Research For Ann.docx

Chapter I INTRODUCTION Math learning difficulties are common, significant and worthy of serious instructional attention in both primary and secondary education. Many students struggle learning about Math, its concepts, equations, and formula. Students may response to repeated failure with withdrawal of effort, lowered self-esteem, and avoidance between behaviors. In addition, significant Math deficits can have serious consequences on the management of everyday life as well as on job prospects and promotion. There are many ways on how to cope up with students’ difficulties in learning Mathematics in order for the teachers and schools to help students with learning difficulties to improve their achievement in Mathematics. One way would be, cooperative group learning whereas pupils can work in a small group (4-6 members) and the teacher would encourage them to discuss and solve problems, then the teacher would move from one group to another, giving assistance and encouragement, ask thoughts provoking questions as the need arise. In that way, students would also learn to work together as a team fostering cooperation rather than competition and this will develop students’ social interaction and their problem solving abilities. Another way would be laboratory approach or this can be defined as “learning by doing”. This would often involve the students to play and manipulate concrete objects in structured situations. In that way, students are able to proceed at their own rate, build readiness for the development of more abstract concepts and the use of this concrete objects or manipulatives are useful for younger children. Educational research indicated that the most valuable learning occurs when students actively construct their own mathematical understanding, which is often accomplished through the use of manipulatives.

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In support, Seedfelt and Wasik, (2006) added that “In order to have opportunities to learn math, children need firsthand experiences related to math, interaction with other children and adults concerning these experiences and time to reflect on the experiences.” Another study by Sutton and Krueger (2002) had been found that the use of manipulatives in teaching Mathematics do not only allow students to construct their own cognitive models for abstract Mathematical ideas and processes, they also provide a common language with which to communicate these models to the teachers and other students. And also, manipulatives have the additional advantage of engaging students and increasing both interest in and enjoyment of Mathematics. Students who are presented with the opportunity to use manipulatives report that they are more interested in Mathematics. Long-term interest in Mathematics translates to increased Mathematical ability. According to Schweinle, Meyer, and Turner (2006) the experiences that students have in the classrooms, motivationally and emotionally, are crucial factors that affect their attitudes, behaviors, and achievement. The purpose of this study is to determine whether the use of manipulatives has an effect in the academic performance of Grade V sections II and III in the lesson of fractions. This paper aims to promote the use of manipulatives in teaching Mathematics in public schools due to the reason that public schools were given less the chance to use manipulatives, after the intervention of manipulatives, the researchers will conduct an interview to the experimental group of students about their experiences when taught using the manipulatives. The researchers will equip the students and the teachers the specific manipulatives to be used in teaching Math specifically in the lesson about Addition of Similar and Dissimilar Fractions. Statement of the problem

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This study aims to determine the effects of using manipulatives in the academic performance of learning fractions. Specifically, the study will seek to answer the following questions: 1. What is the profile of the Grade V section 2 and 3 pupils? 2. What is the mean of the academic performance of the pupils on fractions under the traditional approach when grouped according to: a. Sex b. Age 3. What is the mean of the academic performance of the experimental group pupils on fractions when grouped according to: a. Sex b. Age 4. Is there a significant difference in the scores between the pre-test of traditional and the experimental group? 5. Is there a significant difference in the scores between the post test of traditional and the experimental group? 6. Is there a significance difference between the pre test and post test of each group? 7. What are the experiences of the pupils whose been taught using the manipulatives?

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Hypothesis 1. There is no significant difference between the pre-test scores of pupils under the traditional approach and the experimental group. 2. There is no significant difference between the post-test scores of pupils under the traditional approach and the experimental group. Scope and Limitation of the Study The study generally focused on the effects of using manipulatives in the academic performance of pupils on fractions of Grade 5 section sections 2 and 3 at Alfredo Montelibano Sr. Elementary School presently enrolled during the conduct of the study. A pre-test and post test were used in gathering the information to answer our research questions. The topic is limited to Adding Similar and Dissimilar Fractions for Grade 5 pupils only. Hence, there is no assurance that the result may also be applicable to other topics in fractions and grade levels. The manipulative used is specifically fraction circles. Significance of the study This study aims to determine the effects of the use of manipulatives in the academic performance of the operations in teaching fractions. Specifically, the study will be proved significant and beneficial to the following: Department of Education. The findings of the study done positive results, the Department of Education can recommend the use of manipulatives to educational institution in order to

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improve the academic performance of the students, specifically the grade 5 pupils in elementary school. Administrators. The result will shed light and provide additional manipulatives not only in the field of fraction but on other topics and subjects as well. Curriculum Planner. The findings will be use as a proof and a way that curriculum planner must include the use of manipulatives in the curriculum in the field of Mathematics in elementary schools. Math Teachers. The result will help them on the effective way of teaching concepts and operations of fraction with the help of manipulatives. Students. The result of this study will provide the students meaningful experiences and reinforce the learning with the use of manipulative in teaching fractions. The findings of the study would let them understand and appreciate how manipulatives can be a great help to improve their academic performance on fractions. Future Researchers. The result will serve as a basis for more comprehensive study for the future researches. The study will serve as a reference for future researchers who wants to further examine the use of manipulatives in teaching fractions and as well as in other areas of mathematics such as in problem solving, numbers and integers, place values system and whole numbers operations.

Theoretical Framework 5

The study was anchored from two influential educational theories of John Dewey and Jean Piaget. The following theories will be used as basis of our study that will either comprehend or not to the findings. The proponents recommended the use of concrete objects that enables students’ to actively participate in the lesson for an effective learning. First is John Dewey’s theory in education known as Experiential Learning or Learning by Doing. Dewey believes, “there is an intimate and necessary relation between the processes of actual experience and education.” Dewey argued that children need assistance from teachers in developing a concrete understanding of the world. With that, students need educational experiences which enable them to become valued, equal, and responsible members of society that suggests the use of hands on materials to develop experiential education. Experience is at the heart of the educational process, indeed education is defined exclusively in terms of the extent to which it develops. In addition, Jean Piaget’s cognitive developmental learning theory states that active learners who master the concepts by progressing through the three levels of knowledge: concrete, pictorial, and abstract. At the concrete level, tangible objects, such as manipulatives, are used to approach and solve problems. Almost anything students can touch and manipulate to help approach and solve a problem is used at the concrete level. At the pictorial level, representations are used to approach and solve problems. These can include drawings (e.g., circles to represent coins, pictures of objects, tally marks, number lines), diagrams, charts, and graphs. These pictures are visual representations of the concrete manipulatives. It is important for the teacher to explain this connection. At the abstract level, symbolic representations are used to approach and solve problems. These representations can include numbers or letters.

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In summary, the use of manipulatives enables students to explore concepts at first or concrete level of understanding. When students manipulate objects, they are taking the necessary first steps toward building understanding and internalizing math problems, processes and procedures. Therefore these theories in education highly influenced to provide experiential learning to students with the use of manipulatives. Thus, such manipulatives were used in the study.

Operational Framework

Grade V-2

Grade V-3

Pre-test

Use of Manipulatives

Traditional Approach 7

Post-test

Interview

The diagram above shows the flow of the procedure on how to conduct the study and answer the problem by conducting an experimental research in Grade 5 sections 1 and 2 of Alfredo Montelibano Sr. Elementary School wherein a pre-test will be answered by both groups. A lesson will be conducted afterwards about adding similar fractions and dissimilar fractions with the intervention of manipulatives in one group and a traditional classroom instruction with the same topic on the other. A post-test will be given then after the lesson and this time one group of the students will be answering the post-test using a manipulative and the other without. To determine if there is a significant difference between the pre and post-test. An interview with 10 chosen students in the experimental group who had two extreme scores in post-test to identify their experiences in using manipulatives. Definition of terms The following terms are used in this study. Hence, they are defined conceptually and/or operationally. Academic Performance. Refers to how well a student is accomplishing his or her tasks and studies (http://www.studymode.com, 2015). This refers as a basis in the increase or decrease in students’ individual pre-test and post-test scores. 8

Effect. This refers to the ability to cause a change in thought, action or behavior through non-coercive and transparent means. Wu, M.,(2012). Operationally, effect refers on how manipulative affect the pupils’ academic performance in dealing with fractions. Fractions. This refers to a part of a whole in which the denominator represents the total number of parts in a whole unit, and the numerator represents the number of parts shaded or counted and a particular representation of a rational number, a rational number being any number that can be expressed as a quotient of two integers a/b. Park, J., (2012). Operationally, fraction refers as the main topic to be used. Manipulatives. This refers to the materials that represent explicitly and concretely mathematical ideas that are abstract. They have visual and tactile appeal and can be manipulated by students through hands-on experiences (Moyer, 2001). Operationally, manipulatives specifically fraction circles are tangible objects used as an instrument or material in teaching fractions. Traditional Approach. This refers to as ancient formal teaching approach which involves the directed flow of information from teacher as sage to student as receptacle Raine, D.,Collett, J., (2001). It is a product-centered, reading is passive, bound to a specific context focus on forms, schema is not considered. Lyons, L.,(n.d). Operationally, it is a lecture and discussion method to be used in conducting the lesson on the control group. CHAPTER II REVIEW OF RELATED LITERATURE

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This chapter presents both related literature and studies that support the problem of the study. Academic Performance Venkatesh K.and Karimi, A.(ret. fr.http://medind.nic.in/jak/t10/i1/jakt10i1p147.pdf) , stated that good academic performance is very important not only not only to students and their parents, but also to institutions of learning, educationists and any progressive. The quality of students’ academic performance is influenced by wide range of environmental factors rather simply teacher factors and psychological factors within the learners such as motivation and the self, rather than simply by ability. Students' academic performance consists of his scores at any particular time obtained from a teacher- made test. Fractions One of the building blocks in mathematics is fraction. Students who understand fractions find it easy for them to apply it in other field in mathematics. Fractions are one of the most important topics students need to understand in order to be successful in algebra and beyond, yet it is an area in which U.S. students struggle. National Assessment for Educational Progress (NAEP) test results has consistently shown that students have a weak understanding of fraction concepts (Sowder & Wearne, 2006; Wearne & Kouba, 2000). Moreover, in order to be able to develop the understanding about fractions, students need to know how it come up with that solution and how the process done. Teachers do discussion and let the students do some exercises to be able to fully understand the lesson in fractions. Fractions teaching have embraced attention of mathematics teachers and educators worldwide due to the fact

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that many learners seem to have problems in understanding the concept of fractions. Jamilah Yusof (2013) Another study states that report on students’ understanding of the concepts of fractions, such as, finding equivalent fractions, identifying fractions, and the importance of using equal parts also take account of the research on students understanding of the operations on fractions, in particular multiplying and dividing and the use of manipulatives to teach and learn before mentioned fraction concepts (Valle, 2006). Difficulties in Fraction McNamara and Shughnessy (2010) stated that fractions are difficult because it was written in a unique way and students over generalize their whole-number knowledge. The concept of fractions is a complex one and it takes time, combined with a rich range of experiences and appropriate mathematical models, for children to develop a deep and rigorous understanding (Jennie Pennant and Liz Woodham 2013). It is important for a teacher to help students see how fractions are alike and different from whole numbers. Common misapplications of whole-number knowledge to fractions leads students to a misconception on how fractions concepts should be understand. First, students think that the numerator and denominator are separate values and have difficulty seeing them as a single value 3

(Cramer & Whitney, 2010). It is hard for them to see that 4 is one number. . Second, students think 1

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that a fraction such as 5 is smaller than a fraction such as 10 because 5 is less than 10. Conversely, students may be told the reverse—the bigger the denominator, the smaller the fraction. Teaching 1

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such rules without providing the reason may lead students to over generalize that 5 is more than10.

