Validated Practices for Teaching Mathematics to Students with Learning Disabilities: A Review of Literature.

teaching sequence while simultaneously teaching 12 FOCUS ON EXCEPTIONAL CHILDREN SEPTEMBER 1998 TABLE 2 Problem-Solving Studies Dependent Intervention Citation Subjects Setting Design Measure Results Direct Wilson & Sindelar, 62 Elementary Group CriterionDirect instruction groups scored Instruction 1991 resource room, office comparison referenced better than sequence-only control space, media center, group and cafeteria Manipulative Baker, 1993 46 Elementary Group CriterionComparable results for strategy + Devices and research-only comparison referenced drawing and strategy-only groups Drawings class Huntington, 1995 3 High school Multiple CriterionCSA was effective for teaching resource room probe referenced algebra word problems; skills were maintained for 9 weeks Jitendra & Hoff, 1996 3 Private elementary Multiple CriterionSchema-based diagrams improved school for students probe referenced problem-solving skills with learning disabilities March & Cooke, 1996 3 Elementary general Multiple CriterionManipulative devices improved education classroom baseline referenced word-problem skills Miller & Mercer, 1993a 67 Elementary selfPreCriterionTwo studies: Posttest results showed contained class and posttest, referenced improvements for multiplication and resource room no control subtraction group Walker & Poteet, 70 Middle school Group CriterionNo differences between diagrammatic 1989-1990 resource room comparison referenced instruction and traditional key-word method Strategy Case, Harris, & 4 Elementary selfMultiple CriterionStrategy instruction improved Instruction Graham, 1992 contained class baseline referenced problem-solving in addition and subtraction Cassel & Reid, 1996 2 Elementary Multiple CriterionBoth students mastered the strategy resource room baseline referenced and maintained gains at 6 and 8 weeks Hutchinson, 1993 20 Junior high Repeated CriterionStrategy instruction was effective for resource room (ABAB) and referenced improving algebra scores group comparison Lambert, 1997 76 High school Group CriterionNo difference between strategy group resource room comparison referenced and textbook group Montague, 1992 6 Middle school Multiple CriterionCombination of cognitive and research-only class baseline referenced metacognitive strategies was more effective than either strategy alone Montague, Applegate, 72 Junior high researchGroup CriterionStrategies improved problem-solving & Marquard, 1993 only class comparison referenced acquisition and maintenance; students compared well to normally achieving peers basic computation skills. In the first study, 54 elementary students with learning disabilities were taught multiplication facts using three concrete-level lessons involving manipulative devices, three semiconcrete-level lessons involving drawings and tallies, and between 10 and 15 abstract-level lessons depending on student absences. Also included in this instructional process were two mnemonic devices: one for computation and one for problem solving. The lessons were scripted and followed a four-step instructional process (advance organizer, demonstrate and model, guided practice, independent practice). The word problems gradually increased in difficulty from one lesson to the next. Prior to these instructional lessons, the students were unable to solve any word problems. After the lessons, the overall posttest mean score was 84%. An analysis of the different types of posttest problems resulted in a mean score of 97% for problems without extraneous information, 94% for problems with extraneous information, and 60% for createyour-own problems. Students needed additional practice to become proficient in making up their own word problems. In the second study, 13 elementary students with learning disabilities were taught subtraction using the same instructional format (three concrete-level lessons, three serniconcrete-level lessons, one mnemonic device lesson, 15 abstract-level lessons, four-step instructional process, and graduated word problem sequence). Prior to these instructional lessons the students were unable to solve any word problems. After the lessons, the overall mean score was 92%. Analysis of the various types of problems resulted in a mean score of 100% for problems without extraneous information, 86% for problems with extraneous information, and 91 % for create-your-own word problems. Huntington (1995) examined the effect of a concretesemiconcrete-abstract teaching sequence on the algebraic problem-solving performance of three high school students with learning disabilities. A multiple-probe design was used to investigate the three phases of instruction. During the concrete instructional phase, students represented problems with manipulative devices. During the semi-concrete instructional phase, students represented problems with manipulative devices and drew pictures of the representations. During the abstract instructional phase, students represented problems with manipulative devices, drew pictures of the representations, wrote algebraic equations, and solved the equations. Results of this study indicated that the CSA teaching sequence was effective. All three students reached 100% accuracy during three consecutive sessions. Moreover, these skills were maintained at 9 weeks. 