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Self-regulation strategies to improve mathematical problem solving for students with learning disabilities.

Abstract. This article provides a review of research in cognitive strategy instruction for improving mathematical problem Mathematical problem may mean two slightly different things, both closely related to mathematical games:
general meaning
a question that can be answered with the help of mathematics ; formal meaning : any tuple (S, C( ), r
 solving for students with learning disabilities (LD). The particular focus is on one of the salient components of this instructional approach--self-regulation. Seven studies utilizing this approach for teaching problem solving problem solving

Process involved in finding a solution to a problem. Many animals routinely solve problems of locomotion, food finding, and shelter through trial and error.
 to students with LD were previously evaluated to determine its status as evidence-based practice. The results of this evaluation are described, and the self-regulation The term self-regulation can signify
  • in systems theory: homeostasis
  • in sociology / psychology: self-control
  • in educational psychology: self-regulated learning
  • Self-Regulation Theory (SRT) is a system of conscious personal health management
 component embedded Inserted into. See embedded system.  in the cognitive routine for each of the studies is presented. The article concludes with a discussion of several principles associated with research and practice in strategy instruction and some practical considerations for implementation in schools.


This article provides a review of research in strategy instruction for improving mathematical problem solving for students with learning disabilities (LD) with a focus on one of the salient components of this instructional approach--self-regulation. Research has consistently shown that students with LD are poor self-regulators who benefit from strategy instruction that incorporates self-regulation training (Graham & Harris Harris, Scotland: see Lewis and Harris. , 2003; Wong n. 1. A field. , Harris, Graham, & Butler, 2003).

Self-regulation, the ability to regulate one's cognitive activities, underlies the executive processes and functions associated with metacognition Metacognition refers to thinking about cognition (memory, perception, calculation, association, etc.) itself or to think/reason about one's own thinking. Types of knowledge  (Flavell, 1976). Metacognition has to do with knowledge and awareness of one's cognitive strengths and weaknesses as well as self-regulation, which guides an individual in the coordination of that awareness while engaged in cognitive activities (Wong, 1999). Self-regulation strategies, such as self-instruction, self-questioning self-ques·tion·ing
Scrutiny of one's own feelings, actions, and motivations.
, self-monitoring, self-evaluation, and self-reinforcement, help learners gain access to cognitive processes Cognitive processes
Thought processes (i.e., reasoning, perception, judgment, memory).

Mentioned in: Psychosocial Disorders
 that facilitate learning, guide learners as they apply the processes within and across domains, and regulate their application and overall performance of a task.

Swanson's (Swanson, 1999; Swanson & Sachs-Lee, 2000) meta-analyses of 30 years of both group and single-subject intervention studies intervention studies, the epidemiologic investigations designed to test a hypothesized cause and effect relation by modifying the supposed causal factor(s) in the study population.
 conducted with students with LD revealed that direct instruction and strategy instruction were the two most effective instructional approaches, particularly when combined, for teaching students with LD across academic domains (i.e., reading, writing, and mathematics).

Interventions were considered direct instruction if they contained the following components: (a) drills and probes, (b) repeated feedback, (c) rapidly paced instruction, (d) individualized instruction Individualized instruction is a method of instruction in which content, instructional materials, instructional media, and pace of learning are based upon the abilities and interests of each individual learner. , (e) breaking the task down into a sequence of steps, (f) pictorial diagrams, (g) small-group instruction, and (h) direct questioning by the teacher (Swanson, 1999).

In contrast, strategy instruction focuses on processes; for example, metacognition or self-regulation. The following procedures characterized char·ac·ter·ize  
tr.v. character·ized, character·iz·ing, character·iz·es
1. To describe the qualities or peculiarities of: characterized the warden as ruthless.

 strategy instruction: (a) systematic and direct explanations and/or and/or  
Used to indicate that either or both of the items connected by it are involved.

Usage Note: And/or is widely used in legal and business writing.
 verbal descriptions of the performance of a task; (b) verbal modeling, questioning, and demonstrations by the teacher of the steps and processes in the cognitive routine; (c) systematic prompts and cues to use the processes, strategies, and procedures; and (d) cognitive modeling The term cognitive model can have basically two meanings. In cognitive psychology, a model is a simplified representation of reality. The essential quality of such a model is to help deciding the appropriate actions, i.e.  using "think aloud" to model task completion or problem solving (Swanson, 1999).

Although the two instructional approaches were found to operate independently, they share many components and procedures, such as drill and repetition REPETITION, construction of wills. A repetition takes place when the same testator, by the same testamentary instrument, gives to the same legatee legacies of equal amount and of the same kind; in such case the latter is considered a repetition of the former, and the legatee is entitled , distributed practice, task analysis, small-group instruction, and strategy cues, all of which were found to increase the predictive power The predictive power of a scientific theory refers to its ability to generate testable predictions. Theories with strong predictive power are highly valued, because the predictions can often encourage the falsification of the theory.  of treatment effectiveness. Direct instruction was associated more with effective instruction for teaching basic skills such as decoding de·code  
tr.v. de·cod·ed, de·cod·ing, de·codes
1. To convert from code into plain text.

