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By Edgar L. Edwards

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The results lead to the problem of showing that the two different strings of symbols (the two rules) are mathematically equivalent. 38 ALGEBRA Fon Evianvoma Length Surface Area for n Rods in Staircase 2 2 (4n + 1) 3 2(5 n + 2) 4 2(6n + 3) m 2[(m+2)n+(m—l)]=2mn+4n+2m-2 Surface Area For Staircase Constructed With Rods of Length N0- in Staircase (Rods >< Surface Area — Overlap) 2 2(4m+2)-2(m—l) 3 3(4m+2)-4(m-1) 4 4(4m +2)—6(m—l) n [n(4m+2)—2(n—l)(m—l)]=2mn+4n+2m-2 Fig. 3 Within a single problem situation, students have practiced gathering and organizing data, looking for pattems, analyzing and generalizing pattems, and using algebraic language.

Silver and Kilpatrick (1987) relate that the problem variations should be progressive. After the students have solved a problem, we could change the context of the problem and pose it again. Next, we could change the data in the problem. A few lessons later we could use the technique of reversibility by giving the result and asking for the given portions of the problem situation. Also, we could make the problems more complicated by requiring multiple operations, extraneous data, and insufficient data.

Consistently, classes of students are not motivated to solve a problem. If the students are not interested, little value is realized in proceeding with an explanation of a solution. Predictions, guesses, conjectures, and confusion can each lead to discussions and defense of positions on processes and solutions. The content requires discussion of the processes involved and the various ways to solve the problem. Discussion gives students a means of articulating aspects of a situation, which, according to Pimm (1987), helps the speaker to clarify thoughts and meanings.

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