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Institute
Ioannis Agaliotis, Ph.D.
For student with learning disabilities (LD) often have difficulty mastering customary or standard algorithms for computing operations (Fleischner & Manheimer, 1997). Because of these difficulties, many authorities have suggested they be taught alternative algorithms (e.g., Foley & Cawley, 2003; Rondolph, 1996; Randolph & Sherman, 2001).
Several authorities have argued that alternative algorithms facilitate students’ conceptual and skill development, especially for place value and regrouping (e.g. Bryant, Hartman, & Kim, 2003; Cawley & Parmar, 1992). They also facilitate the understanding of standard algorithms, thus allowing students to acquire a variety of computational techniques and increase their mathematical efficiency (Sheffield & Cruikshank, 2000). Below are examples of alternative algorithms for each computing operation: addition, subtraction, multiplication, and division.
The Partial Sums Algorithm of Addition
In this algorithm, when the sum of the ones column is equal to or greater than 10, it is recorded as a two-digit number at the bottom of the ones and tens columns (in the above addition 9 + 8 = 17). Then, the tens column is added and the sum is recorded below the sum of the ones column (in the above addition 5 tens + 3 tens equals 8 tens or 80). Finally, the two partial sums are added (in the above addition 17 + 80 = 97).
The Low Stress Algorithm of Subtraction
In the low stress algorithm of subtraction (Hutchings, 1975), the renaming of the minuend (the larger number) and its writing between the original minuend and the subtrahend (the smaller number) takes place before subtracting individual digits. Teachers should remind students that renaming is necessary each time the subtrahend in each column is greater than the minuend.
The Drop Notation Algorithm of Multiplication
The drop notation algorithm of multiplication is another low stress algorithm introduced by Hutchings (1976). One of its advantages is that students can work only with multiplication facts as the algorithm foresees that both digits of the products are immediately written (so the amount of the information committed to memory during computation is drastically reduced and regrouping is unnecessary). Due to place value requirements though, the second digit is "dropped" on the next line, and is recorded in a "staircase" manner. After their calculation, the products are added vertically.
The Expanded Notation Algorithm of Division
The expanded notation algorithm of division (Cawley & Parmar, 1992) gives students an opportunity to show place value representations of numerals and to calculate answers.
Using Alternative Algorithms
Alternative algorithms differ from standard ones in their cognitive requirements. Thus, alternative algorithms may more effectively accommodate the individual preferences and needs of some students with LD.
Of course, teachers unfamiliar with alternative algorithms may have difficulty. Difficulty is not a surprise. Like anything else, mastery takes study and practice.
Teachers may also question the utility of alternative algorithms with students with LD. Clearly, if they struggle with standard algorithms they need an alternative. As using any algorithm is a matter of convention—a way of using symbols to express the meanings of arithmetic operations with real objects—the decisive factor in choosing an algorithm should not be tradition, but the degree to which the algorithm fits students’ information processing preferences and helps them better develop their mathematical reasoning and computing. If standard algorithms are unsuccessful, alternatives make sense.
References Bryant, D., Hartman, P., & Kim, S. (2003). Using explicit and strategic instruction to teach division skills to students with learning disabilities. Exceptionality, 11(3), 151-164.
Cawley, J. F., & Parmar, R. S. (1992). Arithmetic programming for students with disabilities: An alternative. Remedial and Special Education, 13(3), 6-18.
Fleischner, J. & Manheimer, M. (1997). Math interventions for students with learning disabilities: myths and realities. School Psychology Review, 26(3), 397-414.
Foley, T., & Cawley, J. (2003). About the mathematics of division: Implications for students with disabilities. Exceptionality, 11, 131-149.
Hutchings, B. (1975). Low-stress subtraction. The Arithmetic Teacher, 22, 226-232.
Hutchings, B. (1976). Low-stress algorithms. Reston, VA: National Council of Teachers of Mathematics.
Randolph, T., & Sherman, H. (2001). Alternative algorithms: Increasing options, reducing errors. Teaching Children Mathematics, 7( 8), 480-484.
Rondolph, P. (1996). Multicultural mathematics and alternative algorithms. Teaching Children Mathematics, 3, 128-133.
Sheffield, L., & Cruikshank, D. (2000). Teaching and learning elementary and middle school mathematics. New York: Wiley & Sons.
Ioannis Agaliotis, Ph.D.
is a Lecturer 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.
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