Analysing Classroom Assessment

Hassan Basarally 806007430 EDME 2006

Name: Hassan Basarally I.D.: 806007430 Faculty: Humanities and Education Department: Liberal Arts Course Code: EDME 2006 Course Name: Classroom Testing and Evaluation Lecturer: Ms. Melville Academic Year: 2007 – 2008 Semester: 2 Date of Submission: 07/04/08

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Hassan Basarally 806007430 EDME 2006

Table of Contents INTRODUCTION BACKGROUND PURPOSE OBJECTIVES CONTENT TEST CONSTRUCTION TABLE OF SPECIFICATION ITEM SELECTION TEST ITEMS MARKING RUBRIC INTERPRETATION APPENDIX I APPENDIX II

Introduction Assessments and tests are the primary methods of evaluation in the education system. However the concepts of assessment, test and evaluation are not clearly understood by those in the school system. A test “connotes the presentation of a standard set of questions to be answered” (Mehrens & Lehmann, 1991, p. 4). Assessment is often used interchangeably with test but according to Mehrens and Lehmann this is not the case. Assessment is the use of both 2

Hassan Basarally 806007430 EDME 2006

“formal and informal data-gathering procedures and the combining of the data in a global fashion to reach an overall judgement” (1991, p. 4). Both tests and assessment are therefore used to evaluate students. Evaluation is the process of making a value judgement based on information from one or more sources. It must be noted that evaluation cannot exist truly of the entire student but of a certain aspect such as academic performance, behaviour and attitude. The project undertaken was to test the mathematical concept of fractions in a Form One (1) class. The topics under fractions tested were proper and improper fractions, mixed numbers and their conversions, fraction equality and simplification, fraction arrangement and fraction computation. The test conducted was summative in nature. This type of assessment provides information about student learning at the end of a unit of instruction. Summative assessment is not the only type of assessment available to the teacher. Among others are formative, alternative and performance. Formative assessment “refers to assessment that is specifically intended to provide feedback on performance to improve and accelerate learning” (Sadler, 1998, p. 77). Alternative assessment includes the non traditional forms of assessment such as portfolios and projects. The performance assessment requires the student to demonstrate the knowledge learnt.

Background The test was administered at Williamsville Junior Secondary on Tuesday 18th March, 2008. The school is located on Main Road, Williamsville. It has large catchment area which consists of Princes Town, Gasparillo, Marabella and Williamsville and its environs. The test was one (1) hour long, it started at 9: 50 a.m. and ended at 10: 50 a.m. The target group was Form One (1), given the designation 1.1 by the school. The target group was composed of eighteen (18) students; six (6) boys and twelve (12) girls. The age group of the students ranged from twelve (12) to thirteen (13) years old. This age group was selected because of the cognitive 3

Hassan Basarally 806007430 EDME 2006

development stage in which they were experiencing. Piaget calls this the formal operation stage. During which “learners display an increased abilityto abstract and deal with ideas independent of his or her own experience” (Wadsworth, 1996, p. 84). This is important to the choice of subject,

mathematics. Mathematics has many abstract and practical concepts, which both describe fractions. The test calls on students to test what they have learnt in a systematic fashion to determine its validity. In summary, it was found that the students did not fully grasp the topic of fractions, especially the concepts of Lowest Common Multiple (LCM) and division of fractions. The texts used in the teaching period and for the formulation of questions were: Layne, C. E. et. al., (1997). STP Caribbean Mathematics. (New Edition). Cheltenham: Stanley Thomas Publications. Pramanand, Harbukhan & Seegobin, (1997). Integrated Mathematics for Caribbean Schools: a problem solving approach. Trinidad: Daiken Publishers Limited. The classroom conditions for the test were appropriate. The classroom was well lit with all the lights working. There was adequate ventilation but not too much wind. The room was spacious enough for the students to be able to be well spaced from each other. There was a table for students to place their bags so as not to be forced to hold them while doing the exam. Noise level was to a minimum and extra stationary was provided by the test invigilator.

