幾何證明分段作業的學習效率分析研究
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(2) 62. Journal of Education & Psychology March, 2014, Vol. 37 No. 1, pp. 61-94. Analysis of Learning Efficiency on Geometry Proof by Using Segmentation Tai-Yih Tso*. Feng-Lin Lu**. Abstract. In this study, we adopt cognitive load theory and take Thales’ theorem as an example to investigate how expert and novice’s understanding on geometric proof interacts with perception of cognitive load and further investigate their learning efficiency. We follow mathematics structure and the construct of reasoning with three levels of organization (Duval, 1998) to process segmentation, which forms two learning versions, the segmented version and the non-segmented one. According to 79 experts’ and 66 novices’ performance of comprehension and perception of cognitive load, we utilize measurement of learning efficiency provided Paas and van Merriënboer (1993) to analyze experts’ and novices’ learning efficiency under different learning versions. The results are as follows: Firstly, experts’ performance of comprehension is irrelevant with their perception of cognitive load. However, novices’ performance of comprehension is positively correlates with their willingness to read and confidentiality. Secondly, confidence can be considered to be a reference to examine students’ comprehension on geometric proof. Thirdly, lowering the complexity of tasks will promote experts’ learning *. Tai-Yih Tso: Associate Professor, Department of Mathematics, National Taiwan Normal University ** Feng-Lin Lu (Corresponding Author): PhD Student, Department of Mathematics, National Taiwan Normal University E-mail: fenglin.lu@gmail.com Manuscript received: 2013.01.15; Revised: 2013.09.13; Accepted: 2013.10.01.
(3) 63. efficiency under each level and novices’ learning efficiency under local level.. Keywords: geometry proof, learning efficiency, cognitive load theory.
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(28) 88 教育與心理研究 37 卷 1 期. 明之學習成效. (三) 降低作業複雜度能提升專家 在各理解層次的學習效率與 生手在局部層次的學習效率 從專家和生手在不同版本下的學 習效率分析顯示，專家在分段版本下的 學習效率顯著高於在未分段版本下的學 習效率，說明降低作業複雜度對專家來 說，他們不論是在微觀、局部或是整體 的理解層次，學習效率都會比在未分段 版本下來得好。而生手在分段版本下， 其局部層次的學習效率會顯著高於未分 段版本的學習效率，由此可知，透過分 段方式能夠提高生手在局部層次的學習 效率，但對微觀與整體層次的學習效率 並無顯著效果，表示或許還需要輔以其 他教學策略來幫助生手對微觀與整體層 次的學習。. 二、建議 根據研究結果，由於學生的信心 與理解表現具有很高的相關性，因此， 本研究認為幾何證明的分段呈現能讓學 生較有信心看懂，可以做為教學設計上 的一個基本指標。同時，可以透過詢問 學生對於是否理解一個幾何證明的信心 程度（例如你有多少把握看懂這個證 明？）來做為學生在幾何證明學習成效 上的參考訊息。此外，學生在閱讀幾何 證明時，如何輔助學生，激發他們對看 懂證明的信心有待做進一步的研究分. 析。. 參考文獻 左台益、呂鳳琳、曾世綺、吳慧敏、陳明 璋、譚寧君（2011）。以分段方式降低 任務複雜度對專家與生手閱讀幾何證 明的影響。教育心理學報，43（閱讀 專刊），291-314。 【Tso, T. Y., Lu, F. L., Tzeng, S. C., Wu, H. M., Chen, M. J., & Tan, N. C. (2011). Impact of reducing task complexity by segmentation on experts’ and novices’ reading comprehension of geometric proof problems. Bulletin of Educational Psychology, 43(Special Issue on Reading), 291-314.】. 張俊彥（2008）。TIMSS 2007國際數學與 科學教育成就趨勢調查國家報告。取 自http://www.dorise.info/DER/download _T2007/resault/TIMSS-2007-full_ver.pdf 【 Chang, C. Y. (2008). National report of Trends in International Mathematics and Science Study in 2007. Retrieved from http://www.dorise.info/DER/download_T2 007/resault/TIMSS-2007-full_ver.pdf】. 葉明達、柳賢（2007）。建立判讀理解層 級：高中生進行數學論證判讀活動困 難 之 探 討 。 教 育 與 心 理 研 究 ， 30 （3），79-109。 【Ye, M. D., & Leou, S. (2007). To establish validating comprehension level: Discuss the difficulties that senior high students faced in argumentation validation activity. Journal of Education and Psychology, 30(3), 79-109.】. Ayres, P. L. (2001). Systematic mathematical errors and cognitive load. Contemporary Educational Psychology, 26, 227-248. Ayres, P. L. (2006). Impact of reducing intrinsic cognitive load on learning in a mathematical domain. Applied Cognitive Psychology, 20, 287-298..
