In general education classrooms, adaptations and modifications in mathematics instruction are appropriate for all students, not just students with LD. Teachers of mathematics will find that simple changes to the presentation of mathematical concepts enable students to gain a clearer understanding of the process rather than a merely mechanically correct response. For many teachers with limited or no preparation for working with students with LD, inclusion of students with mathematics disabilities may create concern.
This article provides information on how to adapt and modify mathematics instruction to promote success and understanding in the areas of mathematical readiness, computation, and problem-solving for students with math disabilities. It also presents techniques that promote effective mathematics instruction for these students. Ariel stresses the need for all students to develop skill in readiness, computation, and problem-solving skills. As illustrated below, adaptations and modifications can be implemented to help students succeed in all three areas.
According to Ariel , students with LD must acquire a general developmental readiness, and b conceptual number readiness. General developmental readiness includes ability in the areas of classification, one-to-one correspondence, seriation, conservation, flexibility, and reversibility. Knowledge of the student's level of general readiness allows the teacher to determine how adaptations and modifications must be enacted to allow for the student to progress.
For some students, mathematics readiness instruction may need to include the development of language number concepts such as big and small and smallest to largest; and attributes such as color, size, or shape. Instruction, review, and practice of these concepts must be provided for longer time periods for students with mathematics disabilities than for other students. Conceptual number readiness is essential for the development of addition and subtraction skills Ariel, Practice and review with board games or instructional software are effective ways to develop conceptual number readiness for students with mathematics disabilities.
Manipulatives, such as Cuisenaire rods and Unifix math materials e. Adaptations and modifications in the instruction of computational skills are numerous and can be divided into two areas: memorizing basic facts and solving algorithms or problems. Basic Facts. Two methods for adapting instruction to facilitate recall of basic facts for students with math disabilities include a using games for continued practice, and b sequencing basic facts memorization to make the task easier.
Further, McCoy and Prehm suggest that teachers display charts or graphs that visually represent the students' progress toward memorization of the basic facts. Sequencing fact memorization may be an alternative that facilitates instruction for students with LD. The x2 and x5 facts are next, adding 28 to the set of memorized facts. The x9s are introduced next, followed by doubles such as 6 x 6. The remaining 20 facts include 10 that are already known if the student is aware of the commutative property e.
New facts should be presented a few at time with frequent repetition of previously memorized facts for students with LD. Solving Algorithms.
Computation involves not only memorization of basic facts, but also utilization of these facts to complete computational algorithms. In the addition process, McCoy and Prehm present three alternatives to the standard renaming method for solving problems, including expanded notation see Figure 1 partial sums see Figure 2 , and Hutchings' low-stress algorithm see Figure 3. Further, arrays that use graph paper to allow students to plot numbers visually on the graph and then count the squares included within the rectangle they produce.
Arrays can be used in combination with partial products to modify the multiplication process, thereby enabling students with math disabilities to gain further insight into the multiplication process. Providing adaptations is often very effective for helping students with mathematics disabilities successfully use facts to solve computational problems. Salend lists suggestions for modifying mathematics assignments in computation.
These suggestions are shown in Table 1. Further adaptations and modifications in computational instruction include color coding of the desired function for the computation problem Ariel, , either ahead of time by the teacher or during independent practice by the student. This process serves as a reminder to the student to complete the desired function and also may be used as an evaluation device by the teacher to determine the student's knowledge of the mathematical symbols and processes they represent.
Matrix paper allows students a physical guide for keeping the numbers in alignment Ariel, , thus decreasing the complexity of the task and allowing the teacher and student to concentrate on the mathematical process. In simplifying the task, the teacher then can identify problems in the student's understanding of the process rather than in the performance of the task. Finally, modeling is another effective strategy for helping students solve computational problems. For example, Rivera and Deutsch-Smith cited in Salend, recommend the use of the demonstration plus permanent model strategy, which includes the following three steps designed to increase skill in comprehending the computation process: a the teacher demonstrates how to solve a problem while verbalizing the key words associated with each step in solving the computation problem; b the student performs the steps while verbalizing the key words and looking at the teacher's model; and c the student completes additional problems with the teacher's model still available.
Other modeling examples provided by Salend include the use of charts that provide definitions, correct examples, and step-by-step instructions for each computational process. Lack of critical thinking skills compounds problem-solving difficulties. Several cognitive and meta-cognitive strategies can be used effectively. For example, recommends the use of six problem-solving strategies that students can monitor on an implementation sheet. Students verbalize the steps while completing the problem and note their completion of the steps on the monitoring sheet.
