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PAGE 1 Mathematics S t rategy Instruction (SI) for Middle School S t udents with Learning Disabilities Paula Maccini and Joseph Gagnon PAGE 2 Mathematics Strategy Instruction (SI) for Middle School Students with Learning Disabilities y Paula Maccini and Joseph Gagnon ne of the greatest ch allenges teachers currently face with students who are struggling Left vel, y, ) ne effective approach to assisting middle school youth with LD in accessing challenging l ction hat Is a Strategy and What are the Key Features? B O academically is how to provide access to the general education curriculum. The No Child Behind Act of 2001 and the Individuals with Disab ilities Education Act of 2004 support the assertion that all children, in cluding those with disabilities, should have access to the same curriculum. Furthermore, the National Council of Teachers of Mathem atics (2000) supports providing all youth equal access to mathematical concepts. However, students with disabilities in general, and those with learning di sabilities (LD) at the middle school le often have difficulty meeting academic content standards and passing state assessments (Thurlow, Albus, Spicuzza, & Thompson, 1998; Thur low, Moen, & Wiley, 2005). Specificall students with LD often have difficulties with mat hematics, including basic skills (Algozzine, OShea, Crews, & Stoddard, 1987; Cawley, BakerKroczynski, & Urban, 1992), algebraic reasoning (Maccini, McNaughton, & Ruhl, 1999) and problemsolving skills (Hutchinson, 1993; Montague, Bos, & Doucette, 1991). Many of these students str uggle with how to (a approach math problems; (b) make effective decisions; and (c) carry out the chosen plan (Maccini & Hughes, 2000; Maccini & Ruhl, 2000). O mathematical concepts is to provide strategy instruction (SI). This brief defines strategy instruction, identifies key features of effective strategi es, and identifies key components necessary for instructing youth in the use of a strategy. In addition, we provide a practica example for the use of a math instructional strategy that can be app lied to a variety of concepts and settings, and provide some key consi derations when using strategy instru in mathematics classes. W uence of needed actions but also ices to help students remember the strategy (e.g., a First Letter ther (b) Strategy steps that use familiar word s stated simply and concisely and begin with (c) the (d) e cognitive abilities (i.e., the (e nitoring problem solving performance (Did I check my answer? ) (Lenz, Ellis, & Scanlon, 1996). A strategy refers to, a plan that not only specifies the seq consists of critical guidelines and rules relat ed to making effective decisions during a problem solving process (Ellis & Lenz, 1996, p.24). Some f eatures that make strategies effective for students with LD are: (a) Memory dev Mnemonic, which is created by forming a word from the beginning letters of o words); action verbs to facilitate student involvement (e.g., read the problem carefully); Strategy steps that are sequenced approp riately (i.e., students are cued to read word problem carefully prior to solv ing the problem) and lead to the desired outcome (i.e., successfully so lving a math problem); Strategy steps that use prompts to get students to us critical steps needed in solving a problem); and ) Metacognitive strategies that use prompts for mo PAGE 3 strategies combine several of these features. Some cini & Hughes, 2000; Maccini & Ruhl, 000) firstletter mnemonic that can help students recall the sequential steps from familiar The steps for STAR include: m; Figure 1). eachers can use selfmonitoring forms or structured worksheets to help students remember nd organize important steps and substeps. For example, students can keep track of their STAR is an example of an empirically validated (Mac 2 words used to help solve word problems involving integer numbers. (a) Search the word problem; (b) Translate the proble (c) Answer the problem; and (d) Review the solution (see T a problem solving performance by checking off ( ) the steps they completed (e.g., Did I check the reasonableness of my answer? ). What Is Strategy Instruction and What are the Key Components in Math? trategy instruction involves teaching strategies that are both effective (assisting students ire the hine ents with LD (see S with acquiring and generalizing information) and efficient (helping students acqu information in the least amount of time) (Lenz et al., 1996, p. 6). Student retention and learning is enhanced through the systematic use of effective teaching variables (Rosens& Stevens, 1986). That is, certain teaching variables (i.e., review, teacher presentation/modeling, guided practice, independent practice, feedback, and cumulative review) are both effective and efficient for teaching math to secondary studGagnon & Maccini, in press, for a complete