2. To convert from a scrambled electronic signal into an interpretable one.

 and math fact recall, as opposed to strategy instruction, which was associated more with effective instruction in higher order learning (e.g., reading comprehension Reading comprehension can be defined as the level of understanding of a passage or text. For normal reading rates (around 200-220 words per minute) an acceptable level of comprehension is above 75%.  and mathematical problem solving) that utilized higher order skills such as metacognition, self-monitoring, rule learning, and self-awareness self-awareness
Realization of oneself as an individual entity or personality.
 (Swanson, 1999; Swanson & Sachs-Lee, 2000).

Likewise, Kroesbergen and van Luit (2003), in their meta-analysis meta-analysis /meta-anal·y·sis/ (met?ah-ah-nal´i-sis) a systematic method that takes data from a number of independent studies and integrates them using statistical analysis.  of mathematics intervention studies conducted with students with disabilities, found that self-instruction, a self-regulation strategy, as a component of instructional models, is most effective generally for mathematics learning, but direct instruction appeared more effective for basic skills acquisition.

Following a comprehensive search of the literature, seven intervention studies were located that investigated the effects of cognitive strategy instruction on mathematical problem solving for students with disabilities. The five single-subject design and two group-design studies were evaluated individually using previously identified quality indicators to determine whether they qualified as "high quality" or "acceptable" and then to determine if the instructional practice, in this case, cognitive strategy instruction for improving mathematical problem solving, qualified as "evidence-based" or "promising" (Gersten et al., 2005; Homer Homer, principal figure of ancient Greek literature; the first European poet. Works, Life, and Legends

Two epic poems are attributed to Homer, the Iliad and the Odyssey.
 et al., 2005).

For the single-subject studies, the benchmarks included (Homer et al., 2005):

1. Sufficient description of the participants and setting

2. Sufficient description of the measures and measurement procedures, including interrater agreement

3. Sufficient description of the intervention A procedure used in a lawsuit by which the court allows a third person who was not originally a party to the suit to become a party, by joining with either the plaintiff or the defendant.  and procedures for determining

fidelity of implementation

4. Sufficient description of the baseline The horizontal line to which the bottoms of lowercase characters (without descenders) are aligned. See typeface.

baseline - released version
 phase and evidence of a pattern prior to intervention

5. At least three demonstrations of experimental effect, explanations of how internal and external validity External validity is a form of experimental validity.[1] An experiment is said to possess external validity if the experiment’s results hold across different experimental settings, procedures and participants.  were controlled, and established social importance and cost-effectiveness cost-effectiveness

pertaining to cost-effective.

cost-effectiveness analysis
a comparison of the relative cost-efficiencies of two or more ways of performing a task or achieving an objective.
 of the intervention

For the group-design studies, the benchmarks included (Gersten et al., 2005):

1. Research based on previous studies or a compelling argument for its importance

2. Sufficient description of the participants, setting, attrition Attrition

The reduction in staff and employees in a company through normal means, such as retirement and resignation. This is natural in any business and industry.

, and intervention agents

3. Sufficient description of the intervention, procedures for determining fidelity of implementation, and differences between treatment and control groups

4. Sufficient description of the measures and technical adequacy and data collection procedures

5. Sufficient description of the analytic an·a·lyt·ic or an·a·lyt·i·cal
1. Of or relating to analysis or analytics.

2. Expert in or using analysis, especially one who thinks in a logical manner.

3. Psychoanalytic.
 procedures with emphasis on the power analysis, unit of analysis, and variability in the sample

These studies were then reviewed using the benchmarks to determine the quality of the research and, ultimately, to draw conclusions as to whether cognitive strategy instruction is evidence-based or at least promising (Montague The name Montague can refer to the following: People
  • Andrew Jackson Montague
  • Bruce Montague
  • Charles Edward Montague, British author
  • Ed Montague (baseball player)
  • Ed Montague (umpire), son of the baseball player
 & Dietz, in press).

The remainder of this article provides a summary of the results of the review, describes the self-regulation component embedded in the cognitive routine for each of the studies, reviews several principles associated with research in strategy instruction, and offers some guidelines guidelines, a set of standards, criteria, or specifications to be used or followed in the performance of certain tasks.
 for implementation.

Results of the Literature Review

Montague and Dietz (in press) evaluated five single-subject studies: Montague and Bos 1. (operating system) BOS - Basic Operating System.
2. (tool) BOS - A data management system written at DESY and used in some high energy physics programs.
3. (programming) BOS - The Basic Object System.
 (1986); Case, Harris, and Graham (1992); Montague (1992); Hutchinson (1993); and Cassel and Reid (1996); and two group studies: Montague, Applegate, and Marquard (1993); and Chung and Tam (2005). The studies were rated by three independent raters to determine (a) whether each study met each of the quality indicators listed above; (b) whether each individual study met the criteria for "high quality" research; and (c) whether, as a body of work, the research met the standards for deeming the practice "evidence-based."

Single-subject design studies. For the single-design studies to meet the standards, the body of research must have included at least five studies that met minimally acceptable methodological criteria, documented experimental control, appeared in peer-reviewed journals peer-reviewed journal Refereed journal Academia A professional journal that only publishes articles subjected to a rigorous peer validity review process. Cf Throwaway journal. , were conducted by at least three different researchers across at least three geographical locations, and had at least 20 participants across studies.

