A review of literature on mathematics instruction for elementary students with learning disabilities.

Introduction

The majority of students with learning disabilities (LD) are taught in the general education classroom today because of the trend toward inclusive services in special education. However, there is limited literature on teaching math for such students compared to the literature on teaching reading and language arts. Some very basic modifications could increase the success for students with LD and also help others in the classroom that are at risk for having learning problems. In fact, students with LD benefit from many of the same teaching strategies that help all children learn mathematics, but perhaps with modified pacing and depth. The purpose of this article is to overview LD characteristics, review the literature on math and LD at the elementary level, and then summarize some key principles for practice in general education K-5 classes. These ideas address the need for an overlap in the fields of mathematics education and learning disabilities.

Learning Disabilities

Children that are labeled LD must meet several criteria. First of all, there must be a discrepancy between potential and performance. In other words these students are underachieving in mathematics, reading, writing, or some other academic area. This problem is presumed to be related to a central nervous system disorder, which suggests that LD is an intrinsic problem and therefore cannot be completely controlled by the children. In addition, students with LD exhibit processing problems such as memory and perception. Finally, the children do not have other primary disabilities such as mental retardation or emotional disturbance (Lerner, 2000).

Many of the characteristics that children with LD exhibit interfere significantly with their mathematics performance. It is useful for teachers to be aware of these potential problems because often teachers can intervene appropriately once they are able to detect the types and causes of errors for a given student. Processing problems make mathematics learning particularly difficult. Visual processing problems, for example, could lead to children losing their places when using the textbook, confusing numbers like 15 for 51, and using a number line in reverse. They could even make errors by working certain problems from left to right (as they are taught in reading) instead of right to left (Miller and Mercer, 1997). Some children would have trouble focusing on certain aspects of a problem; the entire page of work would look like a confused mass instead of a problem in a particular sequence. They might confuse the signs for the operations and have trouble with manipulative materials for money, measurement, and time. Lining up work for place value accurately could also be difficult and therefore lead to errors with the calculations (Bley and Thornton, 2001).

Auditory processing problems, on the other hand, could lead to difficulty understanding the teacher's explanations. Even if the material were on the appropriate level, a student with auditory deficits would have no chance to succeed if the oral directions and instruction were not clear to him. Because their auditory processing problems make the oral explanations and examples confusing, students often act as if their math assignments are much too difficult.

Motor processing problems will often cause handwriting difficulties, which can affect math performance. There is a great emphasis particularly in the elementary textbooks today on writing for mathematics. These students will make many errors copying the problems from their texts and will probably have great difficulty learning to write their numbers accurately (Miller and Mercer, 1997). For example, a very bright math student could get the wrong answers to complex math problems because he cannot read his own writing.

Memory problems interfere greatly in math work. A common example is that children will have even more difficulty than usual memorizing their math facts. They will have difficulty with long procedures; they may forget certain parts of the problem as they are working. They would tend to have trouble with multi-step math problems because they cannot remember all of the steps (Bley and Thornton, 2001).

Language difficulties make word problems particularly difficult. Students with language disorders may have trouble converting from numbers to words (Miller and Mercer, 1997). They would have difficulty with vocabulary, listening to math lessons presented orally, and putting the math ideas into their own words (Bley and Thornton, 2001). Even at the kindergarten level, the vocabulary such as order, more than, and capacity, can be confusing for students.

Many students with LD have cognitive strategy deficits and therefore have difficulty selecting appropriate strategies for problems (Miller and Mercer, 1997). For example, they could be very confused about selecting guess and check, working backward, and drawing picture strategies for word problems. These students also have difficulty organizing information and generalizing what they have learned to similar examples and problems (Lerner, 2000). Students with LD might master skills when taught in isolation but might not see any connections to using those same strategies as part of a subsequent, more difficult problem in the next unit of study.

Finally, there are many students with LD that exhibit minor social emotional problems. Some of the more typical examples that affect math performance are attention deficits, hyperactivity, and impulsivity. These students will have trouble completing all of the steps in a math problem such as division of three and four digits or working long enough to figure out pattern activities. They will have difficulty paying attention to all of the directions that are needed for success. They often have low self-esteem and therefore do not even attempt difficult problems. They may be so embarrassed about their problems that they are afraid to ask for help when needed or admit that they do not understand.