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Hassan Basarally 806007430 EDME 2006

Purpose of the Test The purpose of the test was: Evaluate the performance of a sample of pupils in a specified secondary school in Trinidad and Tobago. To determine whether the teaching objectives were met. To determine the strengths and weaknesses of the students in the topic of fractions. To determine which areas need to be re taught. To provide the teacher with possible areas for instruction improvement. Group students according to performance on the test. Assist in educational research. Assist in the evaluation of the researcher‘s ability to apply learnt concepts from EDME2006.

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Hassan Basarally 806007430 EDME 2006

Objectives Objectives are essential to the successful classroom experience. An absence of objectives results in“the educator [being] vulnerable to externally imposed prescriptions, to fads and frills, to authoritarian schemes” (Ornstein & Hunkins, 2004, p. 9). This results in the teacher having no set

strategy for teaching and not being able to determine whether anything was achieved. Objectives help “a teacher plan instruction, guide student learning, and provide a criteria for evaluating student outcomes” (Mehrens & Lehmann, 1991, p. 4). At the end of the teaching period students will be able to: 1. Identify types of fractions. 2. Identify the components of a fraction. 3. Match fractions with their diagrammatical representations. 4. Infer fractions of a whole from given word or diagram problems. 5. Select fractions of equal value. 6. Simplify given fractions to their lowest value. 7. Arrange fractions in ascending or descending order. 8. Compute missing numerator or denominator values in order to show fractions of equal value. 9. Convert mixed numbers to improper fractions and vice versa. 10. Solve computing problems involving addition, subtraction, multiplication and division of fractions.

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Hassan Basarally 806007430 EDME 2006 11. Appreciate that fractions express quantities that are used in daily living by using them to show portion allocation.

Content The content of the test included the following areas: 1. Improper fractions

2. Proper fractions. 3. Mixed numbers 4. Mixed number conversion. 5. Simplification of various types of fractions. 6. Fraction equality. 7. Ascending order of fractions. 8. Descending order of fractions. 9. Fraction computation.

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Test Construction Using the equation: weeks per topictotal weeks x # of questions, the amount of questions per objective was determined. This was to ensure that the amount of questions were proportional to the amount of teaching time. Topic

Teaching Time (weeks)

Projected amount

Actual amount

of questions

of questions

Proper / Improper

1.5

5.625

7

Fractions Conversion / Mixed

0.5

1.825

2

Fractions Fraction Equality /

1.5

5.625

5

Simplification Fraction Arrangement Fraction Computation Total

0.5 4 8

1.825 15 30

2 14 30

Table 1: Calculation of Questions per Topic It should be noted that the discrepancy between the projected and actual amounts of questions are normal due to factors such as teaching time per topic. (A list of the questions and the corresponding objectives is found in the Table of Specification). Example of calculations: weeks per topictotal weeks x # of questions 1.58 x 30=5.625 weeks per topictotal weeks x # of questions 0.58 X 30=1.825 weeks per topictotal weeks x # of questions 48 x 30=15 8

Hassan Basarally 806007430 EDME 2006

Table of Specification Objective Dimension Content Dimension (Im)proper Fractions/ Fraction Identifications

Knowled

Comprehens

ge

ion

(1,2,9,1

(11,12,14)

0)

III

IV

3&6

Application

Analy sis

Tota l

7

1&4 Conversion/Mixe d Fractions

(3)

(19)

I

I

3 &2

5

2

9

Hassan Basarally 806007430 EDME 2006

Fraction Equality/Simplifi cation

(13)

(4,5,20,21)

I

IV

3, 6 & 8

6, 7 & 8

Fraction Arrangement

5 (22,2 3)

2

II Fraction computation (+ x

÷)

Total

4

5

(6,7,8,24,25,26,27,2

9 (15,

8,29,30)

16,17

VIII

, 18)

2, 10, & 11

IV

15

10 6

14

30

Table 2: Table of Specification Key: The numbers in parenthesis () represent the question number on the test. The numbers in Roman numerals represent the number of items on the test. The plain numbers represent the corresponding objective. For example, (1, 2, 9, 10) IV 1 means that questions 1, 2, 9 and 10 fall under the content dimension listed, there are IV (4) questions of this type in the test and these questions test objective number one (1) in the list of objectives.