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(30) 90 教育與心理研究 37 卷 1 期. Larkin, J., McDermott, J., Simon, D. P., & Simon, H. A. (1980). Expert and novice performance in solving physics problems. Science, 208, 1335-1342. Lin, F. L., Cheng, Y. H., & linfl team (2003). The competence of geometric argument in Taiwan adolescents. International Conference on Science and Mathematics Learning, 12, 16-18. Longo, G. (2012). Theorems as constructive visions. In G. Hanna & M. de Villiers (Eds.), Proof and proving in mathematics education (Vol. 15, pp. 5166). New York, NY: Springer. Mayer, R. E., & Moreno, R. (2003). Nine ways to reduce cognitive load in multimedia learning. Educational Psychologist, 38(1), 43-52. Moreno, R. (2004). Decreasing cognitive load for novice students: Effects of explanatory versus corrective feedback in discovery-based multimedia. Instructional Science, 32, 99-113. Niss, M. (2002). Mathematical competencies and the learning of mathematics: The Danish KOM project. Retrieved from http://www7.nationalacademies.org/mse b/mathematical_competencies_and_the_l earning_of_mathematics.pdf Paas, F., & van Merriënboer, J. (1993). The efficiency of instructional conditions: An approach to combine mental effort and performance measures. Human Factors, 35(4), 737-743. Paas, F., & van Merriënboer, J. (1994). Variability of worked examples and transfer of geometrical problem-solving skills: A cognitive-load approach. Journal of Educational Psychology,. 86(1), 122-133. Raman, M. (2003). Key ideas: What are they and how can they help us understand how people view proof? Educational Studies in Mathematics, 52, 319-325. Reigeluth, C. M. (1983). Instructional design theories and models: An overview of their current states. Hillsdale, NJ: Lawrence Erlbaum. Selden, A. (2012). Transitions and proof and proving at tertiary level. In G. Hanna & M. de Villiers (Eds.), Proof and proving in mathematics education (Vol. 15, pp. 391-420). New York, NY: Springer. Selden, A., & Selden, J. (2003). Validations of proofs considered as texts: Can undergraduates tell whether an argument proves a theorem? Journal for research in mathematics education, 34(1), 4-36. Sweller, J. (2010). Element interactivity and intrinsic, extraneous, and germane cognitive load. Educational Psychology Review, 22(2), 123-138. Sweller, J., Ayres, P., & Kalyuga, S. (2011). Cognitive load theory. New York, NY: Springer. Sweller, J., van Merriënboer, J., & Paas, F. (1998). Cognitive architecture and instructional design. Educational Psychology Review, 10(3), 251-296. Van Gerven, P. W. M., Paas, F., van Merriënboer, J., Hendriks, M., & Schmidt, H. G. (2003). The efficiency of multimedia learning into old age. The British Journal of Educational Psychology, 73(4), 489-505. van Merriënboer, J., & Sweller, J. (2005). Cognitive load theory and complex learning: Recent developments and.
(31) 幾何證明分段作業的學習效率分析研究 91. future directions. Educational Psychology Review, 17, 147-177. Weber, K. (2001). Student difficulty in constructing proof: The need for strategic knowledge. Educational Studies in Mathematics, 48, 101-119. Weber, K., & Mejia-Ramos, J. (2011). Why and how mathematicians read proofs: An exploratory study. Educational Studies in Mathematics, 76(3), 329-344. Yang, K. L., & Lin, F. L. (2008). A model of reading comprehension of geometry. proof. Educational Studies in Mathematics, 67(1), 59-76. Zaslavsky, O., Nickerson, S. D., Stylianides, A. J., Kidron, I., & Winicki-Landman, G. (2012). The need for proof and proving: Mathematical and pedagogical perspectives. In G. Hanna & M. de Villiers (Eds.), Proof and proving in mathematics education (Vol. 15, pp. 215-229). New York, NY: Springer..
(32) 92 教育與心理研究 37 卷 1 期. 附錄. 幾何證明閱讀理解測驗. 1.請依據下列敘述「作 EF ⊥ AB 且F在 AB 上」，在圖一中畫出 EF 。. 圖一. HJJG. 2.如圖二， BH ⊥ DE ，請用一個算式表示 + EDB 的面積。. 圖二. 3.請依據圖三回答下列問題： (1) 請說明「因為 DE // BC ，所以 BH = CK 。」 這個敘述成立的理由。 理由：. (2) 若 BH = 3 ，則 CK =__________。. 圖三.
(33) 幾何證明分段作業的學習效率分析研究 93. 4.如圖四，在 + ACD 中，E在 AC 上， 試證. + EAD面積 AE 。 = + ECD面積 EC. 證明：. 圖四. 5.如圖五， DE // BC ，. HJJG. HJJG. HJJG. H、K在 DE 上且 BH ⊥ DE ， CK ⊥ DE ， 試證 + EDB面積 =+ ECD面積 。 證明：. 圖五.
(34) 94 教育與心理研究 37 卷 1 期. 6.如圖六，已知 DE // BC ，請證明. AD AE = 。 DB EC. （下面附有參考敘述，可供參考作答。） 證明過程：. 圖六. 註： 下面14個敘述都是正確的敘述，不需要再證明（參考圖六）。你可以選取一些 適當的敘述（或代號）來結構出. AD AE = 的證明過程。 DB EC.
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