Testing and Evaluation in Mathematics

7/27/2019 Testing and Evaluation in Mathematics

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INSTITUT PENDIDIKAN GURU

KAMPUS GAYA Peti Surat 10491, 88805 Kota Kinabalu, SabahTel: 088-210646 Faks: 088-217124

TESTING AND EVALUATIONIN

MATHEMATICS

NAME : ARNA BINTI JUNAIDE

IC NUMBER : 851207-12-5130

GROUP : KPLI MATH (NOVEMBER 2009)

CODE AND SUBJECT : MATEMATIK

LECTURER’S NAME : EN BUSTAM BIN DAMAN

DATE :15 APRIL 2011





7. ANALYSIS AND INTERPRETATION OF ITEM/QUESTION

21

7.1 Difficulty Index (D.I)

21

7.2 Discrimination Index (D.s.I)

23

7.3 Calculation to find Difficulty Index and Discrimination Index

25

7.4 Difficulty Index Analysis

7.5 Conclusion for the items Difficulty Index Analysis

29

7.6 Discrimination Index (D.s.I) Analysis

30

7.7 30% a high-score group and 30% of low-score group

32

7.8 Conclusion for the items Discrimination Index Analysis

32

8. ANALYSIS AND INTERPRETATION OF TEST RESULT

33

8.1 Mean 33

8.2 Median

34

8.3 Mode 35

9. CONCLUSION

36

1.0 INTRODUCTION



Teaching and learning process is a very effective educational system

to see the student’s achievements. It is also known as an activity to help

student to achieve the learning outcome that have been set earlier.

1.1 TEACHING OBJECTIVES

Teaching objective is a process that changing students’ behaviour and

it can be achieved by teachers in one learning process. This changing can be

achieved through teaching and learning activities that are well planned. All

the changes or progress from the student must be assessed from time to

time.

Continuous assessment will determine the effectiveness of teaching

and learning. The students’ progress or change also has to be evaluated and

record time to time. This clearly shows the existence of the relationship

between the objective of teaching, testing, measurement and evaluation.

1.2 MEANING OF TEST

Test is a systematic way to measure changes in an individual’s

behaviour. Referring this to a class, a test is conducted to measure changes

in behaviour related to pupil’s learning activities. It is, in general, a

measuring instrument to obtain information regarding pupil’s achievement inthe cognitive, psychomotor or affective fields.

Test is an important part in learning and teaching process. It is also

one of the techniques to obtain the process of the pupils changing. To

implement the evaluation, teacher must plan and construct a test first. There

have same criteria in construct a question for a test. To legislate a question

paper, it have to follow the specification that have been set which is the test

must be well manage include the whole of the syllabus. This is to ensure all

student have been test using the relevant question from the syllabus and

also from the teaching objectives.

Test also help teacher to find out whether the pupils have successfully

mastered certain skill, and acquired the knowledge which have been learned

before. The test that use to determine the level of achievement of students

is implement by using a paper test. It is divide into two category which is;

1. Objective test



2. Subjective test

1.3 MEANING OF MEASUREMENT

Measurement is a process or system used to determine the

quantitative value of a certain test in learning. This quantitative value isusually expressed in a fixed unit of number, in terms of percentage or grade,

by using a related measuring instrument.

1.4 MEANING OF EVALUATION

Evaluation is defined as a process to determine, obtain and provide

useful information so as to make objective consideration for follow-up action.

In the context of education, evaluation is a system or a process which covers

activities to collect information relating to strategy of teaching-learningactivity and to make analysis and interpretation of result, with the aim to

take appropriate follow-up action such as replanning for more effective

teaching-learning activity thereafter.











Objective : converting a number in a certain base to a number

in

another base.

Level of Difficulty : Hard

Item no 8 : Express 12435 as a number in base eight.

A. 3068

B. 3608

C. 6038

D. 6308 (Level: Understanding)

9. Item number : 9

Topic : Number Bases

Subtopic : 1.1 Number bases in Two,Eight and Five

Objective : Addition of two binary numbers

Level of Difficulty : Easy

Item no 9: 1011012 + 111012 =

A. 10010102

B. 10110102

C. 10111102

D. 11110002 (Level: Understanding)

10.Item number : 10



Topic : Number Bases

Subtopic : 1.1 Number bases in Two, Eight and Five

Objective : Subtraction of two binary numbers


Item no 10: 1010102 – 1112 =

A. 100112

B. 1000112

C. 1010012

D. 1011012 (Level:

Knowledge)

11.Item number : 11

Topic : Graphs of Functions II

Subtopic : 2.1 Graph of function

Objective : Drawing the graphs of functions


Item no 11: Which of the following graphs represents y = 4-X2?