When applying the standards and criteria developed by Homer et al. (2005) to evaluate the quality of the research, the five single-subject design studies stood up well. All used researcher-developed interventions, which, although similar in many respects, varied somewhat with regard to the cognitive and metacognitive components. All interventions produced positive outcomes for individual students. Performance improved, although some students did not meet the criterion for mastery. Most students showed maintenance over time and maintained use of the strategy in classroom settings. However, there was evidence that performance declined over time without distributed review and practice. An overall analysis of the studies as a group concluded that the practice--cognitive strategy instruction--is evidence-based and does improve mathematical problem solving for students with mathematical disabilities.

Group-design studies. For the two group-design studies to meet the standards, the body of research must have included at least four acceptable studies or two high-quality studies that supported the practice. In addition, to be considered evidence-based, the weighted effect size must have been significantly greater than zero; for "promising," there must have been at least a 20% confidence interval confidence interval,
n a statistical device used to determine the range within which an acceptable datum would fall. Confidence intervals are usually expressed in percentages, typically 95% or 99%.
 for the weighted effect size that was greater than zero.

The two group studies did not meet the criteria for either evidence-based or promising practice due to methodological issues. The primary problems for both studies included a lack of procedures to measure treatment fidelity and limited information regarding the technical adequacy of the outcome measures. This suggests that group studies designed to test the effectiveness of this practice need to be more rigorous and designed with the quality indicators in mind. All raters agreed that the interventions for both studies were described clearly and the results were positive.

Self-Regulation Components of Cognitive Strategy Instruction

The goal of cognitive strategy instruction is to teach learners multiple cognitive and metacognitive processes and strategies to facilitate and enhance performance in academic domains (e.g., mathematical problem solving) as well as nonacademic domains (e.g., social problem solving). The processes and strategies range from simple to complex depending on task difficulty and context of the task.

Students with LD characteristically are poor strategic learners and problem solvers and manifest manifest 1) adj., adv. completely obvious or evident. 2) n. a written list of goods in a shipment.

MANIFEST, com. law. A written instrument containing a true account of the cargo of a ship or commercial vessel.
 strategy deficits and differences that impede im·pede  
tr.v. im·ped·ed, im·ped·ing, im·pedes
To retard or obstruct the progress of. See Synonyms at hinder1.

[Latin imped
 performance, particularly on tasks requiring higher level processing. These students need explicit instruction in selecting strategies appropriate to the task, applying the strategies in the context of the task, and monitoring their execution. They have difficulty abandoning and replacing ineffective strategies, adapting strategies to other similar tasks, and generalizing strategies to other situations and settings. Instruction aims to develop strategic learners who have an effective and efficient repertoire Repertoire may mean Repertory but may also refer to:
  • Repertoire (theatre), a system of theatrical production and performance scheduling
  • Repertoire Records, a German record label specialising in 1960s and 1970s pop and rock reissues
 of strategies and are motivated mo·ti·vate  
tr.v. mo·ti·vat·ed, mo·ti·vat·ing, mo·ti·vates
To provide with an incentive; move to action; impel.

, self-directed, and self-regulating.

In contrast to direct instruction, which is didactic di·dac·tic
Of or relating to medical teaching by lectures or textbooks as distinguished from clinical demonstration with patients.
 and grounded in behaviorism behaviorism, school of psychology which seeks to explain animal and human behavior entirely in terms of observable and measurable responses to environmental stimuli. Behaviorism was introduced (1913) by the American psychologist John B. , the theoretical foundation of cognitive strategy instruction considers both behavioral behavioral

pertaining to behavior.

behavioral disorders
see vice.

behavioral seizure
see psychomotor seizure.
 and cognitive theory Conitive theory may refer to:
  • Theory of cognitive development, Jean Piaget's theory of development and the theories which spawned from it.
  • Two factor theory of emotion, another cognitive theory.
; that is, information processing information processing: see data processing.
information processing

Acquisition, recording, organization, retrieval, display, and dissemination of information. Today the term usually refers to computer-based operations.
 and developmental theory. Instruction focuses on cognitive processes, such as visualization Using the computer to convert data into picture form. The most basic visualization is that of turning transaction data and summary information into charts and graphs. Visualization is used in computer-aided design (CAD) to render screen images into 3D models that can be viewed from all , and metacognitive or self-regulation strategies, such as self-questioning. Cognitive strategy instruction teaches students to think and behave like good problem solvers and strategic learners. A cognitive routine is taught using explicit instruction, an instructional model that consists of very structured and organized lessons, appropriate cues and prompts, guided and distributed practice, cognitive modeling, interaction between teachers and students, immediate and corrective cor·rec·tive
Counteracting or modifying what is malfunctioning, undesirable, or injurious.

An agent that corrects.

 feedback on performance, positive reinforcement positive reinforcement,
n a technique used to encourage a desirable behavior. Also called
positive feedback, in which the patient or subject receives encouraging and favorable communication from another person.
, over-learning, and mastery.

