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The potential for developing algebraic thinking from purposeful guessing and checking

Two contrasting problem solving episodes are presented in which a prealgebra student uses systematic guessing and checking to solve algebra word problems with the underlying structure: "For a given y, m, and b, find x such that y=mx+b." The first episode illustrates how the student initially engages in systematic guessing and checking organized in a table (a "Guess and Check chart") to converge to the solution to a given word problem. The second episode describes the student's activity in a later problem solving session where he discovers a deterministic and essentially algebraic algorithm (linear interpolation) for solving word problems of this same form. The evolution of the student's thinking from purposeful guessing and checking to an algebraic algorithm illustrates that this activity structure can potentially offer a context in which the meaningful development of the fundamental concepts of variable, function, and rate of change can be engaged.