher think-alouds.
significant and positive on measures of
word problems, operations, or accuracy, To be considered systematic, mathematics
the effects cannot be attributed to a sin- instruction should gradually build profi-
gle component of these multicomponent ciency by introducing concepts in a logical
interventions. order and by providing students with nu-
merous applications of each concept. For
example, a systematic curriculum builds
55. Darch, Carnine, and Gersten (1984); Jiten- student understanding of place value in
dra et al. (1998); Fuchs et al. (2003a); Tournaki
an array of contexts before teaching pro-
cedures for adding and subtracting two-
56. Darch, Carnine, and Gersten (1984); Jitendra
et al. (1998); Schunk and Cox (1986); Tournaki digit numbers with regrouping.
57. Darch, Carnine, and Gersten (1984); Fuchs Explicit instruction typically begins with
et al. (2003a); Jitendra et al. (1998); Schunk and a clear unambiguous exposition of con-
Cox (1986); Tournaki (2003); Wilson and Sindelar cepts and step-by-step models of how
��
Darch, Carnine, and Gersten (1984). 61. Darch, Carnine, and Gersten (1984); Jiten-
59. Darch, Carnine, and Gersten (1984); Fuchs dra et al. (1998); Tournaki (2003); Schunk and
et al. (2003a); Jitendra et al. (1998); Tournaki Cox (1986).
(2003). 62. Darch, Carnine, and Gersten (1984); Jitendra
60. �� et al. (1998); Tournaki (2003).
Tournaki (2003). 63. ��
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