摘 要

本文通過建立數(shù)學(xué)模型對(duì)教學(xué)質(zhì)量進(jìn)行評(píng)估，應(yīng)用馬爾科夫鏈分析法考慮學(xué)生的原始狀態(tài)，在同一標(biāo)準(zhǔn)下把 、 教師學(xué)生的初始成績(jī)分成五個(gè)等級(jí)，確定出狀態(tài)空間，然后根據(jù)施教后的成績(jī)，求出一步轉(zhuǎn)移概率，建立一步轉(zhuǎn)移概率矩陣，最后根據(jù)馬爾科夫鏈的遍歷性求出極限分布 ．對(duì)每一等級(jí)賦予分?jǐn)?shù)，對(duì)教學(xué)效果的定量指標(biāo)加權(quán)平均，由所得的數(shù)學(xué)期望來評(píng)價(jià)教師的教學(xué)質(zhì)量，這種方法克服了學(xué)生原有基礎(chǔ)存在差異的問題.轉(zhuǎn)移概率矩陣 集中反映了教學(xué)質(zhì)量、教學(xué)條件、學(xué)風(fēng)及社會(huì)環(huán)境等等因素的影響， 的極限狀態(tài)說明，這些因素穩(wěn)定時(shí)，教學(xué)效果在這些條件的影響下可能達(dá)到的程度，且這一可能達(dá)到的程度與學(xué)生原有的基礎(chǔ)無關(guān)．由極限分布 可以對(duì)極限狀態(tài)下各等級(jí)人數(shù)進(jìn)行預(yù)測(cè)，通過分析使教師獲取教學(xué)的反饋信息，以對(duì)自己的教學(xué)方式進(jìn)行反思和調(diào)整，促進(jìn)教學(xué)水平的提高．
對(duì)馬氏鏈模型進(jìn)行改進(jìn)，在求得一步轉(zhuǎn)移概率矩陣后，定義學(xué)生成績(jī)“進(jìn)步度”，即學(xué)生進(jìn)步的程度（退步時(shí)進(jìn)步度為負(fù)值），把握學(xué)生成績(jī)的整體進(jìn)步、退步情況，同時(shí)也消除了基礎(chǔ)差異的因素．通過計(jì)算進(jìn)步矩陣和效率值又能對(duì)教學(xué)效果進(jìn)行評(píng)價(jià)，且與馬氏鏈模型所得的結(jié)論一致，同時(shí)還能得到一些很有參考價(jià)值的教學(xué)反饋信息．

關(guān)鍵詞：馬爾科夫鏈；轉(zhuǎn)移概率矩陣；遍歷性；教學(xué)質(zhì)量評(píng)估；極限分布










ABSTRACT

In this thesis, we study a mathematical model which can eva luate the quality of teaching. We analysis the original state of the student by Markov chain, In the same standard, we divided the students who tought by teacher A and teacher B respectively into five levels by initial achievement, then determine the state space. We calculate the one step Transition probability matrix according to the student’s achievement that after teaching, and obtain the Limit distribution by Markov chain Ergodicity. The quantitative index of teaching weighted average by given every level scores, so we can eva luate the teaching quality according to the mathematical expectation. This method overcome the problem of student’s basis difference, therefore, Transition probability matrix reflects the quality of teaching, teaching conditions, style of study and social environment, and so on. Limit distribution shows teaching effectiveness may reach when these factors are stable, and it have nothing to do with the students’ basis difference. By analyzing the feedback information, teachers can adjust their teaching style, and improve the teaching quality.
To improve the Markov chain model, we define ‘progress degree’,which is the extent of the student progress (the progress degree is negative when regress). From the progress degree we can learn the overall progress of students achievement, and it also eliminates the factor of basis difference. We can eva luate the teaching quality by calculate the progress matrix and the value of efficiency, and it’s conclusion is similar to the model of Markov chain, we can also get some valuable feedback information.
Keywords: Markov chain; Transition probability matrix; Ergodicity; Teaching eva luation; Limit distribution





目 錄

前 言 1頁
第一章 馬氏鏈基本理論 2頁
1.1馬爾科夫過程及其概率分布 2頁
1.1.1馬爾科夫性及其馬爾科夫過程 2頁
1.1.2馬兒科夫鏈及轉(zhuǎn)移概率 2頁
1.1.3轉(zhuǎn)移概率矩陣 3頁
1.2遍歷性 3頁
第二章 問題的分析 5頁
第三章 模型的建立與求解 6頁
3.1模型的建立 6頁
3.2實(shí)例求解 8頁
3.3成績(jī)的分析及結(jié)論 9頁
3.4模型的改進(jìn) 11頁
第四章 模型的評(píng)價(jià) 13頁
參考文獻(xiàn) 14頁
附 錄 15頁
致 謝 17頁