THE VARIABLE IN LINEAR INEQUALITY:

COLLEGE STUDENTS' UNDERSTANDINGS

Carole P. Sokolowski

Merrimack College

csokolowski@merrimack.edu

This doctoral study examined six undergraduate students' conceptions of variable in linear inequality. Subjects completed a written algebra test and participated in unstructured interviews in which they solved and interpreted multiple representations of six linear inequality problems. A strong understanding of the concept of inequality was exhibited by subjects with an advanced conception of the variable as a varying quantity and also by the subject with an elementary conception of the variable as an evaluated unknown. Results of the analyses also indicated that an advanced conception of variable as a varying quantity appears to be necessary in order to algebraically model, solve, and interpret solutions to linear inequality problems.

There is currently a call for all levels of algebra instruction to emphasize the development of algebraic thinking. Understanding the many and sometimes complex uses of the algebraic variable and understanding the fundamental mathematical relations of functions, equations, and inequalities are necessary conditions for this important goal of algebra instruction. Although much research has investigated students' understandings of variables in equations and functions, little has been done with inequalities. In addition to being important mathematical relations, inequalities form a rich context in which to study the concept of the variable as it is understood by undergraduate students, who have had years of experience with school algebra. Variables in inequalities are manipulated as when solving equations; however, variables in inequalities usually represent sets of numbers, which makes them similar to variables in functions.

The purpose of this doctoral study was to investigate college students' conceptions of the variable in linear inequality. Sfard and Linchevski's (1994) operational-structural theory of reification formed the fundamental theoretical framework for describing subjects' understandings of the mathematical concepts of variable and inequality. Küchemann's (1978; 1981) six categories of uses of the variable (evaluated, ignored, used as an object, a specific unknown, a generalized number, or a varying quantity) provided the framework to classify subjects' levels of understanding of the concept of variable.

Methodology

The sample for this study consisted of six college students enrolled in a variety of mathematics courses at a four-year New England college. All six subjects had successfully completed high school mathematics through precalculus; five of the six subjects had successfully completed at least one calculus course. Data were gathered for this study using a quantitative and a qualitative instrument. The Chelsea Diagnostic Algebra Test (Brown, Hart & Küchemann, 1985) a valid and reliable paper and pencil test, was used to quantitatively classify subjects' levels of understanding of the variable in expressions, equations, and functions into four hierarchical levels, Level 1 being the most elementary to Level 4, the most advanced.

The qualitative instrument was an Inequality Test designed by the researcher which consisted of six linear inequality problems which were presented in prose, symbolic, and graphical forms. The prose versions of the Inequality Test problems were modeled after the inequality word problems used by Goodson-Espy (1994) in her investigation of college students' abilities to solve linear inequality word problems. The characteristics of the Inequality Test problems were graduated in situational context, mathematical structure, domain, and type of solution. With respect to mathematical structure, Problems 1 and 2 were of the general form, ax + b > c; Problems 3 and 4 were modeled by the general form, ax + b > cx + d; Problems 5 and 6 were similar to Problems 3 and 4, but their mathematical models required the use of parentheses. With respect to the problem solutions, Problems 1 through 4 resulted in solution sets, but Problems 5 and 6 resulted in contradictions and had no solution.

The Inequality Test was administered to each subject in three sections during an unstructured, video-taped interview. In Section I, the subject was asked to talk aloud while solving six prose problems. During Section II, the subject was asked to solve and interpret each of the symbolic representations of the prose problem relationships from Section I. Finally, in Section III, the subject was asked to use given Cartesian coordinate graphs and the corresponding Section I prose problems to again solve and interpret each problem and its solution. Table 1 illustrates the three presentation forms of one of the Inequality Test problems.

Table 1

Three Presentation Forms of Linear Inequality Problem #3

Section I: Prose
Two fitness centers in one town are competing for business. The Fitness Factory is offering new 6-month memberships for $125 plus a fee of $3.25 per visit. Slimnastics, Inc. is offering new 6-month memberships for $75 plus a fee of $4 per visit. Both centers have similar equipment and trained personnel. How many visits would a new member have to make during the 6-month period for Slimnastics to be the better choice?
Section II: Symbolic	Section III: Graphical
75 + 4x < 125 + 3.25x

