SO BẰNG KẾT QUẢ TRẮC NGHIỆM MÔN TOÁN BẰNG LÍ THUYẾT ĐÁP ỨNG IRT

TẠP CHÍ ĐẠI HỌC SÀI GÒN Số 6 - Tháng 6/2011

121
SO BẰNG KẾT QUẢ TRẮC NGHIỆM MƠN TỐN
BẰNG LÍ THUYẾT ĐÁP ỨNG IRT

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
Việc so sánh các điểm trắc nghiệm của các đề thi trắc nghiệm khác nhau là một vấn đề
rất quan trọng trong việc đo lường và đánh giá năng lực của thí sinh. Nếu hai thí sinh làm
hai đề trắc nghiệm khác nhau thì làm sao so sánh điểm của hai thí sinh đó với nhau. Trong
bài viết này, chúng tơi trình bày ngắn gọn khái niệm so bằng trong lí thuyết trắc nghiệm cổ
điển và hiện đại. Chúng tơi tiến hành thực nghiệm thơng qua hai đề trắc nghiệm khác nhau
nhưng có một số câu hỏi bắc cầu, thực hiện quy trình so bằng kết quả tính tốn từ hai đề
trắc nghiệm nhờ phần mềm phân tích trắc nghiệm VITESTA.
ABSTRACT
Comparing the scores of different multiple-choice tests is very important in measuring
and assessing students ‘ability. Since it’s not easy to compare the scores of two students on
two different multiple-choice tests, we present in this article with a few words the concept
of equal comparison in classical test theory (CTT) and item response theory (IRT).
Through two different multiple-choice tests with some bridging questions and with the help
of the software VITESTA which analyzes multiple-choice tests, we worked on the process
of balancing the results of the two tests.


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       



       
   
 so sánh
 


       
        
       

        

        


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

 



        



   


1. 

1.1. Lí thuyết trắc nghiệm cổ điển
(Classical test theory – CTT)
a. 





122
      

       

        
        
        

y
= a 
x
+ b, 
y
= a 
x

(
x
, 
y


x
, 
y




()
y
xy
x
yx




  



      
y
y
x
x
y
x







b. 
(1) 

(2)        


(3)        


(4)        

c. 
    



       

    y   

1.2. Lí thuyết trắc nghiệm hiện đại
(Item response theory – IRT)
a. 
      





        


         




       
      n cùng 1

         
   

      


b.       


*) =
P() khi ta thay  * =  
= b + 

a
.

a = 1.
*  


Trường hợp 1: Trường hợp 2 nhóm thí
sinh làm 1 đề trắc nghiệm
 



 


123
Cách 1: Chuẩn hoá độ khó  
        


        
      
ang



Cách 2: Chuẩn hoá các giá trị năng lực.
 
A
, a
A
     


B

, a
B



b
A
= b
B
+ ; a
A
=

B
a

và * = 
B
+ 


Vì ,      
        
        

Trường hợp 2: Trường hợp 1 nhóm thí
sinh làm 2 đề trắc nghiệm
        
      
         

 
và 1)

        

Trường hợp 3: Trường hợp nhiều
nhóm thí sinh làm nhiều đề trắc nghiệm
       
       





STT





1

nhóm.

cho cùng 1 nhóm thí sinh.


m



2











3


các 










4

có các thí sinh
chung.


















124
2.     
 
1. 


2. 
        
        
       
       
      
  


3. 
         

 Ta 







       
        
        




  
        
      






 
         

      

3. 




3.1. Mô tả dữ liệu
 -2.dat,
toan1.dat, toan2.dat, toan-2.key và file

 


i.
      