(Level: Understanding)

12.Item number : 12


Subtopic : 2.1 Graph of function

Objective : Finding the value of y or x from the graph


Item no 12: Which graph represents y = -1/2 X – 1?




13.Item number : 13


Subtopic : 2.2 solving the equation graphically

Objective : Finding the value of y or x by using the equation

graphically.

Level of Difficulty : Medium

Item no 13: which graph represent y=X2 – X – 6?







B. 4 D. 16 (Level:

Synthesis)

16.Item number : 16


Subtopic : 2.1 Graph of Functions

Objective : Sketching the graph of a linear function


Item no 16: The diagram shows a Cartesian plane.

Which of the straight lines, A, B, C or D, represents the graph of y = -2x -5?


17.Item number : 17


Subtopic : 2.1 Graph of Functions



Objective : Sketching the graph of a linear function


Item no 17: The following points satisfy the equation y = 3X – 1 except

A. (-2, -7)

B. (0, -1)

C. (1, -2)

D. (2, 5)


18.Item number : 18


Subtopic : 2.3 Region representing inequalities in two

variables

Objective : Identifying the shaded region in the graph




Item no 18:

The inequality represented by the shaded region in the graph is

A. y< -2x + 2

B. y> -2x + 2

C. y ≤ -2x + 2

D. y ≥ -2x + 2 (Level:

Knowledge)

19.Item number : 19



variables

Objective : Identifying the shaded region in the graph


Item no 19: The diagram shows a Cartesian plane.



Which of the regions, A, B, C or D, satisfies the inequalities x + y ≤ 4 and y ≥

x?


20.Item number : 20



variables

Objective : Shading the region representing the given

inequalities


Item no 20:



Which of the following sets of inequalities define the shaded region in the

diagram?

A. 4x + 3y ≥ 12, x – y ≤2, y < 3

B. 4x + 3y ≤ 12, x – y ≥2, y < 3

C. 4x + 3y ≥ 12, x – y ≥2, y ≤ 3

D. 4x + 3y ≥ 12, x – y >2, y ≤ 3

(Level:

Application)

21.Item number : 21

Topic : Transformations III

Subtopic : 3.1 Combination of Two Transformations

Objective : Identifying the centre of the rotation.


Item no 21: The diagram shows two right-angled triangles, P and Q, drawn

on a grid of



squares.

Q is the image of P under an anticlockwise rotation of 90. Which of the pointsA,B,C or D, is the centre of rotation?


22.Item number : 22



Objective : Combination of Two isometric transformations


Item no 22: The diagram shows two trapeziums, P and Q, drawn on a grid of

squares.







25.Item number : 25





Item no 25: The diagram shows a quadrilateral PQRS drawn on a grid of

squares.

Given W is a reflection in the line x= 1 and T is a translation ( ). Find the

coordinates of the image of point Q under the combined transformation TW.

A. (-2, 5)

B. (1,5)

C. (5,-2)

D. (5, 1) (Level: Application)

26.Item number : 26







Item no 27: Given V is a reflection in the line x=2 and U is a translation ( ).

Find the coordinates of the image of point (-2, 4) under the combined

transformation UV.

A. (-1, 4)

B. (-1, 9)

C. (4,-1)

D. (9,-1) (Level: Understanding)

27.Item number : 27



Objective : understanding the single transformation reflection


Item no 28: The diagram shows five quadrilaterals drawn on a grid of squares.

Which of the quadrilaterals, A,B,C or D, is not the image of quadrilateral W

under a reflection? (Level: Analysis)



28.Item number : 28



Objective : Understanding the transformation of enlargement


Item no 27: The diagram shows five quadrilaterals drawn on a grid of

squares.

Which of the quadrilaterals, A,B,C or D, is the image of quadrilateral P under

an enlargement?

(Level: Analysis)

29.Item number : 29



Objective : Understanding the transformation of enlargement




Item no 29: The diagram shows five triangles drawn on a grid of squares.

Which of the triangles, A,B,C or D, is the image of triangle P under anenlargement with a scale factor of 2?

(Level: Analysis)

30.Item number : 30





Item no 30: In the diagram, trapezium PTUV is the image of trapezium PQRS

under a rotation followed y an enlargement.

Given that the area of trapezium PQRS is 13.5 cm2, find the area, in cm2, of

the shaded region.



