Fractions In The Mathematics Curriculum Education Essay
This chapter reviews the relevant literature related to the study. It will explore the overview of the topic and theories that are related and relevant to the study. The study can be divided into two main themes-, the issue and the intervention. The issue here refers to learning problems associated with “Fraction” which includes the place of the subtopic in the Brunei Mathematics Curriculum, review of previous research on students’ difficulty and underlying theories related to the topic. The intervention describes how the instruction will be carried out, what are the relevant instructional type of intervention used, the theory behind the choice, including previous research on the choice of intervention. These are then referred to the Brunei SPN-21 curriculum framework.
2.1 Issue: “ Fractions”
2.1.1 Fractions in the Mathematics Curriculum
Fractions are first introduced to pupils in Year 2 in Brunei mathematics curriculum. The curriculum keeps revisiting the topic of fractions at different depth up to secondary education. The syllabus content for O Level Mathematics outlined the expected outcome in this topic. Students are expected to be able to use the language and notation of simple vulgar and decimal fractions and percentages in appropriate contexts, recognize equivalence and perform calculations by suitable methods, with and without a calculating aid in involving fractional operations (Cie,2012). The Curriculum Development Department (CDD), Ministry of Education, Brunei Darussalam has outlined the learning outcomes that should be attained by students at each level as shown in Table 1.
Placement of Fraction in the Brunei Syllabus
Understand the ideas of ‘fraction’ as a part of a whole
Use fraction chart to name fractions with denominators up to 10.
Name simple fractions based on fraction diagrams.
Shade simple fractions on given diagrams.
Demonstrate that when all fractional parts of a whole are included the result equals one whole. Read and write for example;
= 1 , = 1 , = 1
Use fraction chart and number line to recognize and name fractions with denominators greater than 10.
Shade or colour fractions with denominators greater than 10.
State the numerator and the denominator of a given fraction.
Compare like fractions.
Order like fractions in order of size.
Compare unit fractions .(S)
Arrange unit fractions in order of size. .(S)
Use diagrams or fraction chart to recognise equivalent fractions. (S)
Compare unlike fractions. (C)
Arrange unlike fractions in order of size. (C)
Add and subtract like fractions within one whole.
Determine equivalent fractions of a given fraction with denominator ≤ 10
Reduce a given fraction to its simplest form
Compare and order fractions with denominators <20 and <100. (M&C)
Convert improper fractions into mixed numbers & vice versa (S)
Addition and subtraction of like fractions with results >1
Add and subtract related fractions(C)
Solve word problems. (S&C)
Add and subtract related fractions
Add and subtract unlike fractions
Multiply fractions (include mixed numbers) by a 1-digit whole number
Divide fractions (include mixed numbers) by a 1-digit whole number (C)
Interpret fraction as division
Solve word problems (M, S&C)
Apply concept of fraction in estimating answers in computations. (e.g. less than 2; is slightly more than 52, etc.) (C)
Understand fraction as part of a set
Find a fraction of a set
Divide fractions (including mixed numbers) by a whole number (C)
Multiply a fraction by another fraction (S)
Divide a fraction by another fraction (S)
Solve word problems (M, S&C)
Perform operations on fractions without / with the use of the calculator.
Apply fraction as part of a set and as a division of two numbers in various contexts.
Recognise place values of digits in a given decimal.
Convert between fractions and decimals.
Compare fractions and/or decimals using words and symbols: <, >, ≤, ≥ and =
After Year 7, fraction is incorporated into other topics such as algebra, everyday mathematics and measurements. The content of the topic is designed by using Bruner’s spiral curriculum. Jerome Bruner, a renowned psychology in the constructivist theory, believed that any subject could be taught at any stage of development in a way that fit the child's cognitive abilities. Spiral curriculum refers to the idea of revisiting basic ideas over and over, building upon them and elaborating to the level of full understanding and mastery. As shown in Table 1, the topic is revisited from Year 2 to Year 7 in different depth.
2.1.2 Students’ difficulty in learning and understanding Fraction
It is well documented that fractions are among the most complex mathematical concepts that children encounter in their years in primary education (Newstead & Murray, 1998, Bezuk, Cramer & Streetfland (1991)). Hartung (1958) acknowledged the complexity of the fraction concept that cannot be grasped all at once. He also cited that knowledge of fractions must be acquired through a long process of sequential development. This is probably one of the reason why in our curriculum framework, the topic is being taught in stages from as early as when the children are in Year 2, and is developed as they grow older to a more complex form.
