数学建模,如何客观、合理的评价学生学习状况

-.z.如何客观、合理的评价学生学习状况摘要现行的以考试成绩衡量学生学习状况的方法比拟主观，且评价方式单一，忽略了不同根底水平的同学的进步程度，为了鼓励优秀学生努力学习取得更好的成绩，同时鼓励根底相对薄弱的学生树立信心，不断进步，我们需要建立一个客观，合理的评价学生状况的数学模型。考虑到以上情况，本文通过以下几步来到达目的。步骤一：通过分析题目所给198名学生的整体成绩情况，包括大一两个学期每个学期的整体平均成绩、及格率、方差、标准差等多项指标有关，通过所给数据，得到图表。分析数据充分理解学生的学习情况，更有利于以下两个模型的进展，为模型的建立提供参考。步骤二：对于全面、客观、合理的评价学生的学习状况，我们采用了二个模型：模型一：利用黑尔指数法求得的进步分数和层次分析法进展评价：设定适当的权系数，使最终成绩更为合理。本专业为工科类专业，应更加重视专业学习能力，因此专业课程所占权系数较高，成绩也能更好的选拔专业能力强的学生。同时为了鼓励进步学生，进步分也占有局部权限，能够起到很好的鼓励作用。为此我们设置：最终成绩Y=0.55*专业课程+0.4*其他课程+0.05*进步分数。模型二：采用成绩标准化模型对成绩进展评价：采用对数变换将负偏态的成绩分布正态化，并用Matlab进展了正态检验。从而学生成绩的差距分布更为合理，成绩偏低的学生变换后将处于中等位置，得到适当的鼓励，改变了负偏态分布中较多学生成绩集中在高分段或低分段的现象。然后，将正态分布归一化为标准正态分布，消除每个学期评价考核体系的不稳定性因素，得到每个学生各学期的“有效成绩〞。并基于"有效成绩"提出了等级评定子模型，确定了等级分数线，更清楚的说明了每个学生在整体位置。关键词：黑尔指数层次分析成绩标准化有效成绩问题重述现行的评价方法相比照拟局限、主观、有失公允，只能对学习根底好的学生产生鼓励作用，而不能对所有学生尤其是后进学生起到鼓励作用，这种评价弊端开场被越来越多的人关注。评价学生学习状况的目的是鼓励优秀学生努力学习取得更好的成绩，同时鼓励根底相对薄弱的学生树立信心，不断进步。然而，现行的评价方式单纯的根据“绝对分数〞评价学生的学习状况，忽略了根底条件的差异；只对根底条件较好的学生起到促进作用，对根底条件相对薄弱的学生很难起到鼓励作用。请通过科学合理的分析评价方法，设计适宜的数学模型，对学生学习状况做出全面、客观、合理的整体评价。问题分析根据统计学知识，这198名学生的整体情况应包括两个学期整体的平均成绩、极差、中位数、方差、标准差、偏度等多项指标。通过整理所给数据，可以得到两个学期所需的相关指标，并进展正态性检验。可以由此对学生的整体情况进展全面、直观的说明。评价一个学生的学习情况可以有很多方面的因素，如智力因素、学习环境、学习根底等。但我们只有学生两学期的考试成绩，如果考虑很多因素就不合现实，具有不可操作性，因此，只着眼于考试成绩和进步状况两类因素。模型一：为了科学的评价学生的学习情况，必须按照学校专业的特点，制定合理的学习情况评价体系。由美国学者T.L.Satty提出的层次分析法把人的思维层次化、数量化，并用数学方法为分析评价或决策提供定量依据。我们借用该方法建立评价生成绩的层次构造如下。为了鼓励根底薄弱但是进步很大的学生，我们设置了进步分工程，在最终成绩的评定中占有一定比例。我院为工科类院，注重专业素养的培养，因此专业课程成绩十分关键。学习状况的评价第一层学习状况的评价进步得分专业课程其他课程进步得分专业课程其他课程第二层在第二层以下为第三层，是具体的课程或科目。设定适当的权系数，使最终成绩更为合理。模型二：为了使得学生之间成绩的差距分布更为合理，原来成绩偏低的学生经过变换后处于中等位置，从而使他们会得到适当的鼓励，树立信心，不断进步，并改变负偏态分布中有较多同学集中在高分段或低分段的情况，鼓励成绩较低的学生努力学习取得更好的成绩，有必要将负偏态分布的学生成绩通过数学手段变换为正态分布，而且变换成正态分布后，还会对数据处理带来极大的方便。由于每个学期的评价体系存在一定的波动，例如考核中不可防止的难易程度的变化等因素会使各学期之间的同一学生成绩缺少一定的比拟性。为了消除学期之间的差异，为此将正态分布再经过变换为标准正态分布，使得同一学生在不同学期的成绩具有更可靠的可比性。由此我们最终得到了标准化的成绩，称之为“有效成绩〞，并运用该成绩对学生的学生状况进展评价。模型假设1.假设总成绩总分都是100分；2.假设两个学期的学生人数未发生改变；3.假设每位同学的学习考试环境一样；4.假设每位同学的学习能力根本保持不变；5.假设数据中的零是特殊情况导致的；6.假设影响学生成绩的因素主要有真实成绩与进步程度。