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Lastly, Students mistakenly use the operation “rules” for whole numbers to compute with fractions—for example,

1 2

+

1 2

=

2 4

. Students who make these errors do not understand fractions.

Until they understand fractions meaningfully, they will continue to make errors by over applying whole‐number concepts (Cramer & Whitney, 2010; Siegler et al., 2010). National Council of Teachers of Mathematics (NCTM) states that students in middle school should acquire a deep understanding for fractions and be able to use them competently in problem solving (Naiser et. al, 2004). In order for this deep understanding to occur and for the students’ learning to be lasting, multiplying and dividing fractions need to be taught at the concrete level Aids of the Teachers Difficulties in Fractions A study found that there were differences in the way mathematics is taught in the United States versus the higher achieving countries. Teachers in the United States teach most of their lessons using a procedurally based method while teachers in the higher performing countries teach mostly conceptually based lessons (TIMSS, 2003). The most effective way to help students reach higher levels of understanding is to use multiple representations, multiple approaches, and explanation and justification (Harvey, 2012; Pantziara & Philippou, 2012). According to Georgiou, Zahn and Meria (as cited in the study of Rizk, 2011), “the heart of experiential learning lies in reflectively observing concrete experience and actively experimenting with abstract conceptualizations”. Students learned through experiences by feeling the objects and how it is being manipulate helps students gain an abstract concept.

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Another way would be, laboratory approach or this can be defined as “learning by doing”. This would often involve the students to play and manipulate concrete objects in structured situations. In that way, students are able to proceed at their own rate, build readiness for the development of more abstract concepts and the use of this concrete objects or manipulatives are useful for younger children. A study had been found that the use of manipulatives in teaching Mathematics do not only allow students to construct their own cognitive models for abstract Mathematical ideas and processes, they also provide a common language with which to communicate these models to the teachers and other students. And also, manipulatives have the additional advantage of engaging students and increasing both interest in and enjoyment of Mathematics. Students who are presented with the opportunity to use manipulatives report that they are more interested in Mathematics. Long-term interest in Mathematics translates to increased Mathematical ability. (Sutton and Krueger, 2002) Manipulatives John van de Walle and his colleagues (2013) define a mathematical tool as, “any object, picture, or drawing that represents a concept or onto which the relationship for that concept can be imposed. Manipulatives are physical objects that students and teachers can use to illustrate and discover mathematical concepts, whether made specifically for mathematics (e.g., connecting cubes) or for other purposes (e.g., buttons). Moore (2013) provides an example of this structure in action. In this approach,built on Bruner’s (1966) work, students first use concrete materials to solve problems and look for patterns and generalizations. As students need to record their work, they do so first by sketching pictures

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(representations) of the manipulative models and then finally move to using abstract (and more formal) mathematical notations for their work. In 2013, the National Council of Supervisors of Mathematics (NCSM) issued a position statement on the use of manipulatives in classroom instruction to improve student achievement. Moreover, in order to develop every student’s mathematical proficiency, leaders and teachers must systematically integrate the use of concrete and virtual manipulatives into classroom instruction at all grade levels. According to the theory of experiential education revolves around the idea that learning is enhanced when students acquire knowledge through active processes that engage them (Hartshorn and Boren, 1990). Manipulatives can be key in providing effective, active, engaging lessons in the teaching of mathematics. Stein and Bovalino (2001) reported that manipulatives can be important tools in helping students to think and reason in more meaningful ways. By giving students concrete ways to compare and operate on quantities, such manipulatives as pattern blocks, tiles, and cubes can contribute to the development of well-grounded, interconnected understandings of mathematical ideas.

Advantages of Manipulatives

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The order of manipulative use appears to impact the development of conceptual understanding and the students’ ability to transition to abstract algorithms (Hunt, 2011). In order to fully understand one must know the concepts and how it is transfer in abstract. In 2007, Allen found that it has been claimed that the usage of a manipulative not only increases student achievement, but also allows them to improve their conceptual understanding and problem solving skills. It is easy for them to solve any equations or problem solving with the aid of manipulatives. According to Resnick and Omanson (1988) as cited in the study of Uttal (2003) manipulative use can facilitate children's acquisition and fluid use of mathematics concepts. For example, children who regularly used Dienes Blocks acquired flexibility in subtraction skills such as borrowing. Many children who began the year with little or no knowledge of subtraction were able to perform well with the Dienes Blocks by the end of the year. In particular, many children's understanding of the borrowing procedures in subtraction, as evidenced by their Dienes Blocks constructions, increased substantially throughout the year. It proves that the use of manipulatives in teaching increases the understanding of students thus enable them to outgrown others who doesn’t use manipulatives. Raphael &Walstrom (1989) states that it is found, for the most part, that when manipulatives are used properly to teach the concepts of multiplying and dividing fractions, students outperform those students who do not use manipulatives (p.173). In addition, manipulative use also gets the students more engaged in the lesson and provides a way for students to represent their thinking (Naiser, Wright, &Capraro, 2004).

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According to Naiser et al., (2004) teachers must find a variety of strategies, including the use of manipulatives, to use in the classroom for teaching multiplying and dividing fractions. To add up, “Manipulatives can be used to induce understanding of concepts in the middle school curriculum when they are used properly” (Moyer & Jones, 2004, p. 32). Glidden (2002) also believes when used properly, manipulatives are valuable tools for helping students understand mathematical concepts and operations, and their use should be part of every teacher’s practice. Naiser, Wright &Capraro (2004) found out that manipulatives also provide a good method for teachers to understand what their students are thinking. Naiser et al. reported that by watching what the students are doing with the manipulatives and how they construct meaning, a teacher can tell if a student has conceptual understanding or not. Kieren (1988) recommends that instruction on multiplying and dividing fractions build on children’s intuitive understanding of the concepts of fractions and on interaction with objects rather than merely following procedures. “Teachers should help students generalize symbolic algorithms from their experiences with real world problem contexts, manipulatives and pictures” (Bezuk& Armstrong, 1992, p. 729). This will allow students to construct their own mathematical understanding, which is vital to a satisfactory foundation for future understanding (NCTM, 1989). Similar to what Raphael and Wahlstrom found, Clements (1999) found that when teachers, who are knowledgeable about manipulative use, teach mathematics using manipulatives the students usually outperform students who do not use manipulatives. Clements reports that these benefits hold true no matter what the grade level, ability or topic. He also writes that students’ attitudes toward mathematics improved when knowledgeable teachers taught them with concrete materials.

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May (1994) states that the best way for students to learn multiplication and division fractions is to have them create a model that gives meaning to the operation. She believes that the best way to do that is to provide them with good activities and the right kind of manipulative. CRA (Concrete, Representational & Abstract) is a “highly touted” instructional method for teaching mathematics to students with learning disabilities (Witzel, 2001). Witzel has developed a workshop to implement the CRA model in any mathematics classroom. This sequence of instruction, starting with concrete, then moving to pictorial representation and then on to abstract symbols, has many benefits for students (Witzel, 2006). Witzel states “Three of the benefits of the CRA model are students develop mathematical concepts, a greater procedural understanding and it makes mathematics fun.” Disadvantages of Manipulatives According to Resnick and Omanson (1988) as cited in the study of Uttal (2003) stated that the acquisition of mathematical concepts from manipulatives and children's transfer (or lack of transfer) to written representations. Their study included a wide array of methods, ranging from intensive interviews of individual children to reaction time measures of children's processing of numerical information. In 2003, Uttal stated that it is often assumed that manipulatives should be interesting and attractive to be effective. However, it is stated in the review of his research on symbolic development strongly suggests that attractive manipulatives may sometimes be counterproductive; they may cause children to focus on the superficial properties of the manipulatives as objects rather than on their relation to written representations (p.111). Students can be attractive

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with the manipulatives but it may not help them to understand the concept or they may be amazed and just focus on the manipulatives. Teachers’ attitudes often undercut potential benefits. A number of teachers use hands-on activities solely to pique students’ interest and add variety and fun to a lesson, failing to leverage manipulatives to enhance mathematical understanding (McNeil and Jarvin, 2007). Students may perceive manipulatives as a hands-on learning activity as play time. There are many articles, books, videos and workshops that train educators how to use manipulatives to teach concepts. The National Council of Teachers of Mathematics (NCTM) organization is an exceptional resource for finding articles and workshops on using manipulatives as well as multiplying and dividing fractions. Even if the activity or lesson found is not focused on fractions, or even mathematics for that matter, most of the hands-on activities can be modified to use with whatever concept we are teaching. There is a plethora of resources out there in the form of articles written by teachers for teachers that give detailed descriptions of activities that can be used and adapted. For example, there is an article called “Putting the pieces together”, in which the author details five different activities from grades k-8 (Naylor, 2003). Another article gives hands-on ideas and discusses the importance of students verbally sharing what they have learned (Krech, 2000). There is a website that has several virtual manipulatives the students can use online to learn the concepts of multiplying and dividing fractions (National Science Foundation, 2005). According to the National Council of Teachers of Mathematics (NCTM, 2000), when it is time to teach fractions, teachers feel a level of frustration not unlike that of the students. Some teachers lack the understanding of the concepts of fractions, so it is difficult to show students how 18

to use the manipulatives. Some teachers find it hard to admit they do not know something that relates to their field, so they are not even asking co-workers for help. NCTM also states “effective teaching requires understanding of what students know” (2000, p.11). Since some teachers do not feel competent using fractions in the classroom, students miss out on this exceptional teaching tool. Most teachers are willing to make the necessary changes to their teaching methods but they need the necessary professional development experiences. In addition, through these experiences teacher are willing and able to make changes to their practices (Andreasen, Swan, & Dixon, in press).

