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 Depdendent Variable

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 Dependent Variable

 Number of inequalities to solve: 23456789
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# SOLVING LINEAR EQUATIONS

In this research, students’ strategies for solving linear equations were examined.
Of particular interest was the strategy referred to as “change of variable” or CV. CV was
found when students rewrote terms such as 3(x+2) + 6(x+2) as 9(x+2). There are very
few research studies which attempt to understand students’ strategies to solve linear
equations (e.g., VanLehn and Ball, 1987; Pirie and Lyndon, 1997). In prior research in
this area, researchers have not commented on the use of the CV strategy to solve linear
equations.

157 students who had completed 6th grade participated in five one-hour problem-solving
sessions on linear equation solving. Students were given a pretest and then a
short lecture (20 minutes) in which the researcher introduced four different steps for
solving equations (adding to both sides of equation, multiplying both sides of the
equations by the same constant, distributing, and combining variables or constants) to
solve linear equations. After that, students worked to solve a series of linear equations for
three one-hour sessions.

25% of students used CV at least one question throughout pretest and posttest. An
analysis of the time that students spent solving each problem indicated that students who
used the CV strategy spent less time than students who did not use CV. For example, the
average time for all CV users was 3 min 41 seconds to solve CV questions throughout the
pretest and posttest, while the average time for non-CV users to solve the same questions
is 4 min 53 seconds, a difference that is significant. Not only did students who use CV
solve problems faster, but they also used fewer steps to solve each problem, on average.
While CV users typically solved problems in 3-5 transformations, non-CV users solved
the same problems in 4-7 steps. The use of CV enabled solvers to solve problems both
quicker and in fewer steps. Finding shorter and quicker solution paths is not only
important for solution efficiency but it reduces the chance of error (VanLehn and Ball,
1987). Non-CV users had significantly higher rates of error on CV questions as compared
to CV users.

CV is an example of an innovative strategy for solving linear equations, but it has
received little attention in prior research on linear equation solving. This study represents
an initial attempt to investigate the prevalence and use of CV among beginning algebra
learners