符号说明与名称解释4.1符号说明i第i个学生，i=1,2…199j第j个指标，j=1,2,…7学生第学期的成绩n=1,2学生每学期的进步程度学生的综合评定指数为实际学习成绩为学习成绩进步度表示第个学生第学期的成绩表示第个学生第学期的进步度4.2名称解释黑尔指数转换:用指数方差确定进步幅度和难度，并根据上下分的进步幅度，以不同的难度权重，最后根据粗测验获得的“进步分〞的多少来进展评价。层次分析：由美国学者T.L.Satty提出的层次分析法把人的思维层次化、数量化，并用数学方法为分析评价或决策提供定量依据。成绩标准化模型：采用对数变换将负偏态的成绩分布正态化的模型。模型的建立与求解5.1对成绩进展分析利用所给13级大一成绩进展统一，得到了学生成绩总体分布的情况如下图。数据处理时把成绩分为四个等级。80分及以上为优秀，70分到80分之间为良好，60分到70分之间为合格，小于60分为不及格。从结果可看出，成绩优秀和良好的同学居多。但是第二学期优秀人数下降，良好人数上升。不及格人数也上升。说明专业课程难度增加，学生需要更努力才能更上专业课难度的上升。利用E*cel对统计的数据进展进一步分析，得到模型需要的相关数据。如下：E*cel对统计后数据进一步分析计算得到的表格如下：由上表可以得出：一、二、学期的偏度为负，说明呈负偏态分布，即分数小于平均分的学生比大于平均分的学生多。同时我们还可以得到以下结论：〔1〕、两个学期的及格率均在97%以上，可以肯定大局部学生的学习能力；〔2〕、第二学期的标准差较第一学期的大，说明第二个学期的分数较为分散，学生的差距较大；〔3〕、两个学期中，分数大于90分的学生比拟少，所以该学校应该加强尖子生这块的培养；〔4〕、两个学期的总平均成绩在75分左右，学生的总体学习情况良好。5.2.评价学生的学习情况5.2.1所给成绩数据处理进步得分的计算：黑尔指数转换:用指数方差确定进步幅度和难度，并根据上下分的进步幅度，以不同的难度权重，最后根据粗测验获得的“进步分〞的多少来进展评价。根据原始的黑尔指数表格回归分析出T分与进步分公式，根据黑尔指数换算表回归分析出T分与相应的进步分y的关系：y=0.09966*1.0473T,得到进步分yij。先是利用公式，其中*ij是第i个学生第j学期的成绩，j,*j分别每个学期的学生的总体的平均分与标准差。利用这个公式将成绩转换为T分。将两学期两次额成绩分别转换成T分，然后将T分转换为进步分，见附录。MATLAB计算程序见附录以下表是**前二十位学生为例所求的T分和进步分层次分析法:我们借用该方法建立评价生成绩。为了鼓励根底薄弱但是进步很大的学生，我们设置了进步分工程，在最终成绩的评定中占有一定比例。我院为工科类院，注重专业素养的培养，因此专业课程成绩十分关键。在层次分析法中，同一层中的各项成绩对上一层的奉献程度不是均等的，带有不同的权重，总成绩按加权平均计算。以下为大一两学期科目所对应字母设置：第一学期高等数学ⅠA计算机工程制图大学计算机根底Ⅰ电路分析abcd思想道德修养与法律根底体育电子测量技能训练军训和军事理论Ⅰ大学外语ⅠAefghi第二学期体育马克思主义根本原理概论大学生职业开展与就业指导Ⅰ大学外语ⅠB中国近现代史纲要概率论与数理统计模拟电路大学物理ⅡC语言程序设计EFGHIABCD最终成绩Y=0.55*专业课程+0.4*其他课程+0.05*进步分数以下权系数的设置根据本小组同学对本专业了解。专业课程和其他课程的分配也是本小组观点，不代表学院分配。由于能力有限抑或是对本专业了解还不够深入，权限系数的设置有待优化。专业课=一二学期平均第一学期专业课成绩=0.3*a+0.2*b+0.2*c+o.3*d第二学期专业课成绩=0.2*a+0.3*b+0.2*c+o.3*d其他课程=一二学期平均第一学期其他课程成绩=0.2*e+0.25*f+0.2*g+o.1*h+0.25*i第二学期其他课程成绩=0.25*E+0.2*F+0.1*G+o.25*H+0.2*I以下表是**前二十位学生为例所求的最终成绩所有数据见附录成绩标准化模型原始成绩的标准化下面讲述如何将原始成绩变换为标准化的成绩。第一步：原始成绩的正态化及其检验假设(i=1,2,……612)为612个学生的*一学期的原始成绩,由将偏态分布变换为正态分布的对数变换法，令：此时这些学生的变换成绩yi满足正态分布。由于该函数是单调递减函数，原始成绩高的反而变换成绩低，为了与传统习惯保持一致，再经过下述变换，此时的为正态化之后的成绩。从频次直方图可以看出根本符合正态分布。为了进一步验证成绩分布是否为正态分布，我们用matlab进展了正态性检验，检验程序见附录。检验结果如下图，从图中可以看出实际观测值与期望值在中央横线的一段，坐标点落在中央横线附近，在中央横线的两端则有一定的偏离，但绝大局部偏离值均小于0.05，仅有个别点偏离较大。