探討幾何問題中的情境及相關變因對解題影響之研究

The influence of problem presentation formats and other relevant variables on the performances of mathematically gifted students in solving Euclidean geometry problems

黃明瑩(Ming-Ying Huang)

指導教授 ： 譚克平

國立臺灣師範大學/理學院/科學教育研究所/碩士(1999年)

摘要

本研究之目的主要有三：一是瞭解學生在以數學符號方式描述的幾何問題與以文字符號方式描述的幾何問題中的解題表現之差異，二是進一步探討與學生解題表現相關的變因有哪些，三則是瞭解學生對於題目類型的看法。本研究的研究對象為21位國中數學較資優的學生，但主要以國二學生為主，且這群學生大部分均曾有代表台灣參加國際性數學競賽的經驗。本研究所涉及的幾何問題之範圍則以國中的幾何基本教材為主，所採的題目型式包括利用幾何性質求解的計算題與作圖題。在研究設計上，研究者依據各次上課的主題而設計二種不同類型的幾何問題，一種是以數學符號方式描述的題目，並附有圖形，另一種則是以文字符號來描述故事情境的題目，而圖形則依題目之內容及性質，有些題目有附圖形，有些題目則不附圖形。其中以文字符號方式描述的題目類型又可再分為兩種，一種為平鋪直敘式，一種為劇本對話式。此外，故事中的人名有採用非本研究對象之名字，及採用本研究對象之名字兩種，情境則分為學生較熟悉與不熟悉的。此種設計方式在於瞭解題目內容、題目敘述方式、圖形、主角名字等變項與學生解題表現的關係。另一方面，為了探討學生本身的解題能力與其解題表現之關係，本研究設計了一份幾何基本能力前測試題，學生在此試題中的得分即代表其解題能力之高低。由於本研究所採用之樣本很小，故在資料分析上以描述性統計及質性分析為主。
本研究的研究結果顯示學生在數學符號題中的解題表現略優於在故事情境題中的解題表現，而如果學生已成功解出數學符號題，則其能成功解出同構之故事情境題的百分比並不高，但反之則不然，即如果學生已成功解出故事情境題，則其能成功解出同構之數學符號題的比例則相對較大。整體來看，學生在發現題目同構性方面的比例並不高。與學生解題表現相關的變因則有學生的先備知識(解題能力)、題目的呈現方式及圖形的正確性等。例如解題能力愈高的學生，在這兩種題型中的解題表現沒有太大的差異，而解題能力愈低的學生，則在數學符號題中的解題表現明顯優於在故事情境題中的解題表現。至於學生在情意方面的表現，雖然大體上，學生在數學符號題中的解題表現略優於在故事情境題中的解題表現，但大部分的學生還是比較喜歡有故事背景的題目，覺得有故事背景的題目比較可以引起解題興趣，以及幫助瞭解題目的要求。本研究中還發現，在故事情境題中，有些學生會考慮到真實情況中其它會影響所求答案的變項，所以故事方式描述的題目提供了與生活環境之間產生聯結的可能性。
根據上述的研究結果與發現，如欲將幾何與生活產生聯結，故事情境題為一可採用的方式，而教師以故事情境之題目教學時，應與學生生共同討論現實情境中所存在的變因、從哪一個面向來看題目，以及如何轉換成數學模式來分析題目等，而在情境的選擇上，應盡量選擇學生所熟悉的情境，如果要使用學生的名字為故事中的主角時，最好事先徵求學生的同意，以免引起反效果。此外，在圖形的呈現上，教師也須特別留意圖形的長短比例及盡量避免以特殊例子取代較高階的概念。至於日後的研究方向，則可再加入其它的變項，例如學生的閱讀能力、認知風格、性別、教師的教學方式與題目的長短等變因，以及將研究對象推廣為一般國中生。

關鍵字

情境學習 ； 幾何

並列摘要

The main purposes of this study are threefold. The first one is to find out if there are differences between the way students solve Euclidean geometry problems that are given either in a story context or in terms of the conventional symbolic format. The second one is to identify if there are relevant variables that may affect students' performance in solving geometry problems. The third one is to find out what attitudes the students had regarding the presentation formats of the problems they have to solve.
Our research sample are twenty one seventh to ninth graders who are gifted in mathematics. Most of them have participated in various international mathematics competition. This research focuses mainly on geometric concepts that are relevant to the curriculum of the junior high level. Various topics that involve computational, derivational and constructional skills are covered in the study. We design two types of problems that are isomorphic to each other. Of which, one is presented in the form of mathematical symbols and figures only, while the other is by way of stories with or without figures. The story-type problems are further divided into the straight forward descriptive and the dramatic types. The purpose behind this design is to investigate into the relationship between students' performance in relation to the context which the problems are presented in. The result will shed some light on the role of situated learning so far as junior high Euclidean geometry is concerned. In addition, a pretest is used to measure their prior knowledge in geometry. Nevertheless, due to the small sample size, data analysis is mainly by way of descriptive statistics and qualitative analysis.
The result shows that our students perform relatively better on the problems presented in mathematical symbols. However, there are not many students who can first solve the mathematical symbol problems, and then also solve the corresponding isomorphic problems that are presented with a context. On the contrary, there is a high percentage of students who can first solved the problems in a contextualized format and then succeed in solving their isomorphic counterparts. Furthermore, it is noticed that not many students are able to tell that the problems presented in two different formats are actually isomorphic to each other.
With respect to the second purpose, it is found that the prior knowledge of students, the presentation formats of problem, and the accuracy of the figures are some relevant variables that may affect students' performance. More specifically, students with better mathematical skill perform equally well on problems in either presentation formats. However, students with weaker skill perform relatively well on problems presented in mathematical symbols only. Although on the whole, students are better in solving problems presented in mathematical symbols, most of them prefer to solve problems with a storyline. They indicate that problems with a storyline are more interesting and enable them to understand the problems better. Nevertheless, it is found that some students may consider extraneous variables in real life which they may bring in to solve problems with a context. This reveals that students can make connection between problem solving and real life situation.
Based on this study, it is quite acceptable to present geometry problems in more real-life format. It is suggested that teachers should discuss with their students various variables that can be identified in real life. This will enable the students to transform the story situation into a more realistic mathematical model. Besides, teacher should pay attention to the figures they provide in problems, especially with respect to the length of lines and the ratio between lines. This will prevent the students from mistaking special cases as a general principle. In the future, other variables, such as students' comprehensive ability on reading, students' gender, teachers' teaching methods and the length of presentation formats can be included for further study. A large sample size as well as extending the study to more general classrooms should also be considered.

並列關鍵字

situated learning ； Geometry

參考文獻

教育部(民83)：國民中學課程標準。

王春展(民86)：專家與生手間問題解決能力的差異及其在教學上的啟示。教育研究資訊, 5(2), 80-92。

劉繕榜(民89)：國中數學資優生尺規作圖表現之探討。台北市：國立臺灣師範大學科學教育研究所碩士論文。

陳儀君(民87)：國中學生在現實情境中解決測量問題的解題思維與互動歷程之研究。台北市：國立臺灣師範大學教育研究所碩士論文。

侯鳳秋(民87)：適性CAI中個人化文意範例對國小學生解題數學文字題之影響。花蓮市：國立花蓮師範學院國民教育研究所碩士論文。