Self-regulation strategies to improve mathematical problem solving

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"How can I solve fhis problem?" "FAST DRAW will belpme organize my problem solving and remember all thethings I need to do in order to successfully completea word problem." "Oops, I made a mistake, so I need tocorrect it." Four third and fourth graders reached masteryon several types of addition and subtraction problemsand maintained performance over time.Hutchinson (1993). Hutchinson's study (1993) targetedthree types of algebra problems: relational problems,proportion problems, and two-variable twoequationproblems. Twelve secondary school studentswere taught a strategy that included two types of selfregulationcomponents. The first consisted of a series ofself-questions for representing the problems and a secondseries of self-questions for solving the algebra problems.The self-questions for representing algebra wordproblems were as follows:l.Have I read and understood each sentence? Arethere any words whose meaning I have to ask?2. Have I got the whole picture, a representation, forthe problem?3. Have I written down my representation on theworksheet? (goal, unknown(s), known(s), type ofproblem, equation)4. What should I look for in a new problem to see it isthe same kind of problem?The self-questions for solving algebra word problemswere:l.Have I written an equation?2. Have I expanded the terms?3. Have I written out the steps of my solution on theworksheet? (collected like terms, isolated unknown(s),solved for unknown(s), checked myanswer with the goal, highlighted my answer)4. What should I look for in a new problem to see if itis the same kind of problem?The second self-regulation component was a structuredworksheet with the following prompts: (a) Goal,(b) What I don't know, (c) What I know, (d) I canwrite/say this problem in my own words. Draw a picture,(e) Kind of problem, (f) Equation, (g) Solving theequation, (h) Solution, (i) Compare to goal, and (j)Check. Using this strategy, students improved substantiallyin algebra problem solving, had significantlyhigher posttest scores than a comparison group, andmaintained performance over time.Chung and Tam (2005). The last study, conducted byChung and Tam (2005) with 30 Chinese students withmild intellectual disabilities, used a modification ofMontague's (1992) cognitive routine. The researchers'variation included the following five steps:l.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.5. Check the answer.The self-regulation component was an adaptation ofMontague's (1992) SAY, ASK, CHECK procedure (seeFigure 1). Students were randomly assigned to (a) conventionalinstruction, (b) worked example instruction,or (c) cognitive strategy instruction. Students in theworked example and cognitive strategy instructiongroups outperformed students receiving conventionalinstruction on immediate and delayed measures oftwo-step addition and subtraction problems.In sum, these studies had a common goal: to improvemathematical problem solving for students with LDusing an instructional approach that promotes strategicand self-regulated learning. Self-regulation is integral tocognitive strategy instruction as it directs and guidesstudents in the application of the problem-solvingprocess and is essential to effective and efficient mathematicalproblem solving.The concluding sections of this article focus onguidelines for future research in strategy instructionand on practical considerations for implementing cognitivestrategy instruction in today's schools. Eightprinciples of instruction discussed by Swanson (1999),derived from the literature on cognitive, learning, andmemory, serve as guidelines for implementing andevaluating cognitive strategy instruction and should beconsidered in future research investigating "evidencebasedpractices." That is, interventions must be notonly effective but also efficient, and teachers must considerthe practices to be feasible and usable in typicalclassroom settings. Swanson's principles provide guidancein selecting and implementing evidence-basedpractices.Principles of Strategy InstructionPrinciple 1: Instruction must operate on the lawof parsimony. As discussed, most cognitive strategyinstruction programs have multiple components (see,for example, Cassel & Reid, 1996). In essence, theyare packages of content, strategies, and procedures.Determining the components of instruction that bestpredict student performance is a challenge for interventionresearchers. Montague et al. (1993) attemptedto address this question by separating the cognitive andmetacognitive components in their routine and concludedthat both were required, particularly for maintainingperformance over time. Swanson's review(1999) suggests that the best of the instructional programsinclude (a) teaching a few critical strategies well;(b) teaching students to monitor their learning and performance;(c) teaching students how, when, and whereto use the strategies to promote generalization; (d) integratingstrategy instruction into the general curricu-Leaming Disability Quarterly 42
