Abstract
| Mathematics is always evolving. It always undergoes changes through the years. It undergoes paradigm shift when new elements are adapted into the teachings. Teachers are always finding new ways to teach maths, whether senior or young teachers who are competent in ICT. Senior teachers who have fossilised look very confident with their ways, but for teachers who have just graduated from teacher college, they are out there finding new ways, just to teach Maths the effective and fun way. Whether a teacher is teaching 4A or 4G (the last class)in streaming schools, they must adapt their lesson plans for both competency levels to make the students able to achieve the objectives before the bell rings. The current Maths methodology has evolved, the only thing that is consistent is evolution. The methodology presented in this assignment is interesting and designed to achieve its objectives. In the first part, it is presented the errors the students make when they perform multiplication process, the second part how we can improve our ways to suit their needs, then this paper. |
1.0 Introduction
I am a teacher in Sekolah Kebangsaan Air Baruk. I teach Mathematics in 4 Nilam there. The problem with my sample students are they do not understand the question’s requirements, they do not understand the Maths terms used in the questions, I always think, why do they not able to remember the times-table and lastly, they cannot perform the regrouping process when they multiply. So, I have done a non structural survey in my class, These errors are evident from the sample answers I have done and given to them, and also from the exercises I have done with them in the classroom. My first analysis that I have arrived to is they are quite lazy to understand the question that are presented as case studies, they do not want to put forth their initiative to master multiplication. Here are examples of the errors I have found in their answers. This problem happens because they do not revise their lessons during the weekends. This may also come from me, as perhaps I need to adapt my teaching styles according to their competency level. Even though I detect many errors that my students have done, I just focus on 4 main elements, that are to make them understand the questions and the terms used, make them remember the times-table and they be able to perform regrouping for questions that require regrouping. To solve their problems an innovation where I combine language with Mathematics is done. From my observation, it is found that the teacher can grasp the student’s attention by using realia, real things that they find in their lives. I also run away from the normal writing using pencil on papers, I use cards, so they need not write to perform the correct method and arrive at the correct answers. The students become more interested and the teacher can identify student’s area of interests and strengths.General Objective is I want to increase the students’ skills in multiplication among the students of 4 Nilam through relia and kinaesthetic skills theory and specific objective is by the end of the research, 100 % students will be able to perform multiplication effectively. Change the students’ perception that Mathematics is a difficult subject, and increase their confidence in themselves and the sense of responsibility towards themselves. Create a good interaction between the students and students and teachers.
2.0 MISTAKES
2.1 Do not understand the question’s requirements.
They answer not according to the requirements, because the question requires them to multiply but the pupils answer the question using addition, subtraction and dividing numbers. Students also do not understand what the question means, although the question is in both languages. They also do not understand the technical terms, when it is presented as a case study, then they cannot understand it. They are able to solve the problems when they are presented in numbers only. Students do not answer according to the question requirement, the question requires the students to answer using multiplication but the students answered using addition and subtraction. The students are lazy in reading the questions even though the questions are bilingual, and this makes them not understand.
2.2 Misunderstanding on terms.
The participating students performed best for word problems in Bahasa Melayu. However, most of the Standard 4 students from urban schools faced fewer problems in learning mathematics. This could be due their higher proficiency in English language as their parents are more conscious about the education of their children and there is better opportunity to learn English either in schools or outside of schools as compared to the low English proficiency of students from rural schools.
Students perceived that they were not ready to learn mathematics through English. However, they were very positive towards learning of mathematics. The participating teachers perceived that limited English proficiency is one of the reasons why students cannot follow the mathematics lessons. As a result, teachers need to explain the lessons in Bahasa Melayu because the message can then be delivered successfully. A moderate number of teachers perceived that they were ready to teach mathematics through English. Majority of the teachers agreed that students’ achievement in mathematics would improve if they do a lot of revision after school. As a whole, teachers showed good attitude towards mathematics and English.
Students do not understand the English terms and perhaps they do not read the Malay translation. For example, in question number 2 for all question sets. They cannot relate the word weekdays, they think there are 7 days in a weekday, from Sunday to Saturday., but they do not remember it’s called only week. The error is visible in all question sets 1 until 4.
The terms are presented on cue cards, and they are extracted from case study questions. For example,
Ali has 235 stamps and he gives them equally to his three siblings. How many does one has?
