研究建議

依據本研究的結論與限制，對未來的研究提出以下建議：

壹、後續研究上的建議

由於本研究的研究對象係屬北市重點發展學校的學童，家長有一定的社經背 景，因此學生普遍在相關認知學習上有一定水準的表現，研究結果只能呈現該校 部分學童在列聯表認知上的狀況，因此建議可以將相關研究跨及其他區域、縣市，

如此將可以更明確、更全面的瞭解臺灣國小學童在問題解決上的能力與表現。透 過研究結果也希望將來能將相關研究朝更低年級或更小年齡的兒童來進行，以確 認我國學童在比率推理、機率與列聯表解讀上的認知發展，如此將有助於相關課 程在國小階段實施的設計、安排與依據。

此外，本研究主要是透過紙筆測驗的方式，來獲得學童在列聯表解讀上的能 力與認知，建議未來的研究也可以透過大量的晤談，來進一步深入去探討學童解 題的步驟與策略，有助於確認學童資料解讀的認知，並對課程設計提供更積極的 建議。也建議未來在測驗工具的編製上，能同時考量題數與題目選項的適切性，

盡量避免學童因為猜測的關係而使得研究結果產生偏誤；甚至在試題的呈現上也 能考量到學童認知發展的階段，如此將可以初步從中獲知學童迷思概念的發生階 段。而在測驗的實施上，也可以嘗試將試題電腦化，透過明亮鮮豔的試題呈現，

將可為提高學童的參與率與作答興趣，更能避免學童對於冗長紙本測驗的排斥感。

在試題的分析上，也建議可以將學生的作答反應視為是學生的能力表現來加以分 析，而不再只是將學生的作答反應當作是總分來加以分析與判讀。

貳、課程安排上的建議

根據本研究之結果，建議九年一貫數學領域的課程至少可在國小一年級即進

58

行分數等分概念上的教學；簡易的簡單比例式（A：B＝C：B）的學習也可以提 早到國小三年級來進行，如此分階段螺旋式的課程學習也許將有助於學童對於後 續相關概念的熟悉度與適應性，也避免了直到五年級才學習比率概念的衝擊與學 習上的挫敗感。本研究在機率方面的分析結果，與 NCTM 上的課程安排，建議 我國至少可在二年級的學童就進行機率概念上的學習，特別是在判斷事件發生可 能性的判斷與事件發生機會大小的比較。未來期待可將機率概念上的學習落實在 小學教育中，從生活應用的角度去接觸、學習機率，而不需要等到九年級才從理 論的角度去學習機率課程，不僅無法提高學生的學習興趣，更容易產生學生學習 機率相關概念的挫折。

59

參考文獻

Andrew, L. (2009). Experimental probability in elementary school. Teaching Statistics, 31 (2), 34–36.

Bar, V. (1987) Comparison of the development of ratio concepts in two domains.

Science Education, 71(4), 599-613.

Batanero , C., Godino, J.D. , Vallecillos, A. , Green, D. R., & Holmes, P. (1994).

Errors and difficulties in understanding elementary statistical concepts, International Journal of Mathematical Education in Science and Technology, 25(4), 527-547.

Batanero,C., Estepa, A., Godino, J.D. , & Green, D. R. (1996).Intuitive Strategies and Preconceptions about Association in Contingency Tables. Journal for Research in Mathematics Education, 27(2), 151-169.

Behr, M. J., Wachsmuth, I., Post, T. R., & Lesh, R. (1984). Order and Equivalence of Rational Numbers: A Clinical Teaching Experiment. Journal for Research in Mathematics Education. 15(5), pp. 323-341.

Behr, M. & Post, T. (1992). Teaching rational number and decimal concepts. In T. Post (Ed.), Teaching mathematics in grades K-8: Research-based methods (2nd ed.) (201-248). Boston: Allyn and Bacon.

Cramer, K., & Post, T. (1993). Connecting Research To Teaching Proportional Reasoning. Mathematics Teacher, 86(5), 404-407.

delMas, R. C. (2002). A Review of the Literature on Learning and Understanding Probability. The Annual Meeting of the American Educational Research Association New Orleans, LA.

Falk, Ru., Falk, Ra., & Levin, I. (1980). A potential for learning probability in young children. Educational Studies In Mathematics, 11(2), 181–204.

Fujimura, N. (2001). Facilitating Children's Proportional Reasoning: A Model of

60

Reasoning Processes and Effects of Intervention on Strategy Change. Journal of Educational Psychology. 93(3), 589-603.

Godino, J. D., & Batanero, C. (1998). Clarifying the meaning of mathematical objects as a priority area of research in mathematics education. Mathematics Education as a Research Domain: A Search for IdentityNew ICMI Studies Series Volume 4, 177-195.

Hiebert, J., & Tonnessen, L. H. (1978). Development of the Fraction Concept in Two Physical Contexts: An Exploratory Investigation. Journal for Research in Mathematics Education ,9(5), 374-378.