        

 
          
21.
3.2. Thực hiện quy trình so bằng
và các kết quả


     
         
      
sau:

125
 
- 04:31

| |
¦ Câu| b | MSE (*)|
| + + |
¦ 1¦ -0.80386¦ 0.14598¦
¦ 2¦ -0.61153¦ 0.14029¦
¦ 3¦ -0.96071¦ 0.15244¦
¦ 4¦ -0.64649¦ 0.14116¦
¦ 5¦ -0.79136¦ 0.14554¦
¦ 6¦ -0.26756¦ 0.13537¦
¦ 7¦ -0.47535¦ 0.13759¦
¦ 8¦ -0.72991¦ 0.14352¦
¦ 9¦ -0.55414¦ 0.13903¦
¦ 10¦ -0.39799¦ 0.13651¦
¦ 11¦ -0.82910¦ 0.14690¦
¦ 12¦ -0.50892¦ 0.13816¦
¦ 13¦ -0.18146¦ 0.13508¦
¦ 14¦ -0.65824¦ 0.14146¦
¦ 15¦ -0.81644¦ 0.14644¦
¦ 16¦ 0.40706¦ 0.14212¦
¦ 17¦ -0.96071¦ 0.15244¦

¦ 18¦ -0.59997¦ 0.14002¦
¦ 19¦ 0.56689¦ 0.14672¦
¦ 20¦ 0.57965¦ 0.14714¦
¦ 21¦ 0.23327¦ 0.13843¦
¦ 22¦ -0.52017¦ 0.13837¦
¦ 23¦ 0.17716¦ 0.13753¦
¦ 24¦ -0.79136¦ 0.14554¦
¦ 25¦ -0.00921¦ 0.13553¦
¦ 26¦ -0.75429¦ 0.14429¦
¦ 27¦ 0.41900¦ 0.14242¦
¦ 28¦ 0.30169¦ 0.13972¦
¦ 29¦ -0.74207¦ 0.14390¦
¦ 30¦ -0.64649¦ 0.14116¦
| |

| |
¦ Câu| b | MSE (*)|
| + + -|
¦ 1¦ -0.80314¦ 0.15322¦
¦ 2¦ -0.35972¦ 0.13855¦
¦ 3¦ -0.76222¦ 0.15135¦
¦ 4¦ -1.10958¦ 0.17131¦
¦ 5¦ -0.68317¦ 0.14805¦
¦ 6¦ -0.80314¦ 0.15322¦
¦ 7¦ -0.81701¦ 0.15389¦
¦ 8¦ -0.57039¦ 0.14403¦
¦ 9¦ 1.19561¦ 0.17668¦
¦ 10¦ -0.17209¦ 0.13572¦
¦ 11¦ -0.61975¦ 0.14569¦
¦ 12¦ 0.19259¦ 0.13550¦

¦ 13¦ -0.41667¦ 0.13978¦
¦ 14¦ 1.00955¦ 0.16327¦
¦ 15¦ -0.65756¦ 0.14706¦
¦ 16¦ 0.26863¦ 0.13633¦
¦ 17¦ -0.02165¦ 0.13479¦
¦ 18¦ -0.84512¦ 0.15527¦
¦ 19¦ 0.29054¦ 0.13664¦
¦ 20¦ 0.03176¦ 0.13474¦
¦ 21¦ 0.32361¦ 0.13714¦
¦ 22¦ -1.14462¦ 0.17388¦
¦ 23¦ -0.02165¦ 0.13479¦
¦ 24¦ 0.30154¦ 0.13680¦
¦ 25¦ 1.07389¦ 0.16754¦
¦ 26¦ -0.91776¦ 0.15912¦
¦ 27¦ 0.49355¦ 0.14072¦
¦ 28¦ 0.06381¦ 0.13478¦
¦ 29¦ -0.64488¦ 0.14659¦
¦ 30¦ -0.43975¦ 0.14033¦
| |





   
         

        u



        



 


126
:


       
        
     
        
      

       

       
  


 và













 

| Câu 3 | -0.9607065 | Câu 3 | -0.7622227 |
| Câu 8 | -0.7299129 | Câu 8 | -0.570385 |
| Câu 11 | -0.8291038 | Câu 11 | -0.6197521 |
| Câu 15 | -0.8164439 | Câu 15 | -0.6575601 |
| Câu 18 | -0.5999659 | Câu 18 | -0.8451234 |
| Câu 21 | 0.2332707 | Câu 21 | 0.3236081 |


 - -0.5219058
 0.4333358  0.4260145



127




 

        








       
        





       











128
* :
       

         

        
        


 
 





* :