The six steps are:. Further, Mercer identifies the components necessary for students to engage in successful problem-solving. According to Merger, the problem-solving process involves 10 steps, which can be expanded into learning strategies to enable students with math disabilities to be more effective in solving word problem. The 10 steps are:. They encourage teachers to think about how to alter instruction while maintaining the primary purpose of mathematics instruction: Competence in manipulating numbers in the real world.
Their suggestions include:. This section examines effective instructional techniques that the general educator can incorporate into the classroom for all learners, and especially for students with math disabilities. Providing enough time for instruction is crucial. Instructional time is brief, often consisting of a short modeling of the skill without a period of guided practice.
By contrast, small-group practice where students with math disabilities complete problems and then check within the group for the correct answer, use self-checking computer software programs, and receive intermittent teacher interaction are positive modifications for increasing time for mathematics instruction. Additionally, time must be provided for students to engage in problem-solving and other math "thinking" activities beyond the simple practice of computation, even before students have shown mastery of the computational skills.
Polloway and Patton suggest that the components of effective instruction play an important role in the success of students with disabilities in general education mathematics instruction. One suggested schedule for the class period includes a period of review of previously covered materials, teacher-directed instruction on the concept for the day, guided practice with direct teacher interaction, and independent practice with corrective feedback.
During the guided and independent practice periods, teachers should ensure that students are allowed opportunities to manipulate concrete objects to aid in their conceptual understanding of the mathematical process, identify the overall process involved in the lesson i. Teaching key math terms as a specific skill rather than an outcome of basic math practice is essential for students with LD Salend, The math terms might include words such as "sum," "difference," "quotient," and "proper fraction," and should be listed and displayed in the classroom to help jog students' memories during independent assignments.
Varying the size of the group for instruction is another type of modification that can be used to create an effective environment for students with math disabilities. Large-group instruction, according to McCoy and Prehm , may be useful for brainstorming and problem-solving activities. Small-group instruction, on the other hand, is beneficial for students by allowing for personal attention from the teacher and collaboration with peers who are working at comparable levels and skills.
This arrangement allows students of similar levels to be grouped and progress through skills at a comfortable rate. When using grouping as a modification, however, the teacher must allow for flexibility in the groups so that students with math disabilities have the opportunity to interact and learn with all members of the class see Rivera in this series for cooperative learning information.
Salend recommended that new math concepts be introduced through everyday situations as opposed to worksheets. With everyday situations as motivators, students are more likely to recognize the importance and relevance of a concept. Further, everyday examples involve students personally in the instruction and encourage them to learn mathematics for use in their lives.
Changing the instructional delivery system by using peer tutors see Miller et al.
Progression of Computational Skills
Adaptations and modifications of reinforcement styles or acknowledgment of student progress begin with teachers being aware of different reinforcement patterns. By concentrating on the process of mathematics rather than on the product, students may begin to feel some control over the activity. In addition, teachers can isolate the source of difficulty and provide for specific accommodations in that area.
For example, if the student has developed the ability to replicate the steps in a long division problem but has difficulty remembering the correct multiplication facts, the teacher should reward the appropriate steps and provide a calculator or multiplication chart to increase the student's ability to obtain the solution to the problem. The mathematical ability of many students with LD can be developed successfully in the general education classroom with proper accommodations and special education instructional support.
Second, allow students to respond to easier practice items orally rather than in written form to speed up the rate of correct responses. In a math lesson on estimating area, for example, give students the homework task of calculating the area of their bedroom and estimating the amount of paint needed to cover the walls.
Have students plot the number of assignments turned in on-time in green, assignments not turned in at all in red, and assignments turned in late in yellow. During large-group math lectures, teachers can help students to retain more instructional content by incorporating brief Peer Guided Pause sessions into lectures.
Students are trained to work in pairs. At one or more appropriate review points in a lecture period, the instructor directs students to pair up to work together for 4 minutes. During each Peer Guided Pause, students are given a worksheet that contains one or more correctly completed word or number problems illustrating the math concept s covered in the lecture.
The sheet also contains several additional, similar problems that pairs of students work cooperatively to complete, along with an answer key.
Student pairs are reminded to a monitor their understanding of the lesson concepts; b review the correctly math model problem; c work cooperatively on the additional problems, and d check their answers. The teacher can direct student pairs to write their names on the practice sheets and collect them as a convenient way to monitor student understanding.
Response cards can increase student active engagement in group math activities while reducing disruptive behavior. The teacher instructs at a brisk pace. The instructor first poses a question to the class. Students are given sufficient wait time for each to write a response on his or her response card. The teacher then directs students to present their cards.
