本研究在商高定理單元中，使用九年一貫數學領域能力指標，分析學生面對
文字題及圖文題，解題時所使用的相關數學概念，並透過Schoenfeld 數學解題歷
程，分析學生在兩種表徵題的解題歷程順序及時間差異。四位研究對象為高雄市
某完全中學國中部二年級數學程度及表達能力佳的學生，將商高定理題目分為
「形」、「面積」、「數」三類型，以放聲思考法及半結構性晤談來蒐集資料，並採
三角檢證法進行原案分析。
研究結果發現，第一，數學概念展現上，在「形」的商高定理題目中，學生
展現的解題概念種類的確因題目表徵不同而有所不同；而「面積」、「數」的題目
中，除了文字題因畫圖輔助解題，需比圖文題多出幾何的概念外，其餘解題概念
因題目表徵不同而造成的差異較少，但學生使用的解題方法不同會造成使用概念
種類及順序上的不同。總體來說，數學概念種類可用使用概念次數來彌補，而且
文字題使用的解題概念種類會比圖文題多。第二，解題時間中，學生在Schoenfeld
前三個解題階段所花的時間，文字題是圖文題的1.6 倍；整個解題所花的總時間，
文字題是圖文題的1.25 倍。在解題階段上，學生面對文字題時，較多會經歷探
索階段，才能往下解題，而且在解題階段的往返次數較多。第三，無論是文字題
或圖文題，學生答對題數最多的，其概念使用次數最多、解題階段往返次數最少；
反之亦然。
教學建議方面，商高定理的教學順序中，研究者建議教師活用教具，先由「形」
的面積拼補概念導出商高定理公式，再將公式應用在各類型「數」的題目中，學
生有了以上「形」及「數」的商高定理概念後，教師再從「面積」的角度去解釋
並應用商高定理。另外，在佈題時，教師需發展要運用許多數學概念的問題情境
並涵蓋不同題目表徵，以加強學生數學概念的活用及兼顧不同學生的需求。

The aim of this study is to analyze students’ mathematics concepts in solving
Pythagorean Theorem problems presented in two different representations (word
problems and word problems with diagrams). The investigators employed the
mathematics competence indicators in Grade 1-9 Integrated Curriculum in developing
such problems. In analyzing data, the investigator used Schoenfeld’s method in
depicting their problem-solving processes, with attention to students’ sequence and
difference in time consumption. Four eight grade students with good competence in
mathematics and expressions from a secondary school were selected as research
subjects. Problems related to Pythagorean Theorem were divided into three types:
Shape, Area, and Number. Data were collected using thinking aloud method and
semi-structured interview, and triangulation was further applied in protocol analysis.
The research results revealed 3 findings: (1) For the “Shape” type problems,
students’ problem-solving concepts varied with different problem representation. For
the “Area” and “Number” types of problems (without diagram), students were
required to use their geometric concept when processing word problems. Students”
use of problem-solving concepts would not significantly vary with problem
representation types. However, students’ use of problem-solving methods would
affect the types and priorities of concepts used. Generally, the types of mathematics
concepts could be made up by the frequency of concepts used, and more types of
problem-solving concepts would be used for word problems representation than for
word problems with diagrams representation. (2) In terms of the time consumed in the
first three problem-solving stages of Schoenfeld, the time required to solve word
problems was 1.6 times of that required to solve word problems with diagrams. In
terms of the total time consumed, the time required to solve word problems was 1.25
times of that required to solve word problems with diagrams. In the problem-solving
stages, students needed to explore the problem first when dealing with word problems
before they could go on to solve the problem, and such repetition was more frequent
when they dealt with word problems. (3) For both type of problem representations,
there is a higher number of correctly-answered problems. This finding indicated that
a higher frequency of problem-solving concepts and less repetition in the
problem-solving stage were required; and vice versa.
As to the sequence of Pythagorean Theorem concepts to be taught, the
investigator suggest teachers to start with the concept of area filling in the “Shape”
type of problems to derive Pythagorean Theorem, and further apply the formula to
- III -
solving “Number” problems. After students have acquired basic competency in
“Shape” and “Number” Pythagorean Theorem problems, teachers could explain and
introduce this theorem from the perspective of “Area”. Finally, in problem posing,
teachers were also advised to apply various contexts; covering all kinds of
representations of problems that enhance students’ utilization of mathematics
concepts; and to cater for various needs of students.