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Item Selection The test consisted solely of Objective Type Items (see Appendix I). There are two types of objective type questions, Supply Type and Select Type. The Supply Type includes True/False questions, short answer and completion statements. True/ False questions allow a large amount of subject matter to be covered. In addition it benefits poor readers, however like the multiple choice question pupil’s scores “may be unduly influenced by good or poor luck in guessing” (Mehrens & Lehmann, 1991, p. 123). “Short answer items are particularly useful in mathematics and sciences, where computational answers are required or where a formula or equation is to be written” (Mehrens & Lehmann, 1991, p. 112). The Select Type includes the multiple choice and matching items. The benefit of this type of question is that it is more economical in correction time. In addition as more questions can be asked it builds a broad base of knowledge. “Because of the lessened amount of time needed for pupils to respond to a objective items many questions can be asked in a prescribed examination period and more adequate content sampling can be obtained, resulting in higher reliability and better content validity” (Mehrens & Lehmann, 1991, p. 107). The matching items are useful as “the major advantage of the matching exercise is its compact form, which makes it possible to measure a large amount of related factual material” (Linn & Miller, 2005, p.181). Despite this Objective Type Items tend to not measure higher mental processes. The test contained features of a Power Test as the students were allocated sixty (60) minutes in addition the questions were ordered. “Items on a power test have different levels of difficulty usually arranged in a hierarchy from knowledge level (easy) to increasing difficulty” (Notar, C. E. et. al., 2004). The test paper had the following composition of questions:

Short Answer

Supply Type True/False

Completion

Select Type Multiple

Matching

Total

Statements

Choice

Items 11

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12

4

2

8

4

30

Table 3: Test Item Selection

Test items NAME CLASS DATE SUBJECT

Mathematics - Fractions

DURATION OF EXAM

1 hour

INSTRUCTIONS: Fill in your name, class and date on the cover sheet. 12

Hassan Basarally 806007430 EDME 2006 Read all instructions carefully. Answer all thirty (30) questions. Write answers in pen or pencil. All answers are to be written in this paper. All working can be done in the working column or the blank side of the question paper. If you have any questions during the test, please raise your hand. This paper consists of four (4) pages.

Directions -In each of the following questions, Nos. 1 – 8, choose the best option by shading either (A), (B), (C) or (D).

(1) Which of the following is an improper fraction?

[1mark]

(A) 2¼ (B) 56 (C) 32 (D) 12

(2) Which of the following is a proper fraction? [1mark] (A) 14 (B) 3½ (C) 107

(D) 0.5 13

Hassan Basarally 806007430 EDME 2006

(3) What mixed number does Diagram 1 represent? [1mark]

Diagram 1 (A) 4⅔

(B)5¼ (C)5¾ (D)6¼ Working Column (4) What is the missing number, x, in the equality: 45= x50

[1mark]

(A) 100 (B) 40 (C) 12 (D) 10

(5) When simplified, 1218 is equal to: (A) (B) (C) (D)

[1mark]

46 35 23 14

(6) What is 23

+

27 ?

[1mark]

(A) 421 (B) 410 (C) 35 (D) 2021 14

Hassan Basarally 806007430 EDME 2006

(7) What is 2 45 + 7 310 ?

[1mark]

(A) 1110 (B) 12 (C) 9 710

(D)

10 110

(8) What is 3¾ × 10?

[1mark]

(A) 14 (B) 38 (C) 3⅔ (D) 36 ______________________________________________________________________________

Directions – Complete the following two statements:

(9) The top number in a fraction is called the _____________________. [1mark]

(10) The bottom number in a fraction is called the ____________________. [1mark] ______________________________________________________________________________

Directions -Match the shaded diagrams in Column A with their fractions in Column B by drawing a line to connect them.

[1 mark

each]

Column A

Column B 15

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(11) 16

(12)

38

(13)

14

(14)

13

16

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34 Directions Tick the appropriate box, True or False, for each of the following statements: [1 mark each]

Working Column (15)

3½ ÷ 1 = 3½ × 1

[ ]

[ ]

True

(16)

False

½ × (¼ + ⅛) = half of 38

[ ] True

(17)

½+½÷2=¾

(18) 12× 14

=

12

×

41

[ ] False

[ ]

[ ]

True

False

[ ]

[ ] 17

Hassan Basarally 806007430 EDME 2006 True

False

______________________________________________________________________________

Directions – Work out each of the following in the working column provided. Place your answer in the space marked: Answer: _______ for each question.

