發展問題知識－以數學能力低學童對乘除法算式為例

The Knowledge of Developing Problems: An Example of the Students with Lowly Mathematical Abilities in the Equations of Multiplication and Division

10.7060/KNUJST.200712.0099

馬秀蘭(Hsiu-Lan Ma)

知識 ； 乘除法 ； 發展問題擬題 ； 算式 ； 數學能力 ； knowledge ； multiplication and division ； developing problems posing problems ； equation ； mathematical ability

高雄師大學報：自然科學與科技類

23期（2007 / 12 / 01）

99 - 123

繁體中文

本文旨在探討學童對乘除法算式發展問題歷程中涉及的知識，作者修正Mayer(1992)之解題知識成「發展乘除問題知識」，它指出學生若要成功發展問題必須具有策略性（檢視列式）、基模（情境、結構類型）、語意（事物表示）和語文（正確中文表之）四種知識；研究對象為兩位國小六年級數學能力較低之學童。結果發現在發展問題時，學生之認知結構可透露出此四種知識，且學童涉及的知識及內涵會受算式數字型式（如乘數是小數，非整數之被除數大於或小於除數）影響。其中1.乘法算式：為配合乘數是整數或非整數，有人用離散量或連續量事物，以「單位量（數）×單位數（量）」比例因子結構；有人保握住單位數為整數之累加模式，總用離散量事物，以「單位量×單位數」比例因子結構。有人在乘數是純小數之多步驟算式因不知何者是單位量（數），及在乘數是小數之單步驟算式因不識表示物相關資料，而發展不正確問題。2.除法算式：為配合非整數之被除數大或小於除數，學童大多以離散量或連續量事物，以「總數量÷單位數」之等分除結構。有人在被除數小於除數之單步驟算式因錯誤檢視列式或不識表示物相關資料、在小數被除數大於除數算式因錯誤檢視列式及不識表示物、或在有餘數算式因不完整檢視列式，而發展不正確問題。上述結果可作教師瞭解學童數學概念與文字表徵之參考。

The purpose of this paper was to explore the knowledge the students developed problems in the equations of multiplication and division. The knowledge of ”developing multiplication and division problems” was revised from Mayer's (1992) knowledge of solving problems by the author of this paper. It denoted that the students need to have four kinds of knowledge, strategic, schematic, semantic, and linguistic knowledge, when developing proper problems. The subjects were the two sixth graders with lowly mathematical abilities. The findings were the following. The students' cognition structure can reveal the knowledge above. Their knowledge and the content of knowledge would be influenced by different numbers (multiplier is decimal, non-integer dividend is more or less than divisor). 1. Multiplicative equation: Some students applied the structure of rate factor with discrete or continued objects to adjust integer or non-integer multipliers. Among multiple-step equations with multiplier between 0 and 1 and single-step equations with decimal multiplien, some students failed because they did not know which one was the number in each subset or the number of sets that can be made nor did they understand the related information of representative objects. 2. divisive equation: Most students applied the structure of partition division with discrete or continued objects to adjust non-integer dividend being more or less than divisor. Some students failed because the following reasons. Among single-step equations with dividend being less than divisor, equations with the decimal dividend being more than divisor, and equations with remainder, some students did not correctly view the equation or understand the related information of representative objects, correctly view the equation or understand the representative objects, or completely view the equation respectively. These findings might be a reference to understand students' mathematical concept and word representation for teachers.

