Students' approaches to mathematical tasks using software as a black-box, glass-box or open-box

摘要

Three mathematical software modes are investigated in this thesis: black-box software showing no mathematical steps; glass-box software showing the intermediate mathematical steps; and open-box software showing and allowing interaction at the intermediate mathematical steps. The glass-box and open-box software modes are often recommended over the black-box software to help understanding but there is limited research comparing all three. This research investigated students' performance and their approaches to solving three mathematical task types when assigned to the software boxes. Three approaches that students may undertake when solving the tasks were investigated: students' processing levels, their software exploration and their self-explanations. The effect of mathematics confidence on students' approaches and performance was also considered. Thirty-eight students were randomly assigned to one of the software boxes in an experimental design where all audio and video data were collected via a web-conference remote observation method. The students were asked to think-aloud whilst they solved three task types. The three task types were classified based on the level of conceptual and procedural knowledge needed for solving: mechanical tasks required procedural knowledge, interpretive tasks required conceptual knowledge; and constructive tasks used both conceptual and procedural knowledge. The results indicated that the relationship between students' approaches and performance varied with the software box. Students using the black-box software explored more for the constructive tasks than the students in the glass-box and open-box software. These black-box software students also performed better on the constructive tasks, particularly those with higher mathematics confidence. The open-box software appeared to encourage more mathematical explanations whilst the glass-box software encouraged more real-life explanations. Mathematically confident students were best able to appropriate the black-box software for their conceptual understanding. The glass-box software or open-box software appeared to be useful for helping students with procedural understanding and familiarity with mathematical terms.