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。

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.

一、中文部份
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。
古明峰(民87)：數學應用題的解題認知歷程之探討。教育研究資訊, 6(3), 63-77。
古明峰(民88)：加減法文字題語意結構、問題難度及解題關係之探討。新竹師院學報, 12, 1-25。
吳坤銓(民86)：國小學生認知能力、問題解決能力與創造傾向之相關研究。高雄市：國立高雄師範大學教育學系碩士論文。
邱上貞(民78)：初探解題歷程的理論教學。特殊教育季刊, 31, 1-4。
邱美虹、劉嘉茹、周金城、劉家祺(民88)：認知師徒制對學生概念改變的影響。第十五屆科學教育學術研討會暨第十二屆科學教育學會年會研討會手冊。彰化市：國立彰化師範大學。
邱貴發(民85)：情境學習理念與電腦輔助學習─學習社群理念探討。台北市：師大書苑。
李美瑜(民83)：情境學習與去情境學習環境對國二學生物理壓力概念學習成效影響之研究。台北縣：私立淡江大學教育資料科學研究所碩士論文。
林碧珍(民78)：國小學生數學解題的表現及其相關因素的研究。台北市：國立臺灣師範大學數學研究所碩士論文。
林麗娟(民86)：情境學習與動機。視聽教育雙月刊, 38(4), 18-27。
紀惠英(民80)：國小六年級學生數學應用問題表徵類型與同構性之研究。台北市：國立臺灣師範大學特殊教育研究所碩士論文。
施郁芬、陳如琇(民85)：情境脈絡與學習遷移。教學科技與媒體, 29, 23-31。
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。
徐新逸(民84a)：如何借重電腦科技來提昇問題解決的能力？─談「錨式情境教學法」之理論基礎與實例應用(上)。教學科技與媒體, 20, 25-30。
徐新逸(民84b)：如何借重電腦科技來提昇問題解決的能力？─談「錨式情境教學法」之理論基礎與實例應用(下)。教學科技與媒體, 21, 47-51。
徐新逸(民85)：情境學習在數學教育上之應用。教學科技與媒體, 29, 13-22。
徐新逸(民87)：情境學習對教學革新之回應。研習資訊, 15(1), 16-24。
唐淑華(民84)：語文理解能力對解答數學應用題能力之實驗研究。八十四年度師範學院教育學術論文發表會。屏東市：國立屏東師範學院。
曹宗萍(民77)：高屏地區國小兒童四則問題的解題過程表現及其相關因素之研究。屏東師院學報, 1,51-116。
張春興(民83)：心理學。台北市：東華。
張景媛(民83)：國中生數學學習歷程統整模式的驗證及應用：學生建構數學概念的分析及數學文字題教學策略的研究。台北市：國立臺灣師範大學教育心理與輔導研究所博士論文。
許家驊(民88)：數學認知監控與改變型數學文字題錯誤偵測作業在促進國小低年級學生數學解題監控能力上之應用。高雄市政府公教人力資源發展中心編，新典範數學(pp.135-183)。高雄市：高雄市政府公教人力資源發展中心。
教育部(民82)：國民小學課程標準。
教育部(民83)：國民中學課程標準。
陳其英(民88)：國一數學資優生問題同構轉化能力及外在表徵對解題影響之研究。台北市：國立臺灣師範大學科學教育研究所碩士論文。
陳國泰(民86)：錨式情境教學法的理論架構與應用。教育資料文摘, 40(3), 146-158。
陳龍安(民77)：數學動動腦─數學創造思考教學研究。台北市：心理。
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。
陳慧娟(民87)：情境學習理論的理想與現實。教育資料與研究, 25, 47-55。
黃敏晃譯(民74)：數學解題。國教月刊，32(7.8)，40-52。
黃敏晃(民87)：數學年夜飯。台北市：心理。
詹秀美(民78)：問題解決能力的訓練與評量。資優教育季刊, 32, 13-16。
楊家興(民84)：情境教學理論與超媒體學習環境。教學科技與媒體, 22, 40-48。
楊榮祥(民81)：1992國際數理教育評鑑IAEP─我們能做什麼？科學教育月刊, 149, 2-31。
熊召弟(民85)：真實的科學認知環境。教學科技與媒體, 29, 3-12。
鄭晉昌(民82)：電腦輔助學習的新教學設計觀─認知學徒制。教育資料與圖書館學, 31(1), 55-66。
鄭麗玉(民82)：認知心理學。台北市：五南。
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。
鍾邦友(民83)：情境式電腦輔助數學學習軟體製作研究。台北市：國立臺灣師範大學教育研究所碩士論文。
羅浩原(民86)：生活的數學。台北市：九章。
譚寧君(民81)：兒童數學態度與解題能力之分析探討。台北師院學報, 5, 619-688。
Polya著(1957)，閻育蘇譯(民80)：怎樣解題。台北市：九章。
Polya著(1962)，九章編輯部編譯(民78)：數學發現(1、2冊)。台北市：九章。
二、英文部份
Anand, P. G., & Ross, S. M. (1987).Using computer-assisteded instruction to personalize arithmetic materials for elementary school children. Journal of Educational Psychology, 79(1), 72-78.
