Exploring the link between preservice teacher's conception of proof and the use of dynamic geometry software

School Science and Mathematics, May 2002 by Pandiscio, Eric A

This case study investigated how secondary preservice mathematics teachers perceive the need for and the benefits of formal proof when given geometric tasks in the context of dynamic geometry software. Results indicate that preservice teachers are concerned that after using dynamic software high school students will not see the need for proofs. The participants stated that multiple examples are not equivalent to a proof but, nonetheless, questioned the value of formal proof for high school students. Finally, preservice teachers found the greatest value of geometric software to be in helping students understand key relationships within a problem or theorem. Participants also tended to study a problem more deeply with the software than without it.

This study seeks to connect two well-established research strands. One is the role of proof in geometry, and the other is the link between teachers' subject matter knowledge and pedagogical content knowledge. Of particular interest is how secondary preservice teachers perceive the need for and the benefits of formal proof when given geometric tasks in the context of dynamic geometry software. Recent scholarship on intuitive reasoning, geometric proofs, and dynamic software provide the intellectual foundation for the study. In addition, statements such as, "Proof offers powerful ways of developing and expressing insights" (National Council of Teachers of Mathematics, 2000, p. 56), and "Students' ownership of abstract mathematical ideas can be fostered through technology" (p. 25), speak to the importance of the type of research undertaken in this study.

Building on the work of Eves (1972), who conjectured that mathematicians determine truth by methods that are originally intuitive and empirical, and Lakatos (1976), who suggested that the deductive format in which proofs are presented is misleading, Battista and Clements (1995) posited that students should learn the meaningful use of proof by avoiding formal proof and focusing instead on justifying ideas and building visual and empirical foundations for later work. One way to do this is with dynamic geometry software, such as Cabri and Geometer's Sketchpad, which facilitate the making and testing of conjectures. Bershadsky and Zaslavsky (1999) investigated how such dynamic environments impacted students' awareness of the intuitive, visual aspects of geometric situations and described how this awareness, in turn, was reflected in students' understanding ofthe ideas under study. Crisan (1999) claimed that the use of mathematical software both challenges and enriches teachers' subject matter knowledge, as well as their pedagogical content knowledge.

Building upon this body of extant research, this report offers a case study of four preservice teachers as they solved two geometric problems and then attempted to create formal proofs that generalized their results. The problems were chosen to be unfamiliar in their specifics, yet based on traditional Euclidean concepts. Thus, the preservice teachers were faced with issues of proof in areas that challenged their subject matter knowledge and were asked to do so in a software environment that also challenged their pedagogical content knowledge. The most striking result is that all four participants saw dynamic software as a tool to make sense of proofs, but not necessarily as a tool that is helpful to create proofs.

Subjects and Methodology

The project was based on a case study approach. As described by Gall, Borg, and Gall (1996), and Lincoln and Guba (1985), the case study approach is widely and appropriately used in education. It is particularly suitable for producing a detailed description of a phenomenon (Gall, Borg, & Gall, 1996, pp. 544, 549). In this case study, according to Gall, Borg, and Gall's framework, the phenomenon is geometric proof, and the specific case is the manner in which dynamic software impacts prospective teachers' conception of the need for and ability to produce rigorous proofs.

The rationale for using a case study design is further strengthened by the type of research question posed. As Yin (1994) suggested, when the research question is of an explanatory variety, seeking to shed light on how subjects act within a complex environment, a case study is both suitable and helpful. In addition, Yin advocated the use of case study when the behavior of the participants is of interest to the researcher, yet cannot be carefully and precisely controlled. Merriam (1998) agreed, and further recommended using a case study approach when the goal of the researcher is to understand a complex issue, often involving individuals in a social context, and to generate hypotheses to explain events and outcomes. The researcher uses an inductive approach that results in a rich, comprehensive description. Given that the goals ofthe present study are to explain the behavior of preservice teachers when they encounter technology in the context of mathematical proof, a case study design provides robust analysis of the participants' experiences.