lum; and (e) providing ongoing supervised studentfeedback and distributed practice.Principle 2. The use of effective instructional strategiesdoes not necessarily eliminate processing differencesin students. In other words, research mustinclude measures of both cognitive processes and strategiesas well as academic measures and groups of studentswith and without LD to determine the impact notonly on academic performance but also on the cognitiveand metacognitive processes and strategies underlyingperformance. To illustrate, Hutchinson (1993) useda "think aloud" procedure and a metacognitive interviewto ascertain growth in students' understandingand use of strategies during algebra problem solving.Principle 3: Instructional strategies serve differentpurposes. By this, Swanson (1999) noted that certaincomponents of instruction and combinations of componentshave a differential impact on performance indifferent domains. That is, in his synthesis, no one combinationof instructional components was responsiblefor outcomes across domains.Principle 4: Comparable performance does notmean students use comparable processes or strategies.Students with LD may perform as well as nondisabledstudents on some tasks but may use different processesor strategies to achieve the goal. These students mayhave a repertoire of strategies that may be effective onrelatively simple tasks or tasks that present little difficulty,but on more complex activities, such strategiesmay not apply or may not be sufficient. Identifying thestrategies students have and use appropriately and thosethey need to be successful on more difficult tasks is anadditional challenge for intervention research.A simple informal procedure like the MathematicalProblem Solving Assessment (Short Form) (Montague,1992) provides information about students' perceptionof ability; attitude toward mathematics and math problemsolving; and knowledge, use, and control of mathproblem-solving processes and strategies. Informalmeasures like this provide insight into students' knowledgeof strategies and their ability to apply them appropriatelyon tasks like math problem solving that requirehigher order processing.Principle 5: Strategies must be considered in relationto a student's knowledge base and capacity.Whether students will benefit from various types andlevels of strategy instruction may depend on their cognitivecharacteristics such as intellectual ability or memorycapacity. Thus, successful strategy instruction mustconsider the match between the strategy and learnercharacteristics. Therefore, assessing students prior tocognitive strategy instruction in a domain like mathematicsis important to determine if they have the competenciesto benefit from instruction as designed.Otherwise, modifications to the cognitive routine aridinstructional procedures may be necessary. For example,Montague et al. (1993) excluded sixth graders fromthe group study because they had not met the criterionfor mastery in the earlier single-subject study(Montague, 1992). The researchers concluded thatsixth-grade students may not be maturationally readyfor the comprehensive cognitive routine as designedand that the routine should be modified for youngerlearners.Principle 6: Comparable instructional proceduresmay not eliminate performance differences. Thisrelates to the idea that students with LD may learn touse a strategy as well as their nondisabled counterpartsbut may still not perform as well on an academic task.To reach the performance level of peers, some studentswith LD need additional intervention.Principle 7: Good instructional approaches for studentswith LD are not necessarily good approaches fornondisabled students and vice versa. This principle isvery important. The cognitive strategy instruction interventionsdescribed in this article were developed specificallyfor students with LD with knowledge of theircognitive and behavioral characteristics. It is importantto remember that students with LD are not performingas well as their nondisabled peers for a variety of reasons,so the challenge for intervention researchers is todescribe not only the characteristics of students but alsohow these characteristics interact with the componentsof cognitive strategy instruction.Principle 8: Instructional strategies as taught do notnecessarily generalize to other situations, settings,and tasks. Evidence suggests that as children acquiresimple strategies, the strategies undergo modificationor transformation as they are applied to other and moredifficult tasks, thus allowing generalization of strategyuse (Pressley, Brown, El-Dinary, & Allferbach, 1995).Students with LD may not possess the cognitive mechanismsto facilitate strategy transformation, or if theydo, may fail to use the mechanisms appropriately toadapt and modify strategies to perform more efficiently.If students are expected to generalize strategyuse to other situations, settings, and tasks, then instructionmust include procedures to promote generalization.Implications for PracticeIn conclusion, cognitive strategy instruction to improvemathematical problem solving for students withLD appears to qualify as an evidence-based practice.The primary question regarding implementing cognitivestrategy instruction is: How, when, and by whomshould cognitive strategy instruction be provided forstudents with LD?Volume 31, Winter 2008 43
Page 2 and 3: descriptions of the performance of
Page 4 and 5: important information? What is the
Page 8: Let's first consider the ideal cond

Self-regulation strategies to improve mathematical problem solving

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