Selfregulation 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:
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 evidencebased practice. The results of this evaluation are described, and the selfregulation The term selfregulation can signify
********** 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 approachselfregulation. Research has consistently shown that students with LD are poor selfregulators who benefit from strategy instruction that incorporates selfregulation training (Graham & Harris Harris, Scotland: see Lewis and Harris. , 2003; Wong n. 1. A field. , Harris, Graham, & Butler, 2003). Selfregulation, 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 selfregulation, which guides an individual in the coordination of that awareness while engaged in cognitive activities (Wong, 1999). Selfregulation strategies, such as selfinstruction, selfquestioning selfques·tion·ing n. Scrutiny of one's own feelings, actions, and motivations. , selfmonitoring, selfevaluation, and selfreinforcement, 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 & SachsLee, 2000) metaanalyses of 30 years of both group and singlesubject intervention studies intervention studies, n.pl 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) smallgroup instruction, and (h) direct questioning by the teacher (Swanson, 1999). In contrast, strategy instruction focuses on processes; for example, metacognition or selfregulation. 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. 2. strategy instruction: (a) systematic and direct explanations and/or and/or conj. 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, smallgroup 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. 3. 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 200220 words per minute) an acceptable level of comprehension is above 75%. and mathematical problem solving) that utilized higher order skills such as metacognition, selfmonitoring, rule learning, and selfawareness selfawareness n. Realization of oneself as an individual entity or personality. (Swanson, 1999; Swanson & SachsLee, 2000). Likewise, Kroesbergen and van Luit (2003), in their metaanalysis metaanalysis /metaanal·y·sis/ (met?ahahnal´isis) 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 selfinstruction, a selfregulation 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 singlesubject design and two groupdesign 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 "evidencebased" 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 singlesubject 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 costeffectiveness costeffectiveness pertaining to costeffective. costeffectiveness analysis a comparison of the relative costefficiencies of two or more ways of performing a task or achieving an objective. of the intervention For the groupdesign 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. Notes: , 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 adj. 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 evidencebased or at least promising (Montague The name Montague can refer to the following: People Surnames
The remainder of this article provides a summary of the results of the review, describes the selfregulation 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, n.pl 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 singlesubject 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 "evidencebased." Singlesubject design studies. For the singledesign 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 peerreviewed journals peerreviewed 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 singlesubject design studies stood up well. All used researcherdeveloped 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 practicecognitive strategy instructionis evidencebased and does improve mathematical problem solving for students with mathematical disabilities. Groupdesign studies. For the two groupdesign studies to meet the standards, the body of research must have included at least four acceptable studies or two highquality studies that supported the practice. In addition, to be considered evidencebased, 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 evidencebased 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. SelfRegulation 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. 2. 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 hinder^{1}. [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:
tr.v. mo·ti·vat·ed, mo·ti·vat·ing, mo·ti·vates To provide with an incentive; move to action; impel. mo , selfdirected, and selfregulating. In contrast to direct instruction, which is didactic di·dac·tic adj. 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:
information processing Acquisition, recording, organization, retrieval, display, and dissemination of information. Today the term usually refers to computerbased 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 computeraided design (CAD) to render screen images into 3D models that can be viewed from all , and metacognitive or selfregulation strategies, such as selfquestioning. 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 adj. Counteracting or modifying what is malfunctioning, undesirable, or injurious. n. An agent that corrects. corrective, n 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. , overlearning, 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 selfregulation component. The studies included a total of 142 students ranging in age from 84 to 167 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 thirdgrade 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 sevenphase model with specific selfregulation components. In the 1986 study, selfregulation was embedded in a script; for example, A selfquestioning 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 selfinstruct, 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 selfquestioning, 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 n. 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. v. the problem, they ASK themselves, "Have I underlined the important information? What is the question? What am I looking for?" Finally, CHECK is the selfmonitoring 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´mulat) 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 routinethe 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 selfregulation 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 singlesubject 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 812; on the posttest post·test n. 