- VII -
圖目次
圖2-2-1 Schoenfeld 之解題基模大綱…………………………………………15
圖2-2-2 Glass與Holyoakh問題解決模式………………………………………17
圖2-3-1 Lesh, Post與Behr表徵關係圖 ………………………………………..23
圖2-3-2 Lesh, R., Landau, M.與Hamilton, E.解題過程 ………………………. 24
圖2-4-1 周髀前幾頁 …………………………………………………………..30
圖2-4-2 周髀的”弦圖”…………………………………………………………31
圖2-4-3 趙爽注周髀算經之弦圖……………………………………………... 31
圖2-4-4 勾股定理課程流程……………………………………………………33
圖4-2-1-(1) 小諺足球場文字題解題歷程…………………………………………81
圖4-2-1-(2) 小諺足球場圖文題解題歷程 ……………………………………81
圖4-2-2-(1) 小諺池塘文字題解題歷程 …………………………………………82
圖4-2-2-(2) 小諺池塘圖文題解題歷程 …………………………………………82
圖4-2-3-(1) 小諺八邊形文字題解題歷程…………………………………………84
圖4-2-3-(2) 小諺八邊形圖文題解題歷程 ………………………………………84
圖4-2-4-(1) 小諺直線距離文字題解題歷程 ……………………………………85
圖4-2-4-(2) 小諺直線距離圖文題解題歷程 ……………………………………85
圖4-2-5-(1) 小諺纜車文字題解題歷程 …………………………………………86
圖4-2-5-(2) 小諺纜車圖文題解題歷程 …………………………………………87
圖4-2-6-(1) 小愛足球場文字題解題歷程 ………………………………………88
圖4-2-6-(2) 小愛足球場圖文題解題歷程 ……………………………………88
圖4-2-7-(1) 小愛池塘文字題解題歷程 ………………………………………89
圖4-2-7-(2) 小愛池塘圖文題解題歷程……………………………………………90
圖4-2-8-(1) 小愛八邊形文字題解題歷程…………………………………………91
圖4-2-8-(2) 小愛八邊形圖文題解題歷程…………………………………………91
- VIII -
圖4-2-9-(1) 小愛直線距離文字題解題歷程………………………………………92
圖4-2-9-(2) 小愛直線距離圖文題解題歷程………………………………………92
圖4-2-10-(1) 小愛纜車文字題解題歷程……………………………………………93
圖4-2-10-(2) 小愛纜車圖文題解題歷程……………………………………………94
圖4-2-11-(1) 小淨足球場文字題解題歷程…………………………………………95
圖4-2-11-(2) 小淨足球場圖文題解題歷程…………………………………………95
圖4-2-12-(1) 小淨池塘文字題解題歷程……………………………………………96
圖4-2-12-(2) 小淨池塘圖文題解題歷程……………………………………………96
圖4-2-13-(1) 小淨八邊形文字題解題歷程…………………………………………98
圖4-2-13-(2) 小淨八邊形圖文題解題歷程…………………………………………98
圖4-2-14-(1) 小淨直線距離文字題解題歷程………………………………………99
圖4-2-14-(2) 小淨直線距離圖文題解題歷程………………………………………99
圖4-2-15-(1) 小淨纜車文字題解題歷程 …………………………………………100
圖4-2-15-(2) 小淨纜車圖文題解題歷程 …………………………………………100
圖4-2-16-(1) 小聰足球場文字題解題歷程 ………………………………………102
圖4-2-16-(2) 小聰足球場圖文題解題歷程 ………………………………………102
圖4-2-17-(1) 小聰池塘文字題解題歷程 …………………………………………103
圖4-2-17-(2) 小聰池塘圖文題解題歷程 …………………………………………104
圖4-2-18-(1) 小聰八邊形文字題解題歷程 ………………………………………105
圖4-2-18-(2) 小聰八邊形圖文題解題歷程 ………………………………………105
圖4-2-19-(1) 小聰直線距離文字題解題歷程 ……………………………………106
圖4-2-19-(2) 小聰直線距離圖文題解題歷程 ……………………………………107