description). Example of Strategy Instruction in Secondary Math: T he example below demonstrates a classroom lesson incorporating the firstletter mnemonic es the previously noted strategy ) l ake sure they have the rerequisite skills and vocabulary relevant to the appropriate math concept(s) and to make what a strategy, STAR (Maccini, 1998). This strategy incorporat features and effective teaching components to help teach the information efficiently and effectively. In addition, the strategy incorporates the concretesemiconcreteabstract (CSAinstructional sequence, which gradually advances to abstract ideas using the following progression: (a) concrete stage (i.e., threedimensional representation in which students manipulate objects to represent math problems); (b) semiconcrete (i.e., twodimensionarepresentation in which students draw pictures of the math problem); and (c) abstract (i.e.students represent the problem using numerical symbols). Prior to the lesson, the teacher should pretest students to m p sure the strategy is needed. The teacher then introduces the strategy and describesstrategy is, including the rationale for learning the specific instructional strategy and where and when to apply it. After an explanation, the teacher asks students to explain the purpose of the strategy, how it will help them solve word problems, and how to use the strategy. PAGE 4 Students should memorize the steps of the mnemonic strategy an d related substeps for ease of recall by using a rapidfire rehearsal. This rehearsal technique involves first calling on individual students (or throwing a ball to student s) and having them stat e a strategy step, then repeating the process with ot her students in the class. The rehearsal becomes faste students become more fluent with the steps and rely less on teacher prompts or written prompts. r as igure 1: Instructional Steps for a Classroom Lesson Adv izer of the strategy to help: n to the new lesson; R with word problems teger numbers. We used our Algebra tiles to demonstrate the ur F 1. Provide an ance Organizer The teacher provides an advance orga n (a) relate previously mastered informatio (b) state the new skill/informati on that is to be presented; and (c) provide a rationale for learning the new information. Teacher: Yesterday, we used the problem solving strategy, STA involving in problem and our STAR worksheets to keep track of the steps. Today, we are going to use the strategy and draw pictures to demonstrate the problems on o worksheets. This will be us eful because we will not al ways have the math tiles available to help us solve subtraction problems involving integer numbers. It is important to learn how to solve these problems in order to solve many realworld problems, including money and exchange problems, temperature differences, and keeping track of yardage lost or gained in a game. 2. Provide Teacher Modeling of the T ta e ents ers problem states, On a certain morning in College lly, and write down what negative cancel each other. I can cancel and +8, which results in +9 remaining. Strategy Steps he teacher first thinks aloud while modeling the use of the strategy with the rget problems. Then the teacher checks off the steps and writes down th responses on an overhead version of the structured worksheet, while the stud write their responses on individual structured worksheets. Next, the teacher models one or two more problems while gr adually fading his or her assistance and prompts and involving the students vi a questions (e.g., What do I do first?) and written responses (i.e., having students write down the problems and answ on their structured worksheet). Teacher: Watch and listen as I solve the problem using the STAR strategy and the structured worksheet. The Park, Maryland, the high temperature was 8 F, and the temperature increased by 17 F by the afternoon. What was the tem perature in the afternoon that day? (See Figure 2 for a copy of the structured worksheet). S : Okay, so the first step in the STAR strategy is for me to search the word problem. That means I need to read the problem carefu I know and what I need to find. In th is problem, I know that I have two temperatures and I need to find the temperature by the afternoon. T : My next step is to translate the problem in to picture form. First, Ill draw 8 tiles in the negative area and then Ill draw 17 tiles in the positive area. A : Then I need to answer the problem. I know one positive and one Therefore, the answer is +9. R : Finally, I need to check my answer. OK, Ill reread the word problem and check the reasonableness of my answer. Yes, my answer is +9F and it is a reasonable answer. PAGE 5 3. Provide Guided Practice Tvag their structured worksheets. Guidance is gradually faded the students perform the task with few prompts from the teacher. he teacher provides many opportunities for the students to practice solving a riety of problems usin until 4. Provid e Independent Student Practice Students perform additional problems without teacher prompts or assistance, and the teacher monitors student performance. 