後設認知架構與執行性數學寫作對微積分變化率應用問題解題的影響

Metacognitive Instruction and Transctional Mathematical Writing in Calculus Related-rates Problem-solving

10.7060/KNUJST.200812.0065

辛靜宜(Ching-I Hsin)

後設認知架構 ； 執行性數學寫作 ； 微積分 ； 應用問題解題 ； Metacognitive instruction ； transactional mathematical writing ； calculus ； problem-solving

高雄師大學報：自然科學與科技類

25期（2008 / 12 / 01）

65 - 86

繁體中文

Many students of technological universities cannot solve problems on the secondary educational level, and even have no intention to try. To help them deal with the problem, the researcher implemented metacognitive instruction and transactional mathematical writing, The researcher also conducted an experimental research with the aid of interviewing. The research findings are as follows. When solving not-so-difficult applied problems, most of the students could solve the problems. However, there is a by-peak phenomenon in post-testing. Some students were almost ignorant, but the others were almost excellent. By ANCOVA, the researcher found that the performance of the comparative group was worse than the other two groups in the post-test (p＜.001), because the students in the comparative group could set up unknown variables than those in the other two groups. But the learning sheets help students indirectly. The learning sheets display each process of problem-solving clearly, so students could memorize the problem-solving processes easily, and then they earned some credits. However, the learning sheets cannot help students learn more except for mastering the processes of problem-solving. This finding is consistent with Burks' research finding (1993) for junior high school students' experiences of mathematical writing in Calculus related-rates problem-solving.

Many students of technological universities cannot solve problems on the secondary educational level, and even have no intention to try. To help them deal with the problem, the researcher implemented metacognitive instruction and transactional mathematical writing, The researcher also conducted an experimental research with the aid of interviewing. The research findings are as follows. When solving not-so-difficult applied problems, most of the students could solve the problems. However, there is a by-peak phenomenon in post-testing. Some students were almost ignorant, but the others were almost excellent. By ANCOVA, the researcher found that the performance of the comparative group was worse than the other two groups in the post-test (p＜.001), because the students in the comparative group could set up unknown variables than those in the other two groups. But the learning sheets help students indirectly. The learning sheets display each process of problem-solving clearly, so students could memorize the problem-solving processes easily, and then they earned some credits. However, the learning sheets cannot help students learn more except for mastering the processes of problem-solving. This finding is consistent with Burks' research finding (1993) for junior high school students' experiences of mathematical writing in Calculus related-rates problem-solving.