圖4-2-20-(1) 小聰纜車文字題解題歷程 …………………………………………108
圖4-2-20-(2) 小聰纜車圖文題解題歷程 …………………………………………108
圖5-2-1 商高定理建議教學順序流程圖…………………………………… 126

一、中文部分
吳昭容（1990）：圖示對國小學童解數學應用問題之影響。國立台灣師範大學心
理學研究所碩士論文。
杜佳真 (1999)。數學文字題的表徵教學策略。科學教育研究與發展季刊，15，
59-67。
李俊彥（2004）： 不同題目表徵型式的面積問題對國三學生解題表現之探討。
國立高雄師範大學數學教育研究所碩士論文。
李靜瑤（1994）：高雄市國二學生數學解題歷程之分析研究。國立高雄師範大學
數學教育研究所碩士論文。
林文生 ( 1996)。一位國小數學教師佈題情境及其對學生解題交互影響之分析研
究。國立台北師範學院碩士論文。
林明哲（1990）：國中學生數學解題行為之分析研究。國立彰化師範大學科學教
育研究所論文。
林美惠（1997）：題目表徵型式與國小二年級學生加減法解題之相關研究。國立
嘉義師範學院國民教育研究所碩士論文。
林碧貞（1990）。從圖形表徵與符號表徵之間的轉換探討國小學生的分數概念。
新竹師院學報，4，295-347。
倪玫娟（2004）：一位國小教師實施乘法教學的行動研究。國立新竹師範學院數
理教育碩士論文。
莊松潔（1995）：不同年級學童在具體情境中未知數概念及解題歷程之研究。國
立中山大學教育研究所碩士論文。
唐淑華（1989）：「語文理解課程」對增進國一學生數學理解能力與解答應用問
題能力之實驗研究。國立台灣師範大學教育心理與輔導研究所碩士論文。
徐文鈺 (1996 )。不同擬題教學策略對兒童分數概念、解題能力與擬題能力之影
響。國立台灣師範大學教育心理與輔導研究所博士論文。
130
梁宗巨（1995）：《數學歷史典故》。台北：九章出版社。
梁淑坤（1996）：從佈題探談數學科教科書的評鑑。教師之友，第37 卷，第4
期，23-28 頁。
《國民中學數學教師手冊第三冊》（2007）。台南：南一書局，25-44。
《國民中學數學課本第三冊》（2006）。台南：南一書局。
《國民中學數學教師手冊第三冊》（2007）。台北：康軒文教事業股份有限公司，
41-48。
陳立倫（2000）：兒童解答數學文字題的認知歷程。國立中正大學心理研究所碩
士論文。
陳啟明 ( 1990)。不同題目表徵型式及相關因素對國小五年級學生解題表現之影
響。國立嘉義大學國民教育研究所碩士論文。
陳美芳（1995）：「學生因素」與「題目因素」對國小高年級兒童乘除法應用問
題解題影響的研究。國立台灣師範大學教育心理與輔導研究所博士論文。
教育部（2003）：國民中小學九年一貫課程綱要。台北：教育部。
黃敏晃（1991）：淺談數學解題。教與學，23，2-15。
黃瑞琴（1994）：《質的教育研究方法》。台北：心理出版社。
張欣怡 (1997)。地球科學不同課文表徵教材對學習表現之研究。國立台灣師範大
學科學教育研究所碩士論文 。
張春興（1989）：《張氏心理學辭典》。台北：東華書局。
游自達(1995)。數學學習與理解之內涵-從心理學觀點分析。初等教育研究集
刊，3，31-45。
楊德清、洪素敏（2003）。創意教學∼「三角形」的教學活動，科學教育研究與
發展季刊，32，41-54。
蔣治邦（1994）：由表徵觀點探討新教材數與計算活動的設計。國民小學數學科
新課程概說（低年級），60-76。台北：台灣省國民教師研習會。
劉秋木(1996)：《國小數學科教學研究》。台北：五南。
謝淡宜（1998）：小學五年級數學資優生與普通生數學解題時思考歷程之比較。
131
台南師院學報，31，225-268。
謝毅興 (1991)。兒童解數學應用問題的策略。國立台灣大學心理學研究所碩士論
文。
二、英文部分
Barnett, J. ( 1984 ). The study of syntax variables. In G. A. Goldin & C. E. McClintock ( Eds. ), Task variables in mathematics problem solving, 23-68. Philadelphia, Pennsylvania : The Franklin Institute Press.
Bishop, A. J. ( 1989 ). Review of research on visualization in mathematics education . Focus on Learning Problems in Mathematics , 11 ( 1 ), 7-15.