The phrase gives them equally:
| gives them equally |
The students are given 4 cards which contain 4 different symbols, they are symbols for divide, multiply, addition and subtraction. They show to the class what is the symbols that is meant by the phrase gives them equally.
Second example;
Mr Salleh divides equally RM 5 to his 5 sons. How much will one son gets?
2.3 Do not remember times-table
The students know the question’s requirements but they face problems to calculate because they do not remember the times-table. The times-table for 2,3 and 4 that they have learn t in standard One. It makes them to write the mistake. The student very lazy to read or memories the times- teble because they think the times-table is not important in mathematics or life.
2.4 Error in regrouping
The student’s error is visible in the value of tens, they do not add multiplication answer with the value on top of the tens number. They just write the answer for the multiplication and ignore the number supposed to be added to the tens number.
There are many different ways to perform multiplication. Usually, we perform the task of multiplication by the standard form, which is writing the numbers to be multiplied in vertical order.
| 7 | 0 | 2 |
| | 1 | 1 | |
| | 2 | 3 | 4 |
| x | | | 3 |
Algorithm here is written in a standard written method, which is commonly used in the class. Multiplication in regrouping in ones will place ones inside and tens in put up. This type will not use bigger space and less numbers to provide the answers. Students do not add the numbers on top of the tens and hundreds, which will be a mistake. Perhaps it is because the number 1 is smaller than the normal size , that makes them not see the importance.
The sample of the mistake is visible at second question set, third question and in the first question set, question no 4.
3.0 Improvements
3.1 Improvements for not understanding the question’s requirements
Ask students to read the questions many times, and identify the English words used in the question, so the students will understand the requirements. Ask students to underline the key words so they know the requirements.
3.2 Improvements for misunderstanding on terms
Students understand English words. Teacher discuss with students topics related to the question, and stress out to the students about week, month and year. Guide the student to take out the terms or underline them the important terms in the question
3.3 Improvements for inability to memorise times-table
The basic facts of multiplication involve the products of any two 1-digit whole numbers. We shall concentrate ourselves on teaching half on the basic facts to our Year 2 students, that is, till 9 x 5 = 45. As we can organize in a similar manner. In fact organizing the basic facts into a table could help our students to remember and master the basic facts systematically and easily. Before helping students to construct a table of basic facts of the basic facts of multiplication, we should make sure that they understand them well.
Here are some of the important basic multiplication facts: Multiplying by 0 : Any number multiplied by 0 equals 0. Multiplying by 1 : Any number multiplied by 1 equal number. Multiplying by 2 and 3 are as follows:
| TWO | ||||
| 0 x 2 = 0 | 1 x 2 = 2 | 2 x 2 = 4 | 3 x 2 = 6 | 4 x 2 = 8 |
| 5 x 2 =10 | 6 x 2 =12 | 7 x 2 =14 | 8 x 2 =16 | 9 x 2 =18 |
THREE | ||||
| 0 x 3 = 0 | 1 x 3 = 3 | 2 x 3 = 6 | 3 x 3 = 9 | 4 x 3 = 12 |
| 5 x 3 =15 | 6 x 3 =18 | 7 x 3 =21 | 8 x 3 = 24 | 9 x 3 = 27 |
Multiplying by 4 and 5.
| FOUR | ||||
| 0 x 4 = 0 | 1 x 4 = 4 | 2 x 4 = 8 | 3 x 4 =12 | 4 x 4 =16 |
| 5 x 4 =20 | 6 x 4 =24 | 7 x 4 =28 | 8 x 4 =32 | 9 x 4 =36 |
FIVE | ||||
| 0 x 5 = 0 | 1 x 5 = 5 | 2 x 5 = 10 | 3 x 5 = 15 | 4 x 5 = 20 |
| | | | | |
There are 4 important things that a teacher must practice at school, to help the students to remember the basic factor in multiplication more easily and accurately. Besides the above the teacher can use other technique more effectively. Teacher must ask random question and they practice by filling empty zero box. The teacher uses 3 minutes for each teaching session
I systematize a 5 minute times-table time, the students will recite the times-table for five minutes before a maths class is on the go. If I feel it takes time, I will stand by the class that I teach Maths during the assembly, and when they disperse, they say out the multiplication facts when they are walking to their class. This way will make my students not to waste time talking.