All the studies included in Montague and Dietz's (in press) review focused on teaching a specific cognitive routine for mathematical problem solving that includes a self-regulation component. The studies included a total of 142 students ranging in age from 8-4 to 16-7 years. Most of the participants were identified with learning disabilities (N = 110), while two identified participants as having mild intellectual disabilities (Cassel & Reid, Chung & Tam, 2005). Montague used additional preset preset Cardiac pacing A parameter of a pacemaker that is programmed permanently when manufactured  criteria for participation that included average intelligence, at least a third-grade reading level, and facility with the four basic math operations using whole numbers and decimals.

Montague et al. (1986, 1992, 1993). Montague's cognitive routine (Montague & Bos, 1986; Montague, 1992; Montague et al., 1993) is a seven-phase model with specific self-regulation components. In the 1986 study, self-regulation was embedded in a script; for example, A self-questioning technique such as "What is asked?" or "What am I looking for Looking for

In the context of general equities, this describing a buy interest in which a dealer is asked to offer stock, often involving a capital commitment. Antithesis of in touch with.
?" was used to provide focus on the outcome.

The two later studies specified a SAY, ASK, CHECK routine for each of the seven processes taught. SAY requires students to self-instruct, which helps students identify and direct themselves as they solve the problem. For example, when reading the problem, students SAY "Read the problem. If I don't understand it, read it again." ASK refers to self-questioning, which promotes internal dialogue that helps to systematically analyze the problem information and regulate execution of the cognitive processes. When students paraphrase par·a·phrase  
1. A restatement of a text or passage in another form or other words, often to clarify meaning.

2. The restatement of texts in other words as a studying or teaching device.

 the problem, they ASK themselves, "Have I underlined the important information? What is the question? What am I looking for?" Finally, CHECK is the self-monitoring strategy that promotes appropriate use of specific strategies and encourages students to monitor their performance throughout the problem solving process. When students formulate formulate /for·mu·late/ (for´mu-lat)
1. to state in the form of a formula.

2. to prepare in accordance with a prescribed or specified method.
 a visual representation of the problem, they CHECK "the picture against the problem information."

Figure 1 presents the entire routine--the seven processes and the corresponding SAY, ASK, CHECK component for each. Students are required to memorize mem·o·rize  
tr.v. mem·o·rized, mem·o·riz·ing, mem·o·riz·es
1. To commit to memory; learn by heart.

2. Computer Science To store in memory:
 the processes and become familiar with the self-regulation component. After students understand what the processes are and can recite them from memory, the teacher uses process or cognitive modeling to demonstrate how good problem solvers approach a mathematical problem. Students are then required to "think aloud" as they solve practice problems. Finally, they become the "teacher," modeling how good problem solvers think and behave.

In the two single-subject studies (Montague & Bos, 1986; Montague, 1992), the strategy use of six secondary and six middle school students improved substantially, and strategy maintenance was evident. However, the two sixth graders did not meet the mastery criterion, suggesting that the comprehensive cognitive routine may have been developmentally beyond their ability. For the group study, 72 middle school students were taught in groups of 8-12; on the posttest post·test  
A test given after a lesson or a period of instruction to determine what the students have learned.
, they performed to the level of a group of nondisabled students.

Graham and Harris (2003). The Self-Regulated Strategy Development model (SRSD SRSD self-regulated strategy development
SRSD Southern Regional School District
; Graham & Harris, 2003), designed in the early 1980s to improve composition skills of students with LD, was the basis for the intervention studies by Case et al. (1992) and Cassel and Reid (1996). This model includes the basic components of all cognitive strategy instructional routines. The model consists of six stages to guide instruction: (a) develop and activate background knowledge by providing the knowledge and skills needed to acquire and apply strategies and procedures for problem solving, (b) discuss the strategy by looking at the student's current performance and explaining the strategies and how they will help the student improve their problem solving, (c) model the strategy using "think aloud" to demonstrate how giving oneself instructions helps regulate strategy use during problem solving, (d) have students memorize the strategy steps and self-statements, (e) support strategy use by providing guided practice using scaffolded instructional techniques, and (f) monitor students' performance until they can use the specific math problem-solving and self-regulation strategies independently.

Case et al. (1992). The variation of the SRSD model in the study by Case et al. (1992) included preskill development; conferencing See teleconferencing.  regarding each student's current performance level, metastrategy information, and commitment to learning the strategy; discussing the problem-solving strategy; modeling the strategy and self-instructions; mastery of the strategy steps; collaboratively practicing the strategy and self-instructions; independent performance; and generalization gen·er·al·i·za·tion
1. The act or an instance of generalizing.

2. A principle, a statement, or an idea having general application.
 and maintenance components. As part of the package, instructional goals were set collaboratively by the student and the teacher, followed by a discussion of the importance of the strategy and the self-regulation strategies (self-assessment, self-recording, and self-instruction).

The strategy was introduced using a small chart listing the following five steps:

1. Read the problem out loud.

2. Look for important words and circle them.

3. Draw pictures to help tell what is happening.

4. Write down the math sentence.

5. Write down the answer.

Four students with LD in grades 5 and 6 progressed from learning to apply the strategy with simple addition problems to subtraction subtraction, fundamental operation of arithmetic; the inverse of addition. If a and b are real numbers (see number), then the number ab is that number (called the difference) which when added to b (the subtractor) equals  problems. Students' performance on the addition problems remained high after instruction. On the subtraction problems, student performance increased dramatically, and students were able to discriminate dis·crim·i·nate  
v. dis·crim·i·nat·ed, dis·crim·i·nat·ing, dis·crim·i·nates

 between addition and subtraction problems, thus minimizing selection of the wrong operation.

