Conjecturing in a Computer Microworld: Zooming out and Zooming In
Zazkis, Rina, Sinclair, Nathalie, Liljedahl, Peter, Focus on Learning Problems in Mathematics
Teachers and researchers agree that making and testing conjectures is an invaluable part of mathematical learning. This belief is reflected in the NCTM's Reasoning and Proof standard, which states that students at all grade levels should have opportunities to make and investigate mathematical conjectures (NCTM, 2000). Conjecturing--activity of making, testing and interpreting conjectures--is seen as a major pathway to discovery, and is thus indispensable in problem posing and solving. As such, teachers are encouraged to support students' conjectures through collaborative verification, rather than treat them as statements that prompt haste judgment.
In this article we focus on conjectures and conjecturing in computer-based environments. Some researchers have argued that such environments can be particularly supportive of student conjecturing by providing empirical access to mathematical properties and relationships (Cuoco & Goldenberg, 1996; Hadas, Hershkowitz, & Swartz, 2000; Sfard, 1991). When using computers students are capable of performing calculations more quickly and accurately than can be done with paper and pencil. This allows students to effortlessly gather greater amounts of reliable data, in which patterns may be more easily discerned. In addition, many computer environments are more powerful than calculators, and capable of working flexibly with larger sets of data, thus offering alternative and multiple representations of information. As such, various kinds of patterns and relationships are made more accessible, depending on the representations chosen. Most importantly perhaps, well-designed computer-based environments can be created to provide learners with instant, pedagogically useful feedback, frequently visual in nature, that can be used by learners to evaluate or reflect on their own actions (Goldenberg, 1989; Noss & Hoyles, 1996). As a result, some learners adopt a greater sense of agency in their mathematical learning as authority is transferred from the teacher to the computer (Hewitt, 2001).
Our goal in this article is twofold. First, we take a broader view, "zoom out", of conjecturing and situate it within the larger cluster of mathematical activities with which it interacts. We thus aim to provide a better account of the motivations for the effects of conjecturing in student mathematical inquiry than is currently available in the research literature, where conjecturing is frequently seen as a precursor to proof. Second, we examine the details, "zoom in", of the student conjecturing in the context of a specific computer-based problem situation. We examine particular elements of participants' interactions with the problem, including the triggers that shaped their conjectures and the ways in which conjecturing guided their problem solving. In addition to providing a fine-grain analysis of student conjecturing, "zooming in" allows us to understand better the role a teacher can play in supporting productive conjecturing in computer-based environments.
Situating This Report
Participants and setting
This report is part of a larger study that investigates the interaction of a group of preservice teachers with a web-based microworld referred to as "Number Worlds" (the microworld is introduced below). The study investigated the mathematical experiences and understandings of 90 preservice elementary school teachers who interacted with the microworld using a set of given mathematical tasks. These tasks provided the participants with the opportunity to explore concepts related to elementary number theory, such as factors, divisors, divisibility, prime numbers, and reexamine their understanding of these concepts in an environment that supported experimentation and visualization. A comprehensive description of various aspects of this study is found in Sinclair, Zazkis and Liljedahl (2003). In this article we focus only on 20 participants (out of 90), who volunteered to participate in a clinical overview about their Number Worlds experiences, and on one mathematical problem given during the interviews. …