Inequality Test transcripts were coded with Küchemann's uses of the variable, methods of solving the problems, and indices of strength of understanding of the concept of inequality. In order to detect trends, tables were prepared to represent each subject's uses of variable, methods of solving and interpreting the problems, and strength of understanding of inequality, for each problem across all presentation forms and for each presentation form across all problems. Narratives were prepared describing each subject's methods of solving the Inequality Test problems, describing relationships between subjects' performances on the Algebra Test with their methods of solving the Inequality Test problems, and describing subjects' uses of variables for each of the three presentation forms for each pair of similar problems (Problems 1 and 2, 3 and 4, 5 and 6). Finally, Sfard and Linchevski's (1994) theory of reification as it was applied to algebraic thinking provided a framework for interpreting the results of the study. Their theory specifies three phases of algebraic thinking: operational, fixed-value algebra, and functional algebra, which correspond, respectively, to the stages of interiorization, condensation, and reification of the concept of variable. Sfard and Linchevski's (1994) indicators of these phases of algebraic thinking were used to synthesize and frame the results of the study with respect to subjects' use of variables and their understanding of inequality. A case study for each subject was produced.

Results and Conclusions

Table 2 illustrates subjects' levels of use of the variable as revealed by the Algebra Test; their methods, arithmetic (did not use a variable) or algebraic (used a variable), of solving the prose and symbolic representations of the Inequality Test problems; their overall level of understanding of inequality; their overall use of the variable in the context of linear inequality problems; and their phase of algebraic thinking (operational, fixed-value algebra, or functional algebra) as determined by analyses of the Inequality Test interviews.

Table 2

Summary of Results of Analyses of Algebra Test and Inequality Test

Subject	Level of Use of the Variable	Method of Solving Prose Problems	Method of Solving Symbolic Problems	Level of Understand-ing of Inequality	Use of Variable in Linear Inequality	Phase of Algebraic Thinking
Eric	1	Arithmetic	Arithmetic (Guess & check)	Moderately strong	Evaluated	Operational
Carl	3	Arithmetic	Algebraic	Weak	Used as object	Operational Pseudostructural
Angie	3	Arithmetic	Algebraic	Moderate	Specific Unknown	Operational
Bob	4	Arithmetic	Algebraic	Moderately strong	Specific Unknown	Fixed-value Algebra
Denise	4	Algebraic	Algebraic	Strong	Varying Quantity	Fixed-value Algebra
Frank	4	Algebraic	Algebraic	Strong	Varying Quantity	Functional Algebra

The six undergraduate subjects in this study exhibited a wide range of conceptions of variable. Eric, the one subject who tested at Level 1 understanding of the variable, was at an operational phase of algebraic thinking. He used arithmetic guess-check-and-revise procedures to solve the prose and the first two symbolic problems (of the form ax + b > c). Notably, when he reached the third and subsequent symbolic problems (of the form ax + b > cx + d), he said that he could not solve them. He appeared to be mired in what Sfard and Linchevski (1994) describe as the 'process-product dilemma.' He saw the left side of a problem, such as 215 + 3x > 374, as a prescriptive process and, without hesitation, used a guess-and-check procedure to evaluate the variable and compared his guesses with the number, 374. His inability to use these same arithmetic methods on Symbolic Problem 3 (illustrated in Table 1) indicated his inability to see that the products of his calculations could then be compared with each other. Filloy and Rojano(1989) describe a cognitive gap between students' abilities to solve equations of the general form ax + b = c and equations of the form ax + b = cx + d. They call this gap the 'didactic cut.' Eric did not use algebraic methods to solve the former type of inequality. Yet his ability to solve Symbolic Problems 1 and 2 arithmetically, combined with his revelation that he could go no further when he met inequalities of the latter form, indicate that a 'didactic cut' appears to also exist for inequalities.

Despite his inability to work with variables unless he evaluated them, Eric displayed a rather strong understanding of the concept of inequality. In a manner consistent with Kieran's (1988) findings, Eric appeared to be interiorizing the concept of variable and strengthening the relational concept of inequality through his use of guess-check-and-revise procedures.

Two subjects, Carl and Angie, were classified at Level 3 by their performance on the Algebra Test, and analyses of the interview data revealed that they were at Sfard and Linchevski's operational phase of algebraic thinking. However, their work on the Inequality Test problems revealed different understandings of variables. Carl appeared to use variables as specific unknowns only in that manner described by Sfard and Linchevski (1994) as "pseudostructural." He efficiently and correctly solved Symbolic Problems 1 through 4, but was unable to interpret the meanings of either the variables or the critical endpoints in his solutions and could not interpret the meaning of the loss of variables in Symbolic Problems 5 and 6, which had no solutions. It appeared that Carl had never interiorized the concept of variable beyond that of an object to manipulate.