 
- 04:31

¦ |
Câu| b| SE (*) |
| + + |
¦ 1¦ -0.80386¦ 0.14598¦
¦ 2¦ -0.61153¦ 0.14029¦
¦ 3¦ -0.90908¦ 0.15244¦
¦ 4¦ -0.64649¦ 0.14116¦
¦ 5¦ -0.79136¦ 0.14554¦
¦ 6¦ -0.26756¦ 0.13537¦
¦ 7¦ -0.47535¦ 0.13759¦
¦ 8¦ -0.69777¦ 0.14352¦
¦ 9¦ -0.55414¦ 0.13903¦
¦ 10¦ -0.39799¦ 0.13651¦
¦ 11¦ -0.77205¦ 0.14690¦

¦ 12¦ -0.50892¦ 0.13816¦
¦ 13¦ -0.18146¦ 0.13508¦
¦ 14¦ -0.65824¦ 0.14146¦
¦ 15¦ -0.78462¦ 0.14644¦
¦ 16¦ 0.40706¦ 0.14212¦

|
¦ Câu | b | MSE (*)|
| + + |
¦ 1¦ -0.89838¦ 0.15322¦
¦ 2¦ -0.45495¦ 0.13855¦
¦ 3¦ -0.90908¦ 0.15135¦
¦ 4¦ -1.20482¦ 0.17131¦
¦ 5¦ -0.77841¦ 0.14805¦
¦ 6¦ -0.89838¦ 0.15322¦
¦ 7¦ -0.91225¦ 0.15389¦
¦ 8¦ -0.69777¦ 0.14403¦
¦ 9¦ 1.10037¦ 0.17668¦
¦ 10¦ -0.26733¦ 0.13572¦
¦ 11¦ -0.77205¦ 0.14569¦
¦ 12¦ 0.09735¦ 0.13550¦
¦ 13¦ -0.51191¦ 0.13978¦
¦ 14¦ 0.91431¦ 0.16327¦
¦ 15¦ -0.78462¦ 0.14706¦
¦ 16¦ 0.17339¦ 0.13633¦
TOÁN TOÁN 1

ANFA = 1
BETA = 0.09523785

  | DIF 

| Câu 3 | -0.9607065 | Câu 3 | -0.8574606 | -0.103246 | -0.90908|
| Câu 8 | -0.7299129 | Câu 8 | -0.6656229 | -0.06429005 | -0.69777|
| Câu 11 | -0.8291038 | Câu 11 | -0.71499 | -0.1141138 | -0.77205|
| Câu 15 | -0.8164439 | Câu 15 | -0.7527979 | -0.06364602 | -0.78462|
| Câu 18 | -0.5999659 | Câu 18 | -0.9403612 | 0.3403953 | -0.77016|
| Câu 21 | 0.2332707 | Câu 21 | 0.2283702 | 0.004900411 | 0.23082 |







129
¦ 17¦ -0.96071¦ 0.15244¦
¦ 18¦ -0.77016¦ 0.14002¦
¦ 19¦ 0.56689¦ 0.14672¦
¦ 20¦ 0.57965¦ 0.14714¦
¦ 21¦ 0.23082¦ 0.13843¦
¦ 22¦ -0.52017¦ 0.13837¦
¦ 23¦ 0.17716¦ 0.13753¦
¦ 24¦ -0.79136¦ 0.14554¦
¦ 25¦ -0.00921¦ 0.13553¦
¦ 26¦ -0.75429¦ 0.14429¦
¦ 27¦ 0.41900¦ 0.14242¦
¦ 28¦ 0.30169¦ 0.13972¦
¦ 29¦ -0.74207¦ 0.14390¦
¦ 30¦ -0.64649¦ 0.14116¦

| |
¦ 17¦ -0.11689¦ 0.13479¦
¦ 18¦ -0.77016¦ 0.15527¦
¦ 19¦ 0.19530¦ 0.13664¦
¦ 20¦ -0.06348¦ 0.13474¦
¦ 21¦ 0.23082¦ 0.13714¦
¦ 22¦ -1.23986¦ 0.17388¦
¦ 23¦ -0.11689¦ 0.13479¦
¦ 24¦ 0.20630¦ 0.13680¦
¦ 25¦ 0.97865¦ 0.16754¦
¦ 26¦ -1.01300¦ 0.15912¦
¦ 27¦ 0.39831¦ 0.14072¦
¦ 28¦ -0.03143¦ 0.13478¦
¦ 29¦ -0.74012¦ 0.14659¦
¦ 30¦ -0.53498¦ 0.14033¦
|



