第五章 結論與建議
第二節 建議
根據本研究的研究結果及發現，提出下列建議，以作為教學上及後續 研究上之參考。
一、 在理論與研究方面
1. 本研究僅採用潛在類別分析將受試者進行分群並探究其各群組所採用 解題規則之特徵，未來可嘗試配合其他分析方法進行比例問題的解題 規則探究其次序性及結構性。
2. 本研究僅就受試者之作答結果進行潛在類別分析，未來可配合晤談深 入了解受試者各種不同的解題策略，對學童的認知結構可以獲得更進 一步的分析。
3. 本研究係採用潛在類別分析，找到最佳分群數，並探究各群組所採用 解題規則之特徵。若採用其他分群之分析方法，是否也能得到相同的 群數及群組特徵，及使用何種分群分析方法，所找到的群組是最佳的 分群，並未有相關探討。研究者建議，未來可針對不同分群分析方法，
進行彼此間的優劣比較。
二、 教學與實務方面
1. 本研究將受試者進行分群，可供教學者針對教材內容在教學現場做驗 證，亦可提供教師找到最適合的群組來進行認知診斷與補救教學策略 的參考。
2. 教學者可以運用潛在類別分析方法，依據受試者的反應組型，分析受 試者隸屬的潛在類別，並作教學上學童同質性或異質性分組教學的參 考依據。
3. 除了可提供教師分析學童的認知結構外，亦能伴隨著診斷訊息與補救
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教學的功能，以協助教師更能了解學童的認知結構，進行有效的診斷 與補救教學。
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參考文獻
中文部分
王郁琮、溫福星 (2012)。國中生學校學習與家庭關係困擾之群體異質性分 析：以 IRT Mixture Model。教育心理學報，44，185-205。
何意中 (1988)。國小三、四、五年級學生比例推理之研究。花蓮師院學報，
2，387-433。
李幸娟 (2012)。四年級學童直覺法則與其後設認知之探討。國立臺中教育 大學數學教育學系碩士論文，未出版，臺中市。
吳毓瑩、呂玉琴(1997)。潛在類別分析對兒童等值分數概念結構之解析。
行政院國家科學委員會專題研究計劃精簡報告( 編號：
NSC-85-2511-S-152-007)。
吳毓瑩、林原宏 (1996)。潛在類別分析取向的除法概念結構。測驗學刊，
43，345-358。
林原宏、游森期 (2006)。次序理論取向的解題規則階層結構分析及其結構 圖比較之探究。測驗學刊，53，239-260。
林惠雅 (2008)。國小學童母親信念、教養目標和教養行為之類型初探：兼 論其與子女學業表現之關聯。應用心理研究季刊，37，181-213。
林福來、郭汾派、林光賢 (1986)。國中生的比例概念發展。科教月刊，87，
14-42。
周筱亭、黃敏晃 (2002)。國小數學教材分析：比(含線段圖)。臺北：國立 教育研究院籌備處。
林碧珍 (2010)。比與比值初始概念的教學初探。新竹教育大學教育學報，
27，127-160。
林瑋詩 (2007)。次序理論取向的比例問題解題規則階層分析及其結構圖比 較之研究。國立臺中教育大學數學教育學系碩士論文，未出版，臺中 市。
82
翁宜青、劉祥通 (2003)。一位國小三年級學生解簡單式比例問題之研究。
科學教育研究與發展季刊，31，32-53。
洪藹鈺 (2008)。小數概念之潛在類別分析及其縱貫研究-以國小五年級學 童為例。國立臺中教育大學數學教育學系碩士論文，未出版，臺中市。
涂金堂 (2003)。認知診斷評量的探究。南師學報，37，67-97。
教育部 (2009)。九年一貫課程數學課程綱要。臺北市：教育部。
郭生玉 (1989)。心理與教育測驗。臺北市：精華書局。
莊玉如 (2005)。國小四年級學童比例問題解題表現之研究。國立臺中教育 大學數學教育學系碩士論文，未出版，臺中市。
莊嘉坤 (1995)。 國小學生科學態度潛在類別的分析研究。屏東師院學報，
8，111-135。
陳竹村、林淑君、陳俊瑜 (2002)。國小數學教材分析：比(含線段圖)。臺 北：國立教育研究院籌備處。
陳宇雯 (2008)。應用潛在類別分析國小五年級學生的分數概念。國立新竹 教育大學人資處數學教育碩士論文，未出版，新竹市。
陳惠萍 (2007)。國小高年級學生在相關性問題之解題規則階層結構與分群 探討。國立臺中教育大學數學教育學系碩士論文，未出版，臺中市。
張育綾 (2008)。潛在類別分析在國小五年級學童四則運算規則之縱貫研究。
國立臺中教育大學數學教育學系碩士論文，未出版，臺中市。
傅宗聖 (2007)。國小六年級學童比例問題之表現概況與解題策略分析。國 立臺南大學數學教育學系碩士論文，未出版，臺南市。
楊錦連 (1999)。國小高年級兒童解決比例問題之研究。嘉義大學國民教育 研究所碩士論文，未出版，嘉義縣。
劉秋木 (1996)。國小數學科教學研究。臺北：五南圖書出版公司。
83
劉祥通、周立勳 (1999)。國小比例問題教學實踐課程之開發研究。國立臺 中師範學院數理學報，3(1)，3.1-3.25。
