# Higher Expectations to Better Outcomes for Children with Disabilities

Mathematics Instruction for Secondary Students with Learning Disabilities By: Eric D. This article will discuss techniques higher Expectations to Better Outcomes for Children with Disabilities have been demonstrated to be effective with secondary students who have learning disabilities in mathematics.

Secondary students with learning disabilities generally make inadequate progress in mathematics. Their achievement is often limited by a variety of factors, including prior low achievement, low expectations for success, and inadequate instruction. This discussion considers six factors that predictably confound efforts to increase the effectiveness of instruction. Each of the factors is particularly relevant in the case of instruction for secondary students with LD. For secondary students with disabilities, the adequacy of instruction in mathematics will be judged not merely on how quickly basic skills can be learned. Students must also acquire generalizable skills in the application of mathematical concepts and problem solving.

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In earlier years, many professionals readily accepted that individual psychological differences accounted for failure to learn in school. By the time students with LD become adolescents, they have typically endured many years of failure and frustration. They are fully aware of their failure to achieve functional skills in the operations and applications of mathematics. They found that students’ judgments of their ability to solve specific types of mathematics problems were useful predictors of their actual ability to solve those problems.

Specific student estimates of self-efficacy were more accurate predictors of performance than prior experience in mathematics. For secondary students with LD, expectations to fail to learn mathematics skills can be an important obstacle. Although students’ perceptions of their self-efficacy originally develop from their experiences with success and failure in instruction, those expectations later can become factors that prevent low-achieving students from attempting to learn, or persevering in trying to apply, mathematical concepts and skills. Quality of instruction is dependent on two elements of curriculum design: organization of content and presentation of content.

Secondary students are expected to master skills in numeration and mathematical operations and to be able to apply those skills across a broad range of problem solving contexts. An important part of the business of education is selecting and organizing examples to use in instruction such that students will be able to solve problems they encounter outside of instruction. Unfortunately, commercial math curricula frequently do not adequately manage the selection or organization of instructional examples. A second deficiency is an inadequate sampling of the range of examples that define a given concept. If some instances of a concept are under represented in instruction or simply not included in instruction, students with LD will predictably fail to learn that concept adequately. If inadequate selections of instructional examples are provided, the range of a concept is not illustrated and students form limited or erroneous conceptualizations. To learn correct conceptualizations, students must be taught which attributes are relevant and which are irrelevant.

If sets of instructional examples consistently contain attributes that are irrelevant to a concept, then students will predictably learn misconceptualizations that may seriously hinder achievement. It is not uncommon to find that presentations of misleading variables have inhibited mathematics achievement. Such misconceptualizations will confuse students unless their exposure to potentially misleading cues is carefully managed. Initial sets of instructional examples should not contain key words. Secondary students must learn to deal with complex notations, operations, and problem solving strategies. Presenting the instructional examples to the student. Ambiguities that enter the design process will result in predictable misunderstandings.

The premise that curriculum quality is related to the degree to which concepts and skills are explicitly taught is being debated in the current movement to reform mathematics education. We encourage practitioners to follow this debate in the professional literature, because there is likely to be a subsequent effect on classroom practice. However, for students with LD, we believe that this dictum summarizes the empirical literature: More explicit instruction results in more predictable, more generalizable, and more functional achievement. If we do not explicitly teach important knowledge and skills, these objectives will not be adequately learned.

The need for explicitly identifying and teaching important mathematical concepts, skills, and relationships is apparent in the persistent failure of students with LD to deal adequately with common fractions, decimal fractions, percentages, ratios, and proportions. A clear example of the link between explicit instruction and expected student performance can be seen in states that use proficiency tests as criteria for promotion or graduation. Curricula should be organized so that instruction of specific skills and concepts is tightly interwoven around critical concepts. In summary, the content of instruction and its organization play critical roles in determining its quality and outcomes. Unfortunately, commercial math curricula frequently stop well short of providing adequate opportunities to learn to solve mathematics problems that involve the contexts of work and everyday situations.

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Successful student performance on algorithms and abstracted word problems does not always result in competent real, life mathematical problem solving. LD may not be managing instruction effectively. Alternatives to worksheet instruction and didactic lectures have been investigated in empirical studies. The approaches to managing instruction that we will discuss are direct instruction, interactive instruction, peer-mediated instruction, and strategy instruction. The effectiveness of the first three of these approaches has been documented across a variety of curriculum areas with secondary students in general, remedial, and special education programs.

During each phase the teacher works to maintain high levels of active student involvement, successful acquisition, and progress through the curriculum. A brief statement, such as, “Look up here. We are going to begin,” is generally adequate. The teacher reminds the students of what was accomplished in the previous lesson and sets the goal for the current session. For example, “Yesterday we learned how to calculate the area of squares and rectangles. Review pertinent achievements from previous instruction. State the goal of the lesson.

Prompt the students to perform the skill along with you. Check the students’ acquisition as they perform the skill independently. Review the accomplishments of the lesson. Preview the goals for the next lessons. With the first phase of instruction the teacher models the task. Several prompted trials are usually necessary before students can be expected to respond independently. After students respond correctly on several independent test trials, the teacher should have them complete practice examples.

If the students are practicing newly acquired skills, they should be supervised for at least the first few practice examples. If they do not need assistance completing their practice exercises, the teacher may close the lesson. First, he or she reviews what was learned during the current lesson, where there may have been difficulty, and where performance may have been particularly good. The review may also include a brief statement of how learning in the current session extended what was already known. Second, the teacher provides a brief preview of the instructional objectives for the next session. Practice activities are essential components of mathematics instructional programs. Students with LD will generally need more practice and practice that is better designed than students without LD, if they are to achieve adequate levels of fluency and retention.

