摘要

應用電腦化類比解題策略輔助國小學生數學學習

林國賓

本研究嘗試以電腦化類比解題策略輔助國小學生學習數學文字題。研究首先發展應用類比解題策略的數學輔助學習系統，接著由國小學生實際使用以評估其成效。學習系統的內容包括國小六年級的「公因數、公倍數」、「分數的除法」、「以符號代表數」、「等式與數量關係」及「速率」等單元。
學習系統是以類比策略循序漸近的協助學生解題，類比策略的使用依序分為表徵、回想、映對、及應用等階段。「表徵階段」要求學生指出已知數及未知數以確認題意，「回想階段」提示學生以問題表徵（已知數及未知數）為線索回想類似題，「映對一階段」提供學生已做過的類似題題目幫助學生思考解法，「映對二階段」進一步提供欲解問題與類似題問題表徵的比對，最後的「應用階段」則直接給予學生類似題解法。希望學生經由系統逐步的協助後終能解題，並能習得類比解題技巧。
系統實際使用評估是以33位國小六年級學生為對象，學生共使用系統進行六週十二次的解題練習。綜合分析學生解題記錄、學生問卷、教師問卷、與課堂觀察紀錄得到以下結論：（1）系統在提供類似題題目（映對一）階段對學生的學習最有幫助，（2）學生對系統的使用持正面的態度，（3）系統在使用介面、回想策略、及類似題庫建立等方面仍待改善。

關鍵詞：類比解題、數學文字題、電腦輔助學習

Abstract
The Use of Computerized Analogical Problem Solving Strategy
to Facilitate Elementary School Students Learning Mathematics Word Problems
By
Kuo-Pin Lin
This study tried to use computerized analogical problem solving strategy to help elementary school students learn mathematics word problems. The study began with developing a computer-assisted instruction system, then a field-test was implemented to assess the effect of this system. The contents of the learning system include: ‘common factor and common multiple’, ‘division of fractional number’, ‘representing the number by symbol’, ‘equation and number relation’, and ‘speed’.
This learning system uses analogical strategy to gradually help students solve problems. The strategy in sequence is representation, retrieval, mapping and using stage. ‘The representation stage’ requires students to point out known numbers and unknown numbers in order to confirm the meaning of the question. ‘The retrieval stage’ reminds students to recall the similar problem by the clue of the question representation (known numbers and unknown numbers). ‘The mapping I stage’ offers the similar question that students had done before to help students to ponder over the solution. ‘The mapping II stage’ compares the representation of the question to be solved with the representation of the similar question. ‘The using stage’ offers the solution to the similar problem directly. We hope students can solve problems and learn the skill of analogical problems solving through the progressive aid of the system.
Thirty-three sixth-grade elementary school students participated in the 12 problems solving exercises during six weeks. We get the following conclusions by synthetical analysis of students’ solving problems records, students’ questionnaire, teacher’s questionnaire, and the classroom observation: (1) ‘The mapping I stage’ offers the similar question that students had done before is the most helpful to students’ learning, (2) students showed positive attitude toward the use of computer-assisted system, and (3) the interface, retrieval strategy, and the analogical questions database of the system need to be improved in the future.

Keywords: analogical problem solving, mathematics word problems, computer-assisted instruction.