Experts have outlined a number of reasons to explain students’ difficulty with fractions. Some researchers had point out the causes for the low performance in this topic (e.g Kerslake, 1986,Hart, 1988, Domoney, 2002, Hannula, 2003). One of the predominant factors contributing to the complexities is the fact that fractions comprise a multifaceted notion encompassing five interrelated sub-constructs which are part-whole, ratio, operator, quotient, and measure (Brousseau, Brousseau & Warfield, 2004; Kieren, 1995; Lamon, 2001). It has been suggested that children should develop an integrated understanding of different sub constructs (Post, Cramer, Bejr,Lesh, & Harel,1993).
Other factors which contribute to the students’ weakness in fractions is that fractions cannot be counted and there are infinite numbers of fractions between any two fractions, as reported by Robert Siegler (2010). Students tend to memorize formulas or algorithms instead of understanding them. Students also have a difficulty in incorporating concept into practice, example is that students do not know why addition and subtraction require a common denominator. Although being exposed to the computing of fractions from primary school, students in secondary school still make significant error in the addition and subtraction of fractions (Wan, 2002).
Studies have also established that students’ difficulties are mainly due to lack of conceptual understanding of fraction itself. Students had good procedural understanding of fractions as this had been the method taught to them since primary school. (Moss & Case, 1999). The development of conceptual understanding involves seeing the connections between concepts and procedures, and being able to apply mathematical principles in a variety of context.(BOS NSW, 2002).
A number of recent research studies in Brunei Darussalam have confirmed that pupils in schools are drilled into application of rules and formulas at the expense of mathematical understanding (Veloo and Lopez-Real, 1994; Wong and Veloo 1996; Clements, 2002; Lim, 2000; Khoo 2001; Norjum & Veloo, (2003); Veloo and Ali Hamdani, 2005). This is further supported by a report on error analysis on students’ performance in PMB 2008. The report revealed that students were mostly drilled to do mathematical rules without understanding (MOE, 2008). Study in Brunei on Primary 5 and Primary 6 pupils, had identified some common error patterns, namely grouping error, basic fact error, defective algorithm, incorrect operation and careless error. (Yusof & Malone,2002). The study also reported that although the students’ achievement in the post test had improved but their performance on fraction work remained unsatisfactory particularly on basic operations.
Various studies in Brunei primary schools (Clements (1999), Fatimah (1998), Jabaidah (2001), Leong, Fatimah & Sainah (1998) Raimah (2001) ) also revealed that pupils in the upper primary school find fractions to be extremely difficult and most of them had no relational understanding of fraction concept. Suffolk and Clements (2003) studied students in Form 1 and Form 2 from 27 secondary schools in Brunei also found out that many students were experiencing serious difficulties with elementary fractions tasks. Another study by Zurina (2003) involving Form 4 (N-Level) students discovered that students had very poor knowledge and understanding of fractions and decimals. The major contributing factors were that teacher spent large amount of time on preparing students for high-stake examination, therefore the traditional ‘drill and practice’ method was mostly employed by teachers. She further commented that teaching and assessment methods were not generating towards the desired quality of students.
Despite being a difficult and complex topic, fraction is one of the main topics in the Brunei Mathematics syllabus, and is being taught formally as early as in Year 2. Wu (1999), cited that fraction understanding is vital to a student’s transformation from computing arithmetic calculations to comprehending algebra. In Year 7, students are expected to know and understand the sub-constructs of fractions, and are able to perform operations using fractions fluently. They should have acquired the conceptual and procedural understanding of fractions.
Addition and subtraction of fractions was first introduced to pupils in Brunei in Year 3 and continued in different depth to secondary education. Although students have been exposed to computing of fractions as early as in Year 3, they still make significant errors in addition and subtraction of fractions in secondary school (Wan,2002). Samsiah (2002) in her study on Primary 6 pupils in Brunei Darussalam found out that pupils don not acquire accurate procedural knowledge for carrying out fraction operations and she further recommended that teaching and learning environment need to be created which are conducive to a healing process. Common errors in addition and subtraction of fractions is the ‘classic error’ of adding or subtracting the numerator and denominator. This implies that students were thinking of fractions in a disjointed rather than holistic manner.
Students’ difficulties in fractions could be seen as a global phenomena as being discussed. Recognizing the difficulties and acknowledging the importance of fraction in mathematics education makes this study of great significance to the researcher and mathematics teachers.
2.1.3 Students’ confidence
“Concentration, Confidence, Competitive urge, Capacity for enjoyment”
Confidence is a state of being certain whether the hypothesis or prediction is correct or that a chosen course of action is the best or most effective. According to Jones (2001), the self-confidence is the assurance that a person has in his or her own abilities. Self-confidence is also defined as the sureness of feeling that you are equal to the task at hand. This sureness is characterised by absolute belief in ability. Bandura (1986) reasoned that the most important source of information on students’ confidence comes from the mastery experience. The term mastery experience implies that individuals are to reflect on and evaluate their own performance. Self-confidence is extremely important in almost every aspect of our lives, yet so many people struggle to find it. Sadly, this can be a vicious circle: People who lack self-confidence can find it difficult to become successful. In order to develop confidence in Mathematics, students need to be provided with opportunity to use mathematics in a real context.