1. 馬秀蘭(2004)。數學乘除問題情境發展之研究－以佈告欄為管道。科學教育學刊，12(1)，53-81。
   連結：
2. 馬秀蘭(2002)。透過電腦網路來發展數學加減法問題之研究。科學教育學刊，9(4)，375-399。
   連結：
3. 蔡文標、許天威、蕭金土(2003)。影響國小數學低成就學生數學成就的相關因素之研究。特殊教育學報，17，1-37。
   連結：
4. Badian, N.(1999).Persistent arithmetic, Reading or Arithmetic and Reading Disability.Annuals of Dyslexia,49,45-70.
5. Baroody, A. J.(1993).Problem solving, reasoning, and communication (K-8): Helping children think mathematically.New York:MacMillan.
6. Bell, A. W.,Fischbein, E.,Greer, G. B.(1984).Choice of operation in verbal arithmetic problems: The effects of number size, problem structure and context.Educational Studies in Mathematics,15,129-147.
7. Booth, L. R.(1986).Child-method in secondary mathematics.Educational Studies in Mathematics,12,301-316.
8. Cai. J.(2000).Mathematical thing involved in US and Chinese students` solving process-constrained and process-open problem.Mathematical Thinking and Learning,2,309-340.
9. Cai. J.(1995).Journal for Research in Mathematics Education Monograph Series.Reston, VA:National Council of Teachers of Mathematics.
10. Cai. J.,Hwang. S.(2002).Generalized and generative thinking in US and Chinese students`mathematical problem solving and posing.Jounal of Mathematical Behavior,21,401-421.
11. Charles, R.,Lester, F.(1982).Teaching problem solving: What, why and how.Palo Alto, CA:Dale Seymour.
12. Davydov, V. V.,L. P. Steff (Ed.)(1991).Psychological abilities of primary school children in learning mathematics.Reston, VA:National Council of Teachers of Mathematics.
13. Ellerton, N. F.(1986).Children`s made up mathematics problems a new perspective on talented mathematicians.Educational Studies in mathematics,17,261-271.
14. Fischbein, E.,Deri, M.,Nello, M. S.,Marino, M. S.(1985).The role of implicit models in solving verbal problems in multiplication and division.Journal for Research in Mathematics Education,16,3-17.
15. Fuchs, L.,Fuchs, D.(2001).Principles for the prevention and Intervention of mathematics difficulties.Learning Disabilities Research & Practice,13(3),126-137.
16. Greer, B.,D. A. Grouws (Ed.)(1992).Handbook of research on mathematics teaching and learning.New York:Macmillan publishing company.
17. Greer, B.,Mangan, C.,J. M. Moser, (Ed.)(1984).Proceedings of the sixth annual meeting of North American Chapter of the international group for the psychology of mathematics educational.Madison, WI:Wisconsin Center for Education Research, University of Wisconsin.
18. Greer, B.,McCann, M.,F. Furinghetti (Ed.)(1991).Children`s word problems matching multiplication and division calculations.Proceedings of the Fifteen International Conference for Psychology of Mathematics,Assisi, Italy:
19. Healy, C. C.(1993).Creating miracles: A story of student discovery.Berkeley, CA:Key Curriculum Press. Miracles.
20. Hegarty, M.,Mayer, R. E.,Monk, C. A.(1995).Comprehension of arithmetic word problems: A comparison of successful and unsuccessful problem solvers.Journal of Educational Psychology,87(1),18-32.
21. Kilpatrick, J.,A. H. Schoenfeld (Ed.)(1987).Cognitive science and mathematics education.Hillsdale, New Jersey:Lawrence Erlbaum Associates.
22. Knuth, E. J.(2002).Fostering Mathematical Curiosity.Mathematics Teacher,95(2),126-130.
23. Krutetskii, V. A.(1976).The psychology of mathematical abilities in school children.Chicago:The University Chicago Press.
24. Lerner, J.(1993).Learning Disabilities: Theories, diagnosis, and teaching strategies.Boston, MA:Houghton Mifflin.
25. Ma, H. L.(2005).Bulletin board systems: another supporting channel for helping students work on mathematics.Paper presented in International Conference on Education, Redesigning Pedagogy: Research, Policy, Practice,Singapore:
26. Ma, H. L.,Wu, D. B.,G. Dhompongsa,F. M. Bhatti,Q. Kristen (Eds.)(2006).The role of pattern in the algebraic concept learning via Internet.Proceedings of Thailand International Conference on 21st Century Information Technology in Mathematics Education,Thailand:
27. Mayer, R. E.(1992).Thinking, Problem Solving, Cognition.New York:W. H. Freeman and Company.
28. Mayer, R. E.,Lewis, A. B.,Hegarty, M.,J. I. D. Campbell (Ed.)(1992).The nature and origins of mathematical skills.Amsterdam:Elsevier.
29. National Council of Teachers of Mathematics(1989).Curriculum and evaluation standards for school mathematics.Reston, VA:NCTM.
30. National Council of Teachers of Mathematics(2001).Curriculum and evaluation standards for school mathematics.Reston, VA:NCTM.
31. National Council of Teachers of Mathematics(1991).Professional standards for teaching mathematics.Reston, VA:NCTM.
32. Nohda, N.,T. Kawaguchi (Ed.)(1984).The heart of 'open-approach' in mathematics teaching.Proceedings of ICMI-JSME regional conference on mathematical education,Tokyo:
33. Reitman, W.(1965).Cognition and thought.New York:Wiley.
34. Riley, M. S.,Greeno, J. G.,Heller, J. I.,H. P. Ginsburg (Ed.)(1983).The development of mathematical thinking.New York, NY:Academic Press.
35. Rudnitsky, A.,Etheredge, S.,Freeman, J. M.,Gilbert, T.(1995).Learning to solve addition and subtraction word problems through a structure-plus-writing approach.Journal for Research in Mathematics Education,26(5),467-486.
36. Schloemer, C. G.(1994).University of Pittsburgh.
37. Schwartz, J.,J. Hiebert,M. Behr (Eds.)(1988).Number concepts and operations in the middle grades.Reston, VA:National Council of Teachers of Mathematics.
38. Silver, E. A.(1994).On mathematical problem posing.For the Learning of Mathematics,14(1),19-28.
39. Silver, E. A.(1995).The nature and use of open problem in mathematics education: Mathematical and pedagogical perspectives.International Reviews on Mathematics Education,2,67-72.
40. Silver, E. A.,Cai, J.(1996).An analysis of arithmetic problem posing by middle school students.Journal for Research in Mathematics Education,27(5),521-539.
41. Silver, E. A.,J. Hiebert (Ed.)(1986).Conceptual and procedural knowledge: The case of mathematics.Hillsdale, New Jersey:Lawrence Erlbaum Associates.
42. Smith, C. R.(1994).Learning disabilities: The interaction of learning, task, and setting.Needham Heights, MA:Allyn & Bacon.
43. Spanos, G.,National Clearinghouse on Literacy Education, W.(1991).Cultural Considerations in Adult Literacy Education.ERIC Digest.
44. Stovanova, E.,Ellerton, N. F.(1996).Talking mathematics: Supporting children`s voices.Portsmouth, NH:
45. Swanson, H. L.,Cooney, J. B.,Brock, S.(1993).The influence of working memory and classification ability on children's word problem solution.Journal of Experimental Child Psychology,55,374-395.
46. Tsubota, E.(1987).On children`s problem posing (grade 1 to 3).Tokyo:Japan Society of Mathematics Education.
47. Usiskin, Z.,Bell, M.(1983).Applying arithmetic: A handbook of applications of arithmetic. Part II: Operation.IL:Chicago University.
48. Weir, S.(1987).Cultivating Minds.NY:Harper & Row..
49. Whitin, P.(2004).Promoting problem-posing explorations.Teaching Children Mathematics,11(4),180-186.
50. Wilson, J. W.,Fernandez, M. L.,Hadaway, N.,P. S. Wilson (Ed.)(1994).Survey of research in mathematics education: Secondary School.Reston, VA:National Council of Teachers of Mathematics.
51. Winograd, K.(1990).Greeley, Colorado,University of Northern Colorado.
52. 王文科(2001)。教育研究法。台北市:五南圖書出版公司。
53. 李承華(2002)。碩士論文(碩士論文)。台北市，國立台北師範學院數學教育研究所。
54. 林碧珍(1991)。國小兒童對於乘除法應用問題之認知結構。新竹師院學報，5，221-288。
55. 徐文鈺(2002)。台北市，國立台灣師範大學教育心理與輔導研究所。
56. 馬秀蘭(2002)。學校能力低之學生所發展之問題情境探討。中華民國第十八屆科學教育學術研討會
57. 馬秀蘭、林思行(2003)。論文載於九十二學年度師範學院教育學術論文發表會論文集。國立臺南師範學院。
58. 張景媛(1994)。學業低成就學生的教學策略。測驗與輔導，127，2598-2600。
59. 梁淑坤(1997)。擬題能力之評量：工具之製作。國科會補助專案報告。
60. 梁淑坤(1993)。「擬題」的研究及其在課程的角色。國民小學數學研討會
61. 梁淑坤(1995)。師範生擬題型為之研究。國科會補助專案報告。
62. 陳美芳(1995)。台北市，國立台灣師範大學。
63. 陳淑琳(2002)。屏東市，國立屏東師範學院數理教育研究所。
64. 游麗卿(1999)。除法概念形成歷程中的錯誤分析對教學的啟示。高雄市:復文圖書出版社。
65. 劉秋木(1996)。國小數學教科教學研究。臺北:五南圖書出版公司。