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MATHEMATICS INSTRUCTION FOR
STUDENTS WITH LEARNING DISABILITIES
OR DIFFICULTY LEARNING MATHEMATICS
A Guide for Teachers
Madhavi Jayanthi
Russell Gersten
Scott Baker
Instructional Research Group
2008
MATHEMATICS INSTRUCTION FOR
STUDENTS WITH LEARNING DISABILITIES
OR DIFFICULTY LEARNING MATHEMATICS
A Guide for Teachers
The authors would like to express their appreciation to Becky
Newman-Goncher and Kelly Haymond for their contributions to
this publication.
This publication was created for the Center on Instruction by
Instructional Research Group. The Center on Instruction is
operated by RMC Research Corporation in partnership with
the Florida Center for Reading Research at Florida State
University; Instructional Research Group; the Texas Institute for
Measurement, Evaluation, and Statistics at the University of
Houston; and the Meadows Center for Preventing Educational
Risk at the University of Texas at Austin.
The contents of this document were developed under
cooperative agreement S283B050034 with the U.S.
Department of Education. However, these contents do
not necessarily represent the policy of the Department of
Education, and you should not assume endorsement by the
Federal Government.
Editorial, design, and production services provides by
RMC Research Corporation.
Preferred citation:
Jayanthi, M., Gersten, R., Baker, S. (2008).
Mathematics
instruction for students with learning disabilities or difficulty
learning mathematics: A guide for teachers.
Portsmouth, NH:
RMC Research Corporation, Center on Instruction.
To download a copy of this document, visit www.centeroninstruction.org.
CONTENTS
1 INTRODUCTION
5 RECOMMENDATIONS
13 LIST OF RECOMMENDATIONS
14 REFERENCES
v
INTRODUCTION
Historically, mathematics instruction for students with learning disabilities and
at-risk learners has not received the same level of consideration and scrutiny
from the research community, policy makers, and school administrators as the
field of reading. A recent review of the ERIC literature base (Gersten, Clarke, &
Mazzocco, 2007) found that the ratio of studies on reading disabilities to
mathematics disabilities and difficulties was 5:1 for the years 1996–2005. This
was a dramatic improvement over the ratio of 16:1 in the prior decade. Even
though this is far from a large body of research, sufficient studies exist to
dictate a course of action.
Recently, the Center on Instruction conducted a meta-analysis on the topic
of teaching mathematics to students with learning disabilities (Gersten, Chard,
Jayanthi, Baker, Morphy, & Flojo, 2008). A meta-analysis is a statistical method
by which research studies on a particular method of instruction are summarized
to determine the effectiveness of that instructional method. A meta-analysis
helps combine findings from disparate studies to determine the effectiveness
of a particular method of instruction.
In the meta-analysis on teaching mathematics to students with learning
disabilities (LD), only studies with randomized control trials (RCTs) and high
quality quasi-experimental designs (QEDs) were included. In an RCT, the study
participants (or other units such as classrooms or schools) are randomly
assigned to the experimental and control groups, whereas in a QED, there
is no random assignment of participants to the groups.
Seven Effective Instructional Practices
Based on the findings of the meta-analysis report, seven effective instructional
practices were identified for teaching mathematics to K–12 students with
learning disabilities. In describing these practices, we have incorporated
recommendations from
The Final Report of the National Mathematics Advisory
Panel
(National Mathematics Advisory Panel, 2008) as well. This report
specified recommendations for students with learning disabilities
and
for
students who were experiencing difficulties in learning mathematics but were
not identified as having a math learning disability (i.e., at-risk). The seven
1
effective instructional practices in this document are supported by current
research findings. Other instructional practices may be effective, but there is, at
present, not enough high quality research to recommend their use at this time.
Some of the recommendations listed later in this document (e.g., teach
explicitly and use visuals) are age-old teaching practices. While there is nothing
new about these practices, research continues to validate them as effective
instructional practices for students with learning disabilities and at-risk students,
and continued use is warranted. Other instructional methods recommended
here, such as using multiple instructional examples and teaching multiple
strategies, have also been endorsed in studies that focused on reform-oriented
mathematics instruction in general education classes (e.g., Silver, Ghousseini,
Gosen, Charalambous, & Strawhun, 2005; Rittle-Johnson & Star, 2007). This
alignment of teaching methods between special education and general
education enables students with learning disabilities to learn meaningfully
from general education curricula in inclusive classrooms.