劉祥通 (2004)。分數與比例問題解題分析：從數學提問教學的觀點。臺北：
師大書苑。
樊雪春 (1998)。學生科學迷思概念的法則分析與建構教學取向教學法之實 驗效果研究。國立臺灣師範大學教育心理與輔導研究所博士論文。未 出版，臺北市。
謝如山 (2003)。潛在分類模式在括號概念的應用。教育與心理研究，26，
277-403。
84
英文部分
Akaike. H. (1973). Information theory and an extension of the maximum likelihood principle. 2nd International Symposium on Information Theory ( pp. 267-281).
Akaike. H. (1978). A Bayesian analysis of the minimum AIC procedure.
Annals of the Institute of Statistical Mathematics, 30, Part A. 9-14.
Bozdogan, H. (1992). Choosing the number of component clusters in the mixture model using a new Informational complexity criterion of the inverse-fisher information matrix. In O. Opitz, B. LAusen and R. Klar (Eds.), Information and classification: Concepts, Methods and
Applications (pp. 44-45), New York. Springer-Verlag.
Castle, D. J., Sham, P. C., Hesseley, S. & Murray. R. M. (1994). The subtyping of schizophrenia in men and women: a latent class analysis. Psychological Medicine, 24, 41-51.
Clogg, C. C. (1977). Unrestricted and restricted maximum likelihood latent structure analysis: A manual for users. The Pennsylvania State University:
Population Issues Research Office.
Dempster, A. P., Laird, N. M., & Rubin, D. B. (1977). Maximum likelihood from incomplete data via the EM algorithm. Journal of the Royal Statistical Society (Series B), 39, 1-22.
Fischer, G. H. (1973). The linear logistic test model as an instrument in educational research. Acta Psychologica, 37, 359-374.
Freudenthal, H. (1983). Didactical phenomenology of mathematical structures.
Dordrecht, Holland: D. Reidel Publishing Co.
Goodman, L. A. (1974a). The analysis of systems of qualitative vaariables when som of the variables are unobservable. Part I--A modified latent structure approach. American Journal of Sociology, 79, 1179-1259.
Goodman, L. A. (1974b). Exploratory latent structure analysis using both
85
identifiable and unidentifiable models. Biometrika, 61, 215-231.
Hart, K.M. (1981). Fractions. In K. M. Hart, D. Kerslake, M. L. Brown, G.
Ruddock, D. E. Kuchemann, & M. McCartney (Eds.), Children
understanding of mathematics (pp. 11-16). Oxford, London: Northampton.