Learning Disabilities Modifications

There are a variety of teaching strategies and modifications that are recommended for all children, but are absolutely critical for those students with LD. Without these types of modifications, the non-disabled children will probably still learn even though it would not be as easy. Without these types of modifications, the students with LD will almost certainly fail to learn.

Advance organizers are recommended for many subjects and all types of learners; they are particularly useful for students with learning disabilities because of their difficulty focusing and processing information that make lessons seem confusing. These introductions to the lessons could be in oral or written form. They help prepare the students for lessons by emphasizing the purpose and justification. They also review prior knowledge and provide prompts and clues (Miller, Strawser, and Mercer, 1996).

At the beginning of the lesson, it is important to review prerequisite skills. Since many students with learning disabilities have memory problems, they may need even more time to review the previously taught concepts. The prerequisites must be mastered or the new material will not make sense and will become even more confusing to these students (Carnine, 1997; Brigham, Wilson, Jones, and Moisio, 1996). It is important to be aware that reviewing prerequisites for a student with LD may mean a great deal more than just going over the material from the previous week. It could involve even reminding students of important related concepts and rules from the previous years! In many cases a math lesson for students with LD might require some skills from three or four years ago that needed reviewing. Although this lengthy review and possibly re-teaching takes much time, it is necessary if these students are going to move forward. Sometimes the review would also be valuable for the non-LD students to reinforce these learnings that others may have forgotten or don't have at the automatic level.

Another important strategy for mathematics instruction that is valuable for students with LD is focusing on the major concepts that apply to many problems and provide foundations and relations to other skills in math. These "big ideas," such as division and proportion, are useful because they generalize to several types of problems. Therefore it is important to prioritize and teach the most important concepts, and then review them frequently (Carnine, 1997; Brigham, Wilson, Jones, and Moisio, 1996).

In addition, the math lessons taught should be meaningful. Realistic applications to life make mathematics instruction more effective (Goldman and Hasselbring, 1997). Since students with LD have trouble generalizing, practice and examples in real-life situations are critical for transfer and use in math situations outside of the classroom.

Modeling procedures is important in mathematics instruction (Miller, Butler, and Lee, 1998). These demonstrations may have to be repeated additional times for students with LD, at a slower pace, and with extra clues and reminders. The modeling should emphasize step-by-step procedures with clear explanations of each step (Carnine, 1997). It is often necessary to repeat demonstrations several times before the material is clear to the students with LD; however, once the procedures are understood through step-by-step modeling, the students can work independently to become proficient.

In most of the literature on elementary math and particularly elementary math and LD, the use of concrete examples to illustrate calculations and problem-solving is recommended (Brigham, Wilson, Jones, and Moisio, 1996; Miller, Butler, and Lee, 1998; Bley and Thornton, 2001). An appropriate sequence would be to present new skills and procedures initially in a concrete manner with manipulative materials, then make a transition to pictures to clarify, and finally explain the abstract presentation. Often students who have learning disabilities may need even more time and practice at the concrete and picture stage. Students may need to use concrete materials to add and subtract even though they can perform fairly difficult grade level problems that are presented by the teacher and in textbooks. It is important, however, to consider that manipulative materials may not always be the answer for students with LD. For students with severe visual motor problems, blocks and counters could interfere; they could drop them, miscalculate, and generally be distracted by the concrete materials. These students might work more accurately when the new skills and procedures are presented orally and if they have the opportunity to talk through the strategies.

Practice and review of new material is important for instruction at any level and in any subject if mastery is expected. Students who have learning disabilities need even more practice, and it needs to be broken down into smaller steps (Bley and Thornton, 2001; Brigham, Wilson, Jones, and Moisio, 1996). It is particularly important that students with LD clearly understand the assignment before independent class work and homework is given. The practice for LD should be very easy at first with lots of guidance provided to ensure motivation and possible success. Then gradually the work can become more challenging (Miller, Strawser, and Mercer, 1996; Carnine, 1997). Because of the memory problems associated with learning disabilities, the practice needs to be spaced over a long period of time even after the typical review is finished. Sometimes it would be helpful if prompts were provided based on the students' deficits. For example, if there are visual spatial deficits, problems can be separated or presented one at a time. Students could be given a sample problem to keep in their notebooks as a model to follow. The teachers would not have to take extra time for this modification if the students are taught to make these adjustments themselves (Bley and Thornton, 2001). They can be taught to draw lines on their math problems to separate for accurate place value work and to draw circles around each problem to separate for clarity.