Mathematical Knowledge
Current mathematics researchers emphasize three areas of mathematical
abilities (e.g., Kilpatrick, Swafford, & Findell, 2001; Rittle-Johnson & Star, 2007;
Bottge, Rueda, LaRoque, Serlin, & Kwon, 2007). They are:
• procedural knowledge,
• procedural flexibility, and
• conceptual knowledge.
Procedural knowledge
refers to knowledge of basic skills or the sequence
of steps needed to solve a math problem. Procedural knowledge enables a
student to execute the necessary action sequences to solve problems (Rittle-
Johnson & Star, 2007).
Procedural flexibility
refers to knowing the many different ways in which
a particular problem can be solved. Students with a good sense of procedural
flexibility know that a given problem can be solved in more than one way,
and can solve an unknown problem by figuring out a possible solution for
that problem.
2
Conceptual knowledge
is a grasp of the mathematical concepts and ideas
that are not problem-specific and therefore can be applied to any problem-
solving situation. Conceptual understanding is the over-arching understanding
of mathematical concepts and ideas that one often refers to as a “good
mathematical sense.”
It is reasonable to extrapolate from this small but important body of
research that such an emphasis would also benefit students with disabilities
and at-risk students. Recent studies have attempted to address reform-oriented
math instruction in special education settings. Researchers such as Woodward
(e.g., Woodward, Monroe, & Baxter, 2001) and Van Luit (Van Luit & Naglieri,
1999) have endeavored to address the issue of procedural flexibility in their
research by focusing on multiple strategy instruction—a recommendation in
this document, as mentioned earlier. Bottge (e.g., Bottge, Heinrichs, Mehta, &
Hung, 2002) has addressed the issue of procedural knowledge and conceptual
understanding by means of engaging, real-life, meaningful problem-solving
contexts; however the limited number of studies precludes any
recommendations at this time.
Effective Instruction at Each Tier
The current focus on Response to Intervention (RTI) as a tiered prevention and
intervention model for struggling mathematics learners also calls for evidence-
based instructional methods (Bryant & Bryant, 2008). While RTI models can
have three or more tiers (the most common being three tiers), they all share
the same objectives. For example, Tier 1 instruction, with an emphasis on
primary prevention, requires teachers to provide evidence-based instruction
to all students. Tier 2 focuses on supplemental instruction that provides
differentiated instruction to meet the learning needs of students. Tier 3
emphasizes individualized intensive instruction. The ultimate goal of the RTI
model is to reduce the number of students in successive tiers and the number
of students receiving intensive instruction. The groundwork for the success
of this model is the effectiveness of the instruction provided in Tier 1.The
evidence-based instructional strategies identified in this document need
to be part of the teaching repertoire of Tier 1 teachers. These validated
techniques, when implemented soundly, can effectively bring about student
gains in mathematics.
3
Our document guides K–12 teachers of students with disabilities and at-risk
students in their selection and use of effective mathematics instructional
methods. For each of the seven recommendations, we explain what works,
describe how the practice should be done, and summarize the evidence
supporting the recommendation.
4
RECOMMENDATIONS
Recommendation 1:
Teach students using
explicit instruction
on a regular basis.
Explicit instruction, a mainstay feature in many special education programs, includes teaching
components such as:
• clear modeling of the solution specific to the problem,
• thinking the specific steps aloud during modeling,
• presenting multiple examples of the problem and applying the solution to the problems, and
• providing immediate corrective feedback to the students on their accuracy.
When teaching a new procedure or concept, teachers should begin by modeling and/or thinking
aloud and working through several examples. The teacher emphasizes student problem solving
using the modeled method, or by using a model that is consonant with solid mathematical
reasoning. While modeling the steps in the problem (on a board or overhead), the teacher should
verbalize the procedures, note the symbols used and what they mean, and explain any decision
making and thinking processes (for example, “That is a plus sign. That means I should…”).
Teachers should model several problems with different characteristics (Rittle-Johnson & Star,
2007; Silbert, Carnine, & Stein, 1989). A critical technique is assisted learning where students
work in pairs or small groups and receive guidance from the teacher. During initial learning and
practice, the teacher provides immediate feedback to prevent mistakes in learning and allows
students to ask questions for clarification.