Cassel and Reid (1996). Cassel and Reid (1996) used similar procedures to teach the strategy: preskill development, initial conference, discussion of the problem-solving strategy and self-regulation procedures, modeling the strategy and self-instructions, strategy mastery, collaborative practice, independent practice, and maintenance.

The strategy consisted of the following nine steps and the acronym acronym: see abbreviation.

A word typically made up of the first letters of two or more words; for example, BASIC stands for "Beginners All purpose Symbolic Instruction Code.

1. Read the problem out loud.

2. Find and highlight the question, then write the label.

3. Ask what are the parts of the problem, then circle the numbers needed.

4. Set up the problem by writing and labeling the numbers.

5. Reread Verb 1. reread - read anew; read again; "He re-read her letters to him"
read - interpret something that is written or printed; "read the advertisement"; "Have you read Salman Rushdie?"
 the problem and tie down the sign (decide if you use addition or subtraction).

6. Discover the sign (recheck the operation).

7. Read the number problem.

8. Answer the number problem.

9. Write the answer and check by asking if the answer makes sense.

The teacher modeled strategy use using self-talk self-talk,
n in behavioral medicine, internal monologues that can have a positive or negative influence upon the individual.
 and self-questioning; for example, "What is it I have to do?" "How can I solve this problem?.... FAST DRAW will help me organize my problem solving and remember all the things I need to do in order to successfully complete a word problem." "Oops, I made a mistake, so I need to correct it." Four third and fourth graders reached mastery on several types of addition and subtraction problems and maintained performance over time.
Figure 1. Math problem-solving processes and strategies.

READ (for understanding)

Say:      Read the problem. If I don't understand, read it again.
Ask:      Have I read and understood the problem?
Check:    For understanding as I solve the problem.

PARAPHRASE (your own words)

Say:      Underline the important information. Put the problem in my
          own words.
Ask:      Have I underlined the important information? What is the
          question? What am I looking for?
Check:    That the information goes with the question.

VISUALIZE (a picture or a diagram)

Say:      Make a drawing or a diagram. Show the relationships among
          the problem parts.
Ask:      Does the picture fit the problem? Did I show the
Check:    The picture against the problem information.

HYPOTHESIZE (a plan to solve the problem)

Say:      Decide how many steps and operations are needed.
          Write the operation symbols (+, -, x, and /).
Ask:      If I.... what will I get? If I.... then what do I
          need to do next? How many steps are needed?
Check:    That the plan makes sense.

ESTIMATE (predict the answer)

Say:      Round the numbers, do the problem in my head, and write
          the estimate.
Ask:      Did I round up and down? Did I write the estimate?
Check:    That I used the important information.

COMPUTE (do the arithmetic)

Say:      Do the operations in the right order.
Ask:      How does my answer compare with my estimate? Does my
          answer make sense?
          Are the decimals or money signs in the right places?
Check:    That all the operations were done in the right order.

CHECK (make sure everything is right)

Say:      Check the plan to make sure it is right. Check the
Ask:      Have I checked every step? Have I checked the computation?
          Is my answer right?
Check:    That everything is right. If not, go back. Ask for
          help if I need it.

From Solve It! A Practical Approach to Teaching Mathematical Problem
Solving Skills by M. Montague, 2003, Reston, VA: Exceptional
Innovations. Copyright by Exceptional Innovations. Reprinted with

Hutchinson (1993). Hutchinson's study (1993) targeted three types of algebra algebra, branch of mathematics concerned with operations on sets of numbers or other elements that are often represented by symbols. Algebra is a generalization of arithmetic and gains much of its power from dealing symbolically with elements and operations (such as  problems: relational problems, proportion problems, and two-variable two-equation problems. Twelve secondary school students were taught a strategy that included two types of self-regulation components. The first consisted of a series of self-questions for representing the problems and a second series of self-questions for solving the algebra problems. The self-questions for representing algebra word problems were as follows:

1. Have I read and understood each sentence? Are there any words whose meaning I have to ask?

2. Have I got the whole picture, a representation, for the problem?

3. Have I written down my representation on the worksheet? (goal, unknown(s), known(s), type of problem, equation)

4. What should I look for in a new problem to see it is the same kind of problem?

The self-questions for solving algebra word problems were:

1. Have I written an equation?

2. Have I expanded the terms?

3. Have I written out the steps of my solution on the worksheet? (collected like terms, isolated unknown(s), solved for unknown(s), checked my answer with the goal, highlighted my answer)

4. What should I look for in a new problem to see if it is the same kind of problem?

The second self-regulation component was a structured worksheet with the following prompts: (a) Goal, (b) What I don't know Don't know (DK, DKed)

"Don't know the trade." A Street expression used whenever one party lacks knowledge of a trade or receives conflicting instructions from the other party.
, (c) What I know, (d) I can write/say this problem in my own words. Draw a picture, (e) Kind of problem, (f) Equation, (g) Solving the equation, (h) Solution, (i) Compare to goal, and (j) Check. Using this strategy, students improved substantially in algebra problem solving, had significantly higher posttest scores than a comparison group, and maintained performance over time.