3.4 Improvements for mistake of not regrouping the numbers
Teacher gives a series of exercise related to regrouping so students become familiar. Stress to the students that every time there is the value multiplication calculation must be added with first number of the two digit number, based on the place value. Teacher also gives extra work so students become familiar
4.0 Activities
Example of the mistakes are in question set 1 on question number 1,2 and 3.
4.1.1 Activity named Mix and Match
- Teacher arranges mix and match game.
- The students are briefed on how to play the game.
- Teacher gives cards containing mathematical terms. Teacher flashes a card containing a phrase, and the students will select and show a maths term that carries the same meaning.
- Teacher gives marks for the teams with the most answers correct.
(The terms are multiply, divide, addition, subtraction, multiplicand, equals division and others. The phrases are such as give away, how much left, how much taken, How many days are there in a week day ? , How many days are there in a weekend, and How many days are there in a week ?. Teacher also has questions such as :What is the meaning of Balance ? What is the maths process involved if a question has this phrase ? Added more, Increase the volume and decrease the volume)
4.1.2 Question in Cloze passage
1. Teacher gets the students into groups of 5.
2. They are given a cloze passage.(refer below)
3. When they arrive at the underlined phrases, they must find the word that carries the equivalent definition to the phrase.
4. They show the word or task that carries the similar meaning to the phrases underlined.
This is the passage:
Salleh has given away 4 out of 5 marbles he has to Kalam. How many does he has now ? Kamal has added 2 marbles to his marbles. How many does he has now ? Kamal lost 3 marbles. How many does he has now ? 6 students went and see Kamal and each give him two marbles. How many does he has now ? Kamal give away equally all his marbles to three friends. How many marbles does Kamal has now ? How many marbles do each of his friend has ?
4.2.1 Misunderstanding on terms
Students are given the time to utter the multiplication facts
4.2.2 The Animal Farm
Activity
- The teacher helps them by showing to them animals that represent a value
- The students are shown 3 kinds of pictures.(refer below)
- Teacher asks a question such as : what is 4 X 5 ? and gives them 5 seconds to answer. The students answers and makes the sound and act like a cat if they answer in more than 5 seconds or they did not answer at all. If they did not know the answer, teacher shows 2 replicas of a cat. The students will guess the number must be 20, because 2 cats equals to ten times two equals to twenty.
Picture of a cat, a hen and a bird. Each carries a different value from one another. The Cat has 10 in value, the hen has 5 in value and the bird carries 1 in value.
4.3 Inability to memorise times-table
Activity
- Students are divided into groups of 5.
- They are given a set of questions to answer. They are given the cards containing numerical values.(refer below)
- When they are given a question, they collect the cards; according to the amount it is multiplied. Then, they do addition for all the values of the cards. What is 4 X 5 ?
(refer below)
- They take 5 cards and count the total values and they answer twenty.
Cut these cards out
| 4 | 4 | 4 | 4 | 4 |
| 4 | 4 | 4 | 4 | 4 |
| 4 | 4 | 4 | 4 | 4 |
+ + + +
=4 =8 =12 =16 =20
4.4 Students do not regroup the numbers
Activity
- Students are separated into groups of 5. They are given 5 questions. They are taught on the algorithm of multiplication (see below).
- They attempt the first question what is two hundred thirty four times three ? They arrange the cards on the rubber mat so it represents a written work of the algorithm.
- The students present how do they settle the maths problem on the white board.
- Teacher checks the answers and rearranges the numbers if there are mistakes.
(refer attachment 1)
This method prevents the students from forgetting to add the number on top of the tens and hundreds numbers. It uses a different solution on the whole. Instead of writing the numbers small on top, this time it uses the box system . The students do not add any numbers on top, because there is no small numbers on top of the tens and hundreds. They must understand to write the value of the digit correctly. The digit value will be multiplied and the answers will be written inside the box provided.
8.0 Conclusion
After doing this action research, I see improvements in the results, where the students’ interest in learning Maths, especially multiplication improves and their mars also gets better. I propose the four methods to other teachers and it is hoped that this action research will benefit other classes as well. At the same time teachers must requires the students to master the times-table as well because times-table is important is mathematics and life.