可见，学期1,2的成绩呈现正态分布。第二步：将正态分布标准化因为已是正态分布，因而可由正态分布转化为标准正态分布的相关公式，将转化到服从标准正态分布，得：有效成绩i=1，2，……612均值为，方差为σ2=。这就是我们所定义的有效成绩。下表是我们应用E*CEL，由两学期原始成绩计算的有效成绩(由于篇幅有限成绩列表均只列出局部成绩，所有成绩见附录)。图为有效成绩的频次分布直方图，可以看出它已很好的符合正态分布。标准正态化程序见附录。由于篇幅有限，以下为**前二十位同学的有效成绩：第一学期第二学期**正态分布成绩有效成绩正态分布成绩有效成绩0001153.6149-12.39978461142.1672-9.3109900740002154.1093-0.971879522142.69040.647288080003154.0695-1.891844374142.4548-3.8369824340004153.95-4.654050399142.4926-3.1175196780005154.1508-0.012619689142.67490.3522702830006154.1352-0.373208928142.71811.1745134330007154.21071.371950527142.6188-0.7155038080009154.1720.47741184142.93235.2514690540010154.42286.274577287143.211710.569402760011154.32974.122599204142.86964.0580744820012154.1508-0.012619689142.4487-3.9530862130013154.21631.501392818142.94735.5369701480014153.9205-5.335933894142.3891-5.0874772260015154.0551-2.224695979142.75911.954883090016153.9246-5.241163646142.4579-3.7779788750017154.29283.269666967142.88374.326445510018154.1404-0.253012515142.69040.647288080019154.0409-2.552924644142.3748-5.3596549350020154.46557.261574754142.92735.156302023此时应用有效成绩已经能够对学生的学习情况进展公平、合理的评价，因为原始分数没有比拟的参照点，故而不可比。而有效成绩以学生整体的平均分数作为比拟的基准，以标准差作为单位，而且它的根本形式都是平均数为零、标准差为1。因而无论不同学期成绩的平均分和标准差多么不同，一经转换为均值为零和标准差为1的标准分数，则不同学期成绩所处的相对地位是平行的，从而有了可比性。这时学生学习状况的评定不再是简单的绝对分的比拟，名次的提高，也即进步成为了决定学生成绩的重要因素。有的同学虽然总分不高，但是根据有效成绩却排名靠前，充分肯定了学生的进步。图5第三步：基于有效成绩的等级评定在将原始成绩化为符合标准正态分布的数据之后，我们将建立一种评分制度——标准化分数为根底的成绩标准化评价模型。：为总体算术平均值：为总体标准差我们根据落入和外的概率来确定成绩的等级，取落人的概率为68％，落入外的概率为10.075％，落入余下的概率为21.5％，则可确定优秀、不合格各占50％，良好、台格各占10.075％，中等占69.35％。各等级的对应分数线为概率等级分数线〔DP〕，经对数变换的成绩数据复原成原始分数，即为各等级的分数。经计算可得各等级分数线表，如下所示：以第一学期的为例，原始分数高于85的认定为优秀，而低于71.11的认定为不合格，但在这种评定标准中优秀与不合格所占比例较小，大局部的人集中在中等层次，从来相对于一般的评定标准更多的人上升到了合格或中等。但这一评定标准是建立在整体成绩为正态分布的根底上的，从而当出现因为试卷人为的过于简单而了导致大量学生的成绩偏高时，运用该评定方法则可以提高不合格或合格的分数线，维持整体的正态分布，从而保证了评价的合理、公平。模型的评价与推广模型的评价：学生学习状况评价模型:模型一：层次分析法优点：层次分析法可以提高学生学习情况综合评价指标权重值的科学性和可信性，从而能够很好地反映学生的实际学习情况，防止了传统的将各项分数相加求和的不合理性做法，从而使教育管理者能更好的了解学生学习状态，有效的实施教学管理。特别对于专业性要求很高的工科专业，此方法更有利于选拔专业素养好的学生。同时也对进步大的学生有所鼓励，激起他们对学习的兴趣。