RELATED STUDIES Fractions Fractions are among the most complex mathematical concepts that children encounter in their years in primary education. One of the main factors contributing to this complexity is that fractions comprise a multifaceted notion encompassing five interrelated sub constructs (partwhole, ratio, operator, quotient and measure). During the early 1980s a theoretical model linking the five interpretations of fractions to the operations of fractions and problem solving was proposed. Since then no systematic attempt has been undertaken to provide empirical validity to this model. Charalambous and Pantazi (2005) on their study, Revisiting a theoretical model on fractions: Implications for teaching and research provides support to the assumption that mastering the five interpretations of fractions contributes towards acquiring proficiency in the

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operations in fractions. In teaching fractions, teachers need to scaffold students to develop profound understanding of the different interpretations of fractions since an understanding could uplift students’ performance in task related to the operations of fractions. Thereby, instead of rushing to give students with different algorithms to execute operations in fractions, lend them to support that teachers should emphasize more on the conceptual understanding of fractions. The study also suggests that teaching the different operations in fractions should be directly linked to specific interpretations of fractions. In particular, the findings of study indicate that teaching of equivalent fractions could be reinforced by learning ratios whereas the operator and the quotient sub constructs could support developing conceptual understanding of the multiplicative operations on fractions. Hurrell (2013) on his study, Effectiveness of teacher professional learning: Enhancing the teaching of fractions in primary schools cited the importance of teaching and learning fractions. Fractions: are core component of being numerate; are an integral part of understanding division; provide an insight into children’s understanding of numbers and number operations; impact upon the strands of measurement and space; and are necessary for achieving success in algebra. Teachers and schools should ensure that fractions receive the attention they deserve. The study also mentioned the teacher’s perception of the importance of fractions and teacher’s perception of the status of fractions in the schools and the curriculum and results concluded that fractions are important for it is used to learn other mathematical concept and the other one with the assertion that teachers hold the teaching and learning of fractions as being important but this status can be further raised through attendance at well-structured They perceive that the importance of fractions has not been made explicit by their schools). This perception further extends to the education

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system in which they teach, through the lack of explicit guidance in the curriculum documents respectively. Academic Performance Academic performance determines whether learning occurred successfully or not. Students vary in academic performance across the different subjects and even elective courses from elementary, secondary and tertiary level. Academic performance may be in a form of a written output or performance based activity. Most students have difficulty in excelling in their academics especially in the field of mathematics. Ross (2008) on her study, The effect of mathematical manipulatives materials on third grade students’ participation, engagement and academic performance found positive relationship between student work samples and student participation and engagement. The teacher used mathematical manipulatives materials as a supplement to aid conceptual understanding of addition and subtraction. A pre-test and post test and student samples were used by the researcher to determine effects on academic performance. Data showed that student’s academic performance increased however the relationship between academic performance and manipulatives was found to require further research and study Luke (2012) on her study, The impact of manipulatives on students’ performance on money word problems recommended the use of physical objects as a representation and problem solving as a teaching method in mathematics. Three populations were included in the study namely adults who struggle with numeracy, children with learning disability, and children who are typically developing. Results indicate that none of the participants performed better with manipulatives than they performed without manipulatives. Researcher suggested for more research to understand the 21

impact of manipulative use in mathematics instruction for adults who struggle with numeracy, children with a learning disability, and children who are typically developing. Indeed academic performance does not measure the intelligence of the student to a particular topic of a subject but shows that maybe there are factors that lead to the misconceptions thus giving poor performance and factors like if it was encountered before by students thus prior knowledge comes in to play leading to better performances of other students. Difficulties in Fraction However clear the objectives for learning fractions, the mathematics education literature is resounding in its findings that understanding fractions is a challenging area of mathematics for North American students to grasp (National Assessment of Educational Progress, 2005). Students also seem to have difficulty retaining fractions concepts (Groff, 1996). Educators and researchers agree that most students encounter significant problems and misconceptions when learning fractions (Behr, Lesh, Post & Silver, 1983; Carraher & Schliemann, 1991; Hiebert 1988 to name just a few). Hasemann (1981) provided four possible explanations about why children find fractions so challenging: 1) fractions are not used in daily life regularly; 2) the written notation of fractions is relatively complicated; 3) ordering fractions on a number line is exceedingly difficult; and, 4) there are many rules associated with the procedures of fractions, and these rules are more complex than those of natural numbers. Other researchers have taken up further study of some of these explanations. Moss and Case (1999) agreed that notation is one factor that could be linked to children‟s difficulties with fractions but they also pointed to several other complications: 1) Too much time is devoted to teaching the procedures of manipulating rational numbers and too little time is spent teaching their 22

conceptual meaning; 2) Teachers do not acknowledge or encourage spontaneous or invented strategies, thereby discouraging children from attempting to understand these numbers on their own (Confrey, 1994; Kieren, 1992; Mack, 1993; Sophian & Wood, 1997) and, 3) When introduced, rational numbers are not sufficiently differentiated from whole numbers (e.g., the use of pie charts as models for introducing children to fractions (Kieren, 1995). Aids of the Teachers Difficulties in Fractions Students are challenged when learning fractions and problems often persist into adulthood. Teachers may find it difficult to remediate student misconceptions in the busy classroom, particularly when the concept is as challenging as fractions has proven to be. Although teaching and learning initial fractions is a multifaceted process, leaders in both education and government are becoming increasingly concerned with these immutable results and might welcome suggestions in teaching approaches that could help reverse these trends. McGuire (2004) on his study, Exploring an Interdisciplinary Strategy for Teaching Fractions to Second Graders incorporated music and movement to see if second grade students could learn to transform visual, aural, and kinesthetic rhythm experiences into mathematical symbols in order to equate and add fractions with unlike denominators. Results showed that the experimental group as a whole achieved gain scores that were significant thus providing evidence that the incorporation of music and movement was successful in a way. An important detail is that neither any of the group of students had been exposed to fractions containing traditional notion therefore success of some students in learning to perform calculations was notable. According to Bruce & Ross (2009) on their study, Conditions for effective use of interactive on-line learning objects: The case of a fractions computer-based learning sequence, used 23

technology-assisted learning is exploring additional ways of enhancing student understanding with challenging math concepts such as fractions. Results showed that students controlled pacing of the learning object tasks led to more effective learning. Students who performed fractions through learning the sequence methodologically without skipping task benefited better over-all both in terms of student achievement measure and self-reported feelings of success than students who did not do the fractions. Introductory task are directly connected with learning goals to helped students become mentally prepare for the learning object task. When introductory tasks were unrelated, students fail to see the relevance and connection thus becoming derailed and struggling to associate to the learning activities. Technology-oriented learners were confident in using the intervention even they began with very low understanding of fractions concepts. An assumption was made that students who struggle with math concepts but have high technology facility might benefit more from online learning objects as a method of instruction. Advantages of Manipulatives Manipulatives provide genuine and authentic experience for students. Their use is highly encourage for all children in all levels, not just only for preschoolers but also for the gifted and those with special needs. Manipulatives are constructed to allow children to learn naturally through play and exploration. Ojose & Sexton (2009), on their study, The Effect of Manipulative Materials on Mathematics Achievement of First Grade Students pointed out the need for teachers to recognize the importance and impact of hands-on activities and manipulatives for students in all grades. Ruzic & O’Connell (2001) found that long-term use of manipulatives has a positive effect

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on student achievement by allowing students to use concrete objects to observe, model, and internalize abstract concepts. Raphael & Wahlstrom (1973) on The Influence of Instructional Aids on Teaching Mathematics Achievement mentioned that, for the most part, when manipulatives are used properly to teach concepts of multiplying and dividing fractions, students outperform those students who do not use manipulatives. Similar to the findings of Raphael and Wahlstrom, Clements (1999) found that when teachers who are knowledgeable about manipulative use, teach mathematics using manipulatives the students usually outperform students who do not use manipulatives. Clements reports that these benefits hold true no matter what the grade level, ability or topic. He also writes that students’ attitudes toward mathematics improved when knowledgeable teachers taught them with concrete materials. In addition, Naiser et al (2004) on Teaching Fractions: Strategies used for Teaching Fractions to Middle Grade students mentioned on their research that using manipulatives to teach multiplying and dividing fractions made the lessons more active and provided an effective way for the students to represent their thinking. They also found that teachers were able to understand student thinking by observing them use manipulatives where as with paper and pencil the students “genuine thinking” is not insured. Students who were exposed to a procedural lesson prior to a conceptual lesson scored significantly lower on an assessment than students who were only taught the conceptual lesson (Philipp & Vincent, 2003). Allen (2007) on her study, Action based research study on how using manipulatives will increase students’ achievement in Mathematics stated that students increased their skills and 25

showed more interest and enjoyment when learning was done through the use of manipulatives. They were active and confident with their math skills. Manipulatives provide a concrete way for students to link new, often abstract information to already solidified and personally meaningful networks of knowledge, thereby allowing students to take in the new information and give it meaning. Boggan, Harper & Whitmire (n.d.) on their study, Using manipulatives on Teaching Mathematics concluded that elementary teachers who use manipulatives to help teach math can positively affect student learning. Students at all levels and of all abilities can benefit from manipulatives. They also agreed as to what Mathematician Seymour Papert think of manipulatives and it is “object to think with”. Disadvantages of Manipulatives Uttal (2003) on his study ON THE RELATION BETWEEN PLAY AND SYMBOLIC THOUGHT: The Case of Mathematics Manipulatives concluded four important points on the use of manipulatives. First point is that manipulatives cannot be an end in themselves. Manipulatives can facilitate some particular types of mathematical reasoning. The common problem is that this knowledge if often disconnected from other (written) representations. Therefore, manipulatives can be part of a specific integrated system of instruction. The problem is not that manipulatives do not work; the problem is that they sometimes have been presumed to have instant magic effect on learning. Like all instructional techniques, rnanipulatives have their advantages and disadvantages. Second point is manipulatives do not obviate teachers. Manipulatives is no better substitute than a teacher but what Uttal is driving out is that students should be guided accordingly to make discovery with the manipulative. Without such guidance, manipulatives may do much harm as

26

good. The challenge for the teacher is to figure out when, and how, to introduce and reinforce correspondences between the manipulative representation of a mathematics problem and its corresponding written representation. Third point is effective manipulative use takes time. Children need to learn as to as to how each manipulative works. There were also researches on symbolic development that pointed out that children lose interest in using manipulatives as objects in themselves. Manipulatives are interesting objects in themselves at first and therefore potential relation to written representations may be difficult for the children to perceive. However extensive practice of the use of manipulatives may be a part of the daily routine so that the students may be able to focus on what the manipulatives are intended to represent. Fourth point is that attractive or interesting manipulatives may not always be the best. A review on symbolic development suggests that attractive manipulatives sometimes becomes counterproductive: students pay attention on the aesthetic appeal rather than its relation to written representations. A limited set of manipulatives used consistently throughout elementary years by Japanese teachers. On the other hand, American teachers’ uses variety of manipulatives thus having variety of representational materials causes’ confusion and it makes more difficult for students to use the objects for representational solution of the mathematical problem. Synthesis Literature and studies revealed that there are factors that greatly affect a child’s capacity upon understanding mathematical concepts and performing the operations specifically in adding fractions. Factors that affect academic performance of a child on fractions whether the manner of teaching was taught procedurally or conceptually. Though it is seen as a problem, the researchers believe that these problems can be still addressed through the conception of using manipulatives. It aims to improve academic performance of a child in mathematics for it is believe that learning 27

fraction is the foundation of other mathematical skills with the concept of having an interactive classroom instruction, utilization of authentic assessment and even technological integration.

