臺灣博碩士論文加值系統

本研究旨在探討高雄縣壽齡國小一至六年級學童，對於平面幾何圖形概念之發展，以及三角形角度張開大於135度時，是否會產生迷失概念。為探討壽齡國小與高雄縣其他各地區的國小學童，對幾何圖形的辨識能力是否有差異。因此以高雄縣新甲、橋頭、興達、等國小一至六年級學童，共計488人為一般組型，作為壽齡國小與一般組型之比較。首先以Clements等學者（1999）發展的相關研究為主，收集的資料以單因子變異數分析及事後考驗（Scheffe法）進行分析，探討城鄉、年齡、性別之間的差異性。實驗效益分析以壽齡國小為例，壽齡國小的研究樣本一至六年級共282人，分成實驗組與對照組，實驗組於一般教學後，進行本研究之補救教學，對照組除了接觸學校正常課程外，不另外進行教學，進一步比較各年級之間的差異。研究者針對9位學生進行訪談，瞭解學童慣用語彙作為編製晤談結果試卷，在對770位進行測試，以瞭解學童於視覺期對圖形的特徵，思維型態轉變方式，及其對平面幾何圖形的認知概念。
本研究有以下幾點發現：
一、 三角形角度張開超過135度時，角度越大，學生越容易產生迷思現象。惟各學校的五年級學生，在此向度答對率普遍高於其他各年級。顯示年齡並不會影響其圖形答對率高低，而是與學童的成熟度有關，結果顯示城鄉、性別並無顯著差異。
二、 各年級學童對於圓形、長方形與正方形比較容易掌握。年級越高，學生越能掌握圖形的特徵，且城鄉之間達.05顯著差異。三角形部份，各年級學生依不同圖形的變化，答對率會出現不同的現象，各年級答對率反而與年齡無關。
三、 新甲、橋頭、興達等三所國小男、女生於圓形（F= 2.731）、正方形（F=.801）、三角形（F=.3212）、長方形（F=.497）等四種圖形皆無顯著差異（p＞.05）。壽齡國小各年級的實驗組、對照組的前測與延後測的男、女生除三年級未達顯著差異（p＞.05），其餘各年級的男、女生達.05顯著差異，而且實驗組成績優於對照組。
四、 壽齡國小在隱藏式圖形部分實驗組一升二年級學童，實施補救教學前後之各答對率分別為18.33與23.60達.05顯著差異，顯示協助學童澄清圖形的概念、分類與性質，有助增進學習效能。對照組二升三年級學生答對率才由15.90提高至22.96，達到實驗組一升二年級的水準，顯示比實驗組慢一年，此與van Hiele教學理論相印證。

This research was to investigate how the students of Shou-Ling Elementary School, Kaohsiung Hsien, developed their concept about geometric shapes and whether they would feel confused when the angle of a triangle was larger than 135 degrees. In order to find out if there was a significant difference in the ability to recognize geometric shapes among the students of Shou-Ling Elementary Schools and those from other elementary schools in Kaohsiung Hsien, the subjects of this research included the first graders to the sixth graders of Hsin-Chia, Chiao-Tou, and Hsing-Ta Elementary Schools, Kaohsiung Hsien, totally 488 students forming the general group in contrast with the students of Shou-Ling Elementary School. First, the relevant research made by Clements, et al. (1999) was taken for reference. Then, the data gathered were analyzed by means of the single-factor variant analysis and subsequent testing (Scheffe method) so as to investigate the differences between the city and country, age, and sex. The experimental efficiency analysis was based on the students of Shou-Ling Elementary School. Totally 282 students from the first grade to the sixth grade of Shou-Ling Elementary School were divided into two groups — experimental group and control group. Make-up teaching of this study was given to the experimental group after normal curricular activities, whereas no extra teaching was offered to the control group aside from normal school courses. Later, a further comparison was made to see the differences between each grade. 9 students were interviewed by the researcher so as to understand the commonly used words among students, and the result was made into the test paper. Then, a test was given to 770 students to get a better knowledge about the students’ thinking pattern about the characteristics of shapes at the visual level, and their cognition about geometric shapes.
The findings of this research are shown as follows:
1. When the angle of a triangle was larger than 135 degrees, the larger the angle became, the more confused the students would feel. Generally, the fifth graders obtained the highest percentage in answering correctly than other graders in each school regarding this condition. This suggested that the students’ maturity, not their age, would influence the percentage of answering correctly. And the result showed no significant difference between the city and country, and sex.
