V.2 #1 Mathematics - Curriculum Design as a Factor of Effective Instruction for Students with Mathem
The mathematical achievement of students with learning disabilities (LD) seems to be strongly influenced by the way in which mathematical concepts and skills are introduced to them, namely by the design of their curriculum (Grossen & Carnine, 1996; Kelly, Gersten, & Carnine, 1990).
Two oft-used mathematics curriculum designs are “strand curriculum” and “spiraling curriculum.” Strand curriculum organizes lessons around multiple concepts and skills. Each is addressed five to ten minutes per lesson and is revisited day-after-day until students have achieved mastery. Spiraling curriculum introduces numerous skills in a single graded book and systematically reintroduces them in higher level books.
Carnine (1997) argued that curriculum—like spiral curriculum—that introduces new mathematical concepts in a rapid rate, with insufficient time for explanations, practice, and review, overwhelms students with LD. It produces failure. Similarly, Miller and Mercer (1998) argued that spiral curriculum is unsuitable for students with mathematics disabilities; it superficially covers many skills without securing skill mastery. These arguments were supported by Crawford and Snider (2000), who examined the effects of strand and spiral curriculum designs on a wide range of students, including students with LD. Across groups of students, strand organization produced higher achievement.
To improve the quality of mathematical instruction for students with LD, Carnine proposed that curriculum design emphasize five principles.
Use Big Ideas. Organize the content around central ideas that will make subordinate concepts and skills easier to learn and more meaningful. For example, teach the principle of commutativity before teaching combinations of addition.
Use Conspicuous Strategies. Teach guidelines that help student acquire and use their knowledge on similar tasks. For example, teach students to draw pictures depicting the situation described in a problem before algorithmically elaborating the problem).
Use Time Efficiently. Emphasize important, meaningful objectives. Spread instruction on complex content over several days. Organize lessons around strands. For example, teach only “paraphrasing” and “picture drawing” as problem solving strategies; for several days, have students practice these on different problems; spend about ten minutes a day reteaching the strategies.
Communicate Explicitly. Make strategies clear; teach the students whatever background knowledge they need, but lack; offer consistent task-feedback. For example, present the meaning of commutativity with different materials, in all representational forms [concrete, iconic, abstract]; help students identify their errors and figure out how to correct them.
Provide Appropriate Practice and Review. Give students the time they need to practice, so they reach an automatic level of response. Give them abundant opportunity to integrate new and older knowledge. For example, teach students to count by 5’s until they can do it quickly and accurately. Teach students to add double digits (e.g., 25+13) until they can do it quickly and accurately.
Thornton, Langrall, and Jones (1997), drawing on group and case studies of students with LD, suggested that the mathematical abilities of these students, especially their higher order thinking and problem solving, can be accommodated and capitalized upon, when instruction is based on these principles.
Provide a Broad and Balanced Mathematics Curriculum. Use problem-driven instruction that emphasizes number sense and estimation, data analysis, spatial sense and geometric thinking, patterns and relationships.
Engage Students in Rich and Meaningful Problem Tasks. Use these to raise other problems, raise other questions, discuss alternative solutions, and make multiple connections.
Accommodate the Diverse Ways Children Learn. Vary instruction to meet children’s differences, such as the amount of time they need to understand a problem, the kind and the multitude of the representations they need to grasp the meaning of a concept, the classroom grouping format that works best for different children.
Encourage Students to Discuss and Justify their Problem Solving Approaches. This helps students take ownership of their learning and gain credibility with their peers.
Although much more research is needed to validate the suggestions in this column, experience suggests that they hold promise for helping students with LD to succeed in mathematics.
Carnine, D. (1997). Instructional design in mathematics for students with learning disabilities. Journal of learning Disabilities, 30, 2, 130-141.
Crawford, D. & Snider, V. (2000). Effective mathematics instruction: The importance of the curriculum. Education and Treatment of Children, 23, 2, 122-143.
Grossen, B. & Carnine, D. (1996). Considerate instruction helps students with disabilities achieve world class standards. Teaching Exceptional Children, 28, 4, 77-81.
Kelly, B., Gersten, R., & Carnine, D. (1990). Student error patterns as a function of curriculum design: Teaching fractions to remedial high school students and high school students with learning disabilities. Journal of Learning Disabilities, 1, 23-29.
Miller, S. & Mercer, C. (1998). Educational aspects of mathematics disabilities. In D. Rivera (Ed), Mathematics education for students with learning disabilities (pp. 81-96). Austin, TX.: Pro-Ed.
Thornton, C., Langrall,, C., & Jones, G. (1997). Mathematics instruction for elementary students with learning disabilities. Journal of Learning Disabilities, 30, 2, 142-150.
Ioannis Agaliotis, Ph.D. is Assistant Professor of Instructional Methodology for Students with Special Educational Needs in the Department of Educational and Social Policy of the University of Macedonia of Thessaloniki, Greece. Dr. Agaliotis is co-editor of the journal Insights on Learning Disabilities: From Prevailing Theories to Validated Practices, published by LDW®. He has presented at national and international conferences and has published articles and books on inclusive education, assessment and instruction for students with mild disabilities, mathematics for students with special needs, and academic and social support for students with learning disabilities.