Title

國中數學資優生尺規作圖表現之探討

Translated Titles

Investigation into the performance of junior high mathematically gifted students in straightedge and compass construction problems

Authors

劉繕榜

Key Words

尺規作圖 ; 幾何證明 ; 圖形 ; straightedge and compass construction ; geometry proof ; graph

PublicationName

臺灣師範大學科學教育研究所學位論文

Volume or Term/Year and Month of Publication

1999年

Academic Degree Category

碩士

Advisor

譚克平

Content Language

繁體中文

Chinese Abstract

本研究旨在探討國中數學資優生在(一)解作圖題時會產生什麼錯誤類型;(二)先前解過相關問題對解作圖題有何影響;(三)解題時的模式以及輔助問題對學生解作圖題的助益;(四)作圖能力與其在解幾何證明題表現之間的相關性;(五)在作圖後證明中所發生的錯誤類型為何。本研究是從民國八十八年九月至八十九年二月間以幾何教學實驗方式進行,是在學校教師正式教幾何之前所進行的幾何課程,全程用十二次共三十六小時教學。教學內容是由一群數學教育研究者協同計畫而成的,內容包含全等三角形、相似三角形、尺規作圖、圓以及切線性質等知識。而本研究的研究對象乃是二十一位國中數學資優生,他們在小學階段時均曾代表台灣參與國際性數學競賽;至於教學方式則是以小組教學的方式進行。 為了要搜集資料回答本研究所提出的問題,研究者設計了以下三項評量工具:一為教學前尺規作圖基本能力測驗;二是教學後作圖問題評量;三是晤談中作圖問題測驗。根據研究對象在教學前尺規作圖基本能力評量工具中的表現,本研究先將學生為區分高、中、低作圖能力者,以探討他們在教學期間解作圖問題時之表現。至於他們在教學後作圖問題上的表現,以及作圖後晤談的資料,則分別以量及質的方式進行資料分析。 本研究發現學生在解作圖題時,部分資優生會犯兩類型錯誤:一是基本概念的錯誤;另一則是圖形的錯誤引導。當中圖形的錯誤引導又可分為「草圖的錯誤引導」、「給定圖形的錯誤引導」等。而研究者認為部分學生發生該兩類型錯誤可能原因是由於他們(1)存有迷思概念;(2)未能建立起邏輯的聯繫,學生「內在驗證」的能力不足。這推論是根據分析學生在純證明題表現以及其對圖形操作反應的資料而得。在晤談的過程當中亦引證出如果缺乏「內在驗證」的能力,則會影響他們的解題表現。 除此之外,本研究中的學生在作圖問題上的表現、作圖後的證明以及解純證明題之表現統計上的顯著相關。例如學生在作圖問題上有好的表現,他們在解作圖後的證明,以及解純證明題時也會有好的表現等。並且高、中、低作圖能力的三組學生,彼此間在作圖後證明及純證明題之表現呈顯著差異,其中高作圖能力組的學生在幾何課程中會傾向有好的證明能力表現。另外亦發現學生在作圖後證明中表現出如下的弱點:有加入題目未說明的條件、循環論證以及只以圖式的證明而未提供文字說明等。 研究結果另外顯示部分學生解作圖題會受到先前解過作圖題中的圖形影響。再者,學生所回憶的知識內容之不同,會在解作圖題彰顯出想法的個別差異。在本研究透過深入分析其中三位學生在晤談時的紀錄,並按照他們解題時的個別差異,整理出學生解作圖題時,約可分為三種類型,分別是「連結類型」、「操作和驗證類型」以及「臆測和推理類型」。其他的資料則顯示本研究所設計之輔助問題對解作圖問題有一定的幫助。 本研究對教學方面提出一些建議。教師在教導尺規作圖單元時,可引導學生多利用畫草圖或動手操作的方式進行解題,並且加強學生的基本幾何知識以應用在解作圖題上,譬如圓與切線性質等。同時教師在教學上可針對部分作圖題設計安排一系列的輔助問題,並鼓勵學生嘗試解輔助問題以增加其解題經驗。對於已經存有迷思概念的學生,本研究建議採用認知衝突策略,並善用作圖後的證明以及加強學生的證明能力。 至於日後的研究方向,由於本研究只針對數學的資優生,因此研究者建議以後能廣泛地收集一般學生在作圖表現、作圖後證明及純證明題目之表現,同時進行量與質的分析,藉以瞭解一般學生對於在作圖題的表現等等是否與數學資優生有別,也可為幫助一般學生學習尺規作圖提供寶貴的資料。另一方面,由於現階段對尺規作圖方面的實徵研究比較缺乏,建議以後能有更多的研究對尺規作圖單元作教學效益的研究,為日後幫助學生學習幾何知識、幾何證明以及尺規作圖時提供更深入具體的幫助。

