本研究的主要目的在於以貝氏網路的方法論著手，發展學生模式與證據模式，建立一個以貝氏網路為基礎的錯誤類型與子技能認知診斷模式，並以國小五年級數學學習領域中「小數」單元作為特定應用領域，而後評估此貝氏網路認知診斷模式應用於實際數學學習診斷的有效性。在評估貝氏網路認知診斷模式應用之有效性的過程中，本研究將探討加入子技能變項、測驗資料型態、分類決斷值以及訓練樣本大小等四個因素對貝氏網路認知診斷模式診斷正確率之影響，以進一步了解如何建立最佳的貝氏網路認知診斷模式，使其未來能與電腦科技結合，發展以貝氏網路為基礎的適性診斷系統，以便更有效地應用於實際教學情境，進而提昇學生的學習成效。
研究結果發現：
一、應用貝氏網路認知診斷模式於數學學習領域中國小五年級「小數」單元，可建構出一套能同時診斷錯誤類型及子技能，適用於單元教學的認知診斷模式。
二、專家診斷結果與訪談診斷具有統計上的一致性，因此，以專家診斷結果作為本研究模式評估之效標是適當的。
三、貝氏網路認知診斷模式應用於國小五年級「小數」單元內四種評量重點，在錯誤類型與子技能部分皆可達到良好的診斷結果，惟子技能的診斷正確率低於錯誤類型。
四、加入子技能的貝氏網路認知診斷模式對錯誤類型的診斷正確率有些許提昇，但影響的程度很小。
五、以多選項記分資料作為證據的貝氏網路之診斷正確率最佳，以二元記分資料為證據的貝氏網路次之，以二種IRT選答率為證據的貝氏網路診斷正確率最差。
六、使用由訓練樣本所選取的動態分類決斷值之診斷正確率優於採用固定式的分類決斷值。
七、在使用動態分類決斷值的前提下，訓練樣本之大小對錯誤類型與子技能的診斷正確率影響程度均不大。
最後，研究者針對研究結果提出建議，作為本研究貝氏網路認知診斷模式在教學應用以及未來研究的參考。

The main purpose of this study is to develop the cognitively diagnostic model based on Bayesian networks that include the student model and the evidence model, and to explore the efficiency of using Bayesian networks for modeling assessment data and identifying bugs and sub-skills of decimals after students have learned the related contents. Four factors are involved in this study: the effects of using sub-skill nodes in addition to bug nodes, the effects of various scoring type of assessment data, the effects of various cut-points, and the effects of setting various size of training sample. By varying these factors to assess the effectiveness of the generated Bayesian network models work in predicting the existence of bugs and sub-skills in order to find the best cognitively diagnostic model based on Bayesian networks. The results of this study will lay a foundation for building a computerized adaptive diagnostic mathematics test based on Bayesian networks for elementary schools successively.
The major findings of this study are summarized as follows:
1.The cognitively diagnostic model based on Bayesian networks can be constructed and employed in the diagnosis of bugs and sub-skills of decimals in the fifth grade mathematics of elementary school successfully.
2.Experts’ diagnostic results are agreed with interview statistically, so, it is suitable to assess the effectiveness of the generated Bayesian network models using experts’ diagnostic results as criterion.
3.The results show that using the cognitively diagnostic model based on Bayesian networks to diagnose the existence of bugs and sub-skills of students can get good performance, but the sub-skill prediction rates are lower than the bug prediction rates.
4.The increased bug prediction rates for adding sub-skill nodes are minimal.
5.The prediction rates for employing specific multiple-choice answer information are better than binary-answer Bayesian network. The prediction rates for employing the probability of item response theory information are worst.
6.The dynamic cut-points based on the response data of training samples can get better performance than the fixed cut-points.
7.The value of prior distribution is the important factor in Bayesian networks estimation. In this study, the prediction rates of bugs are increasing with the size of training sample, but the effects are minimal.
Based on the findings, some suggestions regarding instructional application and future research also offer.

