Remembering Word Problems of Math

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An examination of recall protocols provides a second useful approach to the study of mathematical understanding (Hegarty et al., 1995; Mayer, 1982). In our research, we asked student to read a series of arithmetic word problems. For each problem, students were assigned to treatment groups that required them either to write down the key information, draw a picture, construct a diagram, or compute an answer. Then they were given memory tests such as cued recall and recognition.

            Mayer (1981) has shown that word problems consist of assignment statements and relational statements. Assignment statements specify a numerical value for a variable, such as “At Lucky, butter costs 65 cents per stick”. In this case, the value is 65 and the variable is the cost (in cents) per stick at vons. The assignment can be expressed as LUCKY = 65. A relational statement expresses the quantitative relation between two variables such as “This is two cents less per stick than butter at vons.” In this case, the relation can be expressed as an equation, LUCKY = 2 + VONS, where LUCKY is the cost (in cents) per stick as Lucky and VONS is the cost (in cents) per stick at Vons.

            Prediction 3 : students make more errors in remembering relational statements than in remembering assignment statements. The construction of situation model requires special attention be paid to relations among the variables, particularly as expressed in the relational statements. Therefore, if a student uses a problem model approach, then the student should remember the actual relation between the two variables described in a relational statement. For example, the crucial relation in the relatiomal statement in the butter problem is that butter at vons cosrs more that butter at lucky. In contras, if a student uses a direct translation strategy, a primary goal is to assign values to variables. Therefore, thestudent is less likely to correctly remember relational statements. It follows that when students make errors in remembering word problems, the will be more likely to make errors in remembering relations than assignments.

            In order to test this prediction, we asked college students to read and later to recall to recall a series of eight problems. The student had 2 minutes to read each problem, and was asked to draw a picture, write an equation, or summarize the main information. As predictes, students made 3 times as many errors in recalling relational statements than in recalling assignment statements that had appeared in the problems. Furthermore, an analysis of errors revealed 20 cases in which students remembered a relation as an assignment, but only one case in which an assignment was recalled as a relation. For example, one student changed the relational statement “The steamer’s engine drives in still water at a rate of 12 miles per house than the rate the current,” to an assignment statement, “its engines push the boat at 12 mph in still water.” These result suggest that students have more difficulty in representing, storing, or retrieving relations (or any combination thereof) than assignments, and point to the difficulty some students may face in using a problem model approach on problems invcving relational statements.

            Prediction 4: successful problem solvers are more likely to remember the relation between two variables and less likely to remember the exact wording of the relational term than are the unsuccessful problem solvers. We can make a more specific prediction concerning the retention performance of unsuccessful and successful problem solbers. We define successful problem solvers as those who use a correct solution plan in solving a set of word problems and unsuccessful problems as those who make errors in solving a set of word problems. If successful students use a direct translation approach, we expect them to remember the key word (e.g., “less” or “more”), but not necessarily to remember the correct relations among the variables (e.g., that butter costs more at vons than lucky), especially students use a problem model approach, we expect them to remember the correct relations among the variables, but not necessarily to remember the exact wording of the key word.

            To test this prediction, we asked college students to solve a series of 12 word problems that contained four targer problems, which were two-step problems with relational statements such as the butter problem. Then we asked students to recall the problems and to take a recognition test.