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**V.1 #1 Mathematics - Teaching Arithmetic Combinations to Students with Learning Disabilities: What Teachers Should Know**

The term "arithmetic combinations" as well as the equivalent, but older term "basic math facts" refer to (a) the number combinations in addition and subtraction that result in sums and differences less than 20, and (b) the multiplication tables through 9x9 with their corresponding division facts. Research shows that students with learning disabilities (LD) are often weak in retrieving such arithmetic combinations from memory and using suitable counting strategies for figuring out combinations they forget (e.g., Garnett & Fleischner, 1983; Geary, 1993; Zeleke, 2004).

According to Bley and Thornton (2001), acquiring arithmetic combinations presupposes these skills.

Counting up and down from a given number from 1 to 10, after stating the number that precedes it and the number that follows it. If, for example, the given number is 6, the student is expected to name "5" and "7" as the preceding and the following numbers respectively, and then count these sequences aloud: "6, 7, 8, 9, 10" and "10, 9, 8, 7, 6."

Skip-counting up and down by 2s, 3s and the like from 1 to 20. For example, "4, 8, 12, 16, 20."

Stating the number that is bigger or smaller by 1, 2, 3 and the like from a given number. For example, upon being asked, "Which number is bigger than 4 by 2?," the student should answer "6."

Having a good number sense, especially with the numbers 1 to 20.

Another prerequisite of memorizing and using arithmetic combinations is the student's ability to understand (a) the way arithmetic combinations can be represented and translated in concrete, semi-concrete, verbal, and numerical forms, and (b) the arithmetic principles, like the commutativity principle in addition and multiplication or the role of zero in the same operations (Gerster, 1999; Scherer, 1999).

Successful memorization of arithmetic combinations depends also on the form and the sequence in which the combinations are presented. Research has established that presenting the arithmetic combinations to students with LD according to the magnitude of the result does not help them memorize and retrieve the combinations; this is because they tend to see each combination as a separate bit of information rather than an integral part of a coherent system (Bley & Thornton, 2001; Grisseman & Weber, 1990). Considering that students with LD often have memory problems (Geary, 1994), asking them to remember large amounts of seemingly unrelated information becomes an instructional shortcoming, especially when teaching them combinations without first teaching them the properties of and the relations inherent in the arithmetic system (Gerster, 1999). It's like holding a pound of sugar in your hands: it's a problem if the granules are loose, but not if they're in a sealed bag. Analogously, presenting arithmetic combinations in a form and sequence that would help the students understand the consistency of the arithmetic system, and would allow them to acquire the information in distinctive, related clusters would aid memory and substantially increase the students' chances of mastering the combinations (Agaliotis, 2004).

Teachers can help students memorize and recall mathematical facts by systematically teaching the relations between combinations. Moreover, knowing relationships can help students with LD calculate results when their memorization and recall fails. To this end, easier and already mastered combinations should be used consistently to help students acquire unknown and more difficult facts. For example, acquiring the combinations "4x2" and "8x2" can be facilitated by knowing the combination "2x2," combined with knowing the relations among "2," "4," and "8" (Grisseman & Weber, 1990).

The combined use of these techniques is found in the following proposal for grouping and presenting the 100 arithmetic combinations of addition: "zero combinations - 19 facts" (1+0 / 0+1, 2+0 / 0+2….), "1 combinations - 17 facts" (2+1 / 1+2, 3+1 / 1+3....), "2 combinations - 15 facts" (2+2, 3+2 / 2+3, 4+2 / 2+4....), "the doubles - 7 facts" (3+3….9+9), "9 combinations - 12 facts" (9+3 / 3+9, 9+4 / 4+9....), "10 sums - 4 facts" (6+4/ 4+6, 7+3 / 3+7), "known combinations plus 1 - 14 facts" (based on the doubles and the 10 sums, combinations like 4+3 / 3+4, 8+3 / 3+8.... can be acquired, for example based on the double combination 3+3, the student can learn the combination 3+4), "separate sums - 12 facts" (5+3 / 3+5, 6+3 / 3+6....8+6 / 6+8) (Agaliotis, 2004).

Miller and Mercer have proposed two strategies for acquiring basic facts. They are DRAW (1991) and SOLVE (1993). DRAW stands for these steps: D = Discover the (calculation) sign, R = Read the problem (recognize the numbers involved), A = Answer or draw and check (if the answer does not come automatically, make a drawing to find it), and W = Write the answer. SOLVE stands for S = See the sign, O = Observe and answer (if unable to answer keep going), L = Look and draw, V = Verify your answer, and E = Enter your answer. After students are taught the steps of the strategy, they use them to find the results and learn the basic facts. According to Miller, Strawser, and Mercer (1996), these strategies help students with LD acquire the basic facts.