Chapter III METHODOLOGY In this chapter presents the research methods and procedures used in the study. This part of the study describes the research design, area, participants and variables. The instrument used to collect the data including methods implemented to maintain validity and reliability of the instruments and how the data will be gathered and interpreted are also included in this part of the study. Research Design The study on determining the effect of manipulatives in the academic performance of pupils on fraction used a mixed method design for it involves both quantitative and qualitative in gathering data. The study used a Quasi-Experimental Design to obtain the quantitative data and a Narrative Design for the qualitative data. Quasi-experimental design involves selecting groups, upon

which

a

variable

is

tested,

without

any

random

pre-selection

processes 28

(https://explorable.com/research-designs, 2008). Narrative Design research is a term that subsumes a group of approaches that in turn rely on the written or spoken words or visual representation of individuals. These approaches typically focus on the lives of individuals as told through their own stories. The emphasis in such approaches is on the story, typically both what and how narrated (https://explorable.com/research-designs, 2012) is. Lastly, a Pretest and Posttest design was utilized in the study in order to determine whether there are changes before and after the experimental manipulation. Pretest-posttest was considered especially in a quasi-experimental design to examine the effects. The Respondents of the Study The respondents of the study will include the whole population of Grade V sections 2 and 3 of Alfredo Montelibano Elementary School presently enrolled during the conduct of the study. The Grade V section 2 pupils will serve as the experimental group wherein they will be taught using manipulatives and this section has a total of 23 students which is composed of 12 boys’ ages 10-15 and 11 girls’ ages 9-12. The Grade V section 3 pupils will serve as the controlled group wherein they will be taught traditionally and this section has a total of 27 students which is composed of 16 boys’ ages 9-15 and 11 girls’ ages 9-12. Both group are already grouped heterogeneously. The Research Instrument The researchers design self-made test questionnaires for the data gathering process to get the quantitative data. The primary aim of the questionnaire is to determine whether there is a significant difference between the tests given. A pre-test (Appendix C) will be given before the intervention of manipulatives. The pre-test is validated by three (3) Math experts and it is 29

composed of 10 items multiple choice test consisting of a mixture of addition of similar and dissimilar fractions and one word problem for 5 points. Then, a lesson will be conducted to the controlled group (Grade V section III) and to the experimental group (Grade V section II) with the help of a lesson plan (Appendix D and E) as a guide to conduct the lesson. This lesson plan is a detailed lesson plan following a 4A’s format wherein it shows the flow of the lesson, starting off with a motivational activity, then moving on with the lesson proper and/ with the intervention of manipulatives specifically fraction circle (Appendix I) and values integration will also be processed, and lastly, an assessment will be given afterwards to test whether the groups of students have understood the lesson. Another set of assessment will be given and this will serve as the posttest (Appendix F) which will be given afterwards. The post-test is a self-made test which is composed of 10 items multiple choice tests consisting of a mixture of addition of similar and dissimilar fractions and one word problem for 5 points. Both pre-test and post-test will have the same test questions but the test questions will be shuffled. Afterwards, a narrative interview will be conducted to the pupils under the experimental group. This is to determine their experiences when they are taught using the manipulatives. Research Validity This study aimed to determine the effects of the use of manipulatives in the pupils’ academic performance in fractions. To determine the effect of the manipulatives, a test was constructed which contain a 10 item questions and one word problem. The lesson plans (controlled and Experimental lesson plan) made by the researchers were also validated by three Math teachers from Negros Occidental High School.

30

The researchers controlled the possible threats of the validity of the results by ensuring that all the research instruments used in the study were validated by at least three persons who were experts in the subject matter and in making questionnaires. The researchers used the Good and Skates Criteria for validating the researcher instruments.

Data Gathering Procedure The researchers undertook the following steps in conducting the data gathering: 1. The researchers constructed a letter addressed to the principal of Alfredo Montelibano elementary School and had it signed by the research class adviser. 2. The letter was submitted to the principal. 3. After the letter was approved, the researchers gathered the profile of the two sections in Grade five. Profiling included gathering of information such as the age, number of pupils per section, sex and summative test of the pupils. 4. The researchers used the BEC Curriculum guide for Mathematics subject for the Math subject and used the competencies as guide in the construction of the test questionnaires to be used in the pre test and post test. 5. The pretest was validated by three validators who are experts in the field of math.

31

6. The researchers constructed the lesson plans to be used in the traditional approach and in the intervention (manipulatives) to the experimental group. The researchers aligned the construction of the lesson plan in the BEC curriculum guide for Math. 7. The lesson plans (both traditional and experimental) was validated by same validators of the pre test. 8. The researchers conducted the pre test to the controlled and experimental group. 9. The next day, one of the researchers conducted a lesson on the controlled group using a traditional approach simply in a discussion/lecture method. Also the researcher conducted a lesson to the experimental group with the intervention of manipulatives. 10. After every lesson, the researchers conducted the post test to the controlled and experimental group. 11. The researchers interviewed 10 pupils who were from the experimental group, five with high extremes and five with low extremes about their own experiences in the use of manipulatives. 12. For this study, threats to internal and external validity were few. Threats to internal validity with regard to maturation, students may develop or change during the experiment, and these changes may affect their scores between the administering the pre test and post test. Participants in the study were grade V students, same grade and same age bracket. Mortality threat, individual may drop out of a study or experiment due to any number of reasons. Students who did not attend the classes anymore or who transferred to other schools were eliminated from the study.

32

Data Analysis The gathered data were tabulated, grouped and presented in tables and graphs in the next chapter. The following statistical measures were used to analyze the data. Quantitative 1. Mean This is used to measure the pre-test and post-test by adding up the scores and divide it by the number of respondents. 3. Frequency This is used to determine the number of responses to the survey with regards to the questionnaires. 4. Percentage This is the representation of frequency as it is a part of a hundred.

Where: P – percentage n – number of respondents

33

N – total number of respondent 100 – constant 5. Median This is used to rank the ages and scores of the respondents from lowest to highest in terms of scores while in ranking the age it is from youngest to oldest to get the middle most number in the set. 6. Standard Deviation This is a statistic that tells how tightly all the various examples are clustered around the mean in a set of data. 7. T-test This is a statistical significance indicates whether or not the difference between two groups’ averages most likely reflects a “real” difference in the population from which the groups were sampled (statwing.com). This will be performed to compare the two variables, controlled and experimental group base on their academic performance. Also, narrative inquiry will be used to identify the responses of the students about their experiences on using manipulatives. Reliability The Kuder and Richardson Formula 20 will be used to check the internal consistency of measurements with dichotomous choices. It is equivalent to performing the spilt methodology on all combinations of questions and is applicable when each question is either right or wrong. A

34

correct questions scores 1 and an incorrect question scores 0 (Zaiontz, 2014). The reliability test will be conducted to 50 pupils of Alfredo Montelibano Sr. Elementary School presently enrolled during the conduct of the study. The reliability result of the study was 0.7.

Chapter 4

PRESENTATION, ANALYSIS AND INTEPRETATION OF DATA

This chapter presents, analyzes and interprets the gathered data of the study. It is divided into three parts: (1) Profile of the Respondents, (2) Descriptive Data Analysis, and (3) Inferential Data Analysis. Part One, Profile of the Respondents, presents the percentage of respondents surveyed grouped according to sex and age. Part Two, Descriptive Data Analysis, shows the results and discussion of quantitative data that were gathered and analyzed. Part Three, Inferential Data Analysis, presents the results and discussion from the statistical tests performed to test the hypotheses presented at the beginning of the study.

35

Profile of Participants In this study, pupils who were included came from the Grade V sections 2 and 3 of Alfredo Montelibano Sr. Elementary School and all of them are enrolled when this study was conducted. Table 1 shows the distribution of the sex and age of the participants of this study by group – experimental and control group.

In both groups, majority of the respondents are boys having

59.3% and as to age, both groups had 9 years old as the youngest participant, while the oldest age in the control group was 15 years old while that of the experimental group was 14 years old but majority are of age 10 having 37%. Table 1 Profile of Participants by Group, Sex and Age

Experimental Control Group Variable Category Sex

Age

Group

As a Whole

f

%

f

%

F

%

Boys

16

59.3

12

52.2

28

56.0

Girls

11

40.7

11

47.8

22

44.0

Total

27

100.0

23

100.0

9

1

3.7

2

8.7

3

6.0

10

10

37.0

10

43.5

20

40.0

11

7

25.9

4

17.4

11

22.0

50 100.0

36

12

5

18.5

4

17.4

9

18.0

13

2

7.4

2

8.7

4

8.0

14

0

0.0

1

4.3

1

2.0

15

2

7.4

0

0.0

2

4.0

Total

27

100.0

23

100.0

Mean

11.2

10.9

11

SD

1.5

1.3

1.4

Median

11

10

11

50 100.0

Descriptive Analysis The data gathered as well as those analyzed through descriptive statistics are presented in the following paragraphs in answer to the different specific problems enumerated in the study.

On the level of Academic Performance of pupils under the traditional approach when grouped according to (a) sex (b) age. The respondents level of Academic Performance was assessed.