HodnikČadež, T., & Škrbec, M. (2011). Understanding the Concepts in

Probability of Pre-School and Early School Children. Eurasia Journal of Mathematics, Science & Technology Education, 7(4), 263-279.

Hunting, R. P., & Sharpley, C. F.(1988). Fraction Knowledge in Preschool Children.

Journal for Research in Mathematics Education, 19(2), 175-180.

Koerber, S., Sodian, B., Thoermer, C., & Nett, U. (2005). Scientific reasoning in young children. Preschoolers’ ability to evaluate covariation evidence. Swiss Journal of Psychology, 64, 141–152.

Lamon, S. J. (1993). Ratio and Proportion: Connecting Contect and Children’s Thinking. Journal for Research in Mathematics Education. 24(1), 41-61.

Lamon, S. J. (1999). Teaching fractions and rations for understanding essential content knowledge and instructional strategies for teachers. Mahwah, New Jersey：

Lawrence Erlbaum Associates, Pumlushers.

Langrall, C. W., & Swafford, J. (2000). Three Balloons for Two Dollars: Developing Proportional Reasoning. Mathematics Teaching in the Middle School, 6(4), 54-61.

Ledesma, E. F. R. (2010). Using an Interactive Computer System to Support

the Task of Building the Notions of Ratio and Proportion. Creative Education, (2),

61

115-120.

Noelting, G. (1980). The Development of proportional reasoning and the ratio concept (PART I - DIFFERENTIATION OF STAGES). Educational Studies in Mathematics, 11, 217-253.

Polaki, M. V. (2002). Using Instruction to Identify Key Features of Basotho

Elementary Students' Growth in Probabilistic Thinking. Mathematical Thinking and Learning. 4(4), 285-313.

Reiss, Barchfeld, Lindmeier, Sodian, & Ufer (2011). Interpreting scientific evidence:

primary student's understanding of base rates, sampling procedures, and contingency tables. Proceedings of the 35th Conference of the International Group for the Psychology of Mathematics Education, 4, 4-33. Ankara, Turkey:

PME.

Rodrigo, M. J., Castaneda, J., & Camacho, J. (1999). Mental Models in Predictive Reasoning with Perceptual and Semantic Base Rates: A Developmental Perspective. European Journal of Cognitive Psychology, 11 (4), 499- 529.

Ruffman, T., Perner, J., Olson, D. R., & Doherty, M. (1993). Reflecting on scientific thinking: Children’s understanding of the hypothesis-evidence relation. Child Development, 64, 1617–1636.

Shtulman, A., & Carey, S. (2007). Improbable or impossible? How children reason about the possibility of extraordinary events. Child Development, 78, 1015 – 1032.

Singh, P. (2000). Understanding the concepts of proportion and ratio constructed by two grade six students. Educational Studies in Mathematics. 43, 271-292.

Sophian, C., & Wood, A. (1997). Proportional Reasoning in Young children : The Parts and the Whole of It. Journal of Educational Psychology, 89(2), 309-317.

Spinillo, A. G. & Bryant, P. (1991). Children's Proportional Judgments: The Importance of "Half". : Child Development, 62(3), 427-440.

62

Tourniaire, F. (1986). Proportions in elementary school. Educational Studies in Mathematics, 17, 401-412.

Tversky, A. & Kahneman, D. (1974). Judgment under Uncertainty: Heuristics and Biases. Science,185, 1124-1131.

Wallsten, T. S. & Fillenbaum, S., & Cox, J. A. (1986). Base rate effects on the interpretations of probability and frequency expressions. Journal of Memory and Language, 25(5), 571–587.

Yost, P. A., Siegel, A. E., & Andrews, J. M. (1962). Nonverbal Probability Judgments by Young Children. Child Development, 33(4), 769-780.

沈明勳、劉祥通(2002)。分析學生解比例問題文獻－國小數學課程與教學的建議。

科學教育研究與發展季刊，27，81-96。

林福來、黃敏晃、呂玉琴（1996）。分數啟蒙的學習與教學之發展性研究。科學 教育學刊，4(2)，161-196。

周玉秀（2006）。 從PISA 看數學素養與中小學數學教育。科學教育月刊，293，

2-21。

洪繼賢、劉祥通(2004)。一位小五學生解分數單位量等分割問題的表現。

科學教育研究與發展季刊，35，53-75。

教育部（2007）。國民中小學九年一貫課程綱要。台北：教育部。

陳欣民、劉祥通（2002）。從兒童迷思概念之文獻分析談機率單元的教學與課程。

科學教育研究與發展季刊，26，40-51。

蘇國樑（1995）。國小兒童統計概念分析之研究(I)：資料分佈概念之分析。

空大人文學報，4，205-226。

美國數學教師學會 (National Council of Teachers of Mathematics)：

http://www.nctm.org/

紐西蘭國家教育部數學與統計課程的學習標準（Ministry of Education, The New Zealand Curriculum online）：

63

http://nzcurriculum.tki.org.nz/National-Standards/Mathematics-standards/The-sta ndards