Multiple Solution Strategies for Linear Equation Solving

1. Multiple Solution Strategies for Linear Equation Solving Beste Gucler and Jon R. Star, Michigan State University Introduction An algorithm is a procedure that guarantees to lead to a solution when all steps are applied correctly and in a predetermined order. Although an algorithm (referred to here as the “standard algorithm” or SA) exists for solving linear equations, its use does not always lead to the most efficient solution (VanLehn and Ball, 1987; Star, 2001). For example, several possible solution strategies for solving an equation are shown in Table 1: Table 1: Several possible solution strategies for a linear equation The first strategy is the SA. This algorithm consists of expanding the parentheses, combining the similar terms (variables and constants), moving the constant to isolate the variable, and then dividing to get the value for the variable. The second strategy (“change in variable” or CV) uses an alternative in which (x+2) was treated as a unit and then combined in the first step. The last strategy uses both CV and another transformation (“divide not last” or DNL) in which the equation is divided by 8 as an intermediate step, rather than as a final step (as is the case in the other two strategies). We will consider this strategy (CV & DNL) which uses both CV and DNL, to be the most efficient, given that it involves the application of the fewest transform Of interest in the present research is how students learn to use and be flexible in their use of multiple strategies for solving linear equations. We were particularly interested in the effect of direct instruction of multiple strategies on students’ ability to be flexible. We consider flexibility as the knowledge of multiple solution strategies and the selective choice among them to fit the particular situation. Method 153 sixth-grade students participated in the study. In an initial one-hour session, students completed a pretest and were then introduced to the steps that could be used to solve equations. Students then spent three one-hour sessions working individually through a series of linear equations (similar to the one in Table 1). In the last session, students completed a posttest. 81 of the students received an eight-minute presentation on how to use CV and DNL (the “strategy instruction” or SI condition). Three worked examples (one for each of the three strategies illustrated in Table 1) were shown to SI students; strategies were demonstrated without any discussion of their relative efficiency. The remaining 72 students saw no examples of solved equations (the “strategy discovery” or SD condition). Results Although the SI and SD conditions had a similar effect on students’ use of SA, all of the students who used CV in the post-test were in the SI condition (see Figure 1). However, even after receiving a demonstration of the most efficient strategy (CV & DNL), all students who used CV in the post-test only did so using strategy CV. We interpret these results to suggest that these students were able to initiate the most efficient strategies only through direct instruction, which is consistent with work by Schwartz and Bransford (1998). References Schwartz, D. L., & Bransford, J. D. (1998). A time for telling. Cognition and Instruction, 16(4), 475-522. Star, J. R. (2001). Re-conceptualizing procedural knowledge: Innovation and flexibility in equation solving. Unpublished doctoral dissertation, University of Michigan, Ann Arbor. VanLehn, K., & Ball, W. (1987). Understanding algebra equation solving strategies (No. PCG-2). Pittsburgh: Carnegie-Mellon University. Contact Information Jon R. Star, jonstar@msu.edu; Beste Gucler, guclerbe@msu.edu. College of Education, Michigan State University, East Lansing, Michigan, 48824. This poster can be downloaded at www.msu.edu/~jonstar.