“Strategy instruction” is a promising intervention for helping students with learning disabilities (LD) improve their ability to solve mathematical problems. Strategies for problem solving strategies may range from specific heuristics (methods used by solvers to present word problems in new ways that facilitate understanding) to broad guidelines:
Specific heuristics aim directly at helping students understand and solve a problem. Two heuristics are paraphrasing the problem and drawing a picture depicting the situation.
Broad guidelines help students activate and monitor domain specific heuristics (e.g. self-monitoring, self-evaluation). An example is asking students involved in solving a problem to first to predict the answer, then to carry out the calculations, and finally to compare their prediction and the result of their calculation (Butler, Beckingham, & Lauscher, 2005).
The FAST DRAW Strategy: A Domain Specific Heuristic
FAST DRAW is a domain specific heuristic that Mercer and Miller (1992) used to teach mathematics to elementary students with learning problems. Specifically, FAST DRAW was used to teach these students to remember multiplication facts and solve word problems.
The initials FAST DRAW stand for:
Find what you’re solving for.
Ask yourself, what are the parts of the problem.
Set up the numbers.
Tie down the sign.
Discover the (calculation) sign.
Read the problem (recognize the numbers involved).
Answer or draw and check (if the answer does not come automatically make a drawing that helps you find it).
Write the answer.
FAST DRAW uses a concrete-to semiconcrete-to abstract sequence of activities. In concert with this sequence, it successively uses tally marks, drawings, simple words, and traditional paragraphs to present problems. Importantly, difficulty is controlled. The difficulty of problems is increased gradually, following this sequence: Problems are presented as simple computations, then presented in complete sentences, then presented with extraneous information, then presented as traditional word problems. In the final phase, students are asked to create their own problems and to apply multiplication to real–life word problems.
Conceptually, this strategy is sound. Thus, it’s not surprising that Mercer and Miller’s (1992) field-tests found that the students—despite their learning problems—improved their problem solving skills.
Two Broad Guidelines Strategies
Case, Harris, and Graham (1992) used a Broad Guideline Strategy to successfully teach fifth and sixth graders with LD to solve one-step addition (joining, combining) and subtraction (separate, comparison, missing addend, combining) word problems. Their strategy asked students to:
But this was not all. They also implemented a self-instructional strategy (defining the problem, devising a plan, applying the strategy, self-evaluating and self-reinforcing) to regulate strategy use. Their instructional procedures included:
teaching students to identify important cue words commonly found in word problems.
having teachers meet with individual students to discuss their performance on baseline probes, the importance of learning the strategy, and the student’s commitment to the task.
modeling the strategy and self-instructions.
having students master the strategy steps.
engaging students in guided practice.
asking students to show how they independently perform the strategy and use the self-instructions.
giving students ample opportunities to apply the strategy so they become proficient and can generalize it.
Like FAST DRAW, a domain specific heuristic, Case, Harris, and Graham’s Broad Guideline Strategy helped students improve their ability to solve word problems. Moreover, the researchers found that the students generalized the strategy across settings and that they and their teachers developed a positive view of the strategy and self-instructions.
Similarly, Montague and her colleagues found that a combination of cognitive and metacognitive strategies (Broad Guideline Strategies) helped middle- and high-school students with LD improve their ability to solve mathematical problems. On the basis of several studies, Montague concluded that teaching students both cognitive and metacognitive strategies is superior to teaching them only a cognitive or metacognitive strategy. (See Montague, 1992; Montague, Applegate, & Marquard, 1993; Montague & Bos, 1986). Below is an overview of Montague’s cognitive and metacognitive strategies.
Montague’s cognitive strategy teaches students to:
read the problem to understand it.
paraphrase the problem by translating the information into their own words.
visualize it by drawing a picture or diagram).
hypothesize how to solve it by making a plan to solve it.
predict the answer.
carry out the calculations.
evaluate the results.
Montague’s metacognitive strategy teaches students to:
self-instruct (say—e.g. “I must read the problem carefully. If I don’t understand I have to read it again”).
self-question (ask—“Have I read and understood the problem?”).
self-monitor (check—e.g. “As I solve the problem I check if I have understood what it says”).
Butler, D., Beckingham, B., & Lauscher, H. (2005). Promoting strategic learning by eighth-grade students struggling in mathematics: A report of three case studies. Learning Disabilities Research and Practice, 20, 156-174.
Case, L., Harris, K., & Graham, S. (1992). Improving the mathematical problem-solving skills of students with learning disabilities: Self-regulated strategy development. The Journal of Special Education, 26, 1-19.
Mercer, C. & Miller, S. (1992). Teaching students with learning problems in math to acquire, understand, and apply basic math facts. Remedial and Special Education, 13, 19-35.
Montague, M. (1998). Cognitive instruction in mathematics for students with learning disabilities. In D. Rivera (Ed), Mathematics education for students with learning disabilities (pp. 177-200). Austin, TX: Pro-Ed.
Montague, M. (1992). The effects of cognitive and metacognitive strategy instruction on the mathematical problem solving of middle school students with learning disabilities. Journal of Learning Disabilities, 25, 230-248.
Montague, M., Applegate, B., & Marquard, K. (1993). Cognitive strategy instruction and mathematical problem solving performance of students with learning disabilities. Learning Disabilities Research and Practice, 8, 223-232.
Montague, M. & Bos, C. (1986). The effect of cognitive strategy training on verbal math problem solving performance of learning disabled adolescents. Journal of Learning Disabilities, 19, 26-33.
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.