According to
The Final Report of the National Mathematics Advisory Panel
(National
Mathematics Advisory Panel, 2008), explicit systematic instruction improves the performance of
students with learning disabilities and students with learning difficulties in computation, word
problems, and transferring known skills to novel situations. However, the panel noted that while
explicit instruction has consistently shown better results, no evidence supports its
exclusive
use
for teaching students with learning disabilities and difficulties. The panel recommends that all
teachers of students with learning disabilities and difficulties teach explicitly and systematically on
a regular basis to some extent and not necessarily all the time.
Summary of Evidence to Support Recommendation 1
Meta-analysis of Mathematics Intervention Research for Students with LD
COI examined 11 studies in the area of explicit instruction (10 RCTs and 1 QED). The mean effect
size of 1.22 was statistically significant (p<.001; 95% CI = 0.78 to 1.67).
National Mathematics Advisory Panel
The panel reviewed 26 high quality studies (mostly RCTs) on effective instructional approaches for
students with learning disabilities and low-achieving students. Explicit Systematic Instruction is
identified in
The Final Report of the National Mathematics Advisory Panel
as one of the defining
features of effective instruction for students with learning disabilities (National Mathematics
Advisory Panel, 2008).
RCT = Randomized control trial. QED = Quasi-experimental design. CI= Confidence interval.
5
Recommendation 2:
Teach students using
multiple instructional examples
.
Example selection in teaching new math skills and concepts is a seminal idea that is strongly
emphasized in the effective instruction literature (e.g., Ma, 1999; Rittle-Johnson & Star, 2007;
Silbert, Carnine, & Stein, 1989). Teachers need to spend some time planning their mathematics
instruction, particularly focusing on selecting and sequencing their instructional examples. The
goal is to select a range of multiple examples of a problem type. The underlying intent is to
expose students to many of the possible variations and at the same time highlight the common
but critical features of seemingly disparate problems. For example, while teaching students to
divide a given unit into half, a variety of problems can be presented that differ in the way the
critical task of half is addressed in the problems (i.e., use the symbol for half; use the word half,
use the word one-half; etc.) (Owen & Fuchs, 2002).
Multiple examples can be presented in a specified sequence or pattern such as concrete to
abstract, easy to hard, and simple to complex. For example, fractions and algebraic equations can
be taught first with concrete examples, then with pictorial representations, and finally in an
abstract manner (Butler, Miller, Crehan, Babbitt, & Pierce, 2003; Witzel, Mercer, Miller, 2003).
Multiple examples can also be presented by systematically varying the range presented (e.g.,
initially teaching only proper fractions vs. initially teaching both proper and improper fractions).
Sequencing of examples may be most important during early acquisition of new skills when
scaffolding is needed for student mastery and success. The range of examples taught is probably
most critical to support transfer of learned skills to new situations and problems. In other words, if
the teacher teaches a wide range of examples, it will result in the learner being able to apply a
skill to a wider range of problem types. Both of these planning devices (sequence and range)
should be considered carefully when teaching students with LD.
Summary of Evidence to Support Recommendation 2
Meta-analysis of Mathematics Intervention Research for Students with LD
COI examined 9 studies on range and sequence of examples (all RCTs). The mean effect size of
0.82 was statistically significant (p<.001; 95% CI = 0.42 to 1.21).
National Mathematics Advisory Panel
The panel reviewed 26 high quality studies (mostly RCTs) on effective instructional approaches
for students with learning disabilities and low-achieving students. The panel recommends that
teachers, as part of explicit instruction, carefully sequence problems to highlight the critical features
of the problem type (National Mathematics Advisory Panel, 2008).
6
Recommendation 3:
Have students
verbalize decisions and solutions
to a math problem.
Encouraging students to verbalize, or think-aloud, their decisions and solutions to a math problem
is an essential aspect of scaffolded instruction (Palincsar, 1986). Student verbalizations can be
problem-specific or generic. Students can verbalize the specific steps that lead to the solution of
the problem (e.g., I need to divide by two to get half) or they can verbalize generic heuristic steps
that are common to problems (e.g., Now I need to check my answer). Students can verbalize the
steps in a solution format (First add the numbers in the
units
column. Write down the answer.
Then add numbers in the
tens
column…) (Tournaki, 2003) or in a self-questioning/answer format
(What should I do first? I should…) (Pavchinski, 1998). Students can verbalize during initial
learning or as they are solving, or have solved, the problem.