On the other hand, Angie's Level 3 performance on the Algebra Test was consistent with her understanding of variables as specific unknowns on the Inequality Test. However, Angie appeared to be stuck in an operational mode because she could not use a variable to represent the unknown quantity in the prose problems and, indeed, attempted to use two different input values to determine output values for some of the problems. In Prose Problem 3 (illustrated in Table 1), she stated that she would like to make the two fitness centers' fees equal, but that she could not do that. This appeared to be because she did not see that the processes which produced the two fees could also be considered quantities. She was bothered by the 'process-product dilemma' (Sfard and Linchevski, 1994) in her attempts to solve the prose problems, which indicated that she had not yet made the transition to fixed-value algebraic thinking.

Among the three subjects who were classified at Level 4 understanding of the variable, there were also differences in their understanding of variables in linear inequalities. Bob exhibited a strong conception of variable as a specific unknown. His inability to interpret his solution steps or the resulting contradictions in the symbolic versions of Problems 5 and 6 of the Inequality Test suggested that he was at the fixed-value phase of algebraic thinking and had not made the transition to functional algebra. Bob was a junior Computer Science major and had developed considerable facility with symbolic algebra. Yet, like Eric, Carl and Angie, he did not generate algebraic models of the prose problems on the Inequality Test, but struggled to solve them arithmetically.

Denise and Frank, both classified at Level 4 understanding of the variable, were the only subjects who used algebraic methods in order to solve the prose versions of the Inequality Test problems. Both used variables as generalized numbers and as varying quantities. However, Denise had some difficulty interpreting her answers to Symbolic Problems 5 and 6 (the contradictions), and she relied on a guess-and-check process in several of the prose and symbolic problems in order to determine the correct direction of the solution set and to solve Prose Problem 6. This showed a strong understanding of inequality, but it was an indication that she had not yet reified the variable at its highest level and was still at the fixed-value algebra phase of algebraic thinking. Only Frank appeared to have made the transition to the functional algebra phase of algebraic thinking, as defined by Sfard and Linchevski (1994).

It is important to restate the fact that all of these subjects, except Eric, had successfully completed a traditional calculus course. Yet, only Frank, a senior mathematics major, consistently made connections among the representations of the inequality problems and displayed a deep and robust understanding of the variable. Undergraduate students are assumed to have attained a minimal level of understanding with respect to functions and their applications in order to succeed in many, if not all, college mathematics courses. The results of this study indicate that, despite a displayed proficiency with symbol manipulation, many college students do not demonstrate that level of algebraic thinking which Sfard and Linchevski call functional algebra, which is characterized, in this study, by an ability to use variables flexibly, to generate algebraic models of word problems, and to clearly interpret the resulting solutions.

References

Brown, M., Hart, K., & Küchemann, D. (1985). Chelsea diagnostic mathematics tests and teacher's guide. Windsor, Great Britain: NFER-NELSON Publishing Company Ltd.

Filloy, E., & Rojano, T. (1989). Solving equations: The transition from arithmetic to algebra. For the learning of mathematics, 9(2), 19-25.

Goodson-Espy, T. (1994). A constructivist explanation of the transition from arithmetic to algebra: Problem solving in the context of linear inequality. Paper presented at the Sixteenth annual meeting of the North American chapter of the international group for the psychology of mathematics education, Baton Rouge, LA.

Kieran, C. (1988). Two different approaches among algebra learners. In A. Coxford & A. Shulte (Eds.), The ideas of algebra, K-12, (pp. 91-96). Reston, VA: NCTM.

Küchemann, D. (1978). Children's understanding of numerical variables. Mathematics in school, 7(4), 23-26.

Küchemann, D. (1981). Algebra. In K. M. Hard (Ed.), Children's understanding of mathematics: 11-16, (pp. 102-119). London: John Murray.

Sfard, A., & Linchevski, L. (1994). The gains and pitfalls of reification-The case of algebra. Educational studies in mathematics, 26, 191-228.

Sokolowski, C. (1997). An investigation of college students' conceptions of variable in linear inequality. Doctoral dissertation. Boston, MA: Boston University.