When preparing teachers to use technology to teach, it is imperative that they understand the specifics of the technology tools they are using. Even more important, teachers need to know how to design and implement mathematical tasks that support students’ learning while using these tools. The importance for teachers to select and implement worthwhile mathematical tasks to engage all students in learning mathematics has been emphasized as a significant pedagogical activity by the National Council of Teachers of Mathematics (NCTM, 1991) and shown by researchers to be a critical component of effective mathematics classrooms (Stein, Smith, Henningsen & Silver, 2000). Some studies have examined how teachers design tasks for students that utilize technology (e.g., Laborde, 2002; Sinclair, 2003) or implement technology-based tasks with students (Author1 & Other, 2005; Author2, 2005). And while there has been research that has focused on the implementation of non-technology based tasks in mathematics classrooms (e.g., Stein, Grover, Henningsen, 1996), few studies have considered how teachers design and implement mathematical tasks with students while using technology.  The purpose of the current study is to examine the ways in which prospective mathematics teachers’ design and implement mathematical tasks using technology. In particular, this study focuses on prospective secondary teachers’ design and implementation of geometry tasks using The Geometer’s Sketchpad with middle school students enrolled in a high school geometry course.

Academic tasks and questions have long been a focus of research on mathematics learning (e.g., Doyle, 1983, 1988; Friedman, 1976; Nicely, 1976). Some researchers have focused on the types of tasks that are provided to students in textbooks, while others have considered the questions teachers pose as tasks are implemented with students. It is important to consider both the task as presented in a textbook as well as what the teacher says and does and how it impacts the cognitive demand of the task on which the student works. Research has showed that the teacher plays a critical role in maintaining, lowering, or increasing the cognitive demand of mathematical tasks (Stein, Smith, Henningsen, & Silver, 2000). Therefore, in our research we adopted the definition of “task” presented by Others & Author 1 (2002). That is: (a) tasks are guided by a goal and require more than repeating a previously memorized fact or procedure; (b) tasks might be posed in a curriculum, by a researcher, teacher, or student; (c) tasks may be explicitly stated or implicitly understood by the student; and (d) while a student is working on a task, subtasks may emerge such that tasks overlap. This definition of task enabled the researchers to identify tasks and to follow the ways in which the task was modified while the teachers implemented it with students.

Methods

Six participants for the study were selected from a class of twenty prospective teachers (PSTs) who were enrolled in a junior/senior level mathematics methods course, at a large public university in a southeast, that focuses on the use of technology in the teaching and learning of mathematics. The participants were paired and as part of the course assigned to design, share and implement a thirty-minute, exploratory task using The Geometer’s Sketchpad (GSP) to be implemented with a middle school student enrolled in a high school geometry course. There were three components to the task:  (1)  a pre-constructed GSP file, (2) a Word document for the middle school student, posing the task, including questions related to the task and/or simple technology directions for using GSP, and (3) a separate Word document for the pair of prospective teachers, describing what they anticipated students doing during the task and including a list of questions to ask the student during the thirty-minute session to guide them to the new discovery without giving it away; the document also described how the task could be adjusted for a student at a lower-level or extended  for a higher-level student. The pairs worked together to design the task, and then they were separated and paired with different PSTs during a class in which they piloted their task and received feedback about how it could be modified. The participant pairs worked together after the feedback session to make modifications to their tasks. The following class period they implemented their tasks individually with a single middle school student. After this session, the prospective teachers gathered to reflect on the experience. These piloting, teaching, and reflective sessions were videotaped and analyzed.

Findings and Discussion

The data were analyzed by first examining the written artifacts that were submitted by all PSTs in the class. These artifacts included pre-constructed GSP sketches, task sheets, and questions they anticipated from students and possible responses. From the analysis, codes related to PSTs uses of the technology (measures, dragging, animation, computation, construction) characterization of the task (closed task, inductive reasoning task, open-middled task, exploration task), and the nature of the questions posed were generated (e.g., describe, predict, generalize, justify). Some codes were created while others were based on existing research.  Initial analysis of the written artifacts revealed that all of the PSTs developed tasks that were either inductive reasoning, open-middled (NCTM, 1999 - one solution, but multiple solution paths) or exploration. None of the PSTs developed a task that was coded as closed. The pre-constructed sketches developed by the PSTs all made use of dragging, computation and measures. Generally, the PSTs wanted to students to identify an invariance in the sketch related to a geometric theorem that they wanted students to “discover.” The questions that were posed engaged students in predicting, generalizing, and describing. There were very few questions that involved justification.

In addition to the written artifacts, videos of the six PSTs implementing the tasks with students will be analyzed. These videos will focus on how the teacher presents and supports student work on the task. The questions that are posed, how they are sequenced, and the PST’s role in directing or supporting the student’s work (Kimball, 1994) will be examined. Findings from this study will be shared during the presentation.