2. It was easier for students to recognize the circle, square, and rectangle. The higher grade the students were in, the more easily the students could recognize the characteristics of shapes. And there was a significant difference of .05 between the city and country. As for the triangle, the percentage of answering correctly varied when the shape changed differently. And the percentage of answering correctly had nothing to do with the age of students.
3. No significant difference (p > .05) existed among boy and girl students of Hsin-Chia, Chiao-Tou, and Hsing-Ta Elementary Schools concerning the circle (F = 2.731), square (F = .801), triangle (F = .3212), and rectangle (F = .497). As for the pre-test and post-test given to the experimental group and the control group in each grade of Shou-Ling Elementary School, there was a significant difference of .05 in all grades except for the third grade (p > .05). And the experimental group performed better than the control group in the tests.
4. Regarding the hidden pictures, the first-turn-second graders in the experimental group of Shou-Ling Elementary School had different percentages of answering correctly before and after the make-up teaching was given (18.33 and 23.60, respectively), exhibiting a significant difference of .05. And this suggested that assisting the students to get a better understanding about the shapes and their classification and property would improve the students’ learning efficiency. However, there was an increase from 15.90 to 22.96 in the percentage of answering correctly for the second-turn-third graders in the control group, and that was just the level of the first-turn-second graders in the experimental group, indicating that the control group was one year slower in learning than the experimental group. And this result was the same as shown in the teaching theory of van Hiele.

第一章 緒論
第一節 研究動機 ……………………………………………………. 1
第二節 研究目的 ……………………………………………………. 3
第三節 研究問題 ……………………………………………………. 4
第四節 研究範圍和限制 …………………………………………… 4
第五節 名詞界定 ……………………………………………………. 6
第二章 文獻探討
第一節 兒童圖形概念的發展 ……………………………………… 9
第二節 van Hiele夫婦理論之視覺期 …………………………….. 18
第三節 prerecognitive level…………………………………………. 20
第三章 研究方法
第一節 研究流程 …………………………………………………… 25
第二節 研究對象 …………………………………………………… 27
第三節 研究工具 …………………………………………………… 30
第四節 資料處理與分析 …………………………………………... 32
第五節 幾何圖形補救教學之設計 ……………………………….. 35
第四章 研究結果與討論
第一節 不同年級學童於三角形角度變化的迷失 ………………. 43
第二節 平面基本圖形於城鄉學習成效的差異 …………………. 55
第三節 學童在視覺期對圖形迷失概念之探討 …………………. 68
第四節 各年級學生對隱藏式圖形的認知概念 …………………. 72
第五章 結論與建議
第一節 結論 …………………………………………………………. 83
第二節 建議 …………………………………………………………. 87
參考書目
一、 中文部分 ……………………………………………………… 89
二、 西文部分 ……………………………………………………… 90
附錄A ……………………………………………………………………… 99
附錄B ………………………………………………………………………102
附錄C ………………………………………………………………………108
附錄D ………………………………………………………………………113

一、中文部分：
王文科譯（民85）：皮亞傑式兒童心理學與應用。台北市：心理出版社。
皮亞傑著，王憲鈿譯（1990）：發生認識論。商務印書館。
皮亞傑著（1972）：從青少年到成人的智慧發展。見「人類發展」（英文版），15：1-12。
李其維（民84）：皮亞傑心理邏輯學。嘉義：揚至文化事業股份有限公司。
林碧珍 (民82)：兒童「相似性」概念發展之研究－長方形。新竹師院學報，6期，P333∼377。
林軍治 (民81)：兒童幾何思考之VAN HIELE 水準分析研究－－VHL、城鄉、年級、性別、認知型式與幾何概念理解及錯誤概念之關係。台中市：書桓出版社。
邱皓政（民89）：量化研究與統計分析。台中市：五南圖書出版社。
吳德邦（民84）：范析理(van Hiele)模式對我國師範學院學生在非歐幾何學的學習成就與幾何思考層次之研究(英文撰寫、中文摘要)。台中師院學報，9，443∼474。
吳德邦（民87）：國中學生van Hiele幾何思考層次之研究。文章載於「行政院國家科學委員會科學教育發展處」編印，八十七年度數學教育專題研究計畫成果討論會摘要，P91∼96。台北市：國家科學委員會。
吳德邦（民88）：台灣中部地區國小學童范析理幾合思考層次之研究－筆試部分。八十八學年度師範學院教育學術論文發表會論文集。
國立編譯館（民86）：數學教學指引第一冊。台北市：台灣書店。
教育部（1975）：國民小學課程標準。台北市：正中書局。