缺点：仍在一定程度受主观因素的影响，各项指标权重确实定方式有待进一步的改良。模型二：成绩标准化模型优点：通过标准化过程，使学生成绩呈正态分布，让一些成绩靠后的学生能进入中间水平，同时各个学期的成绩经过标准化之后具有了可加性，相加的最终结果能正确的反映学生的整体水平，而不是在绝对分数中只靠几次突出的成绩就能提高得到好的名次，从而更加公平、合理。缺点：正态化的方法还要进一步探讨，从而让结果能有更好的正态性。参考文献周凯等人编著"数学建模竞赛辅导教程"大学2009年8月第一版;启源，金星，叶俊，数学建模(第三版)[M],：高等教育，2003，224-229。茆诗松，程依明，濮晓龙，概率论与数理统计教程，：高等教育，2004.7中庚，数学建模的方法及其应用[M],：高等教育，2006。于广华，信梅，考试成绩的描述统计分析及其评定[A]，1-2页，2000，No.7。马引弟,黎延海,基于模糊层次分析法的高校课堂教学质量综合评价模型，1-3页，2009年第八卷第3期。翁洁静，标准分制度在成绩统计过程中的应用[A]，1-3页，〔2009〕04-0027-03。附录：MATLAB程序第一学期原始数据第二学期原始数据T分、进步分的计算程序正态化计算程序标准正态化计算程序附录一T分和进步分第1学期第2学期**T分进步分T分进步分000118.18940.23123.530.2957000248.92430.956152.3481.12000346.97040.873541.21380.6695000440.62590.651543.17870.7331000550.87811.046451.69311.0866000650.15371.01253.49421.1809000753.55651.184349.2370.97000951.84411.094261.35361.6981001061.85491.737969.37682.4603001158.43011.483559.2251.539001250.87811.046440.88630.6594001353.7981.197661.84481.7371001438.93540.602637.61150.5668001546.2460.844855.13161.2737001639.17690.609341.37750.6745001756.98121.387459.71621.5743001850.39521.023352.3481.12001945.52150.816936.79290.5458002063.32581.860161.18891.6853002148.92430.956146.78090.8659002257.22271.40357.91511.4486002357.94721.450763.97341.9166002435.75220.520140.55880.6495002540.38440.644347.92710.913002654.03941.2145.14350.8028002749.16580.966845.30730.8089002834.54470.491935.15550.506002955.02731.267652.67551.137003042.82120.721139.41270.616003145.2580.80753.49421.1809003230.1540.401635.48290.5137003326.48780.33929.86130.3962003456.23481.340354.47661.2357003563.56731.88163.80971.9022003658.69361.501659.881.5863003742.82120.721152.83921.1457003854.03941.21160.04371.5983003958.69361.501659.55251.5624004044.05060.763250.21941.015004148.92430.956151.20181.0622004261.61341.718651.52931.0784004355.26881.281851.52931.0784004446.72890.863851.36561.0702004541.61380.68245.96220.8337004656.98121.387463.6461.8878004750.87811.046456.44151.3532004856.73971.37256.27771.343004964.05021.923466.26582.1308005055.02731.267656.27771.343005153.55651.184351.85681.0948