Mathematical Performance of the Control Group

37

Table 2 shows the distribution of the pre-test scores of the control group by sex. With a total score of 15, pupils mean score result was (8.7) as a whole. The performance of the boys and girls seemed the same. Where boys having a mean score of (8.6) level as well as the girls with (8.7). Table2 Distribution of Pre-test Scores of the Control Group by Sex

Boys

Girls

As a Whole

Pre-test

F

%

f

%

F

%

5

2

12.5

1

9.1

3

11.1

7

2

12.5

0

0.0

2

7.4

8

3

18.8

2

18.2

5

18.5

9

2

12.5

5

45.5

7

25.9

10

5

31.3

3

27.3

8

29.6

11

2

12.5

0

0.0

2

7.4

Total

16

100.0

11

100.0

27

100.0

Mean

8.6

8.7

8.7

SD

1.9

1.4

1.7

Median

9.0

9.0

9.0

38

The post-test scores of the control group by sex are shown in Table 3. The girls were higher by one point as far as the mean and median post-test scores. With girls leveled average for having (11.0) mean score and boys leveled having (9.9) mean score. Table 3 Distribution of Post-test Scores of the Control Group by Sex

Boys

Girls

As a Whole

Post-test

f

%

f

%

F

%

6

1

6.3

0

0.0

1

3.7

39

7

1

6.3

1

9.1

2

7.4

8

2

12.5

0

0.0

2

7.4

9

3

18.8

1

9.1

4

14.8

10

2

12.5

0

0.0

2

7.4

11

3

18.8

4

36.4

7

25.9

12

3

18.8

4

36.4

7

25.9

13

1

6.3

1

9.1

2

7.4

Total

16

100.0

11

100.0

27

100.0

Mean

9.9

11.0

10.3

SD

2.0

1.7

1.9

Median

10.0

11.0

11.0

(Table 4) The pre-test scores of the control group shown in the when they were grouped by median age (11 years old), showed almost the same means and equal medians. The pupils with ages 11 and below got a mean score of (8.6) while pupils with ages 12 and above got (8.9). Table 4 Distribution of Pre-test Scores of the Control Group by Age

11 and Below

12 and Above

As a Whole

40

Pre-test

f

%

f

%

F

%

5

2

11.1

1

11.1

3

11.1

7

2

11.1

0

0.0

2

7.4

8

3

16.7

2

22.2

5

18.5

9

5

27.8

2

22.2

7

25.9

10

5

27.8

3

33.3

8

29.6

11

1

5.6

1

11.1

2

7.4

Total

18

100.0

9

100.0

27

100.0

Mean

8.6

8.9

8.7

SD

1.7

1.8

1.7

Median

9.0

9.0

9.0

The post-test scores of the control group by age had the exactly the same medians and almost the same means (Table 5). Ages 11 and below pupils with the (10.2) mean score and a mean score of (10.7) for pupils ages 12 and above similarly. Table 5 Distribution of Post-test Scores of the Control Group by Age

11 and Below

12 and Above

As a Whole

41

Post-test

F

%

F

%

f

%

6

1

5.6

0

0.0

1

3.7

7

2

11.1

0

0.0

2

7.4

8

1

5.6

1

11.1

2

7.4

9

3

16.7

1

11.1

4

14.8

10

1

5.6

1

11.1

2

7.4

11

3

16.7

4

44.4

7

25.9

12

6

33.3

1

11.1

7

25.9

13

1

5.6

1

11.1

2

7.4

Total

18

100.0

9

100.0

27

100.0

Mean

10.2

10.7

10.3

SD

2.1

1.5

1.9

Median

11.0

11.0

11.0

Mathematical Performance of the Experimental Group The data gathered as well as those analyzed through descriptive statistics are presented in the following paragraphs in answer to the different specific problems enumerated in the study.

42

On the level of Academic Performance of pupils under the experimental approach when grouped according to (a) sex (b) age. The respondents level of Academic Performance was assessed.

Table 6 shows the pretest scores of the experimental group by sex.

The girls seemed to

have higher scores than the boys. Specifically, the girls had a higher mean score of (5.7) as contrasted with the boys with (4.3) and with a mean score of (5.0) as a whole. Table 6 Distribution of Pre-test Scores of the Experimental Group by Sex

Boys

Girls

As a Whole

Pre-test

f

%

f

%

F

%

0

1

8.3

0

0.0

1

4.3

2

1

8.3

0

0.0

1

4.3

3

2

16.7

0

0.0

2

8.7

4

2

16.7

2

18.2

4

17.4

5

2

16.7

1

9.1

3

13.0

6

3

25.0

6

54.5

9

39.1

7

1

8.3

2

18.2

3

13.0

Total

12

23

100.0

100.0

11

100.0

43

Mean SD Median

4.3

5.7

5.0

2

1.0

1.7

4.5

6.0

6.0

Table 7 shows the distribution of post-test scores of the experimental group. The performance of both the girls and boys seemed to be the same for the boys have a total mean score of (7.4) leveled as well as the girls for having a total mean score of (7.9) and (7.7) as for the whole. Table 7 Distribution of Post-test Scores of the Experimental Group by Sex

Boys

Girls

As a Whole

Post-test

F

%

f

%

f

%

4

2

16.7

0

0.0

2

8.7

5

1

8.3

2

18.2

3

13.0

6

2

16.7

0

0.0

2

8.7

7

1

8.3

3

27.3

4

17.4

8

1

8.3

2

18.2

3

13.0

9

2

16.7

1

9.1

3

13.0

44

10

2

16.7

2

18.2

4

17.4

11

1

8.3

1

9.1

2

8.7

Total

12

100.0

11

100.0

23

100.0

Mean

7.4

7.9

7.7

SD

2.4

2.0

2.2

Median

7.5

8.0

8.0

Table 8 shows the distribution of the scores of the experimental group by age. The basis for grouping the participants was the median age (11 years old). The mean mathematical performance of the two age groups seemed the same leveled both with a mean score of (7.8) for ages 11 and below and (7.4) for pupils whose ages are 12 and above. Table 8 Distribution of Pre-test Scores of the Experimental Group by Age

11 and Below

12 and Above

As a Whole

Pre-test

f

%

F

%

F

%

0

0

0.0

1

9.1

1

4.3

2

1

8.3

0

0.0

1

4.3

3

0

0.0

2

18.2

2

8.7

45

4

4

33.3

0

0.0

4

17.4

5

2

16.7

1

9.1

3

13.0

6

8

66.7

1

9.1

9

39.1

7

1

8.3

2

18.2

3

13.0

Total

16

133.3

7

23

100.0

63.6

Mean

7.8

7.4

7.7

SD

1.9

2.8

2.2

Median

7.5

8.0

8.0

The post-test scores of the control group were relatively the same as well as they are both been labeled based on their mean scores where 11 and below got (7.8) and (7.4) for the pupils ages 12 and above. Table 9 Distribution of Post-test Scores of the Experimental Group by Age

11 and Below

12 and Above

As a Whole

Post-test

f

%

F

%

F

%

4

0

0.0

2

28.6

2

8.7

5

3

18.8

0

0.0

3

13.0

46

6

1

6.3

1

14.3

2

8.7

7

4

25.0

0

0.0

4

17.4

8

2

12.5

1

14.3

3

13.0

9

2

12.5

1

14.3

3

13.0

10

3

18.8

1

14.3

4

17.4

11

1

6.3

1

14.3

2

8.7

Total

16

100.0

7

100.0

23

100.0

Mean

7.8

7.4

7.7

SD

1.9

2.8

2.2

Median

7.5

8.0

8.0

Inferential Data Analysis The data gathered as well as those analyzed through inferential statistics are presented in the following paragraphs in answer to the different specific problems enumerated in the study.

On significant difference of the pre-test scores of pupils under the traditional and the experimental group.

The tests’ significant difference was assessed. Pre-test Scores by Group

47

Table 10 shows the distribution of the pre-test scores of the experimental and the control group. To determine whether there was a significant different in the means of the two groups, ttest for independent samples was employed. The test showed a p value less than 0.001, indicating that the means of the two groups (8.7) vs. (5.0) were statistically significantly different. In fact, the boxplots in Fig. 1 confirms this difference. Table 10 Distribution of Pre-test Scores by Group

Experimental Control Group

Group

As a Whole

Pre-test

F

%

f

%

F

%

0

0

0.0

1

4.3

1

2.0

2

0

0.0

1

4.3

1

2.0

3

0

0.0

2

8.7

2

4.0

4

0

0.0

4

17.4

4

8.0

5

3

11.1

3

13.0

6

12.0

6

0

0.0

9

39.1

9

18.0

7

2

7.4

3

13.0

5

10.0

8

5

18.5

0

0.0

5

10.0

48

9

7

25.9

0.0

0.0

7

14.0

10

8

29.6

0.0

0.0

8

16.0

11

2

7.4

0.0

0.0

2

4.0

Total

27

100.0

23

100.0

50

100.0

Mean

8.7*

5.0*

7.0

SD

1.7

1.7

2.5

Median

9.0

6.0

7.0

* t-test is highly significant, t = 7.63, df = 48, and p < 0.001

Experimental Control Experimental Control

Fig. 1. Boxplots Showing the Pre-test Scores of Experimental and Control Groups Post-test Scores by Group 49

The data gathered as well as those analyzed through inferential statistics are presented in the following paragraphs in answer to the different specific problems enumerated in the study.

On significant difference of the post-test scores of pupils under the traditional and the experimental group. The tests’ significant difference was assessed. Table 11 shows the distribution of post-test scores of the experimental and control groups. When a formal t-test was conducted to determine whether the mean post-test scores between the two groups were significantly different, it generated a p-value less than 0.001, indicating that the means were different.

This result implied that the control group had post-test mean that were

significantly higher than the experimental group. In fact, this difference is confirmed in the boxplots of the two groups in Fig. 2.

Table 11 Distribution of Post-test Scores by Group

Control Group

Experimental Group

As a Whole

Post-test

F

%

F

%

f

%

4

0

0.0

2

8.7

2

4.0

5

0

0.0

3

13.0

3

6.0

6

1

3.7

2

8.7

3

6.0

50

7

2

7.4

4

17.4

6

12.0

8

2

7.4

3

13.0

5

10.0

9

4

14.8

3

13.0

7

14.0

10

2

7.4

4

17.4

6

12.0

11

7

25.9

2

8.7

9

18.0

12

7

25.9

0.0

0.0

7

14.0

13

2

7.4

0.0

0.0

2

4.0

Total

27

100.0

23

100.0

50

100.0

Mean

10.3*

7.6*

9.1

SD

1.9

2.2

2.4

Median

11.0

8.0

9.0

51

* t-test is h ighly significant, t = 4.615, df = 48, and p < 0.001

Control 2. Boxplots Showing the Post-test Scores of Experimental and Control Groups Experimental Fig.