【中文部分】
艾如昀（1994）。國小學生處理小數的歷程與困難。國立中正大學心理研究所碩士論文，未出版，嘉義。
江愛華（2002）。國小五年級小數診斷教學之研究。國立屏東師範學院數理教育研究所碩士論文，未出版，屏東。
江愛華、劉曼麗（2002）。國小學童「小數比較大小」的錯誤類型之探討。科學教育，15，22-29。
吳昭容（1996）。先前知識對國小學童小數概念學習之影響。國立台灣大學博士論文，未出版，台北市。
吳裕益、黃桂君（1995）。能力診斷分析模式之建立與效度驗證。測驗年刊，43輯，33-52。
李坤崇（1998）。人性化、多元化教學評量－從開放教育談起。載於高雄市政府公教人力資源發展中心主編：多元教學評量，91-134。高雄市：高雄市政府公教人力資源發展中心。
李俊儀（2005）。以貝氏網路為基礎的電腦化適性測驗選題策略-以國小數學科診斷測驗為例。國立台中師範學院數學教育研究所碩士論文，未出版，台中市。
余民寧（1997）。教育測驗與評量---成就測驗與教學評量。台北市：心理出版社。
余民寧（2003）。線上認知診斷評量模式之研究：以國小數學科低成就學生為對象。行政院國家科學委員會專題研究計畫成果報告（NSC91-2521-S-
004-011）。台北市：國立政治大學。
宋德忠、陳淑芬、張國恩（1998）。電腦化概念構圖系統在知識結構測量上的應用。測驗年刊，45(2)，37-56。
杜建台（1996）。國小中高年級學童小數概念理解之研究。國立台中師範學院國民教育研究所碩士論文，未出版，台中市。
林世華、梁仁楷、劉子鍵（1998）。認知設計系統的建構與試題輔助產生引擎的運作:以二度空間視覺化測驗為例。師大學報，43（1），17-31。
林軍治（1986）。兒童數學學習之心理基礎。台灣省教育廳。
林垣圻（2006）。以貝氏網路為基礎的適性測驗電腦化的可行性評估-以國小數學科診斷測驗為例。國立台中師範學院數學教育研究所碩士論文，未出版，台中市。
周筱亭（1990）。電子計算器對於國民小學小數運算學習之影響（I）。台灣省國民學校教師研習會。
周筱亭、方美娥、秋子虎（1988）。國民小學數學課程研究資料（75、76年度工作報告第二輯）。台灣省國民學校教師研習會。
施淑娟（1997）。認知網路評量模式及其實例應用之研究－以「分數的加法」單元為例。國立台中師範學院國民教育研究所碩士論文，未出版，台中市。
郭孟儒（2002）。國小五年級學童小數迷思概念及其成因之研究。國立屏東師範學院數理教育研究所碩士論文，未出版，屏東。
許天維、劉湘川（1994）。數學教育指標研究－國小四至六年級技能與解題能力的學習進展指標（Ι）。行政院國家科學委員會專題研究計畫成果報告（NSC84-2511-S-142-004）。台中市：國立台中師範學院。
許雅菱（2005）。貝氏網路在教育測驗分析上的應用。國立台中師範學院測驗統計研究所碩士論文，未出版，台中市。
陳永峰（1998）：國小六年級學童小數知識之研究。國立屏東師範學院國民教育研究所碩士論文，未出版，屏東。
黃桂君（1996）。聽覺障礙學生與普通學生分數減法演算法則之分析。新竹師院學報，第九期，363-390。
黃桂君（2002）。學習困難學童基本數學概念診斷教學系統之建構。行政院國家科學委員會專題研究計畫成果報告（NSC90-2614-S-134-001）。新竹市：國立新竹師範學院。
張國恩、侯文娟（1995）。Application of Neural Network for Implementing a Practical Student Model . Proceedings of the National Science Council(Part D) , 5(2), 133-142。
趙育倫（1997）。以無參數IRT理論為基礎的規則空間。國立台中師範學院國民教育研究所碩士論文，未出版，台中市。
劉曼麗（1998）。國小數學教學實踐課程開發研究－小數認識及加減部份。行政院國家科學委員會專題研究計畫成果報告（NSC87-2511-S-153-
011）。屏東：國立屏東師範學院。
劉曼麗（1999）。國小學童小數知識現況之研究。行政院國家科學委員會專題研究計畫成果報告（NSC88-2511-S-153-007）。屏東：國立屏東師範學院。
劉曼麗（2001）。國小學童的小數知識研究。屏東師院學報，第十四期，823-858。
劉曼麗（2002）。台灣地區國小學童小數概念研究（II）：國小學童「小數與小數運算」概念量測工具發展研究。行政院國家科學委員會專題研究計畫成果報告（NSC90-2521-S-153-003）。屏東：國立屏東師範學院。
劉湘川（2000a）。點二系列相關試題鑑別指數之值譜分析及其在IRT上之應用。測驗統計年刊，第八輯，1-20。台中市：國立台中師範學院。
劉湘川（2000b）。多元計分三參數試題選項分析固定效應模式。測驗統計年刊，第八輯，21-36。台中市：國立台中師範學院。
劉湘川（2001a）。相關加權核平滑化無參數試題選項特徵曲線估計法及其IORS整合模式。第五屆華人社會心理與教育測驗學術研討會。C5.1 ，1-10。台北市：中國測驗學會、台灣師範大學。
劉湘川（2001b）。核平滑化試題選項特徵曲線與選項關聯結構整合擴充模式。測驗統計年刊，第九輯，24-38。台中市：國立台中師範學院。
劉湘川（2002）。高階相關比累進加權核平滑化試題選項分析綜合模式。測驗統計年刊，第十輯，197-218。
簡茂發（1999）。多元化評量之理念與方法。教師天地，99期，11~17。