Chung and Tam (2005). The last study, conducted by Chung and Tam (2005) with 30 Chinese students with mild intellectual disabilities, used a modification of Montague's (1992) cognitive routine. The researchers' variation included the following five steps:

1. Read the problem out loud.

2. Select the important information.

3. Draw a representation of the problem.

4. Write down the steps for doing the computation Computation is a general term for any type of information processing that can be represented mathematically. This includes phenomena ranging from simple calculations to human thinking. .

5. Check the answer.

The self-regulation component was an adaptation of Montague's (1992) SAY, ASK, CHECK procedure (see Figure 1). Students were randomly assigned as·sign  
tr.v. as·signed, as·sign·ing, as·signs
1. To set apart for a particular purpose; designate: assigned a day for the inspection.

 to (a) conventional instruction, (b) worked example instruction, or (c) cognitive strategy instruction. Students in the worked example and cognitive strategy instruction groups outperformed students receiving conventional instruction on immediate and delayed measures of two-step addition and subtraction problems.

In sum, these studies had a common goal: to improve mathematical problem solving for students with LD using an instructional approach that promotes strategic and self-regulated learning The term self-regulated can be used to describe learning that is guided by metacognition, strategic action (planning, monitoring, and evaluating personal progress against a standard), and motivation to learn . Self-regulation is integral to cognitive strategy instruction as it directs and guides students in the application of the problem-solving process and is essential to effective and efficient mathematical problem solving.

The concluding sections of this article focus on guidelines for future research in strategy instruction and on practical considerations for implementing cognitive strategy instruction in today's schools. Eight principles The Eight Principles are one of the basic ways Chinese medicine has to diagnose. It uses the following eight divisions of symptoms:
  • Yin or Yang (yin-yang 陰陽)
  • Superficial or internal (li-biao 表裡)
  • Cold or hot (han-re 寒熱)
 of instruction discussed by Swanson (1999), derived from the literature on cognitive, learning, and memory, serve as guidelines for implementing and evaluating cognitive strategy instruction and should be considered in future research investigating "evidence-based practices." That is, interventions must be not only effective but also efficient, and teachers must consider the practices to be feasible and usable USable is a special idea contest to transfer US American ideas into practice in Germany. USable is initiated by the German Körber-Stiftung (foundation Körber). It is doted with 150,000 Euro and awarded every two years.  in typical classroom settings. Swanson's principles provide guidance in selecting and implementing evidence-based practices.

Principles of Strategy Instruction

Principle 1: Instruction must operate on the law of parsimony law of parsimony
See Ockham's razor.

Noun 1. law of parsimony - the principle that entities should not be multiplied needlessly; the simplest of two competing theories is to be preferred
. As discussed, most cognitive strategy instruction programs have multiple components (see, for example, Cassel & Reid, 1996). In essence, they are packages of content, strategies, and procedures. Determining the components of instruction that best predict student performance is a challenge for intervention researchers. Montague et al. (1993) attempted to address this question by separating the cognitive and metacognitive components in their routine and concluded that both were required, particularly for maintaining performance over time. Swanson's review (1999) suggests that the best of the instructional programs include (a) teaching a few critical strategies well; (b) teaching students to monitor their learning and performance; (c) teaching students how, when, and where to use the strategies to promote generalization; (d) integrating strategy instruction into the general curriculum; and (e) providing ongoing supervised su·per·vise  
tr.v. su·per·vised, su·per·vis·ing, su·per·vis·es
To have the charge and direction of; superintend.

[Middle English *supervisen, from Medieval Latin
 student feedback and distributed practice.

Principle 2. The use of effective instructional strategies does not necessarily eliminate processing differences in students. In other words Adv. 1. in other words - otherwise stated; "in other words, we are broke"
put differently
, research must include measures of both cognitive processes and strategies as well as academic measures and groups of students with and without LD to determine the impact not only on academic performance but also on the cognitive and metacognitive processes and strategies underlying performance. To illustrate, Hutchinson (1993) used a "think aloud" procedure and a metacognitive interview to ascertain growth in students' understanding and use of strategies during algebra problem solving.

Principle 3: Instructional strategies serve different purposes. By this, Swanson (1999) noted that certain components of instruction and combinations of components have a differential impact on performance in different domains. That is, in his synthesis, no one combination of instructional components was responsible for outcomes across domains.

Principle 4: Comparable performance does not mean students use comparable processes or strategies. Students with LD may perform as well as nondisabled students on some tasks but may use different processes or strategies to achieve the goal. These students may have a repertoire of strategies that may be effective on relatively simple tasks or tasks that present little difficulty, but on more complex activities, such strategies may not apply or may not be sufficient. Identifying the strategies students have and use appropriately and those they need to be successful on more difficult tasks is an additional challenge for intervention research.

A simple informal procedure like the Mathematical Problem Solving Assessment (Short Form) (Montague, 1992) provides information about students' perception of ability; attitude toward mathematics and math problem solving; and knowledge, use, and control of math problem-solving processes and strategies. Informal measures like this provide insight into students' knowledge of strategies and their ability to apply them appropriately on tasks like math problem solving that require higher order processing.

