Solving Problems on Functions: Role of the Graphing Calculator

Article excerpt

Abstract

To study the roles that the graphing calculator plays in solving problems about functions, a small quasi-experimental study was conducted with four pairs of undergraduate students solving problems with and without the graphing calculator. The analysis of the protocols of the sessions did not reveal major differences that could be attributed to the presence or absence of the tool but indicated differences in strategies used with each problem that could be explained in terms of the nature of the knowledge at stake and to students' availability of that knowledge. The study suggests a model for conducting research that looks for explaining the effects of technology in learning and instruction.

Graphing calculators have become part of high school mathematics classrooms. A survey of calculator usage in high schools commissioned by the College Board (Dion et al., 2001) indicated that graphing calculators are either required or allowed in at least 87% of the mathematics classes offered in high schools (p. 430). This imposes an interesting challenge to both college mathematics teachers and to mathematics educators who are responsible of preparing future mathematics teachers, as many of their students may come with experience with graphing calculators from their high school. A review of the research involving graphing calculators at the undergraduate level shows at least two types of studies. On the one hand, there are studies that investigate the impact of introducing graphing calculators in the classrooms on students' motivation, attitude, achievement, and retention (Hennessy, 1997; Hollar & Norwood, 1999: Quesada & Maxwell, 1994; K.B. Smith & Schotsberger, 1997). On the other hand there are studies that investigate students' understanding of the content or their discursive practices in the classroom in relation to the representations offered by the graphing calculators (Dick, 2000; Kaput, 1992; Roschelle, Pea, Hoadley, Gordin, & Means, 2000; Ruthvem. 1990; Shumway, 1990; Slavit, 1994). Both types of studies work under the assumption that the immediate availability of multiple representations of mathematical objects facilitate the process of making connections among those representations which in turn produces more robust or connected learning (Hiebert & Carpenter, 1992; Schoenfeld, 1987). However, using the graphing calculator efficiently in the classroom or documenting what actually is done with the tool has proven to be more difficult to accomplish. Teachers' beliefs and how students organize themselves to work on problems, have been cited as reasons why implementations with graphing calculators do not work as expected (Demana, Schoen, & Waits, 1993; Simmt, 1997).

In this article I want to suggest that the nature of the tasks, students' previous mathematical knowledge, and their experiences with graphing technology-independently of the availability of the graphing calculator-shape the collaborative construction of solutions among pairs of students. The present study was carried out to investigate the roles that the graphing calculator played when students had controlled access to it in a problem solving session. Studies that look at large effects of introducing the graphing calculator in classrooms (e.g., contrast overall achievement of a group of students when technology is present vs. not present) overlook the fact that the curriculum that is offered to each group is not comparable, and therefore it is not possible to conclude that differences in achievement, attitudes, or retention could be attributed only to the presence of the graphing calculators. And studies that look very closely at what happens when graphing calculators are used in the classroom, can not attribute results to the presence of the graphing calculator because there is not much knowledge about the particular aspects related to how the graphing calculator is used in specially crafted situations or about how problems are solved without the graphing calculator. …