Bibliography
- Dr Mahmood Othman. (2011).HMBT3103 Teaching Mathematics in Year Four. Kuala Lumpur . Open University Malaysia
- Wan Yusof Wan Ngah et al. (2005). Mathematics Teachers Guidebook Year 4. Kuala Lumpur. Dewan Bahasa Dan Pustaka
- http://www.curiousmath.com/index.php?name=News&file=article&sid=10
- http://edhelper.com/multiplication.htm
- http://www.mathleague.com/help/wholenumbers/wholenumbers.htm
- http://www.coolmath4kids.com/times-tables/number-monster-times-tables-multiplication.htm
- http://www.mad4maths.com/math_help_multiplication/
- http://www.mathcats.com/grownupcats/ideabankmultiplication.html
Attachment 1
| x | 200 | 30 |  4 | |
| 3 | 600 | 90 | 12 | |
| | 234 x 3 = 600 + 90 + 12 = 702 | | ||
| | | |||
| | | | | |
Introduction
The level of competency among standard four students are different from each other. This assignment aims to expose to the latest techniques that has been proven in schools. These techniques are entertaining and educational, in other words they are edutaining. The first activity tackles the problem the students are careless when they subtract. Students must be able to identify the words and phrases that are used in the question and determine what algorithm is needed. solve 4 major problems in solving division task. In division, the students use the times-table and subtraction method. It also tests students on the skills of recognising small and big numbers.
Students are introduced to the concept of subtraction using the formula. They are shown on how to solve the dividing number on the white board. They are given a problem, and they have numbers from 1 to 9 printed on cards, and they arrange the numbers according to the correct method. They use the buttons to represent zero. The exercises are shown, and they In this assignment, the aim is to expose to the latest techniques that has been proven in schools. These techniques are entertaining and educational. The errors are visible at question sample no 1, 2 and 3.
First error: The students are careless when subtracting between 2 numbers.
They comprehend the questions and they arrange the numbers in a normal way, but the error happens when they did not subtract correctly, they become careless. This makes them to make an error. They arranged the numbers according to the standard method, but they subtract upwards. This error happens when they have done an error when they multiply the number. When they cannot find the correct way, they do not make correction, instead they carry on subtracting. For example, 1526 / 20= They still do this 15-20=5. The standard method is to subtract 152 to 140, which is the nearest and a lower value than 150(152-140=12).
The pupils understand the concept of division, they divide the numbers according to the normal way, it is just that they are do not subtract correctly, such as 23-3= 19, 15-3=9 and 18-4=8. It makes the error and then the error will affect the method .It happened when the pupils is careless because they want to finish up answering quickly. A simple subtraction such as 26-3 will be answered as 24. The problem with dividing is it will involve only subtraction and multiplication, and there is no addition, but if the student makes an error in the process of dividing, then it will not get the correct answer. This is perhaps they are hasty in doing the work, or they subtract mentally but they are unable to complete the task.
Second error: Students do not understand the concept.
Students come to school with varied language experience. While some students utilize Standard English, others may use a local dialect or speak a different language at home. There will be some children who speak very little even though they may talk a lot at home. One of the errands of teachers is to extend their use of language. Speech not only facilitates the child’s successful manipulation of objects but also control the child’s own behaviour. In this case, misunderstanding will happen, and the students are not able to grasp a question’s requirements. Students have a low competency level and they do not understand the concept of the question.
The question requires the students to answer by division, but the students answer by addition and subtracting. Students also do not understand what the question means, although the question is in Bahasa Malaysia and English, they are in both languages. They also do not understand the technical terms, when it is presented as a case study, and that makes them not to be able to understand it. They are able to solve the problems when they are presented in numbers only.
When they read a case, such as Mr Salleh divides his RM 5 for all of his 5 sons. How much will one son gets? Instead of dividing the amount RM using the normal way, they multiply RM 5 with 5= RM 25. The second sample is Ali has 235 stamps and he gives them equally to his three siblings. How many does one has? In this case the student just use the concept of subtraction, that is 235-3= 232, so he thinks each siblings get 232 stamps which is illogical. The second error is they do not understand the meaning of the technical terms, for example, they understand the meaning of the words, they can read either in Bahasa or English or both, but they do not get what the question wants. This situation makes them not be able to fulfil the questions’ requirements. They must divide between 2 numbers, but they happened to add together, subtract from one another and multiply between the two numbers
Third error: The students did not write zero number at the end of the answer.
