Doctor of Education
Faculty of Education
Ekanayake, Madduma Bandara, The development and evaluation of a web-based learning environment for proof-type problem solving in geometry among secondary students, D Ed thesis, Faculty of Education, University of Wollongong, 2004. http://ro.uow.edu.au/theses/390
The major purpose of this study was to address the instructional needs of proof-type geometry problem solving. It was designed to address two research questions:
(Q1). What are the predictive indicators of successful proof-type geometry problem solving? (Q2). Based on needs with an emphasis on formative evaluation, what is one design solution to support students solving proof-type problems in geometry?
The overall study focused on a learning need assessment in the first phase of the study (Study 1) and a development process to translate instructional needs identified into a supportive instructional environment for proof-type geometry problem solving in the second phase (Study 2).
The review of literature revealed that proof-type geometry problems have different learning requirements compared to other mathematical problems types. The solution process for proof-type geometry problems demands the adoption of a non-algorithmic approach in which students could activate problem-solving strategies that are domain specific. These strategies include heuristics such as using auxiliaries (parallel lines, bisectors and perpendiculars), alternative proving ethods (indirect proof, reductio ad absurdum, method of contradiction). Equally important are the role of domain-general strategies during the solution of proof-type geometry problems such as working backward and logical inferencing. The literature review suggested that geometry content knowledge, general processes, and mathematical reasoning could be potential predictive indicators of successful proof-type geometry problem solving. However, the relative importance of these variables during the construction of geometry proofs had not been subjected to an empirical evaluation.
Study 1 takes up the above issue by determining the relative importance of these variables in proof type geometry problem solving. Data were collected from 166 Sri Lankan students on three independent variables: Geometry Content Knowledge (GCK), General Problem-Solving processes (GPS) and Mathematical Reasoning Skills (MRS); and a dependent variable Proof-Type Geometry problem-solving (PTG). The relationship among these variables was examined through a multiple linear regression analysis procedure. This analysis showed that geometry content knowledge, general problem-solving processes, and mathematical reasoning are predictive indicators of successful proof-type geometry problem solving. Among these variables, geometry content knowledge was found to be the most influential one followed by general problem-solving processes and mathematical reasoning.
Three experts participated in a series of meetings to translate the above findings into a support framework for helping students learn to solve proof-type geometry problems in Study 2. This development process resulted in a conceptual model consisting of three major components: Remedial, Instructional and Problem Solving. The Remedial Component was suggested to address the learning needs related to geometric reasoning development, the Instructional Component focused on the development of content knowledge related to Euclidian deductive system, and the Problem-Solving Component was designed to facilitate proof-type geometry problem-solving skills among students who have the prerequisite geometric content knowledge and reasoning skills.
An iterative development process of design, development, review and revision was used to translate the Problem-Solving component into a Web-based, prototype learning environment in Part I of Study 2. This prototype, titled ANGEL (A Non-linear Geometry Environment for Learning), contained problem sets, process guidance, worked examples, diagram support and embedded content knowledge as core structural elements. Hyperlinked metacognitive supports were incorporated to facilitate the problem-solving process through guidance provided by general problem-solving processes such as analysis, representation, planning and use of knowledge retrieval by accessing embedded content. Although technology driven learning environments are mainly for student-technology interactions, ANGEL has additional advantages as it was designed for classroom use with teacher intervention to enhance social interactions: teacher-student and student-student that promote learning and construction.
The usability of ANGEL was tested in a constructivist collaborative learning environment. Six students selected from an Australian high school solved a series of proof-type geometry problems in pairs in a two-hour problem-solving session with the help of ANGEL. During their problem-solving attempts, data were collected in the form of student verbalization of the solution process, observation of problem-solving attempts, and written workings in the workbook. Having completed the problem-solving session, the students were interviewed to collect data on how they perceived ANGEL as a learning tool. The qualitative data analysis showed that the target group of students accepted ANGEL as a learning tool and that students enjoyed using ANGEL in problem solving. These patterns of results suggest that ANGEL works as designed and assists students to construct knowledge related to proof-type geometry problem solving.