Adaptations and modifications come in many forms. Students with learning disabilities (LD) are increasingly receiving most of their mathematics instruction in general education classrooms. Studies show that these students benefit from general education mathematics instruction if it is adapted and modified to meet the individual needs of the learners (Salend, 1994). Adaptations and modifications come in many forms. They can be as simple as using graph paper to help student with mathematics disabilities keep columnar addition straight or as complex as solving calculus equations with calculators. To ensure effective instruction, adaptations and modifications for instruction are necessary in the areas of lesson planning, teaching techniques, formatting content, adapting media for instruction, and adapting evaluation (Wood, 1992).
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. Additionally, adapting and modifying instruction for students creates a more positive atmosphere that encourages students to take risks in problem-solving, which strengthens student understanding of the concept (McCoy & Prehm, 1987).
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.
How can general education teachers facilitate the learning of mathematical skills?
Ariel (1992) 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 (1992), 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,1992). 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.g.,100 block trays) allow students with math disabilities to visualize numerical concepts and engage in age-appropriate readiness skills (see Lambert in this series for additional suggestions about manipulatives).
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. Beattie and Algozzine (cited in McCoy & Prehm, 1987) recommend the use of dice rolls, spinners, and playing cards to give students extra practice with fact memorization and to promote interest in the task by presenting a more game-like orientation. Further, McCoy and Prehm (1987) 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. For example, in teaching the multiplication facts, Bolduc (cited in McCoy & Prehm, 1987) suggest, starting with the xO and x1 facts to learn 36 of the 100 multiplication facts. 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.g., 4 x 7 = 7 x 4). 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. An algorithm is a routine, step-by-step procedure used in computation (Driscoll, 1980 cited in McCoy & Prehm, 1987). In the addition process, McCoy and Prehm (1987) 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). Subtraction for students with mathematics disabilities is made easier through the use of Hutchings’ low-stress subtraction method (McCoy & Prehm, 1987) (see Figure 4) where all renaming is done first. Multiplication and division (McCoy & Prehm, 1987) can be illustrated through the use of partial products (see Figure 5) . 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 (1994) lists suggestions for modifying mathematics assignments in computation. These suggestions are shown in Table 1.
|Figure 1. Expanded Notation||29 =||2 tens and 9 ones|
|+43 =||4 tens and 3 ones|
|Step one: Add the ones and tens.||6 tens and 12 ones|
|Step two: Regroup the ones, if necessary||6 tens and (1 ten 2 ones)|
|Step three: Put the tens together.||(6 tens and 1 ten) and 2 ones|
|Step four: Write the tens in a simpler way.||7 tens and 2 ones|
|Step five: Write the answer in number form.||72|
|Figure 2. Partial Sums|
(sum of the ones) 14
(sum of the tens) 90
|Figure 3. Hitchings’ Low-Stress Algorithm|
Problem: 45 + 77 + 56 + 83 + 27 + 39 =
|77||1)||Add 5 + 7 and record 12, put the “1” above the tens.|
|56||2)||Add 2 + 6 and record 8, no tens to carry.|
|83||3)||Add 8 +3 and record 11, put the “1” above the tens.|
|27||4)||Add 1 +7 and record 8, no tens to carry.|
|39||5)||Add 8 + 9 and record 17, put the “1” above the tens.|
|6)||Add 3 + 4 and record 7, no tens to carry.|
|7)||Add 7 + 7 and record 14, put the “1” in the hundreds|
|8)||Add 4 + 5 and record 9, no hundreds to carry.|
|9)||Add 9 + 8 and record 17, put the “1” in the hundreds.|
|10)||Add 7 + 2 and record 9, no hundreds to carry.|
|11)||Add 9 + 3 and record 12, put the “1” in the hundreds.|
|12)||Add the hundreds place.|
|Figure 4. Hutchings’ Low-Stress Subtraction Algorithm|
|1) Rewrite the tens and ones places.|
|2) Determine if renaming is necessary.|
|3) Rewrite the hundreds, tens and ones places.|
|4) Determine if renaming is necessary.|
|5) Renaming is necessary to complete subtraction in the hundreds place. Rewrite the number in the hundreds place.|
|6) Complete subtraction with renaming already accomplished.|
|Figure 5. Partial Products|
|1) 2 * 3 = 6|
|2) 2 * 20 = 40|
|3) 10 * 3 = 30|
|4) 10 * 20 = 200 / 276|
|Table 1. Tips for Modifying Mathematics Computational Assignnments.|
Further adaptations and modifications in computational instruction include color coding of the desired function for the computation problem (Ariel, 1992), 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, 1992), 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, 1994) 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 (1994) include the use of charts that provide definitions, correct examples, and step-by-step instructions for each computational process.