簡茂發、劉湘川（1993）。八十一學年度國民教育階段學生基本學習成就評量國小組試題編製及抽測結果報告。台中市：國立台中師範學院。

【外文部分】
Almond, R. G., & Mislevy, R. J. (1999). Graphical models and computerized adaptive testing. Applied Psychological Measurement, 23,223-237.
Anderson, J. R.(1993). Rule of the mind. Hillsdale, NJ: Lawrence Erlbaum Associates.
Andrich, D. (1978). A binomial latent trait model for the study of Likertstyle attitude questionnaires. British Journal of Mathematical and Statistical Psychology, 31, 84-98.
Andrich, D. (1982). An extention of the Rasch model for ratings providing both location and dispersion parameters. Psychometrika, 47, 105-113.
Ashlock, R. B.(1990). Error patterns in computation：A semi-programmed approach (5th ed).Columbus,Ohio：Charles E.Merrill.
Birnbaum, A. (1968). Some latent trait models and their uses in inferring an examinee's ability. In F. M. Lord., & M. R. Novick, Statistical theories of mental test scores. (pp.395-479). Reading, MA : Addison-Wesley.
Bock, R.D. (1972). Estimating item parameters and latent ability when responses are scored in two or more nominal catagories. Psychometrika, 37, 29-51.
Britton, B. K., & Tidwell, P. (1995). Cognitive structure testing: A computer system for diagnosis of expert-novice differences. In P. Nichols, S. Chipman & R. Brennan. (Eds.) Cognitively Diagnostic Assessment. Hillsdale, NJ: Lawrence Erlbaum Associates.
Brown, J. S., & Burton, R. R. (1978). Diagnostic Models for Procedural Bugs in Basic Mathematical Skill. Cognitive Science, Vol.2, No.2, pp. 155-192.
Castillo, E., Gutierrez, J. M., & Hadi, A. S. (1997). Expert systems and probabilistic network models. New York: Springer.
Embretson, S. E. (1991). A Multidimensional latent trait model for measuring learning and change. Psychometrika, 56, 495-515.
Gelman, A., Carlin, J., Stern, H., & Rubin, D.B. (1995). Bayesian data analysis. London: Chapman & Hall.
Heckerman, D., Mamdani, A., & Wellman, M. P. (1995). Real-world applications of Bayesian networks. Communications of the ACM, 38.
Henrion, M., & Druzdzel, M. J. (1990). Qualitative propagation and scenario-based approaches to explanation of probabilistic reasoning. Proceedings Sixth Conference on Uncertainty in Artificial Intelligence, 10-20. Cambridge.
Hiebert, J., & Wearne, D. (1985). A model of students’ decimal computation procedures. Cognition and Instruction, 2(3&4),175-205.