Experiences of the Pupils Using Manipulative The data gathered as well as those analyzed are presented in the following paragraphs in answer to the different specific problems enumerated in the study.

On the experiences of students whose been taught using manipulatives Students

Responses

Extreme High

52

Student 1

●

Nasadyahanko kay first time komakatilawgamit sang circle-circle na gin pagamitni Sir Miguel.

Student 2

●

Nabudlayankogamay kay bag-o langkoyakagamit sang ginpagamitni Sir. Pero hapos man langgalisaulihina.

Student 3

●

Kanami mag lesson kay Sir Miguel

Student 4

●

Nalingawko mag kapyot sang mgabilog-bilogkag mag utod sang mgapapel

Student 5

●

Kanami sang lesson ni Sir Miguel taposginhatagyalangsaamonangbilognga may mga color.

Extreme Low

Student 6

●

Unanabudlayanko mag-inchindisamganumeropero sang gin lesson ni Sir Miguel angpartisa fraction dawnakabalokogamay.

Student 7

Student 8

●

Bag-o langkokagamit sang mgabilognapapelna may mga colors.

●

Indi guidko mayo kainchindi

●

Gahatagsi Sir Miguel damonga example peronabudlayankodyapon.

●

Kabudlay mag-inchindi. Nabudlayanguidkohalinsasugodastasamgapapelnaginghatagni Sir Miguel.

Student 9

●

Nahuyakomamangkot kay Sir Miguel kung

53

paanomaggamitperosadyahanko

Student 10

●

Nalingawkokagnanami-an sa lesson peroindiguidkoyakainchindi.

As seen on the table, pupils whose been taught using the manipulatives with high extreme and low extreme scores results in their post-test was interviewed and it shows through their responses that most of the pupils enjoyed the use of manipulatives as well as the strategies of the teacher. They loved to have hands-on activities which only proves that manipulatives were not yet introduced to them or maybe, was not often used especially that they are into public schools where mostly lacked facilities and resources to be found. They find learning fractions fun using manipulatives and they were able to understand more the concepts on fractions. However, some students despite of the manipulatives used for them to stay active and experience a different hands-on activity still doesn’t understand how to used it, they are more confused with the usage of manipulatives and it confuses them much more in learning fractions. Hence, despite of the different responses the researcher was able to find out that manipulatives somehow improve their academic performance.

Chapter 5 SUMMARY, CONCLUSIONS AND RECOMMENDATIONS

54

This chapter includes the summary of the study, its findings, the conclusions that were drawn based on the findings and its recommendations Summary The study’s main objective was to improve the pupil’s academic performance of fractions by the use of manipulatives. Through a mixed-method design, the studies were conducted on 2 different sections of grade 5 pupils of Alfredo Montelibano Elementary School. The implementation lasted for 5 non-consecutive days, where the researchers tested the effectiveness of manipulatives by applying the strategy to one section and using the Traditional Approach in teaching the other section. The behavior and test results of the students were recorded and observed by the researchers and were subjected to the appropriate statistical tools. The presentation, analysis and interpretation of the data gathered were presented in Chapter 4 of the study. Based on the summary of data found in Chapter 4, the following were the findings of the study: (1) The researchers conducted a profiling in both sections and identified that in both groups, there were more boys than girls. As to age, both groups had 9 years old as the youngest participant, while the oldest age in the control group was 15 years old while that of the experimental group was 14 years old.

55

(2) On the mean of Academic Performance of pupils under the traditional approach when grouped according to sex, the result of scores for pre-test of the boys have a mean of 8.8 and a mean of 8.7 for the girls. (3) On the mean of Academic Performance of pupils under the traditional approach when grouped according to sex, the result of scores for post-test of the boys have a mean of 9.9 and a mean of 11.0 for the girls. (4) On the mean of Academic Performance of pupils under the traditional approach when grouped according to age, the result of scores for pre-test of pupils with ages 11 and pupils have a mean of 8.6 while pupils with ages 12 and above have a mean of 8.9. (5) On the mean of Academic Performance of pupils under the traditional approach when grouped according to age, the result of scores for post-test of pupils with ages 11 and below have a mean of 10.2 while pupils with ages 12 and above have a mean of 10.7. (6) On the mean of Academic Performance of pupils under the experimental approach when grouped according to sex, the result of scores for pre-test of the boys have a mean of 4.3 and a mean of 5.7 for the girls. (7) On the mean of Academic Performance of pupils under the experimental approach when grouped according to sex, the result of scores for post-test of the boys have a mean of 7.4 and a mean of 7.9 for the girls. (8) On the mean of Academic Performance of pupils under the experimental approach when grouped according to age, the result of scores for pre-test of pupils with ages 11 and below have a mean of 7.8 while pupils with ages 12 and above have a mean of 7.4.

56

(9) On the mean of Academic Performance of pupils under the experimental approach when grouped according to age, the result of scores for post-test of pupils with ages 11 and below have a mean of 7.8 while pupils with ages 12 and above have a mean of 7.4. (10) The results of the scores between the pre-test of the control and the experimental groups show that there is a highly significant difference in their scores based on the data that was gathered. In the pre-test, the mean score of the control group is 8.7 while the experimental group got a mean score of 5.0. (11) The results of the scores between the post-test of the control and the experimental groups show that there is a highly significant difference in their scores. The control group gained 10.3 in their mean score while the experimental group gained 7.6 in their mean score. To compare the post-test results from the pre-test results, the control group has a difference of 1.6 and a difference of 2.6 for the experimental group. Therefore, the use of manipulatives indeed had an effect in teaching fractions. (12) Based on the interview that the researchers conducted, some pupils’ who were under the experimental group find the use of manipulatives as an enjoyable activity. They said that the use of paper and fractions circles wherein they were able to divide it by themselves was very interesting for it was their first time to use those things. They also find the teacher nice. However, some of the students who got the lowest scores also find the use of manipulatives as an interesting thing but, they still have a difficulty understanding the lesson even though it was taught to them using manipulatives. Some are confused on how to use the manipulatives at the same time; they were hesitant to ask the teacher.

Conclusion

57

Based on the findings of the study, the researchers present the concluding statements: “I hear and I forget. I see and I remember. I do and I understand” (Confucius). School is cool when learning is fun. Young children are naturally passionate about learning and they enjoy it while manipulating objects. Sadly, most students especially in public schools were deprived of using or experiencing hand-on activities. This is the reason why the researchers choose to study about manipulatives which will really help the pupils in understanding math concepts. The researchers conducted test both before and after the intervention in order to know the effect on the academic performance of pupils. The pre-test mean score results of the control group when grouped according to sex shows that between the two sexes both seem to have the same performance while on the post test results of the same group it show that this time, girls performed better. When the control group was grouped according to age, their pre-test scores shows that pupils ages 11 and below and pupils with ages 12 and above almost got the same mean scores with a difference of 0.3 while on the post test, pupils with ages 12 and above differ a mean score of 0.5 to pupils ages 11 and below. On the other hand, the pre-test mean score results of the experimental group when grouped according to sex shows that between the two sexes girls had a higher mean score than the boys while on the post test results of the same group it show that this time, girls only differ with a mean score of 0.5 from the boys. When the experimental group was grouped according to age, their pretest scores shows that pupils ages 11 and below differ a mean score of 0.4 from pupils with ages 12 and above while on the post test, pupils with ages 11 and below still differ a mean score of 0.4 from pupils ages 12 and above.

58

The result of the scores between the pre-test and the post-test of the pupils under the control group and the experimental group has a significant difference based on the data collected therefore rejected both hypotheses . The control group had higher mean scores from the pre-test to the post test. However, both groups shows an increase in their mean scores but on the difference of the increase of the two groups, the experimental group had an increase of 2.6 in their mean score while a mean of 1.6 for the control group. Thus, the students under the control group and the experimental group had increased their knowledge on fractions. Lastly, based on the interview that we had pupils learned how to use manipulatives thus increased their understanding in the lesson about fractions. However, some pupils wasn’t able to grasps the real intention of manipulatives for they are used in a traditional classroom approach. Implications Manipulatives can be key in providing effective, active, engaging lessons in the teaching of mathematics (Hartshorn and Boren, 1990). Another way would be, laboratory approach or this can be defined as “learning by doing”. This would often involve the students to play and manipulate concrete objects in structured situations. (Sutton and Krueger, 2002). In 2013, the National Council of Supervisors of Mathematics (NCSM) issued a position statement on the use of manipulatives in classroom instruction to improve student achievement. Ruzic & O’Connell (2001) found that long-term use of manipulatives has a positive effect on student achievement by allowing students to use concrete objects to observe, model, and internalize abstract concepts. This study adds to the existing mathematics education literature on the use of mathematical manipulatives, it implies that both are good approach in teaching fractions. The students using the 59

manipulatives performed as well as the students in the controlled group based on the results of the post test scores. The experimental group students did show an increased in achievement based on posttest scores results. The researchers only focused on the gained of the two different groupsTherefore, researchers implied that there was an effect in the use of manipulatives in teaching fractions in their academic performance.

Recommendations Based on the result of the study, the following recommendations are set forth: 1.

The researchers also suggest that at the beginning of the class, teacher should review the previous lesson and give diagnostic test to pupils. So that teacher can determine if pupils had already absorb and comprehend the lesson given to them base on the score that they will get and teacher can check the class standing of the pupils.

2.

Teacher of mathematics are recommended to deepen the discussion and exercises in the field of Mathematics. Therefore, teachers should have knowledge of the subject matter and the use of manipulative as well.

3.

Researcher found out that the strategies used by the teacher in conducting the lesson have a great impact to pupils understanding. Therefore, teacher should ensure that the strategies being used in class are appropriate.

4.

The researchers found out that the prior knowledge of the pupils towards fraction is not fully comprehended and there are weakness observed. Therefore, teacher should focus on giving continuous drills to pupils to enhance and strengthen the pupils foundation and their understanding in dealing with fraction 60

5.

Manipulative (fraction circle) is found effective in teaching fraction concepts. Furthermore, future researchers may use (other manipulative) aside from fraction circle as a tool for their study and come up with other manipulative that can suit to the concept in teaching fraction, not only in addition but also in subtraction, multiplication and division. And to broaden the range of manipulative in Mathematics and not to limit the manipulative in fraction but in the other building blocks of the field as well.

6.

The researchers suggest the use of manipulative in teaching fraction because it does not only enhance pupils’ attentiveness towards the duration of class but it help pupils in processing the lesson and the operations.

7.

