Abstract:

The National Council of Teachers of Mathematics (NCTM, 2000) in the United States, emphasizes the use of multiple representations in the teaching of mathematics to support different ways students think about manipulating mathematical objects. The use of dynamic visualization in the teaching of mathematics provides an exciting way to view mathematics in ways not possible in the past. With approximately fifty-percent of the human brain dedicated to the body’s visual faculties (Zimmerman and Cunningham, 1991), this underscores the importance of exploring ways to use visual and dynamic representations in the teaching and learning of mathematics.

This study uses technology-generated (e.g. Geometer’s Sketchpad, GeoGebra) dynamic visual representations of transformations of functions in order to determine how dynamic visualizations influence mathematics instructors’ content knowledge and pedagogical content knowledge on the topic of transformations of functions. Presmeg (1986) defines a visual image as a mental scheme depicting visual or spatial information and we adapt this definition to define a dynamic visual image as a mental scheme depicting families of visual or spatial information. Whereas most definitions of visual imagery refer the imagery in the “absence of a perpetual object,” we agree with Presmeg (1986) and Piaget and Inhelder (1971) that it is possible to have a visual image in the presence of an object.

We used smart pen technology to individually interview and record the work of three secondary school mathematics teachers from a large urban school district in the southwestern United Stated and three mathematics graduate students from a mid-sized (33,000 students) state university in the southwestern United States. Participants encountered three static questions regarding parameter changes for elementary functions encountered in first- and second-year algebra in high school in the United States. In a series of related questions, we posed questions that students might ask related to the latter. After this, participants viewed dynamic visualizations related to all static questions encountered. As they viewed animations, they were asked to explain what they were seeing. Participants were given the opportunity to revise the answers to the static questions based upon their observations. The interview culminated in a dynamic visualization grounded in an applications-based context.

Findings suggest that the dynamic visual images introduced misconceptions regarding some transformations and the graduate students tended to acquire the misconceptions with higher frequency than the secondary school teachers. The descriptions given when viewing the animations revealed a focus on horizontal movement when the transformation acted vertically and vertical movement when the transformation acted horizontally. Gestures, written work, and verbal explanations consistently revealed the misconceptions formed or solidified by the animations. Some evidence supports the idea that the secondary school teachers were less prone to forming misconceptions from the animations due to their teaching experiences with high school students.

Keywords:

Transformations of functions, dynamic visualization, teacher knowledge.