If most or all of the class has the correct answer, the teacher praises the group. If more than one quarter of the students records an incorrect answer on their cards, however, the teacher uses guided questions and demonstration to steer students to the correct answer. The teacher models behavioral expectations for open, interactive discussions, praises students for their class participation and creative attempts at problem-solving, and regularly points out that incorrect answers and misunderstandings should be celebrated—as they often lead to breakthroughs in learning.
The teacher uses open-ended comments e. As with any problem classroom behavior, a first offense requires that the student meet privately with the instructor to discuss teacher expectations for positive classroom behavior. If the student continues to put down peers, the teacher imposes appropriate disciplinary consequences. The instructor first verifies that the student has the necessary academic competencies to learn higher-level math content, including reading and writing skills, knowledge of basic math operations, and grasp of required math vocabulary.
The teacher presents new math content in structured, highly organized lessons. In turn, students are encouraged to think aloud when applying the same strategy—first as part of a whole-class or cooperative learning group, then independently. The teacher observes students during process modeling to verify that they are correctly applying the cognitive strategy. Students get regular performance feedback about their level of mastery in learning the cognitive strategy. That feedback can take many forms, including curriculum-based measurement, timely corrective feedback, specific praise and encouragement, grades, and brief teacher conferences.
Once the student has mastered a cognitive strategy, the teacher structures future class lessons or independent work to give the student periodic opportunities to use and maintain the strategy. Journaling is a valuable channel of communication about math issues for students who are unsure of their skills and reluctant to contribute orally in class.
At the start of the year, the teacher introduces the journaling assignment, telling students that they will be asked to write and submit responses at least weekly to teacher-posed questions. Students are encouraged to use numerals, mathematical symbols, and diagrams in their journal entries to enhance their explanations.
The teacher provides brief written comments on individual student entries, as well as periodic oral feedback and encouragement to the entire class on the general quality and content of class journal responses. Teachers will find that journal entries are a concrete method for monitoring student understanding of more abstract math concepts. To promote the quality of journal entries, the teacher might also assign them an effort grade that will be calculated into quarterly math report card grades.
To create such a checklist, the teacher meets with the student. Together they analyze common error patterns that the student tends to commit on a particular problem type e. For each type of error identified, the student and teacher together describe the appropriate step to take to prevent the error from occurring e. These self-check items are compiled into a single checklist.
Students are then encouraged to use their individualized self-instruction checklist whenever they work independently on their number or word problems. As older students become proficient in creating and using these individualized error checklists, they can begin to analyze their own math errors and to make their checklists independently whenever they encounter new problem types.
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Teachers can best promote students acquisition and fluency in a newly taught math skill by transitioning from massed to distributed practice. Teachers can program distributed practice of a math skill such as reducing fractions to least common denominators into instruction either by a regularly requiring the student to complete short assignments in which they practice that skill in isolation e. A comparison of the methods that high and low-achieving math students typically use to prepare for tests suggests that struggling math students need to be taught 1 specific test-review strategies and 2 time-management and self-advocacy skills.
Among review-related strategies, deficient test-takers benefit from explicit instruction in how to take adequate in-class notes; to adopt a systematic method to review material for tests e. Teachers can efficiently teach effective test-preparation methods as a several-session whole-group instructional module. Three strategies can help students to learn essential math vocabulary: preteaching key vocabulary items, modeling those vocabulary words, and using only universally accepted math terms in instruction.
Math vocabulary provides students with the language tools to grasp abstract mathematical concepts and to explain their own reasoning. Then have students engage in cooperative learning or individual practice activities in which they too must successfully use the new vocabulary—while the teacher provides targeted support to students as needed.
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References Armendariz, F. Using active responding to reduce disruptive behavior in a general education classroom. Journal of Positive Behavior Interventions, 1 3 , Baxter, J. Writing in mathematics: An alternative form of communication for academically low-achieving students. Bryan, T. Teacher-selected strategies for improving homework completion. Carnine, D. Instructional design in mathematics for students with learning disabilities. Journal of Learning Disabilities, 30, Caron, T. Learning multiplication the easy way.
The Clearing House, 80, Chard, D. Vocabulary strategies for the mathematics classroom.
Download Math Computation Skills Strategies Level 3 Math Computation Skills Strategies
Encouraging "math talk" in the classroom. Middle School Journal, 29 5 , Hawkins, J. The effects of independent and peer guided practice during instructional pauses on the academic performance of students with mild handicaps. Hong, E.
Assessment for common misunderstandings
Test-taking strategies of high and low mathematics achievers. Journal of Educational Research, 99 3 , Lambert, M. Effects of response cards on disruptive behavior and academic responding during math lessons by fourth-grade urban students. Journal of Positive Behavior Interventions, 8 2 , Montague, M. Cognitive strategy instruction in mathematics for students with learning disabilities. Solve it!