教育部（1993）：國民小學課程標準。台北市：台捷國際文化實業股份有限公司。
郭為藩（民59）：現代教育的心理學基礎，刊於台灣教育輔導月刊第二時卷第十一期，第16頁。
楊憲明（民，86）：學習障礙。國立台南師範學院特殊教育中心。
劉好 (1998，5月)：平面圖形教材的處理。載於甯自強主編：國立嘉義師範學院八十六學年度數學教育研討會論文暨會議實錄彙編，(PP.33-分析研究--從van Hiele理論的觀點來看。文章發表於直中華民國第十四屆科學教育學術研討會暨第十一屆科學教育學會年會，高雄市國立高雄師範大學，民國８７年12月18-19日。載於中華民國第十四屆科學教育學術研討會暨第十一屆科學教育學會年會會議記錄及短篇論文彙編，718-732頁。
莎莉•溫德寇絲•歐次著，黃慧真譯(民79)：兒童發展。台北：桂冠圖書股份有限公司。
劉錫麒（民62）：我國兒童保留概念的發展。國立台灣師範大學教育研究所集刊，16輯，P97∼160。
鄭昭明（民82）：認知心理學。台北：桂冠圖書股份有限公司。
鄭麗玉（民82）：認知心理學。台中市：五南圖書出版社。
盧銘法（民85）：國小中高年級幾何概念之分析研究-以van Hiele幾何思考水準與試題關聯結構分析為探討基礎。國立台中師範學院國民教育研究所碩士論文(未出版)。
譚寧君（民82）：兒童的幾何觀：從van Hiele幾何思考的發展模式談起。國民教育，33（5/6），頁12-17。
二、英文部分：
Anderson, J. R. (1983). The architecture of cognition. Cambridge, MA: Harvard University Press.
Anderson, J. R. (1985), Cognitive psychology and its implications (2nd ed.). New York: W. H. Freeman.
Baynes, Joyce Frisby(1998). The development of a van Hiele-based summer geometry program and its impact on student van Hiele level and achievement in high school geometry(Academic achievement). Unpublished doctoral dissertation, Columbia University Teachers College.
Boke, H. (1975). Piaget’s mountains revisited－Changes in egocentric landscape. Developmental Psychology, 11, 240－243.
Breen, John Joseph (1999). Achievement of van Hiele level two in geometry thinking by eighth-grad students throughnthe use of geometry computer-based guided instruction. Unpublished doctoral dissertation, University of South Dakota.
Carpenter, T. P., Corbitt, M. K., Kepner, H. S., Lindquist, M. M., & Reys, R. E. (1980). Results of the second National Assessment of Educational Progress mathematics assessment: Secondary school. Mathematics Teacher, 73, 329-338.
Clements, D. H. (1992). Elaboraciones sobre los niveles de pensamiento geomgtrico [Elaborations on the levels of geometric thinking]. In A. Gutierrez (Ed.), Memorias del tercer Simposio Internacional Sobre Investigacion en Educacion Matematica (pp. 16-43). Valencia, Spain: Universitat De Valencia.
Clements, D. H., & Battista, M. T. (1992a). The development of a LOGO-based elementary school geometry curriculum (Final report to the National Science Foundation for Grant MDR-8651668). Buffalo: State University of New York at Buffalo and Kent, OH: Kent State University.
Clements, D. H., & Battista, M. T. (1992b). Geometry and spatial reasoning. In D. A. Grouws (Ed.), Handbook of research on mathematics teaching and learning (pp. 420-464). New York: Macmillan.
Clements, D. H., Battista, M. T,, Sarama, J., & Swaminathan, S. (1997). Development of students'' spatial thinking in a unit on geometric motions and area. The Elementary School Journal, 98, 171-186.
Clements, D. H., Swaminathan, S., Hannibal, M. A. Z. & Sarama, J. (1999). Young children’s concepts of shape. Journal for Research in Mathematics Education, 2, 192-212.
Coxford, A. F. (1978). Research directions in geometry. In R. A. Lesh & D. B. Mierkiewicz(Eds.), Recent research concerning the development of spatial and geometric concepts(pp.323-332). Colunbus, OH: ERIC/SMEAC. (ERIC Reproduction Service No. ED 159062)
Flavell, J. H. (1976) Metacognitive aspects of problem solving.
In L. B. Resnick (Ed.), The natural of intelligense. Hillsdale, N. J: Erlbaum.
Flavell, J. H. (1978) Metacognitive development. In J. M. Scandura, & C. J. Brainerd (Eds.) Structural process theories of complax human behavior. Alphen a.d. Rijn: Sijthoff & Noordhoff.