Brenner, M. E., Herman, S., Ho, H. Z. & Zimmer, J. M. (1999). Cross-National Comparison of Representational Competence. Journal for Research in Mathematics Education , 30 (5), 541-547.
Brownell, W. A. (1942). Problem solving. In N. B. Henry (Ed.), The psychology of learning (41st Yearbook of the National Society for the Study of Education, part 2) (pp. 415-443). Chicago: University of Chicago Press.
Bruner ,J . S .(1966).Toward a theory of instruction. Cambridge, MA: Harward University.
Cramer K. A., Post T. R., & delMas R. C. (2002). Initial fraction learning by fourth- and fifth-grade students: A comparison of the effects of using commercial curricula with the effects of using the rational number project curriculum. Journal for Research in Mathematics Education, 33 (2), 111-144.
Cummin, D.D.（1991）.Childrens interpretions of arithmetic word problem.Congition and Instruction, 8, 261-289.
Davis, R. B.（1984）.Learning mathematics.The Cognitive Science approach to mathematics education. Norwood, New Jersy: Ablex　Publishing corporation.
Dreyfus, T., & Eisenberg , T. (1996). On different facets of mathematical thinking. In R. J. Sternberg & T. Ben-Zeev (Eds.), The nature of mathematical thinking , 253-284. Mahwah, NJ: Erlbaum.
Dufour-Janvier , B. , Bednarz , N. , & Belanger, M. (1987). Pedagogical Considerations concerning the problem of representations. In C. Janvier (Ed.) , Problems of representation in the teaching and learning of mathematics, 109-122. Hillsdale, NJ: Lawrence Erlbaum Associates.
Loomis, E.S., (1940): The Pythagorean Proposition .Council of Teachers of Mathematics, Inc.
Fennell, F. & Rowan, T. (2001). Representation : An Important Process for Teaching and Learning Mathematics. Teaching Children Mathematics, January, 288-292.
Fauvel, John and Jan van Maanen eds.（2000）：“The Pythagorean theorem in different cultures”. History in mathematics education (Dordrecht / Boston / London: Kluwer Academic Publishers), pp. 258-262.
Fraenkel & Norman E. Wallen(1990):How to Design and Evaluate Research in Education.The McGraw-Hill Companies, Inc.500-534.
Gagn’e, E. D（1985）.The cognitive psychology of school learning. NJ: Erlbaum.
Glass, A. L., & Holyoak, K. J. ( 1986 ). Cognition. NY: Random House.
Goldin G.A. （1982）.Department of Mathematical Sciences Northern Illinois University.In F.K. Lester, & J.Garofalo, Mathematical Problem Solving Issues in Research （pp.87-101）. Philadelphia, Pa.: Franklin Institute Press.
Greeno, J. G. (1987). Instructional representations based on research about understanding. In A. H. Schoenfeld (Ed.), Cognitive science and mathematics education (pp.61-88). Hillsdale, NJ: Lawrence Erlbaum.
Hiebert, J., & Carpenter, T. P. (1992). Learning and teaching with understanding.
In Grouws, D. A. (ED.), Handbook of research on mathematics teaching and
learning, 65-97. New York: Macmillan.
Juhani, L. (1995). Working memory and school achievement in Ninth Form. Education Psychology, 15 ( 3 ), 271-281.
Kaput, J. J. ( 1985 ). Representation and Problem Solving: Methodological Issues Related to Modeling . Teaching And Learning Mathematical Problem Solving: Multiple Research Perspectives, 381-397. EA Silver - 1985 - books.google.com.
Kaput, J. J. ( 1987a ). Representation systems and mathematics. In C. Janvier ( Ed. ). Problems of representation in the teaching and learning of mathematics, 19-26. Hillsdale, NJ: Erlbaum.
Kaput, J. J. ( 1987b ). Toward a theory of symbol use in mathematics. In C. Janvier ( Ed. ). Problems of representation in the teaching and learning of mathematics, 159-196. Hillsdale, NJ: Erlbaum.
Kilpatrick, J.(1985).A restropective account of the past 25 years of research and Learning mathematical problem solving.In E.Silver(ED.) Teaching and learning Mathematical problem solving:Multiple research perspectives. Hillsdale, NJ:Lawrece Erlbaum Associates.
Kintsch, W., & Greeno, J.G.(1985).Understanding and solving word arithmetic word problems. Psychological Review, 92, 109-129.
Krutetskii,V.A.(1976).The psychology of mathematical abilities in school children. Chicago:University of Chicago Press.