Working Column

(19)

Convert the mixed number

8¾ to an improper fraction.

[2 marks]

Answer: __________ ______________________________________________________________________________

(20)

Calculate the missing number, x, in: 37= 9x

[2 marks]

Answer: __________ ______________________________________________________________________________

(21)

Simplify the fraction 1656 to its lowest term.

[2 marks]

Answer: ______________ ______________________________________________________________________________

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Hassan Basarally 806007430 EDME 2006 (22)

Arrange the following in ascending order:

[3 marks]

13, 56, 12, 712

Answer: _________________________

______________________________________________________________________________

(23)

Arrange the following in descending order:

[3 marks]

35, 12, 34, 710

Answer: _________________________

(24)

A vase contains 5 pink flowers, 3 red flowers and 7 white flowers. What fraction of the flowers in the vase are:

(a) Red?

Answer: _________ [2 marks]

(b) Pink?

Answer: _________ [2 marks]

(c) Not white?

Answer: _________ [2 marks]

______________________________________________________________________________

(25)

What is

8

45

– 512?

[3 marks]

Answer: _______________ 19

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______________________________________________________________________________

(26)

What is

5

14×

223?

[3 marks]

Answer: _______________

______________________________________________________________________________

(27)

What is

14 ÷ ⅞?

[3 marks]

Answer: _______________

______________________________________________________________________________

(28)

What is

5⅝ ÷ 6¼?

[4 marks]

Answer: _______________

______________________________________________________________________________

(29)

What is

2

7/8

+

1/4

?

[3 marks]

20

Hassan Basarally 806007430 EDME 2006 Answer: _______________

______________________________________________________________________________

(30)

What is

4

2/3

+2

3/4

?

[3 marks]

Answer: _______________

______________________________________________________________________________

Scoring Rubric Question Number 1 2 3 4 5 6 7 8 9 10 11

Answer C A C B C D D D NUMERATOR DENOMINATOR

⅜

Mark 1 1 1 1 1 1 1 1 1 1 1 21

Hassan Basarally 806007430 EDME 2006 12 13 14 15

¼ ⅓

1

16

1 1

1

TRUE FALSE TRUE FALSE

16 17 18

Item #

Total Mark

19

Working

354

2

21

-

(1)

2

3

23

3

24 (a)

2

(b

-

32 + 3 = 35 (½)

3x3 = 9 (½)

-

7 x 3 = 21 (½)

828=414 - (½)

-

(1)

LCM = 12 (1)

Numerators 4,10,6,7 = (1)

34, 71035, 12, (1)

LCM = 20 (1)

-

Numerators 12,10, 15, 14 = - (1)

-

13, 12, 71256, -

15

-

(1)

2

)

8 x 4 = 32 (½)

1656=828 - (½)

27 (1)

22

1

Answer

- (1)

21

1

Breakdown of Mark

2

20

1

⅓

-

total of

Reducing of

flowers = 15 (½)

35 (½)

-

Reducing of

(1)

515

-

(1) (c )

2

815

-

5 + 3=8 (1)

22

Hassan Basarally 806007430 EDME 2006 (1) 25

3

3310

-

(1)

26

3

14

-

8 – 5=3 (1)

-

-

Cancelling - (1)

Improper fraction 214 (1)

3

Numerators 8, 5 (1)

Conversion to

(1)

27

LCM = 10 - (1)

16

-

-

Changing division

(1)

Cancelling - (1)

to multiplication and flip the second fraction - (1) 28

4

910

Conversion of to improper fraction

- (1)

29

3

3 (1)

458-254 - (1)

to multiplication and flip the second fraction - (1)

3⅛

-

LCM and numerators (1)

Answer 1⅛ added to 2 - (1)

7⅝

-

LCM and numerators (1)

Answer 1⅝ added to 6 - (1)

(1)

30

Changing division

Cancelling (1)

Table 4: Rubric

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Interpretation After the papers were corrected by three (3) individuals, two (2) from within the research group and an external party the following scores were earned by the students: Name Anastacia .L Ameer M. Donnelle W. Monique S. Seantel G. Malcom B. Vanessa H. Farisha M. Nerissa J. Akeisha B. O'neil M. Kimberly R. Fareed H. Kevin S. Gizelle J. Janice F Regine A. Damian S.

