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.

SA CV CV & DNL 3(x + 2) + 5(x + 2) = 8 3x + 6 + 5x + 10 = 8 8x + 16 = 8 8x = -8 x = -1

3(x + 2) + 5(x + 2) = 8 8(x + 2) = 8 8x + 16 = 8 8x = -8 x = -1

3(x + 2) + 5(x + 2) = 8 8(x + 2) = 8 x + 2 = 1 x = -1

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.

0 20 40 60 80 100

% of studentsusing CV on atleast one post-test problem

% of studentsusing SA on atleast one post-test problem

Figure 1: Use of SA and CV by condition

SISD