Running head: IDEA PAPER

Improving Problem-Solving Skills for

Middle School Students

with Learning Disabilities (LD)

Anne Brawand

George Mason University

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Improving Problem-Solving Skills for Middle School Students with LD

The National Council of Teachers of Mathematics (NCTM, 2000) has called for

standards that range beyond skills of basic procedural competency. The No Child Left Behind

Act of 2001 (NCLB) frameworks a national initiative to improve the link between elementary

and secondary education and high-stakes testing. According to Miller and Hudson (2007), it is

important to provide balanced instruction across mathematics standards and to address

conceptual, procedural, and declarative knowledge within a comprehensive math curriculum.

General strategies for addressing achievement problems in secondary mathematics include

organizing explicit teaching of important concepts, providing numerous examples of new

concepts that address the overall range of the concepts, direct teaching of relevant cognitive

routines, and systematic teaching of prioritized objectives (Montague & Jitendra, 2007).

Many students with LD exhibit difficulties in the area of memory and general strategy

use, literacy and communication, specific processes and strategies associated with math

problems, and low motivation and affect (Bryant & Bryant, 2008). A number of students may

also have difficulty with the English language and communication aspects of mathematics (Lang

& Pagliaro, 2007). Rao and Mallow (2009) expressed concern that a lack of basic math

knowledge is a common impediment to learning higher-level math for all students, including

those with disabilities, and that students with this deficiency in basic math skills may learn

neither math computation nor higher-order mathematics. However, students who are taught

math skills until they achieve fluency tend to maintain their skills (Axtell et al., 2009).

In 2007, Maccini, Mulcahy, and Wilson extended a previous review completed by

Maccini and Hughes (1997) on math interventions for secondary students with learning

disabilities (LD) in order to determine the nature and focus of current math interventions that

were effective for secondary students with LD. Interventions that produced significant gains for

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secondary students with a learning disability in math included mnemonic strategy instruction,

graduated instructional approach, cognitive strategy instruction involving planning, schema-

based instruction, and contextualized videodisc instruction. Although these interventions focused

on secondary students with disabilities, Maccini et al. (2007) noted that the math skills were

remedial (such as multiplication facts, decimals, integers, and computation skills) versus focused

on secondary mathematics curriculum.

Another problematic area for adolescents with LD is solving word problems. Jitendra,

DiPipi, and Perron-Jones (2002) investigated the effects of schema-based strategy instruction on

math problem-solving for middle school students with LD and found that participants reached

criterion in strategy use and accuracy for solving one-step multiplication and division word

problems by 12 sessions, maintained performance up to 10 weeks after the intervention, and

generalized the strategy to additional types of word problems (PND = 82.3%). The average

increase in mean scores of strategy step use measured before and after the intervention was 66%

on drawing diagrams, and a 32% increase in writing a number sentence. The researchers utilized

a single-subject design in a learning support classroom. Schema-based instruction (SBI) utilizes

diagrams of problem patterns and structure. Schema is the mapping of relationships, with the

purpose of organizing knowledge. The structure similarity or “schemata” of problems is the way

it’s organized or the pattern that guides problem representation. Knowledge of the mathematical

structure of word problems activates relevant patterns that would guide problem representation

necessary to solve problems. A strength of SBI is its focus on identification of problem types,

looking beyond the surface similarities and considering the mathematical structure. Once the

pattern is determined, important information and relations in the problem can be mapped using

schemata diagrams (see Figure 1). The use of schemata diagrams facilitate translation of the

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problem in order to arrive at the solution because they emphasize identification of relationships

among information in a problem. In contrast, simple illustrations of word problem content do not

highlight the mathematical operation to employ (see Figure 2).

