﻿ 基于多级综合模型的教学质量评价

# 基于多级综合模型的教学质量评价Evaluation Model of Multi-Level Teaching Quality

Abstract: The quality of teaching is the soul of a school. The objective, scientific, and impartial evaluation of teaching quality is of great significance to improve teachers’ annual assessment and teaching en-thusiasm. By identifying factor sets, comment sets, weight sets, judgment matrixes, and compre-hensive evaluation, and carring out a certain calculation, we establish a multi-level comprehensive model of teaching quality evaluation in a school. Our goal is to make evaluation more fair, reasonable and scientific.

1. 引言

Table 1. A teacher’s teaching quality quantitative index system

i) 若对任意 $x\in X$ ，有 $\alpha \left(x\right)\le \beta \left(x\right)$ ，则称A包含B，记为 $A\subseteq B$

ii) 若 $A\subseteq B$$B\subseteq A$ ，则称A与B相等，记为A = B。

$C=\left(AUB\right)\left(x\right)=\mathrm{max}\left\{A\left(x\right),B\left(x\right)\right\}=A\left(x\right)\vee B\left(x\right)$ $D=\left(A\cap B\right)\left(x\right)=\mathrm{min}\left\{A\left(x\right),B\left(x\right)\right\}=A\left(x\right)\wedge B\left(x\right)$

$N\left({U}_{{i}_{0}},V\right)=\mathrm{max}\left\{N\left({U}_{1},V\right),N\left({U}_{2},V\right),\cdots ,N\left({U}_{n},V\right)\right\}$

$N\left(A,B\right)=\left(A•B\right)\wedge {\left(A\otimes B\right)}^{c}$

2. 模型方法：因素集、评价集及模糊矩阵

Step 1. 确定被评判对象的因素论域U，取U为教学质量各单项指标的集合， $U=\left\{{u}_{1},\cdots ,{u}_{n}\right\}$

Step 2. 确定评语等级论域V， $V=\left({v}_{1},{v}_{2},\cdots ,{v}_{m}\right)$ 为刻画每一个因素所处的状态的m种决断(即评价等级)。通常评语有V = (优秀，良好，一般，…，较差，差，很差)。

Step 3. 进行单因素评判，构建模糊关系矩阵R。

$R=\left[\begin{array}{cccc}{r}_{11}& {r}_{12}& \cdots & {r}_{1m}\\ {r}_{21}& {r}_{22}& \cdots & {r}_{2m}\\ ⋮& ⋮& & ⋮\\ {r}_{n1}& {r}_{n2}& \cdots & {r}_{nm}\end{array}\right]$ , $0\le {r}_{ij}\le 1$ (1)

Step 4. 确定评判因素权向量 $w=\left({w}_{1},{w}_{2},\cdots ,{w}_{m}\right)$ ，W是U中各因素对被评事物的隶属关系，它取决于人们进行模糊综合评判时的着眼点，即根据评判时各因素的重要性分配权重；

$W=\left({w}_{1},{w}_{2},\cdots ,{w}_{n}\right)$ ,

${W}_{j}=\frac{{{W}^{\prime }}_{j}}{\underset{j=1}{\overset{n}{\sum }}{{W}^{\prime }}_{j}}$ (2)

${u}_{i}\left(i=1,2,\cdots ,n\right)$ ,

$W=\left({w}_{1},{w}_{2},\cdots ,{w}_{n}\right)$ .

Step 5. 选择评价的合成算子，将W与R合成得到 $B=\left({b}_{1},{b}_{2},\text{\hspace{0.17em}}\cdots ,{b}_{m}\right)$

$B=W\circ R=\left({w}_{1},{w}_{2},\cdots ,{w}_{n}\right)\circ \left[\begin{array}{cccc}{r}_{11}& {r}_{12}& \cdots & {r}_{1m}\\ {r}_{21}& {r}_{22}& \cdots & {r}_{2m}\\ ⋮& ⋮& & ⋮\\ {r}_{n1}& {r}_{n2}& \cdots & {r}_{nm}\end{array}\right]$ , (3)

$\circ$ ”为模糊合成算子 $\stackrel{˜}{\otimes }$$\stackrel{˜}{\oplus }$ 是模糊变换的两种运算，具体形式为：