The researchers suggest that school administrators could advice teachers to integrate the use of manipulatives in teaching Mathematics as well as on other subjects that manipulatives are applicable to use.

61

Appendices

Appendix A Letter to the School August 19, 2015

Caridad P Ellera Principal Alfredo M. Montelibano Sr. Elementary School Brgy, Villamonte, Bacolod City

Dear Mrs. Ellera, Greetings in Christ Name!

62

We are 4th year BEED General Education students from the University of Saint La Salle- Bacolod City and we are currently working on our research entitled, “Manipulatives: Its Effect to the Conceptual Understanding of pupils in Fraction”. In line with this, we would like to ask for your amiable authorization in allowing us to conduct our research in your school specifically on your Grade 5 middle section pupils. We believe that your school will be a great venue in providing us with the needed resources for our study. Thus, be assured that we will strongly uphold confidentiality of sensitive matters. Your good office will be given immediate feedback with regards to the result of our findings in this study We would be glad to sit down with you and discuss details regarding the request. For more details you may contact Joan Trisha I. Nabor at 09097749338. We are hoping for your positive response. Respectfully yours,

Angelo Miguel Delos Reyes Joan Trisha I. Nabor Baberlyn P. Picazo EG Marie G. Rectra Relyn Mae B. Saligumba

Noted by: Endorsed by:

Mrs. Chrisalia S. Eriso

Dr. Cynthia Dy

Research Adviser Professor

Elementary Research

Appendix B Letter to the Validators

September 15, 2015 Mr. Jomeo Sumalapao Mathematics Department

63

Negros Occidental High School Bacolod City

Dear Mr. Sumalapao, Greetings in Christ name! We are 4th year BEED General Education student of the University of St. La SalleBacolod City. We are currently working with our research entitled “Manipulatives: Its Effect to the Conceptual Understanding of pupils in Fraction”. With your expertise, We are humbly asking for your permission to validate the attached research instrument, for this study using the attached rating tool. We are hoping for your positive response. Respectfully yours,

Angelo Miguel Delos Reyes Joan Trisha I. Nabor Baberlyn P. Picazo EG Marie G. Rectra Relyn Mae B. Saligumba

Noted by: Ms. Chrisalia S. Eriso Research Adviser

Appendix C

64

Pre-test Questionnaire Mathematics- Fractions Pre-test

Name: ______________________________

Grade & Sec.:_______________

Test I Instructions: Read and understand the following questions. Write the letter of the correct answer in the space provided before the number. _____

1.

What is a fraction? A. Mathematical Operation

B. Numerical Base C. Part of a whole D. All of the above _____

2.

What is the line that separates the numerator from the denominator? A. Bar line B. Fraction line C. Horizontal Line D. Vertical line

65

_____

3.

Given the fraction

, what is 4?

A. Denominator B. Numerator C. Remainder D. Whole Numbers

_____

4.

Add the following fractions:

+

+

A.

B.

C.

D.

_____

5.

+

is equal to?

A.

B.

66

C.

D.

_____

6.

more than

is equal to?

A.

B.

C.

D.

_____ 7.

added by

is equal to?

A.

B.

C.

D.

_____

8.

. What is the sum of the four fractions?

67

A. 2 B. 4 C. 6 D. 8

_____ 9.

is equal to?

A.

B.

C.

D.

_____ 10.

added by

is equal to?

A.

B.

C.

D.

68

Test II Instructions: Analyze and solve the problem. Show the process and box the final answer.

Monday

Tuesday

Wednesday

km

Thursday

km

Friday

km

km

km

Mike undergo for 5-day training for an upcoming marathon. Below is the table showing the distance he covered on the course of his training. How many kilometers did Mike cover during the entire training?

a. What is asked?

b. What is/are the given?

c. What is/are the operation/s to be used?

d. Number sentence

e. Solution

69

Appendix D Traditional Lesson Plan

Lesson: Adding of Similar and Dissimilar Fractions At the end of the lesson, 75% of the students will be able to: K N O

Define fraction;

W L E Identify the kinds fractions (similar and dissimilar fractions); D G E S K I

Perform operations in fractions (addition of fractions);

L L S

70

A T T I

Relate day to day activities that involves fractions.

T U D E

Materials Needed: Chalk and Chalkboard References: Mathematics for Grade 5 Textbook

STRATEGIES

STUDENT’S RESPONSE

ROUTINE ACTIVITIES: PRE-ACTIVITY: Prayer Checking of attendance. Greet the class with enthusiasm REVIEW: A. What is a fraction? A. Fraction is a part of a whole B. What are the parts of fraction? Could you name the 3 parts?

B. Numerator, Denominator and Fraction line

C. Who could give me an example of fraction?

C. ¼, ½ 1/8, ¾ and more

MOTIVATION:

71

So what are the kinds of fractions? What is Similar Fractions? What s Dissimilar Fractions? Could you give me examples of similar fractions? Could you give me examples of dissimilar fractions?

PRESENTATION: In relation to the previous questions, the teacher will discuss how to add similar fractions. Similar fractions can be added if the denominators are the same. Add the numerator and copy the denominator and simplify if needed. In adding dissimilar fractions, denominators must be the same by looking first for the Least Common Multiple (LCM). Rewrite the fractions as equivalent fractions with the LCM as the denominator. Add the numerator and copy the denominator and simplify if needed.

72

ACTIVITY PROPER: Instruction: 

Each student will be given sheets of paper



Process and solution must be written in the piece of paper distributed by the teacher



Ask for volunteers who could answer the questions

Guided Activity 1: Adding of Similar Fractions a.

Present an example of similar fractions for the students to add and say “So are you ready to answer my question? Now I have here similar

a.

On their paper, add the

fractions which are ½ and ½, on your paper add these fractions and give

numerator

me the answer. Who would like to volunteer to show his/her solution on

denominator. A volunteer

the board?”

will show his/her solution on

and

copy the

the board.

b.

Ask the students if what is the answer by saying “So if we add ½ + ½

b.

2/2 or 1

what is the answer?”

Guided Activity 2:Addition of Similar Fractions a.

Present another example of adding similar fractions saying “So are you

a.

ready to answer this one? I have here 2/4 + 1/4. On your paper add the

On

their

numerator

given fractions and give me the answer. Who would like to volunteer

paper and

add

the

copy

the

denominator. A volunteer will

to show his/her solution on the board?”

show his/her solution on the board.

b.

Ask the students if what is the answer by saying “So if we add 2/4 + 1/4 what is the answer?”

b.

¾

73

Guided Activity 3:Addition of Dissimilar Fractions a.

Give another example for the students to answer and this time it is dissimilar fractions and say “I have another example so if ½ plus ¼.

a.

What will you do first to make the fractions similar fractions?”

First look for the Least Common Multiple of the denominators of the two fractions and change them to equivalent fractions. It will be 2/4 + ¼ after looking for the LCM of the two fractions.

b.

After changing the fractions into similar fractions, perform the

b.

On their paper add the

operation and give the answer. Who would like to volunteer to show

numerator

and

copy the

his/her solution on the board?”

denominator. A volunteer will show his/her solution on the board.

c.

c.

¾

What is the answer?

74

POST ACTIVITY: Checking of individual work. ANALYSIS: 1.

2.

3.

4.

In reference to activity number 1, how to add the similar fractions ½ +

Adding

these

½ =?

similar

fractions

In reference to activity number 2, how to add the similar fractions 2/4 +

copy

the

1/4 =?

denominator

and

In reference to activity number, how to add the dissimilar fractions ½ +

add the numerator,

¼ =?

½ + ½ = 2/2 or 1.

In your own understanding, what are similar fractions? What are

1.

2.

dissimilar fractions?

Adding

these

similar

fractions

copy

the

denominator

and

add the numerator, 2/4 + ¼ = 3/4. 3.

Look for the Least Common Multiple first and change the fractions

into

equivalent fractions. After so perform operation asked. ½ + ¼

the being 2/4

+ 1/4 = ¾. 4.

Similar fractions are fractions having the

75

same denominator. Dissimilar fractions are fractions having different denominators.

76

ABSTRACTION: 

How do we add similar fractions?

Steps in adding similar fractions: 1.

Make

sure

the

bottom

numbers (the denominators) are the same. 2.

Add the top numbers (the numerators), put the answer over the denominator.

3.

Simplify the fraction (if needed).

Steps in adding dissimilar fractions:

1. Find the smallest multiple 

(LCM) of both numbers

How do we add dissimilar fractions?

since the denominators of each fraction is different.

2. Rewrite the fractions as equivalent fractions with the LCM as the denominator.

3. Add the top numbers (the numerators), put the answer over the denominator.

4. Simplify the fraction (if needed). 

In eating a pizza and dividing it into slices, in

77

eating chocolates and dividing it to several pieces, in paying at a grocery or supermarket using centavos.



Where can we relate fractions in real life activities?

78

APPLICATION: Instruction: 

Divide the class into groups with each group consisting of 3 members.



Give out an answer sheet and bond papers for each group to work on.



Using those bond papers, fold, cut and label the paper to create fractions and answer the following questions

1.

3/8 + 2/8 + 1/8 =

2.

2/5 + 2/3 =

1.

7/8

2.

14/15

79

ASSIGNMENT:

Steps in subtracting similar fractions: 1.

a.

How do we subtract similar fractions?

Make sure the bottom numbers (the denominators) are the same

2.

Subtract the top numbers (the numerators). Put the answer over the same denominator.

3.

Simplify the fraction (if needed).

Steps

in

subtracting

dissimilar

fractions:

1. Find the smallest multiple b.

How do we subtract dissimilar fractions?

(LCM) of both numbers since the denominators of each fraction is different.

2. Rewrite the fractions as equivalent fractions with the LCM as the denominator.

3. Subtract the top numbers (the numerators), put the answer over the denominator.

4. Simplify the fraction (if needed).

80

Appendix E Experimental Lesson plan

Lesson: Adding of Similar and Dissimilar Fractions At the end of the lesson, 75% of the students will be able to: K N O

Define fraction;

W L E Identify the kinds fractions (similar and dissimilar fractions); D G E S K I

Perform operations in fractions (addition of fractions);

L L S

81

A T T I

Relate day to day activities that involves fractions.

T U D E

Materials Needed: Manipulatives (Fraction Pie), Tape, Bond Paper, Cartolina, Pentel Pen, Scissors References: Mathematics for Grade 5 Textbook

STRATEGIES

STUDENT’S RESPONSE

ROUTINE ACTIVITIES: PRE-ACTIVITY: Prayer Checking of attendance. Greet the class with enthusiasm REVIEW: D. What is a fraction? D. Fraction is a part of a whole E. What are the parts of fraction? Could you name the 3 parts?