Problem-solving can be adapted and modified for students with mathematics disabilities in several different ways (see Kelly & Carnine in this series for additional word problem-solving instructional strategies). Polloway and Patton (1993) note that students with math disabilities improve their problem-solving skills through teacher-directed activities that include (a) having students read or listen to the problem carefully; (b) engaging students in focusing on relevant information and/or significant words needed to obtain the correct answer while discarding the irrelevant by writing a few words about the answer needed (e.g., number of apples), by identifying aloud or circling the significant words in the problem, and by highlighting the relevant numbers; (c) involving students in verbalizing a solution for the problem using a diagram or sketch when appropriate; (d) developing strategies for working through the story problem by writing an appropriate mathematical sentence; and (e) performing the necessary calculations, evaluating the answer for reasonableness, and writing the answer in appropriate terms.
Lack of critical thinking skills compounds problem-solving difficulties. Several cognitive and meta-cognitive strategies can be used effectively. For example, (1992) 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:
- Read and understand the problem.
- Look for the key questions and recognize important words.
- Select the appropriate operation.
- Write the number sentence (equation) and solve it.
- Check your answer.
- Correct your errors.
Further, Mercer (1992) 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:
- Recognize the problem.
- Plan a procedural strategy (i.e., identify the specific steps to follow).
- Examine the math relationships in the problem.
- Determine the math knowledge needed to solve the problem.
- Represent the problem graphically.
- Generate the equation.
- Sequence the computation steps.
- Check the answer for reasonableness.
- Self-monitor the entire process.
- Explore alternative ways to solve the problem.
Hammill and Bartel (in Polloway & Patton, 1993) offer many suggestions for modifying mathematics instruction for students with LD. 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:
- Altering the type or amount of information presented to a student such as giving the student the answers to a story problem and allowing the student to explain how the answers were obtained.
- Using a variety of teacher-input and modeling strategies such as using manipulatives during the instructional phase with oral presentations.
Techniques to enhance mathematics instruction
For students with math disabilities, effective mathematics instruction is the difference between mathematics as a paper-and-pencil/right-answer type of task and an important real-life skill that continues to be used throughout their lifetime. 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.
Increasing instructional time
Providing enough time for instruction is crucial. Too often, “math time” according to Usnick and McCoy (cited in McCoy & Prehm, 1987) includes a long stretch of independent practice where students complete large numbers of math problems without feedback from the teacher prior to completion. 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. Hammil and Bartel (cited in Polloway & Patton, 1993) suggest slowing down the rate of instruction by using split mathematics instructional periods and reducing the number of problems required in independent practice.
Using effective instruction
Polloway and Patton (1993) 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.e., have students talk about “addition is combining sets” when practicing addition problems rather than silent practice with numerals on a worksheet), and write down numerical symbols or mathematical phrases such as addition or subtraction signs.
Teaching key math terms as a specific skill rather than an outcome of basic math practice is essential for students with LD (Salend, 1994) . 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 group size
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 (1987), 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).
Using real-life examples
Salend (1994) 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. Real-life demonstration enables students to understand more readily the mathematical process being demonstrated (see Scott & Raborn in this series for additional ideas). 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. in this series for ideas about peer tutoring); computer-based instruction; or more reality-based assignments such as “store” for practice with money recognition and making change also provide real life math experiences (Hammill & Bartel cited in Polloway & Patton, 1993).
Varying reinforcement styles
Adaptations and modifications of reinforcement styles or acknowledgment of student progress begin with teachers being aware of different reinforcement patterns. Beyond the “traditional” mathematical reinforcement style, which concentrates on obtaining the “right answer,” students with mathematics disabilities may benefit from alternative reinforcement patterns that provide positive recognition for completing the correct steps in a problem regardless of the outcome (McCoy & Prehm, 1987). 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. To this end, teachers should be aware of the necessity for adapting and modifying the environment to facilitate appropriate, engaging instruction for these students. Use of manipulatives is encouraged to provide realistic and obvious illustrations of the underlying mathematical concepts being introduced. Reliance on problem-solving strategies to improve students’ memories and provide a more structured environment for retention of information also is appropriate. Finally, teachers must evaluate the amount of time spent in instruction, the use of effective instructional practices, student progress (see Bryant in this series), and the use of Real-life activities that encourage active, purposeful learning in the mathematics classroom.