Principle 5: Strategies must be considered in relation to a student's knowledge base and capacity. Whether students will benefit from various types and levels of strategy instruction may depend on their cognitive characteristics such as intellectual ability or memory capacity. Thus, successful strategy instruction must consider the match between the strategy and learner characteristics. Therefore, assessing students prior to cognitive strategy instruction in a domain like mathematics is important to determine if they have the competencies to benefit from instruction as designed. Otherwise, modifications to the cognitive routine and instructional procedures may be necessary. For example, Montague et al. (1993) excluded sixth graders from the group study because they had not met the criterion for mastery in the earlier single-subject study (Montague, 1992). The researchers concluded that sixth-grade students may not be maturationally ready for the comprehensive cognitive routine as designed and that the routine should be modified for younger learners.

Principle 6: Comparable instructional procedures may not eliminate performance differences. This relates to the idea that students with LD may learn to use a strategy as well as their nondisabled counterparts but may still not perform as well on an academic task. To reach the performance level of peers, some students with LD need additional intervention.

Principle 7: Good instructional approaches for students with LD are not necessarily good approaches for nondisabled students and vice versa VICE VERSA. On the contrary; on opposite sides. . This principle is very important. The cognitive strategy instruction interventions described in this article were developed specifically for students with LD with knowledge of their cognitive and behavioral characteristics. It is important to remember that students with LD are not performing as well as their nondisabled peers for a variety of reasons, so the challenge for intervention researchers is to describe not only the characteristics of students but also how these characteristics interact with the components of cognitive strategy instruction.

Principle 8: Instructional strategies as taught do not necessarily generalize generalize /gen·er·al·ize/ (-iz)
1. to spread throughout the body, as when local disease becomes systemic.

2. to form a general principle; to reason inductively.
 to other situations, settings, and tasks. Evidence suggests that as children acquire simple strategies, the strategies undergo modification or transformation as they are applied to other and more difficult tasks, thus allowing generalization of strategy use (Pressley, Brown, El-Dinary, & Allferbach, 1995). Students with LD may not possess the cognitive mechanisms to facilitate strategy transformation, or if they do, may fail to use the mechanisms appropriately to adapt and modify strategies to perform more efficiently. If students are expected to generalize strategy use to other situations, settings, and tasks, then instruction must include procedures to promote generalization.

Implications for Practice

In conclusion, cognitive strategy instruction to improve mathematical problem solving for students with LD appears to qualify as an evidence-based practice. The primary question regarding implementing cognitive strategy instruction is: How, when, and by whom should cognitive strategy instruction be provided for students with LD?

Let's first consider the ideal conditions. Instruction should be provided by expert remedial REMEDIAL. That which affords a remedy; as, a remedial statute, or one which is made to supply some defects or abridge some superfluities of the common law. 1 131. Com. 86. The term remedial statute is also applied to those acts which give a new remedy. Esp. Pen. Act. 1.  teachers who understand the characteristics of students with LD. Instruction should be provided to small groups of students (e.g., 8-10 students), who have been assessed to determine if they will benefit from instruction. Instruction should be intense and time-limited, so teachers may wish to remove students from the general education classroom for the duration of strategy instruction and include procedures to ensure that students will generalize strategy use after returning to the class. This requires collaboration Working together on a project. See collaborative software.  between general and special education teachers.

However, the ideal may not be possible for several reasons. First, with the move toward inclusion in most districts, students with LD are being placed in general education mathematics classes often with teachers who have no or limited background teaching these students. Second, teachers may not have the necessary expertise or background in strategy instruction. Therefore, they may need professional development and continued support from a specialist to implement strategy instruction with fidelity. Third, teachers may be pressured by the district to complete the required curriculum and prepare students for state assessments. As a result, they may feel they do not have sufficient time to implement strategy instruction.

The above can be serious impediments IMPEDIMENTS, contracts. Legal objections to the making of a contract. Impediments which relate to the person are those of minority, want of reason, coverture, and the like; they are sometimes called disabilities. Vide Incapacity.
 to implementing evidence-based practices like cognitive strategy instruction for students with LD in typical classroom settings. Obtaining the support of district- and school-level administrators and the commitment of both general and special education teachers is critical to successful implementation of cognitive strategy instruction for students with LD.


Case, L. P., Harris, K. R., & Graham, S. (1992). Improving the mathematical problem-solving skills of students with learning disabilities: Self-regulated strategy development. The Journal of Special Education, 26, 1-19.

Cassel, J., & Reid, R. (1996). Use of a self-regulated strategy intervention to improve word problem-solving skills of students with mild disabilities. Journal of Behavioral Education, 6, 153-172.

Chung, K. H., & Tam, Y. H. (2005). Effects of cognitive-based instruction on mathematical problem solving by learners with mild intellectual disabilities. Journal of Intellectual and Developmental Disability developmental disability
A cognitive, emotional, or physical impairment, especially one related to abnormal sensory or motor development, that appears in infancy or childhood and involves a failure or delay in progressing through the normal
, 30, 207-216.

Flavell, J. H. (1976). Metacognitive aspects of problem solving. In L.B. Resnick (Ed.), The nature of intelligence (pp. 231-245). Mahwah, NJ: Lawrence Erlbaum.