Anderson, J. R., Reder, L. M., & Simon, H. A. (1996). Situated learning and education. Educational Researcher, 25(4), 5-11.
Baranes, R., Perry, M., & Stigler, J. W. (1989). Activation of real-world knowledge in the solution of word problems. Cognition and Instruction, 6, 247-318.
Barnett, J. (1984) The study of syntex variables. In G. A. Goldin & C. E. McClintock (Eds.), Task variables in mathematical Problem solving (pp.23-68). Philadephia: The Franklin Institute Press.
Brown, J.S., Collins, A., & Duguid, P. (1989). Situated cognition and the culture of learning. Educational Researcher, 18(1),32-42.
Brown, J. S., Collins, A., & Newman, S. E. (1989). Cognitive apprenticeship: Teaching the craft of reading, reading, and mathematics. In Resnick(Ed.), Cognition and instruction: Issues and agedas. Hillsdale, NJ: Lawrence Erlbaum.
Burke, G., & Mclellan, H. (1996).The algebra project: Situated learning inspired by the civil rights movement. In H. Mclellan (Ed.), Situated Learning Perspectives(pp.263-278). Engleword Cliffs, NJ: Educational Technology Publications.
Charles, R. I., & Lester, F. K. (1982). Problem solving: What , why, and how. Palo Alto, Calif: Dale Seymour publications.
Choi, J., & Hannafin, M., (1995). Situated cognition and learning environments: roles, structures, and implications for design. Educational Technology Research and Development, 43(2), 53-69.
Cognition and Technology Group at Vanderbilt (1990). Anchored instruction and its relationship to situated cognition, Educational Researcher, 19, 2-10.
Cognition and Technology Group at Vanderbilt (1992).The Jasper series as an example of anchored instruction: Theory, program description, and assessment data. Educational Psychologist, 27(3), 291-315.
Collins, K. F. Romberg, T. A., & Jurdak, M. E. (1986). A technique for assessing mathematical problem-solving ability. Journal for Research in Mathematics Education, 17(3), 206-221.
Cummins, D. D. (1991).Children's interpretations of arithmetic word problems. Cognition and Instruction, 8, 261-289.
DeFranco, T. C., & Curcio, F. R. (1997). A division problem with a remainder embedded across two contests: children's solution in restrictive vs. real-world settings. Focus on Learning Problems in Mathematics, 19(2), 58-72.
Dewey, J. (1910). How we think. Boston: Heath.
Eisenberg, T. (1991). In D. Tall(Ed.), Advanced Marhematical Thinking (pp.14-152). Norwell, MA: Kluwer Academic.
Enright & Beattle,(1989). Problem solving step by step in math. Teaching Exceptional Children, 22, 1-5.
Fawcett, H. P.(1938). The nature of proof. NY: Teachers College, Columbia University.
Fuys, D., Geddes, D., & Tischler, R. (1988). The van Hiele model of thinking in geometry among adolescents. NY: Brooklyn College.
Glass, A. L., & Holyoak, K. J. (1986). Cognition. NY: Random House.
Greeno, J. G. (1978). A study of problem solving. In R. Glaser (Ed.), Advances instructional psychology, Vol. 1. Hillsdale, NJ: Erlbaum.
Griffin, M. M. (1995). You can't get there from here: Situated learning, transfer, and map skills. Contemporary Educational Psychology, 20, 65-87.
Griffin, M. M., & &Griffin, B. W. (1996). Situated cognition and cognitive style: Effects on student's learning as measured by conventional tests and performance assessments. The Journal of Experimental Education, 64(4), 293-308.
Hayes, J. R. (1989). The complete problem solver(2nd ed.). Hillsdale, NJ: Erlbaum.
Hibert, J., Carpenter, T. P., Fennema, E., Fuson, K., Human, P., Murray, H., Olivier, A., & Wearne, D. (1996). Problem solving as a basis for reform in curriculum and instruction: The Case of Mathematics. Educational Researcher, 25(4), 12-21.
Hoffer, A. (1981). Geometry is more than proof. Mathematics Teacher, 74, 11-18.
Keller, B. A. & Hirsch, C. R. (1998). Student preferences for representations of functions. INT. J. MATH. EDUC. SCI. TECHNOL., 29(1), 1-17.
Kilpatrick, J. (1985). A restrospective account of the past 25years of research and teaching mathematical problem solving. In E. Silver(Ed.), Teaching and learning mathematical problem solving: Multiple research perspective. LEA, Hillsdale, New Jersey.
Krulik, S., & Rudnick, J. A. (1987). Problem solving : A handbook for teachers. Boston, London, Sydney, Toronto : Allyn & Bacon.