Many students with learning disabilities are impulsive behaviorally and when faced with multi-
step problems frequently attempt to solve the problems by randomly combining numbers rather
than implementing a solution strategy step-by-step. Verbalization may help to anchor skills and
strategies both behaviorally and mathematically. Verbalizing steps in problem solving may address
students’ impulsivity directly, thus suggesting that verbalization may facilitate students’ self-
regulation during problem solving.
Summary of Evidence to Support Recommendation 3
Meta-analysis of Mathematics Intervention Research for Students with LD
COI examined 8 studies in the area of student verbalizations (7 RCTs and 1 QED). The mean effect
size of 1.04 was statistically significant (p<.001; 95% CI = 0.42 to 1.66).
National Mathematics Advisory Panel
The panel reviewed 26 high quality studies (mostly RCTs) on effective instructional approaches
for students with learning disabilities and low-achieving students. The panel recommends that
teachers, as part of explicit instruction, allow students to think aloud about the decisions they make
while solving problems (National Mathematics Advisory Panel, 2008).
7
Recommendation 4:
Teach students to
visually represent the information
in the math problem.
Visual representations (drawings, graphic representations) have been used intuitively by teachers
to explain and clarify problems and by students to understand and simplify problems. When used
systematically, visuals have positive benefits on students’ mathematic performance.
Visual representations result in better gains under certain conditions. Visuals are more
effective when combined with explicit instruction. For example, teachers can explicitly teach
students to use a strategy based on visuals (Owen & Fuchs, 2002). Also, students benefit more
when they use a visual representation prescribed by the teacher rather than one that they self-
select (D. Baker, 1992). Furthermore, visuals that are designed specifically to address a particular
problem type are more effective than those that are not problem specific (Xin, Jitendra, &
Deatline-Buchman, 2005). For example, in the study by Xin and her colleagues, students first
identified what type of problem they had been given (e.g., proportion, multiplicative) and then
used a corresponding diagram (taught to them) to represent essential information and the
mathematical procedure necessary to find the unknown. Then they translated the diagram
into a math sentence and solved it.
Finally, visual representations are more beneficial if not only the teacher, but both the teacher
and the students use the visuals (Manalo, Bunnell, & Stillman, 2000).
Summary of Evidence to Support Recommendation 4
Meta-analysis of Mathematics Intervention Research for Students with LD
COI examined 12 studies on visual representations (11 RCTs and 1 QED). The mean effect size of
0.47 was statistically significant (p<.001; 95% CI = 0.25 to 0.70).
National Mathematics Advisory Panel
The panel reviewed 26 high quality studies (mostly RCTs) on effective instructional approaches
for students with learning disabilities and low-achieving students. According to
The Final Report
of the National Mathematics Advisory Panel
visual representations when combined with explicit
instruction tended to produce significant positive results (National Mathematics Advisory
Panel, 2008).
8
Recommendation 5:
Teach students to solve problems using
multiple/heuristic strategies
.
Instruction in multiple/heuristic strategies is part of a contemporary trend in mathematics
education (e.g., Star & Rittle-Johnson, in press). Using heuristics shows some promise with
students with learning disabilities. Multiple/heuristic strategy instruction has been used in
addressing computational skills, problem solving, and fractions.
A heuristic is a method or strategy that exemplifies a generic approach for solving a problem.
For example, a heuristic strategy can include steps such as “Read the problem. Highlight the key
words. Solve the problems. Check your work.” Instruction in heuristics, unlike direct instruction,
is not problem-specific. Heuristics can be used in organizing information and solving a range of
math problems. They usually include student discourse and reflection on evaluating the alternate
solutions and finally selecting a solution for solving the problem. For example, in the Van Luit and
Naglieri (1991) study, the teacher first modeled several strategies for solving a computational
problem. However, for most of the lesson, the teacher’s task was to lead the discussion in the
direction of using strategies and to facilitate the discussion of the solutions provided by the
students. Each student was free to select a strategy for use, but the teacher assisted the children
in discussion and reflection about the choices made.
Similarly, in the Woodward (2006) study, students were taught multiple fact strategies. Daily
lessons consisted of introduction of new strategies or review of old strategies. Students were not
required to memorize the strategies. They were, however, encouraged to discuss the strategy
and contrast it with previously taught strategies. For example, students were shown that since 9
X 5 has the same value as 5 X 9, they were free to treat the problem as either nine fives or five
nines. They also were shown that this was equivalent to 10 fives minus one five, and that this
could be a faster way to do this problem mentally. Thus a variety of options were discussed with
the students.