A. 40.5 C. 108

B. 81 D. 121.5

(Level: Analysis)

5.0 SCHEMA / ANSWER DISTRIBUTION

ANSWERS DISTRIBUTION

NO OF

QUESTION

A B C D

1 /

2 /

3 /

4 /

5 /

6 /

7 /

8 /

9 /

10 /

11 /

12 /

13 /

14 /

15 /

16 /

17 /



18 /

19 /

20 /

21 /

22 /

23 /

24 /

25 /

26 /

27 /28 /

29 /

30 /

TOTAL 7 8 9 6





Using these formulae, we can obtain the value of the difficulty Index for a particularitem/question for interpretation and make decision, as illustrated in the following

table;

Difficulty Index

(D.I.)

Interpretation of an

item/question

Decision

0.00 – 0.20 Too difficult Modify

0.21 - 0.40 Difficult Accept

0.41 - 0.60 Average Accept

0.61 - 0.80 Easy Accept

0.81 – 1.00 Too easy Modify

0.00 0.50 1.00

Difficult item Medium Easy



7.2 Discrimination Index

Discrimination Index possesses the power or function to differentiate candidates

between high-score group and low-score group. This index can be calculated by

using the following formulae:

Discrimination Index ( Ds.I.) =

CH = Total number of candidates in the high-score group who answer correctly the

item/question.

CL = Total number of candidates in the low-score group who answer correctly the

item/question

T H+L= Total number of candidates in both groups with high and low score.

The value of Discrimination Index can be utilized to determine whether a certain

item /question is suitable and can be used for a test. Table below illustrates a

general method which is normally used to interpret the suitability of certain

item/question from the calculated value of Discrimination Index.

Discrimination Index

(Ds. I.)

Interpretation of an

item/question

Decision

Ds. I. > 0.40 Positive Discrimination

High

Very suitable and can be

accepted

0.20 < Ds.I. ≤0.40 Positive discrimination

Moderate

Modify the item /

question

0 ≤Ds.I . ≤ 0.20 Positive Discrimination

Low

Item/ question needs to

be reconstructed.

D.I. < 0 Negative Discrimination:

Performance of low-score

group is better than high-

score group.

Item/question is bad and

shoul be removed.

CH – CL

½ T H+L



Interpretation of an item/question based on Discrimination Index

According to research, a good item/question which can be used for a test, usually

has values of difficulty index between 0.40 and 0.60, and discrimination index

of values more than 0.40.

In order to measure the values of difficulty index and discrimination index, we

need to arrange the candidates from the highest score to the lowest score, and

record their performance in every item/question which they answer by marking (/)

as correct and (X) as wrong in a specification table such as table. To determine the

number of candidates in the high-score group CH and the number of candidates in

the low-score candidates (CL), we normally allot about 30% of total candidates as

the high-score group , and about 30% as the low-score group. For example, there

are 19 candidates selected to sit for the test, we calculate the number of candidates

to be included in the high-score group and low-score group respectively as follows;

Number of candidates in the high-score group = 30% x 19 ₌ 5

Number of candidates in the low-score group = 30% x 19 = 5

Table shows arrangement and the values of difficulty index and discrimination

index from 19 candidates who answer 30 items in a form 5 Mathematics objective

test.



7.3 Calculation to find Difficulty Index (D.I) and Discrimination Index

( D.S.I)

ITEM DIFFICULTY INDEX (D.I) DISCRIMINATION INDEX

(D.s.I)

FORMULA

E

Difficulty Index, D.I.=

Numbers of candidates who answer correctly(C)

Total candidates who attempt the

item/question (T)