Heather, J.C. (2008). Investigating Students’ Proportional Reasoning Strategies (Master’s thesis). Available from ProQuest Dissertation and theses database. (UMI No. 1453188)
Hoffer, A. (1992). Ratios and proportional thinking. In T. Post (Ed.), Teaching mathematics in grades K-8 research-based methods (2nd Ed., pp.
303-330).Needham Heights, MA: Allyn and Bacon.
Inhelder, B., & Piaget, J. (1958). The growth of logical thinking from children to adolescence. New York: Basic Books.
Jansen, B. R. J., & Van der Maas, H. L. J. (1997). Statistical test of the rule assessment methodology by latent class analysis. Developmental Review, 17, 321-357.
Jansen, B. R. J., & Van der Maas, H. L. J. (2002). The development of children’s rule use on the balance scale task. Journal of Experimental Child Psychology, 81, 383-416.
Jeong, Y., Levine, S. C., & Huttenlocker, J. (2007). The development of proportional reasoning: Effect of continuous versus discrete quantities.
Journal of Cognition and Development, 8, 237-256.
Karplus, R., & Peterson, R. (1970). Intellectual development beyond
elementary school II: Ratio, a survey. School Science and Mathematics, 70, 735-742.
Karplus, R., Pulos. S., & Stage, E. (1983). Proportional reasoning of early adolescents. In R. Lesh., & M. Landau (Eds.), Acquisition of mathematics concepts and processes. New York: Academic Press.
Kieren, T. E. (1980). Knowing rational numbers: Ideas and symbols. In M.
86
Lindquist (Ed.), Selected Issues in Mathematics Education, Chicago:
National Society for the Study of Education. Reston, VA: NCTM.
Lamon, S. J. (1993). Ratio and proportion: Connecting content and children's thinking. Journal for Research in Mathematics Education, 24, 41-61.
Lamon, S. J. (1994). Ratio and proportion: Cognitive foundations in unitizing and norming. In G. Harel & J. Confrey (Eds.), The development of multiplicative reasoning in the learning of mathematics (pp. 89-122).
Albany, NY: State University of New York Press.
Lamon, S. J. (1995). Ratio and proportion: Elementary didactical
phenomenology. In J. T. Sowder, & B. P. Schappelle (Eds), Providing a foundation for teaching mathematics in the middle grades (pp. 167-198).
Albany, NY: State University of New York Press.
Lazarsfeld, P. F. (1950). The logical and mathematical foundation of latent structure analysis. In S. A. Stouffer, L. Guttman, E. A. Suchman, P. F.
Lawarsfeld, S. A. Star, & J. A. Calusen (Eds.), Measurement and prediction: Studies in social psychology in World WarⅡ (Vol.Ⅳ).
Princeton University Press.
Lesh, R., Post, T., & Behr, M. (1988). Proportional reasoning. In J. Hiebert &
M. Behr (Eds.), Number concepts and operations in the middle grades (pp.
93-118). Reston, VA: NCTM.
Levin, S. (2002). Proportional reasoning: One problem, many solutions! In B.
Litwiller & G. Bright(Eds.), Making sense of fractions, ratios, and proportions: 2002 yearbook. Reston, VA: NCTM.
Lo, J. J., & Watanabe, T. (1995). A fifth grader's attempt to expand her ratio and proportion concepts. Paper presented at the Annual Meeting of the North American Chapter of the International Group for the Psychology of Mathematics Education.
Lo, J. J., & Watanabe, T. (1997). Developing understanding of ratio as measure as a foundation for slope. In B. Litweller, & G. Bright (Eds), Making sense of fractions, and ratio, and proportions (pp.162-175). Reston, VA:
87
NCTM.
Misailidou, C. & Williams, J. S. (2003). Diagnostic assessment of children's proportional reasoning, Journal of Mathematical Behavior, 22, 335-368.
Mislevy, R. J., & Verhelst, N. (1990). Modeling item response when different subjects employ different solution strategies. Psychometrika, 55, 195-215.
National Council of Teachers of Mathematics(2000). Principles and Standards for School Mathematics. Reston, VA: NCTM.
