本研究目的是於小五學生數學教室實施建模活動，教師在設計與實施建模活動時，所遭遇的困難及解決方式，以及實施建模活動後，對國小五年級學生在建模能力和問題解決能力的影響。
本研究採取行動研究法，分為兩個階段實施，第一個階段以每週兩節課總共十週的觀察時間，進行六個建模活動。第二個階段以國小四年級學生為對象，用以改良活動設計。透過建模活動教學設計的進行，以及同儕之間與教師互動的情境下，將學生自己的解題想法外顯化。本研究資料收集的部分包括：學習單、教師教學日誌、教師和學生晤談紀錄，輔以錄影錄音檔資料，再就收集的資料進行分析。
研究者實施建模活動的過程中，遇到下列幾項困難：活動內容不夠真實、學生在學習單上只求寫出做法、聽不清楚其他組別的解題策略、學生無法提供回饋。在學生的質性資料呈現上，本研究有以下三點發現：一、學生所展現的建模能力有：簡化任務、闡明目標、公式化問題、分配變數參數和常數、系統論述、選擇模型、解釋立場與連結回真實情境；二、學生在建模學習過程中，由原先大量收尋數字進行算式的運算，到漸漸使用推理猜測與文字表達等能力解決數字與文字的問題；三、學生解題特徵中，解題策略由單一答案逐漸增加至能提供多個解題策略，以及思考方式由天馬行空至能與數學連結。

關鍵詞： 建模活動、問題解決、教學設計

Using modeling activities, this study is to observe and record the effects on problem- -solving abilities of non-routine problems in 5th-grade students mathematical classrooms. Difficulties that confront the teacher when engaged in the design and implementation of modeling activities, together with the ways to overcome these difficulties, are also investigated in this study. This is a ten-week long action research, consisted of 6 model- -ing activities. Through modeling activities instruction and interaction between other students and the teacher, students wrote down their ideas about problem-solving. The qualitative data included students’ learning sheets, teacher’s journal, interview records, and classroom observations which were video-taped. The following results were obtained:
Throughout their learning processes in modeling, there are some difficulties encoun- -tered: The contents of activity are not close to real life. The students only want to write down the action list on the worksheet, but cannot get clear picture of problem solving strategy from other teams. They are unable to provide feedback. In addition, there are some findings in these qualitative information: First, students’ modeling abilities included stages ranging from making simplifying assumptions, clarifying the goal, formulating the problem, assigning variables, parameters and constants, formulating mathematical statements, selecting a model, interpreting graphical representations, relating back to the real situation. Second, throughout their learning processes in modeling, students moved from looking for numbers in order to use formula for computation in the beginning, toward gradually using abilities such as reasoning speculation and verbal expressions to solve numerical and verbal problems. Third, students’ problem-solving style gradually changes from providing a single solution to offering multiple problem-solving strategies. Forth, students’ thinking methods are from un-logical thoughts to relating with mathematical knowledge.

目 錄
第一章 緒論 1
第一節 研究背景與動機 1
第二節 研究目的與待答問題 4
第三節 名詞解釋 4
第四節 研究限制 5
第二章 文獻探討 7
第一節 建模和建模能力 7
第二節 建模教學實徵研究 24
第三節 問題解決能力 30
第三章 研究方法 37
第一節 研究設計與流程 37
第二節 研究對象 43
第三節 研究工具 44
第四節 資料分析與處理 46
第四章 研究結果 48
第一節 實施建模活動遭遇的困難及解決的方式 48
第二節 小五學生參與建模活動時展現的建模能力 69
第三節 小五學生參與建模活動時展現的解題特徵 89
第五章 結論與建議 99
第一節 結論 99
第二節 建議 101
參考文獻 103




附 錄 次
附錄一 建模教學設計活動一 誰是好心人 109
附錄二 建模教學設計活動二 開心農場 112
附錄三 建模教學設計活動三 樂樂棒球 116
附錄四 建模教學設計活動三 樂樂棒球評量單 120
附錄五 建模教學設計活動四 跳蚤市場 122
附錄六 建模教學設計活動四 跳蚤市場評量單 126
附錄七 建模教學設計活動五 路線規劃 127
附錄八 建模教學設計活動五 路線規劃評量單 130
附錄九 建模教學設計活動六 百分率 131
附錄十 建模教學設計活動六 百分率評量單 134
附錄十一 建模活動教學設計 135
附錄十二 數學能力指標與十大基本能力的關係 145


表 次
表2-1-1 建模教學趨勢 13
表2-1-2 Penrose建模能力 19
表2-1-3 Hall建模能力 20
表2-2-1建模教學相關實徵性研究 28
表2-3-1傳統和建模觀點在問題解決上的差異 31
表3-3-1教學流程 45
表3-4-1 資料編碼介紹表 47
表3-4-2 待答問題與資料來源對照表 47
表4-1-1路線規劃活動遊樂設施名稱 55
表4-1-2誰是好心人分組預測身高 58
表4-1-3跳蚤市場任務學生特色摘錄 60