Jensen, F. V. (1996). An introduction to Bayesian networks. New York: Springer.
Jensen, F. V. (2001). Bayesian networks and decision graphs. New York: Springer.
Lee, Jihyun. (2003). Diagnosis of bugs in multi-column subtraction using Bayesian networks. Unpublished Ph.D., Columbia University.
Johnson, P. J., Goldsmith, T. E., & Teague, K. W. (1995). Similarity, structure, and knowledge: A representational approach to assessment. In P. Nichols, S. Chipman & R. Brennan. (Eds.) Cognitively Diagnostic Assessment. Hillsdale, NJ: Lawrence Erlbaum Associates.
Lauritzen, S. L., & Spiegelhalter, D. J. (1988). Local computations with probabilities on graphical structures and their application to expert systems. Journal of the Royal Statistical Society, Series B, 50(2), 157-224.
Liu, C. L. (2004a) “A Bayesian network-based simulation environment for investigating assessment issues in intelligent tutoring systems. Int. Computer Symposium. Taipei.
Liu, C. L. (2004b) “Using mutual information for adaptive student assessments,” ICALT’04, pp. 585-589.
Lord, F. M.(1980). Application of item response theory to practical testing problems. Hillsdale, NJ : Lawrence Erlbaum Associates.
Martin, J., & VanLehn, K. (1995). A bayesian approach to cognitive assessment. In P. Nichols, S. Chipman & R. Brennan. (Eds.) Cognitively Diagnostic Assessment. Hillsdale, NJ: Lawrence Erlbaum Associates.
Marshall, S. P. (1993). Assessing schema knowledge. In N. Frederiksen, R. J. Mislevy, & I. Bejar(Eds.), Test theory for a new generation of tests(pp.155-180). Hillsdale, NJ: Lawrence Erlbaum Associates.
Masters, G. N. (1982). A Rasch model for partial credit scoring. Psychometrika, 47, 149-174.
Mislevy, R. J. (1994). Evidence and inference in education assessment. Psychometrika, 59, 439-483.
Mislevy, R. J.(1995). Probability-based Inference in Cognitive Diagnosis. In P.Nichols, S.Chipman & R. Brennan. (Eds.) Cognitively Diagnostic Assessment. Hillsdale, NJ: Lawrence Erlbaum Associates.
Mislevy, R. J., & Gitomer, D. H. (1996). The role of probability-based inference in an intelligent tutoring system. User-Modeling and User-Adapted Interaction, 5, 253-282.
Mislevy, R. J., Almond, R.G., Yan, D., & Steinberg, L. S. (1999). Bayes nets in educational assessment: Where do the numbers come from? In K. B. Laskey & H. Prade (Eds.), Proceedings of the Fifteenth Conference on Uncertainty in Artificial Intelligence (437-446). San Francisco: Morgan Kaufmann Publishers, Inc.
Mislevy, R. J., Almond, R., Dibello, L., Jenkins, F., Steinberg, L., Yan, D., & Senturk, D.(2001). Modeling Conditional Probabilities in Complex Educational Assessments. CSE Technical Report 580. Available:http://cresst96.cse.ucla.edu/reports/TR580.pdf
McDonald, R. P. (1967). Non-linear factor analysis. Psychometric Monograph, No.15.
Murphy, K. (2005a). Software Packages for Graphical Models / Bayesian Networks. Retrieved October 31 , 2005, from http://www.cs.ubc.ca/~murphyk/Bayes/
-bnsoft.html.
Murphy, K. (2005b). Bayes Net Toolbox for Matlab. Retrieved November 26, 2005, from http://sourceforge.net/project/showfiles.php?group_id=147099.
Nadaraya, E. A.(1964). On estimating regression. Theory Prob. Applic. 9,141-142.
Naveh-Benjamin, M., Linn, Y. G., & Mckeachie, W. J.(1995). Inferring students’ cognitive structures and their development using the “Fill-in-the-structure”(FITS) technique. In P. Nichols, S. Chipman & R. Brennan. (Eds.) Cognitively Diagnostic Assessment. Hillsdale, NJ: Lawrence Erlbaum Associates.
