The 16 different combinations of modes of presentation and response allow instructional programs to be tailored to both the individual needs of students and the realistic representations of mathematical problem-solving situations. Cawley et al. Over the past 20 years, educators have taken a growing interest in using students to help with each other's academic achievement.

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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. Two classmates may take turns helping each other in one on one practice of skills that have been presented earlier. It could also be an arrangement whereby a higher achieving student helps, or monitors the performance of, a lower achieving student e.

Cooperative learning also involves peers assisting each other, but, instead of pairs of students, cooperative learning groups usually comprise three or more students of differing ability levels.

## Using Manipulatives

Slavin defined cooperative learning as instructional arrangements in which students spend much of their class time working in small, heterogeneous groups on tasks they are expected to learn and help each other learn. The aspects that contribute to the effectiveness of both approaches will be discussed in this section. Slavin's analysis of 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. Second, cooperative incentive structures where two or more individuals depend on each other for a reward that they will share if they are successful themselves did not have significant effects on the achievement of individual students; group rewards for group products did not improve student achievement. Third, group incentive structures were associated with higher achievement if the performance of individual students was accounted for and was reflected in the group rewards.

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The simplest method for managing individual accountability is to average individual scores to determine the reward for group members. A second method is to determine group rewards based on whether, or by how much, individual members exceeded their individual criterion. A third method is to assign each student in a group a unique task.

Slavin warned, however, that although task specialization provides for individual accountability, it must be accompanied by group rewards. Furthermore, task specialization is not appropriate in situations where the instructional goal is for all members of the group to acquire the same knowledge and skills.

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. Slavin cautioned that peer norms and sanctions probably apply only to individual behaviors that are seen by group members as being important to the success of the group.

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For example, he suggested that when rewards are group oriented and there is no individual accountability, group norms may apply to the behaviors of only those group members who are considered by the group as most apt to contribute to the quality of the group product. Several studies of cooperative learning have demonstrated its effectiveness for teaching mathematics skills e. In one particularly relevant study, Slavin and Karweit compared the effectiveness of cooperative learning and direct instruction for teaching mathematical operations from real-life problems to low-achieving students in ninth grade.

They observed higher levels of achievement with cooperative learning than with direct instruction alone. Combining direct instruction with cooperative learning procedures did not produce higher levels of achievement than cooperative learning alone. A caveat, however, is warranted, because a reasonable explanation for the relative superiority of cooperative incentives is that the difficulties the ninth graders in the study experienced were the result of their having forgotten or incompletely learned basic math skills from earlier instruction.

In such cases, incentive structures probably play a more important role than the model-lead-test prompting technique. When teaching new concepts or skills, teachers would be well advised to combine direct instruction and cooperative learning. Students with LD must take an active role in managing their instruction.

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.

Competent students independently select, apply, and monitor strategic procedures to solve complex and novel problems. Dreshler and associates e. 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. Teachers manage the instruction of strategies by overtly modeling strategies and then leading students through their applications. Students verbalize their applications of strategies and monitor their own progress.

The teacher also provides the students with many opportunities to determine which strategies are appropriate, to use the strategies, and to be rewarded for successful applications. 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. They are more apt to attempt to apply knowledge in novel ways and to persevere to solve difficult problems than if they see themselves as ineffective, likely to fail, and dependent on others to solve novel and difficult problems.

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. Because those evaluations have involved only a few areas of math skills, researchers should continue to study the conditions that influence the efficacy of strategy instruction. It is essential that instructional interventions be evaluated frequently.

## Sessions - National Council of Teachers of Mathematics

The academic difficulties of secondary students with LD are diverse and complex. 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. Unless instructional assessments are conducted frequently and with reference to the students' performance on specific tasks, it will not be possible to use the information to make rational decisions for improving instruction.

To an increasing extent, educators have come to the conclusion that traditional standardized achievement testing does not provide adequate information for solving instructional problems, and that a greater emphasis should be placed on data from functional or curriculum-based measurements Reschly, Assessments of instruction should provide data on individual students' progress in acquiring, generalizing, and maintaining knowledge and skills set forth in the curriculum.

Curriculum-based assessment CBA is an approach to evaluating the effectiveness of instruction that has gained substantial attention for its value in the development of effective instructional programs. CBA can be characterized as the practice of taking frequent measure of a student's observable performance as he or she proceeds through the curriculum. As data are gathered, the measures of student performance are organized, usually graphed, and examined to make judgments of whether the student's level and rate of achievement are adequate.

The teacher's reflections on the curriculum-based measures of performance and qualitative aspects of the student's performance may suggest a variety of rational explanations and potentially useful instructional interventions. If the student is making adequate progress, then it is not necessary to modify the program. On the other hand, if progress is inadequate, appropriate interventions might include devoting more time to instruction or practice, engaging the student in higher rates of active responding, slicing back to an easier level of a task, shifting instruction to tasks that are more explicitly or more parsimoniously related to the instructional objective, or changing the incentives for achievement.

If curriculum-based measures appear to indicate that the student is having little or no difficulty meeting instructional criteria, the teacher should consider skipping ahead to more difficult tasks. 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. Such considerations can contribute to improvement in the quality of instruction; however, Fuchs and Deno argued that measurement of subskill mastery is not necessary and that measures of more general curriculum-based measures can provide educators with reliable, valid, and efficient procedures.

Second, preparations for implementing CBA are frequently accompanied by specifications of expectations for instruction, such as how many or at what rate objectives will be learned. Fuchs, Fuchs, and Deno observed that teachers who set clear but ambitious goals for their students tended to obtain higher levels of achievement from their students than teachers who set more modest goals.

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 is no disagreement that all students, including those with LD, should be offered the most effective instruction possible. 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. Sometimes educators' beliefs have positive influences on the development of instruction. For example, if a teacher believes that the source of a student's failure lies in his or her instructional experience, then the teacher may revise the curriculum, allocate more time for instruction, slice back to a less complex level of the task, or systematically and frequently collect achievement data so that the effects of instructional interventions can be monitored.

Proponents of current efforts to reform mathematics education believe that if the quality of instruction is to be improved, then many educators will have to dramatically change their perspectives on how mathematics should be taught e. The National Council of Teachers of Mathematics set forth the following goals for all students: a to learn to value mathematics, b to become confident in their abilities to do mathematics, c to become mathematical problem solvers, d to learn to communicate mathematically, and e to learn to reason mathematically.

We believe that these are worthy goals and that in order to reach them, educators must examine their beliefs; however, beliefs that form the basis of the constructivist approach to mathematics education will be insufficient for guiding the development of mathematics curricula, and may be interpreted by educators as legitimizing inadequate instructional practices.

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.