E. Numerator, Denominator and Fraction line

F.

Who could give me an example of fraction?

F.

1 1 13 , , , 4 2 84

and more

MOTIVATION:

82

So what are the kinds of fractions? What is Similar Fractions? What s Dissimilar Fractions? Could you give me examples of similar fractions? Could you give me examples of dissimilar fractions?

PRESENTATION: In relation to the previous questions, the teacher will discuss how to add similar fractions. Similar fractions can be added if the denominators are the same. Add the numerator and copy the denominator and simplify if needed. In adding dissimilar fractions, denominators must be the same by looking first for the Least Common Multiple (LCM). Rewrite the fractions as equivalent fractions with the LCM as the denominator. Add the numerator and copy the denominator and simplify if needed.

83

ACTIVITY PROPER: Instruction: 

Each student will be given sheets of paper



After successfully dividing the paper according to teacher’s instruction on each specific activity, it will be changed into manipulatives with label



Students will solve the question raise by the teacher using the manipulatives.



Ask for volunteers who could answer the questions

Guided Activity 1: Adding of Similar Fractions a.

Say “I have here a piece of paper. Who could divide this paper and 1 2

give me two piece of papers? “ c.

Will fold the paper into two parts and cut it into two so

b.

Teacher will exchange the cut-out papers with manipulatives labeled as

they will have two pieces of

½ “Now I have here similar fractions which is ½ and ½, add these

½ papers.

fractions with use of this manipulative.” .

d.

Get the manipulative and do it on their table and answer the question after.

Steps: 1.

Show and give to students a fraction pie which is a 2/2. This time give them a fraction pie which is divided into 2 equal parts. Give each

84

student a fraction pie which is divided into 2 equal parts. Make sure that each fraction pie is labeled as 1/2. 2.

With 2 equal pieces, get 2 pieces of 1/2 from the fraction pie.

3.

Demonstrate using your own fraction how to add ½ + ½.

4.

Ask the students to give you the answer using their own fraction pie

e.

c.

2/2 or 1

Will fold the paper into 4 parts and cut it into 4. Another paper will also be given and it will be

c.

Ask the students if what is the answer by saying “So if we add ½ + ½ cut into 4 parts. what is the answer?”

Guided Activity 2:Addition of Similar Fractions a.

Say “I have here a piece of paper. Who could divide this paper so we

d.

Get the manipulative and do it

could have 4 pieces of papers and give me 2/4? Next one is who could

on their table and answer the

divide this paper so we could have 4 pieces of papers and give me a ¼?

question after. Steps:

85

1. b.

Show and give to

Teacher will exchange the cut-out papers with manipulatives labeled as

students a fraction pie

2/4 and 1/4 “Now I have here similar fractions which is 2/4 and 1/4,

which is a 4/4. This time

add these fractions with use of this manipulative.”

give them a fraction pie which is divided into 4 equal parts. Give each student a fraction pie which is divided into 4 equal parts. Make sure that each fraction pie is labeled as 1/4. 2.

With the 4 pieces you have, tell them that you will add only 3 pieces of 1/4 since the operation involved is addition.

3.

Demonstrate using your own fraction pie with the 4 pieces you have in hand and add 3 pieces.

4.

Ask the students to give you the answer using their own fraction pie

e.

¾

86

a.

Will fold the1st paper into 2 parts and cut it into 2. Another paper will also be given and will be folded into 4 parts and cut afterwards.

b.

Get the manipulative and do it on their table and answer the question after.

Steps: 1.

Show to students a fraction pie which is a ½ and 1/4. Make sure that each part is labeled as ½ and 1/4. Let the students follow after you.

2.

With the 2 pieces you have, tell them that you will add the pieces since the operation involved is addition.

3.

Demonstrate using your own fraction pie with

c.

Ask the students if what is the answer by saying “So if we add 2/4 + 1/4

the 2 pieces you have in

what is the answer?”

hand and on the board

Guided Activity 3:Addition of Dissimilar Fractions a.

Say to the class “I have here pieces of paper. Who could divide this

ask them if they could combine the two since

paper into 2 parts so we could have two ½ pieces of paper and on the

87

other piece of paper who could divide it into 4 parts so we could four

they are dissimilar

¼ pieces of paper and give me a ¼?

fractions. 4.

First look for the Least Common Multiple of the denominators of the two

b.

Will exchange the cut-out papers with manipulatives labeled as 1/2 and

fractions

1/4 “Now I have here similar fractions which is 1/2 and 1/4, add these

them

fractions with use of this manipulative and give me the answer.”

fractions.

to

and

change

equivalent After

changing it to equivalent fractions,

add

the

fractions and simplify if needed. c.

c.

¾

So what is the answer?

88

POST ACTIVITY: Checking of individual work. ANALYSIS: 5.

6.

7.

8.

In reference to activity number 1, how to add the similar fractions ½ +

Adding

these

½ =?

similar

fractions

In reference to activity number 2, how to add the similar fractions 2/4 +

copy

the

1/4 =?

denominator

and

In reference to activity number, how to add the dissimilar fractions ½ +

add the numerator,

¼ =?

½ + ½ = 2/2 or 1.

In your own understanding, what are similar fractions? What are

5.

6.

dissimilar fractions?

Adding

these

similar

fractions

copy

the

denominator

and

add the numerator, 2/4 + ¼ = 3/4. 7.

Look for the Least Common Multiple first and change the fractions

into

equivalent fractions. After so perform operation asked. ½ + ¼

the being 2/4

+ 1/4 = ¾. 8.

Similar fractions are fractions having the

89

same denominator. Dissimilar fractions are fractions having different denominators.

90

ABSTRACTION: 

How do we add similar fractions?

Steps in adding similar fractions: 4.

Make

sure

the

bottom

numbers (the denominators) are the same. 5.

Add the top numbers (the numerators), put the answer over the denominator.

6.

Simplify the fraction (if needed).

Steps in adding dissimilar fractions:

5. Find the smallest multiple 

(LCM) of both numbers

How do we add dissimilar fractions?

since the denominators of each fraction is different.

6. Rewrite the fractions as equivalent fractions with the LCM as the denominator.

7. Add the top numbers (the numerators), put the answer over the denominator.

8. Simplify the fraction (if needed). 

In eating a pizza and dividing it into slices, in

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eating chocolates and dividing it to several pieces, in paying at a grocery or supermarket using centavos.



Where can we relate fractions in real life activities?

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APPLICATION: Instruction: 

Divide the class into groups with each group consisting of 3 members.



Give out an answer sheet and bond papers for each group to work on.



Using those bond papers, fold, cut and label the paper to create fractions and answer the following questions

3.

3/8 + 2/8 + 1/8 =

4.

2/5 + 2/3 =

3.

7/8

4.

14/15

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ASSIGNMENT:

Steps in subtracting similar fractions: 4.

c.

How do we subtract similar fractions?

Make sure the bottom numbers (the denominators) are the same

5.

Subtract the top numbers (the numerators). Put the answer over the same denominator.

6.

Simplify the fraction (if needed).

Steps

in

subtracting

dissimilar

fractions:

5. Find the smallest multiple d.

How do we subtract dissimilar fractions?

(LCM) of both numbers since the denominators of each fraction is different.

6. Rewrite the fractions as equivalent fractions with the LCM as the denominator.

7. Subtract the top numbers (the numerators), put the answer over the denominator.

8. Simplify the fraction (if needed).

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Appendix F Posttest Mathematics- Fractions Post-test

Name:

____________________________________ Grade & Sec.:_____Age ___ Sex ___

Test I

Instructions: Read and understand the following questions. Write the letter of the correct answer in the space provided before the number.

_____

1.

What is a fraction? A. Mathematical Operation

B. Numerical Base

C. Part of a whole

D. All of the above

_____

2.

Given the fraction

, what is 4?

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A. Denominator

B. Numerator

C. Remainder

D. Whole Number

_____

3.

What is the line that separates the numerator from the denominator?

A. Bar line

B. Fraction line

C. Horizontal Line

D. Vertical line

_____4.

added by

is equal to?

C.

D.

C.

D.

_____

5.

Add the following fractions:

+

+

C.

D.

96

C.

D.

_____

6.

more than

is equal to?

C.

D.

C.

D.

_____

7.

+

is equal to?

C.

D.

C.

D.

_____ 8.

added by

is equal to?

C.

D.

97

C.

D.

_____ 9.

is equal to?

C.

D.

C.

D.

_____

10.

. What is the sum of the four fractions?

A. 2 B. 4 C. 6 D. 8

Test II

Instructions: Analyze and solve the problem. Show the process and box the final answer.

Monday

Tuesday

km

Wednesday

km

Thursday

km

Friday

km

km

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Mike undergo for 5-day training for an upcoming marathon. Below is the table showing the distance he covered on the course of his training. How many kilometer did Mike cover during the entire training?

a. What is asked?

b. What is/are the given?

c. What is/are the operation/s to be used?

d. Number sentence

e. Solution

Appendix G TOS (Table of Specification)

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Appendix H Pre-Test and Post Test Validation

Juror: _________________________________

Instruction: Please indicate your degree of agreement or disagreement on the statements provided below by encircling the letter which corresponds to your answer. The statements are taken from the criteria for evaluating survey questionnaire set forth by Carter V. Good and Douglas B. Scates. 5 – Strongly Agree 4 – Agree 3 – Undecided 2 – Disagree 1 – Strongly Disagree

CRITERIA

1. The test is short enough that the students will be able to answer it within the allotted time.

OPTIONS:

5

4

3

2

100

1

2. The test is interesting and has an appeal such that the students will be induced to respond to it and accomplish it fully.

5

4

3

2

1

3. The test can obtain some depth to answers and avoid guesswork.

5

4

3

2

1

4. The questions and their alternative responses are neither too suggestive nor too unstimulating.

5

4

3

2

1

5. The test can elicit responses which are definite and not conflicting.

5

4

3

2

1

6. The questions are stated in such a way that the students can understand them clearly.

5

4

3

2

1

5

4

3

2

1

5

4

3

2

1

5

4

3

2

1

7. The questions are formed in such manner to avoid suspicion on the part of the students concerning hidden answers in the test. 8. The test is neither too narrow nor limited in its content. 9. The answers to the problems when taken as a whole, could answer the basic purpose for which the test is designed therefore considered valid.

Comments: ____________________________________________________________________________ ____________________________________________________________________________ ____________________________________________________________________________ ____________________________________________________________________________ ____________________________________________________________________________ _________________________

___________________________________

____________________________

Signature Over Printed Name

Date 101

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