Discrimination Index, D.s.I=

CH – CL

½ T H+L

1

Total correct answer ( C ) =16

Total No of candidates (T )=19

D.I = 0.84

Total no of answers correct

in high-score group (CH) = 5

Total no of answer correct

In low-score group (CL) =3

Total no of candidates in

both group T H+L = 10

D.s.I= 0.4

2

C= 15

T=19

D.I=0.79

CH=5

CL=4

T H+L=10

D.s.I =0.2

3 CH=5



C=18

T=19

D.I=0.95

CL=5

T H+L=10

D.s.I =0

4 C=9

T=19

D.I=0.47

CH=2

CL=2

T H+L=10

D.s.I =0

5 C=13

T=19

D.I=0.68

CH=2

CL=4

T H+L=10

D.s.I =-0.4

6

C=14

T=19

D.I=0.74

CH=4

CL=2

T H+L=10

D.s.I =0.4

7

C=16

T=19

D.I=0.84

CH=4

CL=3

T H+L=10

D.s.I =0.2

8 C=10

T=19

D.I=0.53

CH=5

CL=5

T H+L=10

D.s.I =1



9 C=18

T=19

D.I=0.94

CH=5

CL=4

T H+L=10

D.s.I =0.2

10 C=14

T=19

D.I=0.74

CH=4

CL=4

T H+L=10

D.s.I =0

11 C=13

T=19

D.I=0.68

CH=5

CL=1

T H+L=10

D.s.I =0.8

12 C=10

T=19

D.I=0.53

CH=4

CL=3

T H+L=10

D.s.I =0.2

13 C=7

T=19

D.I=0.37

CH=4

CL=1

T H+L=10

D.s.I =0.6

14 C=6 CH=4



T=19

D.I=0.32 CL=0

T H+L=10

D.s.I =0.4

15 C=9

T=19

D.I=0.47

CH=4

CL=1

T H+L=10

D.s.I =0.6

16 C=8

T=19

D.I=0.42

CH=2

CL=2

T H+L=10

D.s.I =0

17 C=4

T=19

D.I=0.2

CH=3

CL=1

T H+L=10

D.s.I =0.4

18 C=11

T=19

D.I=0.58

CH=4

CL=2

T H+L=10

D.s.I =0.4

19 C=5

T=19

D.I=0.26

CH=3

CL=0

T H+L=10

D.s.I =0.6

20 C=3 CH=3



T=19

D.I=0.16 CL=0

T H+L=10

D.s.I =0.6

21 C=9

T=19

D.I=0.47

CH=4

CL=1

T H+L=10

D.s.I =0.6

22 C=12

T=19

D.I=0.63

CH=4

CL=1

T H+L=10

D.s.I =0.6

23 C=7

T=19

D.I=0.37

CH=4

CL=0

T H+L=10

D.s.I =0.8

24 C=8

T=19

D.I=0.42

CH=4

CL=1

T H+L=10

D.s.I =0.6

25 C=9

T=19

D.I=0.47

CH=5

CL=4

T H+L=10

D.s.I =0.8

26 C=6 CH=3









7.6 Conclusion for the Items Difficulty Index (D.I) Analysis

ITEM DIFFICULTY INDEX (D.I) Remarks

1

3

7

9

0.84

0.95

0.84

0.94

-Almost all student can easily

answer the question correctly.

Based on the result

interpretation of the items, this

question need to modify

because the level are too easy.

14

17

19

0.32

0.21

0.26

- The percent that student were

able to answer the question aretoo low. This item are in

difficult category but are still

accept to test because in

making an item/question for

JPU, there are provide 25% or 9

question for difficult



item/question.

For the item that are in easy, medium and difficult category, it will be

save and store in a question bank so that it can be use anytime when it

needed.

7.7 Discrimination Index (D.s.I ) Analysis

ITEM

S

TOTAL

OF

CORRE

CT

ANSWER

(CH)

TOTAL OF

CORRECT

ANSWER

(CL)