${b}_{k}=\left({w}_{1}\stackrel{˜}{\otimes }{r}_{1k}\right)\stackrel{˜}{\oplus }\left({w}_{2}\stackrel{˜}{\otimes }{r}_{2k}\right)\stackrel{˜}{\oplus }\cdots \stackrel{˜}{\oplus }\left({w}_{n}\stackrel{˜}{\otimes }{r}_{nk}\right)$ $\left(k=1,2,\cdots ,m\right)$

Step 6. 数据比较少时，对一级模糊综合评价结果B作分析，即得结果。

Step 7. 将因素集 $U=\left({u}_{1},{u}_{2},\cdots ,u{}_{n-1},{u}_{n}\right)$ 按照某种属性分成s个子因素集 ${U}_{1},{U}_{2},\cdots ,{U}_{s}$ ，其中 ${U}_{i}=\left\{{u}_{i1},{u}_{i2},\cdots ,{u}_{i{n}_{i}}\right\},i=1,2,\cdots ,s$ ，且满足：

i). ${n}_{1}+{n}_{2}+\cdot \cdot \cdot +{n}_{s}=n$

ii). ${U}_{1}\cup {U}_{2}\cup \cdots \cup {U}_{s}=U$

iii). 对任意的 $i\ne j,{U}_{i}\cap {U}_{j}=\varphi$

Step 8. 重复Step 1~Step 6，对每一因素集 ${U}_{i}$ ，分别做出综合判断。设 $V=\left({v}_{1},{v}_{2},\cdots ,{v}_{m}\right)$ 为评语集， ${U}_{i}$ 中各指标对于V的权重分配是 ${A}_{i}=\left({a}_{i1},{a}_{i2},\cdots ,{a}_{i{n}_{i}}\right)$

${B}_{i}={A}_{i}•{R}_{i}=\left({b}_{i1},{b}_{i2},\text{\hspace{0.17em}}\cdots ,{b}_{im}\right),i=1,2,\cdots ,s$ .

Step 9. 将每个 ${U}_{i}$ 看作一因素，记为

$\stackrel{˜}{W}=\left({\stackrel{˜}{u}}_{1},{\stackrel{˜}{u}}_{2},\cdots ,{\stackrel{˜}{u}}_{s}\right)$ .

$\stackrel{˜}{W}$ 又是一个因素集， $\stackrel{˜}{W}$ 的单因素评判矩阵为

$\stackrel{˜}{R}=\left[\begin{array}{c}{B}_{1}\\ {B}_{2}\\ ⋮\\ {B}_{s}\end{array}\right]=\left[\begin{array}{cccc}{b}_{11}& {b}_{12}& \cdots & {b}_{1m}\\ {b}_{21}& {b}_{22}& \cdots & {b}_{2m}\\ ⋮& ⋮& & ⋮\\ {b}_{s1}& {b}_{s2}& \cdots & {b}_{sm}\end{array}\right]$ , (4)

$W=\left({w}_{1},{w}_{2},\cdots ,{w}_{n}\right)$ ，于是得到二级评判向量

$C=W\circ \stackrel{˜}{R}=\left({c}_{1},{c}_{2},\cdots ,{c}_{m}\right)$ . (5)

2.1. 评价因素、评价等级、因素集的定义

$U=\left({u}_{1},{u}_{2},{u}_{3},{u}_{4},{u}_{5}\right)$

$V=\left({v}_{1},{v}_{2},{v}_{3},{v}_{4},{v}_{5}\right)$ ,

2.2. 评判矩阵

$R={\left({r}_{ij}\right)}_{n×m}=\left[\begin{array}{cccc}{r}_{11}& {r}_{12}& \cdots & {r}_{1m}\\ {r}_{21}& {r}_{22}& \cdots & {r}_{2m}\\ ⋮& ⋮& & ⋮\\ {r}_{n1}& {r}_{n2}& \cdots & {r}_{nm}\end{array}\right]$ , $\left(i=1,2,\cdots ,n;j=1,2,\cdots ,m\right)$

