A Study of the Performance of Polya''s Problem Solving Heuristics on a Group of Seventh and Eighth Grade Mathematically Gifted Students in Learning Geometry Proofs

• Main Authors: Yue-Jen Jan, 詹玉貞
• Other Authors: Tam, Hak - Ping
• Format: Others
• Language: zh-TW
• Published: 2000
• Online Access: http://ndltd.ncl.edu.tw/handle/53033831695890127283

碩士 === 國立臺灣師範大學 === 科學教育研究所 === 88 === The geometry teaching material in this study is arranged mainly according to the Polya''s 4-step problem solving heuristics with a purpose to understand if Polya''s heuristics help guiding students in learning geometry proofs how to start from understanding the question, analyzing it, finding clues to answer from the given conditions, devising a plan to prove and completing the whole proof. This study wants to know if the arrangement of teaching material does upgrade students'' performance in proving geometry. Another purpose for this study is to guide students in applying two-column writing as they prove geometric questions and to see if this style help their performance. I have 21 top junior high students in math as my study object. Most of them have represented Taiwan in international math competitions. They are divided, according to their proving competence showed in pre-tests, into 3 groups: high, middle and low. The measuring tools used for data analysis in this study are eight questions regarding the congruent triangles (I) and (II), circles, the coedged theorem, the coangular theorem, the Ceva''s theorem and interview (I) and (II). Gathered from the data analysis, in the stage of "understanding questions," 85% of the students can grasp the given conditions and respond to the core geometric statement. The remaining 15% have no problem in understanding but have troubles in using the given data for proving the key geometric concept. However all of them can draw the diagram expressed in the question statement and mark proper places. In the stage of devising plan and executing it, the Polya''s 4-step problem solving heuristics does not help the high level students very much. However, to the middle and low level students, it gives much help. The high level students are strong in proving and adapt to the arrangement of the teaching materials without much efforts. They have good performance from the second class on. For the middle and low level students, their teacher has to remind them constantly of how to devise a feasible plan and execute it accurately when they proceed to prove a geometric concept. They depend very much on the teacher''s frequent encouragement and speaking to form such a problem-solving habit. Besides, from the findings, most of the 21 students who follow Polya''s heuristics are correct in their proving steps, while some that do not follow are wrong in their steps. Only one or two students remain correct in their proving steps without following Polya''s heuristics. I gathered that if a teacher wants to help students with correct geometry proofs, s/he has to repeatedly remind students of this heuristics. In the "reflecting" stage, this study intends to see how students extend their diagrams properly according to the question statement, after drawing, devising a plan and proving. From findings on their work, in "reflecting" on the coedged theorem, there are 11 students has such ability and one of them can even draw different figures with sufficient explanations responding to various questions. While the remaining students only mark out the difference with simple, or even little explanations. On co-angular theorem, 17 students have such reflecting power who are able to draw pictures different from the question statement and give proper explanation. Findings in their second class in learning geometry proving show that students have following difficulties in applying "two column" writing geometry proofs. 1.Don''t know how to write the reasons. 2.In quoting the previous proofs, they don''t know how to write the reasons. 3.In writing the geometry proofs, they don''t feel necessary to write the reasons. After this teaching experiment, students make much progress in this area. At the same time, their geometry proof writing become more logical in math with correct proving steps than before. Based on the findings and after-thoughts, this study suggests that teachers make students understand the importance of the stages in understanding the questions, devising a plan, executing the plan and reflecting in their teaching. Students should have time to think in class, while the teacher should help them timely in applying Polya''s 4-step problem solving heuristics in their geometry proofs. For the future study, the program should have longer time frame and take different factors such as various learning styles, meta-cognition etc into consideration for a deep understanding.