DEVELOPING THE STUDENTS’ CONCEPTUAL UNDERSTANDING OF

EQUIVALENT FRACTION BY USING CONCRETE MODELS

KATHIRAVAN S/O KAMALA KARAN

KIRKBY INTERNATIONAL COLLEGE

2014

SUPERVISOR’S APPROVAL

“I agree that I have read this action research report and in my view I think it contains all

the scope and the qualities to nominate with a Bachelor in Teaching (Primary

Mathematics).”

Sign :……………………………………………

Supervisor’s name :……………………………………………

Date :………………………………………….

DECLARATION

“I hereby declare that the writing in this action research is my own expect for quotations

and summaries of other author’s work which have been dully acknowledged.”

Sign :…………………………………………………………….

Author’s name :…………………………………………………………….

Date :…………………………………………………………….

DEDICATION

To my beloved father

Mr. Karunanithe S/O Muniandy

and

To my beloved mother

Mrs. Nagammal D/O Narayanan Nair

ABSTRAK

Penelitian ini adalah tentang pentingnya pemahaman konseptual dalam pecahan setara mendedahkan kepada pecahan wajar yang tepat. Ini menelusuri implikasi dari menggunakan model konkrit dalam pembangunan murid-murid dalam pemahaman konseptual tentang pecahan setara serta kemampuan mereka untuk melakukan operasi dengan pecahan. Model konkrit yang digunakan ialah bulatan pecahan dan jalur pecahan. Para peserta dalam kajian ini terdiri daripada 12 murid yang berprestasi tengah daripada Tahun 4 Anggerik, SJK (T) Bukit Mertajam, Pulau Pinang. Data dikumpulkan daripada ‘pre-test’, ‘post-test’, pemerhatian dan analisis latihan pelajar. Tujuan penelitian ini ialah bagaimana model konkrit membantu mengembangkan pemahaman konseptual pelajar pecahan setara. Penelitian ini telah membuktikan bahawa penggunaan model konkrit telah membangunkan pemahaman konseptual pelajar terhadap pecahan setara. Hal ini kerana model konkrit membantu mereka untuk mengeksplorasi hubungan antara konsep dan pengiraan pecahan setara.

ABSTRACT

This research was about the important of conceptual understanding in express equivalent fractions for proper fractions. It traced the implication of using concrete models on the students’ development in conceptual understanding on equivalent fractions as well as their ability to perform operations with fractions. The concrete models were Fraction circle cut-outs and fraction strips. The participants in the study included 12 middle achievers of Year 4 Anggerik students of SJK (T) Bukit Mertajam, Pulau Pinang. The data were collected from pre-test, post-test, observation and students’ work analysis. Conclusion were drawn about how concrete models help develop the students conceptual understanding of equivalent fractions. The research as proved that the use of concrete models has been developed students’ conceptual understanding of equivalent fractions. This is because the concrete models helped them to explore the connection between the concept and the calculation of equivalent fraction.

ACKNOWLEDGEMENT

I, Gunalatcumy D/O Karunanithe would like to take this opportunity to thank

who made the completion of this action research possible for me. My support system has

grown to include my Mathematics lectures, my supervisor, my classmates and my

students.

First of all, I would like to thank Tuan Haji Abdul Razak bin Othman, my

lecturer of Action Research 1- Primary Mathematics (Methodology) who gave full

information, clear explanation and briefing about Action Research. It gave a clear

picture about the proper way of doing an action research.

Next, my sincere thanks go to Dr. Kim Teng Siang, my supervisor for the Action

Research. He guided me very well and helped me to make my Action Research perfect

and complete. His ability to keep all the details in check, while sharing ideas and

information was awesome.

Then, I would like to thank another lecturer of Action Research 1- Primary

Mathematics (Methodology), Mr.P.Pusparajan. He helped and guided me while I’m

completing my result of findings. He has given me many ideas on how to interpret the

data collected in the report.

I also was fortunate to have a truly dedicated group of friends whose critical

evaluations of the study were of great value during the preparation of this Action

Research. They have helped me through every stage. Each of these friends provided

support and assistance whenever I asked, even though they all had their own busy lives.

Their moral support has been invaluable and I was grateful and greatly appreciated for

all my friends have done for me.

I also want to thank my parents for encouraging me to always do my best. Based

on the values they taught me growing up, I believe in the importance of education.

This Action Research would not have been possible without the teacher and

students who participated in this study. The faculty and students at the school welcomed

me and let me become a part of their learning environment. The students in the study

always worked hard and answered my questions thoughtfully. They made collecting data

a wonderfully exciting and interesting process. I am honored that through this process,

the teachers of the school has become a trusted colleague and friend.

I appreciate the guidance and encouragement from all of these people. I am

thankful for all of them who have become a special part of my life. Their support has

made this work possible. Without the help and support from all of them, I couldn’t have

possibly produced this action research.

Thank you.

TABLE OF CONTENT

CONTENT PageABSTRAK iABSTRACT iiSUPERVISOR’S APPROVAL iiiDECLARATION ivDEDICATION vACKNOWLEDGEMENT viLIST OF APPENDICES viiLIST OF TABLES viiiLIST OF FIGURES ix

1.0 INTRODUCTION

1.1 Introduction1.2 Definition of terms

2.0 FOCUS OF THE STUDY

2.1 Concept of fraction2.2 Concept of equivalent fraction2.3 The problem and causes of the problem2.4 The actions to overcome the problems

3.0 RESEARCH OBJECTIVE AND RESEARCH QUESTIONS

3.1 Research objective3.2 Research questions3.3 Significance of study

4.0 TARGET GROUP

5.0 RESEARCH METHODOLOGY

5.1 Introduction5.2 Action research

6.0 DATA COLLECTION

6.1 Pre-test and post-test6.2 Observation 6.3 Students’ work analysis6.4 Data analysis

7.0 RESULT

8.0 SUMMARY

9.0 SUGGESTION

BIBLIOGRAPHY

APPENDIX

List Of Appendices

Appendix 1 Diagnostic Test..............................................................

Appendix 2 Lesson 1.......................................................................

Appendix 3 Pre-Test.........................................................................

Appendix 4 List of Students Achievement.....................................

Appendix 5 Lesson 2.......................................................................

Appendix 6 Lesson 3.......................................................................

Appendix 7 Lesson 4.......................................................................

Appendix 8 Post-Test.......................................................................

Appendix 9 Observation Form........................................................

List Of Tables

Table 1 Schedule of Action Research..........................................

Table 2 Table of Multiples

Table 3 Table of Divisible

Table 4 The Students Achievement in Diagnostic Test ..........

Table 5 Item Analysis of Pre-Test and Post-Test.....................

List Of Figures

Figure 1 Part /Whole Area Model Simple and Equivalent

Representations for Quarters.

Figure 2 Typical Equivalent Fraction Questions and Answer

Employing Symbolic Representations Only.

Figure 3 Teaching and Learning Process Of Mathematics

Figure 4 The Students’ Achievement in Diagnostic Test.

Figure 5 The Result Of Pre-Test And Post-Test Of 12 Pupils

Figure 6 Error of Students D

Figure 7 Error of Students B

Figure 8 Error of Students K

Figure 9 Error of Students C

Figure 10 Error of Students A

Figure 11 Error of Students J

Figure 12 Correct Answer of Students D

Figure 13 Correct Answer of Students B

Figure 14 Correct Answer of Students K

Figure 15 Correct Answer of Students A

1.0 INTRODUCTION

1.1 Introduction

The Principles and Standards for School Mathematics (NCTM, 2000) emphasize

that students should be given the opportunities to develop conceptual understanding as

well as number sense with fraction. Fraction is a major topic in second grade that

students are first introduced to.

This concept continues throughout their mathematical education. While the teachers

struggle to teach the fraction topic effectively, the students are also struggle to gain a

good conceptual understanding of this topic. As the goal of instruction for this topic is

for the students to be able to visualize fractional amounts accurately and it is necessary

for teachers to express equivalent fraction for proper fraction. Yet most of students have

trouble in expressing equivalent fraction for proper fraction.