Zero is one of the most important concept in maths. Zero is used as a place holder: For example in number 400 ,the zero after 4 tells students that the "tens' column is empty and the zero on the right tells that unit' column is also empty. The only column with any value is the hunderds column. Students would have written 400 as only 4 , which is not the number they meant,if there would not be anything to indicate that two right-hand columns were empty. Students would not be able to distinguish 19081 from 1981 if zero is there.
The students think the zero number do not have a value, valueless, such as when zero stands on its own. They think zero put at the end of a number, such as 300 carry similar value when they are put in front of a number, like 003. The first error is visible in the first question, where the zero number is not written. The solution is there is a game of dividing, where the answers need to be arranged. The zero is represented with buttons, instead of writing the zero number on manila cards.
Fourth error: They divide a small number with a bigger number.
The students divide the first number’s value with the dividend that involved two digits. They cannot understand that a one digit number cannot be done in such a way. When they get a question, such as 1526/20, they divide the first two digits of 1526, that is 15 with 20 and they arrive at the solution, 15 can be subtracted with 20, and they subtract upwards. They receive the first equation to be 5. When they proceed, they subtract 52 with 40, which is the nearest multiplication number to 20, that is 20 X 2= 40.
Improvement:
Improvements for First error: The students are careless when subtracting between 2 numbers.
Stress to the pupils to multiply the correct way, on how to get the nearest value for a number, for example, 5 multiply by what number will get 12, or the nearest and has a lower value from it. arrange the bigger number must be on top of the smaller number. Ask them to count correctly and if possible, use realias such as marbles, straw or sempoa. When it involves a subtraction between 2 digit number, they will find they have to regroup during the subtraction process. In this part, teacher shows the way to subtract, such as 24-16= ?
Improvements for Second error: Students do not understand the concept.
The way to effectively translating and solving word problems is to read the problem fully. Tell students do not start trying to solve anything when they only read half a sentence. They have to get a feel for the whole problem; try first to see what information it has. The students must coin out the phrases that indicate that an operation is required.
The students are given the keywords, teacher stresses on the key words such as difference, balance and remainder, plus and addition, equal sharing, grouping involving dividing and altogether. They are exposed to examples of case studies related to the usage of the keywords. They are also given more exercises to solve the problems, to familiarize the students with problem solving. The students highlight the keywords in the question so they can see clearly the question’s needs.
There are special names for each number in a division: dividend ÷ divisor = quotient. Let us look at the example, 12 is the dividend, 3 is the divisor and 4 is the quotient.
Remind the students about key words such as remainder, difference, sharing, plus and altogether. Vary and add more problem solving so the pupils will become familiar. The students must read the questions carefully, underline the key words in the question.
Improvements for third error: The students did not write zero number at the end of the answer.
Remind the students that a zero has value , and it is essential to put the zero when it is required. A zero before a number has zero value, but a zero after a number has value. Ask every students to solve problems on the board.
Improvements for fourth error: They divide a small number with a bigger number.
While teaching, stress to the students that a one digit number cannot be divided for a 2 digit number. Teacher gives simple questions that involves dividing a one digit number, then dividing a two digit number. Teacher makes group work and peer teaching so the students understand.
Activity for First error: The students are careless when subtracting between 2 numbers
Students are gathered into groups of four. They are given cards containing numbers, each card has a number written on it. They are given 9 situational cases and 1 direct question. They have to arrange the numbers so it represents the question using the number cards.
They solve the problem in groups. When it’s discussion time, every group will show the numbers they have arranged on their desk.
For example:
Salleh has 45 books and he gives 14 books away to his friend. How many books does Salleh has now ?
Arrange the number cards so they represent the standard method of subtraction :
| 4 | 5 |
| - | 4 |
| 4 | 1 |
They are given case studies and they arrange the numbers in the correct order as above.
The Case Studies:
1. Karim has 18 pencils. He gives 11 pencils out as birthday gifts. How many pencils does Mr. Seghers have left?
2. Our teacher, Mrs. Wistrom has 23 cookies. She gives 16 cookies to Mrs. Zarling, the school nurse. How many cookies does Mrs. Wistrom have left?
3. Jalil has 62 marbles. He gives 22 marbles to Dan. How many marbles does Jalil have left?
4. There were 51 fish swimming by the pier. 28 fish swam away. How many fish were left swimming by the pier?
5.Salleh had 100 goldfish. She gave 30 goldfish to her friend, Mary. How many goldfish does Colleen have left?
Activities
- The students are put into groups of four.