Figure 1 Example of Schemata Diagram

Source: Jitendra at al. (2002)

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Figure 2 Pictorial and Schematic Representations

Source: Montague, M. (2004). Math Problem Solving for Middle School Students with Disabilities. Retrieved from: http://www.k8accesscenter.org/training_resources/MathProblemSolving.asp

Butler, Miller, Crehan, Babbitt, and Pierce (2003) sought to determine whether pictorial

representations are just as effective as using concrete manipulatives along with pictorial

representations by comparing the effects of a concrete-representational-abstract (CRA) sequence

to a representational-abstract (RA) sequence for middle school students with mild to moderate

disabilities. Higher mean scores were obtained for the CRA group (M = 69%) compared to the

RA group (M = 63%) on the word problem measure of fractional equivalency (ES = .17), and the

students in the two treatment groups performed as well as the students in the comparison group

of general education students. Both the CRA and RA groups increased their scores from pretest

to posttest by at least 60%. Using the CRA teaching sequence integrates the use of manipulative

devices and pictorial representations into explicit instruction designed to teach important

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concepts (Miller & Hudson, 2007). The students first represent the problem with concrete objects

(manipulatives), then move to the representational (pictures) phase and draw or use pictorial

representations of the quantities, and the final abstract phase involves numeric representations,

instead of pictures.

Schuermann, Deshler, and Schumaker (2009) also employed a problem-solving

intervention, the explicit inquiry routine (EIR), for middle school students with learning

disabilities. The routine successfully increased students’ scores on word problems involving one-

variable equations (PND = 93%), maintained their skills for 11 weeks, and resulted in transfer of

knowledge to textbook word problems. On the concrete manipulation measure, the participants

averaged a 51% increase in scores from pretest to posttest, demonstrating their progress by

concretely illustrating and solving one-variable equations embedded in word problems. The

researchers employed a single-subject design and the intervention was conducted in classroom

space in a charter school. The EIR integrates teaching practices from both general and special

education to engage students in an inquiry process across concrete, representational, and abstract

modes to develop an understanding of the concept (see sample Figure 3). The EIR method

implemented included the instructional components of explicit sequencing, scaffolded inquiry,

and systematic use of varied modes of instruction.

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Figure 3 EIR Student Work Sample

Source: Schuerman et al. (2009)

The components of schema-based instruction (SBI) that include identifying the problem

type to consider the mathematical structure, and then using schemata diagrams to arrive at the

solution promote successful problem-solving instruction for students with LD (Jitendra et al.,

2002). Integration of the use of manipulative devices and pictorial representations into explicit

instruction during the CRA method has also proven to be an effective step to math problem-

solving (Butler et al., 2003; Schuermann et al., 2009). By combining the strength of these

problem-solving techniques, I plan to develop a problem-solving strategy that ensures students

with LD are provided with the appropriate problem-solving tools they need to efficiently solve

word problems.

The problem-solving intervention I plan to implement in math will incorporate structural

and visual components of both the SBI and CRA strategies into the SOLVE method. SOLVE is

the problem-solving paradigm for Key Elements to Mathematical Success (National Training

Network). The SOLVE problem-solving process has five separate steps: S – Study the Problem,

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O – Organize the Facts, L – Line up a Plan, V –Verify the Plan with Action, E – Examine the

Results. My rationale for integrating identification of problem structure from SBI and pictorial

representation from CRA is to improve students’ reasoning ability to arrive at the correct

solution. I will be focusing on middle school math curriculum skills, and the specific population

I intend to target for my dissertation is middle school students with LD. Specific research

questions to explore include:

1) Is a problem-solving strategy that incorporates CRA and SBI components effective in

improving math performance on word problems on operations with fractions for middle

school students with learning disabilities?

2) Do the students maintain the problem-solving skills over time?

3) Do the students generalize their learning to other types of word problems?

Method

Participants

The specific population I will target for my dissertation is middle school students with

LD. I plan to implement this math problem-solving intervention with up to ten students in grades

six to eight. After school system research permission is acquired, phone and/or email contact will

be made with school administrators (i.e., the principal of the school) to provide an overview of

the research (such as found on the Informed Consent documents), to include the approximate

amount of time the students would invest to participate in the research, and inform them that

official approval has been obtained to conduct this research in their school system. Criteria for

student selection include students have the following:

1. An Individualized Education Program (IEP) with mathematics problem-solving skills as a

goal

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2. An attendance record for the previous school year that is consistent (i.e., minimizing the

possibility of attrition due to attendance issues)