5. FeedbackCorrection and elines: ocuments student performance (percent correct); oblems for students to itive feedback. The teacher monitors student performance and provides both positive and corrective feedback using the following guid (a) d (b) checks for error patterns; (c) reteaches if necessary and provides additional pr practice corrections; and (d) closes the session with pos 6. Program for Generalization ew of problems for maintenance over time r students to generalize the her problems (see Figure 3). The teacher provides a cumulative revi (weekly, monthly) and provides opportunities fo strategy to ot Figure 2: Structured, the high temperature was 8F, What was the temperature in the Worksheet of the STAR Strategy P roblem: On a certain morning in College Park, Marylandand the temperature increased by 17F by the afternoon a fternoon that day? Strategy Questions Search the word problem (a) Read the problem c arefully (b) Ask yourself questions: What do ed to find? Tranp Answer the problem Review the Solution lem (b) Ask yourself questions: Does the nse? Why? I know? What do I ne (c) Write down the facts slate the words into an equation in icture form (a) Reread the prob Write a check () after completing each task: and +8, which leavesF + (+17 F) = +9 _______ and +8 and I keep my units of 9 F) _______ ___ ____ I know I have two temperatures answer make se (c) Check the answer I can cancel(8 me with +9 tiles rem a ining, therefore, F I checked my answer (+9 remains when I cancel 8 PAGE 6 W hat are Some Considerations to Keep in Mind when Using Strategy Instruction in M ath Classes? r, 1996; Montague, 1988): word th class via adapting a ved in the STAR strategy , structure but using different story lines) and far There are a few recommendations to keep in mind when using strategy instruction in your ath class (Mille m 1. Recognize student characteristics (c ognitive and behavioral) and preferences. When teaching strategy instruction, be aware of student characterist ics and preferences For example, some students may prefer highlig hting relevant words while reading a problem aloud, while others ma y prefer underlini ng and silently reading the problem. Equally important is the need to recognize student behavioral characteristics, including their selfesteem in math and motivation. For instance, students with low motivation may need additional supports to promote acti ve engagement. Examples include creating individual student math contract s with the targeted math objectives and the goal/criterion and promoting active student involvement by having students l ead discussions while using a strategy (e.g., How did you arrive at your solution?). 2. Promote individualization of strategy instruction (SI) Students should be encouraged to individualize us e of SI in ma strategy learned in class. For exam ple, as processes invol becomes more automatic for students, recalling the first step, Search the word problem may prompt students to read t he problem carefully and to initiate translation into mathematical form (i.e., transla ting words into an equation). 3. Program for generalization It is imperative that both s pecial and general education math teachers program for both near (i.e., maintaining the same f generalization (i.e., incorporating more co mplex problems than the problems in the instructional set) of the SI math strategies in order to promote ret ention and application o strategy use. For example, for near generalization different story lines can be incorporated for generalization (i.e., use of integer numbers with problems involving time zone changes, sea level, and age) in addition to the problems used in the instructional set. For far generalization more complex problems ar e introduced than the problems initially taught in the instruct ional set (e.g., In a certain city, if the difference between the highest and lowest altitude is 155 m and the altitude of the highest point is 900 m above sea level, what is the altitude of the lowest po int?). In addition to its application to problem solving involving integer numbers, the STAR strategy can be generalized across math topics (see Figure 3 for an example involving area). PAGE 7 Figure 3: Area Example Matt is buying walltowall carpeting for his bedroom, which measures 12 feet by 16 feet. If he has $40 to spend, will he have enough money to buy the carpet that costs $2 per square yard? Search the word problem (a) Read the problem carefully (b) Ask yourself questions: What do I know? What do I need to find? (c) Write down the facts Translate the words into an equation in picture form Answer the problem Review the Solution (a) Reread the problem (b) Ask yourself questions: Does the answer make