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。

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.

1.蔡春美（民71）：台東縣山地兒童面積保留與面積測量概念。
國立台灣師範大學科學教育學系碩士論文。
2.蔡春美（民71）：兒童智慧心理學-皮亞傑智慧發展學說。
台北：文景。
3.王文科（民78）：認知發展理論與教育。台北：五南。
4.張平東（民78）：國小數學教材教法新論。台北：五南。
5.劉健飛、張正齊（民78）：數學五千年。台北：曉園。
6.林軍治（民81）：兒童幾何思考之van Hiele 水準之分析。
書恒出版社。
7.劉湘川、劉好、許天維（民82）：我國國小學童對稱概念的發展研
究。國立台中師範學院。
8.陳民澤 譯（民84）：數學學習心理學。台北：九章。
9.譚寧君（民84）：面積概念探討。國民教育，35卷7、8期，14-19頁。
10.熊召弟、王美芬、段曉林、熊同鑫 譯（民85）：科學學習心理學。
台北：心理。
11.劉秋木（民85）：國小數學科教學研究。台北：五南。
12.鍾聖校（民88）：自然與科技課程教材教法。台北：五南。
13.謝展文（民89）：直覺法則對於數學及科學學習的影響-以國小四、
五、六年級學生為對象。國立台灣師範大學科學教育研究所碩士論文。
14.張愛卿（民90）：放射智慧之光—布魯納的認知與教育心理學。
台北：貓頭鷹出版。（p91-204）
15.李其維（民90）：破解『智慧胚胎之謎』-皮亞傑的發生認識論。
台北：貓頭鷹出版。（p191-274）
16.戴政吉（民90）國小四年級學童長度與面積概念之研究。國立屏東師範學院數理教育研究所碩士論文。
17.連信欽（民91）：探討國中生在解不同幾何問題時應用等量公理的情況。國立台灣師範大學科學教育研究所碩士論文。
18.布魯納（民66）：教育的歷程。邵瑞珍譯。台北：五南。
19.Bruner, J S. (1977). The Process of Education: a landmark
in educational theory. Harvad University Press.
20.Burger, W. F., & Shaughnessy, J. M.(1986).Characterizing
the van Hiele levels of development in geometry. Journal
for research in mathematics education, 17, 31-48.
21.Crowley, M. L. (1987). The van Hiele model of the
development of geometric thought. In A. P. Shulte(Ed.),
Learning and teaching geometry, K-12(pp.1-16). Amerian:
The National Council of Teachers of Mathematics.
22.Clements, D. H. & Battista, M T. (1992). Geometry and
spatial reasoning. In D. A. Grouws (Ed.), Handbook of
research on mathematics teaching and learning(pp.437-442).
New York：Macmillan.
23.Fuys, D.; Geddes, D., & Tischler, R. (1988). The van Hiele
model of thinking in geometry among adolescents. American:
The National Council of Teachers of Mathematics.
24.Heath, T.H. (1956). The thirteen books of Euclid’s
Elements. Second Edition. Volume 1. N.Y.: Dover
Publications, Inc.
25.Hoffer, A. (1981). Geometry is more than proof. Mathematics
Teacher, 74(1), 11-18.
26.Sternberg, R. J. (1998).Cognitive Psychology(2nd ed.). New
York:Harcourt Brace College Publishers.
27.Tam H. & Lien, H. C. (2002).On the use of the addition and
the subtraction properties of equality in geometric
argumentation by junior high students in Taiwan. Paper
presented at the 2002 Intervational Conference on
Mathematics: Understanding Proving and Proving to
Understand, Taipei, Taiwan, Nov. 16-19.
28.Piaget J. &Inhelder B., (1960).The Child’s Conception of
Geometry. London and New York.
29.Stavy, R., Tirosh, D., Tsamir, P., & Ronen, H. (1996). The
Role of Intuitive Rules in Science and Mathematics
Education. Unpublished thesis. School of Education,Tel-Aviv
University, Israle.
30.Tirosh, D., & S (1999). The Intuitive Rules Theory and
Inservice Teacher Education. In Fou-Lai Lin(Ed.),
Proceedings of the 1999 International Conference on
Mathematics Teacher Education. Department of Mathematics,
National Taiwan Normal University: Taipei, Taiwan, 205-225.