To find equivalent fractions the students will need to multiply or divide. If students

are still struggling with these skills, they will not be ready to grasp the concept of

equivalent fractions, turning improper fractions into mixed fractions, converting mixed

fractions into improper fractions and so on. So, expressing equivalent fractions for a

fraction accurately play an important role in the computation or algorithms of fractions.

In my experience when I was young and studied in Year Four, I found that I need to

put more effort in the learning and understanding of fractions. This topic seems to be

very different from the other topics and I have difficulties in relating this topic with

others topics.

When I was in Year 4, my Mathematics teacher just prefer to teach the equivalent

fractions with the paper and pencil method. I didn’t get the exact conceptual

understanding of the topic because I only learned about the formulas that use to solve

these fraction questions. I just follow this method and the formula to solve those

questions.

After that, I started to gain the understanding of fraction when I was in Semester

Five of my undergraduate study. In this semester I learned about the Teaching of

Fraction in MTE 3109 course. From that course I realized that equivalent fraction is a

big scope in Fraction and this part needs to be explained very well.

From this course, I also realized that before doing the symbolic representations in

equivalent fractions, the students need to gain the conceptual understanding of these

fractions. I learned about the overall fraction knowledge in this course and I must give

this input to my students.

Lastly, my second practicum in teaching Mathematics to Year 4 Anngerik’s

students of SJK (T) Bukit Mertajam gave me the experience in teaching fraction. When I

helped the students to do revision for their monthly test, the students ask me to give

more explanation for the topic Fraction. They mentioned to me that they found it

difficult in understanding the concepts of Fraction and this topic was the hardest topic

for them to understand clearly.

When I revised this topic with those Year 4 Anggerik’s students, I found that their

main and serious problem was in the Equivalent Fraction. They faced difficulties in

express the equivalent fractions for a proper fraction. Because of this problem they

cannot move on to the next stage of Fraction. This is because equivalent fractions play

an important role in introducing the fractions operations and their algorithms especially

when adding and subtracting with unlike denominators. By this experience, I wonder

and decided to find a suitable action to overcome the problems in expressing or finding

equivalent fractions for any proper fraction.

In conclusion, the above experiences lead me to do my action research on

Equivalent fraction by finding the problem in expressing equivalent fraction for a proper

fraction and the action to the overcome the problem. The action that I planned to do is

‘Touch and Understand’ involving the use of concrete models.

1.2 Definition of terms

1. Develop :

According to the Longman Dictionary of Contemporary English, the word

develop means to grow or gradually change into a stronger or more advanced

state, or to make someone or something to do this.

2. Conceptual understanding :

According to NSW Mathematics curriculum ( Board of studies NSW (BOS

NSW), 2002), the conceptual understanding involves seeing the connection

between concepts and procedures, and being able to apply mathematical

principles in a variety of contexts.

3. Equivalent fractions :

Equivalent fractions means the fractions which have the same value.

4. Concrete models :

Concrete models are an important part of introducing students to a new or more

complex mathematical concept or procedure. It is where the students can touch,

feel and explore the models to understand the concept. Concrete models have

significant effect upon what students learn and the ideas they construct to help

select the appropriate models.

2.0 FOCUS OF THE STUDY

2.1 Concept of Fractions

Fractions play an important role in the school Mathematics program. The topic in

elementary school are thought as rational numbers that can be written as a/b where a and

b are integers with b not equal to zero. A fraction also is a part of a whole or part of a

set. Every fraction has a numerator and a denominator. The numerator is the top number.

It tells how many parts of the whole are being used. The denominator is the bottom

number. It tells us how many parts there in a whole.

The National Council of Teachers of Mathematics (NCTM) (2000), states that

students in middle or secondary school should acquire a deep understanding of fractions

and be able to use them competently in problem solving. However, it seems that just as

students are struggling with fractions, so too are teachers feeling the frustration with

teaching fractions effectively. The National Assessment of Educational Progress reports

show that fractions are “exceedingly difficult for children to master” (NAEP, 2001, p.

5).

Additionally, students are frequently unable to remember prior experiences about

fractions covered in lower grade levels (Groff, 1996). In an effort to increase the

effectiveness of teaching fractions, teachers literatively review and modify the structure

of their lessons on fraction concepts.

This is because fractions are one of the main mathematical concepts that students

continue to struggle with in primary schools. Students do not see fractions as a number

because they usually work with real numbers. Real numbers to students represent a

number in the set {…-4, -3, -2, -1, 0, 1, 2, 3, 4,}; a whole number, or the opposite of a

whole number the numbers they use for counting, and the numbers they see on a number

line in the classroom. Students have difficulty understanding the concepts of Fractions as

numbers between two whole numbers, because they are unable to readily see them.

In Fraction topic, equivalent fraction is one of the important part. Equivalent

fractions involve more than remembering a fact or applying a procedure. It’s mainly

refer to the conceptual understanding.

2.2 Concept of Equivalent fraction

By the time the students enters Year 4, they get the understanding of equivalent

fraction before they learn to add, substract, multiply, and divide fractions. According to

Lamon (1999), the hardest part of learning fractions was understanding that “what looks

like the same amount might actually be represented by different numbers”.

Equivalent fraction refers to two methods. The first method is to factor the

numerator and the denominator. Then, find and eliminate the common factor. The search

for a common factors keeps the process of writing an equivalent fraction to one rule

where top number (numerator) and bottom number (denominator) of a fraction must be

multiplied by the same nonzero number. The second method is to divide the top number

(numerator) and bottom number (denominator) of a fraction to get the simplest form.

Fractions have an infinite number of equivalent fractions that name the same

number or fraction. Equivalence implies similar worth. Thus two fractions are

considered equivalent when they have the same value (BOS NSW, 2002; Skemp, 1986).

That means, equivalent fraction refers to the notation that different fractions can

represent the same amount. For example, 1/2 and 2/4 are different fractions and they

represented the same amounts or quantities.

From the above examples we see that the conceptual understanding of equivalent

fractions involves more than remembering a fact or applying a procedure. It is based on

an intricate relationship between declarative and procedural knowledge; between

fraction interpretation and representation.

Indeed, understanding equivalent fractions is another important prerequisite to

fraction computation and it helps children to evaluate the reasonableness of answers. So,

the conceptual understanding of equivalent fractions appear to be ready to work with the

addition and substraction of fraction.

2.3 The Problems and Causes of The Problem

While teaching Fractions to the Year 4 Anggerik’s students of SJK (T) Bukit

Mertajam, I found many problems that faced by the students in expressing the equivalent

fractions for a proper fraction. Those problems make students difficult to express

equivalent fraction for proper fraction. This problem occurs because of the lack of

conceptual understanding on this topic.

From my experience at the SJK (T) Bukit Mertajam, I realized that the students

are not able to compare the value of proper fractions before they are introduced to

equivalent fractions. For example, students are not able to compare which is greater than

and which is less than when comparing the value of two fractions such as ¼ and ½. This

will make the students not ready for the next step which is equivalent fraction.

For example, the students do not know how to refer to the notation that different

fractions can represent the same amount. For examples, 1/3 and 2/6 are the different

fractions and they represented the same amount. However, to many students these two

are completely different and they have no relation to each other. This is because, they

think that 2/6 must be greater than 1/3, because the numbers are larger.

Beside that the students didn’t get the conceptual understanding on this topic

because most of the teacher didn’t discuss clearly the meaning of the word equivalent.

The teacher also didn’t give a clear explanation about how equivalent fractions are alike

and different.The students have not fully understand that equivalent means the ‘equal

value’. For example, the teacher seldom says that we are writing equivalent fractions

when assigning different names for a specified fraction.

Other than that, student didn’t master the times tables fully. They didn’t

memories the times tables. This make them facing difficulties in multiplying or dividing

the numerator and denominator by the same nonzero number. They also cannot find the

Least Common Denominator (LCD) for the given fractions.