Agaliotis, Dimitrakopoulos, Dimou, Theodorou, Pesli, and Charisi (2003) implemented a program for teaching multiplication combinations to 24 fifth and sixth grade students with LD. They used these steps.

Students were asked to demonstrate their ability (and in case of failure, were taught how) to translate between concrete, pictorial (iconic), and abstract representations of arithmetic combinations. This was done (a) before students were asked to memorize the answers; (b) to help students secure the conceptual understanding of the combinations. Students were also taught the principle of commutativity (e.g. 3x5=5x3) as well as the properties of "0" and "1" in multiplication.

Students were then taught the 100 arithmetic combinations of multiplication in the following groups: "0s" (e.g. 0x3 / 3x0: 19 combinations), "1s" (e.g. 1x6 / 6x1: 17 combinations), "2s" (e.g. 2x7 / 7x2: 15 combinations), "twins" (e.g. 3x3, 4x4: 7 combinations), "5s" (e.g. 5x8 / 8x5: 12 combinations), "9s" (e.g. 6x9 / 9x6: 10 combinations), "3s" (e.g. 3x8 / 8x3: 8 combinations), "4s" (e.g. 4x6 / 6x4: 6 combinations), "6s" (e.g. 6x7 / 7x6: 4 combinations), and "the last ones" (7x8 / 8x7: 2 combinations). Students did not use conventional multiplication tables.

Instruction emphasized both the accuracy of responses and the rapidness of recalling. The aim was to promote accuracy and automaticity.

Maintenance of students' knowledge was checked three weeks after intervention ended.

To enhance generalization, at the end of the program students were given arithmetic problems involving these multiplication combinations.

Instruction took place for four weeks, four times per week, 25 to 35 minutes each time, in small groups. Most students showed major improvement in their mastery of the combinations.

The diversity in cognitive traits characterizing students with LD precludes the adoption of a single approach for teaching arithmetic combinations. Nonetheless, the research suggests that many students with LD can benefit from programs that emphasize (a) conceptual understanding of arithmetic combinations, manifested by students' ability to efficiently use concrete, pictorial, and symbolic representations of the combinations; (b) the teaching of arithmetic principles (like commutativity); (c) the grouping and presenting of combinations in ways that promote memorization and rapid recall, through exploring common characteristics of combinations; and (d) strategies for maintaining and generalizing what was learned.

References

Agaliotis, I. (2004). Learning disabilities in mathematics (4th edition). Athens: Ellinika Grammata (in Greek).

Agaliotis, I., Dimitrakopoulos, Th., Dimou, I. Pesli, V., Theodorou, E. & Charisi, A. (2003). Alternative instruction of basic multiplication facts for students with learning disabilities. Educational Sciences, 3, 53-66 (in Greek).

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

Garnett, K., & Fleischner, J. (1983). Automatization and basic fact performance of normal and learning disabled children. Learning Disability Quarterly, 6, 223-230.

Geary, D. (1993). Mathematical disabilities: Cognitive, neuropsychological, and genetic components. Psychological Bulletin, 114 (2), 345-362.

Gerster, H. D. (1999). Vom zaehlenden Rechnen zur Abrufbarkeit der Basisfakten- ein zentrales Ziel der Praevention und der Foerderung. In B. Ganser (Ed), Rechenstoerungen, diagnose - foerderung - materialien (pp. 173-191). Donauwoerth: Auer Verlag.

Grissemann, H. & Weber, A. (1990). Grundlagen und praxis der dyskalkulietherapie. Bern: Huber Verlag.

Miller, S. & Mercer, C. (1991). Addition facts 0 to 9. Lawrence, KS: Edge Entreprises.

Miller, S. & Mercer, C. (1993). Mnemonics: Enhancing the math performance of students with learning difficulties. Intervention in School and Clinic, 29, 78-82.

Miller, S., Strawser, S., & Mercer, C. (1996). Promoting strategic math performance among students with Learning Disabilities. Learning Disabilities Forum, 21, 2, 34-40.

Scherer, P. (1999). Produktives lernen fuer kinder mit lernschwaechen. Foerdern durch Fordern. Leipzig: Klett.

Zeleke, S. (2004) Learning disabilities in mathematics: a review of the issues and children's performance across mathematical tests. Focus on Learning Problems in Mathematics. 26, 4, 1-19

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.