| In Taiwan, students have considerable experience with tasks requiring geometric calculations with number (GCN) prior to their study of geometric proof (GP). This dissertation examined closely the opportunities provided to Taiwanese students' experiences with GCN and their performance on GCN and GP. Three sequential studies were conducted, corresponding roughly to key aspects of the Mathematical Tasks Framework (MTF); namely, GCN tasks as found in instructional/curricular materials, GCN as enacted by teachers and students, and student performance on paired GCN and GP.;Study One found that GCN used by one Taiwanese mathematics teacher were drawn not only from the textbooks but also from other sources (e.g., tests) and the tasks varied with respect to cognitive complexity, with the tasks additionally included being generally more demanding than those found in the textbooks. The high demand GCN appeared to afford opportunities for Taiwanese students to master the types of knowledge, the reasoning and the problem-solving skills that are essential not only for proficiency with GCN but also for GP. Study Two showed how the teacher sustained the cognitive demand levels by making the diagrams more complex and using gestural moves to scaffold students' visualization of the diagrams so that they could sustain their work on the tasks. Through scaffolded experiences with GCN containing complex diagrams, the teacher appeared to nurture students' competence in constructing and reasoning about geometric relationships in ways that are likely to support their later work with GP. Study Three presented the analysis of Taiwanese students' performance on matched pairs of GCN and GP, which require the same diagrams and geometric properties to obtain solutions. The findings strongly support the hypothesis that students' prior experiences with GCN can support their competence in constructing GP.;Taken together the three studies sketch a plausible pathway through which Taiwanese students might gain high levels of proficiency in creating GP through their experiences with GCN. In addition, the use of a sequence of three studies that examine different aspects of students' experiences with mathematical tasks appears to have utility as a model for other research that seeks to understand cross-national differences in mathematics performance. |