Even the students have mastered the times table, they just use the rules and formula

to complete operational exercises without understanding them. They mechanically use

the rules and formula they have learned, sometimes with success, but often with

incorrect results. They didn’t understand why the rule is so and not otherwise.

Teachers often evaluate competency in the area of equivalent fractions by paper and

pencil methods. If a student can correctly complete the exercise with at least 75% to

100% accuracy, they conclude that the students have understood equivalent fractions

and quickly they move on to addition of fractions.

In this case, the teachers did not pay full attention to the students understanding,

their ability to begin operational tasks based upon a paper pencil methods. The teacher

also does not bother the relationship between what the students can demonstrate using

paper and pencil method and what they can show using manipulatives when dealing with

equivalent fractions problems.

Finally, by going through the problems faced by the students in equivalent

fractions as described above, I realized that the main aspect that bring them to this stage

is the lack of conceptual understanding in equivalent fraction.

In addition to overcome those problems, the students must develop their conceptual

understanding in equivalent fractions topic. The teachers also must understand how

children most successfully learn the subject (Ball, 1993). Teachers are faced with the

responsibility of choosing how they will present the material to their students. Often

teachers seek recommendations from the textbook. However, these resources are not

enough to be used as teaching material. Teachers must be concerned about the action

that can help the students to develop their own conceptual understanding in this topic

and overcome the problem they are facing.

2.4 The actions to overcome the problems

Before I began planning for this research I read many materials from books and

websites in order to prepare a well action research. As I read different sources of

information, I gained many ideas and knowledge regarding my research. This is because,

as a mathematic teacher I have to plan a good action to help my students to develop their

conceptual understanding of the equivalent fraction so that they can master the topic

without any problem. I have gone through few studies that have addressed the actions to

overcome those problems.

According to Bezuk and Bieck (1993), the teacher must discuss the meaning of

the word equivalent as “equal value” and also discuss how equivalent fractions are both

alike and different. This is because, the discussion can help students to generalize the

symbolic algorithm for finding equivalent fractions from their experiences with

manipulatives. It also can help students to realize that different fractions can represent

the same amount.

Next, Jigyel and Afamasaga-Fuata’i (2007) found that the students perceive the

numerator and denominator of a fraction as two unrelated whole numbers. Students were

able to identify equivalent fractions when they were presented geometrically

(particularly circle models compared to rectangular ones). This is because they can help

the students who are struggling and don’t understand onto beginning and developing

concepts quickly. They also help the students to develop mastery of the skills and

concepts in expressing equivalent fractions.

However, misconceptions occurred when students were presented with

equivalent fractions that were presented numerically (e.g. 4/6 and 2/3). Most students

responded that 4/6 and 2/3 are not equal and that 4/6 is double of 2/3. The authors

conclude that these misconceptions require a development and consolidation of students’

understanding fractions using multiple models that clearly represent the relationship

between the numerator and the denominator. In addition, students need to understand the

number of parts and size of parts when comparing fractions (Jigyel & Afamasaga-

Fuata’i, 2007).

Other than that, students are required to: (a) make connections between fraction

models by understanding the sameness and distinctness within these interpretations

(Lesh et al., 1983; NRC, 2001); (b)make connections between the different

representations (Lesh et al., 1983), and (c) show that a fraction represents a number with

many names. The present study examines a small portion of the large body of

knowledge associated with fractions.

The idea is that students’ exposure to concept will cause a spiral effect on

learning which will help to gain the conceptual understanding of other topics. By

gaining the conceptual understanding, they will be better problem solver and thinker.

Instead of the teacher explaining the rules in expressing equivalent fractions for proper

fractions, students who are learning using concrete models will be able to figure out how

to express in equivalent fraction through applying these particular rules because of the

conceptual understanding.

Other than that, students who were presented with tasks that aimed to elucidate

their level of thinking. The demands of the tasks were restricted to identifying symbolic

and pictorial representations and representing fractions using part/whole area and

measure models.

They incorporated “skill” questions that required the recall of a practised routine

or procedure,and “conceptual” questions that required students to apply their knowledge

and explain their actions (Shannon, 1999).

Tasks that incorporate pictorial representations with visual distractors provide

one method of measuring students’ conceptual understanding of equivalent fractions.

Such tasks have been found to highlight the unstable nature of a student’s fraction

knowledge (Ni, 2001; Niemi, 1996). Pictorial representations of part/whole area and

measure models can be described as “simple representations” when the total number of

equal parts in the shape matches the fraction denominator.

They allow students to count the parts (see Figure 1a). The shaded part is

associated with the numerator and the entire shape is associated with the denominator.

Equivalent pictorial representations are visually challenging. They occur when the

number of equal parts of the whole is a multiplicative factor less or greater than the

denominator (Niemi, 1996), as shown in Figures 1b and 1c. The areas of the whole and

shaded part never change, but the number of equal parts into which the whole is divided

can alter dramatically. Thus different fraction names can be offered for the shaded area

and an equivalence set can be identified.

a) simple (b) equivalent – 2 equal part (c) 8 equal parts

Figure 1. Part/whole area model simple and equivalent representations for two quarters.

Equivalent fraction tasks using symbolic notation (see figure 2) are more

cognitively demanding as up to four dimensions are needed to be simultaneously co-

ordinated: the original two-dimensional fraction, 3/8 and its equivalent, 12/32 (English

& Halford, 1995).

Questions that incorporate the interpretation and manipulation of symbolic

notation are ideal for identifying the levels of students’ conceptual understanding of

equivalent fractions.

Evaluation of their responses provides an insight into the students’ thought

patterns, conceptual understanding and procedural knowledge. Teachers who understand

how students develop this knowledge, and are able to help them to see the links between

various representations are providing the most effective fraction programs for students.

Figure 2. Typical equivalent fraction question and answer employing symbolic

representations only.

The study by Van de Walle (2001) is to factor the numerator and denominator.

Then, find and eliminate the common factor. “The search for a common factor keeps the

(a) 3/8 = /32 (b) 3/8 = 12/ (c) 3/8 = /

Answer for a, b and c: 3/8 = 12/32

process of writing an equivalent fraction to one rule: Top and bottom numbers of a

fraction can be multiplied by the same nonzero number” (p. 225). If there are varied

methods to teaching fraction equivalency, then each method must possess some positive

as well as negative aspects. Knowing these aspects can be useful to teachers who are

deciding how to teach fraction equivalency.

From all these studies, I concluded that using concrete models help to overcome the

problems in expressing equivalent fraction for a proper fraction. I make this decision

because most of the studies mentioned that use the concrete models is an appropriate

action to help students to gain the exact conceptual understanding. By the understanding,

the find the way of express equivalent fraction for proper fraction easily.

I also realized that the best strategy for a teacher to use to help students to develop

this conceptual understanding is the use of concrete models. Students should access to

concrete models in the classroom environment such as fraction strips, fraction circle cut-

out, folded paper and number line. Students should be encouraged to use concrete

models as mentioned above to understand the concept of equivalent fractions

independently.

Concrete models serve four purposes for teachers and students. Firstly, they engage

the senses where multi-sensory tools have been shown to dramatically increase

understanding and retention. Many students especially those who had difficulty in

school, need multisensory tools to learn effectively.

Beside that, concrete models help students discover concepts. This is especially true

for math concepts. It can be difficult, for example, to explain how 2/4 = ½, but when

students physically moves shapes, they can prove it for themselves.

Then, concrete models help to keep the students focus. Everyone appreciates a little

variety. By varying activities, a teacher can help students to stay focused and keep

learning. In other hand, the concrete models encourage practice. Students will often

practice more with the concrete models than they will with worksheets.

Finally, I decided to use concrete model such as fraction strips and fraction circle

cut-outs to develop conceptual understanding of equivalent fraction for proper fraction.

By using fraction strips and circle cut-out I will demonstrate to students the concept of

equivalent fractions.

3.0 RESEARCH OBJECTIVE AND RESEARCH QUESTION

3.1 Research objective

3.1.1 General objective

The objective of the proposed study is to develop the students’

conceptual understanding on equivalent fractions.

