Reasoning processes of grade 4-6 students solving two- and three-variable problems
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The goal of this study was to gain insight into the reasoning processes employed by grades 4 through 6 students as they solved two- and three-variable tasks and documented their solution methods. The tasks were presented in three problem settings: weight scales, advertisements, and frame equations. Solution methods were identified as arithmetic (guess, check, and revise) or algebraic (identify relationships, compare equations, and replace variables with numbers.) To determine whether grade level, number of variables, or problem setting were related to solution success and solution methods, the six tasks were administered to 131 grade 4 subjects, 107 grade 5 subjects, and 143 grade 6 subjects from three public elementary schools and one middle school. Subjects' solution success scores on the tasks were compared by grade level, number of variables, problem setting, and solution method. Statistical analyses showed that: 1) there was a significant difference in solution success between fourth and sixth grade subjects; 2) number of variables in a task significantly influenced difference in success scores between fourth and sixth grade subjects; 3) problem setting was a significant factor in solution success for fifth and sixth grade subjects; and 4) subjects who employed algebraic solution methods were significantly more successful than were those who used arithmetic methods. Despite the fact that subjects had not had any formal training in the use of algebraic solution strategies, 20% of solution methods used on all tasks by all subjects were algebraic. At every grade level, algebraic solution methods produced 90% success rates. To elaborate on their reasoning processes, rank tasks by difficulty, and group tasks by "likeness," 25 subjects were interviewed. Solution methods employed by interviewed subjects exhibited common algebraic approaches: subjects identified relationships among unknowns, compared equations to identify differences, and replaced like variables with same numbers. Interviews also revealed that subjects identified the tasks with similar structures despite their different settings.
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