CH – CL

½ T H+L

INTERPRETATION

OF AN

ITEM/QUESTION

DECISION

1 5 3 0.4 Positive

discrimination

Moderate

Modify the

item

2 5 4 0.2 Positive

discrimination low

Item needs

to be

reconstructe

d

3 5 5 0 Positive

discrimination low

Item needs

to be

reconstructe

d

4 2 2 0 Positive Item needs



discrimination low to be

reconstructe

d

5 2 4 -0.4 Negative

discrimination

Item is bad

and should

be removed

6 4 2 0.4 Positive

discrimination

Moderate

Modify the

item

7 2 3 0.2 Positive

discrimination low

Item needs

to be

reconstructe

d

8 5 0 0.1 Positivediscrimination low

Item needsto be

reconstructe

d

9 5 4 0.2 Positive

discrimination low

Item needs

to be

reconstructe

d

10 4 4 0 Positive

discrimination low

Item needs

to bereconstructe

d

11 5 1 0.8 Positive

discrimination high

Very suitable

and can be

accepted

12 4 3 0.2 Positive

discrimination low

Item needs

to be

reconstructe

d

13 4 1 0.6 Positive

discrimination high

Very suitable

and can be

accepted

14 4 0 0.4 Positive

discrimination high

Very suitable

and can be



accepted

15 4 1 0.6 Positive

discrimination high

Very suitable

and can be

accepted

16 2 2 0 Positive

discrimination low

Item needs

to bereconstructe

d

17 3 1 0.4 Positive

discrimination high

Very suitable

and can be

accepted

18 4 2 0.4 Positive

discrimination high

Very suitable

and can be

accepted

19 3 0 0.6 Positivediscrimination high

Very suitableand can be

accepted

20 3 0 0.6 Positive

discrimination high

Very suitable

and can be

accepted



21 4 1 0.6 Positive

discrimination high

Very suitable

and can be

accepted

22 4 1 0.6 Positive

discrimination high

Very suitable

and can be

accepted

23 4 0 0.8 Positive

discrimination high

Very suitable

and can be

accepted

24 4 1 0.6 Positive

discrimination high

Very suitable

and can be

accepted

25 5 1 0.8 Positive

discrimination high

Very suitable

and can beaccepted

26 3 0 0.6 Positive

discrimination high

Very suitable

and can be

accepted

27 5 2 0.6 Positive

discrimination high

Very suitable

and can be

accepted

28 4 1 0.6 Positive

discrimination high

Very suitable

and can be

accepted

29 4 2 0.4 Positive

discrimination high

Very suitable

and can be

accepted

30 2 2 0 Positive

discrimination low

Item needs

to be

reconstructe

d

The table above shows the Discrimination Index for each of the item.

Based on the table above, the discrimination Index of the item almost

accepted by the student except one item which is item number 5 that should

be removed in the question.



6.9 Conclusion for the Items Discrimination Index (D.s.I)Analysis





8.0 ANALYSIS AND INTERPRETATION OF TEST RESULT.

Analysis and interpretation of test or examination result is considered an

important stage in assessment. This stage is normally done by teachers after the

examination, when the test paper of each student was marked and recorded. The

aim of making analysis and interpretation of the students’ result is to obtain the

feedback of the actual performance of the students and to overcome the

weaknesses identified.

The measurement of basic statistics normally used for norm-referencedevaluation is frequency polygon, mean, median, and mode. The results obtained

from the basic statistics will enable teachers to make correct interpretation on the

students’ performance and prepare report for future reference.

8.1 MEAN

SCORE (X) FREQUENCY (f) f X x fx

6 1

1 x 6 6

9 1

1 x 9

9

12 2

2 x 12

32

13 1

1 x 13

13

14 1

1 x 14

14



15 2

2 x 15

30

16 2

2 x 16

32

17 3

3 x 17

51

18 1

1 x 18

18

20 1

1 x 20

20

22 2

2 x 22

44

23 1

1 x 23

23

27 1

1 x 27

27

N = ∑f = 19

∑fx = 319

Frequency distributions of ungroup scores of mathematics test for form 5

ibnu sina

Mean, X = ∑fx = 319 = 16.79

∑f 19



8.2 MEDIAN

Median for grouped or ungrouped data is the value in the middle of an array

of set of data. It is a value that divided the bottom 50% of a group of data

arranged in ascending order, from the top 50% of the data.

If the scores are expresses as single-valued cumulative distributions, then the

formula to find the median is:

Median, M = ( N + 1 ) th score, N = ∑f = Total frequency

2

SCORE X FREQUENCY (f) CUMULATIVE

FREQUENCY

(cf)



6 1 1

9 1 2

12 2 4

13 1 5

14 1 6

15 2 8

16 2 10

17 3 13

18 1 14

20 1 15

22 2 17

23 1 18

27 1 19

N = 19

Median = 19 + 1 = 10 th score

2

:- the median lies the scores of 16

8.3 MODE

In statistics, the mode of a set of numbers, or in a frequency distributions is

the value which has the greatest frequency, i.e.the value which occurs the most

often.

SCORE X FREQUENCY (f)

6 1



9 1

12 2

13 1

14 1

15 2

16 2

17 3

18 1

20 1

22 2

23 1

27 1

The mode is 17, because it occurs as the most number of scores.

A distribution having only one mode and it is called unimodal .

9.0 CONCLUSION



Testing and evaluation need a few more systematic processes from the

beginning to the end of the process. The existences of systematic steps can

easily forming a set of quality tests. This is due to give a test as a part of

the process of teaching and learning. When implement the tests, teachers

must identify the question whether it meet to the objective of teaching and

learning.

When viewed the result of the difficulty index and discrimination index

for the item that have been constructed in this working paper which have

provided this course, the items are well supervised ad appropriately use by

students for testing purposes. This is because, both of the index did not

shows any difficulties and differentiation that will go too far.

So that, every teacher must have sufficient knowledge in the field of

test construction for students to master the skills they have learned before

to be more effectively and the items/question that were appropriately and

challenging for students ability to grow their brain power.



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Testing and Evaluation in Mathematics

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