3.1.2 Specified Objectives

I. To develop the students’ conceptual understanding on equivalent fraction by

using concrete models.

II. To increase the interest of students in express equivalent fractions for proper

fractions.

3.2 Research Questions

I. Do concrete model help to develop the students’ conceptual understanding in

equivalent fraction?

II. Do concrete models increase students’ interest in finding equivalent fractions?

3.3 Significance of the study

Fractional concepts are important building blocks of elementary and middle

school mathematics curriculum. Most of the students have a procedural knowledge of

fractional operations rather than an understanding of underlying concepts. Considering

this statements, conceptual understanding of equivalent fractions is an important task to

be studied.

This action research is important because we can ensure that by using

manipulative materials during teaching and learning process of equivalent fractions, it

will improves and enhances the pupils of Year 4 understand the concept of it in depth.

Apart from that, this will also can become a guide to the teachers to use

manipulative materials in teaching and learning processes of equivalent fractions.

4.0 TARGET GROUP

4.1 Introduction

This chapter discussed about the group which was chosen to conduct the action

research. In this chapter, I explained why I choosed that particular group of my

action research.

4.2 Target Group

A class of Year 4 students of SJK (T) Bukit Meertajam, Pulau Pinang.

I. Number of students : 12 students

II. Class name : 4 Anggerik

III. Number of boys : 5 students

IV. Number of girls : 7 students

V. Level of students : Middle achievers

I had done my research on Year 3 Valluvar of SJK (T) Barathy in this action

research. The students who involved in this research are male and female. I was focused

my research on the middle achievers of the class. There are 5 boys and 7 girls in middle

achievers group. The action was to help the middle achievers to develop the students’

conceptual understanding on equivalent fraction.

I choosed the middle achievers by giving the class a diagnostic test on 9 March

2010. The diagnostic test contains 10 questions. The objective of this test is to test the

understanding in fractions. It 30 minutes test. The diagnostic test consisted of two

learning outcomes:

a) Name and write proper fractions with denominators up to 10.

b) Compare the value of two proper fractions with;

I. The same denominators.

II. The numerator of 1 and different denominators up to 10.

I choose this learning objective because of the important role in understand the

concept of equivalent fractions. I choose the students who get correct five to seven

questions out of ten questions as the middle achievers.

I choose the middle achievers because they are the one who always face

problems in understand and apply correctly the concept of equivalent fractions. The high

achievers mostly will understand the concept easily only by the chalk and talk methods

and they doesn’t need clear picture or visualization of the equivalent fraction. For the

low achievers, they will already have problems in number sense and basic facts of

algorithms. They also might be never has the basic understanding of the fraction.

But, the middle achievers are not too weak on number sense, basic facts of

algorithms and they surely had full understanding in basic concept of fraction. They just

can’t understand the concept of equivalent fraction for proper fraction by the chalk and

talk method. They surely need a clear explanation and clear visualization of the

examples this topic. Then only they can understand fully the concept of equivalent

fraction. So, I choose the middle achievers of the Year 4 Anggerik students for this

action research.

V.0 RESEACH METHODOLOGY

V.1 Introduction

The National Council of Teachers of Mathematics, in its Principles and Standards

for School Mathematics, describes the need for students of all levels to understand and

represent fraction and rational number concepts, recommending that students use a

variety of concrete models to support their thinking about abstract ideas.

By going through the studies that addressed the action to overcome the problems in

express Equivalent fraction for proper fraction, I realised that the students must get a

clear picture of how to develop the conceptual understanding to express equivalent

fraction.

So, by having the studies as a guideline, I used concrete models to express

equivalent fraction for proper fraction which is the best way to solve the problem faced.

The qualitative methods used in this study were designed to collect and analyze data

from the middle achievers of Year 4 Anggerik students. The following sections give

information about how the data was collected and was analyzed.

5.2 Action Research

In addition to having clear and solid understanding of mathematics, teachers must

also understand how children most successfully learn the subject (Ball, 1993). In the

flow of teaching and learning process the important step that to be stressed is the use of

concrete models in express equivalent fraction for proper fraction.

According to learning theory based on psychologist Jean Piaget's research, children

are active learners who master concepts by progressing through three levels of

knowledge--concrete, pictorial, and abstract. The use of concrete models enables

students to explore concepts at the first, or concrete, level of understanding. When

students manipulate objects, they are taking the necessary first steps toward building

understanding and internalizing math processes and procedures.

So, most of the studies and research mentioned the important of concrete models

which help students to make connection between the concrete models and Mathematical

ideas explicit in the development of understanding in equivalent fraction.

Besides that, normally teachers will teach equivalent fractions using symbolic

representation in paper and pencil method. Pupils followed the procedural to solve the

question. Beside using traditional method to teach equivalent fractions, I will use

student-centered method in my class.

This is where I tried to use concrete material such as fraction strips and fraction

circle cut-outs to taught equivalent fraction so that they could develope their conceptual

understanding on this topic. Pupils manipulated those concrete models freely while

express its equivalent fraction.

Pupils have to represented fractions by using fractions strips and followed by

Fraction circle cut-out. When students had enough experience with this concrete models,

they were ready to move to the symbolic representation in expressing the equivalent

fractions. The practice was repeated while pupils were able to write in symbolic

representation. Pupils also understand the concept of equivalent fraction when used

concrete models such as fraction circle cut-outs and fraction strips.

When I taught this topic in class, I followed the following flow of teaching and

learning process (figure 3). In the flow, I stressed more in the second step where the

concrete models were applied.

Figure 3 : Teaching and Learning Process of Mathematics.

This flow of teaching and leaning process applied in the second and third lesson as

mentioned in the following schedule of Action Research.

Express Equivalent Fraction for Proper

Introduction of the concept

Use concrete model .

Use pictorial model.

Symbolic notation or use algorithms

GANTT CHART

Table 1 Schedule of Action Research.

I carried out this schedule of action research for a month. The topic, fraction were

taught during period of my research. For the first week, I gave the diagnostic test to the

students of 4 Anggerik to identify the middle achievers of the class. The diagnostic test

contains 10 questions. The questions were related to the basic concept of fraction which

TIME

ACTIVITY

Week 1 Week 2 Week 3 Week 4

Diagnostic test

Chalk and talk

Pre-test

Express equivalent

fractions for proper

fractions by using fraction

strips and paper strips.

Express equivalent

fractions to its simplest

form by using fraction

strips and paper strips.

Observation and students

work analysis

name and write proper fractions with denominators up to 10 and also compare the value

of two proper fractions. The time allocated for the test is 30 minutes. (Refer to Appendix

1).

After that, I carried out a chalk and talk lesson on equivalent fraction without using

any concrete models or material. The lesson was for one hour. The objective of this

lesson is to make students understand fractions with the same values are determined as

equivalent fraction. The learning outcome of this lesson is:

1. Express and write equivalent fractions for proper fractions.

2. Change any equal fraction by using multiplication.

3. Express equivalent fractions to its simplest form by using division.

In the set induction, I recalled the student’s previous knowledge of compare the

value of proper fractions. I used flash cards of proper fractions to recall students’

information or understanding in ‘Compare value of proper fractions’. I just asked

students to read the fractions and asked them to mention which fraction is greater than

another and which is less. After they answer my questions, I mentioned to them verbally

that compare the value of proper fraction is important to find equivalent fractions.

Then, I moved forward to the lesson development where in step one I introduced the

students to identify equivalent fractions, discover the concept of equivalent fractions,

discover equivalent fractions by multiplication and write the equivalent fractions

properly. I didn’t use any specific concrete models to carry out this step. I just

mentioned to them verbally that ‘the numerator and denominator is multiplied by the

same number it gives an equivalent fraction to the original one’. After that, I guide them

to find equivalent fractions for proper fractions by multiplication.

I also applied the same method to teach them to identify the simplest form for the

given proper fractions. I explained to them the way to find the simplest form where ‘the

numerator and denominator is divided by the same number it gives simplest form to a

proper fractions. I also guide them to find the simplest form by using symbolic method.