English Abstract

The main focus of this study is to find out (1) what kinds of mistakes will mathematically gifted students at junior high level commit while solving construction problems; (2) what kind of influence will previously solved related problems have on students solving construction problems; (3) which kind of models will they use to solve construction problems and the extent of help in the use of auxiliary problem; (4) what extent of relationship is there between students' construction ability and their performance in solving geometry proof problems, and (5) what kinds of mistakes will they make while proving their construction result. This research was carried out in the format of a teaching experiment that lasted from September 1999 to February 2000, right before the geometry units were formally taught in regular classes, the duration of the whole experiment amounted to 36 hours (12 times). The content materials were designed by a group of mathematics education researchers, and included such topics as the equilateral triangle, similar triangle, straightedge and compass construction, circle, tangent, the theorem of Ceva and other topics. The subjects of this research are 21 junior high mathematically gifted students. They had represented Taiwan to join a couple of international mathematical contests when they were in primary school. The teaching experiment was conducted in a small group discussion format. In order to collect data that would throw light on the research questions raised in study, three tools were specially design for this purpose. They were: (1) the basic straightedge and compass construction ability assessment tool administered before the teaching experiment; (2) the construction assessment tool administered during and after the teaching experiment; (3) specially selected problems accompanied with in-depth interview administered at the conclusion of the study. Base on their performance on the pretest, the students were classified as being of high, medium or low construction ability before they entered this study. Their subsequent performances were analyzed with reference to their basic ability. Their performances on the other two instruments were analyzed of solving construction problems after learning and the information by both qualitative and quantitative means. It was found that several of the gifted students encountered two kinds of difficulties when solving the construction problems; namely, the difficulties due to misconception, and difficulties due to misinterpretation of the diagrams. The latter could further be divided into the misinterpretation of the rough sketches, and the misinterpretation of the given diagrams. It was suggested that the reasons why some of the students encountered those difficulties were due to the misconception, that they had, and to their weakness in being able to mentally check their argument. The above observation was based on detailed analysis of their performance in solving pure proof problems and to their responses while attempting various operations on the diagrams. Subsequent interviews further substantiated the diagnosis mentioned above. Besides, it was found that there were statistically significant relationship among the students' performance in solving construction problems, in proving the validity of their constructed diagram and in solving pure proof problems. For example, if a student performed well on construction problems, he/she would, generally speaking, performed well in the other types of problems, and vice versa. It was also found that the high, medium, and low construction ability groups performed significant differently in proving the validity of their constructed diagrams and in solving pure proof problems. More specifically, the students in the high construction ability group performed better than the other groups in solving proof problems. Furthermore, it was found that some of the weaknesses as demonstrated by students in proving the validity of the constructed diagrams included adding in conditions not mentioned in the problems, tautology and in providing proofs without words. Another finding was that some students were influenced by the diagrams in the construction problems that they had previously solved. Moreover, the differences in what be students recalled would manifest into individual differences in the way they solved the construction problems. By analyzing the interviews of three particular students in detail, this research further suggested that there were at least three different approaches to solve construction problems. They were namely, recalling related problem approach, trial and verify approach, and conjecture and inference approach. Other data indicated that the auxiliary problems designed were helpful to some students in solving construction problems. Based on the above findings, it was suggested that while teaching straightedge and compass construction, teachers should encourage the students to draw sketch diagrams or try to operate with hands. He/she should strengthen the students knowledge in geometry that were applicable to solve construction problems. At the same time, the teacher could design a series of auxiliary problems that were related to some construction problems, and encouraged the students to solve them first so as to increase their problem solving experience. As for those students with misconception, it was suggested that teachers should adopt a cognitive conflict strategy, and to encourage students to always prove their constructed diagrams as a means to strengthen the students' ability in construction. As for the future research direction, it was suggested that more research should be directed towards studying the performances of students with various abilities on the kinds of problems discussed in this study. On the other hand, owing to the scarcity of empirical studies in straightedge and compass construction, it was suggested that more research should be conducted along this vein. The knowledge and experiment thus gathered should be of much value in improving students' ability to solve construction problems.

Topic Category 理學院 > 科學教育研究所
社會科學 > 教育學
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