探討國小六年級學生在不同幾何題中應用等量公理解題的情形

The Performance of Sixth Graders on Geometry Problems that are Conducive to the Use of the Equality Axioms

鄭鈐華(Chien-Hua Cheng)

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(2002年)

摘要

本研究所指的等量公理為「等量加等量，總量和仍相等」、「等量減等量，餘量仍相等」的加、減法等量公理。本研究主要有兩個目的，第一個目的是要了解國小六年級學生應用等量公理解加、減法幾何題的情形。第二個目的則是要了解學生是如何辨識出可以應用等量公理來解題，而沒有應用等量公理的學生又會採取那些解題策略及其原因。
本研究主要是採調查法，進行方式則是透過自行編制的紙筆測驗及個別訪談。研究工具的結構是根據Tam & Lien(2002)所指出的等量公理三個成分所設計的。本研究的研究對象，選自台北縣某所國小的一個六年級班級，一共29位。個別訪談時則選取該班中8位不同成就的學生。
為了瞭解學生應用等量公理的表現，本研究在量的分析上，採取描述性統計的方式，而在質的分析上，則以個案深度訪談的方式進行，並且採用資料來源的三角校正法，以期得到最可信的記錄，藉此以瞭解學生在解不同類型的幾何問題時，是否會想到應用等量公理的情形。
本研究的研究結果顯示：(一)不同背景的學生在應用等量公理的表現上有所差異；(二)高成就學生在「a=b，c=d」類型的面積、長度題中較會應用等量公理解題，而在「a=a，c=d」類型的面積題中則多採取其他的解題策略；(三)高成就學生在加法題中比在減法題中，多會應用到等量公理解題；(四)低成就學生在解隱含等量公理的幾何題，不論在那一種題型中，都比較沒有應用到等量公理來解題；(五)學生會藉由注意到題目中提及的相等條件，及發現圖形隱含的相等部分、代入數字做計算、使用符號、或列代數式，而想到應用等量公理解題的過程；(六)沒有應用到等量公理解題的學生，多數會順應直覺法則來解題。

關鍵字

等量公理 ； 幾何原本 ； 直覺法則

並列摘要

The Equality Axioms in this study mean “If equals be added to equals, the wholes are equal.” and “If equals be subtracted from equals, the remainders are equal.”; the former is an additional type, and the latter is a subtractive type. There are two main purposes for this study. The first one is to understand sixth graders’ performance of using the Equality Axioms in solving different addional and subtractive geometry problems. The second is to understand how students recognize to solve problems with Equality Axioms, and what problem solving strategies the other students use and why they do.
This study was a investigation through the written test designed by the researcher, and the way of case study matched with interview. The structure of the instrument was based on Equality Axioms’ three components by Tam & Lien (2002). There were 29 students in this study. They were chosen from a sixth class of an elementary school in Taipei County. And eight different achievers were chosen for the interview.
To understand students’ performance in using Equality Axioms, this study took the way of descriptive statistic in the quantitative analysis, and also adopted the protocol analysis in qualitative analysis. To get the realistic data, we also used the data triangulation. By these methods, we could understand sixth graders’ performance of using Equality Axioms in different geometry problems.
The results in the study were: (a)The performance of different backgrounds’ sixth graders of using Equality Axioms was different. (b)High achievers used Equality Axiom better in area and length problems in 「a=b,c=d」type, and they usually used.other strategies to solve area problems in 「a=a,c=d」type. (c)High achiever used Equality Axioms in additional geometry problems better than subtractive ones. (d)No matter in what geometry problems, low achievers tended to use other strategies to solve problems.(e)Students thought of using Equality Axioms by paying attention to equal conditions, recognizing out the equal parts in the sketch, or trying to use numbers to calculate, use symbles, or write algebra equations. (f)Most of students who didn’t use Equality Axioms usually followed intuitive rules to solve problems.

並列關鍵字

Equality Axiom ； Element ； intuition rule

參考文獻

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