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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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