Fuys, D., Geddes, D., & Tischler, R. (1985). An investigation of the van Hiele model of thinking in geometry among adolescents (Final report of the Investigation of the van Hiele Model of Thinking in Geometry Among Adolescents Project). Brooklyn, NY: Brooklyn College, School of Education.
Fuys, D., Geedes, D., & Tischler, R. (1988). The van Hiele model of thinking in geometry among adolescents.Journal for Research in Mathematics Education Monograph Series, Number 3. Reston, VA： National Council of Teachers of Mathematics.
Geeslin, W, E., & Shar, A. 0. (1979). An alternative model describing children''s spatial preferences. Journal for Research in Mathematics Education, 10, 57-68.
Gibson, E. J., Gibson, J. J., Pick, A. D., & Osser, H. (1962). A developmental study of the discrimination of letter-like forms. Journal of Comparative and Physiological Psychology, 55, 897-906.
Gutierrez, A.,Jaime, D., & Fortuny, J. M. (1991). An alternative paradigm to evaluate the acquisition, 22(3), 237-251.
Hoffer, A. (1981). Geometry is more than proof. Mathematics Teacher, 74(1), 11-18.
Inhelder, B., & Piaget, J. (1958). The Grouth of Logical Thinking from Childhood to Adolescence: an Essay on the Construction of Formal Operational Structures, Trans. A. Parsons and S. Milgram. New York : Basic Books.
Inhelder, B. (1962). Some Aspects of Piaget’s Genetic Appoarch to Cognition, in W. Kessan et al(Eds.), Thought in the Young Child, Monogr. Soc. Res. Child Develpm., 27, p.27.
Koffka, K. (1935). Principles of Gestalt psychology. New York: Harcourt Brace Jovanovich.
Kouba, V. L., Brown, C. A., Carpenter, T. P., Lindquist, M. M., Silver, E. A., & Swafford, J. 0. (1988). Results of the fourth NAEP assessment of mathematics: Measurement, geometry, data interpretation, attitudes, and other topics. Arithmetic
Teacher, 35(9), 10-16.
Lee, Wan-I (1999). The relationship between students’ proof-writing ability and van Hielelevels of geometric though in a college geometry course(college stugents). Unpublished Doctoral dissertation, University of Northern Colorado.
Lehrer, R., Jenkins, M. & Osana, H. (1998). Longitudinal study of children''s reasoning about space and geometry. In R. Lehrer & D. Chazan (Eds.), Designing learning environments for developing understanding of geometry and space (pp. 137-167).
Lodwick, A. R. (1958). An Investigation of the Question whether the Inferences that Children draw in Learning Histry correspond to the Stage of Mental Development that Piaoget Postulates. University of Birmingham Diploma in Education dissertation (unpublished).
Martin, J. L. (1976a). An analysis of some of Piaget’s topological tasks from a mathematical point of view. Journal for Research in Mathematics Education, 7, 8-24.
Martin, J. L. (1976b). A test with selected topological properties of Piaget''s hypothesis concerning the spatial representation of the young child. Journal for Research in Mathematics Education, 7, 26-38.
Mayberry, J. W. (1981). An investigation of the van Hiele levels of geometric thought in undergraduate preservice teachers (Doctoral dissertation, University of Georgia, 1981). Dissertation Abstracts International, 42, 2008A.(Uiversity Microfilms International order No. 8123078)
McClelland, J. L., Rumelhart, D. E., & the PDP Research Group. (1986). Parallel distributed processing: Explorations in the microstructure of cognition. Volume 2: Psychological and biological models. Cambridge, MA; MIT Press.
Molina, D. D. (1990) The teaching of transformational geometry in the secondary schools. Unpublished master’s thesis, The University of Texas at Austin.
Musser, G. L.,Burger, W. F., Peterson, B. E. (2000). Recognizing geometric shapes and definitions. Mathematics for elementary teachers, 12(1), 529-533. New York: John Wiley Sons, Inc.
Neisser, U. (1967). Cognitive Psychology. New York: Appleton-Century-Crofts.
Page, E. I. (1957).Haptic Perception. University of Birmingham Diploma in Education, dissertation (unpubished).
Page, E. I. (1959). Education Review, Vol. 11, No. 2. pp. 115-124.
Peel, E. A. (1959). Experimental examination of some of Piaget''s schemata concerning children''s perception and thinking, and a discussion of their educational significance. British Journal of Educational Psychology, 29, 89-103.