13 Marsh and Cooke ( 1996) examined the effects of using manipulative devices (Cuisenaire rods) to teach students to identify the correct operation to use when solving math word problems. The subjects in their multiple-baseline study were three third-grade boys with learning disabilities. During the baseline phase of this study, the boys received group instruction in analyzing word problems using a verbal questioning technique. After one of the students read the word problem, the teacher asked questions such as, "What are we trying to find?" "Is there any information here that is not needed?" "Is this number needed?" and guided the students through the problem-solving process. During the treatment phase of the study, the students read the word problem and the teacher verbally guided them in positioning their rods to demonstrate what the problem was asking. At the conclusion of each 20-minute lesson in this study, the students were given a 10-item word problem probe to complete without using any manipulative devices. The performance of all three subjects improved after using manipulative devices. Specifically, they improved 58%, 74%, and 77% over baseline. Walker and Poteet (1989-1990) compared the effectiveness of a diagrammatic method and a keyword method for solving math word problems. Their study involved 70 sixth-, seventh-, and eighth-grade students with learning disabilities. Students in the control group received traditional instruction in the keyword method (identifying words such as "altogether," "left," "in all," to help solve the problem). The students were taught to: (a) determine what was to be found based on a keyword method, (b) locate what information was given, (c) write a number sentence from the information given, and (d) find the answer for one-step story problems. Students in the experimental group received diagrammatic instruction. These students were taught to (a) draw a diagrammatic representation of the problem, (b) write number sentences from the representation, and ( c) solve the onestep story problems. After 17 days of instruction, the students were posttested on oneand two-step problems. There was no statistical difference between the two types of instruction; neither approach effectively increased performance from the preto posttest. Baker (1993) investigated the effects of student-generated drawings on the ability of students with learning disabilities to solve two types of word problems-one involving multiplication or repeated addition and the other involving division or repeated subtraction. The 46 thirdthrough fifth-grade students were randomly assigned to the 14 FOCUS ON EXCEPTIONAL CHILDREN SEPTEMBER 1998 control and experimental groups. The control group received instruction in a four-step strategy: (a) read the problem, (b) restate the problem information, ( c) identify the unknown information, (d) generate a problem solution. The experimental group received instruction in the same four-step strategy and, in addition, were encouraged to draw a pictorial representation of the problem prior to solving it. No significant differences were found between the two groups on either type of problem. Moreover, both groups demonstrated improvement from the pretest to the posttest. Jitendra and Hoff (1996) examined the effectiveness of using schema-based diagrams for solving one-step addition and subtraction word problems. Using a multiple probeacross-students design, three elementary students with learning disabilities were taught to analyze word problems and map the critical elements onto preprepared schematic diagrams. The diagrams were designed to illustrate the relationships between the numbers in the word problems. Approximately eight scripted lessons were taught to each student. The lesson format involved teacher-led demonstration and modeling, along with frequent student exchanges to identify the critical elements of the problem and to map them on the appropriate schemata diagram. Results indicated that all three students increased their percentage of correct word problem solutions after the schema-based diagram instruction. The subjects' mean scores prior to instruction were 20.3%, 31.3%, and 26.6%. After instruction their mean scores were 97.6%, 95%, and 95.2%, respectively. Maintenance probes 2 to 3 weeks after instruction ranged from 67% to 95.2%. Use of Strategy Instruction Case, Harris, and Graham (1992) examined the effectiveness of a five-step strategy designed to help students comprehend and devise appropriate solutions for addition and subtraction word problems. The strategy was taught via oneto-one tutoring to four fifthand sixth-grade students with learning disabilit