Then I distributed a worksheet related to this lesson to them to try it out. I was guided

them whenever necessary.

In this lesson, I just stressed on the symbolic representation of equivalent

fractions. Through this way, I got to know their understanding on equivalent fractions

concept before carry out my action. Then, I gave them a pre-test related with the

equivalent fractions. The pre-test was contains 10 questions. First two questions were

representing the pictorial method. The next four questions was where the students must

write the equivalent fraction by using multiplication. The other four questions were

where the students should express the fractions in the simplest form.(Refer to Appendix

2).

After the chalk and talk lesson of the equivalent fraction, I moved to the

important part of the teaching where the use of concrete models to represents the

fractions were given. This is because I realized that to enhance students understanding

on this topic I have to allow the pupils to practice the skill until they gradually mastering

it.

The most suitable concrete model to enable students to express equivalent

fraction for proper fraction is by using fraction strips and fraction circle cut-outs. Most

of the research reviewed documented that concrete models such as fraction strips and

fraction circle cut-outs can facilitate children acquisition and fluid use of conceptual

understanding of equivalent fractions. So, for the second week, I used circle cut-out and

fraction strips to teach the students to express equivalent fractions. The learning

objective of this lesson is same as chalk and talk lesson and the learning outcome were

as below:

1. Express and write equivalent fractions for proper fractions.

2. Change any equal fraction by using multiplication.

I started the teaching and learning process with the set induction. In set induction, I

displayed a fraction board in front of the class. In the fraction board, certain fractional

part was left empty and I selected students to fix the fractional part in correct place.

Then, I guide them to read the fractional part and guided them to look for the equivalent

fractions. This explanation helped students to guess the topic correctly.

In lesson development, first of all I mentioned that 1/2 is equal to 2/4. Then I

explained to them by demonstrated the statement by using circle cut-out. Its help me to

explained the meaning of equivalent where the different fractions represent the same

amount. In the second step of lesson development, I provided three paper strips for each

pair. I asked them to follow my instruction. Pupils listened carefully to my instruction

and followed it correctly.

Then, I provided a group activity to them where they picked up equivalent fraction

correspondently. Each group has four students. Lastly, I provided the worksheet to the

students.

For the third week, I taught the students to express equivalent fraction to its

simplest form for proper fractions by using the suggested concrete models such as

fraction strips and fraction circle cut-outs. I introduced the concept of express equivalent

fraction to its simplest form where the numerator and the denominator must be divided

by the same nonzero number to get the equivalent fraction for the given fraction. In set

induction, I provided fraction strips to students and instructed them express equivalent

fractions to its simplest form by folding method.

In lesson development, I was fully used fractions circle and demonstrated to the

students the way to express equivalent fractions to its simplest form. I provided fractions

circle for each pair where they can explore and apply themselves. Then, I also

emphasize the topic in the game ‘who wants to be a millionaire?’ which has enhance

their understanding. I also has been provided worksheet to them. (Refer to Appendix 4)

For the second and third week lesson, I introduced the language or concept of this

topic in set induction which helped students to recognize the vocabulary related. It is

where I gave the clear explanation and briefing of the equivalent fractions concept by

using fraction chart in second week and fraction strips in third week. I stressed the

meaning of equivalent to students as equal value. I mentioned that equivalent fraction is

the different fractions which represent the same amount of the given proper fraction.

For the fourth week, my lesson was fully refered to the pictorial and symbolic

representation. After the students had mastered the concept of express equivalent

fraction for proper fraction by the concrete model approach, I brought them to the third

step of the teaching and learning process. It is the use of pictorial representation. In this

step I taught the students to convert the concrete models into pictorial representation.

Finally, when students had graps the conceptual understanding of expressing

equivalent fraction for proper fraction, I guided the students to use multiplication table

and division table for the symbolic representation of equivalent fraction. I demonstrated

the method and followed by several students. I mentioned and stressed to the students

that they can find the equivalent fraction by multiplying or dividing the numerator and

denominator by the same number.

I guided them to used a table of multiples to find the equivalent fraction for the

proper fraction and divides tables to express equivalent fractions to its simplest form. In

multiplication table, the multiples in the same column will from equivalent fractions.

So, 1/2 = 2/4 = 3/6 = 4/8.

Table 2 Table of multiples

The multipes for the

numerator and

denominator.

Numerator

Denominator

The divides for the numerator

and denominator.

X2 X3 X4

1 2 3 4

2 4 6 8

÷2 ÷4

4 2 1

8 4 2

So, 4/8 = ½

Table 3 Table of divisibles

Drill and practices will be given as consolidation for the students. After understand

the equivalent process, students were able to express equivalent fraction for proper

fraction.

Lastly, I gave them a post-test to examine their conceptual understanding on

equivalent fractions (Refer to Appendix 5).

6.0 DATA COLLECTION

Numerator

Denominator

6.1 Pre-test and Post-test

In order to identify the development of conceptual understanding in expressing

equivalent fractions of the middle achievers in Mathematics of Year 3 Valluvar pre-test

and post-test were conducted. On the first day of the lesson in express equivalent

fraction for proper fraction, I taught the students by using chalk and talk method. Then, I

let the students to sit for test (pre-test).

Before conduct the test, I make sure that all the questions in the test are based on

the related topic. I also let a Year 3 Mathematics teacher to check on the questions so

that they suit the level of Year 3 students. Using this I had identify the middle achievers

in the class.

Then, I implemented the teaching of expressing equivalent of proper fraction using

concrete models such as fraction strips and fraction circle cut-outs. After the lessons, I

gave them the test (post-test).

The test is almost the same as before. This is to determine the achievement of the

middle achievers before and after the use of concrete models such as fraction bas and

fraction strips in teaching the topic.

6.2 Observation

I divided this observation into two parts that is before and after the use of

concrete models in teaching express equivalent fraction for proper fraction. The purpose

of this observation is to determine whether the interest of students in learning this topic

has increased or not. The observation was based on several criteria such as attention,

expression, participation and attitude when teaching and learning is carried out.

Another Mathematics teacher, Mrs Valliammal was observing the class when I

hold my lesson each week. She observed the students attention, expression, participation

and also their attitude while I was teaching. She also wrote her note and gave me the

note. I go through the notes taken by her each week for the four weeks and explored and

researched the students’ involvement before and while using the concrete models.

The observation help me to got to know the students way of involvement in

teaching and learning process and also identified the role of concrete models in develop

conceptual understanding of equivalent fractions.

6.3 Students’ work analysis.

Besides that, students’ works were collected for data collection. I kept students

worksheet and exercises as a data source. Through this way, I could see the pupils

progressing in understanding the equivalent fractions concept.

6.4 Data analysis

I read and reread my observation notes, pre-test and post-test result and students

work in order to identify any improvement in developing conceptual understanding of

equivalent fraction. I also read through my research notes and analyzed my weakness

and strength of my activities. Through observation and students work analysis I

evaluated the pupil’s conceptual understanding of equivalent fractions.

7.0 RESEARCH FINDING

This session reports the result of the action towards student’s conceptual

understanding on equivalent fractions by using concrete models. It intent to answer my

research questions:-

I. Do concrete model help to develop the students’ conceptual understanding in

equivalent fraction?

II. Do concrete models increase students’ interest in finding equivalent

fractions?

For the first question above, I carried two types of data collection which are pre-

test and post-test and also students work analysis. I also carried out an observation while

my teaching and learning process to answer the second question.

7.1 Sample selection

7.1.1 Diagnostic test

In this study, the data was collected by using diagnostic test, pre-test and post-

test. I carried out the diagnostic test first to choose the middle achievers of the class. The

diagnostic test contains 10 questions. I choose the students who get correct five to seven

questions out of ten questions as the middle achievers. Following Table 1 shows the

result of diagnostic test.

SCORES NUMBER OF STUDENTS ACHIEVEMENT

1 - 4 15 Low achievers

5 - 7 12 Middle achievers

8 - 10 7 High achievers

Table 4 The student’s achievement in Diagnostic test

1 to 4 5 to 7 8 to 1002468

10121416

THE STUDENT'S ACHIEVE-MENT IN DIAGNOSTIC TEST

Scores

Num

ber

of

Stu

dents

Figure 4 The Students’ Achievement in Diagnostic Test.