Piaget, Jean(1928). Judgment and Reasoning in the Child. Trans.
M. Worden, New York: Harcourt, Brace & World. p. 2.
Piaget, Jean(1953). How children From Mathematical Concepts. Scientific American. p. 75.
Piaget, Jean(1956). The child’s conception of space. London: Routledge & K. Paul.
Piaget, J., Inhelder, B., & Szeminska, A. (1960). A child''s conception of geometry. London: Routledge & Kegan Paul.
Piaget, J., & Inhelder, B. (1963). The Child''s Conception of Space. New York: Humanities Press, Inc., p. 97.
Piaget, Jean (1964). De Piaget, Development and Learning, in R.E. Ripple, et al. (Eds.), Piaget Rediscovered, Ithaca N. Y.: School of Education, Cornell University, P.8.-10.
Piaget, J., & Inhelder, B. (1967). The child''s conception of space (F. J, Langdon & J. L. Lunzer, Trans.). New York: W. W. Norton.
Reynolds, A. G. & Flagg, P. W. (1983). Cognitive psychology (2nd. ed.). Boston: Little, Brown and Company.
Rosser, R. A. (1983). The emergence of spatial perspective taking: An information-processing alternative to eqocentrism. Child Development, 54, 660-668.
Rosser, R. A. Ensing, S. S., Glider, P. J., & Lane, S. (1984). An information-processing analysis of children’s accuracy in predicting the appearance of rotated stimuli. Child Development, 55, 2204-2211.
Rosser, R. A. Horan, P. F., Mattson, S. O., & Mazzeo, J. (1984).Reconceptualizing perceptual development: The early emergence of understanding and its limits. Genetic Psychlogy in Monographs, 110, 21-41.
Rosser, R. A. Mazzeo, J., & Horan, P. F.(1984). Reconceptualizing perceptual development: The identification of some dimensions of spatial competence in young children. Contemporary Educational Psychology, 9, 135-145.
Rosser, R. A. Horan, P. F., & Campbell, K. P. (1986). The differential salience of spatial information features in the geometric reproductions of young children. Journal of Genetic Psychology, 147(4), 447-455.
Rosser, R. N. & Cooper, L. A. (1986). Mental images and their transformations. Cambridge, MA: MIT Press.
Rosser, R, A., Lane, S., & Mazzeo, J. (1988). Order of acquisition of related geometric competencies in young children. Child Study Journal, 18, 75-90.
Senk, S., (1989). van Hiele levels and achievement in writing geometry proofs. Journal for Research in Mathematics Education, 3, 309-321.
Sigel, I. E., et al., (1968). Training Procedure for Acquistition 0f Piaget’s Conservation of Quantity: Pilot Study and Its Replication, I. E. Sigel, et al.(Eds.), Logical Thinking in Children, N. Y.: Holt, Rinehart & Winston, PP.295∼308.
Stigler, J. W., Lee, S. Y., & Stevenson, H. W. (1990). Mathematical knowledge of Japanese, Chinese, and American elementary school children, Reston, VA: National Council of Teachers of Mathematics.
Usiskin, Z. P. (1982). van Hiele levels and achievement in secondary school geometry (Final Report of Cogniyive Development and Achievement in Secondary School Geometry Project). Chicago, IL: University of Chicago, Development of Education.(ERIC Reproduction Service No. ED 220228)
Van Hiele, P.M. (1980, April). Levels of thinking: How to meet them, how to avoid them. Pape. presented to the research presession prior to the 58th Annual Meeting of the National Council of Teachers of Mathematics, Seattle.
van Hiele-Geldof, D. (1984). The didactics of geometry in the lowest class of secondary school. In D. Puys, D. Geddes, & R. Tischler (Eds.), English translation of selected writings of Dina van Hiele-Geldof and Pierre M. van Hiele (pp. 1-214). Brooklyn, NY: Brooklyn College, School of Education. (ERIC Document Reproduction Service No. 289697)
van Hiele, P. M. (1986). Structure and insight: A theory of mathematics education. Orlando, FL: Academic Press.
Wallach, L., et al., (1967) Number Conservation: the Roles of Reversibility, Addition－Substration and Misleading Perceptual Cues, Child Development., 38, PP.425∼442.
Wirszup, I. (1976). Breakthroughs in the psychology of learning and teaching geometry. In J. L. Martin &D. A. Bradbard(EDS.), Space and geometry: Papers form a research workshop(pp. 75-97). Columbus, OH: ERIC Center for Science, Mathematics and Environmental Education.(ERIC Document Reproduction service No.132033)