National Research Council (2005). How students learn: Mathematics in the classroom. Committee on How people learn. A Targeted Report for Teachers, M.S. Donovan and J. D. Bransford, Editors. Division of Behavioral and Social Sciences and Educational. Washington, DC: The National Academics Press.
Noelting, G. (1980a). The development of proportional reasoning and the ratio concept: Part I – Differentiation of stage. Education Studies in Mathmatics, 11, 217-253.
Noelting, G. (1980b). The development of proportional reasoning and the ratio concept: Part II- Problem structure at successive stages; problem-solving strategies and the mechanism of adaptive restructuring. Educational Studies in Mathematics, 11, 331-363.
Notenboom, A., Reitsma, P. (2007). Spelling dutch doublets: Children’s learning of a phonological and morphological spelling rule. Scientific Studies of Reading, 11, 133-150.
Rost, J. (1985). A latent class model for rating data. Psychometrika, 50, 37-49.
Rost, J. (1988). Rating scale analysis with latent class models. Psychometrika, 53, 327-348.
Rindskopf, D. (2006). Heavy alcohol use in the “fighting back” survey sample:
separarting individual and community level influences using multilevel latent class analysis. Journal of Drug Issues, 36, 441-462.
88
Ruiz, E. F., & Lupianez, J. L. (2009). Detecting psychological obstacles to teaching and learning the topics of ratio and proportion in sixth grade primary pupils. Electronic Journal of Research in Educational Psychology, 17(7), 397-424.
Siegler, R. S. (1976). Three aspects of cognitive development. Cognitive Psychology, 8, 481-520.
Steinthorsdottir, B. (2006). Proportional reasoning: Variable influencing the problem difficulty level and one’s use of problem solving strategies.
Proceedings of the 30th Conference of the International Group foe the Psychology of Mathematics Education, vol.5 (pp.169-176). Prague:
Charles University.
Sunmgi, A. K. (2009). Thought processes in proportional reasoning (Doctoral Dissertation). Available from ProQuest Dissertation and theses database.
(UMI No. 3385005)
Tatsuoka, K. K. (1983). Rule space: An approach for dealing with
misconceptions based on item response theory. Journal of Educational Measurement, 20, 345-354.
Thom, R. (1975). Structural stability and morphogenesis: An outline of a general theory of models. Reading, MA: Benjamin.
Tourniaire, F., & Pulos, S. (1985). Proportional Reasoning: A Review of The Literature. Educational Studies in Mathematics, 16, 181-204.
Van der Maas, H. L. J., & Molenaar, P. C. M. (1996). Catastrophe analysis of discontinuous development. In A. A. van Eye & C. C. Clogg (Eds.), Categorical variables in developmental research. Methods of analysis (pp.77-105). San Diego: Academic Press.
Van Dooren, W., De Bock, D., Evers, M., & Verschaffel, L. (2006). Pupils’
overuse of proportionality on missing value problems: How numbers may change solutions. Proceedings of the 30th Conference of the International Group for the Psychology of Mathematics Education, vol.5 (pp.305-312).
Prague: Charles University.
89
Van Lier, P. A. C., Verhulst, F. C., & Crijnen, A. A. M. (2003). Screening for Disruptive Behavior Syndromes in Children: The Application of Latent Class Analysis and Implications for Prevention Programs. Journal of Consulting & Clinical Psychology, 71, 353-363.
Vergnaud, G. (1983). Multiplicative structures. In R. Lesh & M. Landau (Eds), Acquisition of mathematics concepts and procedures (pp. 127-174). New York: Academic Press.
Vergnaud, G. (1988). Multiplicative structures. In J. Hievert & M. Behr (Eds), Number concepts and operations in the middle grades (pp. 141-161).
National Council of Teachers of Mathematics and Lawrence Erlbaum Associates.
Yang, X., Shaftel, J., Glasnapp, D., Poggio, J. (2005). Qualitative or
quantitative differences? latent class analysis of mathematical ability for special education students. Journal of Special Education, 38, 194-207.
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