Summary of Evidence to Support Recommendation 5
Meta-analysis of Mathematics Intervention Research for Students with LD
COI examined 4 studies in the area of multiple/heuristic strategy instruction (3 RCTs and 1 QED).
The mean effect size of 1.56 was statistically significant (p<.001; 95% CI = 0.65 to 2.47).
9
Recommendation 6:
Provide
ongoing formative assessment data and feedback
to teachers.
Ongoing formative assessment and evaluation of students’ progress in mathematics can help
teachers measure the pulse and rhythm of their students’ growth and also help them fine-tune
their instruction to meet students’ needs. Teachers can administer assessments to their group
of students and then a computer can provide them with data depicting students’ current
mathematics abilities.
Providing teachers with information regarding their students’ progress in mathematics has
beneficial effects on the mathematics performance of those same students. However, greater
benefits on student performance will be observed if teachers are provided with not only
performance feedback information but also instructional tips and suggestions that can help
teachers decide what to teach, when to introduce the next skill, and how to group/pair students.
For example, teachers can be given a set of written questions to help them use the formative
assessment data for adapting and individualizing instruction. These written questions could
include “On what skill(s) has the student improved compared to the previous two-week period?”
or “How will I attempt to improve student performance on the targeted skill(s)?”
Teachers can respond to these questions and address them again when new assessment
data becomes available (Allinder, Bolling, Oats, & Gagnon, 2000). Teachers can also be provided
with a specific set of recommendations to address instructional planning issues such as which
mathematical skills require additional instructional time for the entire class, which students require
additional help via some sort of small group instruction or tutoring, and which topics should be
included in small group instruction (Fuchs, Fuchs, Hamlett, Phillips, & Bentz, 1994).
Summary of Evidence to Support Recommendation 6
Meta-analysis of Mathematics Intervention Research for Students with LD
COI examined 10 studies on formative assessment (all RCTs). The mean effect size of 0.23 was
statistically significant (p<.01; 95% CI = 0.05 to 0.41).
National Mathematics Advisory Panel
The panel reviewed high quality studies on formative assessment and noted that formative
assessment use by teachers results in marginal gains for students of all abilities. When teachers
are provided with special enhancements (suggestions on how to tailor instruction based on data),
significant gains in mathematics are observed (National Mathematics Advisory Panel, 2008).
10
Recommendation 7:
Provide
peer-assisted instruction
to students.
Students with LD sometimes receive some type of peer assistance or one-on-one tutoring in
areas in which they need help. The more traditional type of peer-assisted instruction is cross-age,
where a student in a higher grade functions primarily as the tutor for a student in a lower grade.
In the newer within-classroom approach, two students in the same grade tutor each other. In
many cases, a higher performing student is strategically placed with a lower performing student
but typically both students work in both roles: tutor (provides the tutoring) and tutee (receives
the tutoring).
Cross-age peer tutoring appears to be more beneficial than within-class peer-assisted learning
for students with LD. It could be hypothesized that students with LD are too far below grade level
to benefit from feedback from a peer at the same grade level. It seems likely that within-class
peer tutoring efforts may fall short of the level of explicitness necessary to effectively help
students with LD progress, whereas older students (in cross-age peer tutoring settings)
could be taught how to explain concepts to a student with LD who is several years younger.
Interestingly, within-class peer-assisted instruction does appear to help low-achieving students
who have learning difficulties in mathematics (Baker, Gersten, & Lee, 2002). One possible
explanation seems to be that there is too much of a gap in the content knowledge between
students with learning disabilities compared to low-achieving students with learning difficulties,
thus enabling peer tutoring to be more beneficial to students at risk than to students with LD.
Summary of Evidence to Support Recommendation 7
Meta-analysis of Mathematics Intervention Research for Students with LD
COI examined 2 studies in the area of cross age peer tutoring (both RCTs).
The mean effect size of 1.02 was statistically significant (p<.001; 95% CI = 0.57 to 1.47).
11
LIST OF RECOMMENDATIONS
1: Teach students using
explicit instruction
on a regular basis.
2: Teach students using
multiple instructional examples.
3: Have students
verbalize decisions and solutions
to a
math problem.
4: Teach students to
visually represent the information
in
the math problem.
5: Teach students to solve problems using
multiple/
heuristic strategies.
6: Provide
ongoing formative assessment data and feedback
to teachers.
7: Provide
peer-assisted instruction
to students.
13
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16