Lankard, B. A. (1995). New ways of learning in the workplace. ERIC Reproduction Service: NO. ED385778.
Lave, J., Murtaugh, M., & Rocha, O. (1984). The dialectic of arithmetic in grocery shopping. In B. Rogoff & J. Lave (Eds.), Everyday cognition: Its development in social context (pp.67-94). Cambridge, MA: Harvard University Press.
Lave, J. (1988). Cognition in practice. Cambridge, England : Cambridge University Press.
Lave, J., & Wenger, E. (1991). Situated learning: Legitimate peripheral participation. Cambridge.
Marshall, S. P. (1995). Schemas in problem solving. Australia: Cambridge University Press.
Marshall, S. P., Pribe, C. A., & Smith, J. D. (1987). Schema knowledge structure for representing and understanding arithmetic story problems. Arilington, VA: Office of Naval Research.
Mayer, R. E. (1992). Thinking, problem solving, cognition. NY : W. H. Freeman and Company.
Mcalloon, A., & Robinson, G. E. (1988). How do you evaluate problem solving? Arithmetic teacher, 35, 44-91.
Merrill, M.D.(1991). Constructivism and instructional design. Educational Technology, 31(5), 45-53.
Miller, G. A., & Gildea, P. M. (1987). How children learn words. Scientific American, 257(3), 94-99.
Muth, K. D. (1991). Effects of cueing on milddle-school students' performance on arithmetic word problems containing extraneous information. Journal of Educational Psychology, 83(1), 173-174.
National Council of Teachers of Mathematics [NCTM](1989). Curriculum and evaluation STANDARDS for school mathematics. Reston, Virginia. NCTM.
Nesher, P., & Hershkovitz, S. (1994). The role of schemes in teo-step problems: Analysis and research findings. Educational Studies in Mathematics, 26, 1-23.
Noddings, N. (1983). Small groups as a setting for research on mathematical problem solving. In E. A. Silver(Ed.), Teaching and learning mathematical problem solving: Multiple research perspectives(pp.345-359). Hillsdale, NJ, London: Lawrence Erlbaum Associates.
Nunes, T., Schlieman, A. D. & Carraher, D. W., (1993). Street mathematics and school mathematics. Cambridge, MA: Cambridge University Press.
Owen, E., & Sweller, J.(1985). What do students learn while solving mathematical problems? Journal of Education Psychology, 77, 272-284.
Resnick, L. B. (1989). Teaching mathematics as an ill-structured discipline. In R. I. Charles, & E. A. Silver(Eds.), The teaching and assessing of mathematical problem solving(pp.32-60). Lawrence Erlbaum Associates.
Riley, M. S., Greeno, J. G., & Heller, J. I. (1983). Development of children's problem solving ability in arithmetic. In H. P. Ginberg (Ed.), The development of mathematical thinking (pp.153-196). Orlando, FL: Academic.
Rogoff, B. (1984).Introduction: Thinking and learning in social context. In B. Rogoff & J. Lave (Eds.), Everyday cognition: Its development in social context (pp. 1-8). Cambridge, MA: Harvard University Press.
Ross, S. M. (1984). Matching the lesson to the student: Alternative adaptive designs for individualized learning systems. Journal of Computer-Based Instruction, 11(2), 42-48.
Schoenfeld, A. H. (1979). Explicit heuristic training as a variable in problem-solving performances. Journal for Research in Mathematics Education, 10, 173-187.
Schoenfeld, A. H. (1985). Mathematical problem solving. Orlando. FL: Academic Press.
Scribner, S. (1984). Studying working intelligence. In B. Rogoff & J. Lave (Eds.), Everyday cognition: Its development in social context (pp. 9-40). Cambridge, MA: Harvard University Press.
Solso, R. L. (1995). Cognitive psychology(4rd ed.). Boston, London, Toronto, Sydney, Tokyo, Singapore : Allyn & Bacon.
Suchman, L. (1987). Plans and situated actions: The problem of human-machine communication. Cambridge: Cambridge University Press.
Verschaffel, L., Corte, E. D., & Lasure S. (1994). Realistic considerations in mathematical modeling of school arithmetic word problems. Learning and Instruction, 4, 273-294.
Whitehead, A.N. (1929). The aims of education. New York: Mac-Millan.
Wyndhamn, J., & Saljo, R. (1997). Word problems and mathematical reasoning ─A study of children's mastery of reference and meaning in textual realities. Learning and Instruction, 7(4), 361-382.
Yancy, A. V. (1981). Pupil generated digrams as strategy for solving word problems for elementary mathematics. Specialist in Education Degree Thesis, Univerity of Louisvile.
Yoshida, H., Verschaffel, L., & De Corte, E. (1997). Realistic considerations in solving problematic word problems: Do Japanese and Belgian children have The same difficulties? Learning and Instruction, 7(4), 329-338.
Young, M. F. (1993). Instructional design for situated learning. Educational Technology Research and Development, 41(1), 43-58.