吳正己、龐能一（1995）：類比在高中職計算機概論教科書的應用分析。師大學報，40期，157-186。
李坤金（2000）：以鷹架輔助坡里亞式思考方式之進行-使用者介面研究。中原大學資訊工程研究所碩士論文。
林文生、鄔瑞香（1999）：數學教育的藝術與實務 : 另類教與學。台北市 : 心理。
林秀鳳（2001）：解題歷程導向之電腦輔助數學應用題解題系統。國立臺灣師範大學資訊教育研究所碩士論文。
林美娟、莊志洋、孫鵬宗、許金葉、黃瀧輝、盧思靜、黃凱澤（2003）：一個數學解題練習系統之設計理念。科學教育學，第十一卷第一期，27-50。
林敏慧（2000）：學習層次和教學策略對國小輕度智障兒童學習不同類型加法應用題成效之影響。國立臺灣師範大學特殊教育研究所碩士論文。
林碧珍（1990）：新竹師院輔導區國小數學科「怎樣解題」教材實施情況調查與學習成效研究。新竹師院學報，3，363-391。
邱美虹、高淑芬（1999）：類比對應對學生建構〝原子結構〞心智表徵之影響。師大學報，第44卷，第1&2期，第31-59頁。
徐順益（1996）：類比解題策略之研究：國中物理力學（II）。行政院國家科學委員會專題研究計畫成果報告（報告編號：NSC 87-2511-S-018-007）。
徐順益（1998）：類比教學對啟動類比解題策略之研究：力學（I）。行政院國家科學委員會專題研究計畫成果報告（報告編號：NSC 87-2511-S-018-014）。
徐順益（1999）：表面表徵與深層表徵相似性的連接關係，對於國中類比學習與解題的影響。行政院國家科學委員會專題研究計畫成果報告（報告編號：NSC 88-2511-S-018-033）。
徐順益（2000）：分析合作學習式的類比學習環並發展物理科教材教授國中二年級學生。行政院國家科學委員會專題研究計畫成果報告（報告編號：NSC 89-2511-S-018-017）。
徐雍智、蔡今中、陳明璋（2002）：數學創意類比與同儕評量及其網路案例設計之初探。師大學報，第47卷，第1期，第1-14頁。
教育部（2003）：教育統計指標之國際比較。上網日期：2005年7月7日。網址：http://www.edu.tw/EDU_WEB/EDU_MGT/STATISTICS/EDU7220001/user3/international.htm
教育部（2004）：國際數學與科學教育成就趨勢調查結果。上網日期：2005年7月7日。網址：http://www5.www.gov.tw/PUBLIC/view.php3?id=93336&sub=63&main=GOVNEWS
陳柏華（1998）：Polya I的導引核心與知識庫：理念與實作。中原大學資訊工程研究所碩士論文。
黃茂華（1998）：Polya I學習軟件與教學執行環境：理念與實作。中原大學資訊工程研究所碩士論文。
謝兆輝（1998）：Polya I的社會學習機制：理念與實作。中原大學資訊工程研究所碩士論文。
Adams, L. T., Kasserman, J. E., Yearwood, A. A., Perfetto, G. A., Bransford, J. D., & Franks, J. J. (1988). Memory access: the effects of fact-oriented versus problem-oriented acquistion. Memory and Cognition, 16(5), 167-175.
Bass, M. (1989). Semantic alignments in mathematical word problem. In D. Gentner, K. J. Holyoak, & B. N. Kokinov. (Eds.), The analogical mind: perspectives from cognitive science. (pp.401-433). Cambridge, MA: MIT Press.
Blanchette, I. & Dunbar, K. (2000). How analogies are generated: the roles of structural and superfical similarity. Memory and Cognition, 28 (5), 108-124.
Bransford, J. D., Franks, J. J., Vye, N. J., & Sherwood, R. D. (1989). Multiple analogies for complex concepts: antidotes for analogy-induced misconception in advanced knowledge acquisition. In S. Vosniadou & A. Ortony (Eds.), Similarity and analogical reasoning. (pp. 498-531). New York: Cambridge University Press.
Catrambone, R. (2002). The effects of surface and structural feature matches on the access of story analogs. Journal of Experimental Psychology: Learning, Memory, and Cognition, 28 (2), 318-334.
Cummins, D. D. (1988). The role of understanding in solving word problems. Cognitive Psychology, 20, 405-438
Flick, L. (1991). Where concepts meet percepts: Stimulating analogical thought in children. Science Education, 75 (2), 215-230.
Gentner, D. (1983). Sructure-mapping: A theoretical framework. Cognitive Science, 7, 155-170
Gentner, D. (1988). Metaphor as structure-mapping: The relational shift. Child Development, 59, 47-59.
Gick, M. L., & Holyoak, K. J. (1980). Analogical problem solving. Cognitive Psychology, 12, 306-355
Gick, M. L., & Holyoak, K. J. (1983). Schema induction and analogical transfer. Cognitive Psychology, 15, 1-38