Worksheets are commonly used to provide practice, but the ones that publishers supply are frequently inadequate. Table 2 provides a list of principles for designing and evaluating practice activities for students with LD. By themselves, the presentation procedures have been demonstrated to provide for efficient instruction across age levels, ability levels, and curriculum domains. Reduce interference between concepts or applications of rules and strategies by separating practice opportunities until the discriminations between them are learned. Make new learning meaningful by relating practice of subskills to the performance of the whole task, and by relating what the student has learned about mathematical relationships to what the student will learn about mathematical relationships. Reduce processing demands by preteaching component skills of algorithms and strategies, and by teaching easier knowledge and skills before teaching difficult knowledge and skills. Ensure that skills to be practiced can be completed independently with high levels of success.

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Current initiatives in both general and special education include two major forms of peer-mediated instruction-peer tutoring and cooperative learning. Both enjoy broad support in the empirical literature. Peer tutoring can take various forms. 46 studies of cooperative learning resulted in three important conclusions regarding which variables contribute to the success of cooperative learning. First, there was no evidence that group work itself facilitated individual students’ achievement.

In summary, individual accountability with group rewards contributes to higher levels of achievement than individualized incentive structures, but group work without individual accountability or group rewards does not contribute to higher achievement than might be obtained with individualized task and incentive structures. They must be able to solve problems independently, because teachers and peers are not always available or able to help them. Not only must students master the information and skills taught in their classes, but they must also successfully apply that knowledge and those skills to solve varied and often complex mathematical problems that they encounter outside instruction. According to their model, effective strategy instruction follows a process that is consistent with the development of curriculum and instruction that we described in the previous sections of this article.

Strategies are selected with reference to the curriculum demands. In the case of mathematics, students confront many quantitative and conceptual relationships, algorithms, and opportunities to apply mathematical knowledge to solve problems. Thus, training for generalization and strategic problem solving can become a ubiquitous part of mathematics curricula. By learning to be active and successful participants in their achievement, students learn to perceive themselves as competent problem solvers. Empirical research on strategy instruction has not been comprehensive in all areas of instruction, but educators should be optimistic and make reasonable attempts to implement strategy instruction. Many studies of strategy instruction have been conducted with secondary students across a variety of academic domains. The academic difficulties of secondary students with LD are diverse and complex.

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Current research on mathematics instruction for students with learning difficulties is not sufficiently developed to provide teachers with precise prescriptions for improving instruction. Therefore, the best educators’ best efforts will frequently be based on reasonable extrapolations. Assessments of instruction should provide data on individual students’ progress in acquiring, generalizing, and maintaining knowledge and skills set forth in the curriculum. The value of CBA as a technique for improving quality of instruction can be attributed to the effects it appears to have on teachers’ instructional behavior. First, the collection of valid curriculum-based measures requires that teachers specify their instructional objectives. Efforts to identify critical instructional objectives may also lead teachers to consider how those objectives should be sequenced for instruction.

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In summary, CBA provides for frequent assessments of student achievement that are directly related to instructional programs. Its use appears to have the effect of more rationally relating instructional decisions to instructional objectives and student difficulties, thus contributing to increased student achievement. There are, however, diverse opinions among educators about the nature of effective instruction. It would be gratifying if empirical research played a bigger role in directing educational practice, but, frequently, beliefs and convictions play more influential roles. Constructivism is an ideology that is becoming increasingly popular in the current mathematics education reform movement.

Its core belief is that knowledge is not transmitted directly from the teacher to the student. Teacher education is a major obstacle, but, compared to the others, it may be the easiest to overcome. Under relatively unstructured conditions in which students play very active roles in the direction of instruction, it is difficult for even the most knowledgeable teacher to facilitate learning adequately and consistently. The second obstacle is that in constructivist approaches, students often persevere through trial-and-error learning.

Over the course of an instructional session, it must be apparent to both the student and the teacher that progress is being made. Two factors limit the probability that progress will be consistently detected and rewarded: difficulty specifying objectives, and complexity of instructional conditions. In summary, the constructivist perspective, though intuitively appealing, is currently unsupported by empirical research and is logically inadequate for the task of teaching adolescents with LD. As a result of frequent failure, and of prolonged instruction on such simple skills, it is generally difficult to motivate them to attempt complex tasks or to persist in independent work. By the time that students graduate or drop out of school, they will have made only the most rudimentary achievements. We strongly believe that efforts to improve mathematics education must be based on empirical research.

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There is already a sufficient body of research to serve as the basis for making substantial improvements in educational practice, but much more is needed. Empirical documentation of effective practice will not be sufficient to effect improvements. We are also convinced that teachers must have at hand effective instructional procedures, materials, and other resources. At the present time they must do much of the work of improving mathematics education themselves. It is the business of commercial publishers to design instructional procedures and curricula. Jones, EdD, is a professor in the Department of Special Education at Bowling Green State University, Bowling Green, Ohio.

His research interests include applications of behavior analysis to the design and delivery of instruction for students with learning difficulties. Rich Wilson, PhD, is a professor in the Department of Special Education at Bowling Green State University. References Click the “References” link above to hide these references. Analysis of mathematics competence of learning disabled adolescents. The Journal of Special Education, 21, 97-107.

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Strategy instruction research comes of age. Special education teachers interpret constructivist teaching. Remedial and Special Education, 15, 267-280. Variables related to the effective instruction of difficult to teach children. Using a demonstration strategy to teach mid-school students with learning disabilities how to compute long division. Journal of Learning Disabilities, 21, 77-81.

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