The nature of geometric tasks that students engage with in classrooms influences the development of their geometric thinking. Although mathematics standards emphasize formal proofs and mathematical reasoning skills, geometric tasks in classrooms remain focused on students’ abilities to recall mathematical facts and use simple procedures rather than conceptual understanding. In order to facilitate students’ high-level mathematical thinking, teachers need to provide sufficient opportunities for students to engage in cognitively demanding mathematical tasks. The use of dynamic geometry software (DGS) in classrooms facilitates conceptual understanding of geometric proofs. The black box approach is a new type of task in which students interact with pre-constructed figures to explore mathematical relationships by dragging and measuring geometric objects. This approach is challenging to students because it “requires a link between the spatial or visual approach and the theoretical one” (Hollebrands, Laborde, & Sträßer, 2008, p. 172).
This study examined how preservice secondary mathematics teachers make choices or create geometric tasks using DGS in terms of cognitive demand levels and how the black box approach influences the way preservice teachers conceptualize their roles in their lesson designs. Three preservice secondary mathematics teachers who took a semester-long mathematics teaching course participated in this qualitative case study. Data include two lesson plans, before and after instructions for geometric DGS tasks, pre- and post-interview transcripts, electronic files of geometric tasks, and reflection papers from each participant.
The Mathematical Task Framework (Stein, Smith, Henningsen, & Silver, 2009) was used to characterize mathematical tasks with respect to level of cognitive demand. A Variety of geometric task types using DGS was introduced to the participants (Galindo, 1998). The dragging modalities framework (Arzarello, Olivero, Paola, & Robutti, 2002; Baccaglini-Frank & Mariotti, 2010) was employed to emphasize the cognitive demand of geometric tasks using DGS. The PURIA model situated the participants’ conceptualized roles in technology use (Beaudin & Bowers, 1997; Zbiek & Hollebrands, 2008).
Findings showed that the preservice teachers only employed geometric construction types on low level geometric DGS tasks, which relied on technological step-by-step procedures students would follow in order to arrive at the same results. The preservice teachers transformed those low level tasks into high level tasks by preparing DGS tasks in advance in accordance with the black box approach and by encouraging students to explore the tasks by posing appropriate questions. However, as soon as they prepared high level DGS tasks with deductive proofs, low level procedure-based tasks followed in their lesson planning. The participants showed positive attitudes towards using DGS to prepare high level geometric tasks that differ from textbook-like procedural tasks. Major factors influencing preservice teachers’ preparation of high level tasks included teachers’ knowledge of mathematics, pedagogy, and technology, as well as ways of using curriculum resources and teachers’ abilities to set appropriate lesson goals.
Findings of this investigation can provide guidelines for integrating DGS in designing high level geometric tasks for teacher educators, researchers, and textbook publishers.