Larkin, J. H. & Simon, H. A. ( 1987 ). Why a diagram is ( sometimes ) worth ten thousand words. Education Psychology, 12 , 65-99.
Lave,J. （1992）.Word problem:A microcosm of theories of learning.In P.
Lesh, R., Landau, M. & Hamilton, E. (1983). Conceptual Models and Applied Mathematical Problem-Solving Research. In R. Lesh & M. Landau (Eds.), Acquisition of Mathematics Concepts & Processes (pp. 263-343). NY: Academic Press.
Lesh, R., Post, T., & Behr, M. (1987). Representation and translation among representation in mathematics learning and problem solving.In C.Janvier(Ed).Problems of representation in the teaching and learning of mathematics,33-40.Hillsdale, NJ: Erlbaum.
Lester, F. K. (1980). Problem solving: Is it a problem? In M. M. Lindquist
（Ed.）, Selected issues in mathematics education,（pp. 29-45）. Berkeley Calif.: McCutchan.
Leung, S. S. (1991). Building connections to the Pythagorean Theorem:An example of teachers treatment of textbook problem. Mathematics Education: Making connections, 41-47.
Lewis, A. B. ( 1989 ). Training students to represent arithmetic word problem. Journal of Educational Psychology, 81, 521-531.
Lewis, A. B., &Mayr, R. E.（1987）.Students miscomprehension of relational statement arithmetic word problems. Journal of education psychology, 79, 361-371.
Marshall, S. P (1987). Schema knowledge structure for representing and understanding arithmetic story problems. First year report, San Diego State University, California, Department of Psychology.
Marshall, S. P., Pribe, C. A., & Smith, J. D. (1987).Schema knowledge structure for representing and understanding arithmetic story problems.(Tech. Rep. Contract No. N00014-85-K-0661). Arilington, VA: Office of Naval Research.
Mayer R.E. ( 1985 ) Implication of Cognitive Phychology for Instruction in Mathematical problem solving. In E.A.Sliver (Ed), Teaching and Learning Mathematical Problem Solving. Hillsdale, NJ：Lawrence Erlbaum Associates.
Mayer, R. E. (1992). Thinking, problem solving, cognition. New York: W.H. Freeman and Company Press.
Mayer, R. E. (1987). Educational Psychology：A cognitive approach. Boston: Little, Brown, and Company.
Moyer, J. C., Sowder, L., Threadgill-Sowder, J., & Moyer, M. B. ( 1984 ). Story
problem formats: draw versus verbal versus telegraphic. Journal for Research in Mathematics Education, 15 ( 5 ), 342-351.
National Council of Supervisors of Mathematics. (1977). Position paper
on basic mathematical skills. Arithmetic Teacher, 25, 19-22.
National Council of Teachers of Mathematics. (1980).Problem solving be the focus of school mathematics in the 1980s. An agenda for action.Palo Alto,Calif. : Dale Seymour Publications.
National Council of Teachers of Mathematics. (1989).Curriculum and evaluation standards for school mathematics. Reston, VA: Author
National Council of Teachers of Mathematics (NCTM) (2000). Principles and standards for school mathematics. Retrieved November 20, 2003 form
http://standards.nctm.org/document/prepost/cover.htm.
Paivio, A.（1991）.Dual coding theory and education.Education Media,24(4),333-339.
Polya, G. (1945). How to solve it. Princeton, New Jersey: Princeton
University Press.
Schoenfeld, A. H. (1985). Mathematical problem solving. New York:
Academic Press.
Sellke, D. H., Behr, M. J., & Voelker, A. M. ( 1991 ). Using data tables to represent and solve multiplicative story problems. Journal for Research in Mathematics Education, 22, 30-38.
Sowder, J. T., & Sowder, L. (1982). Drawn versus verbal formats for mathematical story problems. Journal for Research in Mathematics Education, 13(5), 324-331.
van den Heuvel-Panhuizen, M. (1996). Assessment and realistic mathematics education. Utrecht, The Netherlands: CD-B Press, Freudenthal Institute.
Vasquez, V. M. (1982). Algebra word problems:Exploring high school students conception Through their solution strategis. (Doctoral dissertation, The University of Microfilms)
Webb, L. F., & Sherrill, J. M. ( 1974 ). The effects of differing presentations of
mathematical word problems upon the achievement of preserve elementary teachers. School Science and Mathematics, 74, 559-565.