Using dynamic sketches to enhance secondary preservice teachers' understanding of limit

The purpose of this study was to investigate (a) preservice teachers' understanding of limits, specifically, the formal definitions of the limit of a sequence and a function, (b) preservice teachers' interactions with dynamic sketches of the formal limit definitions, and (c) preservice teachers' changing conceptions of limit during and after instruction using the dynamic limit sketches. The participants in this qualitative study were six preservice secondary mathematics teachers: four fourth-year students in a five-year secondary teacher education program and two post-graduate Masters of Teaching students all attending a public university. The study included two initial interview sessions, two instructional sessions, and one final interview session each audio and videotaped and lasting approximately one hour. During the interviews, participants worked through a variety of limit tasks based on student misconceptions about limits. To control for possible learning during the initial interviews, three students were chosen to participate in the initial and final sessions but not in the instructional sessions. The selection process involved pairing the participants according to their grades in analysis and randomly assigning one participant from each grade bracket to the noninstructional group. For each participant, an individual case study was created providing detailed information about their initial understanding of limits, their interactions with the dynamic sketches during the instructional sessions (for the three in the instructional group), and the improvements in their understanding when completing the analogous final interview tasks. The main finding was that after interacting with the sketches, participants with little understanding of the formal definitions developed an understanding of the gist of the definitions while the participant already competent developed an understanding of the definitions' fine points. Also, dynamically applying the formal definitions to a variety of sequences and functions was instrumental in enhancing participants' understanding of the definitions and in overcoming major misconceptions such as "limit as unreachable" and "limit as boundary." Vital examples included constant and oscillating sequences and functions in addition to nonlinear functions and functions with removable jump discontinuities. These findings have implications for teacher educators as they prepare future calculus teachers and for mathematicians teaching analysis to preservice teachers.