The graph 1 shows that 12 students are the middle achievers. They got scores

from 5 to 7 in the diagnostic test. I choose the middle achievers because they always

have the problem in understanding the concepts. Most of them were able to answer the

questions but done some careless mistakes and the answer was wrong.

7.2 Students’ achievement

7.2.1 Pre-test and Post-Test

As mentioned above, I gave the pre-test and post-test to the 12 middle achievers

of year 4 Anggerik to manipulate the students conceptual understanding on equivalent

fractions. The pre-test was carried out after the first teaching and learning process where

I just used chalk and talk method.

The post-test was carried out after I used the concrete models such as fraction

circle cut-outs and the paper strips. From both of this test, I can compare the middle

achievers’ development in conceptual understanding on equivalent fractions before and

after the use of concrete models.

The both test has same questions. The test paper contains 10 questions. The test

paper where separated into three parts. On the first part, there are two questions which

has pictorial representation. In the first questions the students must express and write

equivalent fraction for proper fraction that given. For the second question, the students

must express equivalent fraction to its simplest form.

At the second part, there are 4 questions where the students need to write the

equivalent fractions for proper fraction by using multiplication. The four questions were

from simple to complex. Third part of the test paper also consist 4 questions where

students need to express the fractions to the simplest form by using division. The

questions also were from simple to complex.

PRE - TEST POST – TEST

PUPIL 1 2 3 4 5 6 7 8 9 10 1 2 3 4 5 6 7 8 9 10

A √ √ √ √ X X √ √ √ X 7 √ √ √ √ √ √ √ √ √ √ 10

B √ X √ X √ X √ X √ √ 6 √ X √ √ √ √ √ √ √ √ 9

C √ √ √ √ √ X √ X X X 6 √ √ √ √ √ √ √ √ √ √ 10

D √ √ √ X X √ √ X X X 5 √ √ √ √ √ √ √ √ √ √ 10

E √ √ √ X √ √ √ X √ X 7 √ √ √ √ √ √ √ √ √ √ 10

F √ √ √ X √ √ √ √ X X 7 √ √ √ √ √ √ √ √ √ √ 10

G X X √ √ X √ √ √ X X 5 √ √ √ √ √ √ √ √ √ X 9

H √ √ √ √ X √ √ √ X X 7 √ √ √ √ √ √ √ √ √ X 9

I X √ √ X √ X √ √ X X 5 √ √ √ √ √ √ √ √ √ √ 10

J √ √ √ √ X √ √ √ √ √ 9 √ X √ √ √ √ √ √ √ √ 10

K √ √ √ √ √ X √ X X X 6 √ √ √ √ √ √ √ √ √ X 9

L X √ √ X √ √ √ √ √ X 7 √ √ √ √ √ √ √ √ √ √ 10

Table 5 Item Analysis of Pre-test and Post-test

From the table of item analysis of pre-test and post-test, I determined that all the

students where wrong about 4 to six wrong in pre-test. Its shows that they were very

poor in conceptual understanding on equivalent fractions before use concrete models.

According to the item analysis of pre-test, question 1 was easy for the students.

This is because only two students get wrong. Question 2 also was easy to the students

because all 10 of 12 students gave correct answer. This might be because the picture

represent the fraction became a guide for the students to apply the fraction.

All the students were correct in question three and seven. This is because the

question already has the symbolic representation as a guide. So, all the students only

need to write the answer by solve the symbolic representation.

Question 4, 5, 6 and 8 seems to be a average question where nearly half of the

students cannot answer it correctly. About 5 to 6 students cannot answer the questions in

pre-test. Question 9 and 10 seem to be quite tough questions to the students. This is

because only 5 students got correct in question 9 and only 2 students correct in question

10.

According to the item analysis of post test, students were done very well in all

questions. Only two students get wrong in question 2 and three students get wrong in

question 10. Other students got full marks.

1 2 3 4 5 6 7 8 9 10 11 120

2

4

6

8

10

12

THE PRE-TEST AND POST-TEST RESULT

pretestpost test

STUDENTS

SC

OR

ES

Figure 5 The result of pre-test and post-test of 12 pupils.

From the above graph, the results shows that students get higher mark in post-

test than in pre-test. There is a good and positive improvement in the students’

achievement in post-test. 7 students out of 12 students overall get full mark, 10 in the

post test. Other 5 students get 9 marks as total.

Three students were get lowest mark in pretest, 5 over ten. But, two of the

students have been scored full marks in post-test and one of them scored 9 marks out of

ten. It was a good development of those students after the application of the concrete

models. The number of students who get 6 out of 10 marks in pre-test was three. One of

them has been scored full marks in post-test and two of them scored 9 out of ten. Five

student where got 7 out of 10 in pre-test. Four of them have been score 10 marks and

one of them got 9 marks. Only one student of the 12 middle achievers get 9 marks in

both test.

The mean of the pre test score is 6.416 and for the post test is 9.583. The

differences between the mean of the pre test and post test is 3.167 (Refer to Appendix).

From this comparison of pre-test and the post-test result, I can proof that the

development of students conceptual understanding on equivalent fractions was improved

a lot by using concrete models such as fraction circle-cutouts and paper strips.

Finally, the errors that made by students in pre-test were similar with the error

that I had explained in the students work analysis. The five students who has one wrong

in post test was because their careless mistakes. They didn’t realize their mistakes

because they were rushing their time to complete the test.

So that, they didn’t rechecked their answers as well as needed. Even their answer

is wrong but their application of symbolic representation was correct. It’s proof that they

understood the concept of equivalent fraction.

From the result of pre-test and post-test, I have been calculated the mean of score

and percentage. The mean of the pre test scores is 6.416 and for the post test is 9.583.

The differences between the mean of the pre test and post test is 3.167.From this

comparison of pre-test and the post-test result, I can proof that the development of

students conceptual understanding on equivalent fractions was improved a lot by using

concrete models such as fraction circle-cutouts and paper strips.

7.3 Students’ work analysis.

Students work document also was one of my data analyses to answer my first

research question. The documents were the students’ worksheets. I go through the

students first worksheet (Refer Appendix ) and the forth worksheet (Refer Appendix ) of

the teaching and learning process. This was to identify their development in the

conceptual understanding on equivalent fractions.

I decided to choose those worksheets because both was the same worksheet

which students tried out by symbolic representation before and after the use of concrete

models in teaching and learning process of express equivalent fractions for the proper

fractions. The analysis helped me to go through the students’ errors in answering the

question related to this topic. By analyzing the both worksheets, I can manipulate and

saw the students’ development in their conceptual understanding on equivalent fractions.

I had compared both worksheets of each student. Their development in

answering the question correctly was improved. The worksheet had ten questions of

equivalent fractions. For the first five questions, the students need to express equivalent

fractions for the proper fractions by using multiplication. The next five questions were

about express equivalent fractions to its simplest form.

From my analyses on the first worksheet, I had determined some regular

mistakes that were done by students in answering the questions. The first worksheets

proved that the students’ conceptual understanding is very poor before using concrete

models. I had determined six regular errors which most of the students done in their

worksheet. The errors are as follow:-

Error 1

The first error that I determined was as below:

Figure 6 Error of Student D

It was the answer of Student D in the first questions. Student E, Student G,

Student I and Student L also had done the same error. According to the students answer

for the first questions, we can see that they only understood that to express equivalent

fraction they needed to use operation of multiplication.

At the first question, I guide the students in the questions solution in find the

equivalent of the numerator as 1 x 2 = , and they must find out the answer of

equivalent numerator as 2. At the denominator part I gave the answer of equivalent

denominator as 4 and they must apply the symbolic representation as 2 x 2 = 4.

But, the students just noticed the answer 4 and didn’t relate it with denominator

of the proper fraction given. The students just wrote down that 1 x 4 = 4 as the answer.