Gersten, R., Fuchs, L.S., Compton, D., Coyne, M., Greenwood Greenwood.

1 City (1990 pop. 26,265), Johnson co., central Ind.; settled 1822, inc. as a city 1960. A residential suburb of Indianapolis, Greenwood is in a retail shopping area. Manufactures include motor vehicle parts and metal products.
, C., & Innocenti, M. S. (2005). Quality indicators for group experimental and quasi-experimental research in special education. Exceptional Children, 71, 149-164.

Graham, S., & Harris, K. R. (2003). Students with learning disabilities and the process of writing: A meta-analysis of SRSD studies. In H. L. Swanson, K. R. Harris, & S. Graham (Eds.), Handbook
For the handbook about Wikipedia, see .

This article is about reference works. For the subnotebook computer, see .
"Pocket reference" redirects here.
 of learning disabilities (pp. 323-334). New York New York, state, United States
New York, Middle Atlantic state of the United States. It is bordered by Vermont, Massachusetts, Connecticut, and the Atlantic Ocean (E), New Jersey and Pennsylvania (S), Lakes Erie and Ontario and the Canadian province of
: Guilford Press.

Horner, R. H., Carr CARR Carrier
CARR Customer Acceptance Readiness Review
CARR Carrollton Railroad
CARR Corrective Action Request and Report
CARR City Area Rural Rides (Texas)
CARR Configuration Audit Readiness Review
CARR Customer Acceptance Requirements Review
, E. G., Halle, J., McGee, G., Odom, S., & Wolery, M. (2005). The use of single-subject research Single Subject Research Designs

aka small-n research designs, quasi-experimental research designs.

This group of research methods is used extensively in the experimental analysis of behavior in both basic and applied settings with both human and non-human
 to identify evidence-based practice in special education. Exceptional Children, 71, 165-179.

Hutchinson, N. L. (1993). Effects of cognitive strategy instruction on algebra problem solving of adolescents with learning disabilities. Learning Disability Quarterly, 16, 34-63.

Kroesbergen, E. H., & van Luit, J.E.H. (2003). Mathematics interventions for children with special needs. Remedial and Special Education, 24, 97-114.

Montague, M. (1992). The effects of cognitive and metacognitive strategy instruction on the mathematical problem solving of middle school students with learning disabilities. Journal of Learning Disabilities, 25, 230-248.

Montague, M. (2003). Solve it! A practical approach to teaching mathematical problem solving skills. Reston, VA: Exceptional Innovations.

Montague, M., Applegate, B., & Marquard, K. (1993). Cognitive strategy instruction and mathematical problem-solving performance of students with learning disabilities. Learning Disabilities Research & Practice, 8, 223-232.

Montague, M., & Bos, C. S. (1986). The effect of cognitive strategy training on verbal math problem solving performance of learning disabled adolescents. Journal of Learning Disabilities, 19, 26-33.

Montague, M., & Dietz, S. (in press). The quality of research in cognitive strategy instruction for teaching mathematical problem solving to students with disabilities. Exceptional Children.

Pressley, M., Brown, R., El-Dinary, P. B., & Allferbach, P. (1995). The comprehension comprehension

Act of or capacity for grasping with the intellect. The term is most often used in connection with tests of reading skills and language abilities, though other abilities (e.g., mathematical reasoning) may also be examined.
 instruction that students need: Instruction fostering constructively responsive reading. Learning Disabilities Research & Practice, 10, 215-224.

Swanson, H. L. (1999). Interventions for students with learning disabilities: A meta-analysis of treatment outcomes. New York: Guilford Press.

Swanson, H. L., & Sachs-Lee, C. (2000). A meta-analysis of single-subject-design intervention research for students with LD. Journal of Learning Disabilities, 33, 114-136.

Wong, B.Y.L. (1999). Metacognition in writing. In R. Gallimore, L. P. Bernheimer, D. L. MacMillan, D. L. Speece, & S. Vaughn (Eds.), Developmental perspectives on children with high-incidence disabilities (pp. 183-198). Mahwah, NJ: Lawrence Erlbaum.

Wong, B.Y.L, Harris, K. R., Graham, S., & Butler, D. L. (2003). Cognitive strategies instruction research in learning disabilities. In H. L. Swanson, K. R. Harris, & S. Graham (Eds.), Handbook of learning disabilities (pp. 383-402). New York: Guilford Press.

MARJORIE MONTAGUE, Ph.D., University of Miami This article is about the university in Coral Gables, Florida. For the university in Oxford, Ohio, see Miami University.

The University of Miami (also known as Miami of Florida,[2] UM,[3] or just The U

Please address correspondence to: Marjorie Montague, School of Education, University of Miami, 5202 University Dr., Coral Gables Coral Gables, city (1990 pop. 40,091), Miami-Dade co., SE Fla., SW of Miami; inc. 1925. Founded at the height of the Florida land boom, Coral Gables is a noted planned city, with tree-lined boulevards and Mediterranean-style buildings. , FL 33146;
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Author:Montague, Marjorie
Publication:Learning Disability Quarterly
Article Type:Report
Geographic Code:1USA
Date:Jan 1, 2008
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