2.2 Intervention and Remediation
According to a study by Kroesbergen and Van Luit (2003), intervention is used to teach basic math skills and problem-solving strategies for students with special needs. Remediation is the effective re teaching of material not previously mastered when it was originally taught, according to a research study for the Southeastern Regional Council for Educational Improvement by Gypsy Anne Abbott and Elizabeth McEntire. A successful remediation strategy covers any prerequisite concepts or skills needed to understand the current objective. Students who did not learn the material the first time it was taught may simply need reteaching or a fresh approach, while students with problems learning may also need modifications to the lessons and assessments, more time to complete assignments or shortened assignments.
In this study, the researcher is doing remediation programme adopting the Learning study strategy in two cycles. First cycle is addressed at improving students’ conceptual understanding of fraction, particularly looking at equivalent fraction which is the pre requisite for addition and subtraction of fraction with different denominators. The second cycle is aimed at improving students’ fluency in doing operation with fractions. The pre-test is administered before the intervention programme is carried out. This is to find out the students’ knowledge of the problem being investigated. The post-test is administered at the end of cycle 2. This is as a measure of the learning which might take place as a result of the intervention.
2.2.1 Learning Study
Learning Study is a process where teachers work collaboratively to plan teaching strategies by focusing on the students’ needs. It builds teacher knowledge about how students develop mathematical understanding. Learning Study aims to advance student learning through building a sequence of learning experiences, reviewing the lessons and evaluating the effectiveness of the learning experiences. It is most effective when supported by an expert to offer constructive advice and support
Learning study is similar to the Japanese Lesson Study (Yoshida,1999 ; Stigler & Hiebert, 1999). It is aimed at improving students’ learning in a cyclic process of planning and revising lesson by a group of teachers. The theory of Variation (Marton, Runesson, & Tsui, 1997) forms the basis of the theoretical framework of Learning Study. According to variation theory, learning is defined as a change in the way a person experiences a particular phenomenon and is associated with a change in discernment in that person’s structure of awareness (Marton & Booth, 1997; Marton & Tsui, 2004; Marton & Pang, 2006). In designing the patterns of variation and invariation, teachers are advised to use the principles of variation, as follows:
The principle of contrast – teacher to give contrasting example (e.g. Fraction and Whole number)
The principle of separation – to test one variable, change the other variable. (e.g to understand relationship of numerator to the value of fraction, vary the numerator and keep the denominator invariant)
The principle of generalization – to generalize a concept, different examples of the same value are given (e.g to generalize the concept of , give all kinds of examples involving say half of an apple, half of an hour etc
The principle of fusion – vary different dimensions simultaneously (e.g. to understand two critical aspect of numerator and denominator, vary both at the same time, systematically)
The main focus of learning in the SPN-21 curriculum is the learner, with emphasis on the teaching and learning for understanding. Learning study is one of the strategies which focus on the teaching and learning for understanding. It is the aim of the Ministry of Education to provide continuous professional development in order to help teachers to improve their understanding of teaching. To support this, Learning study group of secondary school teachers had been set in Brunei to improve teaching and learning of science and mathematics.
Learning Study is a process where teachers work collaboratively to plan teaching strategies by focusing on the students’ needs. It builds teacher knowledge about how students develop mathematical understanding. Learning Study aims to advance student learning through building a sequence of learning experiences, reviewing the lessons and evaluating the effectiveness of the learning experiences. It is most effective when supported by an expert to offer constructive advice and supporTeachers are encouraged to use different approach to their teaching for the improvement of learning in Brunei. Dato Seri Setia Awang Hj Yusoff Hj Ismail, the acting Minister of Education, in his speech at the opening of the World Association of Lessons Studies (Wals) Conference 2010 mentioned on the importance of lesson study and learning study to improve on the teachers’ understanding of their teaching. He further added that the challenge is to ensure that collaborative enquiry trough lesson and learning study takes root in the culture of our school (Brunei Times,2010 Dec)
2.2.2 Use of Manipulative in Mathematic
“I hear and I forget, I see and I remember, I do and I understand”
Mathematics education today are moving towards the facilitation of students’ understanding and conceptualization rather than drill and practice of rote procedures (Heddens,1986). This is in line with the SPN-21 curriculum framework which also give emphasis on the teaching and learning for understanding. One of the ways to promote understanding is by using manipulative.