This mistake of student’s proof that they are very poor in the conceptual understanding

on equivalent fraction. They also didn’t understand the symbolic representation rule of

the equivalent fraction where the numerator and denominator must be multiply by the

same number.

Error 2

Figure 7 Error of Student B Figure 8 Error of Student K

The second error that I determined was at question 5 as shown above. At the

worksheet, I gave a proper fraction as 1/4 and the students need to find out the

equivalent fraction for it. Student B has done this error in the worksheet. Student K also

has done the same error. May be, one of the student copy another students’ answer. Its

might be the student doesn’t know how to answer it because he don’t have enough

understanding on the concept of equivalent fraction.

The students answered the question wrongly. They knew that they must use

operation of multiplication to express and write the equivalent fraction for the given

proper fraction. But, they misconcepted the meaning of equivalent as it should be. They

thought that the equivalent meant about equal value where in equivalent fraction

denominator and numerator must have the same number. So, they multiply the

numerator, 1 with the multiple of 4 and multiply the 4 with the multiple of 1 which gave

them the answer, 4/4.

The students didn’t get the conceptual understanding on this topic because they

failed to understand clearly the meaning of the word equivalent. The previous teacher

might not gave a clear explanation how equivalent fractions are alike and different. The

students no fully understand that equivalent means the ‘equal value’. They didn’t

mastered that the equivalent fractions represent same value even the fractions are

different.

Error 3

Another error that I found out is related to the times table. For example, the

solution of Student C in question 5 as shown below:-

Figure 9 Error of Student C

In this case, the student understood the concept of the equivalent fraction. He had

done the correct symbolic representation where the student had multiply the numerator

and denominator with the same number. But the student had written a wrong answer. He

multiplies the numerator, 1 with 6 and gave the answer as 6. But, for the denominator,

he multiplies the 4 with multiples of 6 and gave the answer as 28. Actually, the answer

must be 24.

The students might be careless when answer the question or he/she might be not

mastered the times table. It might be because the student does not master or memories

the times table correctly.

Error 4

I have found out two different types of error in question 10. The first error was as

below:-

Figure 10 Error of Student A

This problem was done by the Student A. Student F also had done same kind of error

but he had used the multiples of 3. The question 10 has a proper fraction, 4/16. The

students must express the equivalent fraction to its simplest form. So, they need to times

the proper fraction’s numerator and denominator by same number.

But, the students had a misconception where they multiply the proper fraction’s

numerator and denominator by same number to find the equivalent fraction. It showed

that they didn’t really understand the meaning of express equivalent fraction for proper

fraction and express equivalent fraction to its simplest form. In other hand, they also

might confuse whether they should use multiplication or division to find out the simplest

form.

The second error of this question was as below:-

Figure 11 Error of Student J

Student J had done this error. Student H and Student D also have done a same

mistake as this. Those students also understood the concept of express equivalent

fraction to its simplest form. But they didn’t fully master the concept. They can divide

the both numerator and denominator with the same multiple but not able to gave the

lowest term.

Discussion

All the error that done by students in their first worksheet is shows that they

didn’t understand the concept of expressing equivalent fraction correctly. After the

application of concrete models, they have been correct their error and gave correct

answers. Here is some example of students correct answer in second worksheet.

Figure 12 Correct Answer of Student D Figure 13 Correct Answer of Student B

Figure 14 Correct Answer of Student K Figure 15 Correct Answer of Student A

7.3 Observation

As I mentioned above, to answer my second research question I had asked my

friend help to observe my class while I implemented the teaching and learning process. I

provide her an observation form which divided into three parts such as set induction;

lesson development and closure (Refer to Appendix ).

I asked my friend to observe my class during the lesson because it will help me

to recognize students level interest in the class. I can’t observe myself because it’s

difficult to me to observe them while teaching and it also hard for me to take pictures

and take video. So, when she helped me, it became very pleasure for me to carried out

my lesson as planned without any problems.

My friend, Manoranjetam had observed my class for all four lessons that I had

planned and held. She reported her observation notes in the observation form that I have

provided.

I used these observation notes as a resulting summaries to keep a record of the

lessons presented to the whole class, as well as serve as data about the students’

participation in each lesson. It’s helped me to evaluated students understandings.

Observation 1

Lesson : Express and write equivalent fractions for proper fractions by using

multiplication and division.

Method : Chalk and talk

I started my first class with chalk and talk lesson. In this lesson, I taught them about

equivalent fraction without using any concrete models or materials. The lesson was for

one hour. I just used Year 4 Mathematics Text Book, several flash cards of fractions and

worksheet as the materials to teach.

The set induction of this lesson was fully handled by me and the students’

involvement was just in answering my questions. According to Manoranjetam A/P

Mariappen who observed my lesson, only some students read the fractions in the flash

cards. I also realized that when asking students to read the fractions in the flash cards.

The students didn’t show the full attention and some were talking with each other, some

were just quite and one or two students were doing their own work. They didn’t pay full

attention and seems like not interested to answer the questions. Five the twelve students

cannot compare the value of proper fractions shown.

In this both steps, students were seems very bored and some were very sleepy. My

observer also stated the same comment on the observation form. She also mentioned that

this is because the lesson was too teacher centered and the students’ involvement was

very little. So that, the students cant pay full attention and also not willing to try he

questions.

After that, I provided the worksheet to the students and gave them 15 minutes to try

it out. Most of the students can’t answer the questions in worksheet correctly. They

struggle to answer the questions correctly. Most of the students asked me to guide them

to solve the questions because they didn’t have the conceptual understanding. In closure

part, I conclude the lesson by asked some questions verbally which related to the topic.

In this section, only one or two students were answered the questions correctly.

From this chalk and talk lesson, I determined that students did not interested in

teaching and learning process if I didn’t used the concrete models. Their attention,

expression, participation and attitude were not as what I want. They seemed very bored

and sleepy wile the teaching and learning process. They didn’t involve actively in the

teaching and learning process because it was mostly teacher centered.

In this chalk and talk lesson, the students were never given chance to explore the

concepts by using concrete models and there will be no two way interaction between me

and the students. This is because, I only gave explanation and guide them and the

students just answer my questions. There was no discussion about the concept of

equivalent fractions.

Observation 2

Lesson : Express and write equivalent fractions for proper fractions by using

multiplication.

Method : Application of concrete models.

After carried out the chalk and talk class and the pre-test, I started my second

class by using the concrete models. The lesson was for one hour. I used fraction board,

circle cut-out, paper strips, colour pencils, scissors, game task and worksheet as the

teaching materials. The concrete model that I applied in my lesson to develop students’

conceptual understanding on equivalent fractions is circle cut-out and paper strips.

In this second lesson, I did not include ‘express equivalent fractions to its

simplest form’. This is because; the equivalent fraction is included two parts which

express equivalent fractions for proper fractions and express equivalent fractions to its

simplest form.

I separated both subtopics because each one has big scope and needed to be

applied different operation. I separated it because its wont confused the students and also

gave clear pictures to them about the application of different operation for each one.

I introduced the topic by using fractions board. Then, I developed the concept by

used fractions strips to demonstrate to students. I also provided the strips to students

where they can apply by themselves.

For this week, I only taught them to express equivalent fraction for proper

fractions by using the suggested concrete models. I introduced the concept of express

equivalent fraction for proper fractions where the numerator and the denominator must

be multiply by the same nonzero number to get the equivalent fraction for the given

fraction. I also prepared a group activity and worksheet for the students. This lesson also

was held for one hour. (Refer to Appendix 3).

The application of fraction board in set induction helped me to gain students full

attention and also their interest. According to my observer, the students were very

excited and pay full attention to each of my explanation and instruction and followed it

accurately. She also mentioned that they had answered all my questions correctly.

According to her, my set induction was a good starting for the teaching and learning

process.

In lesson development, first of all I mentioned that 1/2 is equal to 2/4. Then I

explained to them by demonstrated the statement by using circle cut-out. Its help me to

explained the meaning of equivalent where the different fractions represent the same

amount. The use of this concrete model helped me to develop the students’ conceptual

understanding easily.