       

 
         





¦ Câu| b | MSE (*)|
| + + |

¦ 1¦ -0.76624¦ 0.14594¦
¦ 2¦ -0.57393¦ 0.14028¦
¦ 3¦ -0.87988¦ 0.10741¦
¦ 4¦ -0.60889¦ 0.14114¦
¦ 5¦ -0.75374¦ 0.14550¦
¦ 6¦ -0.23001¦ 0.13542¦
¦ 7¦ -0.43778¦ 0.13760¦
¦ 8¦ -0.66858¦ 0.10169¦
¦ 9¦ -0.51655¦ 0.13903¦
¦ 10¦ -0.36042¦ 0.13653¦
¦ 11¦ -0.74281¦ 0.10346¦
¦ 12¦ -0.47134¦ 0.13817¦
¦ 13¦ -0.14391¦ 0.13514¦
¦ 14¦ -0.62064¦ 0.14145¦
¦ 15¦ -0.75544¦ 0.10379¦
¦ 16¦ 0.44458¦ 0.14238¦
¦ 17¦ -0.92306¦ 0.15238¦
¦ 18¦ -0.73025¦ 0.10314¦
¦ 19¦ 0.60443¦ 0.14710¦
¦ 20¦ 0.61719¦ 0.14752¦
¦ 21¦ 0.25986¦ 0.09747¦
¦ 22¦ -0.48259¦ 0.13837¦
¦ 23¦ 0.21469¦ 0.13769¦
¦ 24¦ -0.75374¦ 0.14550¦
¦ 25¦ 0.02832¦ 0.13564¦
¦ 26¦ -0.71667¦ 0.14426¦
¦ 27¦ 0.45653¦ 0.14270¦


¦ Câu| b | MSE (*)|

| + + |
28¦ 0.33921¦ 0.13993¦
¦ 29¦ -0.70445¦ 0.14387¦
¦ 30¦ -0.60889¦ 0.14114¦
¦ 31¦ -0.87711¦ 0.15328¦
¦ 32¦ -0.43389¦ 0.13856¦
¦ 33¦ -1.18343¦ 0.17140¦
¦ 34¦ -0.75720¦ 0.14809¦
¦ 35¦ -0.87711¦ 0.15328¦
¦ 36¦ -0.89098¦ 0.15394¦
¦ 37¦ 1.12084¦ 0.17652¦
¦ 38¦ -0.24635¦ 0.13571¦
¦ 39¦ 0.11816¦ 0.13546¦
¦ 40¦ -0.49081¦ 0.13980¦
¦ 41¦ 0.93483¦ 0.16313¦
¦ 42¦ 0.19417¦ 0.13629¦
¦ 43¦ -0.09598¦ 0.13477¦
¦ 44¦ 0.21607¦ 0.13659¦
¦ 45¦ -0.04259¦ 0.13472¦
¦ 46¦ -1.21845¦ 0.17398¦
¦ 47¦ -0.09598¦ 0.13477¦
¦ 48¦ 0.22707¦ 0.13675¦
¦ 49¦ 0.99915¦ 0.16740¦
¦ 50¦ -0.99168¦ 0.15919¦
¦ 51¦ 0.41900¦ 0.14065¦
¦ 52¦ -0.01056¦ 0.13476¦
¦ 53¦ -0.71892¦ 0.14663¦
¦ 54¦ -0.51388¦ 0.14035¦
4.3. Nhận xét

130
        







-    

  

-       
        




       

-       







1. GS.TS Trắc nghiệm và đo lường thành quả học tập, Nxb

2. GS.TS Thống kê ứng dụng trong nghiên cứu khoa học giáo
dục, Nxb 
3. (2009) Trắc nghiệm, đo lường và đánh giá trong Giáo dục.
4. lí 
5. ( Bài giảng môn Đại cương thống kê, 2008.

7. Patrick Griffin (1997). An Introduction to the Rasch Model. Assessment Research
Centre, University of Melbourne.
8. 