Manipulative are physical object help to make mathematical concepts become concrete. Research in many countries supports the idea that the mathematics instruction and students mathematics understanding will be more effective if manipulative materials are used (Canny, 1984; Clements & Battista, 1990; Dienes, 1960; Driscoll, 1981; Fennema, 1972; 1973; Skemp, 1987; Sugiyama, 1987; Suydam, 1984) Allowing students to use concrete objects to observe, model, and internalize abstract concepts will yield a positive effect on students’ achievement (Sowell,1989.,Ruzie and O’Connel,2001) . Manipulative allows students to construct their own cognitive models for abstract mathematical ideas and processes. They are also engaging students and increasing both interest and enjoyment of mathematics. Long term interest in mathematics translates to increased mathematical ability (Suton & Krueger, 2002).
2.2.3 Games in Mathematics Classroom
Games are seen to be fun, not only motivating but ensuring full engagement, particularly through reflection and discussion, on which constructive learning depends (Booker,1996). Games are also valuable for simulating and encouraging mathematical discussion between group of children and between students and teacher (Earnest, 1986). Students may build on their prior knowledge and forms links between the game and their everyday surroundings (Bragg, 2006). Bragg further added that through the use of games, students’ ability to work independently of the teachers and others helps them to build confidence through achieving success in classroom.
Games offer mathematics teachers a way of practicing and reinforcing arithmetic and other mathematical skills, as well as supplementing for drills and practice-problems. Games are seen as a way of presenting “high level” mathematics concepts in a simple and non-threatening way.
Although games has been seen as a beneficial tool in mathematics classroom (Bragg,2006; Booker,2000; Gough,1999; Anily,1990), it is also important to ensure the structure of the game support learning, for learning to take place (Swansed & Marshall, nd). Learning outcomes related to the games should be clearly specified to make the usefulness of games explicit to students (Bragg, 2006).
In this study, the game of “I have.. , who has?” is used. The game is chosen because it involves the whole class and easy to administered. The rule is also very simple In this game, students have to be attentive and at the same time try to figure out the answer that match their cards.
2.2.4 Use of Video song
Music is chosen as another mean of helping students to understand the topic. Music establishes a positive learning state and energizes learning activities. Songs and rhythmic chants invite the students to become active in the learning practice. Music adds an element of fun while helping accentuate the lesson orientation. Songs help stimulate the students' imagination. Music helps ease tension through work that does not feel like typical classroom work. The melody, rhythm and repetition collaborate together as an effective tool in improving students' memories, which in turn will establish good retention of the topic
Music is a “universal” language which promotes reading, creativity, and comprehension skills all at the same time(Wright,2009). Don McMannis, an expert on children’s music, mentioned the positive effect of music on peoples’ emotions and creativity. He also agreed that music is an effective medium for learning and retaining information, in a way that it activates three different centers of the brain at the same time: language, hearing, and rhythmic motor control” (Elias,2009). Music is considered as one of the avenues for learning proposed by cognitive phychologists in the theories of multisensory learning (Harris,2009).
Music is viewed as a multi-sensory approach to enhance learning and retention of academic skills. The music activities used will directly carry the curriculum content that the student is to learn. For example, if the student is to add single digit numbers, the lyrics to the educational song or chant will deal directly with that target skill. Research supports the use of music as a mnemonic device for the learning and recall of new information. Music also plays a role in focusing attention and providing a motivating environment for learning. In addition, educational research confirms that we learn and retain information better when we find it interesting and meaningful.
In this study, a video song from you tube, called the “Mathe Mia Addition of Fractions”, is used. The lyrics of the song summarize how to do addition and subtraction of fractions, from common denominators to unlike denominators and the mixed numbers. After the students have acquired the intended learning objectives, the video song will helps them to recall and retain the information learnt.
It is well documented that fraction is one of the most difficult topic in Mathematics. Fraction has been taught to students in stages; from as early as when they are in Year 2. Understanding how fraction works is needed in life and other field of study. It is therefore important to establish good foundation in this topic.
Lesson study is being practised worldwide and has proven a successful and effective method of enhancing teaching and learning. The ministry of Education has encouraged schools to practice lesson or learning study to help teachers and students in their teaching and learning. Teachers are also encouraged to use of different teaching strategies in enhancing students learning. The use of Manipulative in teaching and learning had been established in the education system. The effectiveness of games in promoting students learning had also been well documented. Games give an alternative way of learning in a fun, enjoyable and non-threatening way, which in turn will boost students’ motivation and confidence. Research had found out the positive impact on the use of music in education, although it is not a very popular means of teaching in the secondary schooling. Through the use of different strategies, students’ learning of the subject might yield positive result.
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