sense? Why? (c) Check the answer _______ _____ __ Matts bedroom measures 12 ft x 16 ft, has $40 to spend, the carpet costs $2/yd 2 I need to first find the total area of the room I know that 3 ft = 1 yd, and (3ft) 2 = (1yd) 2 so 9ft 2 = 1 yd 2 I will ft 2 by 9 to get yd 2 = 21.3 yd 2 The carpet costs $2/yd 2, so I will need $2 x 21.3 yd 2 = $42.60. Matt does not have enou money. gh _______ I checked my answer and it makes senseMatt needs $2.60 more in order to buy the carpet for his room. Area of the room 12 ft x 16 ft = 192ft 2 Conclusions Students with learning disabilities in mathematics often have difficulties deciding how to approach math word problems, making effective procedural decisions, and carrying out specific plans (Maccini & Hughes, 2000; Maccini & Ruhl, 2000). Strategy instruction is an effective method for assisting middle school students with learning disabilities as they complete challenging mathematical problems. To support teacher use of math strategies, this brief defined strategy instruction, and provided key features of effective strategies and instructing youth in the use of a strategy. The practical examples presented illustrate how strategies such as STAR can be applied to a variety of math concepts and can provide the support necessary to ensure student success. PAGE 8 References Algozzine, B., OShea, D. J., Crews, W. B., & Stoddard, K. (1987). A nalysis of mathematics competence of learning disabled adolescents. Journal of Special Education, 21 97107. Cawley, J. F., BakerKroczynski, S., & Urban, A. (1992). Seeking excell ence in mathematics education for students with mild disabilities. Teaching Exceptional Children, 24, 4043. Ellis, E. S., & Lenz, B. K. (1996). Perspectives on instruction in learning strategies. In D. D. Deshler, E. S. Ellis, & B. K. Lenz (Eds.), Teaching adolescents with leaning disabilities (2 nd ed., pp. 960). Denver, CO: Love Publishing Co. Gagnon, J. C., & Maccini, P. (in press). Teacher use of empiricallyvalidated and standardsbased instructional approaches in secondary mathematics. Remedial & Special Education. Hutchinson, N.L. (1993). Effects of cognitive strategy instruction on algebra problem solving of adolescents with lear ning disabilities. Learning Disabilities Quarterly 16, 6463. Individuals with Disabilit ies Education Act of 2004, Pub. L. No. 108446. Lenz, B. K., Ellis, E. S., & Scanlon, D. (1996). Teaching learning strategies to adolescents and adults with learning disabilities Austin, TX: ProEd, Inc. Maccini, P. (1998). Effects of an instructional strategy incorporating concrete problem representation on the introductory algebra performance of se condary students with learning disabilities Unpublished doctoral dissertati on, The Pennsylvania State University, University Park. Maccini, P., & Hughes, C. A. (2000). Effects of a problemsolving strategy on the introductory algebra performance of secondary student s with learning disabilities. Learning Disabilities Research & Practice, 15, 1021. Maccini, P., McNaughton, D., & Ruhl, K. (1999). Algebra instruction for students with learning disabilities: Implications from a research review. Learning Disability Quarterly, 22, 113126. Maccini, P., & Ruhl, K. L. (2000). Effects of a graduated instru ctional sequence on the algebraic subtraction of integers by sec ondary students with lear ning disabilities. Education and Treatment of Children, 23, 465489. Miller, S. P. (1996). Perspectives on math instruction. In D. D. Deshler, E. S. Ellis, & B. K. Lenz (Eds.), Teaching adolescents with leaning disabilities (2 nd ed., pp. 313367). Denver, CO: Love Publishing Co. PAGE 9 Montague, M. (1988). Strategy instruction and mathemat ical problem solving. Reading, Writing, and Learning Disabilities, 4 275290. Montague, M., Bos, C. S., & D oucette, M. (1991). Affective, cognitive, and metacognitive attributes of eighthgrade mat hematical problem solvers. Learning Disabilities Research & Practice, 6, 145151. National Council of Teacher s of Mathematics. (2000). Principles and standards for school mathematics. Reston, VA: Author. No Child Left Behind Act. Reauthorization of the Elementary and Sec ondary Education Act. Pub. L. 107110, 2102(4) (2001). Rosenshine, B., & Stevens, R. (1986). Teachi ng functions. In M. C. Witrock (Ed.), Handbook of research on teaching (3 rd ed., pp. 376391). New York: Macmillan. Thurlow, M., Albus, D., Spicuzza, R., & Thompson, S. (1998). Participation and performance of students with disabilities: Minnesotas 1996 Ba sic Standards Tests in reading and math (Minnesota Report 16). Minneapolis, MN: Universi ty of Minnesota, National Center on Educational Outcomes. Thurlow, M. L., Moen, R. E., & Wiley, H. I. (2005). Annual performance reports: 20022003: State assessment data. Minneapolis, MN: National Center on Educational Outcomes. Retrieved August 17, 2005, from http://education.umn.edu/nceo /OnlinePubs/APRsummary2005.pdf 