2.3. 计算各因素权向量，综合判断，分析

2.4. 模糊分级分类

2.4.1. 模糊相似矩阵求解(标定)

1) 相似系数法：

${r}_{ij}=\frac{\underset{k=1}{\overset{m}{\sum }}|{x}_{ik}-{\stackrel{¯}{x}}_{i}||{x}_{jk}-{\stackrel{¯}{x}}_{j}|}{\sqrt{\underset{k=1}{\overset{m}{\sum }}{\left({x}_{ik}-{\stackrel{¯}{x}}_{i}\right)}^{2}}\sqrt{\underset{k=1}{\overset{m}{\sum }}{\left({x}_{jk}-{\stackrel{¯}{x}}_{j}\right)}^{2}}}$

2) 距离法： ${r}_{ij}=1-c\underset{k=1}{\overset{n}{\sum }}|{x}_{ik}-{x}_{jk}|$

3) 最大最小法：

${r}_{jk}=\frac{\underset{i=1}{\overset{m}{\sum }}\mathrm{min}\left({b}_{ij},{b}_{ik}\right)}{\underset{i=1}{\overset{m}{\sum }}\mathrm{max}\left({b}_{ij},{b}_{ik}\right)}$ , ${b}_{ij}=\frac{{a}_{ij}-{a}_{i\mathrm{min}}}{{a}_{i\mathrm{max}}-{a}_{i\mathrm{min}}}$ (6)

2.4.2. 对评价因素样本集合进行分类(聚类分析)

${R}_{\lambda }\left(j,k\right)=\left\{\begin{array}{l}1,\text{\hspace{0.17em}}当\stackrel{^}{{r}_{jk}}\ge \lambda ,\\ 0,\text{\hspace{0.17em}}当\stackrel{^}{{r}_{jk}}<\lambda .\end{array}$ (7)

3. 模型应用

Case1：对某一教师的教学质量评价(表1)

${A}_{1}=\left(0.3,0.1,0.4,0.2\right)$

${A}_{2}=\left(0.1,0.2,0.3,0.2,0.2\right)$

${A}_{3}=\left(0.3,0.1,0.2,0.1,0.3\right)$

${A}_{4}=\left(0.4,0.2,0.3,0.1\right)$

${B}_{1}={A}_{1}{R}_{1}=\left(0.35,0.364,0.186,0.038,0.042\right)$

${B}_{2}={A}_{2}{R}_{2}=\left(0.27,0.27,0.155,0.2150,0.09\right)$

${B}_{3}={A}_{3}{R}_{3}=\left(0.45,0.16,0.13,0.14,0.12\right)$

${B}_{4}={A}_{4}{R}_{4}=\left(0.50,0.18,0.10,0.17,0.05\right)$

$\begin{array}{c}C=AR=\left(0.30,0.4,0.2,0.2\right)\left[\begin{array}{ccccc}0.35& 0.364& 0.186& 0.038& 0.042\\ 0.27& 0.27& 0.155& 0.215& 0.09\\ 0.45& 0.16& 0.13& 0.14& 0.12\\ 0.50& 0.18& 0.10& 0.17& 0.05\end{array}\right]\\ =\left(0.403,0.2852,0.1638,0.1594,0.0826\right)\end{array}$

Case2：现有5个等级的教学设计模板 $\left({A}_{1},{A}_{2},{A}_{3},{A}_{4},{A}_{5}\right)$ ，待识别教学设计B.反映教学设计质量的因素有六项指标，构成论域U，其中

$\begin{array}{l}U=\left\{{u}_{1}\left(条理清晰\right),{u}_{2}\left(内容简洁\right),{u}_{3}\left(例题典型\right),\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{ }\text{ }{u}_{4}\left(目的明确\right),{u}_{5}\left(习题适当\right),{u}_{6}\left(反思到位\right)\right\}\end{array}$

${U}_{1}=\left(0.8,0.6,0.4,0.2,0.8,0.9\right)$

${U}_{2}=\left(0.9,0.1,0.2,0.4,0.3,0.7\right)$

${U}_{3}=\left(0.6,0.7,0.9,0.8,0.5,0.4\right)$

${U}_{4}=\left(0.2,0.4,0.2,0.2,0.1,0.6\right)$

${U}_{5}=\left(0,0.1,0.2,0.1,0.6,0.8\right)$

$V=\left(0.8,0.5,0.4,0.2,0.3,0.1\right)$

$N\left(V,{U}_{1}\right)=0.8,N\left(V,{U}_{2}\right)=0.7,N\left(V,{U}_{3}\right)=0.6,$