Hofstadter, D. R. (2001). Analgoy as the core of cognition. In D. Gentner, K. J. Holyoak, & B. N. Kokinov (Eds.), The analogical mind (pp. 499-537). London: MIT Press.
Holyoak, K., & Thagard, P. (1990). A constraint-satifaction approach to analogue retrieval and mapping. In K. J. Gilhooly, M. T. Keane, R. Logie, & G. Erdos (Eds.), Lines of Thinking: Reflections on the Psychology of Thought (Vol. 1). New York: Wiley.
Holyoak, K. J., & Thagard, P. (1997). The analogical mind. Retrieved August 23, 2004, from http://cogsci.uwaterloo.ca/Articles/Pages/Analog.Mind.html.
Keane, M. T., Ledgeway, T., & Duff, S. (1994). Constraints on analogical mapping: A comparison of three models. Cognitive Science, 18, 387-438.
Lawson, A. E. (1993). The importance of analogy: A prelude to the special issue. Journal of Research in Science Teaching, 30 (10), 1213~1215.
Mayer, C. D. (1983). Thinking, problem solving, cognition. New York: W. H. Freeman and Company.
Newell, A. (1983). The heuristic of George Polya and its realation to artificial intelligence. In R. Groner, M. Groner, & Walter F. Bischof (Eds.), Methods of heuristics . Hillsdale, NJ: L. Erlbaum Associates.
Novick, L. R. (1988a). Analogical transfer, problem similarity, and expertise. Jounal of Experimental Psychology: Learning, Memory, and Cognition, 14(3), 510-520
Novick, L. R. (1988b). Analogical transfer: processes and individual differences. In D. H. Helman (Ed.), Analogical reasoning (pp. 125-145). Dordrecht, Boston : Kluwer Academic Publishers.
Novick, L. R., & Holyoak, K. J. (1991). Mathematical problem solving by analogy. Journal of Experimental Psychology: Learning, Memory, and Cognition, 17(3), 398-415.
Perfetto, G. A., Bransford, J. D., & Franks, J. J. (1983). Constraints on access in a problem solving context. Memory and Cognition, 11, 24-31.
Polya, G. (1957). How to solve it. Princeton, NJ.: Princeton University Press.
Reed, S. K., Dempster, A., & Ettinger, M. (1985). Usefulness of analogous solutions for solving algebra word problems. Journal of Experimental Psychology: Learning, Memory, and Cognition, 11, 106-125.
Ross, B. H. (1989a). Distinguishing types of superfical similarities: Different effects on the access and use of earlier problems. Journal of Experimental Psychology: Learning, Memory, and Cognition 15, 456-468.
Ross, B. H. (1989b). Remindings in learning and instruction: because wisdom can’t be told. In S. Vosniadou & A. Ortony (Eds.), Similarity and analogical reasoning (pp. 438-469). New York: Cambridge University Press.
Steele, M. M., & Steele, J. W. (1999). DISCOVER: An intelligent tutoring system for teaching students with learning difficulties to solve word problems. Journal of Computers in Mathematics and Science Teaching, 18(4), 351- 359.
Thagard, P. (1988). Dimensions of analogy. In D. H. Helman (Ed.), Analogical reasoning (pp. 3-24). Dordrecht, Boston : Kluwer Academic Publishers.
Vosniadou, S. & Ortony, A. (1989 ). Similarity and analogical reasoning: a synthesis. In S. Vosniadou & A. Ortony (Eds.), Similarity and analogical reasoning (pp. 1-17). New York: Cambridge University Press.