According to my observer, Manoranjetam a/p Mariappen, the students were

involve actively in the teaching and learning process. Each of them tried to answer my

questions and also willing and interested in learning the concept using circle cut-out.

In the second step of lesson development, I provided three paper strips for each

pair. I asked them to follow my instruction. Pupils listened carefully to my instruction

and followed it correctly. They were more excited and could compare and recognized

the fractions that are equivalent fractions.

Then, I provided a group activity to them where they picked up equivalent

fraction correspondently. Each group has four students. The students pay full attention

carefully to the instructions. They actively categories all fractions into correct group and

also analysed all fractions and filled in the blanks with correct equivalent fractions. This

group activity became reinforcement for the students. It’s also helped them to apply

what they had learned and develop their understanding.

Lastly, I provided the worksheet to the students. The students did not asked too

many questions to me. Only certain pupils needed my guides. Its shows that students’

conceptual understanding in express equivalent fractions by multiplication has been

developed. Students able to catch up because almost of the teaching and learning process

was student centered and really gained students’ interest.

Observation 3

Lesson : Express equivalent fractions to its simplest form by using division.

Method : Application of concrete models.

After the teaching and learning process of express and write equivalent fractions

for proper fractions by using multiplication, I carried out the next lesson of express

equivalent fractions to its simplest form by using division. The lesson was for one hour.

I used fraction circle cut-out, paper strips, LCD, Year 4 Mathematics Text and

worksheet as the teaching materials.

The concrete model that I applied in my lesson to develop students’ conceptual

understanding on express equivalent fractions to its simplest form is circle cut-out and

paper strips. The learning objective of this lesson is same as previous lesson and the

learning outcome is express equal fractions to its simplest form by using division.

In set induction, I provided paper strips of 9 equal parts to the each student. Then

I asked them to colour six parts of them. Then, I asked the students compare the paper

strips with another one which have been divided to three parts. I asked the students to

find the relation or connection between the both paper strips. Most of the students can

answer my question as 6/9 is equal to 2/3.

In this session, I found out that the students were very active and involved fully

in the set induction activity. Their participation shows that they were very interested in

using the paper strips. They followed each of my instruction and applied it correctly. My

observer mentioned that, the use of paper strips in this section has really gained all the

students’ interest and attention.

Then, I moved on to the next step, lesson development. In first step of lesson

development, I mentioned the meaning of simplest form as lowest term. I also high

lighted that they must only have one fraction as the simplest form for each equivalent

fraction. I used the fraction circle cut-outs to demonstrate to the students such as the

simplest form of 4/8 is 1/2. Then only, I explained to them that to find simplest form of a

fraction is by divide the numerator and denominator by the same largest divisor. I also

taught them the way to form the simplest form in symbolic method and compare it with

using the circle cut out.

In step two, I provided fraction circle cut-outs to each pair of students and then

written down some proper fractions. I asked the students to find out the simplest form of

those fractions in pair by using the circle cut-out. I called out some students randomly to

show off their answers by using the fraction circle cut-outs.

From the activity, I found out that all the students can understand the concept of

expressing equivalent fraction to its simplest form, and able to found the correct answer

by using the model provided. To enhance their understanding more, I prepared a game,

‘Who wants to be a millionaire?’ which was an individual game. I displayed a question

in slide and asked the students to solve the question by using the circle cut-out that was

provided to them in a minute. I had prepared the students’ name as a lucky draw. Then, I

draw a students name from the lucky draw box and the student gave the answer. I gave a

sticker to the students who provided the correct answers. I repeated up the above steps

for several questions.

According to my observer, the lesson development has gain students attention

and interest fully. They were involved themselves fully and followed each of my

instruction correctly. She also mentioned that, those activities in lesson development

helped my students to emphasize the method of expressing equivalent fractions to its

simplest form. She said that the game was well prepared and the students enjoyed it. It’s

became a good reinforcement to them.

By the use of concrete models in this teaching and learning process, the students

can answer the worksheet individually and not many students needed my guides. The

also could answer my question spontaneously when I showed the flash cards of fractions

and asked then to identify its simplest form.

The teaching and learning process was mostly student centered and it has helped

the students to participate fully in it. From that, they were able to explore each question

by themselves without an interruption. This is because of the concrete models such as

fraction circle cut-outs and the paper strips have helped them to develop their conceptual

understanding on equivalent fractions.

Observation 4

Lesson : Express and write equivalent fractions for proper fractions by using

multiplication and division.

Method : Table of calculation.

The forth teaching and learning method was the last method. The lesson was

mostly same as the first lesson which is the chalk and talk. I prepared the same lesson as

the first one to go through the students development in conceptual understanding of

equivalent fractions after I used the concrete models as planned.

The learning objective and learning outcome of this teaching and learning

process was also same as the first teaching and learning process. I just added the table of

multiple and the table of divisible in this forth lesson in the materials used. This lesson

fully involved the symbolic method of expressing equivalent factions for proper

fractions which used operation of multiplication and division.

In the set induction I still used the flash cards and asked the students to compare

the greater fractions and also the less. The students can compare the fractions easily and

gave correct answer. But, in chalk and talk lesson most of them can’t do so and faced

some trouble. Manoranjetam had mentioned that the students were more advanced in

answer the questions and they paid full attention also.

In Step 1 of the lesson development, I included the table of multiples and the

table of divisible to teach them the calculation method. I also asked them to try out some

questions from the Year 4 Mathematics text Book as practice. They try out it by using

the table method.

Lastly I provided the same worksheet as the first lesson to them to try. The

students did it well and have corrected their previous mistakes. Most of them were

appling the correct method in obtaining their solution. This shows their full understand

in the concept of equivalent fractions.

According to my observer, the students’ involvement in this teaching and

learning process was totally opposite from the first teaching and learning process. They

all were very active, fresh and not seem like bored. They were very excited in

discovering the equivalent fraction by using multiplication and division. They also

didn’t need my help or guide in answering the worksheets.

Lastly, the use of concrete models has helped me tremendously to increase the

student’s interest in finding or expressing equivalent fractions. This is because the

concrete models helped them to explore the connection between the concept and the

calculation of the lesson. The students really very interested in manipulated those

concrete models provided to them and freely expressing equivalent fractions for proper

fractions.

8.0 SUMMARY

The use of concrete models, which were the fraction circle cut-outs and fraction

strips really developed the students’ conceptual understanding of equivalent fractions.

This shows that the research has achieved the research objectives and able to answer the

research questions that were the concrete models were able to develop the students’

conceptual understanding on equivalent fraction by using concrete models and also able

to increase the interest of students in express equivalent fractions for proper fractions.

This action research also proved that concrete models help learning as a

quantitative increase in knowledge. Learning with concrete models is storing

information that can be reproduced. Using the concrete models also help students learn

interpreting and understanding reality in a different way.

The concrete models have been an effective learning tool for the students in

developing their conceptual understanding of equivalent fractions. Its also help improve

students individual performance. Besides that it helps students to become more effective

learners.

Finally this action research helped me find better way to solve the problem face

by the students in learning equivalent fractions. As a future teacher, I also realized that I

must always alert with my students progress in learning by analyze their strength and

weakness and also must be able to overcome the problems faced by my students in

teaching and learning process.

9.0 SUGGESTION

The use of manipulative materials in this action research is applicable for the

particular school. Moreover, the strength of concrete models also may provide

recommendations to other primary school in our country, Malaysia to develop the Year

4 students’ conceptual understanding of equivalent fractions.

Even the use of concrete models gave good result, but there is some students

make some mistakes in their post-test. For example, some students have been wrongly

answered the questions which have pictorial method. They can’t interpret the answers in

the pictorial method. So, next time I should apply the pictorial method after the

application of concrete models.

This is because pictorial method is one of the easy ways to understand the

concept of fractions. This is because some students like to draw and clearly understand

the concept of equivalent fractions. Once students applied the question into the picture

after the use of concrete models that means they understand the basic concept of

equivalent fractions.

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9.0 APPENDICES