3. A recommendation from the mathematics and/or special education teacher indicating the

student is having difficulty solving word problems involving operations with fractions

4. A pre-test with evidence of difficulty with problem-solving to ensure the student matches

the skill set being targeted

5. Willingness to participate in the research as indicated by an Informed Assent

6. Parent Consent to participate in the research

Procedure

The intervention I plan to implement is problem-solving strategy that incorporates CRA

and SBI components effective in improving math performance on fraction word problems for

middle school students with learning disabilities. I will be focusing on middle school math

curriculum skills. I am in the process of reviewing research on math problem-solving, SBI, and

CRA in order to determine the exact instructional steps for my intervention procedure, finalize

the math concept (e.g. operations with fractions), and plan materials to be developed.

Design

I plan to utilize a multiple baseline across participants design for my intervention,

including baseline and treatment phases. Ideally, the participants will have similar learning

histories and exhibit the same target behavior at similar frequencies under similar pre-

intervention conditions (Gast, 2010). I am currently investigating the effectiveness of the

simultaneous prompting intervention for students with LD in the area of Math through a single-

subject pilot study, and feel this design will be best suited for my dissertation topic.

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Data Collection

Data on problem solving accuracy as well as strategy use will be collected with and

without the treatment for each participant and plotted on a graph. A pretest and posttest will be

administered, as well as written probe sheets after designated sessions, in order to measure

accuracy and strategy use in solving word problems involving operations with fractions. I plan to

maintain separate graphs showing the average performance of the group of participants, and

individual graphs that summarize individual performance (Gast, 2010). Participants will also

have the opportunity to indicate a willingness to participate in an interview after the intervention,

and responses will be analyzed to determine student perspectives of intervention's

effectiveness/student satisfaction.

Data Analysis

Visual inspection will be made among the participants' data to determine effectiveness of

intervention/percent of non-overlapping data (PND). Pre-treatment and post-treatment scores

will also be analyzed to measure effectiveness of intervention and acquisition of strategy use.

Validity

To control for internal validity threats, I plan to obtain information about outside events

to control for history threat, watch for patterns/testing effects, stay consistent and measure IOA,

attempt to do in a separate location to control for treatment diffusion, and be sure I have enough

subjects to combat attrition. I will address external validity by using a replications across

participants design (multiple baseline across subjects), and will keep a detailed log on description

of the participants, settings, conditions, procedures, etc. Social validity will be obtained by

measuring the perception of intervention procedures through interviews with direct participants.

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Areas of Expertise Required

Areas of expertise that will need to be represented on my dissertation committee include

proficiency with review of research, knowledge of single-subject research designs, and

experience with intervention research methodology and data analysis. In order to rationalize the

need for my dissertation topic, it’s essential that my proposal include strong literature support. I

would also benefit from the expertise of a researcher who specializes in single-subject design so

I include all possible variations for my intervention to be most effective. Finally, as I pursue

intervention research for my dissertation, I will require guidance with methodological

components to ensure that all areas of validity and data analysis have been adequately addressed.

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References

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using a classwide fluency building procedure for middle school students: A preliminary

study. Psychology in the Schools, 46, 526-538. doi:10.1002/pits.20395

Bryant, B. R., & Bryant, D. P. (2008). Introduction to the special series: Mathematics and

learning disabilities. Learning Disability Quarterly, 31, 3-10.

http://www.highbeam.com/doc/1G1-176203759.html

Butler, F. M., Miller, S. P., Crehan, K., Babbitt, B., & Pierce, T. (2003). Fraction instruction for

students with mathematics disabilities: comparing two teaching sequences. Learning

Disabilities Research & Practice, 18, 99-111. doi:10.1111/1540-5826.00066

Gast, D. L. (2010). ). Single subject research methodology in behavioral sciences. New York,

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learning difficulties. New York: Guiliford.

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multiplication facts by middle school students with cognitive impairment. Education and

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Scheurmann, A. M., Deshler, D. D., & Schumaker, J. B. (2009). The effects of the explicit

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equations. Learning Disability Quarterly, 32, 103-120.