Various self-monitoring strategies have also been shown to be successful because they improve students' independent work habits and achievement levels (Miller, Butler, and Lee, 1998). Students can be taught to record and evaluate their practice performance using graphs and checklists so that they will get immediate feedback and reinforcement on their progress (Miller, Strawser, and Mercer, 1996). Encouraging students to share their ideas on math problem-solving to groups or the whole class can help them think independently and understand math at a deeper level. Students with LD can be taught to analyze their work and figure out effective strategies (Thornton, Langrall, and Jones, 1997; Naglieri and Gottling, 1997).

Finally, teacher guidance and feedback seem to be even more critical for students with LD because of their problems with motivation and processing. This phase of teaching is useful for diagnostic as well as instructional purposes. Teachers can determine the types of errors that students are making and then provide efficient and effective suggestions for correction while they are providing guidance (Miller, Strawser, and Mercer, 1996). This assistance needs to be more extensive at the beginning of new concept presentation and then gradually can be faded to encourage independence in math work (Carnine, 1997).

In conclusion, most of the modifications suggested would be easy to implement and helpful for the whole class, not just students with LD. Students who are at risk for low achievement for any reason would benefit from these recommendations. Even the students who are high achievers would find math clearer and easier as a result of these suggestions. Finally, just the awareness of these typical LD characteristics will often help teachers think through additional modifications for their students.

Summary of LD Modifications for the Elementary Classroom

* Provide advance organizers to introduce purpose of lesson

* Provide additional review of all prerequisites as needed 

* Prioritize, teach, and review major concepts frequently

* Teach generalization and application to real-life situations

* Model sequential procedures at a slow pace and with extra clues

* Present new skills using concrete materials, then pictures, and finally abstract explanations

* Provide additional practice in small steps with sufficient guidance

* Be sure directions are clear before starting independent practice

* Teach students to keep track of their progress with charts and graphs

* Check for error patterns and related corrections when providing guidance

REFERENCES

Bley, N. S. & Thornton, C. A. (2001). Teaching mathematics to students with learning disabilities. Austin: Pro-Ed.

Brigham, F. J., Wilson, R., Jones, E., & Moisio, M. (1996). Best practices: Teaching decimals, fractions, and percents to students with learning disabilities. LD Forum, 21(3), 10-15.

Carnine, D. (1997). Instructional design in mathematics for students with learning disabilities. Journal of Learning Disabilities 30(2), 130-141.

Goldman, S. R. & Hasselbring, T. S. (1997). Achieving meaningful mathematics literacy for students with learning disabilities. Journal of Learning Disabilities 30(2), 198-208.

Lerner, J. (2000). Learning disabilities: Theories, diagnosis, and teaching strategies. Boston: Houghton Mifflin.

Miller, S. P., Butler, F. M., & Lee, K. (1998). Validated practices for teaching mathematics to students with learning disabilities: A Review of literature. Focus on Exceptional Children 31(1), 1-24.

Miller, S. P. & Mercer C. D. (1997). Educational aspects of mathematics disabilities. Journal of Learning Disabilities 30(1), 47-56.

Miller, S. P., Strawser S., & Mercer C.D. (1996). Promoting strategic math performance among students with learning disabilities. LD Forum 21(2), 34-40.

Naglieri, J. A. & Gottling, S. H. (1997). Mathematics instruction and PASS Cognitive Processes: An intervention study. Journal of Learning Disabilities 30(5), 513-20.

Thornton, C. A., Langrall, C. W. & Jones, G. A. (1997). Mathematics instruction for elementary students with learning disabilities. Journal of Learning Disabilities 30(2), 142-150.

Marcee M. Steele

University of North Carolina at Wilmington