圖 次
圖2-1-1 模型世界與真實世界的關係 11
圖2-1-2 模式發展序列 15
圖2-1-3 建模的循環圖 16
圖2-1-4 Burghes和Wood建模七步驟 18
圖3-1-1 研究架構 37
圖3-1-2 教學實驗實施步驟 38
圖3-1-3 研究流程圖 40
圖3-1-4 研究甘梯圖 41
圖3-1-5 腳印圖 42
圖3-1-6 建模活動實施流程圖 43
圖4-1-1 建模教學活動實施解決策略圖 49
圖4-1-2 第二階段學生上台發表小白板 67
圖4-2-1 路線規劃第二組小白板 74
圖4-2-2 路線規劃第四組小白板 75

參考文獻
中文部分
王明慧、柳賢和洪振方（2006）。從數學解題探討數學建模的個案研究。數學與科學教育E-Journal 電子期刊。2010/12/31，取自：
http://140.127.36.251/e-journal/e-journal-95-2.htm
林靜雯和邱美虹（2008）。從認知／方法論之向度初探高中學生模型及建模歷程之知識。科學教育月刊，307，9-14。
胡政德和左台益（2006）。準教師數學建模的個案研究。中華民國第二十二屆科學教育學術研討會，未出版，台北市：國立台灣師範大學。
唐晤容（2009）。數學建模教學對國小六年級學生解題能力提升及數學學習態度改變影響之研究。國立屏東教育大學數理教育碩士論文，未出版，屏東縣。
教育部（2000）。國民中小學九年一貫數學學習領域暫行綱要。台北：教育部。
教育部（2008）。國民中小學九年一貫課程正式綱要。台北：教育部。
楊凱琳（2003）。九年一貫數學能力指標、連結主題之學習與教學的實踐--數學建模。最近瀏覽日期為：2009/04/28，取自： http://www.math.ntnu.edu.tw/~cyc/_private/mathedu/me9/nineyear/
楊凱琳和林福來（2006）。探討高中數學教學融入建模活動的支撐策略及促進參與教師反思的潛在機制。科學教育學刊，14，517-543。
劉錫麒（1993）。合作反省思考的數學解題教學模式及其實徵研究。教育研究資訊雙月刊，1(5)，16-25。