$N\left(V,{U}_{4}\right)=0.4,N\left(V,{U}_{5}\right)=0.3$

Case3：已知物电、信工、土木、食品、农业、生命、经管、数统、化工学院9个学院的16个新进教师上课情况的平均成绩如表2所列。试用模糊聚类分析方法对新进教师上课情况进行评价。

Step 1. 用  分别表示新进教师， $j=1,2,\cdots ,9$ 分别表示物电、信工、土木、食品、农业、生命、经管、数统、化工学院9个学院， ${a}_{ij}$ 表示第j个学院第i个新进老师的平均成绩。

${b}_{ij}=\frac{{a}_{ij}-{a}_{i\mathrm{min}}}{{a}_{i\mathrm{max}}-{a}_{i\mathrm{min}}}$ .

Step 2. 根据标准化数据建立各学院之间新进教师上课成绩指标的相似关系矩阵

Table 2. Average scores of 16 new teachers in class

Table 3. Standardization of average scores

${r}_{jk}=\frac{\underset{i=1}{\overset{m}{\sum }}\mathrm{min}\left({b}_{ij},{b}_{ik}\right)}{\underset{i=1}{\overset{m}{\sum }}\mathrm{max}\left({b}_{ij},{b}_{ik}\right)}$

Step 3. 由平方法可求得传递闭包 $\stackrel{^}{R}={R}^{4}$ ，其结果如表5

Step 4. 分类结果。根据模糊等价关系矩阵 $\stackrel{^}{R}={\left({r}_{ij}\right)}_{9×9}$ 。基于公式(7)，构造并计算 $\stackrel{^}{R}$$\lambda$ 截矩阵。等价传递闭包矩阵中的元素按照从大到小排列，由1降为0，

$\lambda =\left\{1,\text{0}\text{.6137},0.\text{6051},\text{0}\text{.5987},\text{0}\text{.5914},\text{0}\text{.5893,0}\text{.5532},\text{0}\text{.5362},\text{0}\text{.4930}\right\}$ ，写出 ${R}_{\lambda }\left(j,k\right)$ ，按 ${R}_{\lambda }\left(j,k\right)$ 进行分类，

${R}_{\lambda }\left(j,k\right)=1,\text{\hspace{0.17em}}i,j=1,2,\cdots ,9$ .

Table 4. Similar relation Matrix

Table 5. Transmit closure matrix

Figure 1. Classification chart

1) 当 $0.6137<\lambda \le 1$ 时，将9个学院分为8类：{物电}，{信工}，{土木}，{食品}，{农业}，{生命，经管}，{数统}，{化工}。

2) 当时，将9个学院分为7类：{物电}，{信工}，{土木}，{食品}，{农业}，{生命，经管，化工}，{数统}。

3) 当 $0.5987<\lambda \le 0.6051$ 时，将9个学院分为7类：{物电}，{信工}，{土木}，{食品}，{农业，数统}，{生命，经管，化工}。

4) 当 $0.5914<\lambda \le 0.5987$ 时，将9个学院分为7类：{物电}，{信工}，{土木}，{食品}，{农业，数统，生命，经管，化工}。

5) 当 $0.5893<\lambda \le 0.5914$ 时，将9个学院分为2类：{物电，信工，土木，食品}，{农业，数统，生命，经管，化工}。

6) 当 $0.5532<\lambda \le 0.5893$ 时，将9个学院分为2类：{信工，土木，食品}，{物电，农业，数统，生命，经管，化工}。

7) 当 $0.5362<\lambda \le 0.5532$ 时，将9个学院分为2类：{信工，土木}，{物电，食品，农业，数统，生命，经管，化工}。

8) 当 $0.4930<\lambda \le 0.5362$ 时，将9个学院分为2类：{土木}，{物电，信工，食品，农业，数统，生命，经管，化工}。

4. 结束语

w=[0.40.30.20.1];

