Academic journal article
By Tsamir, Pessia; Reshef, Maya
Focus on Learning Problems in Mathematics , Vol. 28, No. 1
We address the question whether to present students with a single method or with a number of methods for solving quadratic inequalities. Twenty 10th graders were presented with three differently sequenced methods, i.e., the graphic, the sign-chart, and the logical connectives method. We examined participants' preferences when solving related tasks, and their success in doing so. Almost all students correctly solved the different inequalities, and most liked being presented with several methods. The method most frequently used was the graphic method, and many students preferred the method they had studied first. Some conclusions and educational implications are drawn.
Students' Preferences When Solving Quadratic Inequalities
When planning instruction, teachers may face the dilemma of whether to present their students with a single method for solving a specific type of mathematical problem or with a number of methods. When a single method is chosen, a question arises about the criteria for this choice. If, on the other hand, several methods are going to be presented, one may wonder whether there is any significance to the order of presentation. A related question is, what should such didactic decisions refer to? For example, should we only consider students' success (as defined by the ability to answer correctly) or should we also attend to the methods students choose either when solving related mathematical tasks or when responding to explicit questions regarding their preference? We address these issues with regard to the process of solving quadratic inequalities.
The literature commonly presents three major methods for solving quadratic inequalities: the graphic method, the sign-chart method, and the logical-connectives method. The graphic method involves the interpretation of graphic representations, e.g., using parabolas to solve quadratic inequalities. The sign-chart method involves finding the zeros of an equivalent equation and using a sign chart to determine the solution of the inequality. The logical-connectives method involves the translation of the inequality into a system of linear inequalities, which are connected to each other by "or" and "and" connectives (see example in Figure 1).
Two approaches for teaching quadratic inequalities can be identified in the literature: the single-method approach, presenting the students with only one method, and the multiple-method approach, presenting the students with any combinations of the two or all three above-mentioned methods for solving this type of inequality. For example, Dreyfus and Eisenberg (1985) preferred the graphic method and provided mathematical and didactical argumentation for their claim:
[FIGURE 1 OMITTED]
this [graphic] approach appears to make the solution of many inequalities easier for average and weaker than average students who have had some experience with graphing functions. It also provides quite a bit of insight into what it means to solve an inequality and in what sense inequalities are related to functions. (p. 653)
McLaurin (1985) and Dobbs and Peterson (1991) identified the sign-chart method as the best method for teaching quadratic inequalities. They explained that "one of the most appealing aspects of sign charts is that they serve as a uniform and relatively easy method for solving what many consider to be more complicated inequalities" (p. 664). In fact, the researchers' claims were actually more far-reaching, since both McLaurin and Dobbs and Peterson suggested that the method they offered provided students with a powerful tool for solving not only quadratic, but any type of inequality. However, their papers report no research that supports their conclusions, which seems to suggest that these conclusions are based on the authors' impressions from their own teaching.
Piez and Voxman (1997), on the other hand, supported the multiple-method approach for solving inequalities. …