Score

Total

53 52

100 100

48

100

46

100

44

100

40

100

40 36 32

100 100 100

29 28

100 100

27 25 24 24 23 22

100 100 100 100 100 100

16

100

Mean 33.833 33

Mode 40 & 24

Media n 30.5

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The range can be found by subtracting the lowest score from the highest one: 53 – 16 = 37. This shows that due to the small range the scores are close together. The mode and the median were lower than the mean, showing that the test was too difficult for the students. The histogram shows the frequency of the class intervals. To further suggest that the items were too difficult, the lowest interval (20-29) had the highest frequency.

Figure 1: Histogram Showing Score Frequency By finding the square root of the square of the mean and dividing it by the amount of students the standard deviation was found. x2N = 33.8333218 = 11.336

Due to the fact that a normal distribution is “a mathematical model-an idealisation-that can be used to represent data collected in behavioural research” a non normal distribution was achieved (Shavelson qtd. in Best & Khan, 1998, p. 352). The result was skewed negatively as shown in Table 4. This meant that the students’ scores were near the low end of the range. According to Best and Khan skewed distribution indicates that “a test... is too easy or hard or an atypical sample (very bright or very low intelligence)” (1998, p. 353). As this project cannot determine whether there was an atypical sample, the interpretation was that the test was too difficult for the students.

Figure 2: Skewed Curve However the student performance can be understood through the use of the Z score. “The conversion of each test score to a sigma score makes them equally weighted and comparable” (Best & Khan, 1998, p. 353). By subtracting the mean from the raw score and dividing it by the standard deviation the Z score is achieved. For example, Seantel G.’s Z score was calculated: 4425

Hassan Basarally 806007430 EDME 2006 33.833311.336 = 0.89. If this score was equivalent or near to a past mathematics test. It would

mean that the performance was typical, resulting in the test not necessarily being too difficult. A previous score was obtained from the teacher and the following calculation was made:4433.833311.336=1.6. This showed that the Z scores were close and that the student

performance was fairly typical. The Item difficulty table showed that generally students did better in the Select Type questions as opposed to the Supply Type. Also called the Item-Achievement Rate it refers to “the percentage or proportion of students who get the individual item correct” (Gallagher, 1998, p. 329). The Item Difficulty Index was low for questions 25- 30 but high for 6 -8 despite both sets testing fraction computation. A possible reason is that the students found difficulty in working the questions and may find the concepts such as LCM difficult to apply. The students scored better in questions in which some visual aide was used such as questions 3, 11 and 12. The exception is question 13 in which students were not able to realise that the fraction could be reduced. The Item Difficulty Index showed that objectives such as: 1. identify types of fractions, 2. identify the components of a fraction and 3. match fractions with their diagrammatical representations were achieved. However, objectives, such as; solve computing problems involving addition, subtraction, multiplication and division of fractions, were not. Therefore students had knowledge and comprehension of the topic but were not able to apply it. Other factors to affect the index include quality on teaching and time on task.

Question Number

Amount of students

Amount of students

Item difficulty

1 2 3 4 5 6 7 8

answering correctly 8 14 13 10 13 2 3 16

answering incorrectly 10 4 5 8 5 16 15 2

index (%) 44 78 72 55 72 11 17 89 26

Hassan Basarally 806007430 EDME 2006

9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 a 24 b 24 c 25 26 27 28 29 30

3 6 17 17 8 17 11 14 3 10 5 15 10 0 1 12 16 1 3 1 3 0 0 1

15 12 1 1 10 1 7 4 15 8 13 3 8 18 17 6 6 17 15 17 15 18 18 17

17 33 94 94 44 94 61 78 17 55 28 83 55 0 5.5 66 89 5.5 17 5.5 17 0 0 5.5

Table 5: Item Difficulty Index The distractors provided met limited success. Some questions had different distractors selected while others did not. Question 7 and 8 had only one distractor selected by the students. While question 7 seems to have an alternative too close to the answer, question 7 shows that students probably did convert the mixed number to an improper fraction before multiplying. Student M. B. G. J. D. W. J. F. D. S. A. L. N. J.