- They are given the number cards and they are given straightforward questions
- They are asked to arrange the numbers so it forms the steps to subtract. They subtract the numbers.
- Teacher checks their work and corrects mistakes.
Activity for Second error: Students do not understand the concept.
They are told a simple formula, everything that is the same item can only be added together or subtracted from., and everything that are now coming from the same category can only be multiplied or divided. Explanations are given as a case study:
Mr Salleh divides his RM 5 for all of his 5 sons. How much will one son gets?
RM 5 is money
5 sons is people.
Both of them are coming from a different category, which is money and people, so the formula that we have learnt just now is, everything that is the same item can only be added together or subtracted from., and everything that are now coming from the same category can only be multiplied or divided. So, the numbers cannot be added together or subtracted from, they can only be divided. This method makes the students to come up with the formula RM 5/5= RM 1.
The second sample:
Ali has 235 stamps and he gives them equally to his three siblings. How many does one has?
235 is the number of the stamps and 3 refers to the amount of siblings that are humans, so they cannot be added like this: 235+3=? or 3 subtracted from 235, which will result like this: 235-3=?. They can be formulated as 235/3= or 235 X 3=.
The question has used the word ‘gives them ‘ so the operation of division is required.
The terms are presented on cue cards, and they are extracted from case study questions. For example,
Ali has 235 stamps and he gives them equally to his three siblings. How many does one has?
The phrase gives them equally:
| gives them equally |
The students are given 4 cards which contain 4 different symbols, they are symbols for divide, multiply, addition and subtraction. They show to the class what is the symbols that is meant by the phrase gives them equally.
Activity for Third error: The students did not write zero number at the end of the answer.
Students are gathered into groups of four. They are given cards that have a number for each card. Students will be given 4 division questions. Starting with the first question, teacher will tell the groups what specific amount of numbers and subtraction symbols will be used. All the numbers and symbols that teacher have told must be arranged into the normal way of division.
They are given a case study.
For example, they are given 2640 / 12. They are given 5 cards containing a zero number each, 5 cards containing the 2 number each, and three cards, each has 4 number. They are given the subtraction symbol (-) three cards.
Activity:
- The class is briefed on the importance of doing division using the normal way, that is from top to bottom.
- They are shown on one sample, how to divide a three digit number with a 2 digit number
- Teacher tells of the division game, where they cannot use a pencil or papers to solve the division problem. Instead, they will use cards .They are given a set of divison cards and 5 division questions. Teacher tells the cards that will be used, and they set aside cards that are not used for question number one.
- They solve the first question by arranging the cards and symbols.
- Teacher looks at their activity.
Cut out the cards
| 0 | 0 | 0 | 0 | 0 |
| 2 | 2 | 2 | 2 | 2 |
| 4 | 4 | 4 | | |
Division symbols to be used:
| ÷ | ÷ | ÷ |
Give questions that require a single or a number of zeros at the end of the answer.
- 4000 ÷4 =
- 2000÷4=
- 100÷4=
- 500÷5=
Ask them to arrange the numbers according to the normal method.
Activity for fourth error: They divide a small number with a bigger number.
They are given all the number cards, but they are given an extra card, which is the ‘ X ’ card. We put the card where a number should not be put. It will avoid the students from putting any numbers in the empty space. Another method is the teacher arranges the X card before the student solve the problem. For example:
They cannot understand how a one digit number can be divided with a two digit. It looks like they try to make the number to be able to subtract. When they see 230 divided with 25, they will just subtract 23 with 25, (23-25) and they put the balance as 2. This makes all the rest of the work as wrong. The solution is to determine which number can be subtracted with another number. They choose a number from category A to be divided with a number from category b.
Activity:
1) Teacher mentions the importance of subtracting correctly.
2) Draw an arrow to link which number in column A can be divided with number in column B.
Sample:
A B
5 714 16
30 1014/7= 7
3) They choose a number from category A with category B and write the question.
Conclusion
It takes quite some time to let students master the skill of division, as it involves subtraction and at the same time it requires the students to master the times-table as well.
References
Dr Mahmood Othman. (2011).HMBT3103 Teaching Mathematics in Year Four. Kuala Lumpur . Open University Malaysia
Wan Yusof Wan Ngah et al. (2005). Mathematics Teachers Guidebook Year 4. Kuala Lumpur. Dewan Bahasa Dan Pustaka
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