w1=[0.20.30.30.2];

w2=[0.30.20.10.20.2];

w3=[0.10.20.30.20.2];

w4=[0.30.20.20.3];

b(1,:)=w1*a([1:4],:);

b(2,:)=w2*a([5:9],:);

b(3,:)=w3*a([10:14],:);

b(4,:)=w4*a([15:end],:)

c=w*b

Case2.类别划分

a=[0.8 0.6 0.4 0.2 0.8 0.9

0.9 0.1 0.2 0.4 0.3 0.7

0.6 0.7 0.9 0.8 0.5 0.4

0.2 0.4 0.2 0.2 0.1 0.6

0 0.1 0.2 0.1 0.6 0.8];

b=[0.8 0.5 0.4 0.2 0.3 0.1];%数据输入

for i=1:5

x=[a(i,:);b];

t(i)=min([max(min(x)) 1-min(max(x))]);

end

t

Case 3： λ截矩阵 R2 =

1 0 0 0 0 0 0 0 0

0 1 0 0 0 0 0 0 0

0 0 1 0 0 0 0 0 0

0 0 0 1 0 0 0 0 0

0 0 0 0 1 0 0 0 0

0 0 0 0 0 1 1 0 0

0 0 0 0 0 1 1 0 0

0 0 0 0 0 0 0 1 0

0 0 0 0 0 0 0 0 1

R3 =

1 0 0 0 0 0 0 0 0

0 1 0 0 0 0 0 0 0

0 0 1 0 0 0 0 0 0

0 0 0 1 0 0 0 0 0

0 0 0 0 1 0 0 0 0

0 0 0 0 0 1 1 0 1

0 0 0 0 0 1 1 0 1

0 0 0 0 0 0 0 1 0

0 0 0 0 0 1 1 0 1

R4 =

1 0 0 0 0 0 0 0 0

0 1 0 0 0 0 0 0 0

0 0 1 0 0 0 0 0 0

0 0 0 1 0 0 0 0 0

0 0 0 0 1 0 0 1 0

0 0 0 0 0 1 1 0 1

0 0 0 0 0 1 1 0 1

0 0 0 0 1 0 0 1 0

0 0 0 0 0 1 1 0 1

R5 =

1 0 0 0 0 0 0 0 0

0 1 0 0 0 0 0 0 0

0 0 1 0 0 0 0 0 0

0 0 0 1 0 0 0 0 0

0 0 0 0 1 1 1 1 1

0 0 0 0 1 1 1 1 1

0 0 0 0 1 1 1 1 1

0 0 0 0 1 1 1 1 1

0 0 0 0 1 1 1 1 1

R6 =

1 0 0 0 1 1 1 1 1

0 1 0 0 0 0 0 0 0

0 0 1 0 0 0 0 0 0

0 0 0 1 0 0 0 0 0

1 0 0 0 1 1 1 1 1

1 0 0 0 1 1 1 1 1

1 0 0 0 1 1 1 1 1

1 0 0 0 1 1 1 1 1

1 0 0 0 1 1 1 1 1

R7 =

1 0 0 1 1 1 1 1 1

0 1 0 0 0 0 0 0 0

0 0 1 0 0 0 0 0 0

1 0 0 1 1 1 1 1 1

1 0 0 1 1 1 1 1 1

1 0 0 1 1 1 1 1 1

1 0 0 1 1 1 1 1 1

1 0 0 1 1 1 1 1 1

1 0 0 1 1 1 1 1 1

R8 =

1 1 0 1 1 1 1 1 1

1 1 0 1 1 1 1 1 1

0 0 1 0 0 0 0 0 0

1 1 0 1 1 1 1 1 1

1 1 0 1 1 1 1 1 1

1 1 0 1 1 1 1 1 1

1 1 0 1 1 1 1 1 1

1 1 0 1 1 1 1 1 1

1 1 0 1 1 1 1 1 1

NOTES

*通讯作者。

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[4] 张媛, 王世真, 朱秀华. 模糊数学用于地表水的综合评价[J]. 大连铁道学院学报, 2004(3): 8-11.

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