英文部分
Ai, X. (2002). Gender differences in growth in mathematics achievement: Three-level longitudinal and multilevel analyses of individual, home, and school influences. Mathematical Thinking and Learning, 4, 1-22.
Alsina, C. (2007). Less chalk, less words, less symbols, more objects, more context, more actions. In W. Blum, P. L.Galbraith, H. W. Henn, & M. Niss. (Ed.). Modeling and applications in mathematics education(pp. 129- 136 ). New York: Springer
Arzarello, F., Pezzi, G., & Robutti, O. (2007). Modelling body motion: an approach to functions using measuring instruments.In W. Blum, P. L.Galbraith, H. W. Henn, & M. Niss. (Ed.). Modeling and applications in mathematics education(pp. 129- 136 ). New York: Springer
Bock, D., Verschaffel, L., & Janssens, D. (2002). The effects of different problem presentations and formulations on the illusion of linearity in secondary school students. Mathematical Thinking and Learning, 4, 65-89.
Bonotto, C. (2007). How to replace word problems with actives of realistic mathe- -matical modeling. In W. Blum, P. L.Galbraith, H. W. Henn, & M. Niss. (Ed.). Modeling and applications in mathematics education(pp.185-192). New York: Springer.
Burghes, D., & Wood, A. D. (1980). Mathematival models in the social, management, and life sciences. New York: Wiley.
Confrey, J., & Maloney, A.(2007). A theory of mathematical modeling in technological settings. In W. Blum, P. L.Galbraith, H. W. Henn, & M. Niss. (Ed.). Modeling and applications in mathematics education(pp.57-68 ). New York: Springer
Doerr, H. M.(2007). What knowledge do teachers need for teaching mathematics th- -rough applications and modeling. In W. Blum, P. L.Galbraith, H. W. Henn, & M. Niss. (Ed.). Modeling and applications in mathematics education(pp.69-78 ). New York: Springer
Elliott, J. (1991). Action research for educational change. Bristoln: Open University
press.
Gagne, E. D. (1985). The cognitive psychology of school learning. Boston, MA: Little, Brown and company.
Gainsburg, J. (2006). The mathematical modeling of structural engineers. Mathema- -tical Thinking and Learning, 8, 3-36.
Galbraith, P.(2007). Beyond the low hanging fruit. In W. Blum, P. L.Galbraith, H. W. Henn, & M. Niss. (Ed.). Modeling and applications in mathematics education(pp.79-88). New York: Springer
Gilbert, S. W. (1991). Model building and definition of science. Journal of Research in Science Teaching, 28, 73-79.
Gravenejer, K.(2007). Emergent modeling as a precursor to mathematical modeling. In W. Blum, P. L.Galbraith, H. W. Henn, & M. Niss. (Ed.). Modeling and app- -lications in mathematics education(pp.137-144 ). New York: Springer
Haines, C. R., Crouch, R. M, & Davis, J. (2000). Mathematical modelling skills: A research instrument. Hatfield: Unversity of hertfordshire, department of mathematics technical report no.55.
Hall, G..(1984). The assessment of modeling projects. In J.S.Berry, D.N. Burghes, I.D.modelling(pp.134-148).Chichester: Ellis Horwood.
Halloun, I. (1996). Schematic modeling for meaningful learning of physics. Journal of Research in Science Teaching, 33, 1019-1047.
Houston, K., Galbraith, P., & Kaiser, G. (2009). History of ICMI. Retrieved November 19, 2009, from http://www.icmihistory.unito.it/ictma.php
Ikeda, T. (1997). A case study of instruction and assessment in mathematical modeling the delivering problem. Teaching and Learning Mathematical modeling (pp. 51-62). Houston, TX: Albion Pub.
Jablonka, C. (2007). The relevance of modeling and applications: relevant to whom and for what purpose. In W. Blum, P. L.Galbraith, H. W. Henn, & M. Niss. (Ed.). Modeling and applications in mathematics education.(pp.193-200). New York: Springer.
Lesh, R., Cramer, K., Doerr, H. M., Post, T., & Zawojewski, J. S. (2003). Model development sequences. In R. Lesh, & H. M. Doerr.(Ed). Beyond constructivism: Models and modeling perspectives on mathematics problem solving, learning, and teaching (pp. 35-58). Mahwah, NJ: Lawrence Erlbaum Associates.
Lesh, R., & Doerr, H. M. (2003). Foundations of models and modeling perspective on mathematics teaching, learning, and problem solving. In R. Lesh, & H. M. Doerr.(Ed). Beyond constructivism: Models and modeling perspectives on mathematics problem solving, learning, and teaching (pp. 3-33). Mahwah, NJ: Lawrence Erlbaum Associates.
Lesh, R., & Lehrer, R. (2003). Models and modeling perspectives on the development of students and teachers. Mathematical Thinking and Learning, 5, 109-129.
Lesh, R., Middleton, J. A., Caylor, E., & Gupta, S. (2008). Designing tasks to engage students in modling complex data. Educational studies in mathematics, 68, 113 -130.
Mayer. R. E. (1992). Thinking, problem solving, cognition. NY: W. H. Freeman and company.
Montague, M. (1992). The effects of cognitive and meta-cognitive strategy instruction on mathematical problem solving of middle school students with learning disabilities. Journal of Learning Disabilities, 25, 230-248.
Mullis, I.V.S., Martin, M.O., & Foy, P. (2008). TIMSS 2007 International mathe- -matics report: findings from IEA’s trends in international mathematics and science study at the fourth and eighth grades Chestnut Hill, MA: TIMSS & PIRLS International Study Center, Boston College. Retrieved December 2,2009, from http://timss.bc.edu/TIMSS2007/mathreport.html
National Council of Teacher of Mathematics. (2000).Principles and standards for school mathematics. Reston, VA: National Council of Teachers of Mathematics.
Organisation for Economic Co-operation and Development. (2003). Test questions – PISA2003. Retrieved November 19, 2009, from http://www.pisa.oecd.org/docu- -ment/38/0,3343,en_32252351_32236173_34993126_1_1_1_1,00.html
Palincsar, A.S. & Brown, A. L. (1984). Reciprocal teaching of comprehension- -fostering and comprehension-monitoring activities. Cognition and Instruction, 1(2), 117-175.
Palm, T. (2007). Features and impact of the authenticity of applied mathematical school tasks. In W. Blum, P. L.Galbraith, H. W. Henn, & M. Niss. (Ed.). Modeling and applications in mathematics education(pp.201-208). New York: Springer.
Penrose, O.(1978). How can we teach mathematival modeling. Journal of mathe- -matical modeling for teachers1(2), 31-42.
Piaget, J.(1969). The child’ s conception of time. London: Routledge & Kegan Paul.
Polya, G. (1945). How to solve it. Princeton, New Jersey: Princeton University Press.
Steen, L. A. & Turner, R. (2007). Developing mathematical literacy. In Blum, W., Galbraith, P. L., Henn, H. W., & Niss,. M.(Ed.). Modeling and applications in mathematics education(pp.285-294). New York: Springer.
Strasser, R. (2007). Everyday instruments: on the use of mathematics.In Blum, W., Galbraith, P. L., Henn, H. W., & Niss,. M.(Ed.). Modeling and applications in mathematics education(pp.171-178 ). New York: Springer
Van Driel, J. H., & Verloop, N. (1999). Teachers’ knowledge of models and model- -ing in science. International Journal of Science in Education, 21, 1141-1153.
Verschaffel, L., De Corte, E., Lasure, S., Van Vaerenbergh, G., Bogaerts, H., & Ratinckx, E. (1999). Learning to solve mathematical application problems: A design experiment with fifth graders. Mathematical Thinking and Learning, 1, 195-229.
von Glasersfeld, E. (1995). Radical constructivism: A way of knowing and learning. Washington, DC: The Falmer Press.
Vygotsky, L. S. (1978). Mind in social: The development of higher psychological processes. Cambridge, MA: Harvard University Press.