Nichols, P. D.(1994). A framework for developing cognitively diagnostic assessment. Review of Educational Research, 64, 575-603.
Pearl, J. (1988). Probabilistic Reasoning in Intelligent Systems: Networks of Plausible Inference. San Mateo, CA: Morgan Kaufmann.
Polk, T. A., VanLehn, K., & Kalp, D. (1995). ASPM2:Progress toward the analysis of symbolic parameter models. In P.Nichols, S.Chipman & R. Brennan. (Eds.) Cognitively Diagnostic Assessment. Hillsdale, NJ: Lawrence Erlbaum Associates.
Ramsay, J.O. (1991). Kernel smoothing approaches to nonparametric item characteristic curve estimation. Psychometrika, 56, 611-630.
Rasch, G. (1960). Probabilistic models for some intelligence and attainment tests. Copenhagen Danmark : Danmark's Paedogogiske Institute for Educational Research. (Chicago : The University of Chicago Press, 1980).
Russell, S., & Norvig, P. (1995). Artificial Intelligence: A modern approach. Upper saddle River, NJ: Prentice Hall Inc.
Sahin, F. (2000). A Bayesian network approach to the self-organization and learning in intelligent agents. Unpublished Ph.D., Virginia Polytechnic Institute and State University, Blacksborgh, VA.
Samejima, F. (1973). Homogeneous case of the continuous response model. Psychometrika, 38, 203-219.
Samejima, F. (1979). A new family of models for the multiple choice item.(Research Report No. 79-4). Knoxville：University of Tennessee, Department of Psychology.
Samejima, F.(1988). Final Report: Advancement of latent trait theory. Final Report of N00014-81-C-0569, Office of Naval Research.
Shafer, G.., & Pearl, J. (1990). Readings in uncertain reasoning. San Mateo, CA: Morgan Kaufmann.
Shih, S. C., & Kuo, B. C. (2005). Using Bayesian Networks for Modeling Students’ Learning Bugs and Sub-skills. Lecture Notes in Artificial Intelligence, 3681, 69-74.
Spiegelhalter, D. J., Dawid, A. P., Lauritzen, S. L., & Cowell, R. G. (1993). Bayesian analysis in expert systems. Statistical Science, 8(3),219-247.
Steinberg, L.S., & Gitomer, D.G. (1996). Intelligent tutoring and assessment built on an understanding of a technical problem-solving task. Instructional Science, 24, 223-258.
Tatsuoka, K. K. (1983). Rule space: An approach for dealing with misconceptions based on item response theory. Journal of Educational Measurement,20(4),345-354.
Tatsuoka, K. K. (1990). Toward an integration of item response theory and cognitive error diagnosis. In N. Frederiksen, R. Glaser, A. Lesgold, & M.Shafto(Eds.). Diagnostic monitoring of skill and knowledge acquisition (pp.453-488). Hillsdale, NJ: Lawrence Erlbaum.
Thissen, D. & Steinberg, L. (1984). A response model for multiple choice items. Psychometrika, 49, 501-519.
Thissen, D. (1991). MULTILOG Version 6 User’s Guide. Mooresville, IN：Scientific Software,Inc.
Vanlehn, K. (1990). Mind bugs: the origins of procedural misconceptions. Cambridge, MA: The MIT Press.
Vomlel, J.(2004a). Building Adaptive Tests using Bayesian networks, Kybernetika, 40, 333-348.
Vomlel, J.(2004b). Bayesian networks in educational testing, International Journal of Uncertainty, Fuzziness and Knowledge Based Systems, 12, 83-100.
Weiss, D.J., & Yoes, M.E. (1991). Item response theory. In R.K. Hambleton & J.N. Zaal (eds.), Advances in educational and psychological testing. Boston: Kluwer Academic Publishers.
Watson, G. S. (1964). Smooth regression analysis. Sankhya A26, 359-372.
Yamamoto, K. (1987). A model that combines IRT and latent class models. Unpublished doctoral dissertation, University of Illinois-Champaign.
Yu, M. (1991a). A two-parameter partial credit model. Unpublished doctoral dissertation, University of Illinois, Urbana, IL.
Yu, M. (1991b). The assessment of partial knowledge. Journal of National Chengchi University, 63, 401-428.