005258.93511.518555.95031.3228005347.45340.893246.12590.8401005454.30291.225949.89190.9998005544.29210.771847.59960.8993005655.99331.325566.26582.1308005741.37230.674439.57640.6207005852.10751.107660.37121.6227005957.70751.434656.1141.3329006037.22310.556734.1730.4835006146.2460.844844.32480.773006248.44130.93549.07320.9627006353.3151.171151.69311.0866006553.3151.171151.03811.0542006657.70571.434658.40631.4818006759.4181.552844.8160.9707006862.09641.757448.41830.934006942.82120.721143.01490.7276007051.36111.0753.82171.1989007140.38440.644350.21941.015007246.72890.863847.59960.8993007353.7891.197650.87441.0462007448.68280.945443.99730.7614007546.2460.844852.02051.1031007658.69361.501665.28332.0362007767.71652.278663.15471.8455007940.14290.637143.17870.7331008044.29210.771838.75770.5976008146.2460.844853.0031.1544008263.80871.902169.21312.4418008316.9820.218526.64110.3414008443.32610.738145.30730.8089008558.18871.46755.78651.3129008655.02731.267657.26021.4054008753.55651.184363.80971.9022008852.10751.107657.58761.4268008951.60261.08247.10840.8791009057.70571.434657.91511.4486009151.84411.094249.56450.9848009263.80871.902155.62281.303009368.44092.356270.5232.5941009459.65951.570259.06131.5274009546.2460.844856.44151.3532009652.8321.145353.0031.1544009761.85491.501664.79211.9905009858.69360.556756.1141.3329009937.22311.296247.27210.8858010055.51031.40350.38311.0227010157.22271.587848.90950.9554010259.9011.923463.80971.9022010364.05021.340364.79211.9905010456.23481.171153.98541.208010553.3150.977647.76330.9061010649.40730.630140.06760.6349010739.90140.854234.8280.4984010846.48740.595350.05571.0074010938.6720.674536.13790.5295011034.54470.491942.19620.7006011142.57970.713140.88630.6594011246.48740.854248.09080.9199011355.51031.296251.36561.0702011455.02731.267650.54691.0305011561.13041.680662.66351.804011652.3491.1248.90950.9554011753.55651.184353.82171.1989011940.14290.637139.90390.6301012063.56731.88151.69311.0866012148.92430.956143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