1. 袁媛(2003)。高中網路數學寫作的實施與其對學生數學態度之影響研究。花蓮師院學報，17，193-210。
   連結：
2. 楊凱琳、林福來(2006)。探討高中數學教學融入建模活動的支撐策略與促進參與教師反思的潛在機制。科學教育學刊，14(5)，517-543。
   連結：
3. 葉明達、柳賢(2003)。高一學生數學合作解題互動歷程與情意分析之因素。花蓮師院學報，16，233-267。
   連結：
4. Arcavi, A.(2003).The role of visual representation in the learning of mathematics.Educational Studies in Mathematics,52,215-241.
5. Bell, E. S.,Bell, R. N.(1985).Writing and mathematical problem solving: Arguments in favor of synthesis.Social science and mathematics,85(3),210-21.
6. Borasi, R.,Rose, B. J.(1989).Journal writing and mathematics instruction.Educational studies in mathematics,20,347-365.
7. Brown, A. L.,F. E. Weiner,R. H. Kluwe (Eds.)(1987).Metacogintion, and understanding.Hillsdale, NY:Erlbaum.
8. Burks, L. C.(1993).Unpublished doctoral dissertation, University of Michigan.
9. Carelle-Elawar, M.(1995).Effects of metacognitive instruction on low achievers in mathematics problems.Teaching and Teacher Education,11(1),81-95.
10. Garofalo, J.,Lester, F. K.(1985).Metacognition, cognitive monitoring, and mathematical performance.Journal for Research in Mathematics Education,16(3),163-176.
11. Hair, J. F. J.,Black, W. C.,Babin, B. J.,Tatham, R. L.(2006).Multivariate data analysis.New Jersey:Prentice Hall.
12. Kapa, E.(2001).A metacognitive support during the process of problem solving in a computerized environment.Educational Studies in Mathematics,47,317-336.
13. Kramarski, B.(2004).Making sense of graphs: Does metacognitive instruction make a difference on students` mathematical conceptions and alternative conceptions?.Learning and Instruction,14(6),593-619.
14. Kramarski, B.,Mevarech, Z. R.,Arami, M.(2002).The effects of metacognitive instruction on solving mathematical authentic tasks.Educational Studies in Mathematics,49,225-250.
15. Mayer, R. E.(1993).Understanding individual differences in mathematical problem solving: Toward a research agenda.Journal Disability Quarterly,16(1),3-5.
16. National Council of Teachers of Mathematics (NCTM)(1989).Curriculum and evaluation standards for school mathematics.Reston, VA:National Council of Teachers of Mathematics.
17. O`shea, J.(2006).Encouraging good mathematical writing.International Journal of mathematical education in Science & Technology,37(1),101-103.
18. Polya, J.(1959).How to solve it.Garden City, NY:Doubleday.
19. Porter, M. K.,Masingila, J. O.(2000).Examining the effects of writing on conceptual and procedural knowledge in Calculus.Educational Studies in Mathematics,42,165-177.
20. Pugalee, D.(2004).A comparison of verbal and writing descriptions of students` problem solving processes.Educational studies in Mathematics,55,27-47.
21. Romberg, T. A.,Carpenter, T. P.,M. Wittrock (Ed.)(1986).Handbook of reaearch in teaching.New York:Macmillan.
22. Rose, B. J.,A. Sterrett (Ed.)(1990).Using writing to teach Mathematics.Washington, DC:Mathematical Association of America.
23. Schoenfeld, A. H.,D. A. Grouws (Ed.)(1992).Handbook of research on mathematics teaching and learning: A project of the national council of teachers of mathematics.New York:Macmillan.
24. Taylor, J. A.,Mcdonald, C.(2007).Writing in groups as a tool for non-routine problem solving in first year university mathematics.International Journal of Mathematical Education in Science and Technology,38(5),639-655.
25. Tougan, P. W.(1993).Unpublished doctoral dissertation, Southern Illinois University.
26. 王民慧、柳賢、洪振方(2006)。高一學生在解題歷程中的數學建模之分析：以指數單元為例。屏東教育大學學報，24，271-310。
27. 余尚芸、謝哲仁、左太政(2006)。美和技術學院學報。24(1)，199-222。
28. 辛靜宜(2006)。電腦動畫視覺化及後設認知架構幫助微積分變化率應用題解題之研究。行政院國家科學委員會專題研究成果報告。
29. 林文忠(2003)。碩士論文(碩士論文)。台北市，國立台灣師範大學資訊教育研究所。
30. 林清山、張景媛(1993)。國中生代數應用題教學策略效果之評估。教育心理學報，27，141-174。
31. 林清山、張景媛(1993)。國中生後設認知、動機信念與數學解題策略之關係研究。教育心理學報，26，53-74。
32. 洪碧霞、林素微、林宜樺、陳沅、陳靜姿(1993)。以多元化的數學評量支持學生的學習進展。教育研究資訊，11(6)，33-63。
33. 莊璧華(2004)。碩士論文(碩士論文)。台北市，國立台北師範學院國民教育學系。
34. 陳怡靖(2003)。數學科教學創新的另一種思維：數學寫作教學的介紹與評析。初等教育學報，8，101-132。
35. 陳彥廷、柳賢(2005)。運用鷹架理論初探國小學生數學學習－以一位五年級學童為例。國立台南大學學報，39(1)，27-44。
36. 潘世尊(2002)。教學上的鷹架要怎麼搭。國立屏東師範學院學報，16，263-294。
37. 凃金堂(1996)。數學解題之研究取向。教師之友，39(3)，38-44。