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 SelfRegulated Strategy Development model (SRSD SRSD selfregulated 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 selfstatements, (e) support strategy use by providing guided practice using scaffolded instructional techniques, and (f) monitor students' performance until they can use the specific math problemsolving and selfregulation 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 problemsolving strategy; modeling the strategy and selfinstructions; mastery of the strategy steps; collaboratively practicing the strategy and selfinstructions; independent performance; and generalization gen·er·al·i·za·tion n. 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 selfregulation strategies (selfassessment, selfrecording, and selfinstruction). 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 a−b 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 v.intr. 1. a. 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 problemsolving strategy and selfregulation procedures, modeling the strategy and selfinstructions, 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. "FAST DRAW." 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 reread 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 selftalk selftalk, n in behavioral medicine, internal monologues that can have a positive or negative influence upon the individual. and selfquestioning; 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 problemsolving 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 relationships? 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 computation. 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 permission. 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 twovariable twoequation problems. Twelve secondary school students were taught a strategy that included two types of selfregulation components. The first consisted of a series of selfquestions for representing the problems and a second series of selfquestions for solving the algebra problems. The selfquestions 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 selfquestions 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 selfregulation 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 selfregulation 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. 2. 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 twostep 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 selfregulated learning The term selfregulated 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 . Selfregulation is integral to cognitive strategy instruction as it directs and guides students in the application of the problemsolving 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:
Principles of Strategy Instruction Principle 1: Instruction must operate on the law of parsimony law of parsimony n. 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 problemsolving 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 singlesubject study (Montague, 1992). The researchers concluded that sixthgrade 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, ElDinary, & 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 evidencebased 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., 810 students), who have been assessed to determine if they will benefit from instruction. Instruction should be intense and timelimited, 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. 2. to implementing evidencebased practices like cognitive strategy instruction for students with LD in typical classroom settings. Obtaining the support of district and schoollevel administrators and the commitment of both general and special education teachers is critical to successful implementation of cognitive strategy instruction for students with LD. REFERENCES Case, L. P., Harris, K. R., & Graham, S. (1992). Improving the mathematical problemsolving skills of students with learning disabilities: Selfregulated strategy development. The Journal of Special Education, 26, 119. Cassel, J., & Reid, R. (1996). Use of a selfregulated strategy intervention to improve word problemsolving skills of students with mild disabilities. Journal of Behavioral Education, 6, 153172. Chung, K. H., & Tam, Y. H. (2005). Effects of cognitivebased instruction on mathematical problem solving by learners with mild intellectual disabilities. Journal of Intellectual and Developmental Disability developmental disability n. 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, 207216. Flavell, J. H. (1976). Metacognitive aspects of problem solving. In L.B. Resnick (Ed.), The nature of intelligence (pp. 231245). 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 quasiexperimental research in special education. Exceptional Children, 71, 149164. Graham, S., & Harris, K. R. (2003). Students with learning disabilities and the process of writing: A metaanalysis of SRSD studies. In H. L. Swanson, K. R. Harris, & S. Graham (Eds.), Handbook
This article is about reference works. For the subnotebook computer, see .
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 singlesubject research Single Subject Research Designs aka smalln research designs, quasiexperimental 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 nonhuman to identify evidencebased practice in special education. Exceptional Children, 71, 165179. Hutchinson, N. L. (1993). Effects of cognitive strategy instruction on algebra problem solving of adolescents with learning disabilities. Learning Disability Quarterly, 16, 3463. Kroesbergen, E. H., & van Luit, J.E.H. (2003). Mathematics interventions for children with special needs. Remedial and Special Education, 24, 97114. 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, 230248. 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 problemsolving performance of students with learning disabilities. Learning Disabilities Research & Practice, 8, 223232. 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, 2633. 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., ElDinary, 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, 215224. Swanson, H. L. (1999). Interventions for students with learning disabilities: A metaanalysis of treatment outcomes. New York: Guilford Press. Swanson, H. L., & SachsLee, C. (2000). A metaanalysis of singlesubjectdesign intervention research for students with LD. Journal of Learning Disabilities, 33, 114136. 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 highincidence disabilities (pp. 183198). 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. 383402). 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), MiamiDade 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 treelined boulevards and Mediterraneanstyle buildings. , FL 33146; MMontague@aol.com 

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