Q1 B C C A A C A

Q2 A A A A A A B

Q3 C C B C B C D

Q4 B A B D B B B

Q5 C C C D A C C

Q6 B B A B B D B

Q7 D C C C C D C

Q8 C D D D C D D

Table 6: Answers to Multiple Choice Questions A

B

C

D 27

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Q1 Q2 Q3 Q4 Q5 Q6 Q7 Q8

3 7 0 1 1 1 0 0

1 1 2 5 0 5 0 0

3 0 4 0 5 0 5 2

0 0 1 1 1 1 2 5

Table 7: Distractor Analysis (Correct answer underlined) The distractor analysis contained students of both the high and low groups. This enabled the investigation of how good the distractors were. This can be done by the following table: Item

1

Item

Item

Difficulty

Discrimination

Index 44

Index 0.3

Alternative

High Group

Low Group

A B C D

0 1 2 0

2 0 0 0

Table 8: Distractor Analysis for Question1 What is learnt is that since only 44% of the students choose the correct answer the item was difficult. This is a flaw in the test as the easier items should come at the beginning of the paper. There appears to be no ambiguity in the stem or alternatives of the item (See Test Items). The Discrimination Index is minimally acceptable, as a high positive figure is required. The distractor D did not work as no student choose it. Distractor A worked as students of the Low group choose it. Due to the small sample the figures cannot give an accurate description of the distractor, generally it worked. From the Item Discrimination Index the questions tend to discriminate in the desired direction. However, there are instances of the index being at zero (0). In question 12 all the students scored, therefore the question may have been too simple. In question 26 all the students 28

Hassan Basarally 806007430 EDME 2006

had the incorrect answer meaning that the fraction computation continues to be a problem area for students.

Question 1 2 12 13 25 26

Number in high

Number in low

group answering

group answering

correctly H 3 3 3 2 2 0

correctly L 1 2 3 0 0 0

H-LN

H–L 2 1 0 2 2 0

D 0.6 0.3 0 0.6 0.6 0

Table 9: Item Discrimination Index By comparing the Item Difficulty Index to the Item Discrimination Index it can be deduced whether a particular item is flawed. This would occur when there is a low Item Difficulty Index and a negative Item Discrimination Index. In short few students got the item correct and those that did were not from the high group. As shown in Table this was not the case for the test. The purposes of the test were fulfilled. It was found that students did understand some of the objectives initially. However in certain areas such as fraction computation and fraction simplification there needs to be revision by the students and if deemed necessary by the teacher, some objectives need to be re taught. It should be noted that the suggestions are made with the limitation of the researcher not actually teaching the class. The matching items questions were well done as these had a higher Item Discrimination Index than the multiple choice items, in which some had poor results in the distractor analysis. The test can be improved through the reduction of multiple choice items (see Appendix II). As evidenced by the Item Difficulty Index students scored better in the select types. Select types such as question 8 and 14 had an index of 89 and 94 respectively. Supply items such as 28 had an index of 0. 29

Hassan Basarally 806007430 EDME 2006

Therefore the test has a certain degree of success and reliability. The Item Discrimination and Difficulty Indices mostly had the desired figures. Improvements need to be made in detractor selection and possibly in instruction to raise the mean.

Works Cited Best, J.W. & Khan J. V. (1998). Research in Education (8th ed.). Gallagher, D. J. (1998). Classroom testing for Teachers. New Jersey: Patience-Hall Inc. Linn L. R. & Miller, M. D. (2005). Measurement and Assessment in Teaching. (9th ed.). New Jersey: Pearson Education Inc. Mehrens, A. W. & Lehmann J. I (1991). Measurement and Evaluation in Education and Psychology (4th ed.). California: Wadsworth/Thomas learning. Ornstein, A. C. & Hunkins F. P. (2004). Curriculum: Foundations, Principles, and Issues (4th ed.). Massachusetts: Allyn and Bacon. Sadler, D. R. (1998) Formative assessment: Revisiting the territory, Assessment in Education, 5(1), pp. 7-83. Wadsworth, B. J., (1996). Piaget’s theory of cognitive and affective development. (5th ed.). New York: Longman Publishers USA.

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